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Journal of Functional Analysis 260 (2011) 2497–2517 www.elsevier.com/locate/jfa Unbounded extensions and operator momen...

Author: A. Connes | D. Stroock (Chief Editors)

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Journal of Functional Analysis 260 (2011) 2497–2517 www.elsevier.com/locate/jfa

Unbounded extensions and operator moment problems E. Albrecht a,∗,1 , F.-H. Vasilescu b a Fachrichtung 6.1 – Mathematik, Universität des Saarlandes, 66041 Saarbrücken, Germany b Laboratoire Paul Painlevé, U.F.R. de Mathématiques, Université de Lille I, 59655 Villeneuve d’Ascq, France

Received 15 October 2007; accepted 26 January 2011 Available online 2 February 2011 Communicated by Alain Connes

Abstract Extension results, expressed in terms of complete boundedness, leading to necessary and sufficient conditions for the solvability of power moment problems with unbounded operator data are given. As an application, necessary and sufficient conditions for the existence of selfadjoint and normal extensions for some classes of commuting tuples of unbounded linear operators are obtained. © 2011 Elsevier Inc. All rights reserved. Keywords: Operator moment problems; Normal extensions; Selfadjoint extensions

1. Introduction Let Ω be a nonempty set, and let Σ be a σ -algebra of subsets of Ω. Let also H be a complex Hilbert space, and let B(H) be the algebra of all bounded, linear operators on H. A fundamental concept in functional analysis, connecting the objects above, is that of spectral measure (sometimes designed as a resolution of the identity, see [9], Definition 12.17). Given a spectral measure E : Σ → B(H), one can associate to each measurable function f : Ω → C a densely defined, closed operator in H, say f (T ), as a result of an integration, and the assignment f → f (T ) enjoys a long list of useful properties (see [9], Theorem 13.24). * Corresponding author.

E-mail addresses: [email protected] (E. Albrecht), [email protected] (F.-H. Vasilescu). 1 The author gratefully acknowledges the hospitality and the support of the Université de Lille I during the preparation

of this work. 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.01.015

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E. Albrecht, F.-H. Vasilescu / Journal of Functional Analysis 260 (2011) 2497–2517

The construction of a spectral measure is not always an easy matter. An important tool to perform such a construction is offered by Naimark’s dilation theorem (see, for instance, [7], Theorem 4.6). Naimark’s theorem shows that the construction a spectral measure, associated to a certain problem, can be often reduced to the construction of a (somewhat more accessible object called) positive measure, i.e., an operator-valued map F : Σ → B(H), assuming positive values, such that Fx,y := F (∗)x, y is a complex-valued measure for all x, y ∈ H, where ∗, ∗ is the inner product of H. Given a positive measure F : Σ → B(H), and a linear space S consisting of Σ-measurable complex-valued functions on Ω, we may consider the following subset of H: DF,S := x ∈ H, f ∈ L2 (Fx,x ), f ∈ S . It is easily seen that DF,S is a subspace of H. Moreover, by replacing, if necessary, the measure F by its compression to the closure of DF,S , we may assume, with no loss of generality, that the space DF,S is dense in H. In that case, we have a map assigning to each function f ∈ S a sesquilinear form sf on D = DF,S , given by sf (x, y) := f dFx,y , x, y ∈ D. Conversely, given an inner product space D, a linear space S consisting of Σ -measurable complex-valued functions on Ω, and a map assigning to each function f ∈ S a sesquilinear form sf on D, a positive measure F : Σ → B(H) such that sf (x, y) = f dFx,y for all x, y ∈ D is sought, where H is the Hilbert space obtained by completing the inner product space D. This is a special type of a moment problem which is described in [14] (where other details and references concerning such problems can be found). The main purpose of this paper is to characterize, in terms of complete boundedness and complete positivity (fruitful concepts introduced by Arveson [1], and extended to more general conditions by Powers [8]), the existence of extensions of some linear maps, defined on subspaces of fractions of continuous functions, whose values are sesquilinear forms on inner product spaces (see Theorem 2.5), following the scalar model initiated in [15] (see also [11]). In the third section, a solution to a moment problem as described above (see Theorem 3.2) is given. This result is then applied to obtain necessary and sufficient conditions for the existence of selfadjoint or normal extensions for certain commuting families of unbounded operators. As a sample, we present a particular case of Theorem 3.3. Let T1 , T2 : D → D be commuting symmetric operators in the inner product space D. The operators T1 , T2 admit commuting selfadjoint extensions if and only if for all l1 , l2 ∈ Z+ , m ∈ N and x1 , . . . , xm , y1 , . . . , ym ∈ D with m l l 1 + T12 1 1 + T22 2 xj , xj 1,

m l l 1 + T12 1 1 + T22 2 yj , yj 1,

j =1

j =1

and for all m × m two variable polynomial matrices p = (pj,k ) with the property supt1 ,t2 (1 + t12 )−l1 (1 + t22 )−l2 p(t1 , t2 )m 1, we have

m

pj,k (T1 , T2 )xk , yj 1,

j,k=1

where ∗ m is the norm in the algebra of m × m complex matrices.


2499

Theorem 3.3 gives a characterization of the existence of selfadjoint extensions whose joint spectral measure is supported by a given set, for some finite families of commuting symmetric operators. The existence of normal extensions is characterized by Theorem 3.4. We note that the present normal extension results are essentially different from the corresponding ones in [13] or [15]. Now, let us describe some of the main tools used in this work. Let Ω be a compact Hausdorff space and let C(Ω) be the algebra of all continuous, complex-valued functions, endowed with the sup-norm f ∞ = supω∈Ω |f (ω)|. As before, let H be a complex Hilbert space, and let B(H) be the algebra of all bounded, linear operators on H. We shall use the following results: Theorem A. Let Ψ : C(Ω) → B(H) be linear, positive and unital. Then Ψ is completely positive and completely contractive. This assertion is essentially Theorem 4 in [10], see also [7], Theorem 3.11 and Proposition 3.6. Theorem B. Let M ⊂ C(Ω) be a subspace with 1 ∈ M. If Φ : M → B(H) is a unital, complete contraction, then there exists a (completely) positive map Ψ : C(Ω) → B(H) extending Φ. This is a consequence of Arveson’s extension theorem (see [1] or [7], Corollary 7.6). Let Q be a family of non-null positive elements of C(Ω). We say that Q is a multiplicative family if (i) 1 ∈ Q, (ii) q , q ∈ Q implies q q ∈ Q, and (iii) if qh = 0 for some q ∈ Q and h ∈ C(Ω), then h = 0. Let C(Ω)/Q denote the algebra of fractions with numerators in C(Ω), and with denominators in the multiplicative family Q, which is a unital C-algebra (see, for instance, [16] for details). This algebra has a natural involution f → f¯, induced by the natural involution of C(Ω). To define a natural topological structure on C(Ω)/Q, for every q ∈ Q we define the space C(Ω)/q := f ∈ C(Ω)/Q; qf ∈ C(Ω) . Obviously, C(Ω)/q ⊃ C(Ω). Setting f ∞,q := qf ∞ for each f ∈ C(Ω)/q, the pair (C(Ω)/q, ∗ ∞,q ) becomes a Banach space (see also [15]). Remark 1.1. In the family Q there is a natural partial ordering, which is reflexive and transitive but not necessarily symmetric, written as q |q for q , q ∈ Q, meaning q divides q , that is, there exists a q ∈ Q such that q = q q. Assuming that the constant function 1 has no divisor in Q \ {1}, the relation q |q becomes symmetric too, but this hypothesis is not necessary for further development. Note also that if q , q ∈ Q and q |q , then C(Ω)/q ⊂ C(Ω)/q with continuous inclusion mapping iq ,q : C(Ω)/q → C(Ω)/q . Indeed, if q = q q and f ∈ C(Ω)/q , then f ∞,q = q qf ∞ q∞ q f ∞ = q∞ f ∞,q . For this reason, C(Ω)/Q = Banach spaces.

q∈Q C(Ω)/q

can be naturally regarded as an inductive limit of

2500


As noticed in [15], the algebras of fractions of continuous functions provide an appropriate framework for the study of positive measures having a certain decay related to the given multiplicative family. The authors are indebted to the referee for several useful comments and bibliographical references. 2. Positive maps on spaces of fractions Let Ω be a compact Hausdorff space, let Q ⊂ C(Ω) be a multiplicative family, and let C(Ω)/Q be the algebra of fractions with numerators in C(Ω), and with denominators in Q. We use throughout the text the notation q −1 to designate the fraction 1/q for any q ∈ Q. In each space C(Ω)/q we have a positive cone (C(Ω)/q)+ consisting of those elements f ∈ C(Ω)/q such that qf 0 as a continuous function. Let D be an inner product space (whose inner product will be denoted by ∗, ∗), and let SF (D) be the vector space of all sesquilinear forms on D. The Hilbert space completion of D will be denoted by H. Definition 2.1. Fix a q ∈ Q. A linear map ψ : C(Ω)/q → SF (D) will be called unital if ψ(1)(x, y) = x, y, x, y ∈ D. We say that ψ is positive if ψ(f ) is positive semidefinite for all f ∈ (C(Ω)/q)+ .

More generally, let Q0 ⊂ Q be nonempty. Let F = q∈Q0 C(Ω)/q, and let ψ : F → SF (D) be linear. The map ψ is said to be unital (resp. positive) if ψ|C(Ω)/q is unital (resp. positive) for all q ∈ Q0 .

Following [8], we shall also use a stronger positivity definition. A linear subspace F = q∈Q0 Fq of C(Ω)/Q (where Q0 ⊂ Q and Fq ⊂ C(Ω)/q for all q ∈ Q0 ) will be called symmetric, if for all q ∈ Q0 and f ∈ Fq we have f¯ ∈ Fq . We denote by M(F ) the linear space of all finite matrices over F , i.e. all matrices (fj,k )j,k∈N such that fj,k = 0 for at most finitely many (j, k) ∈ N2 . The set M(C(Ω)/Q) has the structure of a ∗-algebra in an obvious manner, and it can be identified with q∈Q M(C(Ω)/q). For q ∈ Q, let Kq denote the set of all f = (fj,k )j,k∈N in M(C(Ω)/q) such that for all ω ∈ Ω the matrix (q(ω)fj,k (ω))j,k∈N is positive semidefinite. Then an easy calculation shows that K := q∈Q Kq is a cone, which is admissible in the sense of Powers (see [8], Definition 3.1). Let φ : F → SF (D) be linear. We say that φ is completely positive (in the sense of Powers [8]), if for all matrices f = (fj,k )j,k∈N ∈ M(F ) ∩ K we have ∞ ∞

φ(fj,k )(xk , xj ) 0,

(xj )j ∈N ∈ DN .

(1)

j =1 k=1

Theorem 2.2. Let Q0 ⊂ Q be nonempty, let F = q∈Q0 C(Ω)/q, and let ψ : F → SF (D) be linear and unital. The map ψ is positive if and only if

sup ψ hq −1 (x, x) ; h ∈ C(Ω), h∞ 1 = ψ q −1 (x, x),

q ∈ Q0 , x ∈ D.


2501

If ψ : F → SF (D) is positive, there exists a unique regular positive B(H)-valued measure F on the Borel subsets of Ω such that ψ(f )(x, y) = f dFx,y , f ∈ F , x, y ∈ D. Ω

Proof. Let ψ : F → SF (D) be linear and unital. Set ψ1 = ψ|C(Ω). The map ψ1 is positive if and only if there exists a positive, unital, linear map Ψ1 : C(Ω) → B(H) such that ψ1 (h)(x, y) = Ψ1 (h)x, y, x, y ∈ D. This can be obtained by standard extension arguments, which will be briefly presented for the convenience of the reader. Assuming ψ1 positive, if h ∈ C(Ω) is positive, then 0 ψ1 (h)(x, x) h∞ x2 , because ψ1 is also unital. From this estimate we derive

ψ1 (h)(x, y) h∞ xy,

x ∈ D,

x, y ∈ D,

via the Cauchy–Schwarz inequality. Using the density of D in H and the Riesz theorem concerning the dual of H, we derive the existence of a positive operator Ψ1 (h) ∈ B(H) such that ψ1 (x, y) = Ψ1 (h)x, y, x, y ∈ D. Moreover, the assignment h → Ψ1 (h), h 0, is additive and positively homogeneous. As every function h ∈ C(Ω) is an algebraic combination of four positive functions, we derive easily the general assertion. Conversely, the existence of a positive, unital, linear map Ψ1 : C(Ω) → B(H) such that ψ1 (h)(x, y) = Ψ1 (x), y, x, y ∈ D clearly implies that ψ1 is positive. We use the fact that a linear functional θ : C(Ω) → C is positive if and only if it is continuous and θ = θ (1). Set ψ˜ q (h) = ψ(hq −1 ), h ∈ C(Ω), q ∈ Q0 . Suppose ψ positive and fix an x ∈ D. As C(Ω) ⊂ F and each positive function h ∈ C(Ω) is also positive in C(Ω)/q, the map ψ˜ q is positive on C(Ω). Put ψ˜ q,x (h) = ψ˜ q (h)(x, x), h ∈ C(Ω), which is a positive functional on C(Ω). Hence

ψ˜ q,x = sup ψ hq −1 (x, x) ; h ∈ C(Ω), h∞ 1 = ψ˜ q (1)(x, x) = ψ q −1 (x, x),

q ∈ Q0 ,

which is the stated condition. Conversely, the equality ψ˜ q,x = ψ(q −1 )(x, x) = ψ˜ q (1)(x, x) shows that ψ˜ q,x is positive on C(Ω). Then there exists a positive (Borel) measure μq,x on Ω such that ψ˜ q,x (h) = Ω h dμq,x , h ∈ C(Ω), for all q ∈ Q0 . The relation ψ˜ q1 ,x (hq1 ) = ψ(h)(x, x) = ψ˜ q2 ,x (hq2 ) for all q1 , q2 ∈ Q0 and h ∈ C(Ω) implies the equality q1 μq1 ,x = q2 μq2 ,x . Therefore, there exists a positive measure μx such that μx = qμq,x for all q ∈ Q0 . The equality μx = qμq,x shows the set {ω; q(ω) = 0} must be μx -null. Consequently, μq,x = q −1 μx , and the function q −1 is μx -integrable for all q ∈ Q0 . Moreover, the measure μx is uniquely determined because of the equality ψ(h)(x, x) = Ω h dμx , h ∈ C(Ω). Setting

2502


4μx,y = μx+y − μx−y + iμx+iy − iμx−iy , x, y ∈ D, we have the representation ψ(h)(x, y) = Ω h dμx,y for all h ∈ C(Ω) and x, y ∈ D, via the polarization formula. This shows, in particular, that the map ψ1 = ψ|C(Ω) is (unital and) positive. Therefore, there exists a unital positive map Ψ1 : C(Ω) → B(H) such that ψ1 (h)(x, y) = Ψ1 (h)x, y, x, y ∈ D. It is well known that the map Ψ1 has an integral representation Ψ1 (h) = Ω h dF, where F is a positive B(H)-valued measure F on the Borel subsets of Ω. As Fx,y = μx,y for all x, y ∈ D, and D is dense in H, the measure F is uniquely determined. If f ∈ F is arbitrary, then f = j ∈J hj qj−1 , with hj ∈ C(Ω), qj ∈ Q0 for all j ∈ J, J finite. We can write ˜ ψ(f )(x, x) = hj dμqj ,x = f dμx = f dFx,x ψqj (hj )(x, x) = j ∈J

j ∈J Ω

Ω

Ω

for all x ∈ D, from which we easily derive the formula in the statement. The measure F being positive, the map ψ must be also positive. 2

Remark 2.3. Let F := q∈Q0 C(Ω)/q for a nonempty Q0 ⊂ Q, and let ψ : F → SF (D) be a unital positive map on F . Set ψq = ψ|C(Ω)/q and ψq,x (f ) = ψq (f )(x, x) for all q ∈ Q0 , h ∈ C(Ω)/q and x ∈ D. If ψ˜ q is defined as in the proof of Theorem 2.2, we have the equality ψq,x =

ψq,x (f ) = sup ψ˜ q,x (h) = ψ˜ q,x = ψ q −1 (x, x)

sup

f ∞,q 1

h∞ 1

for all q ∈ Q0 and x ∈ D. We shall need the following fact: Lemma 2.4. Let Q be a multiplicative system on Ω and let q, q1 , q2 ∈ Q be such that q = q1 q2 . Suppose that ψ : C(Ω)/q → SF (D) is a positive, linear map satisfying ψ q −1 (x, x) > 0 and ψ q1−1 (x, x) > 0,

for all x ∈ D \ {0},

so that ∗, ∗q := ψ(q −1 )(∗, ∗) and ∗, ∗q1 := ψ(q1−1 )(∗, ∗) are scalar products on D. Let Dq and Dq1 denote the completions of D with respect to ∗, ∗q and ∗, ∗q1 , respectively. Then there exist uniquely determined linear maps Ψq : C(Ω) → B(Dq ) and Ψq,q1 : C(Ω) → B(Dq1 ) such that, for all h ∈ C(Ω),

Ψq (h)x, y

q

= ψ(h/q)(x, y),

x, y ∈ D,

(2)

and

Ψq,q1 (h)x, y

q1

= ψ(h/q1 )(x, y) = Ψq (hq2 )x, y q ,

x, y ∈ D.

Moreover, the maps Ψq and Ψq,q1 are unital, completely contractive and completely positive.

(3)


2503

Proof. By the positivity of ψ we obtain, for all h ∈ (C(Ω))+ and x ∈ D, ψ(h/q)(x, x) h∞ ψ q −1 (x, x) = h∞ x, xq .

(4)

If h is an arbitrary continuous function on Ω, it can be written in the form h = h1 − h2 + i(h3 − h4 ) with continuous functions satisfying 0 hj h∞ on Ω. Representing the sesquilinear forms (x, y) → ψ(hj /q)(x, y) in polar form, we conclude from (4) that we have for all x, y ∈ D with xq 1 and yq 1,

ψ(h/q)(x, y) 16h∞ . Therefore, there exists a unique operator Ψq (h) ∈ B(Dq ), such that (2) holds. Moreover, the linearity and positivity of ψ imply the linearity and positivity of Ψq . Because of

Ψq (1)x, y

q

= ψ q −1 (x, y) = x, yq ,

x, y ∈ D,

we see that Ψq (1) is the identity operator on Dq . Hence, by Theorem A in the Introduction, Ψq is completely positive and completely contractive. As q1 divides q, we have C(Ω)/q1 ⊂ C(Ω)/q and the restriction of ψ to C(Ω)/q1 is positive. Hence, replacing in the preceding arguments q by q1 , we obtain a unique linear map Ψq,q1 : C(Ω) → B(Dq1 ) such that

Ψq,q1 (h)x, y

q1

= ψ(h/q1 )(x, y) = ψ(hq2 /q)(x, y) = Ψq (hq2 )x, y q

for all h ∈ C(Ω) and x, y ∈ D, and as before, Ψq,q1 is completely positive and completely contractive. 2 Let “≺” be another partial ordering on the set Q. We say that “≺” is multiplicative if q ≺ q implies q |q for all q , q ∈ Q. As usually, a subset Q0 ⊂ Q is said to be cofinal if for every q ∈ Q we can find a q ∈ Q0 such that q ≺ q . In the next statement we shall use the notation ∗ n,∞ to designate the canonical norm of the C ∗ -algebra Mn (C(Ω)) of n × n matrices with entries from C(Ω). Similarly, for further use, the symbol ∗ n denotes the canonical norm in the C ∗ -algebra Mn (C). The norms used in the statement (a) of the next theorem were introduced in Remark 2.3. Theorem 2.5. Let Q be a multiplicative system on Ω endowed with a multiplicative partial ordering “≺”,

and let Q0 be a cofinal subset of Q with 1 ∈ Q0 . Let F = q∈Q0 Fq , where Fq is a vector subspace of C(Ω)/q such that q −1 ∈ Fq ⊂ Fq for all q ∈ Q0 and q ∈ Q0 , with q ≺ q. Let also φ : F → SF (D) be linear and unital, and set φq = φ|Fq , φq,x (∗) = φq (∗)(x, x) for all q ∈ Q0 and x ∈ D. The following two statements are equivalent: (a) The map φ extends to a unital, positive, linear map ψ on C(Ω)/Q such that, for all x ∈ D and q ∈ Q0 , we have: ψq,x = φq,x ,

where ψq = ψ|C(Ω)/q, ψq,x (∗) = ψq (∗)(x, x).

2504


(b) (i) φ(q −1 )(x, x) > 0 for all x ∈ D \ {0} and q ∈ Q0 . (ii) For all q ∈ Q0 , n ∈ N, x1 , . . . , xn , y1 , . . . , yn ∈ D with n φ q −1 (xj , xj ) 1,

n φ q −1 (yj , yj ) 1,

j =1

j =1

and for all (fj,k ) ∈ Mn (Fq ) with (qfj,k )n,∞ 1, we have

n

φ(fj,k )(xk , yj ) 1.

j,k=1

If F is a symmetric subspace of C(Ω)/Q, then (a) and (b) are equivalent to (c) φ is completely positive. Proof. If φ : F → SF (D) extends to a unital, positive ψ : C(Ω)/Q → SF (D) such that ψq,x = φq,x for all q ∈ Q0 and x ∈ D, then ψq,x = φq,x = φ(q −1 )(x, x) = −1 for all q ∈ Q and ψ(q −1 )(x, x), by Theorem 2.2 and Remark 2.3. Because of q−1 0 ∞ q the positivity of ψ, we obtain for all x ∈ D \ {0}: −1 φ q −1 (x, x) = ψ q −1 (x, x) ψ q−1 ∞ (x, x) = q∞ x, x > 0, and thus we have (i). Applying Lemma 2.4, we obtain, for all q ∈ Q, a uniquely determined linear map Ψq : C(Ω) → B(Dq ) satisfying (2), that is completely positive and completely contractive. In particular, for all n ∈ N, x1 , . . . , xn , y1 , . . . , yn ∈ D with n j =1

xj 2q =

n φ q −1 (xj , xj ) 1,

n

j =1

j =1

yj 2q =

n φ q −1 (yj , yj ) 1, j =1

and for all (fj,k ) ∈ Mn (Fq ) with (qfj,k )n,∞ 1, we have

n

n

Ψq (qfj,k )xk , yj q 1 φ(fj,k )(xk , yj ) =

j,k=1

j,k=1

which proves (ii). Hence (a) implies (b). Moreover, if F is a symmetric subspace of C(Ω)/Q and F = (fj,k )j,k=1,...,n ∈ Mn (F ) is positive in the natural order of Mn (C(Ω)/Q), then F ∈ Mn (Fq ) for some q ∈ Q0 and F is positive in Mn (C(Ω)/q), i.e. qF is positive in Mn (C(Ω)). By the complete positivity of Ψq , we obtain for all x1 , . . . , xn ∈ D, 0

n n

Ψq (qfj,k )xk , xj

j =1 k=1

q

=

n n j =1 k=1

Hence, we have shown that, in this case, (a) implies (c).

φ(fj,k )(xk , xj ).


2505

Suppose now that φ : F → SF (D) is a unital, linear map satisfying conditions (i) and (ii). In particular, for all q ∈ Q0 , the sesquilinear form ∗, ∗q := φ(q −1 ) defines a scalar product on D. We write Dq for the completion of D with respect to the corresponding norm ∗ q (see also Lemma 2.4) and still denote the extended scalar product by ∗, ∗q . As φ is unital, we have D1 = H. By (ii), for all h ∈ qFq , the sesquilinear form φ(h/q) extends to a uniquely determined bounded sesquilinear form on Dq . Hence there exists a unique operator Φq (h) ∈ B(Dq ), satisfying φ(h/q)(x, y) = φ q −1 Φq (h)x, y = Φq (h)x, y q for all x, y ∈ D. Note that Φq (1) is the identity operator on Dq . From condition (ii), we conclude that the unital linear map Φq : qFq → B(Dq ) is a complete contraction and hence, by Theorem B in the Introduction, it extends to a completely positive unital map Ψq : C(Ω) → B(Dq ). Thus,

Ψq (qf )x, y

q

= φ(f )(x, y),

f ∈ Fq , x, y ∈ D.

(5)

Applying Lemma 2.4 to the map f → Ψq (qf )∗, ∗q from C(Ω)/q to SF (D), we obtain a unique, unital, completely contractive and completely positive linear map Ψq,q1 : C(Ω) → B(Dq1 ), satisfying

Ψq,q1 (h)x, y

q1

= Ψq (hq2 )x, y q

(6)

and

Ψq,q (h)x, y = Ψq (hq2 )x, y h∞ xq yq , 1 1 1 q q 1

(7)

for all h ∈ C(Ω), x, y ∈ D, q1 , q ∈ Q0 , q2 ∈ Q, q1 q2 = q. For every q ∈ Q0 , we denote by Kq the set of all those families a = a(f, x, y) f ∈C(Ω)/q,x,y∈D ∈ CC(Ω)/q×D×D satisfying

a(f, x, y) qf ∞ xq yq ,

f ∈ C(Ω)/q, x, y ∈ D.

(8)

Endowed with the product topologies, the topological spaces Kq , q ∈ Q0 , and hence K := K q∈Q0 q are compact. For each q ∈ Q0 , let now Hq be the set of all a = (aq )q ∈Q0 ∈ K such that for all q ∈ Q0 with q ≺ q: the map f → aq (f, ∗, ∗) is a positive linear map from C(Ω)/q to SF (D) extending φ|Fq and satisfying aq (f, x, y) = aq (f, x, y),

f ∈ C(Ω)/q , x, y ∈ D.

(9)

2506


Clearly, the sets Hq are closed in K and hence compact. In order to prove q∈Q0 Hq = ∅, it therefore suffices to show that all finite intersections Hq1 ∩ · · · ∩ Hqn with q1 , . . . , qn ∈ Q0 are not empty. As Q0 is cofinal in Q, there exists some q ∈ Q0 such that qj ≺ q for j = 1, . . . , n. Let the operator Ψq ∈ B(Dq ) and, for all divisors q ∈ Q0 of q, the operators Ψq,q ∈ B(Dq ) be as constructed above. We define for all x, y ∈ D, q ∈ Q0 , and all f ∈ C(Ω)/q : aq (f, x, y) :=

0 Ψq,q (q f )x, yq

if q does not divide q, if q divides q

for all f ∈ C(Ω)/q , x, y ∈ D. Notice, that a := (aq )q ∈Q0 ∈ K by (7). Fix an index j ∈ {1, . . . , n} and let q ∈ Q0 satisfy q ≺ qj . Then there are qj , q˜j ∈ Q such that qj = q qj and q = qj q˜j . We conclude from (6) that for all f ∈ C(Ω)/q and x, y ∈ D we have: aq (f, x, y) = Ψq,q q f x, y q = Ψq q qj q˜j f x, y q = Ψq (qf )x, y q = Ψq,qj (qj f )x, y q = aqj (f, x, y), j

so that f → aq (f, ∗, ∗) is a linear map from C(Ω)/q to SF (D) which is positive by the positivity of Ψq,q and extends φ|Fq because of Fq ⊂ Fq and (5). Hence, a := (aq )q ∈Q0 ∈ Hq1 ∩ · · · ∩ Hqn . It follows that there exists some b = (bq )q∈Q0 ∈ q∈Q0 Hq . We define now, for all f ∈ C(Ω)/Q = q∈Q0 C(Ω)/q, x, y ∈ D: ψ(f )(x, y) := bq (f, x, y)

if f ∈ C(Ω)/q.

To see that this is well defined, suppose that f ∈ C(Ω)/qj , j = 1, 2 with q1 , q2 ∈ Q0 . As Q0 is cofinal in Q there exists some q ∈ Q0 such that q1 ≺ q and q2 ≺ q, and so both divide q. As b ∈ Hq , we have for all x, y ∈ D, bq1 (f, x, y) = bq (f, x, y) = bq2 (f, x, y), and the map (x, y) → bq (f, x, y) is a sesquilinear form on D. Thus, ψ : C(Ω)/Q → SF (D) is a well-defined linear map which is easily seen to be positive and extends φ. Given q ∈ Q0 , we see from the fact that b satisfies (8) that φq,x ψq,x x2q = φ q −1 (x, x) = φq,x q −1 φq,x , for all x ∈ D, which completes the proof of (a). Finally, suppose that F is a symmetric subspace of C(Ω)/Q and that the condition in (c) is satisfied. Then φ is a completely positive map on F in the sense of [8]. Note also that every f = f¯ ∈ C(Ω)/Q can be represented as f = h/q with h ∈ C(Ω) real valued and q ∈ Q0 (via the fact that Q0 is cofinal in Q). Setting g = h∞ /q, we have g ∈ Fq , and the difference g − f is positive in Fq (even when g − f is regarded as a matrix). In other words, with the terminology of [8], the space F is cofinal in C(Ω)/Q with respect to the admissible cone K (see Definition 2.1). We conclude from Theorem 3.7 in [8] that φ extends to a completely positive map ψ on C(Ω)/Q, showing that (c) implies (a), via Theorem 2.2. 2


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In the scalar case D = C, we identify SF (D) with C. Using the fact that, in this situation, bounded linear functionals on Fq are automatically completely bounded and the cb-norm coincides with the norm [7, Theorem 3.9], we obtain, as a particular case, Theorem 3.7 in [15]. Corollary 2.6. Let Q be a multiplicative system on Ω endowed with a multiplicative partial ordering “≺” and let Q0 be a cofinal subset of Q with 1 ∈ Q0 . Let F = q∈Q0 Fq , where Fq is a vector subspace of C(Ω)/q such that q −1 ∈ Fq ⊂ Fq for all q ∈ Q0 and q ∈ Q0 , with q ≺ q. A linear functional φ : F → C with φ(1) > 0 extends to a positive linear functional ψ : C(Ω)/Q → C with ψ|C(Ω)/q = φ|Fq for all q ∈ Q0 if and only if φ|Fq = φ(q) > 0 for all q ∈ Q0 . Proof. Without loss of generality we may assume φ(1) = 1. With the remarks above, the statement follows directly from Theorem 2.5. 2 For every q ∈ Q we denote by Z(q) the set {ω ∈ Ω; q(ω) = 0}, that is the zeros of q on Ω. For subsets Q1 of Q we write Z(Q1 ) := q∈Q1 Z(q). Combining Theorem 2.5 with Theorem 2.2, we show now: Corollary 2.7. Suppose that, with the hypotheses of Theorem 2.5, condition (b) is satisfied. Then there exists a positive B(H)-valued measure F on the Borel subsets of Ω such that φ(f )(x, y) =

f ∈ F , x, y ∈ D.

f dFx,y ,

(10)

Ω

For every such measure F and every q ∈ Q, we have F (Z(q)) = 0. Hence, if Q contains a countable subset Q1 with Z(Q1 ) = Z(Q), then F (Z(Q)) = 0. Proof. The existence of F with (10) follows from Theorems 2.5 and 2.2. If F is any such measure and q ∈ Q, since Q0 is cofinal in Q, there exists some q0 ∈ Q0 with q ≺ q0 , and hence Z(q) ⊂ Z(q0 ). Fix an x ∈ D \ {0} and an arbitrary ε > 0. With Aε := {ω ∈ Ω; 0 q0 (ω) ε/φ(q0−1 )(x, x)}, we obtain Fx,x Z(q)

Aε

q0 (ω) dFx,x (ω) q0 (ω)

Aε

ε φ(q0−1 )(x, x)q0 (ω) ε

φ(q0−1 )(x, x)

dFx,x (ω)

q0−1 dFx,x = ε

Ω

and thus Fx,x (Z(f )) = 0 for all x ∈ D, which implies F (Z(q)) = 0. By means of the σ -additivity of the scalar measures, we also obtain the last statement of the corollary. 2

2508


3. Selfadjoint and normal extensions In a classical paper by Fuglede (see [6]) dealing with the multidimensional power moment problem, an operator theoretic characterization of moment multi-sequences in terms of existence of some commuting selfadjoint extensions is given. This is an important motivation to study selfadjoint or normal extensions of some given linear transformations. To fix the terminology, let T1 , . . . , Tn be linear operators defined on a dense subspace D of a Hilbert space H. Assume that D is invariant under T1 , . . . , Tn and that T1 , . . . , Tn commute on D. We say that the tuple T = (T1 , . . . , Tn ) has a selfadjoint (resp. normal) extension if there exists a Hilbert space K containing H as a subspace, and a tuple A = (A1 , . . . , An ) consisting of commuting (in the sense, that the corresponding spectral measures commute) selfadjoint (resp. normal) operators in K such that D ⊂ nj=1 D(Aj ) and Tj x = Aj x, x ∈ D, for all j = 1, . . . , n. Remark 3.1. Our methods give primarily some “dilations” but, as in the proofs of Theorem 3.3 in [3] and Lemma 2 in [5], we can prove that these are actually extensions. Let us explain the meaning of this assertion, giving some direct arguments. Note that if S : D(S) ⊂ H → H is a symmetric operator with SD(S) ⊆ D(S), and if B : D(B) ⊂ K → K is a selfadjoint operator such that H ⊂ K, D(S) ⊂ D(B 2 ) and S k x = P B k x, x ∈ D(S), k = 1, 2, where P is the orthogonal projection of K onto H, then Sx = Bx for all x ∈ D(S). Indeed, we have Sx, Sx = Sx, P Bx = Sx, Bx and Bx, Sx = P Bx, Sx = S 2 x, x = B 2 x, x = Bx, Bx, for all x ∈ D(S). Therefore Sx − Bx2 = Sx, Sx − Sx, Bx − Bx, Sx + Bx, Bx = 0, for all x ∈ D(S). Similarly, if S : D(S) ⊂ H → H is an arbitrary linear operator and if B : D(B) ⊂ K → K is a normal operator such that H ⊂ K, D(S) ⊂ D(B), Sx = P Bx and Sx = Bx for all x ∈ D(S), then Sx = Bx for all x ∈ D(S). Let D be a complex inner product space and let φ : Pn → SF (D) be a linear unital map. We are interested to find a positive measure F on the Borel subsets of Rn , with values in B(H), where H denotes the completion of D, such that φ(p)(x, y) = p dFx,y for all p ∈ Pn and x, y ∈ D, which is, in fact, an operator moment problem (see, for instance, [14]). When such a positive measure F exists, we say that φ : Pn → SF (D) is a moment form and the measure F is said to be a representing measure for φ. When the representing measure F of φ vanishes on the complement of the closed subset K in Rn , we say that φ is a K-moment form. We intend to apply the characterization given by Theorem 2.5. As in [15], we shall use the following framework. Let Zn+ be the set of all multi-indices α = (α1 , . . . , αn ), i.e., αj ∈ Z+ for all j = 1, . . . , n.


2509

Let Pn be the algebra of all polynomial functions on Rn , with complex coefficients. We shall denote by t α the monomial t1α1 · · · tnαn , where t = (t1 , . . . , tn ) is the current variable in Rn , and α ∈ Zn+ . Let (R∞ )n = (R ∪ {∞})n , i.e., the Cartesian product of n copies of the one point compactification R∞ = R ∪ {∞} of the real line R. We consider the family Qn consisting of all rational functions of the form qα (t) = (1 + t12 )−α1 · · · (1 + tn2 )−αn , t = (t1 , . . . , tn ) ∈ Rn , where α = (α1 , . . . , αn ) ∈ Zn+ is arbitrary. The function qα can be continuously extended to (R∞ )n \ Rn for all α ∈ Zn+ . Moreover, the set Qn becomes a multiplicative family in C((R∞ )n ). Set also pα (t) = qα (t)−1 , t ∈ Rn , α ∈ Zn+ . β β Let Pn,α be the vector space generated by the monomials t β = t1 1 · · · tn n , with βj 2αj , j = 1, . . . , n, α ∈ Zn+ . For each p ∈ Pn,α , the rational function p/pα can be continuously extended to (R∞ )n \ Rn , and so it can be regarded as an element of C((R∞ )n ). Therefore, Pn,α is a subspace of C((R∞ )n )/qα = pα C((R∞ )n ) for all α ∈ Zn+ . Fix a closed set K ⊂ Rn . As we are primarily interested in the unbounded case, we will assume, in general, that K is unbounded. We write Kˆ for the closure of K in (R∞ )n , which is ˆ be the set of all functions from Qn , (extended to (R∞ )n and) rea compact space. Let Qn (K) ˆ let A(q) := {α ∈ Zn+ ; ˆ ˆ For q ∈ Qn (K) stricted to K. This is a multiplicative family in C(K). ˆ by defining q ≺ q if for all α ∈ A(q ) qα |K ≡ q}. We introduce a partial ordering in Qn (K) ˆ there exists some β ∈ A(q ) such that β − α ∈ Zn+ . In this case we have q = q qβ−α |K, which shows that the partial ordering “≺” is multiplicative. Notice, that the space Pn (K) of may be regarded as a subspace of the algebra of fracall restrictions to K of polynomials in Pn −1 ∈ P ⊂ C(K)/q ˆ ˆ ˆ ( K). Indeed, with P := and tions C(K)/Q q q α∈A(q) Pn,α |K, we have q n n , we see that P ⊂ P P . Moreover, because of P ⊂ P if β − α ∈ Z Pn (K) = q∈Qn (K) q n,α n,β q q ˆ + whenever q ≺ q . This discussion shows

conditions to apply Theorem 2.5 are fulfilled.

that the required ˆ we have Note also that if s = α∈A(q) sα ∈ Pq = α∈A(q) Pn,α |K for a fixed q ∈ Qn (K),

ˆ ˆ qs = α∈A(q) sα q = α∈A(q) sα qα ∈ C(K). Hence, s ∈ C(K)/q and s∞,q := supt∈Kˆ |sq| = supt∈Kˆ |sqα | for all α ∈ A(q), which provides the natural norm of the space Pq . Let φ : Pn → SF (D) be a unital, linear map. If φ is a K-moment form, then we clearly have ψ(p) = 0 for each polynomial p such that p|K = 0. Conversely, the linear map φ : Pn → SF (D) is said to be K-compatible if it has the property ˜ given that φ(p) = 0 whenever p|K = 0. In such a case, φ induces a linear map on Pn (K), say φ, ˜ ) = φ(p), for all f ∈ Pn (K) and p ∈ Pn with f = p|K. As the map φ˜ is unambiguously by φ(f defined by φ, it will be also denoted by φ. Notice, that K-compatibility can only be violated if K is contained in the set of zeros of a polynomial p = 0. In particular, if int K = ∅, then every linear map φ : Pn → SF (D) is K-compatible. In the case n = 1, every linear map φ : Pn → SF (D) is K-compatible for all unbounded closed subsets K of R It is clear that if φ : Pn → SF (D) is a positive, linear map such that |φ(p)(x, x)| supt∈K |qα (t)p(t)|φ(pα )(x, x) for all p ∈ Pn,α , x ∈ D and α ∈ Zn+ , then φ is K-compatible. Nevertheless, a stronger condition is necessary in order to derive the existence of a representing measure for such a form. Theorem 3.2. Let K ⊂ Rn be closed and unbounded, let D be a complex inner product space and let φ : Pn → SF (D) be a unital, linear map.

2510


The map φ is a K-moment form if and only if (i) φ(pα )(x, x) > 0 for all x ∈ D \ {0} and α ∈ Zn+ . (ii) The map φ is K-compatible and for all α ∈ Zn+ , m ∈ N and x1 , . . . , xm , y1 , . . . , ym ∈ D with m

φ(pα )(xj , xj ) 1,

j =1

m

φ(pα )(yj , yj ) 1,

j =1

ˆ we have and for all f = (fj,k ) ∈ Mm (Pq ) with supt∈K q(t)f (t)m 1, where q = qα |K,

m

φ(fj,k )(xk , yj ) 1.

j,k=1

Proof. Assume first that φ is a K-moment form and let F be a representing measure of φ carried by K. Then we have φ(p)(x, y) = K p dFx,y for all p ∈ Pn and x, y ∈ D. As noticed above, such a map φ is K-compatible. Therefore, it induces a linear and unital map φ : Pn (K) → ˆ and α ∈ A(q), we have for all x ∈ D \ {0}, SF (D). Moreover, if q ∈ Qn (K) φ q −1 (x, x) = φ(pα )(x, x) =

pα dFx,x

K

dFx,x = x2 > 0, K

˜ as pα 1 and F is positive. This shows that condition (i) in Theorem 2.5 holds for φ. ˆ ˆ To prove condition (ii), we define a unital, linear map ψ : C(K)/Qn (K) → SF (D) via the ˆ ˆ and x, y ∈ D. This definition is equation ψ(f )(x, y) = K f dFx,y for all f ∈ C(K)/Q n (K) ˆ ˆ we can find an index α ∈ Zn+ such that h := f qα |Kˆ ∈ correct, since for each f ∈ C(K)/Q n (K) ˆ Then we have C(K).

|f | dFx,x h∞

K

pα dFx,x = h∞ φ(pα )(x, x) < ∞, K

ˆ for all x ∈ D. This shows that each f ∈ C(K)/Q n is integrable with respect to each measure Fx,x , and hence integrable with respect to each measure Fx,y , for all x, y ∈ D, by the polarˆ ization formula. Moreover, the restriction of ψ to the space C(K)/q is clearly positive, for all ˆ ˆ q ∈ Qn (K). Thus, ψ is a unital, positive extension of φ. Setting, as before, ψq = ψ|C(K)/q, ˆ and x ∈ D, we ψq,x (∗) = ψq (∗)(x, x), φq = φ|Pq , φq,x (∗) = ψq (∗)(x, x) for all q ∈ Qn (K) have: φ q −1 (x, x) = ψ q −1 (x, x) = ψq,x φq,x φ q −1 (x, x), via Theorem 2.2. This shows that the map φ : Pn (K) → SF (D) satisfies condition (a) in Theorem 2.5.


2511

Fix α ∈ Zn+ , m ∈ N and x1 , . . . , xm , y1 , . . . , ym ∈ D such that m m φ q −1 (xj , xj ) = φ(pα )(xj , xj ) 1, j =1 m

j =1 m φ q −1 (yj , yj ) = φ(pα )(yj , yj ) 1,

j =1

j =1

ˆ Take f = (fj,k ) ∈ Mm (Pq ) with supt∈K q(t)p(t)m 1. Then, by (ii) in where q = qα |K. Theorem 2.5, we infer that

m

φ(fj,k )(xk , yj ) 1.

j,k=1

Conversely, assume that conditions (i) and (ii) are fulfilled. Because φ is K-compatible, it induces the unital, linear map φ : Pn (K) → SF (D), as noticed before. Moreover, conditions (i) and (ii) in the statement above imply conditions (i) and (ii) in Theorem 2.5. Note also that the function q(1,...,1) is null on the set Kˆ \ K, and this set contains the zeros of any function from ˆ Qn (K). By virtue of Theorem 2.5, and by Corollary 2.7 as well, it follows that the map φ extends to ˆ whose a unital, positive, linear map having a representing measure F˜ on the Borel sets of K, ˆ for all Borel sets B in Rn is support lies in K. Then the measure F given by F (B) := F˜ (B ∩ K) a positive B(H)-valued measure satisfying φ(p)(x, y) = p d F˜x,y = p dFx,y , p ∈ Pn . K

Hence, φ is a K-moment form.

Rn

2

Remark 1. In the statements of Theorem 3.2, the conditions (i) and (ii) may be replaced by similar conditions, in which the multi-index α runs only in a cofinal family in Zn+ (with respect to the partial ordering ξ ≺ η for two multi-indices ξ = (ξ1 , . . . , ξn ), η = (η1 , . . . , ηn ), meaning that ξj ηj , j = 1, . . . , n), which suffices to apply Theorem 2.5. Note also that if the map Pn p → p|K ∈ Pn (K) is injective, then Pq = Pn,α |K where α is ˆ the only multi-index such that A(q) = {α}, for all q ∈ Qn (K). We shall use the following well-known fact: If S = (S1 , . . . , Sn ) is a tuple of (not necessarily bounded) commuting normal linear operators in a Hilbert space K, in the sense that their spectral measures E1 , . . . , En commute, then there exists a unique spectral measure E on Cn , satisfying E(A1 × · · · × An ) = E1 (A1) · · · En (An ) for arbitrary A1 , . . . , An in σ -algebra Bor(C) of all Borel sets in C, and Sj x = Cn zj dE(z) x for all x in the domain D(Sj ) of Sj , j = 1, . . . , n. We call E the joint spectral measure of S and say that E has support in a closed set K ⊂ Cn if E(Cn \K) = 0. In particular, when S = (S1 , . . . , Sn ) consists of commuting selfadjoint operators, the support of their joint spectral measure lies in Rn (details concerning joint spectral measures and integrals can be found in [2]; see also [9] for some details).

2512


Theorem 3.3. Let K ⊂ Rn be closed and let T = (T1 , . . . , Tn ) be a tuple of symmetric, linear operators defined on a dense subspace D of a Hilbert space H. Assume that D is invariant under T1 , . . . , Tn and that T1 , . . . , Tn commute on D. Let φT : Pn → SF (D) be the linear unital map given by φT (p)(x, y) := p(T )x, y ,

p ∈ Pn , x, y ∈ D.

The tuple T admits a selfadjoint extension such that its joint spectral measure has support in K if and only if the map φT is K-compatible and for all α ∈ Zn+ , m ∈ N and x1 , . . . , xm , y1 , . . . , ym ∈ D with m

φT (pα )(xj , xj ) 1,

j =1

m

φT (pα )(yj , yj ) 1,

j =1

ˆ we have and for all p = (pj,k ) ∈ Mm (Pq ) with supt∈K q(t)p(t)m 1, where q = qα |K,

m

φT (pj,k )(xk , yj ) 1.

j,k=1

Proof. Assume first that T admits a selfadjoint extension. Then there exists a Hilbert space K containing H and a tuple A = (A1 , . . . , An ) consisting of commuting selfadjoint operators in K such that D ⊂ D(Aα ) and T α x = Aα x, x ∈ D, for all α ∈ Zn+ . Moreover, the joint spectral measure E : Bor(Rn ) of A vanishes on Rn \ K. If we define F : Bor(Rn ) → B(H) by F (S) := P E(S)|H for all S ∈ Bor(Rn ), where P is the orthogonal projection of K onto H, we have for all p ∈ Pn , x, y ∈ D, p dFx,y = Rn

p dEx,y = Rn

p(z) dE(z) x, y = p(A)x, y

Rn

= p(T )x, y = φT (p)(x, y).

Hence, F is a representing measure for φT which vanishes on Rn \ K and φT is a K-moment form. It now follows from Theorem 3.2 that the condition in the theorem must be satisfied. Conversely, suppose that the condition in the statement holds. First note that, φT (pα )(x, x) = pα (T )x, x x, x for all α ∈ Zn+ . Thus, if the condition in the statement is fulfilled for T , then the map φT satisfies conditions (i) and (ii) of Theorem 3.2. Hence, there exists a positive measure F on the Borel sets of Rn with values in B(H), which is a representing measure for φT , and which vanishes on Rn \ K. By the Naimark dilation theorem [7, Theorem 4.6], there exists a Hilbert space K, a bounded linear operator V : H → K and a selfadjoint spectral measure E on the Borel sets of Rn with values in B(K), such that F (A) = V ∗ E(A)V for all A ∈ Bor(Rn ). Moreover, the measure E also vanishes on Rn \ K. Because of F (K) = 1H and E(K) = 1K , the operator V is an isometry. Hence, identifying H with its isometric image V (H), we see that F (S) = P E(S)|H,

S ∈ Bor Rn ,


2513

where P denotes the orthogonal projection from K onto H. We then obtain selfadjoint operators A1 , . . . , An by D(Aj ) = x ∈ K; |tj |2 d E(t)x, x < ∞ K

and Aj x :=

tj dE(t) x,

x ∈ D(Aj ).

K

As we have 0 φT (|p|2 )(x, x) = K |p(t)|2 dF (t)x, x < +∞ for all p ∈ Pn and x ∈ D, using a well-known argument (see [3] or [5]), we infer that T α x, y = φT t α (x, y) =

t α d F (t)x, y =

K

t α d E(t)x, y = Aα x, y = P Aα x, y ,

K

provided x, y ∈ D. Hence, P Aα x = T α x for all x ∈ D, α ∈ Zn+ , which shows that the selfadjoint tuple A := (A1 , . . . , An ) is a dilation of T = (T1 , . . . , Tn ), and so A is actually a selfadjoint extension of T , via Remark 3.1. 2 Remark 2. When K = Rn , the previous statement becomes much simpler. Indeed, in this case the map φT is automatically Rn -compatible and we have Pq = Pn,α where α is the only multi-index such that q = qα . Therefore, with the notation of Theorem 3.3, the tuple T = (T1 , . . . , Tn ) admits a selfadjoint extension if and only if for all α ∈ Zn+ , m ∈ N and x1 , . . . , xm , y1 , . . . , ym ∈ D with m

m

j =1

j =1

pα (T )xj , xj 1,

pα (T )yj , yj 1,

and for all p = (pj,k ) ∈ Mm (Pn,α ) with supt∈Rn qα (t)p(t)m 1, we have

m

pj,k (T )xk , yj 1.

j,k=1

The particular case n = 2 has been already presented in the Introduction. We now consider situations in (C∞ )n , where C∞ is the one point compactification of the complex plane C. From now on, let Qn denote the family of all functions of the form qα (z) := (1 + |z1 |2 )−α1 · · · (1 + |zn |2 )−αn , with z = (z1 , . . . , zn ) ∈ Cn and α = (α1 , . . . , αn ) ∈ Zn+ . As in the real case, the functions qα extend continuously to (C∞ )n and Qn is a multiplicative family in C((C∞ )n ). For all α ∈ Zn+ , we denote by Tn,α the linear spaces generated by the monomials zξ zη := ξ ξ z11 · · · znn z1 η1 · · · zn ηn such that, for j = 1, . . . , n, we have ξj + ηj < 2αj or ξj = ηj = αj .

2514


n Note, that for all f ∈ Tn,α the function qα f extends continuously to (C ∞ ) and that Tn,α ⊂ Tn,β if αj βj , j = 1, . . . , n. We also consider the linear subspace Tn := α∈Zn+ Tn,α of C((C∞ )n )/Qn . Let now K be an unbounded closed subset of Cn and Kˆ its closure in (C∞ )n . We define (as ˆ denote the set in the discussion before Theorem 3.2) Tn (K) := {f |K; f ∈ Tn }, and let Qn (K) n ˆ of restrictions to K of all functions in Qn (extended to (C∞ ) ), which is a multiplicative set in ˆ let A(q) := {α ∈ Zn+ ; qα |Kˆ = q}. We introduce a partial ordering in this ˆ For q ∈ Qn (K) C(K). set by defining q ≺ q if for all α ∈ A(q ) there exists some β ∈ A(q ) such that β − α ∈ Zn+ . ˆ so the partial ordering “≺” is multiplicative. The space In this case we have q = q qβ−α |K, ˆ ˆ may be regarded as a subspace of the algebra of fractions C(K)/Q Tn (K) n (K). Indeed, with −1 ˆ Tq := α∈A(q) Tn,α |K, we have q ∈ Tq ⊂ C(K)/q and Tn (K) = q∈Qn (K) ˆ Tq . Moreover, because of Tn,α ⊂ Tn,β if β − α ∈ Zn+ , we see that Tq ⊂ Tq whenever q ≺ q . Hence, the conditions to apply Theorem 2.5 are fulfilled. Let now T = (T1 , . . . , Tn ) be a tuple of linear operators defined on a dense subspace D of a Hilbert space H such that Tj (D) ⊂ D and Tj Tk x = Tk Tj x for all j, k ∈ {1, . . . , n}, x ∈ D. In this situation, we may define a unital linear map φT : Tn → SF (D) by

φT zξ zη (x, y) := T ξ x, T η y ,

x, y ∈ D, α ∈ Zn+ ,

(11)

which extends by linearity to the space Tn of all polynomials in z1 , . . . , z1 , . . . , zn which is generated by these monomials. An easy induction proof shows that, for all α, β in Zn+ with β − α ∈ Zn+ , and x ∈ D \ {0}, we have 0 < x, x φT qα−1 (x, x) φT qβ−1 (x, x).

(12)

When the map φT : Tn → SF (D) is K-compatible (that is, φT (p) = 0 if p ∈ Tn and p|K = 0), then it induces a map from Tn (K) into SF (D), for which we keep the same notation. Theorem 3.4. Let K ⊂ Cn be closed and let T = (T1 , . . . , Tn ) be a tuple of linear operators defined on a dense subspace D of a Hilbert space H. Assume that D is invariant under T1 , . . . , Tn and that T1 , . . . , Tn commute on D. The tuple T admits a normal extension having a joint spectral measure whose support lies in K if and only if the map φT : Tn → SF (D) is K-compatible, and for all α ∈ Zn+ , m ∈ N and x1 , . . . , xm , y1 , . . . , ym ∈ D with m

φT qα−1 (xj , xj ) 1,

j =1

m

φT qα−1 (yj , yj ) 1,

j =1

ˆ we have and for all p = (pj,k ) ∈ Mm (Tq ) with supt∈K q(t)p(t)m 1, where q = qα |K,

m

φT (pj,k )(xk , yj ) 1.

j,k=1

Proof. If the condition of the theorem is fulfilled, and so we have a linear and unital map φT : Tn (K) → SF (D) induced by φT , then conditions (i) (by (12)) and (ii) (of Theorem 2.5) are satisfied for φT . Hence, by that theorem and Corollary 2.7, there exists a regular, positive


2515

ˆ such that (10) holds for φT and such that B(H)-valued measure F on the Borel sets of K, F (Kˆ \ K) = 0. Because φT is unital, F (K) is the identity operator on H. As in the proof of the preceding theorem, by the Naimark dilation theorem [7, Theorem 4.6], there exists a Hilbert space K containing H as a closed subspace and a spectral measure E : Bor(K) → B(K) such that F (A) = P E(A)|H for all A ∈ Bor(K), where P denotes the orthogonal projection from K onto H. For j = 1, . . . , n, let Nj be the corresponding normal operators with domains |zj |2 d E(z)x, x < ∞ D(Nj ) := x ∈ K; K

and Nj x :=

x ∈ D(Nj ).

zj dE(z) x, K

Going back to the arguments from [3] and [5], for all x, y ∈ D, j = 1, . . . , n, we have P Nj x, y = Nj x, y =

zj d E(z)x, y =

K

zj d F (z)x, y

K

= φT (zj |K)(x, y) = φT (zj )(x, y) = Tj x, y. Hence, P Nj x = Tj x for all x ∈ D, j = 1, . . . , n. Note also that Tj x2 = φT |zj |2 (x, x) = =

|zj |2 d F (z)x, x

K

|zj |2 d E(z)x, x = Nj x2 ,

K

for all x ∈ D, j = 1, . . . , n, which shows that the tuple N := (N1 , . . . , Nn ) is a normal extension of T = (T1 , . . . , Tn ) (see also the proof in Remark 3.1). Conversely, if T = (T1 , . . . , Tn ) admits a normal extension N := (N1 , . . . , Nn ) with joint spectral measure E having support contained in K, then, for all α ∈ Zn+ , the space D is contained in

α 2 α α

z d E(z)x, x < ∞ . D T ⊂ D N = x ∈ K; K

ˆ ˆ the function f is integrable on K with respect to the It follows that, for all f ∈ C(K)/(q α |K), positive scalar measure dEx,x := dE(∗)x, x. Using the decomposition 4Ex,y = Ex+y,x+y − ˆ ˆ → SF (D), defined by Ex−y,x−y + iEx+iy,x+iy − iEx−iy,x−iy we see that ψ : C(K)/Q n (K) ψ(f )(x, y) := K

f (z) d E(z)x, y ,

ˆ ˆ x, y ∈ D, f ∈ C(K)/Q n (K),

2516


is a linear map which is obviously unital and positive. Moreover, ψ is an extension of φT because N extends T . As the map ψ has support in K, the map φT should be K-compatible. ˆ ψq,x (∗) = ψq (∗)(x, x), φq = φ|Tq , φq,x (∗) = If we set φ = φT , ψq = ψ|C(K)/q, ˆ ψq (∗)(x, x) for all q ∈ Qn (K) and x ∈ D, we have: φ q −1 (x, x) = ψ q −1 (x, x) = ψq,x φq,x φ q −1 (x, x), via Theorem 2.2. This shows that the map φ : Tn (K) → SF (D) satisfies condition (a) in Theorem 2.5. Consequently, by Theorem 2.5, we infer that the condition in the statement is satisfied. 2 For the particular case n = 1, the set Q1 consists of all functions of the form ql := (1 + |z|2 )−l , with z ∈ C and l ∈ Z+ . Because of ql−1 (z) = (1 + zz)l =

l l k=0

k

zk zk

we obtain from Theorem 3.4: Corollary 3.5. Let S : D(S) ⊂ H → H be a linear operator such that SD(S) ⊂ D(S). The operator S admits a normal extension if and only if for all l ∈ Z+ , m ∈ N and x1 , . . . , xm , y1 , . . . , ym ∈ D(S) with m l l j =1 k=0

k

S k xj , S k xj 1,

m l l j =1 k=0

k

S k yj , S k yj 1,

and for all p = (pj,k ) ∈ Mm (T1 ), with supz∈C (1 + |z|2 )−l p(z)m 1, we have

m

φS (pj,k )(xk , yj ) 1.

j,k=1

Instead of working with (R∞ )n and (C∞ )n , we could also have taken the one point compactifications Rn ∪ {∞} and Cn ∪ {∞}. Instead of Qn , one then considers the multiplicative family {q k ; k ∈ Z+ }, where q(t) := 1 + t2 for all t ∈ Rn , respectively t ∈ Cn . Instead of Pn,α and Tn,α the linear hull Tn,k of the space of all polynomials of degree < 2k − 1 and the functions q j , 0 j k, have to be taken. The formulations and the proofs of the corresponding variants of Theorems 3.2, 3.3 and 3.4 are left to the reader. A different characterization of tuples of symmetric operators having selfadjoint extensions can be found in [14]. The actual statement of Theorem 3.4 is more explicit in terms of the given data. The case of one operator, covered by our Corollary 3.5, also occurs in [13], with a completely different approach. For a further characterization for subnormal operators see also Theorem 3 in [12] Let us finally mention that our main result (Theorem 2.5) has been recently extended to a noncommutative context in [4].


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References [1] W.B. Arveson, Subalgebras of C ∗ -algebras, Acta Math. 123 (1969) 141–224. [2] M.S. Birman, M.Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, D. Reidel Publishing Co., Dordrecht, 1987. [3] E. Bishop, Spectral theory for operators on a Banach space, Trans. Amer. Math. Soc. 86 (1957) 414–445. [4] A. Dosi, Local operator algebras, fractional positivity and the quantum moment problem, Trans. Amer. Math. Soc. 363 (2011) 801–856. [5] C. Foia¸s, Décompositions en opérateurs et vecteurs propres. I. Études de ces décompositions et leurs rapport avec les prolongements des opérateurs, Rev. Roumaine Math. Pures Appl. 7 (1962) 241–281. [6] B. Fuglede, The multidimensional moment problem, Expo. Math. 1 (1983) 47–65. [7] V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, Cambridge, 2002. [8] R.T. Powers, Selfadjoint algebras of unbounded operators. II, Trans. Amer. Math. Soc. 187 (1974) 261–293. [9] W. Rudin, Functional Analysis, McGraw–Hill Book Co., New York, 1973. [10] W.F. Stinespring, Positive functions on C ∗ -algebras, Proc. Amer. Math. Soc. 6 (1955) 211–216. [11] J. Stochel, Solving the truncated moment problem solves the full moment problem, Glasg. Math. J. 43 (2001) 335–341. [12] J. Stochel, F.H. Szafraniec, On normal extensions of unbounded operators II, Acta Sci. Math. (Szeged) 53 (1989) 153–177. [13] J. Stochel, F.H. Szafraniec, The complex moment problem and subnormality: A polar decomposition approach, J. Funct. Anal. 159 (1998) 432–491. [14] F.-H. Vasilescu, Operator moment problems in unbounded sets, in: Recent Advances in Operator Theory and Related Topics. The Bela Szökefalvi–Nagy Memorial Volume, in: Oper. Theory Adv. Appl., vol. 127, Birkhäuser Verlag, Basel, 2001, pp. 613–638. [15] F.-H. Vasilescu, Spaces of fractions and positive functionals, Math. Scand. 96 (2005) 257–279. [16] B.L. van der Waerden, Algebra, vol. 2, Frederick Ungar Publ. Co., New York, 1970.


Composition operators on the Newton space Gordon MacDonald a,∗,1 , Peter Rosenthal b,1 a Department of Mathematics and Statistics, University of Prince Edward Island, Charlottetown, PEI, C1A 4P3,

Canada b Department of Mathematics, University of Toronto, Toronto, ON, M5S 3G3, Canada

Received 10 August 2009; accepted 10 January 2011 Available online 26 January 2011 Communicated by D. Voiculescu

Abstract We investigate properties of composition operators Cφ on the Newton space (the Hilbert space of analytic functions which have the Newton polynomials as an orthonormal basis). We derive a formula for the entries of the matrix of Cφ with respect to the basis of Newton polynomials in terms of the value of the symbol φ at the non-negative integers. We also establish conditions on the symbol φ for boundedness, compactness, and self-adjointness of the induced composition operator Cφ . A key technique in obtaining these results is use of an isomorphism between the Newton space and the Hardy space via the Binomial Theorem. © 2011 Elsevier Inc. All rights reserved. Keywords: Newton space; Hilbert space; Composition operator

1. Introduction If μ is a probability measure on the complex plane C with finite moments |z|n dμ(z) < ∞, C

* Corresponding author.

E-mail addresses: [email protected] (G. MacDonald), [email protected] (P. Rosenthal). 1 The authors acknowledge the support of NSERC Canada.

0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.01.011

G. MacDonald, P. Rosenthal / Journal of Functional Analysis 260 (2011) 2518–2540

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we can define a Hilbert space P 2 (μ) as the closure of the set of polynomials in L2 (C, μ). Some well-known examples are: • If dμ(reiθ ) = δ1 (r) dθ is the normalized arclength measure on the unit circle, this construction yields the Hardy space H 2 (D) of functions analytic on the open unit disk D. • If dμ(reiθ ) = π1 χ[0,1] (r)r dr dθ is the normalized area measure on the unit disk, this construction yields the Bergman space A2 (D) of functions analytic on the open disk D. 1 |Γ (x+iy)|2 • If dμ(x + iy) = 2π Γ (2x+2) dy dγ (x), where γ (x) is the discrete measure on the real line with unit masses at {− 12 + n2 : n ∈ Z+ }, this construction yields the Newton space N 2 (P) of functions analytic on the right half-plane P = {z ∈ C: (z) > − 12 }. (A simpler definition of the Newton space will be given below.)

If X is any domain in the complex plane, let A(X) denote the space of all complex-valued functions which are analytic on X. An analytic function φ, which maps X to X, induces a composition operator Cφ on A(X) defined by Cφ f = f ◦ φ. If μ is a measure on C such that P 2 (μ) can be identified with a vector subspace of A(X), we can consider the composition operator Cφ acting on P 2 (μ). The first question that arises is: For which φ does Cφ map P 2 (μ) into P 2 (μ)? Then, assuming it is the case that Cφ maps P 2 (μ) into P 2 (μ), one can ask a number of questions concerning the operator-theoretic properties of Cφ as an operator on P 2 (μ), such as: • For which φ is Cφ a bounded operator? Can we find a simple formula for (or effective bounds on) the norm of Cφ in terms of φ? • For which φ is Cφ normal? Self-adjoint? Unitary? • For which φ is Cφ compact? Hilbert–Schmidt? In a Schatten p-class? • What is the spectral picture for Cφ ? • What is the invariant subspace lattice of Cφ ? Such questions, and others, have been investigated for composition operators acting on spaces of the type P 2 (μ). The study has been particularly fruitful in the case of H 2 (D) (see [2,12,13]). In this paper we begin the investigation of such questions for composition operators on the Newton space N 2 (P). One reason for the success in analyzing composition operators on H 2 (D) is that in that space the unilateral shift is given by the multiplication operator Mw (defined by (Mw f )(w) = wf (w)) 2 which shifts the orthonormal basis {w n }∞ n=0 . Any composition operator acting on H (D) intertwines this unilateral shift and some other multiplication operator (Cφ Mw = φ(Mw )Cφ ) and this link of composition operators to the unilateral shift is very useful in a number of settings. For example, it is the basis of one proof of Littlewood’s subordination principle, which states that if φ is a self-map of D such that φ(0) = 0, then Cφ is a contraction on H 2 (D) (see [13, p. 13]). It is an easy consequence of this that, on H 2 (D), Cφ is bounded for every φ mapping D into D. In the Newton space N 2 (P), the unilateral shift on the orthonormal basis of Newton polynomials also has a link to composition operators. As will be shown in the next section, I − ∗ is itself a composition operator. Exploiting this connection, as well as an isometric isomorphism between N 2 (P) and H 2 (D) related to the Binomial Theorem, we derive a number of results concerning composition operators on the Newton space N 2 (P).

2520


Interest in the Newton space was initially motivated by the question of whether the Cesaro operator is subnormal. The Cesaro operator C is defined on 2 (N) as follows: for a sequence 2 {f (n)}∞ n=1 in (N) 1 f (j ). n n

C(f )(n) =

j =1

In 1971, Kriete and Trutt [8] gave an alternate description of the Cesaro operator as a multiplication operator on a Hilbert space of analytic functions (a version of the Newton space N 2 (P) mentioned above) and showed that the inner product on that space could be implemented via a measure on C, thus showing that the Cesaro operator is subnormal. In subsequent papers [5,6,9,11], the methods for constructing the measure were simplified, and additional structural properties of the Newton space were discovered which, in turn, led to additional information about the operator-theoretic properties of the Cesaro operator. In [3] and [10], similar methods are used to uncover information about related operators known as quantum Casaro operators. Before we begin investigating properties of composition operators on the Newton space N 2 (P), we give a short summary of the basic facts about the Newton space. In most cases we shall give only a brief description of the proofs. For more detailed background and proofs, we direct the reader to [11]. 2. Background on the Newton space If you are unfamiliar with the Newton space, the definition given above may not seem to be the most natural. In fact, the original definition given for the space is much different. For n ∈ N and α ∈ C, let (α)n = (α)(α + 1) · · · (α + n − 1). (Note that the ordinary factorial n! = (1)n .) Then for n = 0, 1, 2, . . . , the n-th Newton polynomial is Nn (z) =

(z)(z − 1) · · · (z − (n − 1)) (−z)n = (−1)n . n! n!

Theorem 2.1. (See Theorem 1.2 in [11].) The Newton polynomials {Nn (z)}∞ n=0 form an orthonor2 mal basis for N (P); therefore N 2 (P) = F (z) =

∞ n=0

an Nn (z): F 2 =

∞

|an |2 < ∞ .

n=0

The standard way to obtain this result (as shown in [11]) is via the Mellin transform. This gives a map from L2 (R+ , e−t dt) to N 2 (P) which maps the Laguerre polynomials to the Newton polynomials. Using the Plancherel Theorem for the Mellin transform, one can construct the measure μ using the orthogonality of the Newton polynomials. In the development of Newton space theory, the formulation given in the above theorem came first, and is the usual beginning point for study of the Newton space, with the measure being constructed later. From that point of view, one can show directly (without resort to the Mellin transform) that the functions described in Theorem 2.1 are analytic on the appropriate right halfplane.


Lemma 2.2. If F (z) =

∞

n=0 an Nn (z)

and

∞

2 n=0 |an |

2521

< ∞, then F is analytic on (z) > − 12 .

Proof. Rather than using the Mellin transform, one can give a direct proof using Raabe’s ratio test (see [7] for a description of the test): ∞ 1 |(−z)n |2 2 F (z) F n!2 n=0

so setting un =

|(−z)n |2 n!2

we have

−1 un |n − 1 − z|2 n2 = = 1 + − 1 . un−1 n2 |n − 1 − z|2 Setting cn =

n2 −1 |n − 1 − z|2

we see that ncn > δ > 1

⇔

n2 δ > +1 2 n |n − 1 − z|

⇔

|n − 1 − z|2 n < 2 n+δ n

⇔ ⇔

n2 − 2n Re(1 + z) + |1 + z|2 δ δ > 1 if and only if Re(1 + z) > series converges for Re(z) > − 12 (and diverges for Re(z) < − 12 ). 2 The backwards unilateral shift has a particularly nice form on N 2 (P). Theorem 2.3. The map : N 2 (P) → N 2 (P) defined by ( F )(z) = F (z) − F (z + 1) is the backwards unilateral shift on the orthonormal basis {Nn (z)}∞ n=0 .

δ 2

and so the above

2522


Proof. Note that N0 (z) = 1, so clearly N0 = 0, and if n > 0 (−z)n (−(z + 1))n − n! n!

−z + n − 1 −z − 1 − Nn−1 (z) = Nn−1 (z). = n n

( Nn )(z) =

2

Corollary 2.4. For z ∈ P, let T (z) = z + 1 (the function which translates by one to the right), and let CT denote the composition operator on N 2 (P) induced by T (so CT (f ) = f ◦ T ). Then CT = I − . This close connection between a particular composition operator and the backwards shift will be exploited to obtain results about general composition operators. Using Theorem 2.3, we can compute the coefficients in the expansion of any function F ∈ N 2 (P) with respect to the Newton polynomials, as follows. Theorem 2.5. (See Theorem 1.3 in [11].) If F ∈ N 2 (P) has expansion F (z) = then, for n = 0, 1, 2 . . . ,

∞

n=0 an Nn (z),

n

n (−1)j F (j ). an = n F (0) = j j =0

Proof. Since all Newton polynomials except N0 (z) = 1 are such that Nn (0) = 0, we must have

F, 1 = F (0). Then, for n = 0, 1, 2 . . . , ∗ an = F, Nn = F, n 1 = n F, 1 n j n n (−1)j CT F, 1 = (I − CT ) F, 1 = j =

n

n (−1)j F (j ) j

j =0

by the Binomial Theorem.

j =0

2

Theorem 2.6. (See Theorem 1.1 in [11].) The space N 2 (P) has reproducing kernel Kλ (z) =

Γ (z + λ + 1) Γ (z + 1)Γ (λ + 1)

for each λ ∈ P. Proof. For λ ∈ P, Kλ (z) =

∞ n=0

Nn (λ)Nn (z) =

∞ (−λ)n (−z)n n=0

(1)n (1)n


2523

which is the Gauss hypergeometric series which has closed form as given in the theorem (see [1]). 2 3. Matrices with respect to the Newton polynomials If ψ is a bounded analytic function on P then it induces a bounded operator Mψ on N 2 (P) defined for F ∈ N 2 (P) by (Mψ F )(z) = ψ(z)F (z) (see below). The following theorem shows that the matrix of such an operator with respect to the Newton polynomial orthonormal basis depends only on the values of ψ at the non-negative integers and gives a formula for the entries of the matrix. Theorem 3.1. If ψ is a bounded analytic function on P, then (1) for k = 0, 1, . . . , n k (k)

k (j ) ψ (z + k − j )Nn−(k−j ) (z), ψNn (z) = j j =0

(2) and for l = 0, 1, 2, . . . , (n+l)

ψNn (z) =

n

n + l (j +l) ψ (z + n − j )Nj (z) j +l j =1

n + l (l) + ψ (z + n). l

Hence the operator Mψ : N 2 (P) → N 2 (P) defined by (Mψ F )(z) = ψ(z)F (z) is a bounded operator on N 2 (P) whose matrix with respect to the Newton polynomials is 0 if i < j, [Mψ ]ij = i ( (i−j ) ψ)(j ) otherwise j

for i, j = 0, 1, 2, . . . . Proof. A function ψ , bounded and analytic on P, has non-tangential limits a.e. with respect to Lebesgue measure on the boundary line of P. Since the restriction of μ to this line is absolutely continuous with respect to Lebesgue measure, we may extend ψ to the closure of P, yielding a function in L∞ (μ), which in turn induces a bounded multiplication operator Mψ on L2 (μ). Since N 2 (P) can be viewed as a subspace of L2 (μ) which is invariant for Mψ , Mψ will be bounded there as well.

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The proofs of formulas (1) and (2) are by induction. The equation in (1) is clearly true when k = 0. In the general case we shall use the fact that ( φNn )(z) = ( φ)(z)Nn (z) + φ(z + 1)Nn−1 (z). Suppose that (1) is true for a certain value k. Then we have k (k+1)

k (j ) φ (z + k − j )Nn−(k−j ) (z) φNn (z) = j j =0

k k (j +1) = φ (z + k − j )Nn−(k−j ) (z) j j =0

+ (j ) φ (z + k − j − 1)Nn−(k−j )−1 (z) k k (j +1) φ (z + k − j )Nn−(k−j ) (z) = j j =0

+

k k (j ) φ (z + k − j − 1)Nn−(k−j )−1 (z) j j =0

=

k+1

j =1

+

(j ) k φ (z + k − j + 1)Nn−(k−j )−1 (z) j −1

k k (j ) φ (z + k − j − 1)Nn−(k−j )−1 (z) j j =0

= φ(z + k + 1)Nn−(k+1) (z) k+1

k k (j ) + φ (z + k − j + 1)Nn−(k−j )−1 (z) + j −1 j j =1

+ (k+1) φ (z) k+1

k + 1 (j ) φ (z + k + 1 − j )Nn−(k+1−j ) (z). = j j =0

The proof of (1) now follows by induction. To prove (2), we use induction on the parameter l, using the case from (1) with k = n as the initial case l = 0. Suppose (2) is true for some value of l. Then n

n + l

(n+l+1) (j +l) φNn (z) = φ (z + n − j )Nj (z) j +l j =0

n + l (l+1) + φ (z + n). l


2525

Then, similarly to the proof of (1), n

(n+l+1)

n + l + 1 (j +l+1) φNn (z) = φ (z + n − j )Nj (z) j +l+1 j =0

n+l n + l (l+1) + + φ (z + n) l+1 l n

n + l + 1 (j +l+1) = φ (z + n − j )Nj (z) j +l+1 j =1

n + l (l+1) + φ (z + n). l+1 By Theorem 2.5, the (i, j ) entry of the matrix for Mφ is ( (i) φNj )(0). Applying (1) and (2) gives the result. 2 Corollary 3.2. (See [8].) The matrix of the operator M

1 z+1

: N 2 (P) → N 2 (P) with respect to the

basis of Newton Polynomials is the Cesaro matrix; that is 0 if i < j, [M 1 ]ij = 1 z+1 i+1 otherwise for i, j = 0, 1, 2, . . . . We can exploit the specific form of the backwards shift on the Newton polynomials as the identity operator minus a composition operator to derive similar matrix formulas for general composition operators. Theorem 3.3. If φ maps P into P, then the matrix of Cφ with respect to the Newton polynomials is [Cφ ]ij =

i (−i)k (−φ(k))j (1)k (1)j k=0

for i, j = 0, 1, 2, . . . . Proof. Note that K0 (z) = N0 (z) = 1 and it is well known (and easily verified) that Cφ∗ Kw = Kφ(w) . Using these facts we see that i

i Cφ∗ Ni = Cφ∗ ∗ N0 = Cφ∗ I − CT∗ K0 i i i i ∗ k ∗ k ∗ k (−1) CT (−1) Kk = Cφ K0 = Cφ k k k=0

=

i

k=0

i (−1)k Kφ(k) . k

k=0

2526


Thus i i (−1)k Nj , Kφ(k) [Cφ ]ij = Cφ Nj , Ni = Nj , Cφ∗ Ni = k k=0

i i

i (−i)k (−φ(k))j (−1)k Nj φ(k) = . = k (1)k (1)j k=0

2

k=0

If ζ ∈ C is such that (ζ ) 0 (where (ζ ) denotes the real part of ζ ) then Tζ (z) = z + ζ maps P into P. Since these form a commutative family of functions (under composition), the corresponding composition operators CTζ form a commutative family of operators, which also commute with the backwards shift since = I − CT1 . Using this fact directly, we obtain a simpler expression for the entries of the matrix of CTζ with respect to the Newton polynomials. Theorem 3.4. If ζ ∈ C is such that (ζ ) 0, then the matrix of CTζ with respect to the Newton polynomials is 0 if i > j, [CTζ ]ij = (−ζ )j −i if i j (1)j −i for i, j = 0, 1, 2, . . . . Proof. For i, j = 0, 1, 2, . . . , ∗ [CTζ ]ij = CTζ Nj , Ni = Nj , CT∗ζ i K0 ∗ = Nj , i CT∗ζ K0 = i Nj , Kζ 0 if i > j, 2 = Nj −i (ζ ) if i j. As is well known, a bounded analytic function ψ : D → C induces a multiplication operator (Mψ f )(w) = ψ(w)f (w). Such an operator Mψ : H 2 (D) → H 2 (D), defined for f ∈ H 2 (D) by n is called an analytic Toeplitz operator. If ψ(w) = ∞ n=0 an w , then the matrix of such an Mψ ∞ n with respect to the basis {w }n=0 is [Mψ ]ij =

0 ai−j

if i < j, if i j.

From the Binomial Theorem, (1 − w)ζ =

∞ (−ζ )n n=0

(1)n

wn .

Thus CTζ acting on N 2 (P) and M ∗ acting on H 2 (D) have the same matrix representations (1−w)ζ (with respect to the standard orthonormal bases of the respective spaces) and thus are unitarily equivalent. Known results about analytic Toeplitz operators yield much information about CTζ .


2527

We need the following simple lemma related to half-angle formulas. Lemma 3.5. For θ ∈ [−π, π], θ θ−sgn(θ)π 1 − eiθ = 2 sin ei( 2 ) 2 where sgn(θ ) =

1 if θ 0, −1 if θ < 0.

Proof. Using half-angle formulas, we can write 1 − eiθ in polar form as follows: 1 − eiθ = 1 − cos θ − i sin θ

θ θ 2 θ = 2 sin − i 2 sin cos 2 2 2

θ θ θ sin − i cos = 2 sin 2 2 2

θ θ θ = 2 sin sgn(θ )(−i) cos + i sin 2 2 2 θ sgn(θ)π θ = 2 sin e−i 2 ei 2 2 θ i( θ−sgn(θ)π ) 2 = 2 sin e . 2 2 Theorem 3.6. If ζ ∈ C is such that (ζ ) 0, then the composition operator CTζ acting on N 2 (P) has the following properties: (1) CTζ is a bounded operator and, if a = (ζ ) (the real part of ζ ) and b = (ζ ) (the imaginary part of ζ ), then

a a b π 2a exp |b| ; exp b arctan CTζ = r(CTζ ) = 2 |ζ | a 2 (2) CT∗ζ is subnormal. Proof. That CT∗ζ subnormal follows immediately from the fact it is unitarily equivalent to an analytic Toeplitz operator. The norm and spectral radius of an analytic Toeplitz operator Mφ is Mφ = r(Mφ ) = sup φ(w): w ∈ D (see [12]), and, by unitarily equivalence, this is also the norm and spectral radius of CTζ . So to establish (1), we must determine the supremum of the function φ(w) = |(1 − w)ζ | for w ∈ D.

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By the Maximum Modulus Principle, this supremum is a maximum and occurs on the boundary of the disk. Setting a = (ζ ) and b = (ζ ), and using Lemma 3.5 with w = eiθ : θ−sgn(θ)π a+ib (1 − w)ζ = 2 sin θ ei( 2 ) 2 a θ θ−sgn(θ)π ib = 2 sin ei( 2 ) 2 θ a −b( θ−sgn(θ)π ) 2 = 2 sin e . 2 The first term achieves its maximum of 2a at θ = ±π (since a 0). The second term approaches its supremum as θ approaches 0 from the right (if b > 0) or from the left (if b < 0), and in either |b|π case the supremum is e 2 . This gives the norm inequality in (1). To get the exact equality, we must find the maximum of the function θ a −b( θ−sgn(θ)π ) 2 f (θ ) = 2 sin e 2 on the interval θ ∈ [−π, π]. This can be done using standard calculus. The only critical numbers of the above function are

a . 0, ±π and 2 arctan b Since f (0) = 0, the maximum doesn’t occur at 0. Note that f (±π) = 2a

a a a b = 2 . and f 2 arctan exp b arctan b |ζ | a

If we show that

2

a |ζ |

a

b 2a exp b arctan a

the theorem will be proven. Canceling 2a from both sides and taking the logarithm of both sides we see this inequality is equivalent to

b a + b arctan 0. a ln √ 2 2 a a +b To see that this inequality is true, set b = λa. Then the inequality is equivalent to

1 h(λ) = − ln 1 + λ2 + λ arctan(λ) 0. 2 As h is an even function of λ it suffices to prove this for λ > 0. Note that h(0) = 0 and h (λ) = arctan(λ) which is positive for λ > 0 so the result is proven. 2


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4. Connecting via the Binomial Theorem In this section we shall be moving between the Newton space N 2 (P) and the Hardy space H 2 (D). For clarity of exposition, we shall be consistent with the convention that in what follows, z shall always represent a complex number in P and w shall always represent a complex number in D. Therefore, functions N 2 (P) shall be expressed in terms of z ∈ P while functions in H 2 (D) shall be expressed in terms of w. We shall also be consistent with using the notation Cf to denote a composition operator induced by f and Mg to denote a multiplication operator induced by g. These operators may be acting on either H 2 (D) or N 2 (P), but the space should be clear from the context (or from the use of z or w as a variable). 2 For n = 0, 1, 2, . . . , let en (w) = w n , so {en (w)}∞ n=0 is the usual orthonormal basis for H (D). 2 2 Let U : N (P) → H (D) be the linear map such that U (Nn ) = en . Then U is an isometry which extends by continuity to an isomorphism from N 2 (P) to H 2 (D). While any two separable Hilbert spaces can be identified via such an isomorphism which maps basis vectors to basis vectors, this particular U is imbued with additional structure via the Binomial Theorem. The map U , while approached via different methods, is actually the inverse of the map used in [8]. 1 For η ∈ D, let kη (w) = 1−ηw , which is the reproducing kernel for H 2 (D). Theorem 4.1. The map U : N 2 (P) → H 2 (D) has the following properties: (1) (2) (3) (4)

For η ∈ D, fη (z) = (1 − η)z is in N 2 (P) and U (fη ) = kη ; For λ ∈ P, gλ (w) = (1 − w)λ is in H 2 (D) and U (Kλ ) = gλ ; Mw U = U ∗ ; For ζ ∈ C with (ζ ) 0, M(1−w)ζ U = U CT∗ζ .

Proof. From the Binomial Theorem (Ufη )(z) = U

∞

ηn Nn (z) =

n=0

=

∞ n=0

ηn w n =

∞

ηn (U Nn )(z)

n=0

1 = kη (w). 1 − ηw

Similarly (U Kλ )(z) = U

∞ n=0

=

∞

Nn (λ)Nn (z) =

∞

Nn (λ)(U Nn )(z)

n=0

Nn (λ)w n = (1 − w)λ = gλ (w).

n=0

The third claim follows directly from the fact that both Mw and ∗ are unilateral shifts on their respective orthonormal bases. The fourth claim follows from the matrix representations obtained above, or from the following. For λ ∈ P,

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(M(1−w)ζ U Kλ )(z) = M(1−w)ζ gλ (w) = (1 − w)ζ (1 − w)λ = (1 − w)ζ +λ = gλ+ζ (w)

= (U Kλ+ζ )(z) = U CT∗ζ Kλ (z), and, since {Kλ }λ∈P spans N 2 (P), the result follows.

2

5. Composition operators induced by linear self-maps It is an easy exercise to characterize the linear functions φ(z) = mz + b which map P into 1 itself: we must have m real and positive and (b) m−1 2 , or m = 0 and (b) > − 2 . Which of 2 these linear functions give rise to bounded composition operators on N (P)? By Theorem 3.6 in the case where m = 1, Cφ is bounded if and only if (b) m−1 2 = 0. If 1 m = 0, Cφ is a rank-one operator and it is easy to see that we only require that (b) > m−1 2 = −2. We begin by showing unboundedness in the case m > 1. Theorem 5.1. If φ(z) = mz + b, with m > 1, then the corresponding composition operator Cφ does not map N 2 (P) into N 2 (P). Proof. If (b) < m−1 2 , then, by the comments above, φ does not map P into P itself, so Cφ z cannot map N 2 (P) into N 2 (P). If (b) m−1 2 , then for each η ∈ D, consider fη (z) = (1 − η) . 1

As we have seen above, fη ∈ N 2 (P) and, by the Binomial Theorem, fη = (1 − |η|2 ) 2 . Now (Cφ fη )(z) = (1 − η)mz+b . Note (Cφ fη ) = (1 − η)mz+b − (1 − η)m(z+1)+b

= (Cφ fη ) 1 − (1 − η)m so

n n (Cφ fη ) = (Cφ fη ) 1 − (1 − η)m . So

n n (Cφ fη )(0) = (Cφ fη )(0) 1 − (1 − η)m and thus ∞ 2 1 − (1 − η)m 2n Cφ fη 2 = (Cφ fη )(0) n=0

which is finite if and only if |1 − (1 − η)m | < 1. If we choose η ∈ D close to −1, this inequality will fail and therefore Cφ does not map N 2 (P) into N 2 (P). 2 Next, we consider the case where m is in the interval (0, 1]. We shall show that the unitary operator in Theorem 4.1 intertwines Cφ on N 2 (P) and a weighted composition operator defined on H 2 (D) (or at least on a dense set of H 2 (D)).


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The composition operator part of this weighted composition operator on H 2 (D) is induced by the map ψm (w) = 1 − (1 − w)m . It is not immediately obvious that ψm even maps D to itself, but this will follow from Lemma 5.3 below. Theorem 5.2. If φ(z) = mz + b, with m in (0, 1] and (b) weighted composition operator acting on H 2 (D), where

m−1 2 ,

let M(1−w)b Cψm be the

ψm (w) = 1 − (1 − w)m . Then U Cφ fη = (M(1−w)b Cψm )∗ Ufη where U and fη are as in Theorem 4.1. Proof. Note that, for fixed η ∈ D,

z (Cφ fη )(z) = (1 − η)mz+b = (1 − η)b 1 − 1 − (1 − η)m = (1 − η)b fψm (η) . Hence U Cφ fη = U (1 − η)b fψm (η) = (1 − η)b Ufψm (η) = (1 − η)b kψm (η) while ∗ ∗ b (M(1−w)b Cψm )∗ Ufη = Cψ∗ m M(1−w) b kη = Cψm (1 − η) kη

= (1 − η)b Cψ∗ m kη = (1 − η)b kψm (η) = (1 − η)b kψm (η) .

2

Thus, for a given m and b, the two operators Cφ and M(1−w)b Cψm are either both bounded or both unbounded. To prove boundedness we need the following technical lemma. Lemma 5.3. Let ψm (w) = 1 − (1 − w)m , m ∈ (0, 1) and θ ∈ [−π, 0) ∪ (0, π]. Then, for any f ∈ H 2 (D), iθ

f ψm e

1 (2 − 2m )|2 sin θ2 |m

f 2 .

Proof. Note that

f ψm (w) = f, kψ

m (w)

and

f kψm (w)

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1 1 = 2 1 − |ψm (w)| 1 − |1 − (1 − w)m |2 1 = . m (1 − w) + (1 − w)m − |1 − w|2m

kψm (w) 2 =

Now if w = eiθ then, using Lemma 3.5 θ i( θ−sgn(θ)π ) 2 . 1 − w = 2 sin e 2 Hence |1 − w|

2m

θ 2m = 2 sin 2

and

θ m (θ − sgn(θ )π)m . (1 − w) + (1 − w) = 2 sin 2 cos 2 2 m

m

So, kψm (w) 2 = = =

(1 − w)m

1 + (1 − w)m − |1 − w|2m 1

|2 sin θ2 |m (2 cos (θ−sgn(θ)π)m ) − |2 sin θ2 |2m 2

1 1 |2 sin θ2 |m 2 cos (θ−sgn(θ)π)m − |2 sin θ2 |m 2

.

We must determine the maximum value of the function g(θ ) =

1 2 cos

(θ−sgn(θ)π)m 2

− |2 sin θ2 |m

on [π, π]. As this function is even that is equivalent to determining its maximum on [0, π]. Letting x = π−θ 2 , we see that, for θ ∈ [0, π],

m 1 = fm (x) = 2 cos(mx) − 2 cos(x) g(θ ) where x ∈ [0, π2 ), and so we must determine the minimum value of fm on [0, π2 ). Claim. For m ∈ (0, 1), fm is non-decreasing on [0, π2 ]. Proof. It suffices to show that the derivative of fm (x) fm (x) = −2m sin(mx) + 2m m cosm−1 (x) sin(x)


2533

is non-negative for x ∈ [0, π2 ]. When x = 0 or x π3 , this is easily verified. When x < π3 , turn the problem sideways, and consider for each fixed x, gx (m) = (2 cos(x))m−1 sin(x) − sin(mx) as a function of m. Note that g(1) = 0 and that

m−1

gx (m) = 2 cos(x) ln 2 cos(x) sin(x) − x cos(mx) < 0 so gx (m) is positive and thus fm (x) is positive as well. (Thanks to Undergraduate Research Assistant Sam Arnold at the University of Prince Edward Island for providing this proof.) 2 Thus fm (x) fm (0) = 2 − 2m and hence g(θ )

1 2−2m

and the result follows.

Theorem 5.4. If φ(z) = mz + b, with m in (0, 1), and (b) > composition operator Cφ is bounded on N 2 (P).

m−1 2 ,

2

then the corresponding

Proof. We consider the operator M(1−w)b Cψm acting on H 2 (D), as in Theorem 4.1 and consider the case where b is in R. If f ∈ H 2 (D), then, using Lemma 5.3 and integral tables to evaluate the last integral, we obtain 1 M(1−w)b Cψm f = 2π

π

2

1 2π

1 − eiθ 2b Cψ f eiθ 2 dθ m

−π

2b π 1 1 2 sin θ f 22 dθ m 2 2−2 |2 sin θ |m 2

−π

1 1 f 22 π 2 − 2m

π

θ 2 sin 2

2b−m dθ

0 π

1 1 f 22 22b−m 2 π 2 − 2m

2

(sin u)2b−m du 0

√ Γ (b − 1 1 f 22 22b−m 2 π π 2 − 2m Γ (b −

m−1 2 ) . m−2 2 )

Since M(1−w)b Cψm is bounded in this case, so is Cφ with the same bound (by Theorem 4.1). To obtain the result for complex b, compose Cφ with CT where T (z) = z + ia and use Theorem 3.6. 2 Note that, in the case where b ∈ R, we obtain the norm bound 1 22b+1 Γ (b − Cφ 2 √ m π 2 − 22m Γ (b −

m−1 2 ) . m−2 2 )

2534


Summarizing, we have the following: Theorem 5.5. For φ(z) = mz + b, the composition operator Cφ : N 2 (P) → N 2 (P) is unbounded if any of the following is true: (1) m < 0 or m > 1, (2) m = 0 and (b) − 12 , (3) m ∈ (0, 1] and (b) < m−1 2 ; and is bounded if any of the following is true: (1) m = 0 and (b) > − 12 , (2) m ∈ (0, 1) and (b) > m−1 2 , (3) m = 1 and (b) m−1 = 0. 2 We are left with the case where m ∈ (0, 1) and φ(z) = mz + conjecture regarding the associated composition operators. Conjecture 5.6. If m ∈ (0, 1) and φ(z) = mz + Cφ : N 2 (P) → N 2 (P) is bounded.

m−1 2 ,

m−1 2 .

We make the following

then the associated composition operator

6. Composition operators induced by automorphisms Next we consider the problem of determining which automorphisms of P give rise to bounded composition operators, and which of these are invertible. It is well known that the automorphisms of the upper half-plane are given by ψ(z) =

az + b cz + d

where a, b, c, d ∈ R and ad − bc > 0. Since U (z) = i(z + 12 ) maps the half-plane P to the upper half-plane, all automorphisms of P are of the form φ(z) = (U −1 ◦ ψ ◦ U )(z) where ψ is as above. Which of these give rise to bounded composition operators on N 2 (P)? As the next theorem shows, it is those that fix infinity. Theorem 6.1. If φ : P → P is an automorphism, then Cφ : N 2 (P) → N 2 (P) is a bounded operator only if φ(∞) = ∞. Proof. Suppose φ does not fix infinity. Then there is some complex number w0 with (w0 ) = − 12 and φ(w0 ) = ∞. With no loss of generality, we may assume w0 = − 12 since, by Theorem 3.6, the vertical shift Tib (z) = z + bi where b = (w0 ) gives rise to a bounded (and invertible) composition operator, and we can compose φ with Tib . Suppose we have done this, but still call the composed function φ. Then, by the remarks preceding the statement of this theorem, we must have that


φ(z) = −i α −

β i(z +

1 2)

2535

1 − , 2

where α and β are real numbers with β > 0. For each n ∈ N, let λn = − 12 + βn , and consider Cφ∗ Kλn 2 Kλn 2

.

We shall show that the above sequence is unbounded, and hence Cφ is unbounded. Following the same rate of growth terminology and notation of [4], we say two functions f (n) and g(n) have the same rate of growth if there exist positive constants k and K so that k|g(n)| |f (n)| K|g(n)| for sufficiently large n, and write f (n) g(n). As per the usual big-oh terminology, we say f (z) = O(g(z)) if there exists a positive constant K such that |f (z)| K|g(z)| when |z| is sufficiently large. First note that Kλn 2 = =

Γ (2λn + 1) Γ (λn + 1)2 Γ ( 2β n ) Γ ( 12 + βn )2

n

(since the denominator converges and the Gamma function has a simple pole at 0 so the numerator is asymptotically equivalent to n). Next consider ∗ C Kλ 2 = Γ (2(φ(λn )) + 1) n φ |Γ (φ(λn ) + 1)|2 =

Γ (2n) |Γ (n +

1 2

− iα)|2

.

Stirling’s formula (see [1]) states that Γ (z) =

2π z

z

1 z 1+O . e z

Applying this formula, we see that 1 (2n)2n Γ (2n) √ n e2n and

2 2n+1 Γ n + 1 − iα 1 n 2 n e2n+1

2536


so that 22n √ → ∞. n − iα)|2

Γ (2n) |Γ (n + Thus Cφ∗ , and hence Cφ , is unbounded.

1 2

2

A striking dichotomy between composition operators on the Hardy space and composition operators on the Newton space arises when we consider the question of which automorphisms give rise to invertible composition operators (that is, composition operators which are bounded with bounded inverse). In the Hardy case, all automorphisms give rise to invertible composition operators, while in the Newton case, as we show in the next theorem, only automorphisms which are vertical translations do. Theorem 6.2. If φ : P → P is an automorphism, then Cφ : N 2 (P) → N 2 (P) is an invertible operator if and only if φ(z) = z + ri for some real number r. Proof. By Theorem 6.1, φ must fix infinity, and so must be a linear automorphism. By Theorem 5.5, in order for Cφ to be bounded we must that φ(z) = mz + b for some m ∈ (0, 1] and (b) = m−1 2 . But we must also have m = 1, or else the composition operator induced by φ −1 (z) = m1 z − mb has coefficient of z greater than 1, and hence gives rise to an unbounded operator by Theorem 5.5. If Cφ was invertible in this case, its inverse would agree with Cφ −1 on a dense set and hence be unbounded. Thus m = 1 and (b) = m−1 2 = 0 and so b is imaginary, b = ri for some r ∈ R and the result is proven. 2 7. Compactness of composition operators Much work has been done to classify the compact composition operators on H 2 (D). The general heuristic idea that an analytic map φ : D → D induces a compact composition operator on H 2 (D) when the range of φ does not get too close to the boundary of the disk too often eventually led to a complete description of such operators in terms of the Nevanlinna Counting function (see [13]). As we have seen, many aspects of composition operators are more complicated in the Newton space N 2 (P) than in the Hardy space H 2 (D). However, it seems that a similar general heuristic holds: φ : P → P induces a compact composition operator on N 2 (P) when the range of φ does not get too close to the line {z ∈ C: (z) = − 12 } too often, however there seems to be additional conditions involving asymptotic behavior at infinity. We give a few preliminary results in that direction. First recall the following standard definition. Definition 7.1. A compact operator T acting on a Hilbert space H is a Hilbert–Schmidt operator if for any orthonormal basis {en }∞ n=1 of H , ∞ n=1

T en 2 < ∞


2537

and the Hilbert–Schmidt norm of T is defined as T HS =

∞

1 2

T en 2

.

n=1

In our case this gives the following characterization of the Hilbert–Schmidt composition operators. Theorem 7.1. If φ : P → P is an analytic map, then Cφ : N 2 (P) → N 2 (P) is a Hilbert–Schmidt operator if and only if Γ (2 Re(φ(z)) + 1) dμ(z) < ∞, |Γ (φ(z) + 1)|2 P

where μ is the measure implementing the inner product on N 2 (P). Proof. From above, Cφ 2HS =

∞ Cφ (Nn )2 , n=0

so Cφ 2HS

∞ Cφ (Nn )2 = n=0

∞

Nn φ(z) 2 dμ(z) = n=0 P

∞ (−φ(z))n (−φ(z))n dμ(z) = n!n! n=0 P

∞ (−φ(z))n (−φ(z))n = dμ(z). n!n! P n=0

This series is 2 F1 (φ(z), φ(z); 1, 1), a Gauss hypergeometric function, which has closed form 2 F1

Γ (2 Re(φ(z)) + 1) φ(z), φ(z); 1, 1 = |Γ (φ(z) + 1)|2

(see [1]) so the theorem is proven.

2

Note that, by Theorem 2.6 Kz 2 = Kz , Kz =

Γ (2 Re(z) + 1) |Γ (z + 1)|2

2538


and so this function is continuous on compact subsets of P. This gives a large class of compact composition operators. Theorem 7.2. If φ : P → P is analytic and Ran(φ) is a compact subset of P, then Cφ : N 2 (P) → N 2 (P) is a Hilbert–Schmidt operator. Proof. The hypothesis imply that the function z → theorem follows from Theorem 7.1. 2

Γ (2 Re(φ(z))+1) |Γ (φ(z)+1)|2

is bounded on P, and so the

We do have some other compact composition operators at hand. When m ∈ (0, 1), the map ψm (w) = 1 − (1 − w)m is such that if w is on the unit circle and w = 1, |ψm (w)| < 1, and so ψ clearly fails to have finite angular derivative at all such w. The map ψm fixes the point 1, and the image of the closed unit disk under ψm , is a “sideways raindrop” shaped region in D ∪ {1} with point at 1 whose boundary is not tangent to the circle at 1. So ψm also fails to have finite angular derivative at w = 1. Since ψm is univalent, the corresponding composition operator Cψm is compact on H 2 (D) (see Corollary 3.21 in [2]). From this we get some compact composition operators on N 2 (P). Theorem 7.3. For m ∈ [0, 1) and b 0, the composition operator Cφ on N 2 (P) induced by φ(z) = mz + b is compact. Proof. If m = 0, Cφ is a rank-one operator. If m ∈ (0, 1), then by Theorem 4.1, Cφ is unitarily equivalent to (M(1−w)b Cψm )∗ acting on H 2 (D). By the comments preceding this theorem, Cψm is compact, and the condition that b 0 implies that M(1−w)b is bounded, hence the result follows. 2 8. Self-adjoint composition operators Suppose Cφ is a composition operator acting on a Hilbert space of analytic functions with reproducing kernel {kλ }. Then Cφ is self-adjoint if and only if Cφ∗ kλ = Cφ kλ (since {kλ } span the space). From this we obtain the following. Proposition 8.1. The composition operator Cφ : N 2 (P) → N 2 (P) is self-adjoint if and only if, for all y and z in P, Γ (z + φ(y) + 1) Γ (z + 1)Γ (φ(y) + 1)

=

Γ (φ(z) + y + 1) . Γ (φ(z) + 1)Γ (y + 1)

Setting z = 0 in the above equation gives, for all y ∈ P,

Γ (y + φ(0) + 1) Γ φ(0) + 1 = Γ (y + 1) and hence we must have φ(0) = 0. Setting z = 1 we obtain Γ (φ(y) + 2) Γ (2)Γ (φ(y) + 1)

=

Γ (φ(1) + y + 1) . Γ (φ(1) + 1)Γ (y + 1)


2539

Using Γ (t + 1) = tΓ (t) and Γ (t¯) = Γ (t), we obtain φ(y) =

Γ (y + φ(1) + 1) Γ (y + 1)Γ (φ(1) + 1)

= Kφ(1) (y) − 1.

Substituting y = 1 shows that r = φ(1) must be a real number, which is, of course, in the interval (− 12 , ∞). It is easy to see that we must have r = φ(1) 0 or else φ will not map P to P (since large real numbers get mapped to numbers less than − 12 ). Also we must have r 1 or else the imaginary axis will not map into P. Thus our only candidates for self-adjoint composition operators are φr (z) = Kr (z) − 1 for r ∈ [0, 1]. When r = 1 we obtain the identity operator and when r = 0 we obtain the rank-one projection onto the constants. For other values of r, self-adjointness implies that the equation Γ (z + φr (y) + 1) Γ (z + 1)Γ (φr (y) + 1)

=

Γ (φr (z) + y + 1) Γ (φr (z) + 1)Γ (y + 1)

must hold for all z, y ∈ P. Consider this equation when z = 2 and y = 3. Note that (r + 2)(r + 1) Γ (2 + r + 1) −1= −1 Γ (2 + 1)Γ (r + 1) 2

φr (2) = Kr (2) − 1 = and φr (3) = Kr (3) − 1 =

(r + 3)(r + 2)(r + 1) Γ (3 + r + 1) −1= −1 Γ (3 + 1)Γ (r + 1) 6

so the right-hand side is Γ (z + φr (y) + 1) Γ (z + 1)Γ (φr (y) + 1)

=

Γ (2 + φr (3) + 1) Γ (2 + 1)Γ (φr (3) + 1)

=

(φr (3) + 2)(φr (3) + 1) 2

=

( (r+3)(r+2)(r+1) + 1)( (r+3)(r+2)(r+1) ) 6 6 2

while the left-hand side is Γ (φr (z) + y + 1) Γ (φr (2) + 3 + 1) = Γ (φr (z) + 1)Γ (y + 1) Γ (φr (2) + 1)Γ (3 + 1) =

(φr (2) + 3)(φr (2) + 2)(φr (2) + 1) 6

=

( (r+2)(r+1) + 2)( (r+2)(r+1) + 1)( (r+2)(r+1) ) 2 2 2 . 6

2540


Setting these equal and simplifying, we see that the above equation simplifies to r 2 (r − 1)(r + 1) = 0, which gives, r = 0 or r = 1 (as r = −1 does not give a map from P to P). Hence, we obtain the following. Theorem 8.2. The only self-adjoint composition operators Cφ : N 2 (P) → N 2 (P) correspond to φ(z) = z or φ(z) = 0. In closing, we note that there is actually an indexed family of Newton spaces. In these spaces, the Newton polynomials are not orthonormal, but orthogonal. For α 0, we define the Newton space Nα as the following Hilbert space of analytic functions: ∞ ∞ 2 2 Γ (n + α + 1) Nα = F (z) = 0. We shall provide sufficient conditions under which Iε,h → Iε in the so-called Mosco sense (see Definition 6.1). In particular, our sufficient conditions consist of separate Γ -lim inf conditions for ψh and φh as well as a suitable joint recovery sequence condition in the same spirit of [20]. The present functionalconvergence results are new even in the gradient flow case p = 2. Before closing this section let us mention that elliptic-in-time regularizations of parabolic problems are classical in the linear case and some results can be found in the monograph by Lions and Magenes [18]. As for the nonlinear case, one has to recall the paper by Ilmanen [16] where the WED is used in order to prove the existence and partial regularity of the so-called Brakke mean curvature flow of varifolds. Apart from the WED formalism, a number of alternative contributions to other variational formulations to nonlinear evolutionary problems, e.g., Brézis–Ekeland’s principle [7,8], have been considered in order to characterize entire trajectories as critical points of functionals (see also [5, 12–15] for linear cases, and [24,25,32,33,11,29–31] for nonlinear cases). The advantages of the WED formalism over former variational approaches are that it relies on a true minimization procedure (plus passage to the causal limit) and that it directly applies to doubly nonlinear evolution equations.

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G. Akagi, U. Stefanelli / Journal of Functional Analysis 260 (2011) 2541–2578

This is the plan of the paper. Section 2 is devoted to enlist and comment assumptions and present some preliminary facts to be used throughout. In Section 3, we prove the existence of strong solutions of the Euler–Lagrange equation (1.2)–(1.4), whereas Section 4 brings to a proof of the coincidence between global minimizers for Iε and strong solutions of the Euler–Lagrange equation. In Section 5, we check for the causal limit ε → 0 and Section 6 is concerned with the functional convergence of the sequence of WED functionals Iε,h as h → 0. A typical example of a doubly nonlinear PDE fitting the current analysis is α(ut ) − ∇ · |∇u|m−2 ∇u = 0 with α monotone, non-degenerate, and polynomially growing at ∞. Details on the respective WED functional approach as well as its approximation by functional convergence are presented in Section 7. Eventually, Appendix A contains a proof of a technical lemma. 2. Assumptions and preliminary material 2.1. Notation, subdifferential, Gâteaux differential Let us collect in the following some preliminary material along with relevant notation. Let ϕ be a proper (i.e., ϕ ≡ ∞), lower semicontinuous and convex functional from a normed space E into (−∞, ∞]. Then the subdifferential operator ∂E ϕ : E → E ∗ of ϕ is defined by ∂E ϕ(u) := ξ ∈ E ∗ ; ϕ(v) − ϕ(u) ξ, v − u E for all v ∈ E with the domain D(∂E ϕ) := {u ∈ D(ϕ); ∂E ϕ(u) = ∅} and obvious notation for the duality pairing. It is known that ∂E ϕ is maximal monotone in E × E ∗ [6,4]. The functional ϕ is said to be Gâteaux differentiable at u (resp., in E), if there exists ξ ∈ E ∗ such that ϕ(u + he) − ϕ(u) = ξ, e E h→0 h lim

for any e ∈ E

at u (resp., for all u ∈ E). In this case, ξ is called a Gâteaux derivative of ϕ at u and denoted by dE ϕ(u). We can naturally define an operator dE ϕ from E into E ∗ . If ϕ is Gâteaux differentiable at u, then the set ∂E ϕ(u) consists of the single element dE ϕ(u). Throughout this paper, we denote by A the graph of a possibly multivalued operator A : E → E ∗ . Hence [u, ξ ] ∈ A means that u ∈ D(A) and ξ ∈ A(u). 2.2. Assumptions Let V and V ∗ be a uniformly convex Banach space and its dual space with norms | · |V and | · |V ∗ , respectively, and a duality pairing ·,· V and let X be a reflexive Banach space with a norm | · |X and a duality pairing ·,· X such that X → V

and V ∗ → X ∗

with densely defined compact canonical injections. Let ψ : V → [0, ∞) be a Gâteaux differentiable convex functional and let φ : V → [0, ∞] be a proper lower semicontinuous convex functional.


2545

Let p ∈ [2, ∞) and m ∈ (1, ∞) be fixed, and introduce our basic assumptions: p

(A1) there exist constants C1 , C2 > 0 such that C1 |u|V ψ(u) + C2 for all u ∈ V ; p

p

(A2) there exist constants C3 , C4 > 0 such that |dV ψ(u)|V ∗ C3 |u|V + C4 for all u ∈ V ; (A3) there exists a non-decreasing function 1 in R such that |u|m for all u ∈ D(φ); X 1 |u|V φ(u) + 1 (A4) there exists a non-decreasing function 2 in R such that m for all [u, η] ∈ ∂X φX , |η|m X ∗ 2 |u|V |u|X + 1 where φX : X → [0, ∞] denotes the restriction of φ on X. Note that, by (A2) and the definition of subdifferential, the continuity of ψ in V also follows. Furthermore, we can also verify by (A3) and (A4) that φX is continuous in X and D(∂X φX ) = X. Moreover, from the definition of subdifferential and (A1), it also holds that

p C1 |u|V dV ψ(u), u V + C2

(A1) with

C2

(A2)

for all u ∈ V

:= C2 + ψ(0) 0. Let us manipulate (A2) in order to get p

ψ(u) ψ(0) + dV ψ(u), u V C3 |u|V + 1 for all u ∈ V

with C3 := ψ(0) + C4 + C3 + 1 0. Similarly, by (A4), we can also obtain φ(u) 3 |u|V |u|m for all u ∈ X X +1 with a non-decreasing function 3 in R. Finally, let us give a precise definition of WED functionals of our main interest. ⎧ ⎨ 0T e−t/ε (ψ(u (t)) + 1ε φ(u(t))) dt 1,p 1 Iε (u) = ⎩ if u ∈ W (0, T ; V ), u(0) = u0 and ψ(u (·)), φ(u(·)) ∈ L (0, T ), ∞ else with an initial data u0 ∈ V and a parameter ε > 0. Then we remark that D(Iε ) = u ∈ Lm (0, T ; X) ∩ W 1,p (0, T ; V ); u(0) = u0 , as the above remarks imply e

−T /ε

T C1 0

T 0

p u (t) dt + V

1 ε1 (uL∞ (0,T ;V ) )

1 e−t/ε ψ u (t) + φ u(t) dt ε

T 0

u(t)m dt − C2 T − T X ε

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C3

T

p u (t) dt + T V

3 (uL∞ (0,T ;V ) ) + ε

0

T

u(t)m dt + T , X

0

where uL∞ (0,T ;V ) := ess supt∈[0,T ] |u(t)|V , for all u ∈ Lm (0, T ; X) ∩ W 1,p (0, T ; V ). 2.3. Coincidence between ∂X φX and ∂V φ The following proposition shows some relationship between ∂V φ and ∂X φX . Proposition 2.1. Let V and X be normed spaces such that X → V with a continuous canonical injection. Let φ be a proper, lower semicontinuous and convex functional from V into (−∞, ∞]. Moreover, let φX be the restriction of φ onto X. If D(φ) ⊂ X, then D(∂V φ) = w ∈ D(∂X φX ); ∂X φX (w) ∩ V ∗ = ∅ ,

(2.1)

and moreover, ∂V φ(u) = ∂X φX (u) ∩ V ∗

for u ∈ D(∂V φ).

(2.2)

Proof. We first note that V ∗ → X ∗ . Let u ∈ D(∂V φ) and f ∈ ∂V φ(u) be fixed. For any v ∈ D(φX ) ⊂ X, noting that D(∂V φ) ⊂ D(φ) ⊂ X by assumption, we find that φX (v) − φX (u) = φ(v) − φ(u) f, v − u V = f, v − u X , which implies u ∈ D(∂X φX ) and f ∈ ∂X φX (u) ∩ V ∗ . Conversely, let u ∈ {w ∈ D(∂X φX ); ∂X φX (w) ∩ V ∗ = ∅} and f ∈ ∂X φX (u) ∩ V ∗ be fixed. For v ∈ D(φ) ⊂ X, it follows that φ(v) − φ(u) = φX (v) − φX (u) f, v − u X = f, v − u V , which gives u ∈ D(∂V φ) and f ∈ ∂V φ(u). Thus (2.1) and (2.2) hold.

2

2.4. Representation of subdifferentials in Lp (0, T ; V ) We provide here a result on the possible representation of the subdifferential of an integral functional. This representation turns out to be useful later on. We set V := Lp (0, T ; V ) and define the two functionals Iε1 , Iε2 : V → [0, ∞] by Iε1 (u) :=

T 0

∞

e−t/ε ψ(u (t)) dt

if u ∈ W 1,p (0, T ; V ), u(0) = u0 , else


2547

(note that ψ(u (·)) ∈ L1 (0, T ) for all u ∈ W 1,p (0, T ; V ) by (A2) ) and Iε2 (u) :=

T 1 −t/ε φ(u(t)) dt 0 εe

∞

if u ∈ Lm (0, T ; X), else.

Then it is obvious that for u ∈ D(Iε ) = D Iε1 ∩ D Iε2 .

Iε (u) = Iε1 (u) + Iε2 (u)

Moreover, Iε1 and Iε2 are proper, (weakly) lower semicontinuous and convex in V. Let us discuss the representation of the subdifferential operator ∂V Iε1 . Define the operator A : V → V ∗ by A(u)(t) = −

d −t/ε e dV ψ u (t) for u ∈ D(A) dt

with the domain D(A) = u ∈ D Iε1 ; dV ψ u (·) ∈ W 1,p 0, T ; V ∗ , dV ψ u (T ) = 0 . We have the following result. Proposition 2.2 (Identification of A). It holds that A = ∂V Iε1 . Proof. It can be easily seen that A ⊂ ∂V Iε1 . Hence it remains to prove the inverse inclusion. Set W := W 1,p (0, T ; V ). Define two functionals J, K : W → [0, ∞] by T J (u) :=

e−t/ε ψ u (t) dt,

0

K(u) :=

0 ∞

if u(0) = u0 , otherwise

and denote by Iε,1 W the restriction of Iε1 on W (hence Iε,1 W = J + K). Then, J is Gâteaux differentiable in W. Indeed, let u, e ∈ W and let h ∈ R. Then, J (u + he) − J (u) = h

T

e−t/ε

ψ(u (t) + he (t)) − ψ(u (t)) dt. h

0

Since ψ is Gâteaux differentiable in V , the integrand of the right-hand side converges to e−t/ε dV ψ(u (t)), e (t) V for almost every t ∈ (0, T ) as h → 0. Now, we easily compute that

ψ(u (t) + he (t)) − ψ(u (t)) dV ψ u (t) + he (t) , e (t) V dV ψ u (t) , e (t) V h

and, by means of (A2), we obtain

2548


ψ(u (t) + he (t)) − ψ(u (t)) h dV ψ u (t) ∗ + dV ψ u (t) + he (t) V

V∗

e (t)

V

p 2C4 C3 p p C3 2 u (t)V + + u (t) + he (t)V + e (t)V ∈ L1 (0, T ). p p p p Hence, by dominated convergence we deduce that J is Gâteaux differentiable in W and J (u + he) − J (u) → h

T

e−t/ε dV ψ u (t) , e (t) V dt

0

=: dW J (u), e W ,

for all e ∈ W,

where ·,·

W stands for the duality pairing between W and W ∗ . Moreover, K is proper, lower semicontinuous, and convex in W, and we have f, e

W = 0 for all [u, f ] ∈ ∂W K and e ∈ W with e(0) = 0. Therefore, since D(J ) = W, we find that ∂W Iε,1 W = dW J + ∂W K with the domain D ∂W Iε,1 W = u ∈ W; u(0) = u0 . Now, from the fact that W ⊂ V and D(Iε1 ) ⊂ W, it follows that ∂V Iε1 ⊂ ∂W Iε,1 W . Let [u, f ] ∈ ∂V Iε1 (hence u(0) = u0 ). Then, T e

−t/ε

dV ψ u (t) , e (t) V dt =

0

T

f (t), e(t) V dt

0

for any e ∈ W with e(0) = 0. Hence the function t → e−t/ε dV ψ(u (t)) belongs to W 1,p (0, T ; V ∗ ) and is such that f (t) = −

d −t/ε e dV ψ u (t) dt

for a.a. t ∈ (0, T ).

Moreover, we can also observe that dV ψ(u (T )) = 0 from the arbitrariness of e(T ) ∈ V . Thus u ∈ D(A) and f = A(u). Consequently, A coincides with ∂V Iε1 . 2


2549

2.5. Integration by parts at Lebesgue points in vector spaces For m, p ∈ (1, ∞), let the space Lm (t1 , t2 ; X) ∩ W 1,p (t1 , t2 ; V ) (t1 , t2 ∈ [0, T ] with t1 < t2 ) be endowed with the norm · Lm ∩W 1,p (t ,t ) given by X

uLm ∩W 1,p (t X

V

V

1 ,t2 )

1 2

:= uLm (t1 ,t2 ;X) + uW 1,p (t1 ,t2 ;V )

and, classically, t2 uLm (t1 ,t2 ;X) =

u(t)m dt X

1/m ,

t1

uW 1,p (t1 ,t2 ;V ) = uLp (t1 ,t2 ;V ) + u Lp (t

1 ,t2 ;V )

.

Moreover, the space Lm (t1 , t2 ; X) ∩ Lp (t1 , t2 ; V ) is analogously defined, and its dual space can be identified with Lm t 1 , t 2 ; X ∗ + L p t 1 , t 2 ; V ∗ = f1 + f2 ; f1 ∈ Lm t1 , t2 ; X ∗ and f2 ∈ Lp t1 , t2 ; V ∗ . Furthermore, the duality pairing between Lm (t1 , t2 ; X) ∩ Lp (t1 , t2 ; V ) and its dual space can be written by t1 f, u

Lm ∩Lp (t1 ,t2 ) = X

V

f1 (t), u(t) X dt +

t2

t1

f2 (t), u(t) V dt

for f = f1 + f2 .

t2

Then it follows immediately that f

p

Lm +LV ∗ (t1 ,t2 ) X∗

f1 Lm (t1 ,t2 ;X∗ ) + f2 Lp (t1 ,t2 ;V ∗ )

for f = f1 + f2 .

(2.3)

In case (t1 , t2 ) = (0, T ), we omit (0, T ) in the notation of the norms and the duality pairing. We shall be needing an integration by parts formula in this functional space setting. Proposition 2.3 (Integration by parts). Let m, p ∈ (1, ∞) and let u ∈ Lm (0, T ; X) ∩ W 1,p (0, T ; V ) and ξ ∈ Lp (0, T ; V ∗ ) be such that ξ ∈ Lm (0, T ; X ∗ ) + Lp (0, T ; V ∗ ). Let t1 , t2 ∈ (0, T ) be Lebesgue points of the function t → ξ(t), u(t) V . Then it holds that

ξ , u Lm ∩Lp (t

X

V

1 ,t2

= ξ(t2 ), u(t2 ) V − ξ(t1 ), u(t1 ) V − )

t2 t1

ξ(t), u (t) V dt.

(2.4)

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Proof. There exist f1 ∈ Lm (0, T ; X ∗ ) and f2 ∈ Lp (0, T ; V ∗ ) such that ξ = f1 + f2 . Put t Fi (t) :=

fi (s) ds

for i = 1, 2.

0

Then, F1 ∈ W 1,m (0, T ; X ∗ ), F2 ∈ W 1,p (0, T ; V ∗ ) and ξ(t) = F1 (t) + F2 (t) + ξ(0). We observe that T−τ

m F1 (t + τ ) − F1 (t) − f1 (t) dt = 0 τ ∗

lim

τ →0

X

0

and the analogue holds for F2 . Hence, we have

ξ , u Lm ∩Lp (t X

V

t2 1 ,t2 )

= lim

τ →0

t1

t2 = lim

τ →0

t1

F2 (t + τ ) − F2 (t) F1 (t + τ ) − F1 (t) dt , u(t) + , u(t) τ τ X V

ξ(t + τ ) − ξ(t) , u(t) dt. τ V

Moreover, since t1 , t2 are Lebesgue points of ξ(·), u(·) V , it follows that t2 t1

t t 2 +τ 1 +τ

ξ(t + τ ) − ξ(t) 1 1 , u(t) dt = ξ(t), u(t) V dt − ξ(t), u(t) V dt τ τ τ V t2

t2 − t1

t1

u(t + τ ) − u(t) ξ(t + τ ), τ

dt V

→ ξ(t2 ), u(t2 ) V − ξ(t1 ), u(t1 ) V −

t2

ξ(t), u (t) V dt

t1

as τ → 0. Thus we obtain (2.4).

2

3. The Euler–Lagrange equation This section brings to a proof of the existence of strong solutions for the Euler–Lagrange equation (1.2)–(1.4) related to the WED functional Iε . Hence, the value of the parameter ε is kept fixed throughout this section. We shall be concerned with the following precise notion of solution. Definition 3.1 (Strong solution). A function u : [0, T ] → V is said to be a strong solution of (1.2)–(1.4) if the following conditions are satisfied:


2551

u ∈ Lm (0, T ; X) ∩ W 1,p (0, T ; V ), ξ(·) := dV ψ u (·) ∈ Lp 0, T ; V ∗ and ξ ∈ Lm 0, T ; X ∗ + Lp 0, T ; V ∗ ,

there exists η ∈ Lm (0, T ; X ∗ ) such that η(t) ∈ ∂X φX u(t) ,

−εξ (t) + ξ(t) + η(t) = 0

in X ∗

for a.a. t ∈ (0, T ), u(0) = u0 and ξ(T ) = 0. Remark 3.2. By definition, ξ belongs to W 1,σ (0, T ; X ∗ ) with σ := min{m , p } > 1. We particularly deduce that ξ ∈ C([0, T ]; X ∗ ), and hence ξ(t) lies in X ∗ at every t ∈ [0, T ]. The main result of this section is the following. Theorem 3.3 (Existence of strong solutions). Assume that (A1)–(A4) are all satisfied. For every u0 ∈ D(φ), the Euler equation (1.2)–(1.4) admits a strong solution satisfying T 0

T

p u (t) dt 1 φ(u0 ) + C T + εψ(0) , ε 2 V C1

φ uε (t) dt φ(u0 ) + C2 T + εψ(0) T + ε

0

T

T

ξε (t), uε (t) V dt,

(3.1)

(3.2)

0

ηε (t), uε (t) X dt − εξε (0), u0 X −

0

T

0

T

εξε (t), uε (t) V

T dt −

ξε (t), uε (t) V dt,

(3.3)

0

ξε (t), uε (t) V dt −φ uε (T ) + φ(u0 ) + εψ(0).

(3.4)

0

The rest of this section is devoted to a proof of Theorem 3.3. The strategy of the proof is quite classical: we introduce suitable approximating problems by replacing φ with its Moreau–Yosida regularization φλ , establish a priori estimates independently of λ, and finally pass to the limit as λ → 0. For the sake of clarity, we split this proof in subsequent subsections. 3.1. Approximating problem Let us start by introducing the following approximate problems for λ > 0: −εξε,λ (t) + ξε,λ (t) + ηε,λ (t) = 0 in V ∗ , 0 < t < T , ξε,λ (t) = dV ψ uε,λ (t) , ηε,λ (t) = ∂V φλ uε,λ (t) in V ∗ , 0 < t < T ,

(3.5) (3.6)

uε,λ (0) = u0 ,

(3.7)

ξε,λ (T ) = 0,

(3.8)

2552


where ∂V φλ is the Yosida approximation of ∂V φ. Here we recall that ∂V φλ coincides with the subdifferential operator of the Moreau–Yosida regularization φλ of φ given by 1 1 2 φλ (u) := inf |u − v|V + φ(v) = |u − Jλ u|2V + φ(Jλ u), v∈V 2λ 2λ where Jλ is the resolvent for ∂V φ (see [4] for more details). We also recall by definition that FV (Jλ u − u) + λ∂V φλ (u) = 0 for all u ∈ V ,

(3.9)

where FV : V → V ∗ denotes the duality mapping between V and V ∗ . Then, a strong solution uε,λ of (3.5)–(3.8) on [0, T ] will be obtained as a global minimizer for the functional Iε,λ : V → [0, ∞] given by ⎧ ⎨ 0T e−t/ε (ψ(u (t)) + 1ε φλ (u(t))) dt 1,p 1 Iε,λ (u) = ⎩ if u ∈ W (0, T ; V ), u(0) = u0 and φλ (u(·)) ∈ L (0, T ), ∞ otherwise. More precisely, we have the following. Lemma 3.4 (Solvability of the approximating problem). For each ε, λ > 0, the functional Iε,λ admits a global minimizer uε,λ on V. Moreover, uε,λ is a strong solution of (3.5)–(3.8) and ξε,λ ∈ W 1,p 0, T ; V ∗

and ηε,λ ∈ Lp 0, T ; V ∗ .

(3.10)

Proof. We observe that Iε,λ is proper, lower semicontinuous and convex in V. Moreover, Iε,λ is coercive in V by (A1), i.e., Iε,λ (u) → ∞

if uV → ∞.

Hence, Iε,λ admits a (global) minimizer uε,λ for each λ > 0. 2 : V → [0, ∞] as We now define the functional Iε,λ 2 Iε,λ (u) :=

T 1 −t/ε φλ (u(t)) dt 0 εe

∞

if φλ (u(·)) ∈ L1 (0, T ), otherwise.

Note that, as p 2, we have T

1 φλ u(t) dt 2λ

0

T

u(t) − v 2 dt + φ(v)T V

0

0 (see [6,4]). Hence, by testing Eq. (3.5) on uλ (t) and integrating over (0, t) we obtain by (3.12) t C1

u (τ )p dτ + φλ uλ (t) λ V

0

φ(u0 ) + C2 T + ε ξλ (t), uλ (t) V + εψ(0) for all t ∈ Lλ . As the set Lλ has full Lebesgue measure, i.e., the measure of (0, T ) \ Lλ is zero, by integrating both sides over (0, T ) again, we deduce that T

φλ uλ (t) dt φ(u0 ) + C2 T + εψ(0) T + ε

0

T

ξλ (t), uλ (t) V dt.

(3.17)

0

Finally, from the above estimates we obtain T

φλ uλ (t) dt C.

(3.18)

0

Since φ(Jλ uλ (t)) φλ (uλ (t)) and ηλ (t) ∈ ∂V φ(Jλ uλ (t)) ⊂ ∂X φX (Jλ uλ (t)), it follows from assumptions (A3) and (A4) that


T

Jλ uλ (t)m dt + X

0

T

ηλ (t)m∗ dt C.

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X

(3.19)

C,

(3.20)

0

Eventually, a comparison in Eq. (3.5) yields εξ

λ Lm +Lp X∗ V∗

which, in particular, implies sup εξλ (t)X∗ C.

t∈[0,T ]

(3.21)

3.3. Passage to the limit From the a priori estimates of Section 3.2, we have, for some not relabeled subsequences, uλ → u

weakly in W 1,p (0, T ; V ),

(3.22)

Jλ uλ → v

weakly in Lm (0, T ; X), weakly in Lp 0, T ; V ∗ , weakly in Lm 0, T ; X ∗ , weakly in Lm 0, T ; X ∗ + Lp 0, T ; V ∗ .

(3.23)

ξλ → ξ ηλ → η ξλ → ξ

(3.24) (3.25) (3.26)

That is −εξ + ξ + η = 0

(3.27)

and the estimate (3.1) follows directly from the bound (3.15). Note that uλ is equicontinuous in C([0, T ]; V ) with respect to λ from the bound (3.15) and put vλ (t) := Jλ uλ (t) − uλ (t). By (3.9) and the monotonicity of ∂V φ, we have

FV vλ (t + h) − FV vλ (t) , Jλ uλ (t + h) − Jλ uλ (t) V 0,

which, together with estimate (3.16), implies

FV vλ (t + h) − FV vλ (t) , vλ (t + h) − vλ (t) V C uλ (t + h) − uλ (t)V .

Hence, the right-hand side goes to zero as h → 0 uniformly for λ > 0. Since V is uniformly convex, thanks to [26], for each R > 0 there exists a strictly increasing function mR on [0, ∞) such that mR (0) = 0 and

mR |u − v|V FV (u) − FV (v), u − v V

for u, v ∈ BR := u ∈ V ; |u|V R .

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Namely, vλ (·) is equicontinuous in C([0, T ]; V ) for λ > 0, and so is Jλ uλ (·). Recalling that X is compactly embedded in V , by Theorem 3 of [27], we deduce from estimate (3.19) that strongly in C [0, T ]; V .

Jλ uλ → v

(3.28)

By the integral estimate (3.18), we have T

uλ (t) − Jλ uλ (t)2 dt 2λ V

0

T

φλ uλ (t) dt 2λC → 0.

0

Therefore, we have u = v and the bound in (3.16) entails that uλ → u

strongly in Lq (0, T ; V )

(3.29)

with an arbitrary q ∈ [1, ∞). Hence, the strong convergences (3.28) and (3.29) yield Jλ uλ (t) → u(t)

strongly in V for all t ∈ [0, T ],

(3.30)

uλ (t) → u(t)

strongly in V for a.a. t ∈ (0, T ).

(3.31)

Since V ∗ is compactly embedded in X ∗ , estimates (3.16) and (3.20) entail ξλ → ξ

strongly in C [0, T ]; X ∗ ,

(3.32)

which also implies ξ(T ) = 0. p Put now p(t) := lim infλ→0 |ξλ (t)|V ∗ , and note that p ∈ L1 (0, T ) by Fatou’s Lemma and (3.16). Then p(t) < ∞ for a.a. t ∈ (0, T ), and for such t ∈ (0, T ) we can take a subsequence λtn → 0 (possibly depending on t) such that ξλtn (t) → ξ(t)

weakly in V ∗ .

(3.33)

We shall now check for the almost everywhere relations η(t) ∈ ∂X φX u(t) ,

ξ(t) = dV ψ u (t) .

Let us start from the former. Define the subset L ⊂ (0, T ) by

L := t ∈ (0, T ); t is a Lebesgue point of the function t → ξ(t), u(t) V , for any sequence λn → 0, there exists a subsequence

λn → 0 such that ξλn (t), uλn (t) V → ξ(t), u(t) V . Note that the convergences (3.31) and (3.33) entail that L has full Lebesgue measure. For arbitrary t1 , t2 ∈ L with t1 t2 , we have


t2

ηλ (t), Jλ uλ (t) X dt =

t1

t2

ηλ (t), uλ (t) V dt − λ

t1

t2

t2

2557

ηλ (t)2 ∗ dt V

t1

ηλ (t), uλ (t) V dt.

(3.34)

t1

On the other hand, from Eq. (3.5) it follows that t2

ηλ (t), uλ (t) V dt =

t1

t2

εξλ (t), uλ (t) V dt −

t1

t2


(3.35)

t1

Moreover, since uλ ∈ W 1,p (0, T ; V ) and ξλ ∈ W 1,p (0, T ; V ∗ ) (hence the function t → ξλ (t), uλ (t) V is differentiable-in-time for almost all t ∈ (0, T )), we note that t2

εξλ (t), uλ (t) V dt = ε ξλ (t2 ), uλ (t2 ) V − ε ξλ (t1 ), uλ (t1 ) V

t1

t2 −

εξλ (t), uλ (t) V dt

(3.36)

t1

(note that (3.34)–(3.36) also hold for any t1 , t2 ∈ [0, T ] with t1 t2 ). By the definition of L,

ξλn (ti ), uλn (ti ) V → ξ(ti ), u(ti ) V

for i = 1, 2

(3.37)

with a subsequence λn → 0 (possibly depending on t1 , t2 ). By a standard argument for monotone operators, it follows from the weak convergences (3.22) and (3.24) that t2 lim inf λ→0

ξλ (t), uλ (t) V

t2 dt

t1

ξ(t), u (t) V dt

(3.38)

t1

(it also follows for any t1 , t2 ∈ [0, T ] with t1 t2 ). Combining these facts and using Proposition 2.3, we deduce that t2 lim sup λn →0

ηλn (t), Jλn uλn (t) X dt

t1

ε ξ(t2 ), u(t2 ) V − ε ξ(t1 ), u(t1 ) V −

t2 t1

εξ(t), u (t) V dt −

t2 t1

ξ(t), u(t) V dt

2558


= εξ , u Lm ∩Lp (t X

V

t2 1 ,t2 )

−

ξ(t), u(t) V dt =

t2

t1

η(t), u(t) X dt.

t1

By exploiting the maximal monotonicity of ∂X φX in X × X ∗ (see also Lemma 1.2 of [6] and Proposition 1.1 of [17]), we conclude that η(t) belongs to ∂X φX (u(t)) for a.a. t ∈ (t1 , t2 ). It also follows that t2 lim

λn →0

t2

ηλn (t), Jλn uλn (t) X dt =

t1

η(t), u(t) X dt.

(3.39)

t1

Moreover, from the arbitrariness of t1 , t2 ∈ L and the fact that (0, T ) \ L is negligible, we also conclude that η(t) ∈ ∂X φX (u(t)) for a.a. t ∈ (0, T ). Here let us prove the energy inequality (3.3). Test ηλ (t) on uλ (t) and integrate over (0, T ). We have T

ηλ (t), uλ (t) V dt =

0

T

εξλ (t), uλ (t) V

T dt −

0

ξλ (t), uλ (t) V dt

0

= − εξλ (0), u0 V −

T


T dt −

0


0

Therefore, we obtain T

η(t), u(t) X dt lim inf

T

λn →0

0

ηλn (t), Jλn uλn (t) X dt

0

T

(3.34)

lim sup λn →0

ηλn (t), uλn (t) V dt

0

= − lim εξλn (0), u0

T

λn →0

V

− lim inf λn →0

εξλn (t), uλn (t) V dt

0

T − lim

λn →0

ξλn (t), uλn (t) V dt

0

− εξ(0), u0 X −

(3.38)

T 0

which leads us to estimate (3.3).


T 0

ξ(t), u(t) V dt,


2559

Let us next check that ξ(·) = dV ψ(u (·)) almost everywhere. Let t1 , t2 ∈ L be again fixed and consider the same sequence λn as before. For notational simplicity, λn will be denoted by λ. We have t2


dt = ε ξλ (t2 ), uλ (t2 ) V − ε ξλ (t1 ), uλ (t1 ) V − (3.36)

t1

t2

εξλ (t), uλ (t) V dt

t1

= ε ξλ (t2 ), uλ (t2 ) V − ε ξλ (t1 ), uλ (t1 ) V −

(3.5)

t2


t1

t2 −

ηλ (t), uλ (t) V dt

t1

ε ξλ (t2 ), uλ (t2 ) V − ε ξλ (t1 ), uλ (t1 ) V −

(3.34)

t2


t1

t2 −

ηλ (t), Jλ uλ (t) X dt =: RHS.

t1

Hence by convergences (3.37), (3.39), and Proposition 2.3, for λn → 0 we have

RHS → ε ξ(t2 ), u(t2 ) V − ε ξ(t1 ), u(t1 ) V −

t2

ξ(t), u(t) V dt −

t1

= εξ , u Lm ∩Lp (t X

V

t2 1 ,t2 )

+

t2

=

η(t), u(t) X dt

t1


t1

(3.5)

t2

t2

ξ(t), u(t) V dt −

t1

t2

η(t), u(t) X dt

t1

εξ(t), u (t) V dt.

t1

Therefore t2 lim sup λ→0

t1

ξλ (t), uλ (t) V

t2 dt t1

ξ(t), u (t) V dt.

(3.40)

2560


Thanks to the demiclosedness of the maximal monotone operator u → dV ψ(u(·)) in Lp (0, T ; V ) × Lp (0, T ; V ∗ ) and Proposition 1.1 of [17], ξ(t) coincides with dV ψ(u (t)) for a.a. t ∈ (0, T ), and moreover, t2 lim

λ→0

ξλ (t), uλ (t) V

t1

t2 dt =

ξ(t), u (t) V dt

for all t1 , t2 ∈ L.

(3.41)

t1

As in (3.17), we can derive by (3.28) and (3.41) that t2

φ u(t) dt φ(u0 ) + C2 T + εψ(0) (t2 − t1 ) + ε

t1

t2

ξ(t), u (t) V dt

t1

for all t1 , t2 ∈ L with t1 t2 . Letting (t1 , t2 ) → (0, T ), we obtain (3.2). By the weak lower semicontinuity of φ in V , it follows from convergence (3.30) that lim inf φλ uλ (T ) lim inf φ Jλ uλ (T ) φ u(T ) . λ→0

λ→0

Hence combining (3.13) with (3.38), we can derive (3.4). This completes a proof of Theorem 3.3. 4. Minimizers of WED functionals In this short section, we are concerned with the existence and characterization of minimizers of the WED functional Iε in V := Lp (0, T ; V ). Our aim is to prove that every minimizer uε of Iε coincides with a strong solution of (1.2)–(1.4) which is a limit of global minimizers uε,λ for Iε,λ as λ → 0, provided that either ψ or φ is strictly convex. Let us start by defining the minimizers of Iε as follows. Definition 4.1 (Minimizer). A function u ∈ V is said to be a minimizer of Iε in V if ∂V Iε (u) 0. The main result of this section is the following. Theorem 4.2 (Existence and characterization of minimizers). Assume (A1)–(A4). For each u0 ∈ D(φ), the strong solution of (1.2)–(1.4) obtained in Theorem 3.3 is a minimizer of Iε in V. Moreover, if either ψ or φ is strictly convex, then the minimizer is unique. Our proof of this theorem is divided into the following two lemmas. Lemma 4.3 (Strong solutions are minimizers). Let uε be a strong solution of (1.2)–(1.4) obtained in Theorem 3.3. Then, uε is a minimizer of Iε in V. Proof. By Lemma 3.4, we have obtained a global minimizer uε,λ ∈ V of Iε,λ , namely, Iε,λ (v) Iε,λ (uε,λ ) for all v ∈ D(Iε ).


2561

By passing to the limit as λ → 0 and using dominated convergence, we get Iε,λ (v) → Iε (v). Moreover, by the weak lower semicontinuity of Iε1 , Iε2 in V, we also deduce from the convergences (3.22) and (3.28) that T lim inf Iε,λ (uε,λ ) = lim inf λ→0

λ→0

1 e−t/ε ψ uε,λ (t) + φλ uε,λ (t) dt ε

0

lim inf Iε1 (uε,λ ) + Iε2 (Jλ uε,λ ) λ→0

Iε1 (uε ) + Iε2 (uε ) = Iε (uε ). Therefore Iε (v) Iε (uε ) for all v ∈ D(Iε ), namely 0 ∈ ∂V Iε (uε ).

2

Lemma 4.4 (Minimizers are unique). Suppose that either ψ or φ is strictly convex in V . Then, for each ε > 0, Iε admits a unique minimizer. Proof. In both cases the functional Iε turns out to be strictly convex in V and the assertion follows. 2 5. The causal limit In this section we ascertain the fundamental issue of the WED approach. Namely, we prove that the minimizers uε of the WED functionals Iε converge as ε → 0. Our main result is the following. Theorem 5.1 (Causal limit). Assume (A1)–(A4) and that either ψ or φ is strictly convex. Let u0 ∈ D(φ) and let uε be a minimizer of Iε on V := Lp (0, T ; V ). Then, there exist a sequence εn → 0 and a limit u such that uεn → u

strongly in C [0, T ]; V , weakly in W 1,p (0, T ; V ) ∩ Lm (0, T ; X),

and u is a strong solution of (1.5)–(1.6). Proof. For each ε > 0, let uε be the unique minimizer of Iε on V. By Theorem 4.2, uε is a strong solution of (1.2)–(1.4) satisfying estimates (3.1)–(3.4). Since uε (0) = u0 , it follows from estimate (3.1) that sup uε (t)V C.

t∈[0,T ]

(5.1)

2562


Furthermore, by assumption (A2), T

ξε (t)p ∗ dt C. V

(5.2)

0

Hence, by taking a suitable (non-relabeled) sequence ε → 0, uε → u

weakly in W 1,p (0, T ; V ), weakly in Lp 0, T ; V ∗ .

ξε → ξ

(5.3) (5.4)

Combining the bounds (3.2), (3.1), and (5.2), we deduce from assumption (A3) that T

uε (t)m dt C. X

(5.5)

0

Hence, one has uε → u

weakly in Lm (0, T ; X).

Moreover, it classically follows from estimates (3.1) and (5.5) that uε → u strongly in C [0, T ]; V , which also implies uε (t) → u(t)

strongly in V for all t ∈ [0, T ]

(5.6)

and u(0) = u0 . By assumption (A4) together with the bounds in (5.1) and (5.5), we have T

ηε (t)m∗ dt C, X

(5.7)

weakly in Lm 0, T ; X ∗ .

(5.8)

0

which implies ηε → η

By using Eq. (1.2) along with the estimates (5.2) and (5.7), we find that εξ m p C. ε LX∗ +LV ∗

(5.9)

Thus εξε → 0 weakly in Lm 0, T ; X ∗ + Lp 0, T ; V ∗ .

(5.10)


2563

Hence ξ + η = 0.

(5.11)

Moreover, for each v ∈ X, it follows from the final condition (1.4) and the latter convergence that

εξε (t), v

t

X

=

t

εξε (t) dt, v

= X

T

εξε (t), v

X

dt → 0,

T

which leads us to εξε (t) → 0 weakly in X ∗ for each t ∈ [0, T ].

(5.12)

We next claim that η(t) ∈ ∂X φX (u(t)) for almost all t ∈ (0, T ). Indeed, by estimate (3.3), T

ηε (t), uε (t) X dt − εξε (0), u0 X −

0

T →−

T

εξε (t), uε (t) V

T dt −

0

ξε (t), uε (t) V dt

0

ξ(t), u(t) V dt.

0

Hence, we have T lim sup ε→0

ηε (t), uε (t) X dt −

0

T

ξ(t), u(t) X dt =

0

T

η(t), u(t) X dt.

0

Therefore, by using the demiclosedness of the maximal monotone operator ∂X φX in Lm (0, T ; X) × Lm (0, T ; X ∗ ) and applying Proposition 1.1 of [17], we conclude that η(t) ∈ ∂X φX (u(t)) for almost all t ∈ (0, T ). Furthermore, since ξ ∈ Lp (0, T ; V ∗ ), by Proposition 2.1, we have η(t) ∈ ∂V φ(u(t)) for almost every t ∈ (0, T ). Let us now check that ξ(t) = dV ψ(u (t)) for almost every t ∈ (0, T ). By passing to the lim sup as ε → 0 into estimate (3.4) with the aid of the strong convergence (5.6) and the lower semicontinuity of φ in the weak topology of V , we obtain T lim sup ε→0

ξε (t), uε (t) V dt − lim inf φ uε (T ) + φ(u0 ) ε→0

0

−φ u(T ) + φ(u0 ) T =

−η(t), u (t) V dt

0

(5.11)

T

=

0

ξ(t), u (t) V dt.

2564


Thus, we have ξ(t) = dV ψ(u (t)) for almost every t ∈ (0, T ). Consequently, u solves the limiting problem (1.5)–(1.6) on [0, T ]. 2 Remark 5.2. If one is interested in proving the convergence of strong solutions uε of (1.2)–(1.4) satisfying energy inequalities (3.1)–(3.4) as ε → 0 to a strong solution of (1.5)–(1.6), the strict convexity of φ and ψ need not be assumed. 6. Mosco-convergence of WED functionals We shall prepare here some convergence result for sequences of WED functionals at fixed level ε. In particular, we present sufficient conditions for the convergence as h → ∞ of the sequence of WED functionals Iε,h given by Iε,h (u) =

⎧ T ⎨ 0 e−t/ε (ψh (u (t)) + 1ε φh (u(t))) dt ⎩

if u ∈ W 1,p (0, T ; V ) ∩ Lm (0, T ; X) and u(0) = u0,h , ∞ otherwise

with initial data u0,h ∈ X, a Gâteaux differentiable convex functional ψh : V → [0, ∞) and a proper, lower semicontinuous convex functional φh : V → [0, ∞] for h ∈ N. Throughout this section, we assume with no further specific mention that the functionals ψh and φh fulfill the general assumptions (A1)–(A4) with constants independent of h. In particular, by letting Z := Lm (0, T ; X) ∩ W 1,p (0, T ; V ), we easily check that Iε,h are bounded from below in Z uniformly for h ∈ N. Hence, the global minimizers uh of Iε,h are bounded in Z for all h ∈ N. Let us now make precise our notion of functional convergence in the following. Definition 6.1 (Mosco-convergence in Z). The functional Iε,h is said to Mosco-converge to Iε in Z as h → ∞ if the following two conditions hold: (i) (Lim inf condition) Let uh → u weakly in Z as h → ∞. Then lim inf Iε,h (uh ) Iε (u). h→∞

(ii) (Existence of recovery sequences) For every u ∈ Z and sequence kh → ∞ in N, there exist a subsequence (kh ) of (kh ) and a recovery sequence (uh ) in Z such that uh → u strongly in Z

and Iε,kh (uh ) → Iε (u)

as h → ∞.

Note that Mosco-convergence is classical (see [3,23]), and corresponds to the usual notion of Γ -convergence with respect to both the strong and the weak topology in Z. Mosco-convergence of the driving functionals arises as the natural requirement in order to deduce the convergence of the related differential problems (see [3, Theorem 3.74(2), p. 388] for gradient flows and [29, Lemma 7.1] for doubly nonlinear evolutions). Our sufficient conditions for Mosco-convergence are stated in the following.


2565

(H0) (Separability of spaces) V and X are separable. (H1) (Lim inf condition for φh in X) Let (uh ) be a sequence in X such that uh → u weakly in X. Then lim inf φh (uh ) φ(u). h→∞

(H2) (Lim inf condition for ψh in V ) Let (uh ) be a sequence in V such that uh → u weakly in V . Then lim inf ψh (uh ) ψ(u). h→∞

(H3) (Existence of joint recovery sequences for φh and ψh in X) Let (kh ) be a sequence in N such that kh → ∞. Let (uh ) be a sequence in X such that uh → u strongly in X

and φkh (uh ) → φ(u)

as h → ∞.

Then, for every v ∈ X, τ > 0, there exists a sequence (vτ,h ) in X such that

ψkh

vτ,h − uh τ

vτ,h → v strongly in X, v−u →ψ , φkh (vτ,h ) → φ(v) τ

as h → ∞.

(H4) (Convergence of initial data) u0,h ∈ X, u0,h → u0 strongly in X and φh (u0,h ) → φ(u0 ) as h → ∞. The reader should notice that we are not requiring for the separate functional convergence φh → ψ and φh → φ here (Γ - or Mosco-). In particular, our proof makes a crucial use of the possibility of finding a joint recovery sequence as of assumption (H3). Let us comment that the occurrence of such joint condition is not at all unexpected. Indeed, a similar joint recovery condition has been proved to be necessary and sufficient for passing to the limit in sequences of rate-independent evolution problems in an energetic form in [20], namely for p = 1. Moreover, let us note that in case p = 2, the concrete construction of an analogous joint recovery sequence is at the basis of the relaxation proof in [10]. The main result of this section is stated as follows. Theorem 6.2 (Mosco-convergence of Iε,h ). Assume (H0)–(H4). Then, the functionals Iε,h Moscoconverge in Z to Iε as h → ∞. We shall provide a proof of this theorem in the next subsection. Still, let us first point out a corollary, whose immediate proof is omitted. Corollary 6.3 (Minimizers converge to a minimizer). Under the assumptions of Theorem 6.2, let uh be a global minimizer of Iε,h for h ∈ N such that ukh → u weakly in Z as h → ∞ along with some sequence kh → ∞ in N. Then u minimizes Iε .

2566


6.1. Proof of Theorem 6.2 We provide here a proof of Theorem 6.2 by establishing conditions (i) and (ii) of Definition 6.1. Condition (ii) of Definition 6.1 is proved in a smooth case first and then generalized. 6.1.1. Lim inf inequality By using Corollary 4.4 of [29], we can derive from (H0)–(H2) that T lim inf

e

h→∞

−t/ε

φh uh (t) dt

0

T

e−t/ε φ u(t) dt

0

if uh → u weakly in Lm (0, T ; X); T lim inf

e

h→∞

−t/ε

ψh uh (t) dt

0

T

e−t/ε ψ u (t) dt

0

if uh → u weakly in Lp (0, T ; V ). Let (uh ) be a sequence in D(Iε,h ) such that uh → u weakly in Z. Then we can take a subsequence (kh ) of (h) such that ukh → u strongly in C([0, T ]; V ) by the compact embedding X → V , and therefore, u(0) = u0 by (H4). It follows that lim inf Iε,h (uh ) Iε (u). h→∞

Thus the Lim inf condition (i) follows. 6.1.2. Recovery sequence for u ∈ C 1 ([0, T ]; X) Let us next prove the existence of recovery sequences of u ∈ D(Iε ) for Iε,h . We first treat the case that u ∈ C 1 ([0, T ]; X) and u(0) = u0 , which also leads us to u ∈ D(Iε ). Our recovery sequence will be constructed from an approximation of u. Let N ∈ N be fixed and set τ := T /N , uiτ := u(iτ ) and t i := τ i for i = 0, 1, . . . , N (i.e., t 0 = 0 and t N = T ). Define the piecewise linear interpolant uˆ τ ∈ D(Iε ) by uˆ τ (t) = ατi (t)uiτ + 1 − ατi (t) ui+1 τ

for t ∈ t i , t i+1 ,

where ατi (t) := (t i+1 − t)/τ , and a piecewise forward constant interpolant u¯ τ ∈ L∞ (0, T ; X) by u¯ τ (t) = ui+1 τ

for t ∈ t i , t i+1 .

As u ∈ C 1 ([0, T ]; X), it follows that

Now, we find that

uˆ τ → u

strongly in W 1,∞ (0, T ; X),

(6.1)

u¯ τ → u

strongly in L∞ (0, T ; X).

(6.2)


T Iε (u) =

e

−t/ε

e

−t/ε

1 ψ u (t) + φ u(t) dt ε

T −t/ε 1 e ψ uˆ τ (t) + φ u¯ τ (t) dt + φ u(t) − φ u¯ τ (t) dt ε ε

2567

0

T = 0

0

T +

e−t/ε ψ u (t) − ψ uˆ τ (t) dt

0

=: I1,τ + I2,τ + I3,τ . Since φ is convex, letting η(t) ∈ ∂X φX (u(t)) and η¯ τ (t) ∈ ∂X φX (u¯ τ (t)), we can exploit (A4) and convergence (6.2) in order to check that T

e−t/ε η(t), u(t) − u¯ τ (t) X dt ε

I2,τ 0

C ε

T

u(t)m dt + T

1/m

X

u − u¯ τ Lm (0,T ;X) → 0 as τ → 0

0

with C = supt∈[0,T ] 2 (|u(t)|V )1/m and T I2,τ

e−t/ε η¯ τ (t), u(t) − u¯ τ (t) X dt ε

0

Cτ − ε

T

u¯ τ (t)m dt + T X

1/m u − u¯ τ Lm (0,T ;X) → 0 as τ → 0

0

with Cτ = supt∈[0,T ] 2 (|u¯ τ (t)|V )1/m , which is bounded as τ → 0 by (6.2). Hence I2,τ = o(1; τ → 0), where we wrote o(1; τ → 0) instead of o(1) in order to enforce which parameter is supposed to be infinitesimal. Moreover, by letting ξ(t) = dV ψ(u (t)) and ξ¯τ (t) = dV ψ(uˆ τ (t)), we use (A2) and convergence (6.1) in such a way that T I3,τ

e−t/ε ξ(t), u (t) − uˆ τ (t) V dt

0

T

C3 0

p u (t) dt + C4 T V

1/p

u − uˆ

τ Lp (0,T ;V )

→ 0 as τ → 0

2568


and T I3,τ

e−t/ε ξ¯τ (t), u (t) − uˆ τ (t) V dt

0

T

− C3

p uˆ (t) dt + C4 T τ V

1/p

u − uˆ

τ Lp (0,T ;V )

→ 0 as τ → 0.

0

Thus, we observe that I3,τ = o(1; τ → 0), and therefore Iε (u) = I1,τ + o(1; τ → 0).

(6.3)

For τ > 0, let us define a difference operator δτ by δτ χ i+1 :=

χ i+1 − χ i τ

for a vector χ i i=0,1,...,N .

Then I1,τ can be written as follows:

I1,τ =

i=0

=

1 e−t/ε ψ uˆ τ (t) + φ u¯ τ (t) dt ε

t i+1 N −1

N −1 i=0

ti

t i+1 i+1 1 i+1 −t/ε + φ uτ . e dt ψ δτ uτ ε

(6.4)

ti

Let (kh ) be a sequence in N such that kh → ∞. Set u0τ,h := u0,kh . Then by (H3) and (H4), we can take a sequence (u1τ,h ) in X such that u1τ,h → u1τ strongly in X, 1 1 uτ,h − u0τ,h u − u0τ →ψ τ , φkh u1τ,h → φ u1τ . ψkh τ τ Hence iterating this process (N − 1) times, we can further obtain (uiτ,h ) in X for i = 2, 3, . . . , N such that uiτ,h → uiτ strongly in X, i i uτ,h − ui−1 u − ui−1 τ,h τ ψkh →ψ τ , φkh uiτ,h → φ uiτ . τ τ

(6.5) (6.6)


2569

Define the piecewise linear interpolant uˆ τ,h ∈ D(Iε,h ) and the piecewise forward constant interpolant u¯ τ,h ∈ L∞ (0, T ; X) as above and, by convergence (6.5), we get uˆ τ,h → uˆ τ

strongly in W 1,∞ (0, T ; X) as h → ∞.

(6.7)

Therefore, for each τ > 0 we can choose hτ ∈ N such that uˆ τ,hτ − uˆ τ Lm ∩W 1,p < τ X

V

and hτ > τ −1 .

Combining this fact with convergence (6.1), we also deduce that uˆ τ,hτ − uLm ∩W 1,p uˆ τ,hτ − uˆ τ Lm ∩W 1,p + o(1; τ → 0) X

X

V

V

τ + o(1; τ → 0), which implies uˆ τ,hτ → u

strongly in Z as τ → 0.

As for the convergence of Iε,kh (uˆ τ,h ), we calculate T Iε,kh (uˆ τ,h ) =

e

−t/ε

1 ψkh uˆ τ,h (t) + φkh u¯ τ,h (t) dt ε

0

T +

e−t/ε φkh uˆ τ,h (t) − φkh u¯ τ,h (t) dt ε

0

=

t i+1 i+1 1 i+1 −t/ε e dt ψkh δτ uτ,h + φkh uτ,h ε

N −1 i=0

T +

ti

e−t/ε φkh uˆ τ,h (t) − φkh u¯ τ,h (t) dt ε

0

= I1,τ,h + I2,τ,h .

(6.8)

Then, by the above-stated convergences (6.6) and (6.4),

I1,τ,h →

t i+1 1 i+1 −t/ε + = I1,τ e dt ψ δτ ui+1 φ u τ τ ε

N −1 i=0

ti

as h → ∞. Hence, it remains to handle I2,τ,h . From the convexity of φh ,

(6.9)

2570


I2,τ,h =

t i+1 N −1 e−t/ε

ε

i=0

ti

t i+1 N −1 e−t/ε

ε

i=0

i+1 ατi (t)φkh uiτ,h + 1 − ατi (t) φkh ui+1 dt τ,h − φkh uτ,h

ti

t i+1

=

i+1 φkh ατi (t)uiτ,h + 1 − ατi (t) ui+1 dt τ,h − φkh uτ,h

N −1 i=0

e−t/ε i ατ (t) dt φkh uiτ,h − φkh ui+1 τ,h . ε

ti

Here, again by (6.6), we get N −1

t i+1

i=0

e−t/ε i ατ (t) dt φkh uiτ,h − φkh ui+1 τ,h ε

ti

t i+1

= oτ (1; h → ∞) +

N −1 i=0

e−t/ε i . ατ (t) dt φ uiτ − φ ui+1 τ ε

ti

Set ητi ∈ ∂X φX (uiτ ) for i = 0, 1, . . . , N . Then, noticing that i i

, φ uiτ − φ ui+1 ητ , uτ − ui+1 ητi X∗ uiτ − ui+1 τ τ τ X X by assumption (A4) and the strong convergence (6.2), we obtain N −1

t i+1

i=0

e−t/ε i ατ (t) dt φ uiτ − φ ui+1 τ ε

ti N −1 1 τ i i ητ X∗ uτ − ui+1 τ X ε 2 i=0

T C

1 0 u¯ τ (t) − u¯ τ (t − τ ) dt + τ u − u τ τ X → 0 as τ → 0. X

τ

Along these very same lines, it is possible to deduce an analogous estimate from below and we conclude that I2,τ,h oτ (1; h → ∞) + o(1; τ → 0).


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Combining now the latter with the convergence (6.9) and the decomposition (6.8), we deduce from (6.3) that Iε,kh (uˆ τ,h ) = I1,τ,h + I2,τ,h Iε (u) + oτ (1; h → ∞) + o(1; τ → 0). Hence, for each τ > 0 we can extract a sequence hτ ∈ N such that Iε,khτ (uˆ τ,hτ ) Iε (u) + τ + o(1; τ → 0)

and hτ > τ −1 .

Thus, one has lim sup Iε,khτ (uˆ τ,hτ ) Iε (u), τ →0

which together with the Lim inf condition (i) implies Iε,khτ (uˆ τ,hτ ) → Iε (u) as τ → 0. 6.1.3. Recovery sequence for general u Let us now discuss the general case u ∈ D(Iε ), i.e., u ∈ Z and u(0) = u0 . Let v(t) := u(t)−u0 for t ∈ [0, T ]. By using a standard mollification argument, we can construct vn ∈ C 1 ([0, T ]; X) for all n ∈ N such that vn → v

strongly in Z as n → ∞

and vn (0) = 0.

Now let wn (t) := vn (t) + u0 for t ∈ [0, T ]. Then wn ∈ C 1 ([0, T ]; X) satisfies wn → u

strongly in Z

and wn (0) = u0 .

By virtue of assumptions (A2) and (A4), the functions u → J (u) and u → Iε2 (u) are continuous in W 1,p (0, T ; V ) and Lm (0, T ; X), respectively (see Section 2.4). Hence, by relabeling the sequence (wn ) and using the continuity of φX and ψ in X and V , respectively, we can say wn − uLm ∩W 1,p < X

V

1 n

1 and Iε (wn ) − Iε (u) < . n

Now, by the above-proved existence of a recovery sequence in the smooth case, for each n ∈ N, we can take a subsequence (khn ) of (kh ) and a sequence (unh ) in Z such that unh → wn

strongly in Z

and Iε,khn unh → Iε (wn )

as h → ∞

at each n ∈ N. Finally, by using a diagonal argument, we can choose a sequence (un ) in Z and subsequence (kn ) of (kh ) such that un → u

strongly in Z

and Iε,kn (un ) → Iε (u)

as n → ∞.

Consequently, the recovery sequence condition (ii) of Definition 6.1 holds.

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7. Applications In this section, we present doubly nonlinear problems and apply the above-detailed abstract theory to them. Let Ω be a bounded domain of RN with smooth boundary ∂Ω. We start with the following doubly nonlinear parabolic equation (DNP): α(ut ) − am u = 0

in Ω × (0, T ),

(7.1)

u=0

on ∂Ω × (0, T ),

(7.2)

u(·, 0) = u0

in Ω,

(7.3)

where α : R → R and am is the so-called m-Laplace operator with a coefficient function a : Ω → R given by am u = ∇ · a(x)|∇u|m−2 ∇u ,

1 < m < ∞.

Here we assume that (a1) u0 ∈ W01,m (Ω), a ∈ L∞ (Ω) and a1 a(x) a2 for a.e. x ∈ Ω with some a1 , a2 > 0. (a2) α is maximal monotone in R. Moreover, there exist p ∈ [2, ∞) and constants C5 , C6 > 0 such that C5 |s|p − where A(s) :=

s 0

1 A(s) C5

p and α(s) C6 |s|p + 1 for all s ∈ R,

α(σ ) dσ for s ∈ R.

Note that α is continuous in R by (a2). In order to recast (DNP) into an abstract Cauchy problem, we set V = Lp (Ω)

and X = W01,m (Ω)

and define two functionals ψ, φ : V → [0, ∞] by ψ(u) =

A u(x) dx,

Ω

φ(u) =

1 m Ω

∞

a(x)|∇u(x)|m dx

if u ∈ W01,m (Ω), otherwise.

Assume (a3) p < m∗ := Nm/(N − m)+ . Then, by the Rellich–Kondrachov compact embedding theorem, we observe that X → V compactly. We find that ψ is of class C 1 in V and dV ψ(u) = α(u). In particular, the bounds in (A1) and (A2) immediately follow from (a2). Furthermore, φX is of class C 1 in X, and


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∂X φX (u) = −am u equipped with the boundary condition u|∂Ω = 0, and conditions (A3) and (A4) hold. Thus, (DNP) is reduced into the abstract doubly nonlinear problem (1.5)–(1.6). The existence of strong solutions of such a problem has been already discussed in [9]. Our current interest lies in the elliptic regularizations (ER)ε of (DNP) of the form: −εα(ut )t + α(ut ) − am u = 0

in Ω × (0, T ),

(7.4)

u=0

on ∂Ω × (0, T ),

(7.5)

u(·, 0) = u0 α u(·, T ) = 0

in Ω,

(7.6)

in Ω

(7.7)

for ε > 0. By applying our abstract theory, in particular, Theorems 3.3, 4.2 and 5.1, we have Theorem 7.1 (WED approach to (DNP)). Under (a1)–(a3), (ER)ε admits a strong solution uε ∈ Lm (0, T ; W01,m (Ω)) ∩ W 1,p (0, T ; Lp (Ω)). Moreover, uε is the unique minimizer of the WED functional Iε : Lp (0, T ; Lp (Ω)) → [0, ∞] given by Iε (u) =

⎧ T ⎨ 0 e−t/ε ( Ω A(ut (x, t)) dx + ⎩

if u(·, 0) = u0 in Ω and ∞ otherwise.

1 m εm Ω a(x)|∇u(x, t)| dx) dt u ∈ Lm (0, T ; W01,m (Ω)) ∩ W 1,p (0, T ; Lp (Ω)),

Furthermore, uεn converges to a strong solution u of (DNP) in the following sense: uεn → u

strongly in C [0, T ]; Lp (Ω) , weakly in Lm 0, T ; W01,m (Ω) ∩ W 1,p 0, T ; Lp (Ω)

along with some sequence εn → 0. We next consider the following sequence of doubly nonlinear problems (DNP)h for h ∈ N: αh (ut ) − amh u = 0

in Ω × (0, T ),

(7.8)

u=0

on ∂Ω × (0, T ),

(7.9)

u(·, 0) = u0,h

in Ω

(7.10)

with functions u0,h : Ω → R, ah : Ω → R and αh : R → R. Aizicovici and Yan [1] proved the convergence theorem for (DNP)h under appropriate conditions on the convergences of u0,h , ah and αh as h → ∞. As in (DNP), let us introduce elliptic regularizations (ER)ε,h of (DNP)h given by −εαh (ut )t + αh (ut ) − amh u = 0

in Ω × (0, T ),

(7.11)

u=0

on ∂Ω × (0, T ),

(7.12)

u(·, 0) = u0,h αh u(·, T ) = 0

in Ω,

(7.13)

in Ω.

(7.14)

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Then, the same conclusions as in Theorem 7.1 hold also for (DNP)h and (ER)ε,h and the corresponding WED functionals given by Iε,h (u) =

⎧ T ⎨ 0 e−t/ε ( Ω Ah (ut (x, t)) dx + ⎩

if u(·, 0) = u0,h in Ω and ∞ otherwise

1 m εm Ω ah (x)|∇u(x, t)| dx) dt 1,m u ∈ Lm (0, T ; W0 (Ω)) ∩ W 1,p (0, T ; Lp (Ω)),

s with Ah (s) = 0 αh (σ ) dσ for s ∈ R. Finally, let us discuss the Mosco-convergence of Iε,h under the following assumptions. (h1) Condition (a1) holds with functions a and u0 replaced by ah and u0,h , respectively, and the respective constants independent of h. Moreover, ah (x) → a(x) for a.e. x ∈ Ω and u0,h → u0 strongly in W01,m (Ω) as h → ∞. (h2) Condition (a2) holds with α replaced by αh and respective constants independent of h. Γ → A as h → ∞, i.e., the following (i) and (ii) hold: Moreover, Ah − (i) for every sequence sh → s as h → ∞, A(s) lim infh→∞ Ah (sh ), (ii) for every s ∈ R, there exists a sequence sh → s such that Ah (sh ) → A(s). More precisely, we can prove Theorem 7.2 (Mosco-convergence of Iε,h ). Assume (a1)–(a2), (h1)–(h2). Then, Iε,h Moscoconverges to Iε on Z := Lm (0, T ; X) ∩ W 1,p (0, T ; V ) as h → ∞. Let (uh ) be the sequence of unique global minimizers for Iε,h . Then, there exists a sequence kh → ∞ in N such that ukh → u weakly in Z as h → ∞, where u minimizes Iε , i.e., u solves (ER)ε . Proof. Let us check (H1)–(H4) for Iε,h as h → ∞. As in [1], by using standard facts in [3], one can check (H1), (H2) and (H4) from (h1) and (h2). So it remains to prove (H3). Let (kh ) be a sequence in N such that kh → ∞. Let u ∈ X and uh ∈ X be such that uh → u strongly in X and φkh (uh ) → φ(u) as h → ∞. Let v ∈ X and τ > 0 be fixed. Set vh := uh + v − u ∈ X. Here we claim that Akh (s) → A(s) as h → ∞ for all s ∈ R. Indeed, by (ii) of (h2), for any s ∈ R we can take a sequence sh → s such that Akh (sh ) → A(s) as h → ∞. Hence by (a2) for αh with C6 independent of h, we have Akh (s) − A(s) = Akh (s) − Akh (sh ) + Akh (sh ) − A(s) C|s − sh | + Akh (sh ) − A(s) → 0 as h → ∞, and obtain a similar estimate from below as well. Thus Akh (s) → A(s) as h → ∞. Furthermore, we can derive ψkh (w) → ψ(w)

as h → ∞ for any w ∈ V .


2575

Indeed, for any w ∈ V , it follows that Akh (w(x)) → A(w(x)) for a.e. x ∈ Ω, and moreover, by (a2) for αh with C6 independent of h, dominated convergence yields ψkh (w) → ψ(w) as h → ∞. Thus vh − uh v−u v−u = ψkh →ψ as h → ∞. ψkh τ τ τ Moreover, since vh → v strongly in X, it follows from (h1) that 1 φkh (vh ) = m

m 1 akh (x)∇vh (x) dx → m

Ω

m a(x)∇v(x) dx = φ(v)

Ω

as h → ∞. Thus, (H3) holds. Consequently, by Theorem 6.2 and Corollary 6.3, we obtain the desired. Note that the precompactness of global minimizers (uh ) is immediate. 2 Appendix A. Rigorous derivations of (3.11) and (3.12) We report here the details of a result used in Section 3.2. At first, let us recall some useful properties of the Legendre–Fenchel transform ϕ ∗ of a proper, lower semicontinuous and convex functional ϕ from a normed space E into → (−∞, ∞] given by ϕ ∗ (f ) := supv∈E { f, v E − ϕ(v)}, for f ∈ E ∗ (see, e.g., [4]): (i) ϕ ∗ is proper, lower semicontinuous and convex in E ∗ ; (ii) ϕ ∗ (f ) = f, u E − ϕ(u) for all [u, f ] ∈ ∂E ϕ; (iii) u ∈ ∂E ∗ ϕ ∗ (f ) if and only if f ∈ ∂E ϕ(u). Moreover, we observe that, whenever ϕ : E → [0, ∞], one has ϕ ∗ (0) = − infv∈E ϕ(v) 0 and ϕ ∗ (f ) −ϕ(0) for all f ∈ E ∗ . Now, our claim reads, Lemma A.1. The inequalities (3.11) and (3.12) can be rigorously justified within the frame of Section 3.2. Proof. Fix an arbitrary constant τ > 0 and define a backward difference operator δτ− by δτ− χ(t) =

χ(t) − χ(t − τ ) τ

for functions χ defined on [0, T ] with values in a vector space and t τ . Test ξλ (t) by uλ (t) and integrate over (t0 , T ) with an arbitrary t0 ∈ Lλ . Since uλ ∈ W 1,p (0, T ; V ) and ξλ ∈ W 1,p (0, T ; V ∗ ), we have T

ξλ (s), uλ (s) V

t0

Moreover, it follows from (3.8) that

T ds = lim

τ →0 t0 +τ

ξλ (s), δτ− uλ (s) V ds.

2576


T

ξλ (s), δτ− uλ (s) V ds

t0 +τ

=

ξλ (T ), δτ− uλ (T ) V

−

ξλ (t0 + τ ), δτ− uλ (t0

T

+ τ) V −


t0 +τ

1 = − ξλ (t0 + τ ), δτ− uλ (t0 + τ ) V − τ

T

ξλ (s), uλ (s) − uλ (s − τ ) V ds.

(A.1)

t0 +τ

Next, we observe that T

1 τ

ξλ (s), uλ (s) − uλ (s − τ ) V ds

t0 +τ

1 = τ 1 = τ

T

ξλ (s), uλ (s) V

t0 +τ T−τ

1 ds − τ

T−τ

ξλ (s + τ ), uλ (s) V ds

t0

ξλ (s) − ξλ (s

+ τ ), uλ (s) V

1 ds − τ

t0

1 + τ

t 0 +τ

ξλ (s), uλ (s) V ds

t0

T

ξλ (s), uλ (s) V ds.

(A.2)

T −τ

Using the fact that uλ (s) ∈ ∂V ∗ ψ ∗ (ξλ (s)), where ψ ∗ denotes the Legendre–Fenchel transform of ψ , and the definition of subdifferentials, we obtain 1 τ

T−τ

ξλ (s) − ξλ (s

+ τ ), uλ (s) V

1 ds τ

t0

T−τ

∗ ψ ξλ (s) − ψ ∗ ξλ (s + τ ) ds,

t0

and, moreover, 1 τ

T

ξλ (s), uλ (s) V

T −τ

1 ds τ

T

∗ ψ ξλ (s) − ψ ∗ (0) ds.

T −τ

Then, going back to Eq. (A.2), one has 1 τ

T t0 +τ

ξλ (s), uλ (s) − uλ (s − τ ) V ds


1 τ

t 0 +τ

1 ψ ξλ (s) ds − τ ∗

t0

t 0 +τ

2577

ξλ (s), uλ (s) V ds − ψ ∗ (0).

t0

Hence, from Eq. (A.1) we can compute that T


t0 +τ

−

ξλ (t0 + τ ), δτ− uλ (t0

1 + τ) V − τ

t 0 +τ

ψ ∗ ξλ (s) ds

t0 t 0 +τ

1 ξλ (s), uλ (s) V ds + ψ ∗ (0) τ t0

→ − ξλ (t0 ), uλ (t0 ) V − ψ ∗ ξλ (t0 ) + ξλ (t0 ), uλ (t0 ) V + ψ ∗ (0) = −ψ ∗ ξλ (t0 ) + ψ ∗ (0) ψ(0). +

Here we used the facts that t0 ∈ Lλ , ψ ∗ −ψ(0), ψ ∗ (0) 0 and the function t → ψ ∗ (ξλ (t)) is (absolutely) continuous on [0, T ] since uλ (t) ∈ ∂V ∗ ψ ∗ (ξλ (t)), uλ ∈ Lp (0, T ; V ) and ξλ ∈ W 1,p (0, T ; V ∗ ). Consequently, T

ξλ (s), uλ (s) V ds ψ(0)

for all t0 ∈ Lλ

t0

and (3.11) follows from the density of Lλ . Let now t ∈ Lλ be fixed. Arguing as above starting again from Eq. (A.1) with t instead of T , we can also verify that t

ξλ (s), δτ− uλ (s) V ds ξλ (t), δτ− uλ (t) V − ξλ (t0 + τ ), δτ− uλ (t0 + τ ) V

t0 +τ

1 − τ

t 0 +τ

t0

1 ψ ξλ (s) ds + τ ∗

t 0 +τ

t0

ξλ (s), uλ (s) V ds + ψ ∗ (0)

→ ξλ (t), uλ (t) V − ψ ∗ ξλ (t0 ) + ψ ∗ (0)

ξλ (t), uλ (t) V + ψ(0) as τ → 0,

and the inequality (3.12) follows.

2

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References [1] S. Aizicovici, Q. Yan, Convergence theorems for abstract doubly nonlinear differential equations, Panamer. Math. J. 7 (1997) 1–17. [2] G. Akagi, U. Stefanelli, A variational principle for doubly nonlinear evolution, Appl. Math. Lett. 23 (2010) 1120– 1124. [3] H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. [4] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, 1976. [5] M.A. Biot, Variational principles in irreversible thermodynamics with application to viscoelasticity, Phys. Rev. 97 (2) (1955) 1463–1469. [6] H. Brézis, M.G. Crandall, A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math. 23 (1970) 123–144. [7] H. Brézis, I. Ekeland, Un principe variationnel associé à certaines équations paraboliques, Le cas dépendant du temps, C. R. Acad. Sci. Paris Sér. A–B 282 (1976), Ai, A1197–A1198. [8] H. Brézis, I. Ekeland, Un principe variationnel associé à certaines équations paraboliques, Le cas indépendant du temps, C. R. Acad. Sci. Paris Sér. A–B 282 (1976), Ai, A971–A974. [9] P. Colli, On some doubly nonlinear evolution equations in Banach spaces, Japan J. Indust. Appl. Math. 9 (1992) 181–203. [10] S. Conti, M. Ortiz, Minimum principles for the trajectories of systems governed by rate problems, J. Mech. Phys. Solids 56 (2008) 1885–1904. [11] N. Ghoussoub, Self-Dual Partial Differential Systems and Their Variational Principles, Springer Monogr. Math., Springer, New York, 2009. [12] M.E. Gurtin, Variational principles in the linear theory of viscoelasticity, Arch. Ration. Mech. Anal. 13 (1963) 179–191. [13] M.E. Gurtin, Variational principles for linear elastodynamics, Arch. Ration. Mech. Anal. 16 (1964) 34–50. [14] M.E. Gurtin, Variational principle for the time-dependent Schrödinger equation, J. Math. Phys. 6 (1965) 1506–1507. [15] I. Hlaváˇcek, Variational principles for parabolic equations, Appl. Math. 14 (1969) 278–297. [16] T. Ilmanen, Elliptic Regularization and Partial Regularity for Motion by Mean Curvature, Mem. Amer. Math. Soc., vol. 108 (520), 1994, x+90 pp. [17] N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math. 22 (1975) 304–331. [18] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, vol. 1, Dunod, Paris, 1968. [19] A. Mielke, M. Ortiz, A class of minimum principles for characterizing the trajectories of dissipative systems, ESAIM Control Optim. Calc. Var. 14 (2008) 494–516. [20] A. Mielke, T. Roubíˇcek, U. Stefanelli, Gamma-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations 31 (3) (2008) 387–416. [21] A. Mielke, U. Stefanelli, A discrete variational principle for rate-independent evolution, Adv. Calc. Var. 1 (4) (2008) 399–431. [22] A. Mielke, U. Stefanelli, Weighted energy-dissipation functionals for gradient flows, ESAIM Control Optim. Calc. Var. (2009), doi:10.1051/cocv/2009043, in press. [23] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math. 3 (1969) 510–585. [24] B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A–B 282 (1976), Aiv, A1035–A1038. [25] B. Nayroles, Un théorème de minimum pour certains systèmes dissipatifs, Variante hilbertienne, in: Travaux Sém. Anal. Convexe, vol. 6, 1976, Exp. No. 2. [26] J. Prüß, A characterization of uniform convexity and applications to accretive operators, Hiroshima Math. J. 11 (1981) 229–234. [27] J. Simon, Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl. (4) 146 (1987) 65–96. [28] E. Spadaro, U. Stefanelli, A variational view at mean curvature evolution for cartesian surfaces, Preprint IMATICNR, 21PV10/19/0, 2010. [29] U. Stefanelli, The Brezis–Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim. 47 (2008) 1615–1642. [30] U. Stefanelli, A variational principle for hardening elastoplasticity, SIAM J. Math. Anal. 40 (2008) 623–652. [31] U. Stefanelli, The discrete Brezis–Ekeland principle, J. Convex Anal. 16 (2009) 71–87. [32] A. Visintin, A new approach to evolution, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 233–238. [33] A. Visintin, Extension of the Brezis–Ekeland–Nayroles principle to monotone operators, Adv. Math. Sci. Appl. 18 (2008) 633–650.


On Bessel integrals for reducible degenerate principal series representations Takuya Miyazaki Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku, Yokohama 223-8522, Japan Received 10 March 2010; accepted 20 January 2011

Communicated by P. Delorme

Abstract We discuss generalized Bessel integrals with nondegenerate characters, which are assigned to irreducible submodules of a reducible degenerate principal series representation of Sp(n, R). Then we give sufficient conditions for their vanishings which are based on the signatures of the nondegenerate characters. This consequently suggests a reasonable correspondence between open GLn (R)-orbits in the set of real symmetric matrices of size n and irreducible submodules of the reducible principal series representations. © 2011 Elsevier Inc. All rights reserved. Keywords: Degenerate principal series representations; Generalized confluent hypergeometric functions; Generalized Bessel integrals

1. Introduction In [12], Shimura studied in detail the confluent hypergeometric functions on tube domains. Here we recall, in particular, the following function in n × n positive definite matrices y ∈ Yn and (α, β) ∈ C2 : ξ(y, h; α, β) =

e−2πi tr(hx) det(x + iy)−α det(x − iy)−β dx,

Sn

E-mail address: [email protected]. 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.01.013

(1.1)

2580

T. Miyazaki / Journal of Functional Analysis 260 (2011) 2579–2597

where Sn denotes the set of all n × n real symmetric matrices, and h ∈ Sn . The integral is convergent if Re(α + β) > n. It was proved in [12] that for every nondegenerate h, the integral (1.1) is continued to a holomorphic function in (α, β) on the whole C2 , that is also real analytic in y. If α − β ∈ 2Z, then we remark that the ξ(y, h; α, β) is related to a generalized Bessel integral assigned to the character ψh (x) = e2πi tr(hx) on Sn and a degenerate principal series representation I (α + β) of Gn = Sp(n, R) induced from the Siegel maximal parabolic subgroup (see Section 2 for this). Consequently, the module structure of I (α + β) has an influence on the analytic behavior of the confluent hypergeometric function. This arouses our interests, in particular, when I (α + β) is reducible. We recall an example. Assume that n is even. Let us fix an even integer k > n and also put 1 Γn (s) = π 4 n(n−1) nj=1 Γ (s − 12 (j − 1)). Concerning the function ξ(y, h; k, 0) with a nondegenerate h ∈ Sn , [12, (1.23)] tells us the following facts: (i) If h ∈ / Yn , then ξ(y, h; k, 0) = 0. m(m−1) n+1 (ii) If h ∈ Yn , then ξ(y, h; k, 0) = 2− 2 Γn (k)−1 det(h)k− 2 e−2π tr(hy) is non-vanishing. On the other hand, we know that the principal series I (k) is reducible and contains a holomorphic k as its submodule. In fact, we find that ξ(y, h; k, 0) is related discrete series representation πn,0 k . Then, we may use the to a generalized Bessel integral assigned to the lowest K-type of this πn,0 k and the open above facts (i) and (ii) for providing a connection between the discrete series πn,0 GLn (R)-orbit Yn in Sn . The module structures of reducible degenerate principal series of Gn were extensively studied by Lee [9]. In particular, all irreducible submodules of the above I (k) are computed, where we k find that certain unitarizable modules πp,n−p , p ≡ k mod 2, 0 p n, not being equivalent k to πn,0 if p < n, are occurring. Hence, it is natural to ask about generalizations of (i), (ii) above k that are concerned with generalized Bessel integrals assigned to those πp,n−p . Our main result will give a partial answer to this problem, which can be summarized as 1.1. Theorem. If h ∈ Sn is nondegenerate and h ∈ / Ωp,n−p , then all generalized Bessel integrals k assigned to the character ψh and the minimal K-type of πp,n−p are vanishing. Here Ωp,n−p denotes the open GLn (R)-orbit in Sn of matrices with the signature (p, n − p). k . It is strongly expected that a This is clearly a generalization of (i) concerning every πp,n−p generalization of (ii) also holds in a natural way, but we are not arriving at the complete understanding of it only with our method. Besides the above examples, we add that our consideration in this paper covers all irreducible submodules of the degenerate principal series I (k) of Gn with the assumptions that n = 2m is even, and that k is any integer satisfying k m. The paper is organized as follows. After preparing some notations, in Section 2 we will give assignment of generalized Bessel integrals Wh (g; s, d)(ϕ) to every K-type of a degenerate principal series representation I (s), s ∈ C, of Gn (see (2.4) for the definition). Here ϕ varies over certain polynomials on Yn which control right K-translations of those integrals. Consequently, we will obtain at s = k the generalized Bessel integrals that are assigned to the minimal K-type of every irreducible submodule of I (k), k m.


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In Section 3 we will state our main results in advance. To explain the contents here we recall [12, (1.29)], that relates ξ(y, h; α, β) to the integral n+1 n+1 e−tr((v−h)y) det(v)α− 2 det(v − 2h)β− 2 dv. Γn (α)−1 Γn (β)−1 v>0n ,v>2h

We will have to find a suitable generalization of this equality for our integrals. Then, Theorem 3.1 will answer this problem. In addition to it, we will state Theorem 3.2, which gives sufficient conditions for the vanishings of the generalized Bessel integrals in our interests. This combined with a result in [9] yields Theorem 1.1. The proofs of these theorems will be given in the last two sections of this paper, after establishing some necessary preliminaries in Sections 4–7. In Section 4 we will recall some basic facts concerning polynomials ϕ which generate an irreducible GLn -module Vκ of finite dimension. In particular, the notion of EP-polynomials by Kushner, Lebow and Meisner [8] will be addressed. In Sections 5 and 6, we will discuss two kinds of expansion formulas related to ϕ ∈ Vκ : the Pieri formula and the generalized binomial formula. For ϕ being O(n)-invariant, these expansions were studied by Bingham [1] and Kushner [6]. Here we discuss their generalizations so that they shall cover the case of other polynomials not being O(n)-invariant in general. Proposition 6.6 will give a key description of the generalized binomial expansion of ϕ(1n + x) for every ϕ ∈ Vκ . The theory by Herz [3] concerning the generalized Laguerre polynomials Lsϕ (y) defined for all polynomials ϕ on Yn will play an important role in this paper. In Proposition 7.1, Lsϕ (y) will be described with a suitable finite sum according to the binomial expansion formula for ϕ ∈ Vκ . The description will be used essentially in the proof of Theorem 3.1 in Section 8. Finally, in Section 9 we will combine some analytic arguments due to [12] and Gindikin [2] with our results to give the proof of Theorem 3.2. 1.1. Notations For the field F = R, or C, we let Fnm denote the set of m × n matrices with entries in F . Let 1n , or 0n , denote the unit, or the zero, matrix in Cnn . For every matrix x = (xij ) ∈ Cnn , x = (xij ) stands for its complex conjugate. We let σ (x), or δ(x), denote the trace, or the determinant, of x, respectively. Also δi (x) denotes the upper left i × i determinant of x; thus δn (x) = δ(x). For n−m x, y ∈ Cnn we write x[y] = t yxy, where t y is the transpose of y. For any x ∈ Cm m and y ∈ Cn−m x 0 n let us write diag(x, y) for the block diagonal matrix 0 y ∈ Cn . We let Sn , or Tn , denote the space of real, or complex, n×n symmetric matrices, respectively. Also let Yn be the subset of all positive definite elements in Sn . The symbol x1 > x2 for matrices x1 , x2 ∈ Sn means that x1 − x2 ∈ Yn . If a nondegenerate matrix h ∈ Sn has p positive and n − p negative eigenvalues, then we say h has the signature sign(h) = (p, n − p). Let k 0 be an integer. A partition κ = (k1 , k2 , . . .) of k is a finite sequence of non-negative integers satisfying ki ki+1 so that |κ| = k, where we put |κ| = all i ki . We define its length l(κ) by the number of parts ki = 0. For every n k we let Pkn denote the set of partitions κ of k with l(κ) n: Pkn = {κ | |κ| = k, l(κ) n}. Given any κ ∈ Pkn and 1 i n, we define κ(i) and κ (i) by κ(i) = (k1 , . . . , ki−1 , ki + 1, ki+1 , . . . , kn )

if ki−1 ki + 1,

κ (i) = (k1 , . . . , ki−1 , ki − 1, ki+1 , . . . , kn )

if ki − 1 ki+1 .

and

2582


n n We understand κ(i) ∈ Pk+1 and κ (i) ∈ Pk−1 as our rule. Also we put

κ[i] = (k1 − ki , . . . , ki−1 − ki ), i−1 . which belongs to P|κ[i]| For partitions λ = (l1 , . . . , ln ) ∈ Pln and κ = (k1 , . . . , kn ) ∈ Pkn we use the symbol λ ≺ κ if the inequalities li ki are satisfied for all 1 i n.

2. Bessel integrals and confluent hypergeometric functions Let us put G = Gn = {g ∈ SL2n (R) | t gJn g = Jn } with Jn = −10 n 10n . It is the real symplectic group of degree n. Take the maximal compact subgroup K = G ∩ O(2n) in the standard way. Let P = MN denote the Siegel maximal parabolic subgroup of G, where we recall that M = {m(a) | a ∈ GLn (R)} and N = {n(x) | x ∈ Sn } with the symbols m(a) =

a 0

0 t a −1

and n(x) =

1n 0

x 1n

.

For any s ∈ C we define a one-dimensional representation δPs of P by setting δPs (p) = |δ(a)|s for p = m(a)n(x) ∈ MN . Then we consider the induced representation I (s) of G obtained from δPs , whose space consists of all functions f ∈ C ∞ (G) such that f (pg) = δPs (p)f (g) are satisfied for all p ∈ P and g ∈ G, on which G acts by right translations. This is called a degenerate principal series representation of G. We notice that its K-type decomposition is multiplicity free, where all non-trivial K-types are parameterized by the highest weights (d1 , d2 , . . . , dn ), which correspond to all non-increasing sequences d1 d2 · · · dn of even integers. Let Hn denote the Siegel upper half space of degree n: Hn = {z ∈ Tn | z − z > 0n }. −1 The group G acts on Hn in the standard way: namely, gz = (az + d)(cz + d) for every ab g = c d ∈ G with n × n block matrices a, b, c, d. We will also use the symbols μg (z) = cz + d

and jg (z) = δ μg (z) .

The subgroup K stabilizes i = i1n and is isomorphic to the unitary group U (n) by the map K r → μr (i) ∈ U (n). Thus also KC is isomorphic to the general linear group GLn (C). For an integer k 0 we let Vk denote the space of all homogeneous polynomials on Tn of degree k, on which GLn (C) acts by τk : [τk (a)f ](z)= f (t aza) for a ∈ GLn (C), f ∈ Vk , and z ∈ Tn . There is the irreducible decomposition τk = κ τκ , correspondingly Vk =

Vκ ,

(2.1)

κ

where κ varies over all partitions in Pkn and τκ denotes the irreducible action τk |Vκ on the subspace Vκ of the highest weight 2κ = (2k1 , . . . , 2kn ). Notice that this decomposition is multiplicity free.


2583

Here we explain our construction of functions belonging to each K-type of I (s). Fix an even integer d and take the module Vκ for every κ ∈ Pkn . Then, one can check that the functions

−s −d/2 Λ(g; s, d)(ϕ) = jg (i) jg (i)−1 jg (i) ϕ μg (i)−1 μg (i)

(2.2)

in g ∈ G and s ∈ C belong to the K-type of I (s) of the highest weight (2k1 − d, 2k2 − d, . . . , 2kn − d)

(2.3)

for all ϕ ∈ Vκ . On the other hand, if we put c ϕ(z) = ϕ(z), then the functions Λ(g; s, d)(c ϕ) belong to the K-type of I (s) of the highest weight (−2kn − d, −2kn−1 − d, . . . , −2k1 − d) for all ϕ ∈ Vκ . Now for any h ∈ Sn we define an integral transform of Λ(g; s, d)(ϕ), ϕ ∈ Vκ , by Wh (g; s, d)(ϕ) =

e−2πiσ (hx) Λ wn(x)g; s, d (ϕ) dx,

(2.4)

Sn

with w = −10 n 10n . Then its right K-translations behave according to the K-module of the highest weight (2.3). Besides this remark, we also have the identities

n+1−s

Wh[a] (g; s, d)(ϕ) Wh n(x)m(a)g; s, d (ϕ) = e2πiσ (hx) δ(a)

for x ∈ Sn and a ∈ GLn (R). So in other words, the functions Wh (g; s, d)(ϕ), ϕ ∈ Vκ , give the generalized Bessel integrals which are assigned to the character ψh and the K-type of I (s) of the highest weight (2.3). Due to this observation and the Iwasawa decomposition G = P K, we find that the functions Wh (g; s, d)(ϕ) in g ∈ G are determined by those values on M. Let us also introduce the generalized confluent hypergeometric function related to the integral (2.4). Let us put ε(x) = 1n − ix for x ∈ Sn . Then, for any h ∈ Sn , ϕ ∈ Vκ , and (α, β) ∈ C2 , we define ξ(h; α, β)(ϕ) =

−α −β e−2πiσ (hx) δ ε(x) δ ε(x) ϕ ε(x)−1 ε(x) dx.

(2.5)

Sn n(n+1)

Here dx stands for a Euclidean measure on Sn which is identified with R 2 . Since ε(x)−1 ε(x) is a symmetric unitary matrix and ϕ is a polynomial, [12, Lemma 1 and (1.23)] implies that this integral is convergent if Re(α + β) > n. Now one can write

2ρ −s

s −d s +d Wh m(a); s, d (ϕ) = δ(a) n ξ h[a]; , (ϕ) 2 2

(2.6)

2584


by putting ρn = n+1 2 . This is convergent for Re(s) > n. In the latter sections, we will mainly discuss the analytic behavior of the generalized confluent hypergeometric function occurring in the right-hand side above. 3. Main results We shall give the statements of our main results. The proofs will be given in Sections 8 and 9. Let us begin with some necessary definitions. To each partition κ = (k1 , . . . , kn ) ∈ Pkn we shall assign a product of Γ -functions defined by Γn (s, κ) = π

1 4 n(n−1)

n 1 Γ s + ki − (i − 1) , 2

s ∈ C.

i=1

In particular, one has Γn (s, (0n )) = Γn (s) (see Section 1) for the trivial partition (0n ) = (0, . . . , 0) ∈ P0n . For every h ∈ Sn , ϕ ∈ Vκ , and (α, β) ∈ C2 , let us define the integral η(h; α, β)(ϕ) = e−2σ (v−h) δ(v)α−ρn δ(v − 2h)β−ρn ϕ(v) dv. v>0n ,v>2h

This is convergent at least when Re(α) > ρn − 1 and Re(β) > ρn . Now our first main result is stated as follows. 3.1. Theorem. Take a κ ∈ Pkn . Then, for every ϕ ∈ Vκ , there exists a collection of suitable polynomials ϕλ ∈ Vλ for all λ ≺ κ, λ ∈ Pln (l k), so that one has the identity ξ(h; α, β)(ϕ) = Γn (β)−1

Γn (α, λ)−1 η(πh; α, β)(ϕλ ).

(3.1)

λ≺κ

This identity holds at least if Re(α) > ρn − 1 and Re(β) > 2ρn − 1. Here the right-hand side is a finite sum. The proof of this assertion will be given in Section 8 by using Proposition 7.1. Next we assume that n is even, and thus let us put n = 2m. We shall take a pair of integers k and p satisfying the parity condition: k ≡ p mod 2, and the inequalities: k m and 0 p n. n Then, we define a partition κ(p) ∈ P(k−m)p by κ(p) = k − m, . . . , k − m, 0n−p ,

(3.2)

p

which has the length l(κ(p)) = p. As a consequence, we find that the functions Λ(g; k, k − p)(ϕ) belong to the K-type of I (k) of the highest weight ( k − n + p, . . . , k − n + p, p − k, . . . , p − k )

p

n−p

for all ϕ ∈ Vκ(p) . Now we shall state our second main result.

(3.3)


2585

3.2. Theorem. For an even integer n = 2m, we take a pair of integers k and p satisfying the above conditions. Assume that h ∈ Sn is nondegenerate. Then one has the following assertions. s+k−p (i) The function ξ(h; s−k+p 2 , 2 )(ϕ) is continued to an entire function in s for every ϕ ∈ Vκ(p) . (ii) If sign(h) = (p, n − p) for the matrix h ∈ Sn , then ξ(h; p2 , k − p2 )(ϕ) = 0 for all ϕ ∈ Vκ(p) . (iii) If sign(h) = (p, n − p) for the matrix h ∈ Sn , then Wh (g; k, k − p)(ϕ) = 0 for all g ∈ G and ϕ ∈ Vκ(p) .

These assertions will be proved in Section 9. We give some remarks on this theorem, while rest of the paper is logically independent of them. Keep the setting above. Firstly, it is known that I (k) is reducible. Now Theorems 4.1 and 5.2 in [9] tell us that the K-type (3.3) of I (k) attached to each pair (k, p) provides the k k minimal K-type of an irreducible G-submodule πp,n−p embedded in I (k). These πp,n−p are all unitarizable, and they exhaust all irreducible submodules of I (k) when p varies. We have in particular that the generalized Bessel integrals Wh (g; k, k −p)(ϕ) are assigned to the minimal Kk type of πp,n−p for all ϕ ∈ Vκ(p) . Then, we may think that Theorem 3.2(iii) suggests a connection k between πp,n−p and the open GLn (R)-orbit Ωp,n−p in Sn , as stated in Theorem 1.1 in particular. k If k > n, then each πp,n−p has the regular infinitesimal character. Hence we shall identify it with a cohomological representation Aqp,n−p (λk ) of G by applying Proposition 6.1 in Vogan and Zuckerman [14]. Here qp,n−p stands for a maximal parabolic subalgebra in sp(n, C) with the abelian unipotent radical and the Levi subalgebra lp,n−p = u(p, n − p) ⊗ C. Consequently, we may think that the above theorem suggests a reasonable correspondence between the open GLn (R)-orbits in Sn and the cohomological representations of G of the above type. 4. EP-polynomials in GLn -representations We first recall some basic facts concerning the analysis on Yn . The reference is Maass [10, Section 6]. When the space Yn is parameterized by yn = (yij ), 1 i, j n, let us define the operator ∂n = (ηij ∂y∂ij ), where ηij is 1 if i = j , 12 otherwise. We let Ln denote the ring of GLn (R)invariant differential operators on Yn . Then it is known that Ln is algebraically generated by r = σ ((yn ∂n )r ) with 1 r n, see the theorem in p. 64 of [10] for this fact. We will need another system of generators of Ln which is given in p. 67 of [10]. We recall the definition. Take a general n × n matrix A = (aij ). For a pair of sequences 1 i1 < · · · < r ir n and 1 j1 < · · · < jr n of length r, we set a subdeterminant of A as ji11 ···i ···jr A = δ((aiα jβ )1α,βr ). We define differential operators Drn , 1 r n, by Drn

=

1i1 ρn − 1, then one has that

e−σ (zv) δ(v)s−ρn ϕ(v) dv = Γn (s, κ)δ(z)−s ϕ z−1

(7.1)

Yn

for every polynomial ϕ ∈ Vκ , κ ∈ Pkn . This may be regarded as a precise version of Herz [3, Lemma 4.1]. For an s ∈ C we recall the function As (yn ) on Yn defined in [3, Section 2]: As (yn )δ(yn )s = Γn (s + ρn )−1 δ(yn )s 0 F1 (s + ρn , −yn ) −1 eσ (yn z) e−σ (z ) δ(z)−s−ρn dz. = (2πi)−n Re(z)=x0 >0n

Indeed it is O(n)-invariant and gives an entire function of s and yn simultaneously, see [3, p. 486]. The notion of generalized Hankel transforms is discussed in [3, Theorem 3.1]. We recall the definition [3, (4.2)] of the generalized Laguerre function Lsϕ (yn ) that is the generalized Hankel transform of e−σ (yn ) ϕ(yn ) for every ϕ ∈ Vκ , κ ∈ Pkn : e−σ (yn ) Lsϕ (yn ) =

Yn

Here we assume that Re(s) > ρn − 1.

As (yn v)δ(v)s−ρn e−σ (v) ϕ(v) dv.

(7.2)


2593

7.1. Proposition. (i) The generalized Laguerre function Lsϕ (yn ) provides a polynomial on Yn for every ϕ ∈ Vκ , κ ∈ Pkn . Thus, it gives a polynomial on Tn . (ii) For every ϕ ∈ Vκ , there exists a collection of suitable polynomials ϕλ ∈ Vλ for all λ ≺ κ, λ ∈ Pln (l k), so that one has the expansion Lsϕ (yn ) = Γn (s, κ)

Γn (s, λ)−1 ϕλ (yn ).

(7.3)

λ≺κ

In particular, Lsϕ (yn ) is continued to an entire function in s. Proof. When Re(s) > ρn − 1, the inversion formula for the generalized Hankel transform gives us (7.4) e−σ (vz) δ(v)s−ρn Lsϕ (v) dv = Γn (s, κ)δ(z)−s ϕ 1n − z−1 Yn

for z ∈ H n , ϕ ∈ Vκ , see [3, Theorem 3.2, Lemma 4.1, and (4.3)]. By Proposition 6.5, one can write ϕ(1n − z−1 ) in the right-hand side in the form −1 ϕ 1n − z−1 = ϕλ z λ≺κ

by choosing polynomials ϕλ ∈ Vλ for all λ ≺ κ. Consequently, (7.1) implies that the right-hand side of (7.4) is equal to Γn (s, κ)

Γn (s, λ)−1

λ≺κ

e−σ (zv) δ(v)s−ρn ϕλ (v) dv.

Yn

Now the inverse Laplace transform is applied to complete the argument.

2

8. Proof of Theorem 3.1 Take a partition κ ∈ Pkn . For every ϕ ∈ Vκ we have the formula

−s e−σ (ε(x)v) δ(v)s−ρn Lsϕ (2v) dv = (−1)k Γn (s, κ)δ ε(x) ϕ ε(x)−1 ε(x) ,

(8.1)

Yn

which is a variant of (7.4). Here we assume that Re(s) > ρn − 1. We now turn to the proof of Theorem 3.1. By the formula (8.1), we get that ξ(h; α, β)(ϕ) is equal to k

−1

(−1) Γn (α, κ)

Yn

e

−σ (v)

α−ρn

δ(v)

Lαϕ (2v) Sn

−β eiσ ((v−2πh)x) δ ε(x) dx dv.

2594


By the Fourier inversion formula [12, (1.23), p. 274], this is further equal to −1 −1 e−2σ (v−πh) δ(v)α−ρn δ(v − 2πh)β−ρn Lαϕ (2v) dv Γn (α, κ) Γn (β) v>0 v>2πh

up to a scalar multiple, where we assume that Re(α) > ρn − 1 and Re(β) > 2ρn − 1. Now let us insert the expansion (7.3), being modified for Lαϕ (2y), into this expression, which completes the assertions of Theorem 3.1. 9. Proof of Theorem 3.2 n Now suppose that n is even, and thus let us put n = 2m. Recall the partition κ(p) ∈ P(k−m)p defined in (3.2) for a pair of integers k ≡ p mod 2, k m, 0 p n. Also we take κ(n − p) ∈ n as its counterpart for the pair k and n − p. P(k−m)(n−p)

9.1. Lemma. For every ϕ ∈ Vκ(p) there exists a suitable polynomial ϕ ∈ Vκ(n−p) so that one has Λ(g; s, k − p)(ϕ) = Λ(g; s, n − p − k) c ϕ . Proof. Both sides of the identity belong to the same K-type (3.3) of I (s). Since the K-type decomposition of I (s) is multiplicity free, we get the conclusion. 2 Fix a nondegenerate matrix h ∈ Sn with sign(h) = (r, n − r), 0 r n. By taking ϕ ∈ Vκ(n−p) in the above lemma for every ϕ ∈ Vκ(p) , we consider the function s + k − n + p s − k + n − p c s −k+p s +k−p , (ϕ) = ξ h; , ξ h; ϕ . 2 2 2 2

(9.1)

Our goal is to show the vanishing of this function at s = k, if r = p. We first prove the vanishing for p < r n. Let us start with the expression in the left-hand side of (9.1). Then, by Theorem 3.1, we find that it suffices to show the vanishings of all the terms Γn

s +k−p 2

−1 Γn

s −k+p ,λ 2

−1 s −k+p s +k−p , (ϕλ ) η h; 2 2

(9.2)

with ϕλ ∈ Vλ , λ ≺ κ(p), which occur in (3.1). Let us reformulate the problem as follows. We take a matrix q ∈ GLn (R) so that one has h[q −1 ] = diag(1r , −1n−r ) and q t q = diag(a, b), where a ∈ Yr and b ∈ Yn−r are diagonal matrices whose entries consist of the absolute values of the eigenvalues of h. Let us

= diag(0r , 1n−r ), and define the domain Xr = {x ∈ Sn | x + εr put εr = diag(1r , 0n−r ), εn−r

and x + εn−r ∈ Yn }. A change of variables shows that η(h; α, β)(ϕ) is equal to the product of δ(2h)α+β−ρn e−2σ (h) and β−ρn t

τλ (q)ϕ (x + εr ) dx e−4σ (q qx) δ(x + εr )α−ρn δ x + εn−r Xr


2595

for every ϕ ∈ Vλ . By replacing τλ (q)ϕ with an arbitrary polynomial ψ on Sn , we define the integral ζr,n−r (g; α, β)(ψ) =

β−ρn

e−4σ (gx) δ(x + εr )α−ρn δ x + εn−r ψ(x + εr ) dx,

Xr

where g = diag(a, b) ∈ Yn with a ∈ Yr and b ∈ Yn−r . This is convergent at least if Re(α) > ρn − 1 and Re(β) > ρn . Now our concern is to show the next assertion. 9.2. Proposition. Keep the setting above, hence in particular, we assume that λ ≺ κ(p). Then, for general polynomials ψ on Sn , the products Γn

s+k−p 2

−1 Γn

s −k+p ,λ 2

−1

s −k+p s +k−p (ψ) ζr,n−r g; , 2 2

(9.3)

are continued to entire functions in s. They are all vanishing at s = k, if p < r n. Proof. We recall the smooth bijection from Xr onto Ur = Yr × Yn−r × Rrn−r that is defined x z in [12, p. 289]. Write X = t z y ∈ Xr in the block decomposition with x ∈ Rrr , y ∈ Rn−r n−r , and z ∈ Rrn−r . Put f = (x + 1r )−1/2 z(y + 1n−r )−1/2 and s = (1r − f t f )1/2 . Then, the bijection Xr (x, y, z) → (u, v, w) ∈ Ur is given by w = s −1 f , u = x − W , and v = y − W , where we put W = w t w and W = t ww. Its inverse map ι is given by x = u + W , y = v + W , and 1/2 z = (1r + u + W )1/2 (1r + W )−1/2 w 1n−r + v + W . We also have that r n−r ∂(x, y, z) = δ(1r + u + W ) 2 δ 1n−r + v + W 2 δ(1r + W )−ρn . ∂(u, v, w) Notice that when u approaches the boundary ∂Yr , the map ι, together with all its derivatives with respect to u = (uij ), is locally uniformly bounded in u. The analogous properties hold also, when we let the variable v approach ∂Yn−r . By the change of variables given above, we obtain the identity

Γr

s + d − n + r −1 s −d s +d , (ψ) ζr,n−r g; 2 2 2 s−d = δ(v) 2 −ρn e−σ (bv) Ig,s (v)(ψ) dv. Yn−r

Here we put d = k − p to shorten the notation, and Ig,s (v)(ψ) = Γr

s +d −n+r 2

−1 δ(u) Yr

s+d 2 −ρn

e−σ (au) Jg,s,v (u)(ψ) du,

(9.4)

2596


and Jg,s,v (u)(ψ) is defined by

s+d −ρn−r

e−σ (aW )−σ (bW ) δ v + 1n−r + W 2

Rrn−r

· δ(u + 1r + W )

s−d 2 −ρr

δ(1r + W )ρn −s ψ ι(u, v, w) + εr dw.

(9.5)

Notice that the maximal absolute value of entries of X + εr , X = ι(u, v, w) ∈ Xr , is bounded by δ(1n + X + εr ) 2r δ(1n−r + v + W )δ(1r + u)δ(1r + W )−1 . We also have estimates δ(1r + W ) δ(1r + u + W ) 2r δ(1r + u)δ(1r + W )r for (u, w) ∈ Yr × Rrn−r , and the analogous ones for δ(1n−r + v + W ). Since ψ|Xr is a polynomial on Xr , we can choose suitable constants A1 , . . . , A7 so that

e−σ (aW )−σ (bW ) δ(1r + W )A6 μ+A7 dw

A1 · δ(1r + u)A2 μ+A3 δ(1n−r + v)A4 μ+A5 Rrn−r

gives a bound of (9.5), μ = Re(s). This is proved to converge for all s ∈ C, see [12, Lemma 2.8]. Then, also because the convergence of (9.5) is locally uniform in s, v, u, the product e−σ (au) Jg,s,v (u)(ψ), together with all its derivatives, provides a rapidly decreasing function in u ∈ Yr for all (s, v) ∈ C × Yn−r , and stays bounded when u approaches ∂Yr . Thus, we can apply Theorem 3.1 in Gindikin [2] to conclude an entire continuation in s of the generalized Riemann–Liouville integral Ig,s (v)(ψ). Again the function e−σ (bv) Ig,s (v)(ψ) of v ∈ Yn−r has the similar properties as above for all s ∈ C. Then we repeat the same argument to conclude that Γn−r

s−d −r 2

−1 Γr

s +d −n+r 2

−1

s −d s +d , (ψ) ζr,n−r g; 2 2

(9.6)

is continued to an entire function in s. Comparing (9.3) with this entire function, we find that the analytic behavior of (9.3) in s can be determined by that of Γn−r ( s−k+p−r ) Γr ( s+k−n−p+r ) 2 2 Γn ( s−k+p 2 , λ)

Γn ( s+k−p 2 )

.

(9.7)

Now this gives an entire function. This is proved by the assumption λ ≺ κ(p), which implies l(λ) p, combined with the definition (3.1) of each factor in (9.7). Thus, we have obtained the desired entire continuation of (9.3). Let us examine the order of (9.7) at s = k more precisely. We let [c] denote the Gauss symbol −1 has the zero of order [ n−p+1 ] at s = k for λ ≺ κ(p). of c ∈ Q. Then, we find that Γn ( s−k+p 2 , λ) 2 ) at s = k is counted as follows: it On the other hand, the order of the pole of Γn−r ( s−k+p−r 2 n−r ] if r p; or [ n−r+1 is equal to [ n−p+1 2 2 ] if r > p and r ≡ p mod 2; or [ 2 ] if r > p and r ≡ p + 1 mod 2. Consequently, we have the vanishing of (9.7) at s = k, hence, (9.3) is also vanishing at the point. This completes the argument of the proof. 2


2597

We now turn to the proof of Theorem 3.2. If p < r n, then we apply Proposition 9.2 to ψ = τλ (q)ϕλ to get the vanishing of (9.2) at s = k for every λ ≺ κ(p). Then, Theorem 3.1 concludes the vanishing of (9.1) at s = k under the same assumption on r. It remains to prove the vanishing when 0 r < p. To this end we shall begin our discussion with the expression in the right-hand side of (9.1). By Theorem 3.1, the right-hand side of (9.1) appears to be a finite sum of Γn

s +k−n+p 2

−1 Γn

s −k+n−p ,λ 2

−1 s −k+n−p s +k−n+p , (ϕλ ) η −h; 2 2

with polynomials ϕλ ∈ Vλ for λ ≺ κ(n − p). Therefore, our problem is reduced to check the behavior at s = k of Γn

s +k−n+p 2

−1 Γn

s −k+n−p ,λ 2

−1

s −k+n−p s +k−n+p , (ψ) ζr,n−r g; 2 2

for λ ≺ κ(n − p) and general polynomials ψ on Sn . Similarly as in the proof of Proposition 9.2, the behavior is controlled by the function ) Γn−r ( s+k−n+p−r ) Γr ( s−k−p+r 2 2 Γn ( s−k+n−p , λ) 2

Γn ( s+k−n+p ) 2

.

(9.8)

We notice that this (9.8) coincides with the function (9.7) after changing the parameter r, or p, with n − r, or n − p, respectively. Consequently, we obtain the desired vanishing of (9.1) when 0 r < p, which completes the proof of Theorem 3.2(ii). The third claim is an immediate consequence of this and (2.6). Thus, the proof is completed. References [1] C. Bingham, An identity involving partional generalized binomial coefficients, J. Multivariate Anal. 4 (1974) 210– 223. [2] S.G. Gindikin, Analysis on homogeneous domains, Russian Math. Surveys 19 (1964) 1–89. [3] C.S. Herz, Bessel functions of matrix argument, Ann. of Math. 61 (1955) 474–523. [4] A.T. James, Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Statist. 35 (1964) 475–501. [5] B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971) 753– 809. [6] H.B. Kushner, On the expansion of Cρ∗ (V + I ) as a sum of zonal polynomials, J. Multivariate Anal. 17 (1985) 84–98. [7] H.B. Kushner, M. Meisner, Eigenfunctions of expected value operators in the Wishart distribution, Ann. Statist. 8 (1980) 977–988. [8] H.B. Kushner, A. Lebow, M. Meisner, Eigenfunctions of expected value operators in the Wishart distribution, II, J. Multivariate Anal. 11 (1981) 418–433. [9] S.T. Lee, Degenerate principal series representations of Sp(2n, R), Compos. Math. 103 (1996) 123–151. [10] H. Maass, Siegel’s Modular Forms and Dirichler Series, Lecture Notes in Math., vol. 216, Springer, 1971. [11] D.St.P. Richards, Applications of invariant differential operators to multiplicative distribution theory, SIAM J. Appl. Math. 45 (1985) 280–288. [12] G. Shimura, Confluent hypergeometric functions on tube domains, Math. Ann. 260 (1982) 269–302. [13] G. Shimura, On differential operators attached to certain representations of classical groups, Invent. Math. 77 (1984) 463–488. [14] D.A. Vogan, G.J. Zuckerman, Unitary representations with non-zero cohomology, Compos. Math. 53 (1984) 51–90.


Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities Philippe Cieutat a , Khalil Ezzinbi b,∗ a Laboratoire de Mathématiques, Université Versailles-Saint-Quentin-en-Yvelines, 45 avenue des États-Unis,

78035 Versailles cedex, France b Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques, B.P. 2390, Marrakech,

Morocco Received 15 March 2010; accepted 4 January 2011 Available online 2 February 2011 Communicated by J. Coron

Abstract In this work, we study the existence of bounded and almost automorphic solutions for evolution equations in Banach spaces. We suppose that the linear part is the infinitesimal generator of a compact C0 -semigroup of bounded linear operators and the nonlinear part is an almost automorphic function with respect to the second argument. We give sufficient conditions ensuring the existence of an almost automorphic solution when there is at least one bounded solution on R+ . We use the subvariant functional method to show that every K-minimizing mild solution is compact almost automorphic. Applications are provided for both heat and wave equations with nonlinearities in several functional spaces. © 2011 Elsevier Inc. All rights reserved. Keywords: Bounded solution; K-mild solution; Almost automorphic solution; Evolution equations; Semigroup; Compact operator; Heat and wave equations


E-mail addresses: [email protected] (P. Cieutat), [email protected] (K. Ezzinbi). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.01.002

P. Cieutat, K. Ezzinbi / Journal of Functional Analysis 260 (2011) 2598–2634

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1. Introduction Our aim is to investigate the existence of bounded and compact almost automorphic solutions to the following evolution equation in a Banach space X x (t) = Ax(t) + f t, x(t)

for t ∈ R,

(1.1)

where A : D(A) ⊂ X → X is the infinitesimal generator of a C0 -semigroup of bounded linear operators on X and f : R×X → X is an almost automorphic function in t uniformly with respect to the second argument. We give sufficient conditions for the existence of compact almost automorphic solutions when Eq. (1.1) admits at least a bounded solution on R+ . Our main sufficient condition is the uniqueness of some solutions with values in some compact set which minimize a functional. Firstly, we study the existence of bounded solutions and secondly we will show the existence of a compact almost automorphic solution for Eq. (1.1). Then we apply our results to nonlinear partial differential equations. We give sufficient conditions ensuring the existence of compact almost automorphic solutions to some heat and wave equations. Even in the almost periodic framework, our results on the wave equation are new (Corollary 6.15). The question of the existence of periodic, almost periodic solutions of evolution equations has been intensively studied. In the literature, several books are devoted to the almost periodic solutions for differential equations and dynamical system. For example, let us indicate the books of Amerio and Prouse [7], Corduneanu [18], Fink [26], Levitan and Zhikov [35] and Zaidman [47]. Bochner introduced the concept of almost automorphy in the literature in [12] as a generalization of almost periodicity. Veech studied almost automorphic functions on groups in [44,45]. More recently, the existence of almost automorphic solutions to ordinary as well as abstract differential equations has been intensively studied. We refer the reader to N’Guérékata’s book [38]. In [37], Massera proved the equivalence between the existence of a periodic solution and the existence of a bounded solution on R+ for a periodic nonhomogeneous differential equation in finite dimensional spaces. For almost periodic or almost automorphic solutions, the situation is more complicated, since one cannot use fixed point theory on Poincaré’s operators. Early concerning almost periodic solutions of partial differential equations were obtained starting in 1950s by the Italian school, Amerio, Biroli, Prouse and others, [1–10,41,42]. Amerio introduced the concept of minimax principle in the literature in his paper [1] as a generalization to the nonlinear case Favard’s theory [23,24]. The minimax principle asserts that under suitable assumptions, the existence of a unique minimizer of the supremum norm among solutions with values in some compact set, is almost periodic. Several applications to partial differential equations are given in [2,3,7]. Then in the spirit of the Italian school, Dafermos, Haraux, Ishii have given important contributions to the question of almost periodic solutions [19,29–31,33]. Haraux [32] and Ishii [33] have established the existence of almost periodic solutions of contractive almost periodic processes on a Banach space. The main result of Ishii [33] ensures the existence of some almost periodic solution, if there exists a solution defined on R+ with a compact range. In [31], Haraux studied the existence of an almost periodic solutions for the following evolution equation x (t) + Ax(t) f (t)

for t ∈ R,

(1.2)

is a maximal monotone operator on R2 and f is almost periodic. He proved that every where A bounded solution on R+ of Eq. (1.2) is asymptotically almost periodic. The dimension two of the

2600


euclidean space is essential for this last result. Moreover the author proved that every bounded solution on R of Eq. (1.2) is almost periodic (Theorem 2.1 in [31]) by using the minimax principle which is also valid for an arbitrary Banach space. The existence of almost automorphic solution of Eq. (1.1) has been extensively studied when the semigroup generated by A has an exponential dichotomy, since in that case Eq. (1.1) has a unique bounded solution on R which is almost automorphic when f is almost automorphic in t and Lipschitzian with respect to the second argument, and the Lipschitz constant is sufficiently small (depending of the constant governing the exponential dichotomy). More details can be found in [38]. In [36], the authors proved the existence of almost automorphic solutions for the following ordinary differential equation d x(t) = Gx(t) + e(t) for t ∈ R, dt

(1.3)

where G is a constant n × n matrix and e : R → Rn is almost automorphic. They proved that the existence of a bounded solution on R+ implies the existence of an almost automorphic solution. The existence of almost automorphic solutions for differential equations in infinite dimensional spaces has been studied by several authors. Recently in [20], the authors established the existence of almost automorphic solutions for functional differential equations of neutral type, they proved that the existence of a bounded solution on R+ implies the existence of an almost automorphic solution. In [39], the author studied the existence of almost automorphic solutions for the following semilinear abstract differential equation d x(t) = Cx(t) + θ (t) dt

for t 0,

(1.4)

where C generates an exponentially stable semigroup on a Banach space Y and θ is an almost automorphic function from R to Y . The author proved that the only bounded mild solution of Eq. (1.4) on R is almost automorphic. In [27] and [28], the authors investigated the existence and uniqueness of an almost periodic solution for Eq. (1.1) when A = 0 and f is dissipative with respect to the second argument and they proposed as application, the following ordinary differential equation in a Banach space E α x (t) = −x(t) x(t) + h(t)

for t ∈ R,

(1.5)

where α 0 and h : R → E is a continuous function, they showed if the input function h is almost periodic then Eq. (1.5) has a unique bounded solution on R which is also almost periodic. Recently, in [21], the authors extended the works [27] and [28] to almost automorphic case, in fact they proved the existence and uniqueness of a bounded solution on R which is compact almost automorphic. In [15] and [22], the authors studied the existence and uniqueness, attractiveness for a pseudo almost periodic automorphic solution for some general dissipative systems in Banach spaces. Let us indicate the contribution of Zaidman on almost periodic and automorphic functions (see for instance [46–49]). Fink introduced in the literature the concept of subvariant functional in [25] to prove the existence of compact almost automorphic solutions for the following ordinary differential equation x (t) = ϕ t, x(t)

for t ∈ R,

(1.6)


2601

where the function ϕ : R × Rn → Rn is compact almost automorphic in t uniformly with respect to the second argument. In the particular case of ordinary differential equations in finite dimensional spaces, the subvariant functional method of Fink is a generalization of the minimax principle of Amerio, in the sense where the almost automorphic solution which is reached by minimizing a function, the so-called subvariant functional, is not necessarily the supremum norm. In [16], an extension in Banach spaces of the main results in [25] is given. In this work, we extend the results of [16,25] to Eq. (1.1) and we use the subvariant functional method to prove the existence of a compact almost automorphic solution, we prove that the existence of a K-bounded mild solution of Eq. (1.1) that minimizes some subvariant functional is compact almost automorphic. Our work is organized as follows: in Section 2 we recall some results on evolution equations and almost automorphic functions. In Section 3 we announce the main results and we give several corollaries. In Section 4, we discuss some results regarding the existence of bounded solutions and solutions that have a compact range. The proofs of the main results are given in Section 5, notably we state the existence of compact almost automorphic solutions through minimizing some subvariant functional. The last section is devoted to several applications for some heat and wave equations in C0 -space and L2 -space. 2. Almost automorphic functions and evolution equations Let us fix our notations and recall some definitions. Throughout this paper, we denote by (X, .) a Banach space. A C0 -semigroup (T (t))t0 is said to be compact if for each t > 0, T (t) is a compact operator, that is T (t) maps bounded sets into relatively compact sets [40]. Let A be the infinitesimal generator of a C0 -semigroup (T (t))t0 . Let x0 ∈ X and f ∈ C(R × X, X) (continuous maps). We say that x is a mild solution on [t0 , +∞) (where t0 ∈ R) of Eq. (1.1) with the initial condition x(t0 ) = x0 , if x ∈ C([t0 , +∞), X) and satisfies t x(t) = T (t − t0 )x0 +

T (t − σ )f σ, x(σ ) dσ

for t t0 .

t0

Remark that for a mild solution on [t0 , +∞) of Eq. (1.1), one has t x(t) = T (t − s)x(s) +

T (t − σ )f σ, x(σ ) dσ

for t s t0 .

(2.1)

s

We say that x is a mild solution on R of Eq. (1.1) for a given f ∈ C(R × X, X), if x ∈ C(R, X) and satisfies Eq. (2.1) for each t s. For some preliminary results on C0 -semigroup, we refer the reader to [40]. Let BC(R, X) be the space of all bounded and continuous functions from R to a Banach space X, equipped with the uniform topology. Let x ∈ BC(R, X) and τ ∈ R. We define the translation function xτ by xτ (s) = x(τ + s)

for s ∈ R.

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A bounded continuous function x : R → X is said to be almost periodic if {xτ ; τ ∈ R} is relatively compact in BC(R, X). Denote by AP(R, X) the set of all such functions. For some preliminary results on almost periodic functions, we refer the reader to [18,26]. A continuous function x : R → X is said to be almost automorphic if for any sequence of real numbers (tn )n , there exists a subsequence of (tn )n , denoted by (tn )n such that y(t) = lim x(t + tn )

(2.2)

lim y(t − tn ) = x(t)

(2.3)

n→+∞

is well defined for each t ∈ R and n→+∞

for each t ∈ R. Denote by AA(R, X) the space of all almost automorphic X-valued functions. Because of point-wise convergence, the function y ∈ L∞ (R, X) (the space of essentially bounded measurable X-valued functions), but it is not necessarily continuous. It is also clear from the definition above that almost periodic functions (in the sense of Bochner [11,26]) are almost automorphic. If the limits in (2.2) and (2.3) are uniform on any compact subset K ⊂ R, we say that x is compact almost automorphic. If we denote AAc (R, X) the space of all compact almost automorphic X-valued functions, then we have AP(R, X) ⊂ AAc (R, X) ⊂ AA(R, X) ⊂ BC(R, X). For some details on almost automorphic functions, we refer to [38]. We say that a mapping f : R × X → X, (t, x) → f (t, x) is almost periodic in t uniformly with respect to x when it satisfies the two following conditions: i) f ∈ C(R × X, X). ii) For all compact subset K of X, for all ε > 0, there exists > 0 such that for all α ∈ R, there exists τ ∈ [α, α + ] such that supf (t + τ, x) − f (t, x) ε. t∈R

Denote by APu (R × X, X) the set of all such mappings. We say that a mapping f : R × X → X, (t, x) → f (t, x) is almost automorphic in t uniformly with respect to x when it satisfies the two following conditions: i) f ∈ C(R × X, X). ii) For all compact subset K of X and for all sequence of real numbers (tn )n , there exists a map g : R × K → X and there exists a subsequence of (tn )n , denoted by (tn )n such that ∀t ∈ R, ∀t ∈ R,

lim sup f (t + tn , x) − g(t, x) = 0,

(2.4)

lim sup g(t − tn , x) − f (t, x) = 0.

(2.5)

n→+∞ x∈K

n→+∞ x∈K


2603

We denote by AAu (R × X, X) the set of all such mappings. An example of almost automorphic function in t uniformly with respect to x is f (t, x) = F (x) + b(t), with F ∈ C(X, X) and b ∈ AA(R, X). We write L(X, X) for the space of linear and bounded maps from X to X. An other example is the following: f (t, x) = A(t)x + b(t), with A ∈ AA(R, L(X, X)) and b ∈ AA(R, X). Remark 2.1. Our definition is different from the one used by Fink in [25]. The author assumed that f is compact almost automorphic in t uniformly with respect to x, that is to mean, for all compact subset K of X and for all sequence of real numbers (tn )n , there exists a map g : R × K → X and there exists a subsequence of (tn )n , denoted by (tn )n such that for all compact subset I of R, one has lim sup sup f (t + tn , x) − g(t, x) = 0,

n→+∞ t∈I x∈K

lim sup sup g(t − tn , x) − f (t, x) = 0.

n→+∞ t∈I x∈K

Remark 2.2. Here we recall a characterization for the almost automorphic functions in t uniformly with respect to x [17, Theorem 3.14]: f ∈ AAu (R × X, X) if and only if ∀x ∈ X, the partial function t → f (t, x) ∈ AA(R, X) and f is uniformly continuous on each compact K of X with respect to t, that is for all compact subset K of X, ∀ε > 0, ∃δ > 0, ∀x1 , x2 ∈ K, one has x1 − x2 δ

⇒

supf (t, x1 ) − f (t, x2 ) ε. t∈R

We have a similar characterization for almost periodic functions: f ∈ APu (R × X, X) if and only if ∀x ∈ X, t → f (t, x) ∈ AP(R, X) and f is uniformly continuous on each compact K of X with respect to t (see [17, Lemma 2.6]). 3. Almost automorphic solution minimizing a subvariant functional In this section, we use the subvariant functional method which has been introduced for the first time by Fink in [25] to prove the existence of almost periodic and compact almost automorphic solutions for some ordinary differential equations. Let K be a compact subset of X. Let CK (R, X) denote the set CK (R, X) = x ∈ C(R, X) for all t ∈ R, x(t) ∈ K . A mapping λK : CK (R, X) → R is called a subvariant functional associated to the compact set K, if λK satisfies the two following conditions: i) λK is invariant by translation: λK (xτ ) = λK (x) for each τ ∈ R, where xτ (·) = x(τ + ·). ii) λK is lower semicontinuous for the topology of compact convergence: if limn→+∞ xn = y uniformly on each compact subset of R, then λK (y) lim infn→+∞ λK (xn ). Remark 3.1. In the above definition of subvariant functional, note that conditions i) and ii) imply the following: iii) if limn→+∞ xτn = y uniformly on each compact subset of R, then λK (y) λK (x).

2604


Let us denote by FK the set of mild solutions x on R of Eq. (1.1) satisfying x(t) ∈ K, for each t ∈ R. A solution x∗ is called a minimal K-valued solution of Eq. (1.1) if x∗ ∈ F K

and λK (x∗ ) = inf λK (x). x∈FK

An example of subvariant functional is λK (x) = supx(t) t∈R

or more generally λK (x) = sup Φ x(t)

where Φ ∈ C(K, R).

t∈R

Another example which is given in [25] is the following λK (x) = sup x(t) − inf x(t), t∈R

t∈R

where x is R-valued. Remark 3.2. Our definition of subvariant functional is a modification of the definition due to Fink. Our definition is slightly stronger than the one used in [25]. Fink defines the subvariant functional by only iii) instead of i) and ii). For this reason, contrary to our results, in [25], it is assumed the existence of a minimal K-valued solution, except in the particular case of the subvariant functional: λK (x) = supt∈R x(t). The following hypotheses will be used in the main results: (H1) (H2) (H3) (H4) (H5)

A is the infinitesimal generator of a C0 -semigroup (T (t))t0 . The C0 -semigroup (T (t))t0 is compact. There exists t0 ∈ R, such that for all R > 0, one has suptt0 supxR f (t, x) < +∞. f ∈ AAu (R × X, X). f ∈ APu (R × X, X).

Theorem 3.3. Assume that (H1)–(H4) hold. In addition, suppose that Eq. (1.1) admits at least a mild solution x0 defined and bounded on [t0 , +∞). i) Then {x0 (t); t t0 } is relatively compact. ii) Let K be a compact subset of X such that {x0 (t); t t0 } ⊂ K. If λK is a subvariant functional associated to the compact set K, then Eq. (1.1) admits at least a minimal K-valued solution. iii) If Eq. (1.1) has a unique minimal K-valued solution, then the minimizing solution is compact almost automorphic. The proof of Theorem 3.3 will be given in Section 5.


2605

Corollary 3.4. Assume that (H1)–(H4) hold. In addition, suppose that Eq. (1.1) admits at least a mild solution x0 defined and bounded on [t0 , +∞). Let K = {x0 (t); t t0 }. If Eq. (1.1) has at most one K-mild solution on R, then this solution is compact almost automorphic. Proof. Take λK (x) = 1.

2

From Corollary 3.4, we easily deduce the following result. Corollary 3.5. Assume that (H1)–(H4) hold. In addition, suppose that Eq. (1.1) admits a unique mild solution which is bounded on R, then this solution is compact almost automorphic. When the C0 -semigroup (T (t))t0 is not compact, Theorem 3.3 becomes Theorem 3.6. Assume that (H1) and (H4) hold. In addition, suppose that Eq. (1.1) admits at least a mild solution x0 defined on [t0 , +∞) such that {x0 (t); t t0 } is relatively compact. i) Let K be a compact subset of X such that {x0 (t); t t0 } ⊂ K. If λK is a subvariant functional associated to the compact set K, then Eq. (1.1) admits at least a minimal K-valued solution. ii) If Eq. (1.1) has a unique minimal K-valued solution, then the minimizing solution is compact almost automorphic. The proof of Theorem 3.6 will be given in Section 5. Now we give some corollaries of the main results in the almost periodic case. For f ∈ APu (R × X, X), we denote by H(f ) the set of functions g : R × X → X such that for each compact subset K in X, there exists a real sequence (tn )n satisfying lim sup sup f (t + tn , x) − g(t, x) = 0.

n→+∞ t∈R x∈K

For g ∈ H(f ), we consider the following equation x (t) = Ax(t) + g t, x(t)

for t ∈ R.

(3.1)

Corollary 3.7. Assume that (H1)–(H3) and (H5) hold. In addition, suppose that Eq. (1.1) admits at least a mild solution x0 defined and bounded on [t0 , +∞). i) Then {x0 (t); t t0 } is relatively compact. ii) Let K be a compact subset of X such that {x0 (t); t t0 } ⊂ K. If λK is a subvariant functional associated to the compact set K, then Eq. (1.1) admits at least a minimal K-valued solution. iii) If for all g ∈ H(f ), Eq. (3.1) has at most a minimal K-valued solution, then the unique minimal K-valued solution of Eq. (1.1) is almost periodic. Corollary 3.8. Assume that (H1) and (H5) hold. In addition, suppose that Eq. (1.1) admits at least a mild solution x0 defined on [t0 , +∞) such that {x0 (t); t t0 } is relatively compact.

2606


i) Let K be a compact subset of X such that {x0 (t); t t0 } ⊂ K. If λK is a subvariant functional associated to the compact set K, then Eq. (1.1) admits at least a minimal K-valued solution. ii) If for all g ∈ H(f ), Eq. (3.1) has at most a minimal K-valued solution, then the unique minimal K-valued solution of Eq. (1.1) is almost periodic. The proof of Corollaries 3.7 and 3.8 will be given in Section 5. 4. Boundedness and compactness of solutions The object of this section is to state some results on the bounded solutions and on the solutions which have a relatively compact range. The following conditions will be used in this section: (C1) K is a compact subset of X. (C2) F : R × K → X is a measurable function. Let I be the interval [t0 , +∞) or the whole real line R. Assume that the function F satisfies, for each a and b ∈ I such that a b: sup sup F (t, x) < +∞.

(4.1)

atb x∈K

We say that u is a K-mild solution on I of u (t) = Au(t) + F t, u(t) ,

(4.2)

if u ∈ C(I, X) and satisfies u(t) ∈ K

for all t ∈ I,

and t u(t) = T (t − s)u(s) +

T (t − σ )F σ, u(σ ) dσ

for s and t ∈ I such that t s.

(4.3)

s

Remark that with (4.1), the map σ → F (σ, u(σ )) ∈ L1 ((s, t), X). Lemma 4.1. Let I be the interval [t0 , +∞) or the whole real line R. Assume that (H1), (C1) and (C2) hold. In addition, we assume that k = sup sup F (t, x) < +∞.

(4.4)

t∈I x∈K

Then there exists a function φ : [0, +∞) → [0, +∞) satisfying lim φ(τ ) = 0,

τ →0

(4.5)


such that every K-mild solution u on I of Eq. (4.2) verifies for all t, s ∈ I , u(t) − u(s) φ |t − s| .

2607

(4.6)

Remark 4.2. Lemma 4.1 means that every K-mild solution of Eq. (4.2) is uniformly continuous with the uniform continuity modulus φ. A consequence of this last lemma is the following result: if u is a K-mild solution on R and (tn )n is a sequence of real numbers, then the family {t → u(t + tn ); n ∈ N} is uniformly equicontinuous on R. Proof of Lemma 4.1. Since (T (t))t0 is a C0 -semigroup, then there exist ω 0 and M 1 such that T (t) Meωt for t 0 (4.7) (cf. [40, Theorem 2.2, p. 4]). Denote by φ : [0, +∞) → [0, +∞) the function defined by φ(τ ) = sup T (τ )x − x + kM x∈K

τ eωσ dσ.

(4.8)

0

Recall that a C0 -semigroup is strongly continuous: T (.)x : [0, +∞) → X is continuous for each x ∈ X, then lim T (t)x = x

t→0

for x ∈ K.

In view of Banach–Steinhaus’ theorem [43, p. 327], we obtain lim sup T (t)x − x = 0.

(4.9)

t→0 x∈K

By (4.8) and (4.9), we deduce (4.5). Since u is a K-mild solution on I of Eq. (4.2), then (4.3) holds, therefore u(t) − u(s) T (t − s)u(s) − u(s) +

t

T (t − σ )F σ, u(σ ) dσ,

s

for each s and t ∈ I such that t s. By using (4.4) and (4.7), we obtain t

u(t) − u(s) sup T (t − s)x − x + kM

eω(t−σ ) dσ,

x∈K

s

then by interchanging s and t, we deduce that u(t) − u(s) sup T |t − s| x − x + kM x∈K

therefore (4.6) holds with the function φ defined by (4.8).

|t−s|

eωσ dσ 0

2

for t, s ∈ I,

2608


Lemma 4.3. Assume that (H1), (C1) and (C2) hold. In addition, we assume that for all t ∈ R, F (t, .) ∈ C(K, X), k = sup sup F (t, x) < +∞.

(4.10) (4.11)

t∈R x∈K

We assume that u is a K-mild solution on R of Eq. (4.2). If there exists a sequence (tn )n of real numbers and there exists a map G : R × K → X such that lim sup F t + tn , x − G(t, x) = 0 for all t ∈ R,

(4.12)

n→+∞ x∈K

then there exists a subsequence of (tn )n denoted by (tn )n such that u(t + tn ) → v(t)

as n → +∞,

(4.13)

uniformly on each compact subset of R, where v is a K-mild solution on R of v (t) = Av(t) + G t, v(t) .

(4.14)

Remark 4.4. i) The mild solution v satisfies v(t) ∈ K for each t ∈ R. ii) G is measurable on R × K and G satisfies (4.10) and (4.11). Proof of Lemma 4.3. If we denote un (t) = u(t + tn ), then for each n ∈ N, un ∈ C(R, X) and satisfies un (t) ∈ K for t ∈ R, therefore for each t ∈ R, the set {un (t); n ∈ N} is a relatively compact subset of X. By Lemma 4.1, the solution u satisfies (4.6), then the sequence of functions (un )n satisfies: un (t) − un (s) φ |t − s|

for t, s ∈ R and n ∈ N,

where φ satisfies (4.5), therefore the sequence (un )n is uniformly equicontinuous on R. In view of Arzela–Ascoli’s theorem [43, p. 312], we can assert that {un ; n ∈ N} is a relatively compact subset of C(R, X) endowed with the topology of compact convergence. From the sequence (tn )n , we can extract a subsequence (tn )n such that there exists v ∈ C(R, X) and (4.13) holds. It remains to prove that v is a K-mild solution on R of Eq. (4.14). Since u is a K-mild solution of Eq. (4.2), therefore for each n ∈ N, one has u(t + tn ) ∈ K t u(t + tn ) = T (t − s)u(s + tn ) +

for all t ∈ R,

T (t − σ )F σ + tn , u(σ + tn ) dσ

(4.15) for t s.

(4.16)

s

By (4.13) and (4.15), we obtain v(t) ∈ K

for all t ∈ R.

(4.17)


2609

From (4.10) and (4.12), it follows that G(t, .) ∈ C(K, X) for each t ∈ R. From the triangle inequality F t + tn , u(t + tn ) − G t, v(t) F t + tn , u(t + tn ) − G t, u(t + tn ) + G t, u(t + tn ) − G t, v(t) sup F (t + tn , x) − G(t, x) + G t, u(t + tn ) − G t, v(t) , x∈K

and by (4.12) and (4.13), we deduce that lim F σ + tn , u(σ + tn ) = G σ, v(σ ) for s σ t,

n→+∞

since G(t, .) ∈ C(K, X) for each t ∈ R, therefore lim T (t − σ )F σ + tn , u(σ + tn ) = T (t − σ )G σ, v(σ ) for s σ t.

n→+∞

Moreover, by using (4.7) we have T (t − σ )F σ + tn , u(σ + tn ) kMeω(t−σ )

for s σ t,

where k is the constant defined by (4.11) and σ → kMeω(t−σ ) ∈ L1 (s, t). In view of the Lebesgue dominated convergence theorem, we obtain t lim

n→+∞

T (t − σ )F σ + tn , u(σ + tn ) dσ =

s

t

T (t − σ )G σ, v(σ ) dσ.

(4.18)

s

Using (4.13), (4.16) and (4.18), we deduce that t v(t) = T (t − s)v(s) +

T (t − σ )G σ, v(σ ) dσ

for t s.

(4.19)

s

The continuous function v satisfies (4.17) and (4.19), therefore v is a K-mild solution of Eq. (4.14). 2 Lemma 4.5. Let K be a compact subset of X. If f ∈ AAu (R × X, X), then the function g defined by (2.4) satisfies i) g : R × K → X is measurable. ii) For all t ∈ R, g(t, .) ∈ C(K, X). Proof. i) is obvious and ii) is a consequence of f (t + tn , .) ∈ C(K, X) and f (t + tn , .) → g(t, .) uniformly on K. 2

2610


Lemma 4.6. (See [16].) Let K be a compact subset of X. If f ∈ AAu (R × X, X), then sup sup f (t, x) < +∞ and

sup sup g(t, x) < +∞,

t∈R x∈K

t∈R x∈K

where g is the function defined by (2.4). Lemma 4.7. Assume that (H1) and (H4) hold. If Eq. (1.1) admits at least a mild solution x0 on [t0 , +∞) such that {x0 (t); t ∈ R} is relatively compact, then there exists a mild solution x on R of Eq. (1.1) such that x(t); t ∈ R ⊂ x0 (t); t t0 .

(4.20)

Proof. Denote by K = {x0 (t); t t0 } the compact subset of X. The mild solution x0 satisfies x0 (t) ∈ K t x0 (t) = T (t − s)x0 (s) +

for all t t0 ,

T (t − σ )f σ, x0 (σ ) dσ

(4.21) for t s t0 .

(4.22)

s

By Hypothesis (H4), we have k = supt∈R supx∈K f (t, x) < +∞ (cf. Lemma 4.6). By using Lemma 4.1 on F : [t0 , +∞) × K → X defined by F (t, x) = f (t, x) for each (t, x) ∈ [t0 , +∞) × K, we obtain the existence of a function φ : [0, +∞) → [0, +∞) satisfying (4.5) and x0 (t) − x0 (s) φ |t − s|

for t, s t0 .

(4.23)

Let (tn )n be a sequence of real numbers such that lim t n→+∞ n

= +∞.

Since f is almost automorphic in t uniformly with respect to x, then there exists a map g : R × K → X and there exists a subsequence of (tn )n , denoted by (tn )n such that for all t ∈ R, we have lim sup f (t + tn , x) − g(t, x) = 0,

(4.24)

lim sup g(t − tn , x) − f (t, x) = 0.

(4.25)

n→+∞ x∈K

n→+∞ x∈K

Given any interval (τ, +∞), for n ∈ N sufficiently large (τ + tn t0 ), the function t → x0 (. + tn ) is defined on (τ, +∞). Moreover (4.21) and (4.23) imply x0 (t + tn ) ∈ K for t τ, x0 (t + tn ) − x0 (s + tn ) φ |t − s| for t, s τ.

(4.26)


2611

Taking τ as a sequence going to −∞ and applying Arzela–Ascoli’s theorem and using a diagonal argument, we can assert that there exist x∗ ∈ C(R, X) and a subsequence of (tn )n such that x0 (t + tn ) → x∗ (t)

as n → +∞,

(4.27)

uniformly on each compact subset of R. Since x0 satisfies (4.22), then for each t s and for n ∈ N sufficiently large, we have t x0 (t + tn ) = T (t − s)x0 (s + tn ) +

T (t − σ )f σ + tn , x0 (σ + tn ) dσ.

(4.28)

s

By using (4.24), (4.26)–(4.28), we deduce that x∗ is a K-mild solution on R of x∗ (t) = Ax∗ (t) + g t, x∗ (t) . The proof is similar to the one given in Lemma 4.3. From Lemmas 4.5 and 4.6, we obtain that the function g : R × K → X satisfies hypotheses (4.10) and (4.11) of Lemma 4.3. Applying Lemma 4.3 on F = g, u = x∗ and the sequence (−tn )n (cf. (4.25)), we obtain the existence of a K-mild solution x of Eq. (1.1). Consequently, x is a mild solution satisfying (4.20). 2 Proposition 4.8. Assume that (H1)–(H4) hold. If Eq. (1.1) admits at least a mild solution x0 defined and bounded on [t0 , +∞): suptt0 x0 (t) < +∞, then i) its range {x0 (t); t t0 } is relatively compact in X, ii) there exists a mild solution x on R of Eq. (1.1) satisfying (4.20). Proof. i) Let θ ∈ (0, 1). Then the mild solution x0 verifies t x0 (t) = T (θ )x0 (t − θ ) +

T (t − σ )f σ, x0 (σ ) dσ

for t t0 + 1.

(4.29)

t−θ

Here x0 is the sum of two terms. The first term satisfies T (θ )x0 (t − θ ) ∈ T (θ )B 0, x0 ∞ ,

(4.30)

where x0 ∞ = suptt0 x0 (t) < +∞ and B(0, x0 ∞ ) is the closed ball with radius x0 ∞ centered at the origin. The set T (θ )B(0, x0 ∞ ) is relatively compact in X, since the operator T (θ ) is compact for each θ > 0. For the second term, by (4.7), we obtain t t T (t − σ )f σ, x0 (σ ) dσ kM eω(t−σ ) dσ, t−θ

with

t−θ

2612


k = sup

sup

tt0 ξ x0 ∞

Using the fact that

t t−θ

eω(t−σ ) dσ = t

θ 0

f (t, ξ ) < +∞.

eωσ dσ , we obtain

T (t − σ )f σ, x0 (σ ) dσ ∈ B 0, δ(θ ) ,

(4.31)

t−θ

where θ δ(θ ) = kM

eωσ dσ. 0

By (4.29)–(4.31), for each θ ∈ (0, 1), we deduce the existence of compact subset Kθ of X such that x0 (t); t t0 + 1 ⊂ Kθ + B 0, δ(θ ) .

(4.32)

For the sequel, we introduce the Kuratowski measure of noncompactness α(.) of bounded subsets B in the Banach space X defined by α(B) = inf{ε > 0; B has a finite cover of balls of diameter < ε}. The Kuratowski measure of noncompactness verifies the two following properties: α(B1 + B2 ) α(B1 ) + α(B2 ), α(B) = 0

⇐⇒

B is relatively compact in X.

For the details of the properties of α(.), see [34, Section 1.4]. From the inclusion (4.32), we obtain the inequality α x0 (t); t t0 + 1 α(Kθ ) + α B 0, δ(θ ) . Since α(B(0, δ(θ ))) 2δ(θ ) and Kθ is a compact subset of X, it follows α x0 (t); t t0 + 1 2δ(θ )

for 0 < θ < 1,

which implies α({x0 (t); t t0 + 1}) = 0 since limθ→+∞ δ(θ ) = 0. From properties of the Kuratowski measure of noncompactness, it follows that {x0 (t); t t0 + 1} is relatively compact in X, then the range of x0 : {x0 (t); t t0 } is also relatively compact in X, because x0 is continuous. ii) It is straightforward from Lemma 4.7. 2


2613

5. Proof of main results The object of this section is to prove Theorems 3.3, 3.6, Corollaries 3.7 and 3.8. Proof of Theorem 3.3. i) It is a consequence of Proposition 4.8. ii) Let δ = infx∈FK λK (x) be the greatest lower bound (infimum) of {λK (x): x ∈ FK } in R = R ∪ {−∞, +∞}. By Lemma 4.7, we obtain the existence of a mild solution x such that {x(t); t ∈ R} ⊂ K, therefore FK is nonempty, so δ exists in R ∪ {−∞}; it follows that there exists a sequence (xn )n with values in FK , such that lim λK (xn ) = δ.

(5.1)

n→+∞

By definition of FK , we have {xn (t); n ∈ N} is a subset of the compact K, for each t ∈ R. By Lemma 4.6, we can assert that supt∈R supx∈K f (t, x) < +∞. By using Lemma 4.1 on F : R × K → X defined by F (t, x) = f (t, x) for each (t, x) ∈ R × K, we obtain that there exists a function φ : [0, +∞) → [0, +∞) satisfying (4.5) such that xn (t) − xn (s) φ |t − s|

for n ∈ N and t, s ∈ I ;

therefore the sequence (xn )n is equicontinuous. In view of Arzela–Ascoli’s theorem, we can assert that {xn ; n ∈ N} is a relatively compact subset of C(R, X) endowed with the topology of compact convergence, therefore there exists a subsequence of (xn )n such that xn (t) → x∗ (t)

as n → +∞,

(5.2)

uniformly on each compact subset of R. Obviously x∗ ∈ CK (R, X). By (5.2), we deduce that f (t, xn (t)) → f (t, x∗ (t)) as n → +∞ uniformly on each compact subset of R. Since xn is a mild solution on R of Eq. (1.1), then we have t xn (t) = T (t − s)xn (s) +

T (t − σ )f σ, xn (σ ) dσ

for t s,

T (t − σ )f σ, x∗ (σ ) dσ

for t s,

s

and letting n → +∞, we have t x∗ (t) = T (t − s)x∗ (s) + s

which shows that x∗ is a mild solution on R of Eq. (1.1). Then x∗ ∈ FK , therefore δ λK (x∗ ).

(5.3)

From (5.2) and the definition of a subvariant functional, we have λK (x∗ ) lim inf λK (xn ). n→+∞

(5.4)

2614


From (5.1), (5.3) and (5.4), we deduce that λK (x∗ ) = δ, therefore x∗ is a minimal K-valued solution: λK (x∗ ) = inf λK (x), x∈FK

(5.5)

which implies the existence of the minimal K-valued solution. iii) Let us denote by x∗ the unique minimal K-valued solution of Eq. (1.1). To check that x∗ is compact almost automorphic, we have to prove that if (tn )n is any sequence of real numbers, then one can pick up a subsequence of (tn )n such that x∗ (t + tn ) → y∗ (t)

as n → +∞,

(5.6)

y∗ (t − tn ) → x∗ (t)

as n → +∞,

(5.7)

uniformly on each compact subset of R. In fact by assumption, we can choose a subsequence of (tn )n such that for all t ∈ R, lim sup f (t + tn , x) − g(t, x) = 0,

(5.8)

lim sup g(t − tn , x) − f (t, x) = 0,

(5.9)

n→+∞ x∈K

n→+∞ x∈K

where g is a map from R × K to X. Since supt∈R supx∈K f (t, x) < +∞ (cf. Lemma 4.6) and (5.6) holds, then by applying Lemma 4.3 with u = x∗ , F = f and the sequence (tn )n we obtain (5.6) where y∗ is a mild solution on R of equation x (t) = Ax(t) + g t, x(t) , which satisfies all hypotheses of Lemma 4.3 (cf. Lemmas 4.5 and 4.6). From (5.6), by the definition of the subvariant λK , we obtain that λK (y∗ ) λK (x∗ ) and with (5.5), we deduce that λK (y∗ ) inf λK (x). x∈FK

(5.10)

Since x∗ is K-valued, by (5.6), we obtain that y∗ is K-valued. Applying again Lemma 4.3 to u = y∗ , F = g, and the sequence (−tn )n , we obtain that y∗ (t − tn ) → z∗ (t)

as n → +∞

(5.11)

(for a subsequence) where z∗ is a mild solution on R of Eq. (1.1), because (5.9) holds. Moreover from (5.11), we deduce that λK (z∗ ) λK (y∗ ) and with (5.10), we obtain λK (z∗ ) inf λK (x). x∈FK

(5.12)


2615

Since y∗ is K-valued, by (5.11), we obtain that z∗ is K-valued, then z∗ ∈ FK . From (5.12), we obtain that λK (z∗ ) = infx∈FK λK (x), therefore z∗ is a minimal K-valued solution of Eq. (1.1). By uniqueness of the minimal K-valued solution of Eq. (1.1), we deduce that x∗ = z∗ , therefore (5.7) is fulfilled, thus x∗ is compact almost automorphic. 2 Proof of Theorem 3.6. In the proof of Theorem 3.3, assumptions (H2) and (H3) are only used to state that the set {x0 (t); t t0 } is relatively compact (cf. Proposition 4.8). 2 Proof of Corollary 3.7. Since APu (R × X, X) ⊂ AAu (R × X, X), then i) and ii) result of Theorem 3.3. For the same reason Eq. (1.1) admits a unique minimal K-valued solution x∗ . To check that x∗ is almost periodic, we have to prove that if (tn )n and (sn )n are two any sequences of real numbers, then one can pick up common subsequences of (tn )n and (sn )n such that ∀t ∈ R,

lim lim x∗ (t + tn + sp ) = lim x∗ (t + tm + sm )

p→∞ n→∞

m→∞

(5.13)

[26, Theorem 1.17, p. 12], instead of (5.6) and (5.7). In fact by Hypothesis (H5), for each compact subset K of X, we can choose common subsequences of (tn )n and (sn )n such that lim lim sup sup f (t + tn + sp , x) − g(t, x) = 0,

(5.14)

lim sup sup f (t + tm + sm , x) − g(t, x) = 0,

(5.15)

p→∞ n→∞ t∈R x∈K m→∞ t∈R x∈K

where g is a function from R × X to X. Remark that g ∈ H(f ) and f ∈ APu (R × X, X). If we denote y∗ (t) = limp→∞ limn→∞ x∗ (t + tn + sp ) and z∗ (t) = limm→∞ x∗ (t + tm + sm ), we deduce that y∗ and z∗ are two minimal K-valued solution of Eq. (3.1) where g is the function defined by (5.14); the proof of this assertion is similar to the one given in Theorem 3.3. By uniqueness of the minimal K-valued solution of Eq. (3.1) for each g ∈ H(f ), we deduce that y∗ = z∗ , therefore (5.13) holds, thus x∗ is almost periodic. This ends the proof. 2 Proof of Corollary 3.8. By using a similar reasoning that in the proof of Corollary 3.7, we deduce Corollary 3.8 from Theorem 3.6. 2 6. Heat and wave equations with nonlinearities In this section, we provide two examples of partial differential equations to illustrate our theoretical results. We give some existence theorems of compact almost automorphic solutions for a heat equation and a wave equation defined on a domain of Rn . We also give some existence results in the almost periodic context. 6.1. Heat equation To apply Theorem 3.3, we consider the following heat equation in a bounded open subset Ω of Rn with a smooth boundary ∂Ω

2616


⎧ n ⎪ ∂2 ⎨ ∂ v(t, x) = v(t, x) + g v(t, x) + h(t, x) ∂t ∂x 2 i=1 ⎪ ⎩ v(t, x) = 0 on R × ∂Ω,

in R × Ω,

(6.1)

where g : R → R and h : R × Ω → R are continuous functions. In order to rewrite Eq. (6.1) in the abstract form, we introduce the space X = C0 (Ω) of all continuous functions from Ω (the closure of Ω) to R vanishing on ∂Ω, endowed with the uniform norm topology. Define the operator A : D(A) ⊂ X → X by

D(A) = x ∈ C0 (Ω) ∩ H01 (Ω); x ∈ C0 (Ω) , Ax = x,

where is the Laplacian operator. Let us denote by λ1 the smallest eigenvalue of − in H01 (Ω) (λ1 > 0 since Ω is bounded). Lemma 6.1. The linear operator A generates a compact C0 -semigroup (T (t))t0 on X such that T (t) M exp(−λ1 t)

for t 0,

(6.2)

2

with M = exp(λ1 |Ω| n (4π)−1 ). Consequently, Hypotheses (H1) and (H2) are satisfied. Proof. (6.2) is a consequence of [14, Proposition 3.5.10, p. 47]. Using a result in [13], we deduce that the C0 -semigroup (T (t))t0 is compact. 2 In order to study the existence of an almost automorphic solution of Eq. (6.1), we suppose that (E1) The function h : R × Ω → R is continuous, h satisfies h(t, ξ ) = 0 on R × ∂Ω and h is C0 (Ω)-almost automorphic, which means that for any sequence of real numbers (sn )n , there exist a subsequence (sn )n and a measurable function k : R × Ω → R such that ∀t ∈ R,

lim sup h(t + sn , ξ ) − k(t, ξ ) = 0

n→∞ ξ ∈Ω

and ∀t ∈ R,

lim sup k(t − sn , ξ ) − h(t, ξ ) = 0.

n→∞ ξ ∈Ω

Remark 6.2. The function t → h(t, .) is in AA(R, X), then it is bounded on R, therefore the function h is bounded on R × Ω. An example of function h satisfying (E1) is the following: h(t, ξ ) = sin

h0 (ξ ) √ 2 + cost + cos 2t 1

for t ∈ R and ξ ∈ Ω,


2617

where h0 ∈ X = C0 (Ω). It is well known that each almost periodic function is uniformly contin1 √ ) is not uniformly continuous (see [38]), then it uous. Since the function t → sin( 2+cost+cos 2t is not almost periodic. Moreover, we suppose that: (E2) g is locally Lipschitz, g satisfies g(0) = 0 and lim sup|r|→+∞

g(r) r

< λ1 .

Let G : X → X be the superposition operator of g defined by G(x)(ξ ) = g x(ξ )

for x ∈ X and ξ ∈ Ω.

(6.3)

for t ∈ R and ξ ∈ Ω.

(6.4)

Let H : R → X be defined by H (t)(ξ ) = h(t, ξ )

Let us denote by f : R×X → X the map defined by f (t, x) = G(x) + H (t)

for t ∈ R and x ∈ X.

(6.5)

If (E1) and (E2) hold, then G ∈ C(X, X) and H ∈ AA(R, X), therefore f satisfies Hypothesis (H4). With these notations, Eq. (6.1) takes the following abstract form x (t) = Ax(t) + f t, x(t)

for t ∈ R.

(6.6)

For the initial value problem

x (t) = Ax(t) + f t, x(t)

for t 0,

x(0) = x0 ,

(6.7)

we have the following result: Proposition 6.3. Assume that (E1) and (E2) hold. Then for every x0 ∈ X, Eq. (6.7) has a unique mild solution x on [0, +∞). Moreover this solution x is bounded on [0, +∞). Proof. By (E1) and (E2), we have H ∈ L∞ (R, X), g is locally Lipschitz, g(0) = 0 and there exists M > 0 such that rg(r) Cr 2 for |r| M, with C < λ1 . The existence and uniqueness of a global mild solution of Eq. (6.7) which is bounded on [0, +∞) results of [14, Proposition 8.3.7, p. 117]. 2 In order to prove the existence of a compact almost automorphic solution, we need to assume the following assumption: (E3) The function r → g(r) − λ1 r is nonincreasing on R.

2618


Remark 6.4. The functions g defined by g(r) = λ1 sin(r) or g(r) = λ1 r − r 3 satisfy (E2) and (E3). If g is locally Lipschitz, g(0) = 0 and the function r → g(r) − λr (λ < λ1 ) is nonincreasing on R, then Hypotheses (E2) and (E3) hold and the function r → g(r) − λ1 r is strictly decreasing. An example of function satisfying (E2) and (E3) such that r → g(r) − λ1 r is not strictly decreasing is the following: g(r) = λ1 |r| on [−π, π] and g is 2π -periodic on R. To prove our result of existence of an almost automorphic solution, we use the two following lemmas. Let us denote by E : X → R the map defined by 1 E(x) = 2

x(ξ )2 dξ.

(6.8)

Ω

It is obvious that E is continuous on X. Lemma 6.5. Let p ∈ C(R, X). i) If w ∈ C(R, X) satisfies t w(t) = T (t − s)w(s) +

T (t − σ )p(σ ) dσ

for t s,

(6.9)

s

then E w(t) E w(s) +

t

p(t, ξ ) − λ1 w(t, ξ ) w(t, ξ ) dξ dσ

for t s.

(6.10)

s Ω

ii) If w ∈ C(R, X) satisfies t T (t − σ )w(σ ) dσ

w(t) = T (t − s)w(s) + λ1

for t s,

(6.11)

s

and t → E(w(t)) is constant on R, then there exists w0 ∈ X such that ∀t ∈ R,

w(t) = w0 .

(6.12)

Proof. To state Lemma 6.5, we use the L2 -theory. Set Y = L2 (Ω) the Lebesgue space of order two from Ω to R, endowed with its usual scalar product (. | .)Y . The associated norm is denoted by | · |Y . Define the operator B : D(B) ⊂ Y → Y by D(B) = H 2 (Ω) ∩ H01 (Ω) and By = y. Denote by (S(t))t0 the C0 -semigroup generated by B in Y . Recall that ∀y ∈ H01 (Ω),

|∇y|2Y λ1 |y|2Y ,

(6.13)


2619

and y ∈ D(B)

and |∇y|2Y = λ1 |y|2Y

⇒

By + λ1 y = 0.

(6.14)

i) Let w ∈ C(R, X) satisfy (6.9). By using the fact that X ⊂ Y with a continuous injection and T (t)φ = S(t)φ for all t 0 and φ ∈ X, we deduce that w ∈ C(R, Y ) and w satisfies t w(t) = S(t − s)w(s) +

S(t − σ )p(σ ) dσ

for t s.

(6.15)

s

Let s ∈ R and T > 0. Assume that w(s) ∈ D(B) and p ∈ C 1 ([s, s + T ], Y ). In this case, we have w ∈ C([s, s + T ], D(B)) ∩ C 1 ([s, s + T ], Y ) and w (t) = Bw(t) + p(t)

for s t s + T .

It follows that 2 1 d w(t)Y = w (t) w(t) Y = Bw(t) + λ1 w(t) w(t) Y + p(t) − λ1 w(t) w(t) Y . 2 dt Hence by Green’s formula and (6.13), we get (By + λ1 y | y)Y = −|∇y|2Y + λ1 |y|2Y 0 for y ∈ D(B), therefore 2 1 d w(t)Y p(t) − λ1 w(t) w(t) Y 2 dt

(6.16)

and by integrating (6.16) on [s, s + T ], we obtain 2 2 1 1 w(t)Y w(s)Y + 2 2

t

p(σ ) − λ1 w(σ ) w(σ ) Y dσ

for s t s + T .

(6.17)

s

In the general case, we consider two sequences (wns )n ⊂ D(B) and (pn )n ⊂ C 1 ([s, s + T ], Y ) such that (wns )n converges to w(s) in Y and (pn )n converges to p in L1 ((s, s + T ), Y ). Denote by wn the corresponding solution of (6.15). It follows that the sequence (wn )n converges uniformly on [s, s + T ] to w and wn satisfies 2 1 2 1 wn (t)Y wns Y + 2 2

t

pn (σ ) − λ1 wn (σ ) wn (σ ) Y dσ

for s t s + T ,

s

then w satisfies (6.17) on each interval [s, s + T ]. Consequently, w satisfies (6.10) on R. ii) Let w ∈ C(R, X) satisfy (6.11). We fix s ∈ R and T > 0. By [14, Proposition 5.1.1, p. 61], we obtain w ∈ C((s, s + T ], D(B)) ∩ C 1 ((s, s + T ], Y ) and

2620


w (t) = Bw(t) + λ1 w(t) for s < t s + T .

(6.18)

Moreover t → E(w(t)) is constant, then d E w(t) = w (t) w(t) Y = 0, dt therefore Bw(t) + λ1 w(t) w(t) Y = 0 for s < t s + T . By using Green’s formula and (6.14), we have Bw(t) + λ1 w(t) = 0 and from (6.18), we deduce that w(t) does not depend of t. Consequently, there exists w0 ∈ X such that (6.12) holds on each interval [s, s + T ]. This ends the proof. 2 Lemma 6.6. Assume that (E1)–(E3) hold. Let u and v be two mild solutions on R of Eq. (6.6). Then the following assertions hold. i) The function t → E(u(t) − v(t)) is nonincreasing on R. ii) If the function t → E(u(t) − v(t)) is constant on R, then there exists w0 ∈ X such that u(t) − v(t) = w0 for t ∈ R, G u(t) − G v(t) = λ1 w0 for t ∈ R.

(6.19) (6.20)

Moreover, for all θ ∈ [0, 1], θ u + (1 − θ )v is also a mild solution on R of Eq. (6.6). Proof. i) Let w = u − v. Then w ∈ C(R, X) and satisfies (6.9) with p(t) = f t, u(t) − f t, v(t) = G u(t) − G v(t) , where f and G are the function defined by (6.3)–(6.5). By Lemma 6.5, we obtain t q(t) q(s) +

φ u(t, ξ ) − φ v(t, ξ ) u(t, ξ ) − v(t, ξ ) dξ dσ

for t s, (6.21)

s Ω

where q(t) = E u(t) − v(t) , φ(r) = g(r) − λ1 r

for r ∈ R.

(6.22)

By (E3), φ is nonincreasing:

φ(r1 ) − φ(r2 ) {r1 − r2 } 0 for r1 and r2 ∈ R,

and from (6.21) and (6.23), we obtain that the function q is nonincreasing on R.

(6.23)


2621

ii) Assume that the function q is constant on R. By (6.21) and (6.23), we obtain that φ u(t, ξ ) − φ v(t, ξ ) u(t, ξ ) − v(t, ξ ) = 0 for t ∈ R and ξ ∈ Ω. Since φ is nonincreasing, it follows φ u(t, ξ ) = φ v(t, ξ )

for t ∈ R and ξ ∈ Ω,

(6.24)

then with (6.3) and (6.22), we deduce that G u(t) − λ1 u(t) = G v(t) − λ1 v(t) for t ∈ R. Since f t, u(t) − f t, v(t) = G u(t) − G v(t) = λ1 u(t) − v(t)

for t ∈ R,

(6.25)

then w satisfies (6.11). Since the function t → E(w(t)) is constant on R, by Lemma 6.5, we deduce the existence of w0 ∈ X such that (6.19) holds on R and by using (6.25), we obtain (6.20). Since φ is nonincreasing, we deduce from (6.24) that φ θ u(t, ξ ) + (1 − θ )v(t, ξ ) = φ u(t, ξ )

for t ∈ R, ξ ∈ Ω and θ ∈ [0, 1].

It follows that φ θ u(t, ξ ) + (1 − θ )v(t, ξ ) = θ φ u(t, ξ ) + (1 − θ )φ v(t, ξ ) for t ∈ R, ξ ∈ Ω and θ ∈ [0, 1], then g θ u(t, ξ ) + (1 − θ )v(t, ξ ) = θg u(t, ξ ) + (1 − θ )g v(t, ξ ) for t ∈ R, ξ ∈ Ω and θ ∈ [0, 1], therefore f t, θ u(t) + (1 − θ )v(t) = θf t, u(t) + (1 − θ )f t, v(t)

for t ∈ R and θ ∈ [0, 1],

consequently θ u + (1 − θ )v is a mild solution of Eq. (6.6). This ends the proof.

2

Proposition 6.7. Assume that (E1)–(E3) hold. Then Eq. (6.6) has at least one mild solution x1 which is compact almost automorphic. If x2 is a mild solution which is only almost automorphic, then there exists w0 ∈ X such that x2 (t) = x1 (t) + w0

for t ∈ R.

(6.26)

Consequently x2 is compact almost automorphic. Moreover, if the function r → g(r) − λ1 r is strictly decreasing on R, then the compact almost automorphic mild solution is unique.

2622


Proof. To prove the existence of a compact almost automorphic solution of Eq. (6.6), we use Theorem 3.3. By Lemma 6.1, assumptions (H1) and (H2) hold. Hypothesis (H4) results of (E1) and (E2). For Hypothesis (H3), since g is locally Lipschitz, then G is Lipschitz continuous on bounded sets of X, therefore G is bounded on the bounded sets of X. It follows for each t ∈ R, for each x ∈ X such that x R, f (t, x) sup G(x) + supH (t) < +∞, xR

t∈R

therefore (H3) holds. By Proposition 6.3, Eq. (6.6) admits at least a mild solution x0 defined and bounded on [0, +∞). By Theorem 3.3, the set {x0 (t); t 0} is compact. Let us denote by K the closed convex hull of the compact set {x0 (t); t 0}. Then K is a convex and compact subset of X. To prove the existence of a compact almost automorphic solution, we use Theorem 3.3 with the subvariant functional λK (x) = supt∈R E(x(t)) associated to the compact set K. It remains to prove the uniqueness of the minimal K-valued solution. Let u and v be two minimal K-valued solutions of Eq. (6.6). Let us denote δ = sup E u(t) = sup E v(t) . t∈R

(6.27)

t∈R

Case 1. Assume that E(u(t) − v(t)) = c for all t ∈ R. Then, by Lemma 6.6, 12 u + 12 v is a mild solution of Eq. (6.6). Moreover 12 u + 12 v ∈ FK , because K is convex set, then by definition of δ, we obtain 1 1 (6.28) δ sup E u(t) + v(t) . 2 2 t∈R By parallelogram law E

1 1 1 1 1 1 u(t) + v(t) + E u(t) − v(t) = E u(t) + E v(t) , 2 2 2 2 2 2

we obtain sup E t∈R

1 1 1 1 1 u(t) + v(t) + c sup E u(t) + sup E v(t) . 2 2 4 2 t∈R 2 t∈R

(6.29)

By (6.27)–(6.29), we obtain δ + 14 c 12 δ + 12 δ, then E(u(t) − v(t)) = c 0 for all t ∈ R. By definition of E, we obtain u = v, therefore Eq. (6.6) admits at most one minimal K-valued solution. Case 2. General case. Let (tn )n be a sequence of real numbers such that limn→+∞ tn = −∞. In fact by Hypothesis (H4), we can choose a subsequence of (tn )n such that for all t ∈ R, we have lim sup f (t + tn , x) − f∗ (t, x) = 0,

n→+∞ x∈K

lim sup f∗ (t − tn , x) − f (t, x) = 0,

n→+∞ x∈K


2623

where f∗ is a map from R × K to X. There exists a subsequence of (tn )n such that u(t + tn ) → u1 (t)

as n → +∞,

(6.30)

u1 (t − tn ) → u2 (t)

as n → +∞,

(6.31)

v(t + tn ) → v1 (t)

as n → +∞,

(6.32)

v1 (t − tn ) → v2 (t)

as n → +∞,

(6.33)

uniformly on each compact subset of R, where u2 and v2 are two minimal K-valued solutions of Eq. (6.6). The proof of this assertion is similar to the one given in the proof of Theorem 3.3, by using two times Lemma 4.3. By (6.30)–(6.33), we deduce that for t ∈ R, lim

lim E u(t + tn − tm ) − v(t + tn − tm ) = E u2 (t) − v2 (t) ,

m→+∞ n→+∞

and lim E u(t + tn ) − v(t + tn ) = sup E u(τ ) − v(τ ) ,

n→+∞

τ ∈R

since the function t → E(u(t) − v(t)) is nonincreasing on R (cf. Lemma 6.6) and limn→+∞ tn = −∞, therefore for all t ∈ R, E u2 (t) − v2 (t) = sup E u(τ ) − v(τ ) . τ ∈R

(6.34)

By (6.34) and Case 1, we obtain that u2 = v2 . By definition of E, we obtain sup E u(τ ) − v(τ ) = 0,

τ ∈R

therefore u = v. In conclusion, we have proved that Eq. (6.6) admits at most a minimal Kvalued solution. In view of Theorem 3.3, we assert that Eq. (6.6) has at least a compact almost automorphic solution. Let x1 be a mild solution which is compact almost automorphic. If x2 is mild solution which is almost automorphic, then by Lemma 6.6, the function t → E(x1 (t) − x2 (t)) is almost automorphic and nonincreasing, therefore it is constant on R. Again by help of Lemma 6.6, we obtain (6.26). If the function r → g(r) − λ1 r is strictly decreasing, the uniqueness of the compact almost automorphic solution results of (6.19) and (6.20). This ends the proof. 2 Now we give a corollary of Proposition 6.7 in the almost periodic case. For that, we need the following hypothesis: (E4) The function h : R × Ω → R is continuous, h satisfies h(t, ξ ) = 0 on R × ∂Ω and h is C0 (Ω)-almost periodic, which means that for all ε > 0, there exists > 0 such that for all α ∈ R, there exists τ ∈ [α, α + ] such that sup sup h(t + τ, ξ ) − h(t, ξ ) ε. t∈R ξ ∈Ω

2624


Corollary 6.8. Assume that (E2)–(E4) hold. Then Eq. (6.6) has at least one mild solution x1 which is almost periodic. If x2 is a mild solution which is almost periodic, then there exists w0 ∈ X such that x2 (t) = x1 (t) + w0

for t ∈ R.

Moreover, if the function r → g(r) − λ1 r is strictly decreasing on R, then the almost periodic mild solution is unique. Proof. The proof of Corollary 6.8 is similar to the one given in Proposition 6.7 by using Corollary 3.7 instead of Theorem 3.3. In fact Eq. (6.6) can be written as x (t) = Ax(t) + G x(t) + H (t)

for t ∈ R,

(6.35)

where G and H are defined by (6.3) and (6.4). Let us denote by H(H ) the set of functions H∗ ∈ AP(R, X) such that there exists a real sequence (tn )n satisfying lim supH (t + tn ) − H∗ (t) = 0.

n→+∞ t∈R

Then for H∗ ∈ H(H ), Eq. (3.1) can be written as x (t) = Ax(t) + G x(t) + H∗ (t)

for t ∈ R.

(6.36)

Since Eq. (6.36) satisfies all the hypotheses of Eq. (6.35), then Eq. (6.36) has a unique minimal K-valued solution for the compact K and the subvariant functional defined in the proof of Proposition 6.7. By using Corollary 3.7, we obtain the existence of a mild solution x1 which is almost periodic. The proof of others assertion of Corollary 6.8 results of Proposition 6.7 since an almost periodic function is almost automorphic. 2 6.2. Wave equation To apply Theorem 3.6, we consider the following wave equation in an arbitrary open subset Ω of Rn ⎧ n 2 ⎪ ∂2 ∂ ⎨ ∂ v(t, x) = v(t, x) + λv(t, x) + h(t, x) v(t, x) + g ∂t ∂t 2 ∂xi2 i=1 ⎪ ⎩ v(t, x) = 0 on R × ∂Ω,

in R × Ω,

(6.37)

where g : R → R, h : R × Ω → R are continuous functions and λ is an arbitrary real number. In order to rewrite Eq. (6.37) in the abstract form, we introduce the Hilbert space X = H01 (Ω) × L2 (Ω), endowed with the inner product denoted by · | ·, associated to the norm x = Ω

∇x1 (ξ )2 + x1 (ξ )2 + x2 (ξ )2 dξ

1 2

for x = [x1 , x2 ] ∈ X.


2625

Define the operator A : D(A) ⊂ X → X by

D(A) = [x1 , x2 ] ∈ X; x1 ∈ L2 (Ω) and x2 ∈ H01 (Ω) , A[x1 , x2 ] = [x2 , x1 + λx1 ],

where is the Laplacian operator. Lemma 6.9. The linear operator A generates a C0 -semigroup (T (t))t0 on X. Moreover the operator A satisfies Ax | x = (1 + λ)

x1 (ξ )x2 (ξ ) dξ

for x = [x1 , x2 ] ∈ D(A).

(6.38)

Ω

Proof. Define the operator B : D(B) ⊂ X → X by

D(B) = D(A), B[x1 , x2 ] = [x2 , x1 − x1 ].

The linear operator B is skew-adjoint on X, therefore B generates a group of isometry [14, Proposition 2.6.9, p. 33]. Define the operator L : X → X by Lx = [0, x1 ]

for x = [x1 , x2 ] ∈ X.

Then L ∈ L(X, X) and A = B + (1 + λ)L, therefore A generates a C0 -semigroup (T (t))t0 on X [40, Theorem 1.1, p. 76]. Since the linear operator B is skew-adjoint on X, then for all x ∈ D(A), Ax | x = (1 + λ)Lx | x, therefore (6.38) holds. 2 In order to study the existence of an almost automorphic solution of Eq. (6.37), we suppose that: (E5) The function h : R × Ω → R is continuous and L2 (Ω)-almost automorphic, which means that for any sequence of real numbers (sn )n , there exist a subsequence (sn )n and a measurable function k : R × Ω → R such that ∀t ∈ R,

lim

n→∞

h(t + sn , ξ ) − k(t, ξ )2 dξ = 0

Ω

and ∀t ∈ R,

lim

n→∞

k(t − sn , ξ ) − h(t, ξ )2 dξ = 0.

Ω

Remark 6.10. The function t → h(t, .) is in AA(R, L2 (Ω)).

2626


Moreover, we suppose that: (E6) g is Lipschitzian continuous. (E7) λ ∈ such that u ∈ H01 (Ω) and u + λu = 0 implies that u = 0. (E8) The function g is nonincreasing on R. Remark 6.11. In the particular case where the domain Ω is bounded, and its boundary ∂Ω is such that the Dirichlet problem: u + λu = 0 in Ω, u = 0 on ∂Ω, has a sequence of eigenvalues {λn : n 1} such that 0 < λ1 λ2 · · · λn λn+1 · · · and limn→∞ λn = ∞, if λ is not an eigenvalue of − in H01 (Ω), then λ satisfies Hypothesis (E7). Let G : X → X be the map defined by G(x)(ξ ) = 0, g x2 (ξ )

for ξ ∈ Ω and x = [x1 , x2 ] ∈ X.

(6.39)

Let H : R → X be defined by H (t)(ξ ) = 0, h(t, ξ )

for t ∈ R and ξ ∈ Ω.

(6.40)

Let f : R×X → X denote the mapping defined by f (t, x) = G(x) + H (t)

for t ∈ R and x ∈ X.

(6.41)

Since (E5) and (E6) hold, then G ∈ C(X, X) and H ∈ AA(R, X), therefore f satisfies Hypothesis (H4). With those notations, Eq. (6.37) takes the following abstract form x (t) = Ax(t) + f t, x(t)

for t ∈ R.

(6.42)

Let q : X → R and Φ : X → R denote the quadratic forms defined by 1 q(x) = 2

x1 (ξ )2 dξ,

(6.43)

Ω

1 Φ(x) = x2 − (1 + λ)q(x). 2

(6.44)

Then, 1 Φ(x) = 2

∇x1 (ξ )2 + x2 (ξ )2 − λx1 (ξ )2 dξ

for x = [x1 , x2 ] ∈ X.

Ω

It is obvious that q and Φ are continuous on X. With Assumption (E7), the quadratic form Φ is not in general positive or negative definite. To prove our result of existence of an almost automorphic solution, we use Theorem 3.6, with the following subvariant functional λK (x) = sup Φ x(t) t∈R

for x ∈ CK (R, X).


2627

To state our result, we use the two following lemmas. We consider the following linear equation: x (t) = Ax(t) + p(t)

for t ∈ R,

(6.45)

where p ∈ C(R, X). Lemma 6.12. Let p = [0, p2 ] ∈ C(R, X). i) If w is a mild solution on R of the linear equation (6.45), then the function t → Φ(w(t)) is of class C 1 on R and d Φ w(t) = p(t) w(t) = dt

p2 (t, ξ )w2 (t, ξ ) dξ

for t ∈ R.

(6.46)

Ω

ii) Moreover, if p(t) = 0 for all t ∈ R, then the function t → Φ(w(t)) is constant on R and the function t → q(w(t)) is of class C 2 on R with d2 q w(t) = 2 2 dt

w2 (t, ξ )2 dξ − 2Φ w(0)

for t ∈ R.

(6.47)

Ω

Proof. Let us denote by bi : X × X → R (i = 1, 2) the two bilinear forms defined by b1 (x, y) =

x1 (ξ )y1 (ξ ) dξ

and b2 (x, y) =

Ω

x1 (ξ )y2 (ξ ) dξ Ω

for x = [x1 , x2 ] and y = [y1 , y2 ] ∈ X. Then b1 and b1 are continuous, b1 is symmetric and satisfies b1 (x, x) = 2q(x) where q is the function defined by (6.43). i) Let w be a mild solution on R of Eq. (6.45). We fix s ∈ R and T > 0. Assume that w(s) ∈ D(A) and p2 ∈ C 1 ([s, s + T ], L2 (Ω)). Then w is a classical solution on [s, s + T ] of (6.45): w ∈ C([s, s + T ], D(A)) ∩ C 1 ([s, s + T ], X) and w (t) = Aw(t) + p(t)

for s t s + T .

It follows that t → q(w(t)) is of class C 1 and d 1 d q w(t) = b1 w(t), w(t) = b1 w(t), w (t) = b1 w(t), Aw(t) + p(t) , dt dt 2 then d q w(t) = dt

w1 (t, ξ )w2 (t, ξ ) dξ = b2 w(t), w(t) ,

Ω

consequently t → Φ(w(t)) is also of class C 1 and

(6.48)

2628


d d Φ w(t) = w (t) w(t) − (1 + λ) q w(t) dt dt = Aw(t) w(t) + p(t) w(t) − (1 + λ) w1 (t, ξ )w2 (t, ξ ) dξ Ω

and by using (6.38), we obtain (6.46) on [s, s + T ]. Then by integrating (6.48) and (6.46) on [s, s + T ], we obtain q w(t) = q w(s) +

t

b2 w(σ ), w(σ ) dσ,

(6.49)

s

Φ w(t) = Φ w(s) +

t

p(σ ) w(σ ) dσ

(6.50)

s

for s t s + T . For the proof of (6.49) and (6.50) in the general case, we choose two sequences (wns )n ⊂ D(A) and (pn )n ⊂ C 1 ([s, s + T ], X) (with pn = [0, pn2 ]) such that (wns )n converges to w(s) in X and (pn )n converges to p in L1 ((s, s + T ), X). Consequently w satisfies (6.49) and (6.50) on each interval [s, s + T ]. Since w, p and b2 are continuous, we deduce that t → q(w(t)), t → Φ(w(t)) are of class C 1 and (6.46), (6.48) hold. ii) Moreover, if p(t) = 0 for all t ∈ R, by (6.46), we deduce that the function t → Φ(w(t)) is constant on R, therefore 1 w(t)2 − (1 + λ)q w(t) = Φ w(0) 2

for all t ∈ R.

(6.51)

We fix s ∈ R and T > 0. Assume that w(s) ∈ D(A). In this case, we have w ∈ C([s, s + T ], D(A)) ∩ C 1 ([s, s + T ], X) and w (t) = Aw(t)

for s t s + T .

(6.52)

From (6.48), it follows that t → q(w(t)) is of class C 2 and d2 q w(t) = b2 w (t), w(t) + b2 w(t), w (t) = b2 Aw(t), w(t) + b2 w(t), Aw(t) , 2 dt then d2 q w(t) = 2 dt

w2 (t, ξ )2 + w1 (t, ξ ) w1 (t, ξ ) + λw1 (t, ξ ) dξ,

Ω

by using Green’s formula, we obtain d2 q w(t) = 2 dt

Ω

w2 (t, ξ )2 − ∇w1 (t, ξ )2 + λw1 (t, ξ )2 dξ.


2629

From (6.51), we deduce that (6.47) holds. By integrating (6.47) on [s, s + T ], we get d d q w(t) = q w(s) + 2 dt dt

t

w2 (σ, ξ )2 dξ dσ − 2(t − s)Φ w(0)

(6.53)

s Ω

for s t s + T . For the proof of (6.53) in the general case, we choose (wns )n ⊂

a sequence s 2 D(A) such that (wn )n converges to w(s) in X. Since the function t → Ω |w2 (t, ξ )| dξ is continuous, we deduce that the function t → q(w(t)) is of class C 2 on each interval [s, s + T ] and satisfies (6.47). 2 Lemma 6.13. Assume that (E5)–(E8) hold. Let u and v be two mild solutions on R of Eq. (6.42). Then the following assertions hold. i) The function t → Φ(u(t) − v(t)) is nonincreasing on R. ii) If the function t → Φ(u(t) − v(t)) is constant on R, then G u(t) = G v(t)

for t ∈ R,

(6.54)

and for all θ ∈ [0, 1], θ u + (1 − θ )v is also a mild solution on R of Eq. (6.42). iii) If u and v are bounded on R and if Φ(u(t) − v(t)) = c for all t ∈ R with c 0, then c = 0 and u = v. Proof. i) Let us denote w = u − v. Then w ∈ C(R, X) and satisfies (6.45) with p(t) = f t, u(t) − f t, v(t) = G u(t) − G v(t) ,

(6.55)

where f and G are the functions defined by (6.39)–(6.41). By Lemma 6.12, we obtain d Φ w(t) = G u(t) − G v(t) w(t) , dt therefore d Φ w(t) = dt

g u2 (t, ξ ) − g v2 (t, ξ ) u2 (t, ξ ) − v2 (t, ξ ) dξ.

(6.56)

Ω

Since g is nonincreasing, it follows that

g(r1 ) − g(r2 ) {r1 − r2 } 0 for r1 and r2 ∈ R,

(6.57)

and from (6.56) and (6.57), we obtain that the function t → Φ(u(t) − v(t)) is nonincreasing on R. ii) Assume that the function t → Φ(u(t) − v(t)) is constant on R. By using (6.56) and (6.57), we obtain g u2 (t, ξ ) − g v2 (t, ξ ) u2 (t, ξ ) − v2 (t, ξ ) = 0 for t ∈ R and ξ ∈ Ω.

2630


By using (E8), we obtain g u2 (t, ξ ) = g v2 (t, ξ ) = 0 for t ∈ R and ξ ∈ Ω, therefore (6.54) holds, then we deduce g θ u2 (t, ξ ) + (1 − θ )v2 (t, ξ ) = g u2 (t, ξ )

for t ∈ R, ξ ∈ Ω and θ ∈ [0, 1].

It follows that G θ u(t) + (1 − θ )v(t) = θ G u(t) + (1 − θ )G v(t)

for t ∈ R and θ ∈ [0, 1],

therefore f t, θ u(t) + (1 − θ )v(t) = θf t, u(t) + (1 − θ )f t, v(t)

for t ∈ R and θ ∈ [0, 1],

therefore θ u + (1 − θ )v is also a mild solution on R of Eq. (6.42). iii) By (6.54) and (6.55), we deduce that w ∈ C(R, X) and satisfies the homogeneous equation associated to (6.45), i.e. p(t) = 0 for all t ∈ R. By Lemma 6.12, the function t → q(w(t)) is of class C 2 on R and satisfies (6.47). By assumption c = Φ(w(0)) 0 and by (6.47), we obtain d2 q(w(t)) 0, then t → q(w(t)) is convex and bounded on R, therefore t → q(w(t)) is condt 2 2

d stant, i.e. dt 2 q(w(t)) = 0. By (6.47), we obtain c = 0 and w2 (t) = 0 for all t ∈ R. We fix s ∈ R and T > 0. Assume that w1 (s) ∈ H01 (Ω) and w1 (s) ∈ L2 (Ω). In this case, w is a classical solution of the homogeneous equation associated to (6.45), we deduce that w1 (t) = w1 (s) for all t ∈ [s, s + T ] and w1 (s) + λw1 (s) = 0. From Hypothesis (E7), we deduce that w1 (t) = 0, thus w(t) = 0 for all t ∈ [s, s + T ]. For the general case, we choose a sequence (wns )n ⊂ D(A) such that (wns )n converges to w(s) in X, then we obtain that w(t) = 0 for each interval [s, s + T ], therefore u(t) = v(t) for all t ∈ R. 2

Proposition 6.14. Assume that (E5)–(E8) hold. In addition suppose that Eq. (6.42) admits at least a mild solution x0 on [t0 , +∞) such that {x0 (t); t t0 } is relatively compact. Then Eq. (6.42) has at least one mild solution which is compact almost automorphic solution. Moreover, if the function g is strictly decreasing on R, then the mild solution which is compact almost automorphic, is unique. Proof. Hypothesis (H1) is a consequence of Lemma 6.9 and Hypothesis (H4) results of (E5) and (E6), therefore all hypotheses of Theorem 3.6 are fulfilled. Let K be the closed convex hull of the compact set x([t0 , +∞)). Then K is a convex and compact. To prove the existence of a compact almost automorphic solution of Eq. (6.42), we use Theorem 3.6 with the subvariant functional λK (x) = supt∈R Φ(x(t)) associated to the compact set K. It remains to prove the uniqueness of the minimal K-valued solution of Eq. (6.42). Let u and v be two minimal Kvalued solutions of Eq. (6.42). Let us denote δ = sup Φ u(t) = sup Φ v(t) . t∈R

t∈R

(6.58)


2631

Case 1. Assume that Φ(u(t) − v(t)) = c for all t ∈ R. Then, by using Lemma 6.13, 12 u + 12 v is a solution of Eq. (6.42). Since K is convex, we obtain 12 u + 12 v ∈ FK , then

1 1 δ sup Φ u(t) + v(t) . 2 2 t∈R

(6.59)

By parallelogram law applied to the quadratic form Φ, we deduce sup Φ t∈R

1 1 1 1 1 u(t) + v(t) + c sup Φ u(t) + sup Φ v(t) . 2 2 4 2 t∈R 2 t∈R

(6.60)

By (6.58)–(6.60), we obtain δ + 14 c 12 δ + 12 δ, then Φ(u(t) − v(t)) = c 0 for all t ∈ R. By Lemma 6.13, we obtain u = v, therefore Eq. (6.42) admits at most one minimal K-valued solution. Case 2. General case. Let (tn )n be a sequence of real numbers such that limn→+∞ tn = −∞. Then there exists a subsequence of (tn )n such that u(t + tn ) → u1 (t)

as n → +∞,

(6.61)

u1 (t − tn ) → u2 (t)

as n → +∞,

(6.62)

v(t + tn ) → v1 (t)

as n → +∞,

(6.63)

v1 (t − tn ) → v2 (t)

as n → +∞,

(6.64)

uniformly on each compact subset of R, where u2 and v2 are two minimal K-valued solutions of Eq. (6.42). The proof of this assertion is similar to the one given in Proposition 6.7. By Lemma 6.13, the function t → Φ(u(t) − v(t)) is nonincreasing on R, from (6.61)–(6.64) and limn→+∞ tn = −∞, we deduce for all t ∈ R, Φ u2 (t) − v2 (t) = sup Φ u(τ ) − v(τ ) . τ ∈R

(6.65)

By (6.65) and Case 1, we obtain that u2 = v2 , then Φ(u2 (t) − v2 (t)) = 0 for all t ∈ R, therefore sup Φ u(τ ) − v(τ ) = 0.

τ ∈R

(6.66)

By taking a sequence of real numbers (tn )n such that limn→+∞ tn = +∞, we state by similar proof that inf Φ u(τ ) − v(τ ) = 0.

τ ∈R

From (6.66) and (6.67), we have for all t ∈ R, Φ u(t) − v(t) = 0,

(6.67)

2632


therefore by Case 1, we deduce that u = v. In conclusion, we have proved that Eq. (6.42) admits at most a minimal K-valued solution. In view of Theorem 3.6, we assert that Eq. (6.42) has at least a compact almost automorphic solution. Assume that g is strictly decreasing. If u and v are two mild solutions which are compact almost automorphic, then the function t → Φ(u(t) − v(t)) is almost automorphic and by Lemma 6.13, it is nonincreasing, therefore it is constant on R. Again by help of Lemma 6.13, we obtain (6.54) and since g is strictly decreasing, then u2 = v2 . Denote w = u − v. It follows that w is a mild solution on R of the linear equation (6.45) and by the same argument as in the end of the proof of Lemma 6.13, we obtain u = v. 2 Now we give a corollary of Proposition 6.14 in the almost periodic case. For that, we need the following hypothesis: (E9) The function h : R × Ω → R is continuous and L2 (Ω)-almost periodic, which means that for all ε > 0, there exists > 0 such that for all α ∈ R, there exists τ ∈ [α, α + ] such that sup t∈R

h(t + τ, ξ ) − h(t, ξ )2 dξ ε.

Ω

Corollary 6.15. Assume that (E6)–(E9) hold. In addition suppose that Eq. (6.42) admits at least a mild solution x0 on [t0 , +∞) such that {x0 (t); t t0 } is relatively compact. Then Eq. (6.42) has at least one mild solution which is almost periodic solution. Moreover, if the function g is strictly decreasing on R, then the mild solution which is almost periodic is unique. Proof. The proof of Corollary 6.15 is similar to the one given in Corollary 6.8 by using Corollary 3.8 instead of Corollary 3.7. 2 Acknowledgments The authors would like to thank the referee for his careful reading of the paper. His valuable suggestions and critical remarks made numerous improvements throughout. References [1] L. Amerio, Soluzioni quasi-periodiche, o limitate, di sistemi differenziali non lineari quasi-periodici, o limitate, Ann. Mat. Pura Appl. 39 (1955) 97–119. [2] L. Amerio, Problema misto e soluzioni quasi-periodiche dell’equazione delle onde, Rend. Sem. Mat. Fis. Milano 30 (1960) 197–222. [3] L. Amerio, Problema misto e quasi-periodicità per l’equazione delle onde non omogenea, Ann. Mat. Pura Appl. 49 (1960) 393–417. [4] L. Amerio, Almost periodic solutions of the equation of Schrödinger type, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. VIII 43 (1967) 147–153. [5] L. Amerio, Almost periodic solutions of the equation of Schrödinger type. II, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. VIII 43 (1967) 265–270. [6] L. Amerio, Almost-periodic functions in Banach spaces, in: The Harald Bohr Century, Copenhagen, 1987, Mat.Fys. Medd. Danske Vid. Selsk. 42 (1989) 25–33. [7] L. Amerio, G. Prouse, Almost-Periodic Functions and Functional Equations, Univ. Ser. Higher Math., vol. VIII, Van Nostrand Reinhold Company, New York, 1971.


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[8] M. Biroli, On the almost periodic solution to some parabolic quasivariational inequalities, Riv. Mat. Univ. Parma IV 5 (1) (1979) 295–303 (1980). [9] M. Biroli, F. Dal Fabbro, Bounded or almost periodic solutions of a wave equation with nonlinear viscosity, in: Differential Equations, Colorado Springs, CO, 1989, in: Lect. Notes Pure Appl. Math., vol. 127, Dekker, New York, 1991, pp. 47–52. [10] M. Biroli, A. Haraux, Asymptotic behavior of an almost periodic, strongly dissipative wave equations, J. Differential Equations 38 (1980) 422–440. [11] S. Bochner, A new approach to almost periodicity, Proc. Natl. Acad. Sci. USA 48 (1962) 2039–2043. [12] S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, Proc. Natl. Acad. Sci. USA 52 (1964) 907–910. [13] S.H. Bruse, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc. 259 (1980) 299–310. [14] T. Cazenave, A. Haraux, Introduction aux Problèmes d’Evolution Semi-linéaires, Ellipses-Edition, Paris, 1990. [15] P. Cieutat, K. Ezzinbi, Existence, uniqueness and attractiveness of a pseudo almost automorphic solutions for some dissipative differential equations in Banach spaces, J. Math. Anal. Appl. 354 (2009) 494–506. [16] P. Cieutat, S. Fatajou, G.M. N’Guérékata, Bounded and almost automorphic solutions of some nonlinear differential equation in Banach spaces, Nonlinear Anal. 71 (2009) 674–684. [17] P. Cieutat, S. Fatajou, G.M. N’Guérékata, Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations, Appl. Anal. 89 (2010) 11–27. [18] C. Corduneanu, Almost Periodic Functions, Wiley, New York, 1968; reprinted by Chelsea, New York, 1989. [19] C.M. Dafermos, Almost periodic process and almost periodic solutions of evolution equations, in: Dynamical Systems, Proceedings of a University of Florida, International Symposium, Academic Press, 1977. [20] K. Ezzinbi, G.M. N’Guérékata, A Massera type theorem for almost automorphic solutions of functional differential equations of neutral type, J. Math. Anal. Appl. 316 (2006) 707–721. [21] K. Ezzinbi, S. Fatajou, G.M. N’Guérékata, Almost automorphic solutions for dissipative ordinary and functional differential equations in Banach spaces, Commun. Math. Anal. 4 (2008) 8–18. [22] K. Ezzinbi, S. Fatajou, G.M. N’Guérékata, Pseudo almost automorphic solutions for dissipative differential equations in Banach spaces, J. Math. Anal. Appl. 351 (2009) 765–772. [23] J. Favard, Sur les équations différentielles linéaires à coefficients presque-périodiques, Acta Math. 5 (1928) 31–51. [24] J. Favard, Sur certains systèmes différentiels scalaires linéai`res et homogènes à coefficients presque-périodiques, Ann. Mat. Pura Appl. 61 (1963) 297–316. [25] A.M. Fink, Almost automorphic and almost periodic solutions which minimize functional, Tôhoku Math. J. (2) 20 (1968) 323–332. [26] A.M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math., vol. 377, Springer-Verlag, Berlin– New York, 1974. [27] E. Hanebaly, Solutions presque périodiques d’équations différentielles monotones, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983) 263–265. [28] E. Hanebaly, Contribution à l’Etude des Solutions Périodiques et Presque-périodiques d’Equations Différentielles non Linéaires sur les Espaces de Banach, Thèse de Doctorat d’Etat, Université de Pau et des Pays de l’Adour, 1988. [29] A. Haraux, Nonlinear Evolution Equations-Global Behavior of Solutions, Lecture Notes in Math., vol. 841, Springer-Verlag, 1981. [30] A. Haraux, Almost periodic forcing for a wave equations with a nonlinear local damping term, Proc. Roy. Soc. Edinburgh Sect. A 94 (1983) 195–212. [31] A. Haraux, Asymptotic behavior of two-dimensional quasi-autonomous almost periodic evolutions equations, J. Differential Equations 66 (1987) 62–70. [32] A. Haraux, Systèmes dynamiques et dissipatifs et applications, Recherches en Mathématiques Appliquées, Masson, Paris, 1991. [33] H. Ishii, On the existence of almost periodic trajectories for contractive almost periodic processes, J. Differential Equations 43 (1983) 66–72. [34] V. Lakshmikantham, S. Leela, Nonlinear Differential Equations in Abstract Spaces, Internat. Ser. Nonlinear Math., vol. 2, Pergamon Press, Oxford, New York, 1981. [35] B.M. Levitan, V.V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, 1982. [36] J. Liu, G.M. N’Guérékata, Nguyen Van Minh, A Massera type theorem for almost automorphic solutions of differential equations, J. Math. Anal. Appl. 299 (2004) 587–599. [37] J.L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950) 457–475.

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[38] G.M. N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic Publishers, New York, 2001. [39] G.M. N’Guérékata, Almost automorphy, almost periodicity and stability of motions in Banach spaces, Forum Math. 13 (2001) 581–588. [40] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. [41] G. Prouse, Soluzioni quasi-periodiche dell’equazione differenziale di Navier–Stokes in due dimensioni, Rend. Semin. Mat. Univ. Padova 33 (1963) 186–212. [42] G. Prouse, Soluzioni quasi-periodiche dell’equazione non omogenea della membrana vibrante, con termine dissipativo quadratico, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 37 (1964) 364–370. [43] L. Schwartz, Topologie Générale et Analyse Fonctionnelle, Hermann, Paris, 1976. [44] W.A. Veech, Almost automorphic functions, Proc. Natl. Acad. Sci. USA 49 (1963) 462–464. [45] W.A. Veech, Almost automorphic functions on groups, Amer. J. Math. 87 (1965) 719–751. [46] S. Zaidman, Almost automorphic solutions of some abstract evolutions equations, Istit. Lombardo Accad. Sci. Lett. Rend. A 110 (1976) 578–588. [47] S. Zaidman, Almost-Periodic Functions in Abstract Spaces, Res. Notes Math., vol. 126, Pitman, Boston, 1985. [48] S. Zaidman, Existence of asymptotically almost-periodic and almost-automorphic solutions for some classes of abstract differential equations, Ann. Sci. Math. Québec 13 (1989) 79–88. [49] S. Zaidman, On the Bohr transform of almost-periodic solutions for some differential equations in abstract spaces, Int. J. Math. Math. Sci. 27 (2001) 521–534.


On the Guionnet–Jones–Shlyakhtenko construction for graphs Vijay Kodiyalam ∗ , V.S. Sunder The Institute of Mathematical Sciences, Chennai, India Received 30 March 2010; accepted 28 January 2011

Communicated by D. Voiculescu

Abstract Using an analogue of the Guionnet–Jones–Shlyakhtenko construction for graphs we show that their construction applied to any subfactor planar algebra of finite depth yields an inclusion of interpolated free group factors with finite parameter, thereby giving another proof of their universality for finite depth planar algebras. © 2011 Elsevier Inc. All rights reserved. Keywords: Subfactor; Planar algebra; Interpolated free group factor

The main theorem of [5] constructs an extremal finite index II 1 -subfactor N = M0 ⊆ M1 = M from a subfactor planar algebra P with the property that the planar algebra of N ⊆ M is isomorphic to P . We show in this paper that if P is a subfactor planar algebra of modulus δ > 1 and of finite depth, then, for the associated subfactor N ⊆ M, there are isomorphisms N ∼ = LF(r) and M∼ = LF(s) for some 1 < r, s < ∞, where LF(t) for 1 < t < ∞ is the interpolated free group factor of [2] and [14]. This can be regarded as yet another proof of the fact – see [14] and [3] – that interpolated free group factors with finite parameter are universal for finite depth subfactor planar algebras. The word ‘universal’ above is used in the sense of [13] where they essentially prove that LF(∞) is universal for all subfactor planar algebras. We shall now outline the structure of this paper. In Section 1 we construct – see Proposition 1 – a graded, tracial, faithful ∗-probability space Gr(Γ ) associated to a finite, weighted, bipartite * Corresponding author.

E-mail addresses: [email protected] (V. Kodiyalam), [email protected] (V.S. Sunder). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.01.018

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V. Kodiyalam, V.S. Sunder / Journal of Functional Analysis 260 (2011) 2635–2673

graph Γ and establish – see Proposition 5 – an isomorphism between Gr(Γ ) and a filtered, tracial, faithful ∗-probability space F (Γ ) – see Proposition 4. Our main interest will be in an associated finite von Neumann algebra M(Γ ) and some of its corners determined by sets of vertices of Γ – specifically the corner M(Γ, 0) (resp. M(Γ, 1)) determined by the set of even (resp. odd) vertices of Γ . The main result in Section 2 asserts – see Theorem 21 – that if Γ is a connected graph with more than one edge, then, M(Γ ) is the direct sum of a II 1 -factor and a finite-dimensional abelian algebra. The goal of Section 3 is to express Gr(Γ, 0) and M(Γ, 0) – see Proposition 26 and Eq. (23) – as amalgamated free products of the corresponding algebras associated to subgraphs with a single odd vertex. In Section 4 we determine the structure of M(Λ, 0) – see Corollary 33 – for a graph Λ with a single odd vertex. The penultimate Section 5 proves – see Theorem 35 – one of our main results: for a connected graph Γ with more than one edge and equipped with its Perron–Frobenius weighting, the algebra M(Γ ) is an interpolated free group factor with finite parameter. The final Section 6 applies this – see Theorems 41 and 42 – to show that the Guionnet–Jones–Shlyakhtenko (henceforth GJS) construction applied to a finite depth subfactor planar algebra yields an inclusion of interpolated free group factors with finite parameters. The main result of this paper has been proved independently by Guionnet, Jones and Shlyakhtenko in [6] where they also determine the finite parameter values of the interpolated free group factors in terms of the index and global index. 1. The global graded probability space associated to a graph The goal of this section is to associate a graded, tracial, faithful ∗-probability space Gr(Γ ) and a von Neumann algebra M(Γ ) to a graph Γ . Recall that a tracial ∗-probability space consists of a unital, complex ∗-algebra A equipped with a trace τ : A → C that satisfies τ (1) = 1 and τ (a ∗ a) 0, for all a ∈ A. It is said to be graded if the algebra A is graded and to be faithful if τ (a ∗ a) = 0 ⇒ a = 0. Throughout this paper, by a graph, we will mean a finite, weighted, bipartite graph which consists of the following data: (i) a finite set V of ‘vertices’ partitioned as V0 V1 – the sets V0 and V1 will be referred to as sets of even and odd vertices respectively, (ii) a finite set E of ‘edges’ equipped with ‘start’ and ‘finish’ maps s, f : E → V and a ‘reversal’ involution ξ → ξ˜ of E intertwining s and f such that s(ξ ) ∈ V0 ⇔ f (ξ ) ∈ V1 , and (iii) a ‘weighting’ which is a function μ : V → R+ normalised such that v∈V μ2 (v) = 1. For us, the main examples of such graphs are the principal graphs of non-trivial II 1 -subfactors of finite depth (where μ is given by the square root of an appropriately normalised Perron– Frobenius eigenvector) and their subgraphs (with the restricted μ appropriately normalised). The construction of Gr(Γ ) involves paths in Γ , notations and definitions for which we discuss briefly. A path ξ in Γ is denoted ξ

ξ

ξ

ξn ξ ξ1 ξ ξ2 ξ ξ3 v0 −→ v1 −→ v2 −→ · · · −→ vnξ ,

ξ

i → vi ) ∈ E, with the notation being self-explanatory. The start and where vi ∈ V and (vi−1 − finish vertex functions on paths in Γ will also be denoted by s(·) and f (·) respectively and the ξ ξ length function by (·), so that s(ξ ) = v0 , f (ξ ) = vn and (ξ ) = n. For 0 i j n, we will

ξ ξi+1

ξ

ξi+2

ξ

ξj

ξ

use notation such as ξ[i,j ] for the path (vi −−→ vi+1 −−→ vi+2 · · · −→ vj ), where the interval refers to the vertex indices. The symbol ◦ will denote composition of paths and will stand for


2637

path reversal. For n 0, the path space Pn (Γ ) associated to the graph Γ is the complex vector space with basis {[ξ ]: ξ is a path of length n in Γ }. We will now define Gr(Γ ) and its structure maps. As a graded vector space, Gr(Γ ) = n0 Pn (Γ ). The multiplication in Gr(Γ ), denoted by •, is given by concatenation on the path basis and extended by linearity: [ξ ] • [η] =

0 if f (ξ ) = s(η), [ξ ◦ η] if f (ξ ) = s(η).

The involution ∗ on Gr(Γ ) is defined by conjugate linear extension of the reversal mapon the path basis, i.e., [ξ ]∗ = [ξ˜ ]. We define a linear functional τ on Gr(Γ ) motivated by the GJS trace. Suppose that [ξ ] ∈ Pn (Γ ) for n 1. Define τ [ξ ] = τT [ξ ] T

where the sum is over all Temperley–Lieb equivalence relations1 T on {1, 2, . . . , n} (so that it is an empty sum, hence vanishes, for n odd) and τT is defined by τT [ξ ] =

{{i,j }∈T : i<j }

δξi ,ξj

ξ 2−|C| μ vC

C∈K(T )

where (i) K(T ) is the Kreweras complement of T – see [12] – which is also a non-crossing ξ ξ partition of {1, 2, . . . , n} and (ii) vC = vc for any c ∈ C (all of which must be equal if the first product is non-zero). When n = 0, we set τ ([(v)]) = μ2 (v). Proposition 1. Gr(Γ ) is a graded, unital, associative, ∗-algebra and τ is a normalised trace on Gr(Γ ). Proof. The only not completely obvious assertion is the traciality of τ , which too follows, after a little thought, from the rotational invariance of the set of all TL-equivalence relations and from the definition of the product in Gr(Γ ). 2 Note that the multiplicative identity of Gr(Γ ) is the element v∈V [(v)] ∈ P0 (Γ ). In view of the fact that the different [(v)], for v ∈ V , are orthogonal idempotents (adding to 1), we will denote [(v)] also by ev . It is useful to observe that an element, say x, of Pn (Γ ) may be regarded as the square matrix, with rows and columns indexed by V , with (v, w) entry given by ev xew (the part of x which is a linear combination of paths beginning at v and ending at w). The proof of positivity and faithfulness of the trace τ involves some work with a different avatar of Gr(Γ ) which we will find very useful. We begin by recalling, from [7], the category epi-TL which we will denote by E. The objects of E are denoted [n] for n 0 and thought of as n-points (labelled 1, 2, 3, . . . , n) arranged on a horizontal line. A morphism in Hom([n], [m]) consists of a rectangle with m-points on the top horizontal line, n-points on the bottom horizontal line and a Temperley–Lieb like tangle in between, subject to the restriction that each of the points 1 These are the non-crossing relations with every class having two elements.

2638


above is joined to a point below. It must be observed that Hom([n], [m]) is non-empty precisely when n − m is a non-negative even integer. Morphisms are composed by vertical stacking. The morphisms in E are generated by those which have a single cap on the bottom line. Let Sin : [n] → [n − 2] (for 1 i < n) denote the generator with the ith and (i + 1)st points on the bottom line capped. Some work shows that all relations among the morphisms are consequences of the relations n Spn−2 Sqn = Sqn−2 Sp+2

(1)

for n − 2 > p q 1. In fact any element of Hom([m + 2k], [m]) is uniquely expressible in the form Sim+2 Sim+4 · · · Sim+2k with 1 i1 < i2 < · · · < ik < m + 2k. (The left end-points of the k k 1 2 caps of the morphism are precisely at the places i1 , i2 , . . . , ik .) Such a morphism will be called non-nested if the caps are ‘not nested’, or equivalently, if ij +1 ij + 2 for each j < k in its ‘canonical decomposition’ as above. It follows that the category E ‘acts’ on the collection of vector spaces Pn (Γ ) in the sense that any element of Hom([n], [m]) yields a vector space homomorphism Pn (Γ ) → Pm (Γ ) with this assignment being compatible with compositions on both sides. Such an action can be defined2 with Sin acting by ξ μ(vi ) [ξ[0,i−1] ◦ ξ[i+1,n] ] Sin [ξ ] = δξi ,ξ

i+1 ξ μ(vi±1 )

(2)

for [ξ ] ∈ Pn (Γ ). More generally, given an arbitrary S ∈ Hom([n], [m]), it specifies a partition of [n] as T ∪ E, where T is the subset of points in [n] that are joined to a point in [m] and E is its complement. It also specifies a Temperley–Lieb equivalence relation ∼ on E. The action of S is then explicitly given by S [ξ ] =

{i,j }∈∼: i<j

ξ

δξi ,ξj

μ(vi ) ξ

μ(vj )

[◦t∈T ξt ],

(3)

where the concatenation is done in increasing order of elements of T and is interpreted as [(f (ξ ))] if T = ∅. (As in the equations displayed above, we shall often identify elements of Hom([n], [m]) with the associated operators from Pn (Γ ) to Pm (Γ ).) The following lemma is a special case (of Proposition 3) which both motivates and is used in the proof of a different expression for S([ξ ]) when S ∈ Hom([2n], [0]). Note that in this case, E = {1, 2, . . . , 2n} and ∼ = S regarded as an equivalence relation. Lemma 2. Let [ξ ] ∈ P2n (Γ ) and S ∈ Hom([2n], [0]) be given by {{1, 2n}, {2, 2n − 1}, . . . , {n, n + 1}}. Then, n μ(vnξ )

ξ S [ξ ] = δξ ,ξ × v2n . ξ μ(v2n ) i=1 i 2n+1−i

2 Thus we are saying that the operators defined by Eq. (2) satisfy the relations (1).


2639

Fig. 1. The element S ∈ Hom([10], [0]).

Proof. We may assume that ξ is a path consistent with S in the sense that ξi = ξj whenever {i, j } ∈ S, since otherwise, both sides of the desired equality vanish. Thus, ξi = ξ 2n+1−i and in ξ

ξ

particular, vi = v2n−i for each i = 0, 1, . . . , 2n. Using Eq. (3), it now suffices to check that n

μ(vi )

i=1

μ(v2n+1−i )

ξ

ξ

ξ

ξ

=

μ(vn ) ξ

.

μ(v2n )

ξ

But substituting vi = v2n−i , we see that the product on the left telescopes to the expression on the right. 2 We next treat the case of a general S. Proposition 3. For [ξ ] ∈ P2n (Γ ) and S ∈ Hom([2n], [0]), μ(vnξ ) S [ξ ] = ξ μ(v2n ) ×

ξ

{i,j }∈S: i<j n

{i,j }∈S: n n. We define φnm to be the (action by the) sum of all elements of Hom([n], [m]) and ψnm to be (−1)n−m times the (action by the) sum of all the non-nested elements of Hom([n], [m]). We now have the following proposition that identifies Gr(Γ ) and F (Γ ). Proposition 5. The maps φ and ψ define mutually inverse ∗-isomorphisms between Gr(Γ ) and F (Γ ) that intertwine the functionals τ and t. The proof uses the following lemma about the Kreweras complement of Temperley–Lieb equivalence relations. Lemma 6. Let S be a Temperley–Lieb equivalence relation on {1, 2, . . . , 2n} and K(S) be its Kreweras complement. Then, for any class C = {a1 , . . . , ak } of K(S) with a1 < · · · < ak , all the ai have the same parity and {ai + 1, ai+1 } ∈ S for each i = 1, . . . , k (where ai + 1 is computed modulo 2n and i + 1 is computed modulo k).


2641

Proof. Induce on n, with the basis case n = 1 following by a direct check. For n > 1 take i 2n − 1 largest so that {i, i + 1} ∈ S. Let T = S|{1,2,...,2n}\{i,i+1} . The Kreweras complement of S is obtained from that of T by adding i + 1 to the class of i − 1 and adding the singleton class {i}. Observe that i + 1 is the largest element in its K(S) class by choice of i. Now by induction, the parity assertion holds and further, the new {ai + 1, ai+1 } that are needed to be shown to belong to S are both {i, i + 1} which is, indeed, in S. 2 Proof of Proposition 5. The proof that the maps φ and ψ define mutually inverse ∗-isomorphisms between Gr(Γ ) and F (Γ ) is nearly identical to that of Lemma 5.1 in [7] and depends essentially only on properties of the category E. We omit it here. The intertwining assertion that needs to be checked is that τ = t ◦ φ on Gr(Γ ). Note that both sides vanish on paths of odd length and that if [ξ ] is a path of length 2n, then, τ ([ξ ]) = T τT ([ξ ]) where the sum is over all Temperley–Lieb relations T on {1, 2, . . . , 2n} while t ◦ φ([ξ ]) = t ( S S([ξ ])) where the sum is over all S ∈ Hom([2n], [0]), since t vanishes on paths of positive length. The natural identification between Temperley–Lieb equivalence relations on {1, 2, . . . , 2n} and Hom([2n], [0]) shows that it suffices to see that τS ([ξ ]) = t ◦ S([ξ ]) for any Temperley–Lieb relation S on {1, 2, . . . , 2n}. Now both these vanish unless s(ξ ) = f (ξ ); so we assume this. Unravelling the definitions, we need to see that under these assumptions, {{i,j }∈S: i<j }

δξi ,ξj

ξ 2−|C| ξ 2 μ vC = μ v2n

C∈K(S)

ξ μ(vi ) δξi ,ξj , ξ μ(vj ) {{i,j }∈S: i<j }

ξ

ξ

with K(S) being the Kreweras complement of S and vC = vc for any c ∈ C. We may further assume that ξ is a path consistent with S in the sense that ξi = ξj whenever {i, j } ∈ S and show the following ξ −2 ξ 2−|C| μ v2n μ vC =

{{i,j }∈S: i<j }

C∈K(S)

ξ

μ(vi ) ξ

.

μ(vj )

ξ S (i) The product on the right may be rewritten as 2n where S (i) is 1 or −1 according i=1 μ(vi ) as i is the smaller or larger element in its S-class. Next, we may regroup this product in terms of ξ classes of K(S) as C∈K(S) c∈C μ(vc )S (c) . Now, as we have observed before, if ξ is consistent ξ

ξ

with S then all vc for c ∈ C are equal (to a vertex denoted vC ) and so this product now becomes ξ c∈C S (c) . Comparing with the product on the left, what needs to be seen is that C∈K(S) μ(vC ) if C is any class of K(S) then c∈C

S (c) =

2 − |C| −|C|

if 2n ∈ / C, if 2n ∈ C.

(6)

To prove Eq. (6), it suffices to see that for any non-external (i.e., not containing 2n) class C of K(S), S (c) is 1 or −1 according as c is the smallest element in C or not, while for the external class, S (c) = −1 for all its elements. But this is an easy consequence of Lemma 6. If C = {a1 , . . . , ak } is a K(S) class for which ak = 2n, the definition of S (together with Lemma 6)

2642


shows that S (a1 ) = 1 while S (ai ) = −1 for i 2. On the other hand, if ak = 2n, then it similarly follows that all S (ai ) = −1, completing the proof of Eq. (6) and consequently of the proposition. 2 An immediate consequence of Proposition 5 is the following corollary. Corollary 7. For any graph Γ , (Gr(Γ ), τ ) is a graded, tracial, faithful ∗-probability space. We recall that if (A, τ ) is a tracial probability space and e ∈ A is a non-zero projection, then the corner eAe is naturally a tracial probability space where the trace is scaled so as to be 1 on e. We will find some corners of Gr(Γ ) to be useful. For a vertex v ∈ V , we denote by Gr(Γ, v) the probability space ev Gr(Γ )ev . Letting e0 = v∈V0 ev – the sum of the projections corresponding to the even vertices – we will denote e0 Gr(Γ )e0 by Gr(Γ, 0). Similarly, with e1 = w∈V1 ew = 1 − e0 , we denote e1 Gr(Γ )e1 by Gr(Γ, 1). The bipartite nature of the graph Γ implies that the odd graded pieces of these graded algebras reduce to zero. In particular, Gr(Γ, v) has as basis {[ξ ]: ξ is a path beginning and ending at v} and is a connected graded algebra, while Gr(Γ, 0) (resp. Gr(Γ, 1)) has as basis {[ξ ]: ξ is a path beginning and ending at an even (resp. odd) vertex}. There are also corresponding notions in the F (Γ ) picture such as F (Γ, v), F (Γ, 0) or F (Γ, 1) and we will use self-explanatory notation such as Pn (Γ, 0) or Pn (Γ, v). Thus, for instance, F (Γ, v) = n0 P2n (Γ, v). We will also tacitly use the fact that the isomorphism of Gr(Γ ) onto F (Γ ) of Proposition 5 takes Gr(Γ, v) to F (Γ, v) for each v ∈ V . In particular, it takes Gr(Γ, 0) to F (Γ, 0) and Gr(Γ, 1) to F (Γ, 1). Consider the Hilbert space H (Γ ) obtained by completing F (Γ ) for its inner-product, which has orthonormal basis given by all 1 [ξ ] {ξ } = √ μ(s(ξ ))μ(f (ξ ))

(7)

where ξ is a path in Γ . Equivalently, it is the Hilbert space direct sum n0 Pn (Γ ) where each Pn (Γ ) has orthonormal basis {ξ } with ξ a path in Γ of length n. We denote its norm by · . We also need the local Hilbert spaces H (Γ, v) which we define to be the completions of F (Γ, v) for their trace norms. Note that F (Γ, v) is a (non-unital) subalgebra of F (Γ ) and that the norm on F (Γ, v) is a scaled version of the norm on F (Γ ). The paths [ξ ] that begin and end at v are an orthonormal basis of F (Γ, v) (while they have norm μ(v) regarded as elements of F (Γ )). We wish to show that the left regular representation of F (Γ ) on itself extends to a bounded representation on H (Γ ). It clearly suffices to see that for a ∈ Pm (Γ ) and b ∈ F (Γ ) there exists a constant C (depending only on a) such that a # b Cb. The proof of Proposition 4.3 in [10] goes over to show that even the following is sufficient (and that we may take C = (2m + 1)K). Proposition 8. For a ∈ Pm (Γ ) and b ∈ Pn (Γ ) there exists a constant K (depending only on a) such that (a # b)t Kb for any t with |m − n| t m + n. Proof. We will work with the orthonormal basis {ξ } rather than the orthogonal basis [ξ ]. Observe that 0 if f (ξ ) = s(η), {ξ } • {η} = 1 {ξ ◦ η} if f (ξ ) = s(η). μ(f (ξ ))


2643

We may assume that a = {ξ }. Suppose that b = η cη {η}, where the sum is over all paths η in Γ of length n. Since (a # b)t is obtained by an application of at most m Sin ’s to cη n {η: f (ξ )=s(η)} μ(f (ξ )) {ξ ◦ η}, it suffices to bound the operator norm of the Si and the Hilbert cη space norm of {η: f (ξ )=s(η)} μ(f (ξ )) {ξ ◦ η}. It is easily checked that the adjoint of Sin is given explicitly by

n ∗ Si {η} = w

η

ρ

ρ : (vi−1 →w)∈E

μ(w) {η[0,i−1] ◦ ρ ◦ ρ˜ ◦ η[i−1,n−2] }, η μ(vi−1 )

and consequently that Sin

η ρ n ∗ μ(w) 2 {η}, Si {η} = →w ∈E ρ : vi−1 − η μ(vi−1 ) w

for all {η} ∈ Pn−2 (Γ ). Thus, for a vertex v, if we define δ(v) =

μ(w) 2 ρ ρ : (v − → w) ∈ E , μ(v) w

and δ = maxv∈V {δ(v)}, then the operator norm of Sin is bounded above by Finally, note that b2 = η |cη |2 while

√ δ.

2 cη 1 1 {ξ ◦ η} |cη |2 = b2 . 2 2 μ(f (ξ )) μ(f (ξ )) μ(f (ξ )) η η } Thus we may take K = max{1,δ μ(f (ξ )) for a = {ξ } ∈ Pm (Γ ) (the reason for the ‘max’ being to allow for the cases δ < 1 and δ 1). 2 m/2

We thus have a bounded left regular representation λ : F (Γ ) → L(H (Γ )) and we set M(Γ ) = λ(F (Γ )) . Similarly, for i = 0, 1, we have the left regular representation λ : F (Γ, i) → L(H (Γ, i)) and we may define M(Γ, i) = λ(F (Γ, i)) . It is easy to see that – see Lemma 4.4 of [10] – each of M(Γ ), M(Γ, 0) and M(Γ, 1) is a finite von Neumann algebra. The goal of the next section is to show that M(Γ ) is ‘almost a II 1 -factor’. 2. Almost II1 -factoriality of M(Γ ) Throughout this section, our standing assumption will be that the graph Γ is connected and has at least one edge. For such a graph, it is clear that Gr(Γ ) is infinite-dimensional. The main results of this section apply only to graphs with at least two edges and show that the von Neumann algebra M(Γ ) is a direct sum of a II 1 -factor and a finite-dimensional abelian algebra (possibly {0}) by analysing the local graded probability spaces Gr(Γ, v) for each vertex v ∈ V . In the analysis of Gr(Γ, v), an action by a certain category that we denote by C(δ) (where δ ∈ C is some fixed non-zero parameter) will be extremely important, so we begin by describing this category.

2644


Fig. 2. The morphism T ([4, 5], [3, 4])85 ∈ Hom([5], [8]).

Its objects are [0], [1], [2], . . . , where we think of [n] as a set of 2n points on a horizontal line labelled 1, 2, . . . , 2n. Note that the objects of C(δ) are denoted by exactly the same notation as objects of E but mean different things. The set Hom([n], [m]) is stipulated to have basis given by all T (P , Q)m n where P ⊆ [m] and Q ⊆ [n] are intervals of equal cardinality, where T ([4, 5], [3, 4])85 ∈ Hom([5], [8]) is illustrated in Fig. 2. The general prescription for T (P , Q)m n is the following. Points below labelled by elements of the sets 2Q and 2Q − 1 are joined to points above labelled by 2P and 2P − 1 in order preserving fashion, and the rest are capped or cupped off without nesting. Composition in C(δ) for the basis elements is as in Temperley–Lieb categories – by stacking the pictures and replacing each closed loop that appears with a multiplicative δ, thus yielding a multiple of another basis element. In order to explicitly write down the composition rule in terms of the T (P , Q)m n , note that there is a unique order preserving bijection fPQ : P → Q and that for p ∈ P , the marked points above that are labelled by 2p − 1 and 2p are joined, respectively, to the points below that are labelled by 2fPQ (p) − 1 and 2fPQ (p). In the example above, for instance, fPQ (p) = p − 1. Then the composition rule is seen to be n n−|Q∪R| T (P , Q)m T (Y, Z)m n ◦ T (R, S)p = δ p, −1 where Y = fPQ (Q ∩ R) and Z = fRS (Q ∩ R). For n 1 let An− ∈ Hom([n], [n − 1]) be the morphism with a single cap in the bottom n left corner (i.e., An− = T ({1, 2, . . . , n − 1}, {2, 3, . . . , n})n−1 n ) and A+ ∈ Hom([n], [n − 1]) be the morphism with a single cap in the bottom right corner (i.e., An+ = T ({1, 2, . . . , n − 1}, n {1, 2, . . . , n − 1})n−1 n ). Similarly, for n 0 let C− ∈ Hom([n], [n + 1]) be the morphism with n a single cup in the top left corner and C+ ∈ Hom([n], [n + 1]) be the morphism with a sinn = T ({2, 3, . . . , n + 1}, {1, 2, . . . , n})n+1 and C n = gle cup in the top right corner. Thus, C− + n n+1 T ({1, 2, . . . , n}, {1, 2, . . . , n})n . (The letters A and C are meant to suggest similarity to ‘annihilation’ and ‘creation’.)

Proposition 9. If δ = 0, the category C(δ) is generated by the set n n A± : n 1 ∪ C± : n0 of morphisms, and presented by the following relations, valid for all n 0:


2645

A1− = A1+ ,

(8)

0 0 = C+ , C−

(9)

n+2 n+1 n+2 An+1 − A+ = A+ A− , n An+1 − C− = δid [n] , n+1 An+2 − C+

(10) (11)

n n+1 = C+ A− ,

(12)

n+1 n n+1 An+2 + C− = C− A+ ,

(13)

n An+1 + C+ = δid [n] ,

(14)

n+1 n n+1 n C− C + = C+ C− .

(15)

More explicitly, suppose a category D has the property that Hom(D, D ) is a complex vector space for every pair (D, D ) of objects. Then, in order to specify a functor from C(δ) to D, it (is n± : Dn → Dn−1 for necessary and it) suffices to find objects Dn ∈ D for n 0 and morphisms A n :D → D ± , for n 0 satisfying the relations (8)–(15) above. n 1 and C n n+1 Proof. It is easy to check that the relations (8)–(15) are satisfied. To see that these are the only relations, we need to first observe that the following identities, for k, l 0, are consequences of them: k+l k+l 1 A1− · · · Ak− Ak+1 + · · · A+ = A+ · · · A+ ,

(16)

k+l−1 k+l−1 k k−1 0 0 C+ · · · C+ C− · · · C− = C+ · · · C+ .

(17)

These two identities are seen inductively to follow from the equations numbered (8) and (10), and from (9) and (15) respectively, from among the above relations. We next describe a ‘canonical form’ for every morphism in C(δ) as a word in the generators, in such a way that if we assign the ‘rank’ 1, 2, 3, and 4 to any C+ , C− , A+ and A− respectively, then if the word contains generators of ranks i and j , with i < j , then the generator of rank i will appear to the left of the one with rank j . For example, the morphism illustrated in Fig. 2 is expressed as 8 7 6 5 4 3 2 3 4 5 C+ C+ C− C− C− A+ A+ A− . T [4, 5], [3, 4] 5 = C+ (The algorithm for arriving at this word is to ‘first list all the caps from left to right, and then all the cups also from left to right’.) Notice that it is only when there are no through strings that there is an ambiguity about whether to choose a + or a − for the A’s and C’s, but this is resolved using Eqs. (16) and (17) above, using which we can demand in the case of no through strings, that all C’s and A’s come with the subscript ‘+’. On the other hand, if there are through strings, the number of through strings can be read off from this word, at the point of transition from C’s to A’s (the number of through strings is exactly twice the superscript of the rightmost C). (By the way, it is in order to lay hands on id[n] that we need the condition δ = 0.) Finally, if the ‘rank ordering’ specified above is violated in any word in the generators, such instances may be rectified uniquely by using (10)–(15). (For example, any instance where

2646


an A− (of rank 4) precedes any generator of rank 3, 2, or 1, is set right by equations numbered (10)–(12).) 2 We use Proposition 9 to get a functor from C(δ(v)) to the category D of C-vector spaces and ρ 2 → w) ∈ E}|( μ(w) C-linear maps. Recall that δ(v) = w |{ρ : (v − μ(v) ) . Let Dn be P2n (Γ, v) and define the action of the generating morphisms as follows on [ξ ] ∈ P2n (Γ, v) n − [ξ ] = C

ξ μ(v1 ) n− [ξ ] = δξ ,ξ ξ A , 1 2 μ(v) [2,2n]

w

ρ

ρ : (v →w)∈E

n + [ξ ] = C

ξ

μ(v2n−1 ) n+ [ξ ] = δξ ,ξ ξ[0,2n−2] , A 2n−1 2n μ(v)

w

μ(w) [ρ ◦ ρ˜ ◦ ξ ], μ(v) ρ

ρ : (v →w)∈E

μ(w) [ξ ◦ ρ ◦ ρ]. ˜ μ(v)

A little calculation now proves the following. Proposition 10. The action by the generators given by the equations above extends to give a well-defined functor from C(δ(v)) to D. n+1 n Note that regarding P2n (Γ, v) as subspaces of H (Γ, v), the maps A − and C− are adjoints n+1 n + and C + . of each other, as are A We now try to determine the structure of the centre of Gr(Γ, v). In particular, we show that it is at most two-dimensional. We will find the following notation useful. For a ∈ F (Γ, v) let [a] = {ξ ∈ H (Γ, v): λ(a)(ξ ) = ρ(a)(ξ )},3 which is a closed subspace of H (Γ, v). Denote by Ω the vector ev ∈ F (Γ, v) ⊆ H (Γ, v), note that this is (cyclic and) separating for M(Γ, v) and thus the operator equation ax = xa is equivalent to the vector inclusion xΩ ∈ [a] for x ∈ M(Γ, v). Our strategy is similar to that in [10], with some differences. We first define two elements c ∈ P2 (Γ, v) and d ∈ P4 (Γ, v). (For notational convenience we do not use the possibly more correct notation cv and dv .) We then show that [c] ∩ [d] is 1-dimensional if δ(v) 1 and 2-dimensional if δ(v) < 1 (assuming that the graph Γ in question has at least two edges). Finally we show that in case δ(v) < 1, the centre of Gr(Γ, v) is actually 2-dimensional and that the cut-down by one of the central projections is just C. Now some simple computations give the desired result that M(Γ ) is either a II 1 -factor or a direct sum of one with a finite-dimensional abelian algebra. In the sequel, we dispense with ‘tilde’s and continue to use the same symbol for morphisms in C(δ) and the associated linear maps between the P2n (Γ, v). 0 (1). Explicitly, Let c ∈ P2 (Γ, v) be the element C− c=

w

ρ : (v →w)∈E

ρ

μ(w) [ρ ◦ ρ]. ˜ μ(v)

n−1 n−2 0 (1). Thus Let c0 = 1 and by c2n for n 1, we will denote the element C− C− · · · C − c2n ∈ P2n (Γ, v) and c2 = c. Note that by induction on n, c2n is seen to be the highest degree 3 As usual, we write ρ(a) = J λ(a)∗ J , with J being the modular conjugation operator.


2647

term of cn in F (Γ, v) and to be a polynomial in c of degree n. Let C ⊆ H (Γ, v) be the closed subspace spanned by all the c2n Ω for n 0. We then have the following crucial result which is the analogue of Proposition 5.4 of [10]. Proposition 11. [c] = C. Before sketching a proof, we state a key lemma used which is the analogue of Lemma 5.6 ⊥ of [10]. By C2n we denote the (1-dimensional) subspace of P2n (Γ, v) spanned by c2n and by C2n ⊥ its orthogonal complement in P2n (Γ, v). Thus if C is the orthogonal complement of C in ⊥. H (Γ, v), then C ⊥ = n0 C2n ⊥ x → z = (c # x − x # c) Lemma 12. For n 0, the map C2n 2n+2 ∈ P2n+2 (Γ, v) is injective with inverse given by

x=

n

n δ(v)−t T [1, n + 1 − t], [t + 1, n + 1] n+1 (z).

t=1 n (x) and C n (x). We omit the proof except to remark that (c # x)2n+2 and (x # c)2n+2 are just C− + We also omit the proof of the next corollary which is the analogue of Corollary 5.7 of [10] with identical proof.

⊥ ⊥ Corollary 13. Suppose that ξ = (x0 , x1 , . . .) ∈ ∞ n=0 C2n = C and satisfies λ(c)(ξ ) = ρ(c)(ξ ), ⊥ i.e., ξ ∈ C ∩ [c]. Then, for m > n > 0 with m − n = 2r, we have xn =

n

n δ(v)−(t+r−1) T [1, n + 1 − t], [t + r, n + r] m (xm )

t=1

n − T [1, n + 1 − t], [t + r + 1, n + r + 1] m (xm ) .

One more result needed in proving Proposition 11 is the following norm estimate which is the analogue of Lemma 5.8 of [10]. Lemma 14. Suppose that x = xn ∈ P2n (Γ, v) ⊆ F (Γ, v) and let y = T (P , Q)m n (x) ∈ P2m ⊆ 1 2 (n+m)−|P | x. F (Γ, v) for some morphism T (P , Q)m . Then y δ(v) n Proof. Consider the linear extension of the map defined by Hom([n], [m]) T (P , Q)m n → 1 δ(v) 2 (n+m)−|P | ∈ C for each n, m 0. The observation is that it is multiplicative on composin tion. For consider T (P , Q)m n ∈ Hom([n], [m]) and T (R, S)p ∈ Hom([n], [p]). Their composition −1 is given by δ(v)n−|Q∪R| T (Y, Z)m p where Y = fPQ (Q ∩ R) and Z = fRS (Q ∩ R). The multiplicativity assertion amounts to verifying that 1 1 1 (n + m) − |P | + (n + p) − |R| = n − |Q ∪ R| + (m + p) − |Y |, 2 2 2 which is easily verified to hold.

2648


Fig. 3. Pictorial representation of d. n Hence it suffices to verify the norm estimate when T (P , Q)m n is one of the generators C± n or A± of the category C(δ(v)). Note that the norm estimate for all these generators is just y 1

n , we have (C n )∗ C n = An+1 C n = δ(v)I n while for An , we have An (An )∗ = δ(v) 2 x. For C± ± ± ± ± ± ± ± n−1 An± C± = δ(v)I n−1 . This proves the norm estimate for generators and completes the proof of the lemma. 2

Proof of Proposition 11. Each c2n , being a polynomial in c, clearly commutes with c and it follows that C ⊆ [c]. To prove the other inclusion, it is enough to see that C ⊥ ∩[c] = {0}. Suppose that ξ = (x0 , x1 , . . .) ∈ C ⊥ ∩ [c]. Note that x0 = 0 since x0 ∈ C0⊥ while C0 = P0 (Γ, 0) = C. By Corollary 13, we have for m > n > 0 with m − n = 2r, xn =

n

n δ(v)−(t+r−1) T [1, n + 1 − t], [t + r, n + r] m (xm )

t=1

n − T [1, n + 1 − t], [t + r + 1, n + r + 1] m (xm ) .

Now, applying the norm estimate from Lemma 14 and using the triangle inequality gives xn 2nxm . Now ξ ∈ H (Γ, v) ⇒ limm→∞ xm = 0 and so xn = 0 for all n > 0. Hence ξ = 0, completing the proof. 2 We now consider the element d ∈ P4 (Γ, v) defined explicitly by d=

w

ρ : (v →w)∈E

ρ

x

ζ : (w →x)∈E

ζ

μ(x) ρ ζ ζ ρ˜ [v − →w− →x− →w− → v]. μ(v)

Loosely, d can be thought of as pictorially represented as in Fig. 3. It must be noted that the action of the category C(δ) on F (Γ, v) may be extended to an action of the entire Temperley–Lieb category on F (Γ, v) if μ2 is a Perron–Frobenius eigenvector for the incidence matrix of Γ , however, this may no longer be possible if the ‘Perron–Frobenius assumption’ is dropped. Since we shall have to address that situation, we do not assume this. Nevertheless, we will see that in situations of interest for us, this pictorial representation will be of heuristic value. We wish to consider a special element of M(Γ, v) defined when δ(v) < 1. Since the proof of existence of this element requires a careful norm estimate, we digress with the necessary lemma. Lemma 15. Suppose the weighting on Γ is such that δ(v) < 1. The sequence of elements xm = m n c ∈ F (Γ, v) ⊆ 4 M(Γ, v) ⊆ L(H (Γ, v)) converge in the strong operator topology. (−1) 2n n=0 n Hence the series ∞ n=0 (−1) c2n defines an element zv ∈ M(Γ, v). 4 This inclusion is, naturally, via λ.


2649

Proof. It suffices to see that the xm are uniformly bounded in norm and that for ξ in a dense in H (Γ, v). Note that if ξ = Ω(= [(v)]) subspace of H (Γ, v), the sequence xm ξ converges n c Ω. Since c Ω2 = δ(v)n , when (−1) – the vacuum vector of H (Γ, v) – then, xm ξ = m 2n 2n n=0 δ(v) < 1, the xm ξ converge in H (Γ, v). It follows that on the dense subspace M(Γ, v) Ω too, we obtain convergence. It remains to prove the norm estimate. Consider the block matrix representing left-multiplication by c2n on the Hilbert space H (Γ, v) with respect to the orthogonal decomposition H (Γ, v) = ∞ n=0 P2n (Γ, v). The definition of multiplication in F (Γ, v) shows that for any path ξ of length 2j ,

c2n # [ξ ] =

min{2j,2n}

k

k

C n− 2 A 2 [ξ ]

k=0

where, to avoid heavy notation, for [ξ ] ∈ P2j (Γ, v) we write C p Aq [ξ ] to mean j −q+p−1

C−

j −q+1

· · · C−

j −q

j −q+1

C− A−

j −1

j

· · · A− A− [ξ ].

Thus the (i, j )-block (note that i, j 0) of the matrix for λ(c2n ) is 0 unless |i − j | n i + j , in which case it is given by C

n−j +i 2

A

n+j −i 2

.

It now follows that the (i, j )-block of the matrix for xm is given by

(−1)n C

n−j +i 2

A

n+j −i 2

.

|i−j |nmin{m,i+j }

Every odd term of this sum (starting with the first which corresponds to n = |i − j |) is equal, except for sign, with the succeeding even term and so the sum vanishes when there are an even number of terms and equals its last term when there are an odd number of terms. Note that the number of terms is min{m, i + j } − |i − j | + 1. We consider two cases depending on whether m i + j or m > i + j . Case I. If m > i + j , the number of terms is certainly odd and so (xm )ij = (−1)i+j C i Aj . Case II. If i + j m the number of terms is odd or even according as m and i + j have the same parity or not and so, in this case, (xm )ij =

0 (−1)m C

m−j +i 2

m+j2 −i

A

if m < |i − j | or (i + j ) − m is odd, otherwise.

2650


For instance, the matrix for x2 is given by ⎛ I −A A2 ⎜ −C CA 0 ⎜ 2 ⎜C 0 CA ⎜ ⎜ 0 0 C2 ⎜ ⎝ 0 0 C2 .. .. .. . . .

0 0 0 A2 0 A2 CA 0 0 CA .. .. . .

⎞ ··· ···⎟ ⎟ ···⎟ ⎟ ···⎟. ⎟ ···⎠ .. .

Observe that the (i, j ) entry of xm is non-zero only if |i − j | m in which case it is of the 1 form ±C p Aq with p + q |i − j |. Since each of A and C has norm bounded above by δ(v) 2 , |t| the diagonal of xm with i − j = t has norm at most δ(v) 2 and so xm itself has norm bounded by ∞ m |t| t 2 2 t=−m δ(v) 1 + 2 t=1 δ(v) . Thus the xm are uniformly bounded in norm, finishing the proof. 2 For reasons that will become clear in Proposition 17, we define (in the notation introduced in Lemma 15), 0 if δ(v) 1, I ev = (1 − δ(v))zv if δ(v) < 1 and evII = ev − evI . The next proposition is the analogue of Proposition 5.5 of [10]; the reader is urged to compare the proof of that proposition with this one. For the rest of this section, Γ will always denote a connected graph with at least two edges. Proposition 16. Suppose that Γ is a connected graph with at least two edges. Then, [c] ∩ [d] has basis {1} if δ(v) 1 and basis {1, zv } if δ(v) < 1. Proof. Suppose that ξ = (x0 , x2 , x4 , . . .) ∈ [c] ∩ [d]. Since ξ ∈ [c] = C by Proposition 11, there exist ∈ C such that x2n = y(2n)c2n and since ξ ∈ H (Γ, v), we have that ξ 2 = ∞ scalars y(2n) 2 n n=0 |y(2n)| δ(v) < ∞. Some computation now shows that for t > 2, the P2t (Γ, v) component of λ(d)([ξ ]) − ρ(d)([ξ ]) is given by (y(2t − 2) + y(2t − 4))(d # c2t−4 − c2t−4 # d)2t . Writing out (d # c2t−4 − c2t−4 # d)2t for t > 2 in terms of the path basis, inspection shows that it cannot vanish since Γ has at least two edges. Thus each y(2t − 2) + y(2t − 4) vanishes for t > 2. Hence if y(2) = y, then all y(4n − 2) = y and all y(4n) = −y (for n 1). Now the norm condition shows that if δ(v) 1, then y = 0 so that ξ is a multiple of (1, 0, 0, . . .), while if δ(v) < 1, then ξ is a linear combination of 1 and zv . 2 We now justify the choice of notation for evI and evII . Proposition 17. Suppose that Γ is a connected graph with at least two edges. Then: (1) If δ(v) < 1, then Z(M(Γ, v)) is 2-dimensional and has basis {evI , evII }. The element evI ∈ M(Γ, v) is a minimal (and central) projection. (2) evII is a minimal central projection in M(Γ, v) and evII M(Γ, v)evII is a II 1 -factor; in particular, M(Γ, v) is a factor if δ(v) 1.


2651

Proof. (1) Since Z(M(Γ, v))Ω ⊆ [c] ∩ [d] and Z(M(Γ, v)) → Z(M(Γ, v))Ω is injective, Proposition 16 implies that Z(M(Γ, v)) is at most 2-dimensional. If δ(v) < 1, it follows from Proposition 16 that Z(M(Γ, v)) is at most two-dimensional and contained in the span of ev (the identity for M(Γ, v)) and evI . We shall regard c2n as an element of ev F (Γ )ev ⊂ F (Γ ) (so that c0 = ev ) and consider a length one path ξ of F (Γ ). Calculation shows that c2n # [ξ ] = c2n • [ξ ] + c2n−2 • [ξ ] and therefore that zv # [ξ ] = 0. Similarly, (or by simply taking adjoints) we find that also [ξ ] # zv = 0. Associativity of multiplication implies that these equations hold even when [ξ ] is a path of length greater than 1 from w to x. Finally, if ξ is a path of length 0, then ξ = ew for some w, and 0 if w = v, zv # [ξ ] = [ξ ] # zv = zv if w = v. It follows easily that zv ∈ Z(M(Γ )) and that zv M(Γ ) = Czv .

(18)

M(Γ )zv = Czv .

(19)

Taking adjoints yields

Deduce that there is some constant γ > 0 such that zv2 = γ zv . By comparing their inner-products (in H(Γ, v)) with ev , we find that ∞ n δ(v) = γ n=0

and hence that γ = (1 − δ(v))−1 . Thus we find that indeed evI is a projection in M(Γ ) which is central and minimal, since evI M(Γ )evI = CevI .

(20)

(2) It is seen from Proposition 16 that in both cases, evII is a minimal central projection. The assumed non-triviality of Γ ensures that M(Γ, v) is an infinite-dimensional but finite von Neumann algebra, it is seen (from the minimality of evI in case δ(v) < 1) that the localisation eII M(Γ, v)eII is a II 1 -factor. 2

2652


Corollary 18. For distinct vertices v and w, we have evI M(Γ )ew = {0} = ew M(Γ )evI . Proof. We only need to prove the non-trivial case when δ(v) < 1. First deduce from Eq. (18) (when δ(v) < 1) and the definition of evI (when δ(v) 1) that evI M(Γ ) = CevI . Hence, evI M(Γ )ew = CevI ew = 0. The second assertion of the corollary is obtained by taking adjoints in the first. 2 Before proceeding to the next corollary to Proposition 17 we digress with an elementary fact about local and global behaviour of von Neumann algebras. Lemma 19. Suppose {pi : i ∈ I } is a partition of the identity element into a family of pairwise orthogonal projections of a von Neumann algebra M. Then the following conditions are equivalent: (1) M is a factor; (2) pi Mpi is a factor for all i ∈ I , and pi Mpj = {0} ∀i, j . Proof. We only indicate the proof of the non-trivial implication (2) ⇒ (1). For this, suppose x ∈ Z(M). Let us write xij = pi xpj . The assumption (2) clearly implies that (i) xij = 0 for i = j (since x commutes with each pi ); and (ii) for each i ∈ I , xii = λi pi for some λi ∈ C. Fix an arbitrary pair (i, j ) of distinct indices from I . By assumption, there exists a non-zero y ∈ M satisfying y = pi ypj . The requirement xy = yx is seen to now imply that λi y = λj y and hence that λi = λj , and this is true for all i, j . Hence, x = i λi pi ∈ C1. 2 Corollary 20. Assume that Γ is a connected graph with at least two edges. Then: II = 0 for all vertices v, w. (1) evII M(Γ )ew (2) If we let eI = v∈V evI , then eI M(Γ )eI = v∈V CevI . (3) If we let eII = v∈V evII , then eII M(Γ )eII is a II 1 -factor.

Proof. (1) Since Γ is connected, we can find a finite path ξ with s(ξ ) = v and f (ξ ) = w. Then, deduce from Corollary 18 that 0 = [ξ ] = ev # [ξ ] # ew II = evII # [ξ ] # ew . I e for all w ∈ V , it follows that (2) First observe that as ew w

v = w

⇒

I I evI M(Γ )ew = evI M(Γ )ew ew

=0 by Corollary 18.


2653

On the other hand evI M(Γ )evI = CevI . The desired assertion follows from the orthogonality of the evI ’s. (3) As has already been observed, ev M(Γ )ev = M(Γ, v) since M(Γ ) (resp., M(Γ, v)) is generated (as a von Neumann algebra) by the set of all finite paths [ξ ] (resp., those paths which start and finish at v, i.e., which satisfy [ξ ] = ev # [ξ ] # ev ). Given this observation, the assertion to be proved is seen to follow from Proposition 17, Lemma 19 and the already established part (1) of this corollary. 2 We have finally arrived at the main result of this section, whose statement uses the foregoing notation and which is an immediate consequence of Corollary 20. Theorem 21. Assume Γ is a connected graph with at least two edges. Then, we have the following isomorphism of non-commutative probability spaces: M(Γ ) ∼ =M ⊕

CevI

2 {v∈V : δ(v) 0 edges, while the vertices vi for i = k + 1, . . . , l are isolated. Thus k 1. We also set μ2 (vi ) = αi and μ2 (w) = β, so that β + li=1 αi = 1. Our goal in this section is the explicit determination of the finite von Neumann algebra M(Λ, 0). We begin with a simple observation. The assignments of Gr(Γ ) or M(Γ ) to a graph Γ clearly take disjoint unions to (appropriately weighted) direct sums. Thus, if Λ˜ denotes the connected component of w in the graph Λ (with normalised restricted μ), then ˜ ⊕ C ⊕ C ⊕··· ⊕ C M(Λ) ∼ = M(Λ) γ

where γ = 1 − when k = 1.

l

i=k+1 αi .

αk+1

αk+2

αl

(25)

˜ 1). We begin by analysing M(Λ, 1) Note that M(Λ, 1) = M(Λ,


2659

Remark 28. We shall adopt the convention that LF(1) = LZ whereas LF(r), for 1 < r < ∞, will be referred to as an interpolated free group factor with finite parameter. Proposition 29. Let Ω be a graph with a single even vertex v and single odd vertex w joined by q > 0 edges. Let μ2 (v) = α and μ2 (w) = β = 1 − α. Then, ⎧ LF(q 2 ) ⎪ ⎪ ⎪ ⎪ ⎨ LF( 2qα − α 2 ) β β2 ∼ M(Ω, 1) = ⎪ C ⊕ LF(2 − ⎪ ⎪ ⎪ ⎩ 1− αq αq β

if if 1 ) q2

if

α β 1 q α β

> q,
1 and 1 α 2 2 ∼ q β q, in which case M(Ω) = LF(1 + 2qαβ − α − β ). Proof. If M(Ω) is a factor, then, so is M(Ω, 1) and it follows from Proposition 29 that

1 q

βα .

β α.

Similarly the factoriality of M(Ω, 0) and Proposition 29 will imply that q1 Thus, q1 α β q. To see that q > 1, it suffices to observe that if q = 1, the already proved inequality shows that α = β and then again by Proposition 29, both M(Ω, 0) and M(Ω, 1) are LF(1) ∼ = LZ so M(Ω) cannot be a factor. For the converse, if q > 1 and q1 βα q, then q = q1 and so at least one of the inequalities among

α β

1 q

and q

α β

must be strict. Hence 2qα α 2 α α − 2= 2q − β β β β 1 .q q = 1,

>

and so M(Γ, 1) is an interpolated free group factor with finite parameter. Similarly, so is M(Γ, 0). By Lemma 19, M(Γ ) is a factor. Now the corner formula for interpolated free group factors – see [2] or [14] – implies that M(Ω) ∼ = LF(1 + 2qαβ − α 2 − β 2 ). 2 ˜ 1). For this, recall that the weighting, say μ, We now wish to analyse M(Λ, 1) = M(Λ, ˜ on Λ˜ 2 2 is given by the normalised restriction of μ. Thus μ˜ (vi ) = ai for 1 i k and μ˜ (w) = b, where k i=1 ai + b = 1 and (α1 : · · · : αk : β) = (a1 : · · · : ak : b).

2660


Proposition 31. With the foregoing notation, ⎧ ⎪ LF( {i: qi b

k

i=1 qi ai ,

k

i=1 qi ai .

b

˜ 1) is an LF(r) for some r 1 iff b In particular, M(Λ,

k

i=1 qi ai .

Proof. By Eq. (24), we have ˜ 1) ∼ M(Λ˜ vi , 1): i = 1, 2, . . . , k ∼ M(Λ, = ∗P0 (Λ,1) = ∗ M(Λ˜ vi , 1): i = 1, 2, . . . , k , ˜ ˜ 1) ∼ where the second isomorphism holds since P0 (Λ, = C. Each M(Λ˜ vi , 1) is determined using Proposition 29. Now computations from [1] – see Proposition 1.7 – and a little calculation finish the proof. 2 Proposition 32. If Λ has a single odd vertex and at least two edges, then M(Λ) ∼ = LF(s) ⊕ A, for some finite s > 1 and a finite-dimensional abelian A. Proof. Notice that Λ˜ satisfies the hypotheses of this proposition and in addition, is connected. ˜ in other words, we may assume In view of Eq. (25), it suffices to prove the proposition for Λ; without loss of generality that Λ is connected. Hence Theorem 21 is applicable and M(Λ) has the form M ⊕ A for some II 1 -factor M and a finite-dimensional abelian A. Now Proposition 31 tells us that some corner of M(Λ, 1) and hence of M(Λ) is an LF(r) for some finite r. On the other hand, the hypothesis that Λ has at least two edges ensures that M(Λ, 1) is not commutative and hence r > 1. This corner is necessarily a corner of M ∼ = LF(s) for some finite s > 1. 2 Corollary 33. Let Λ be any graph with a single odd vertex and non-empty edge set E. Then, M(Λ, 0) ∼ =

LF(s) ⊕ A LZ ⊕ A

if |E| > 1, if |E| = 1

for some s > 1 and finite-dimensional abelian A. Proof. In case |E| > 1, M(Λ, 0) is necessarily non-abelian and the desired assertion is a direct consequence of Proposition 32. When |E| = 1, observe, as in Eq. (25), that M(Λ, 0) = ˜ 0) ⊕ A for some finite-dimensional abelian A. Now the desired result follows from PropoM(Λ, sition 29 applied with Ω being Λ˜ with vertex parity reversed. (This is because the q of Proposition 29 is 1 and the parameter occurring in the LF(·)-factor is 1 in all the three cases considered there.) 2


2661

5. The structure of M(Γ ) In this section, we determine the structure of M(Γ ) for any finite, connected, bipartite graph Γ with Perron–Frobenius weighting. The main technical result used in the proof is Theorem 34 which is a consequence of the results in [4]. Theorem 34. Let M(w), w ∈ V1 be a finite family of tracial von Neumann algebraic probability spaces over a finite-dimensional abelian probability space D. Suppose that each M(w) ∼ = LF(rw ) ⊕ A(w) with 1 rw < ∞ and finite-dimensional abelian A(w) and that M = ∗D {M(w): w ∈ V1 } is a factor. Then, M is either an interpolated free group factor with finite parameter or the hyperfinite (II 1 ) factor. The following theorem is one of the main results of this paper. Theorem 35. Let a connected graph Γ with at least two edges be equipped with the Perron– Frobenius weighting. Then M(Γ ) ∼ = LF(s) for some 1 < s < ∞. Proof. By Corollary 22, M(Γ ) is a II 1 -factor and so, to see that it an interpolated free group factor with finite parameter, it suffices to see that the corner M(Γ, 0) is also one. The hypotheses on Γ ensure that Corollary 33 is applicable to Γw for each odd vertex w. Then it follows from Eq. (23) that the hypotheses of Theorem 34 are satisfied with D = P0 (Γ, 0), M(w) = M(Γw , 0) for w ∈ V1 and M = M(Γ, 0) and so M(Γ, 0) is either an interpolated free group factor with finite parameter or the hyperfinite factor. To conclude the proof, we only need to ensure that M(Γ, 0) is not hyperfinite. For this we consider two cases. Case 1. Suppose some odd vertex w of Γ has degree greater than 1. In this case Corollary 33 shows that LF(rw ) for some rw > 1 is a corner of M(Γw , 0). A corner of a subalgebra of the hyperfinite factor cannot be LF(rw ) (which is not injective). Hence M(Γ, 0) is not hyperfinite. Case 2. Every odd vertex of Γ has degree 1. Thus Γ is the complete bipartite graph K(1, n) for n 2. The Perron–Frobenius weighting on this graph assigns √1 to the odd vertex and 2

√1 2n

to each even vertex. Now, for any odd vertex w, Proposition 29 applied with Ω be√ ing Γw with reversed vertex parity implies that M(Γw , 0) ∼ = C ⊕ LZ, where δ = n. Clearly, P0 (Γ, 0) ∼ = C. Therefore, from Eq. (23),

1−δ −1

δ −1

√ M(Γ, 0) ∼ = ( C ⊕ LZ)∗n ∼ = LF(2 n − 1), −1 1−δ −1

δ

where the second isomorphism is proved in Corollary 16 of [11]. Since n 2, M(Γ, 0) is an interpolated free group factor with finite parameter in this case too. 2 The only connected graph to which Theorem 35 does not apply is the graph A2 which has two vertices joined by a single edge. For completeness, we determine, in the following proposition, the structure of M(Γ ) in this case.

2662


Proposition 36. Let Γ be the A2 graph with a single even vertex v and a single odd vertex w joined by a single edge. Equip Γ with its Perron–Frobenius weighting given by μ2 (v) = 12 = μ2 (w). Then M(Γ ) ∼ = M2 (LZ). Proof. Recall from Section 1 that with Γ being the A2 graph, elements of Gr(Γ ) may be regarded as matrices with rows and columns indexed by the set {v, w} and (p, q) entry (with p, q ∈ {v, w}) being a linear combination of paths from p to q. We shall write Mij for ei M(Γ )ej for 0 i, j 1, where of course e0 = ev (resp. e1 = ew ) denotes the projection onto the subspace H0 (resp. H1 ) of H generated by the set of all paths starting at v (resp. w). Let ξn (resp. ηn ) be the unique path of length n which starts at v (resp. w). Then, H = H0 ⊕ H1 , and also (see Eq. (7)) {{ξn }: n 0} (resp. {{ηn }: n 0}) is an orthonormal basis for H0 (resp. H1 ). Let x = λ(ξ1 ) ∈ M01 . The definition of multiplication in F (Γ ) shows that x{ξn } = 0 and x{ηn } = {ηn+1 } + {ηn−1 } for all n 1 (with {η−1 } = 0). So, if w : H0 → H1 is the (obviously unitary) operator defined by w({ξn }) = {ηn }, we see that x = ws where s ∈ L(H0 ) is the (standard semi-circular) operator given by s{ξn } = {ξn+1 } + {ξn−1 }. It follows that x is injective (since the Wigner distribution has no atoms). It follows that if x = u|x| denotes the polar decomposition of x, then u∗ u = e1 , and similarly one sees that uu∗ = e0 . Now if y ∈ M01 is arbitrary, then y = e0 ye1 and we see that yu∗ = (e0 ye1 )(e0 ue1 )∗ = e0 ye1 u∗ e0 ∈ M00 , and hence y = ye1 = yu∗ u ∈ M00 u. Arguing similarly, we see that the maps a → au,

a → u∗ a,

and a → u∗ au

define linear isomorphisms of M00 onto M01 , M10 and M11 respectively. Finally, it is easy to see that the assignment ' & ' & a a01 u a00 a01

→ ∗ 00 a10 a11 u a10 u∗ a11 u defines an isomorphism of M2 (M00 ) onto M(Γ ). Since M00 ∼ = LZ by Proposition 29, the proof is complete. 2 6. Application to the GJS construction In this section we relate our Gr(Γ ) to the sequence of algebras Grk (P ) of [5]. Let P be a subfactor planar algebra with finite principal graph Γ = (V , E), distinguished vertex ∗ and modulus δ > 1. Thus δ is the Perron–Frobenius of Γ and we let μ2 (·) be the Perron– eigenvalue 2 Frobenius eigenvector normalised so that v∈V μ (v) = 1. Let tr be the normalised picture trace on Pn . Most of the following facts about the tower of algebras (P0+ =)P0 ⊆ P1 ⊆ P2 ⊆ · · · can all be found in [8]. For vertices v, w of Γ , we write Pn (v, w) for the set of paths of length n in Γ beginning at v and ending at w. Similarly we use notation such as Pn (v, ·) for the set of paths of length n in Γ beginning at v. Pn has a basis given by pairs of paths (ξ(+), ξ(−)) in Γ such that ξ(±) ∈ Pn (∗, v) for some vertex v ∈ V . The set Pmin (Z(Pn )) of minimal central projections of Pn is in natural bijection


2663

with {v ∈ V : v = f (ξ ) for some ξ ∈ Pn (∗, ·)}. For such a v, denote the corresponding minimal central projection in Pn by e(v, n) and any minimal projection under e(v, n) by p(v, n). Then, {(ξ(+), ξ(−)): ξ(±) ∈ Pn (∗, v)} are matrix units (meaning (ξ(+), ξ(−))(η(+), η(−)) = ξ(−) δη(+) (ξ(+), η(−)) and (ξ(+), ξ(−))∗ = (ξ(−), ξ(+))) for the matrix algebra e(v, n)Pn . Further, with tr(·) denoting the normalised picture trace on the planar algebra P , we have tr(p(v, n)) = 2 (v) . δ −n μ μ2 (∗) The inclusion of Pn into Pn+1 is given by ρ(+)n+1 ξ(+) ξ(−) ξ(+), ξ(−) → ρ(+), ρ(−) δρ(+)[0,n] δρ(−)[0,n] δρ(−)n+1 v∈V ρ(±)∈Pn+1 (∗,v)

=

ξ(+) ◦ λ, ξ(−) ◦ λ .

(26)

λ∈P1 (f (ξ(±)),·)

The τ -preserving conditional expectation Pn+1 → Pn is given by ξ(±)

2 ξ(+)n+1 μ (vn+1 ) ξ(+), ξ(−) → δξ(−)n+1 . , ξ(−) ξ(+) [0,n] [0,n] ξ(±) δμ2 (vn )

(27)

The Jones projection en ∈ Pn for n 2 is given by

v∈V ξ(±)∈Pn (∗,v)

ξ(+)

ξ(+)

ξ(+)

ξ(−)

ξ(+)n

ξ(−)n

[0,n−2] δξ(−)[0,n−2] δ n−1 δ n−1

ξ(−)

μ(vn−1 )μ(vn−1 ) ξ(+), ξ(−) . ξ(±) δμ2 (vn )

(28)

In these formulae we have written δji for the Kronecker delta. Our main observation is that the construction in [5] of Grk (P ) (after conjugating by suitable powers of the rotation tangle) depends only on the actions of the inclusion, multiplication and right conditional expectation tangles – and not on that of the left conditional expectation tangles. Hence, in principle, ‘Grk (P ) depends only on the graphs and not on the connection’. We first need to note that the action of the category epi-TL or E of Section 1 on Gr(Γ, ∗) is ‘essentially the same’ as that of certain annular tangles on Gr0 (P ). Consider the full subcategory of E consisting only of the objects [0], [2], [4], . . . . We will denote this category by Eev . Any morphism in Eev , say an element of Hom([2n], [2m]), naturally yields an annular tangle with an internal n-box and an external m-box as in the example in Fig. 5 for n = 4, m = 1. This identification of Hom([2n], [2m]), composed with the action of annular tangles on planar algebras is seen to yield an Eev action on {Pn }n0 . (The tangles in the image of Eev are what were called 0-good annular tangles in [10].) We will find it convenient to identify a basis element (ξ(+), ξ(−)) of Gr0 (P ) with the loop based at the ∗ vertex. Equivalently, the loop ξ based at ∗ is identified with ξ = ξ(−) ◦ ξ(+) (ξ[n,2n] , ξ[0,n] ). Proposition 37. The maps {θn : P2n (Γ, ∗) → Pn }n0 defined by μ(v0η ) θn [η] = η η μ(vn ) for [η] ∈ P2n (Γ, ∗), are Eev -equivariant.

2664


Fig. 5. Correspondence between Eev and good Temperley–Lieb tangles.

Fig. 6. The annular tangle Si2n .

Proof. It clearly suffices to verify the intertwining assertion on generators Si2n . Hence we need to check that for [ξ ] ∈ P2n (Γ, ∗) and 1 i < 2n the equality θn−1 Si2n [η] = ZS 2n θn [η] i

holds. There are three cases according as i < n, i = n or i > n. We will do the first case. The third is similar and the second is easier. When i < n the annular tangle Si2n is shown in Fig. 6. The dotted lines are meant to indicate a decomposition as the (right) conditional expectation tangle applied to the result of post-multiplication with a Temperley–Lieb tangle. The Temperley–Lieb tangle in question here is seen to be the product Ei+1 Ei+1 · · · En , where Et = δet . It now follows by induction on n − i using Eqs. (28) and (26) and the multiplication in Pn that Ei+1 Ei+1 · · · En is given by

v∈V ξ(±)∈Pn (∗,v)

ξ(−)

ξ(−)[0,i−1] ξ(−)[i−1,n−2] ξ(−)n−1 ξ(+)i δξ(+)[0,i−1] δξ(+)[i+1,n] δ δ ξ(−) ξ(+) n

ξ(+)

μ(vn−1 )μ(vi

) ξ(+), ξ(−) .

ξ(−) ξ(+) i+1 μ(v )μ(vi+1 ) n


2665

It follows that ZS 2n (η) is given by δ times the conditional expectation onto Pn−1 of the product i

η(+), η(−) ×

ξ(−)

v∈V ξ(±)∈Pn (∗,v)

=

ξ(−)

ξ(−)

[0,i−1] [i−1,n−2] δξ(+)[0,i−1] δξ(+)[i+1,n] δ n−1 δ

ξ(−)n

η(−)

δ

η(−)

λ∈P1 (vn

η(−)i η(−) i+1

,·)

μ(f (λ))μ(vi

ξ(−) ξ(+) ) ξ(+)i μ(vn−1 )μ(vi ξ(+), ξ(−) ξ(−) ξ(+) ξ(+) i+1 μ(v )μ(vi+1 ) n

) λ . η(+), η(−)[0,i−1] ◦ η(−)[i+1,n] ◦ λ ◦

η(−) η(−) μ(vn )μ(vi+1 )

Now use Eq. (27) to conclude that ZS 2n (η) is i

η(−)

λ∈P1 (vn

η(−)

δ

η(−)i η(−) i+1

,·)

μ(f (λ))μ(vi η(−)

μ(vn

)

η(−)

)μ(vi+1 )

λ δη(+) n

η(−)

μ2 (vn ) μ2 (f (λ))

× η(+)[0,n−1] , η(−)[0,i−1] ◦ η(−)[i+1,n] ◦ λ

η(−) η(−) )μ(vi ) η(−)i μ(vn η(+)[0,n−1] , η(−)[0,i−1] η(+) η(−) η(−) i+1 μ(v n−1 )μ(vi+1 ) η η μ(vn )μ(vi ) ηi = δη

η[0,i−1] ◦ η[i+1,2n] . η η i+1 μ(vn+1 )μ(vi+1 )

=δ

n ◦ η(−)[i+1,n] ◦ η(+)

Hence, η η μ(v0η ) μ(v0 )μ(vi ) ηi ZS 2n θn [η] = (η) = δ Z η[0,i−1] ◦ η[i+1,2n] . 2n η η η η

i+1 i μ(vn ) Si μ(vn+1 )μ(vi+1 )

On the other hand, we have by definition, η μ(vi ) ηi Si2n [η] = δη

[η[0,i−1] ◦ η[i+1,2n] ], η i+1 μ(vi+1 )

and thus ηi θn−1 Si2n [η] = δη

i+1 as was to be seen.

η

η

μ(v0 )μ(vi )

η[0,i−1] η η μ(vn+1 )μ(vi+1 )

◦ η[i+1,2n] = ZS 2n θn [η] , i

2

Next, we generalise the path basis expression for the Jones projections to arbitrary Temperley– Lieb tangles. Let T be a Temperley–Lieb equivalence relation on {1, 2, . . . , 2n} also identified with a Temperley–Lieb tangle as in the following example. Say T = {{1, 2}, {3, 8}, {4, 7}, {5, 6}}. The corresponding tangle is shown in Fig. 7. Given such a Temperley–Lieb equivalence relation T we let Tt be the subset of ‘through classes’, Tu be the subset of ‘up classes’ and Td be

2666


Fig. 7. The Temperley–Lieb tangle T = {{1, 2}, {3, 8}, {4, 7}, {5, 6}}.

the set of ‘down classes’ of T , so that T = Tt Tu Td . In this example, Tt = {{3, 8}, {4, 7}}, Tu = {{1, 2}} and Td = {{5, 6}}. Proposition 38. For any Temperley–Lieb equivalence relation T on {1, 2, . . . , 2n}, the element ZT (1) ∈ Pn is given by

v∈V ξ(±)∈Pn (∗,v)

ξ(−)

{i,j }∈Tt : i<j

×

δ

{i,j }∈Td : i>j

i δξ(+)2n+1−j

ξ(+)2n+1−i ξ(+) 2n+1−j

{i,j }∈Tu : i<j

ξ(−) ξ(−)j

δ i

ξ(−)

)

ξ(−)

)

μ(vi

μ(vj

ξ(+) μ(v2n+1−i ) ξ(+), ξ(−) . ξ(+) μ(v2n+1−j )

For instance, for the Temperley–Lieb relation T of Fig. 7, ξ(+) ξ(−)3 ξ(−)4 ξ(−)1 μ(v1ξ(−) ) ) ξ(+)3 μ(v3 δ ξ(+), ξ(−) . ZT (1) = δξ(+)1 δξ(+)2 δ ξ(−)2 μ(v ξ(−) ) ξ(+)4 μ(v ξ(+) ) ξ(±)∈P4 (∗,·) 2 4

Proof of Proposition 38. Suppose that ZT (1) = v∈V ξ(±)∈Pn (∗,v) cξ (ξ(+), ξ(−)). Since the (ξ(+), ξ(−)) are orthogonal (for the inner-product on Pn given by x, y = tr(y ∗ x)) with 2

ξ(±)

) n (ξ(+), ξ(−))2 = δ −n μ μ(v2 (∗) ,

cη δ −n

η(±) ) μ2 (vn ) ( = η(+), η(−) , ZT (1) μ2 (∗) = tr ZT (1)∗ η(+), η(−)

= δ −n × picture trace of ZT (1)∗ η(+), η(−) = δ −n × picture trace of ZT ∗ (1) η(+), η(−) .

Hence, cη =

μ2 (∗) ξ(±) μ2 (vn )

× picture trace of ZT ∗ (1) η(+), η(−) .

(29)

We next compute the picture trace of ZT ∗ (1)(η(+), η(−)). The equivalence relation T ∗ is the one obtained from T be replacing each i by 2n + 1 − i. Regarding T as an element of


2667

Hom([2n], [0]), there is an associated 0+ -annular tangle, which also we will denote by T . The context and the nature of its arguments should make it clear whether we are referring to the morphism T or the associated Temperley–Lieb tangle T or the associated annular tangle T . Some doodling now shows that picture trace of ZT ∗ (1) η(+), η(−) = ZT η(+), η(−) = ZT (η),

(30)

where, clearly, T ∗ is regarded as a Temperley–Lieb tangle and T as an annular tangle. Finally, notice that Proposition 37 says that T ∈ Hom [2n], [2m]

⇒

θ m ◦ T = T ◦ θn .

When m = 0, since θ0 = idC , we see that T = T ◦ θn . i.e., η μ(v0 ) T [η] = T η η μ(vn ) η

=

μ(v0 ) η

μ(vn )

(31)

ZT (η).

However, by Proposition 3, we have μ(vnη ) T [η] = η μ(v0 ) ×

η

η

{i,j }∈T : i<j n

δηji

μ(vi )

η

η

μ(vj )

{i,j }∈T : in<j

η ηi μ(vi ) . δηj η μ(vj ) {i,j }∈T : n 1, and M0 ⊆ M1 be the subfactor constructed from P by the construction in [5]. Then M1 ∼ = LF(s) for some 1 < s < ∞. Since the proof is very similar to that of Theorem 41, we will only sketch the proof giving details where it differs from the previous proof. We first recall some preliminaries from [9]. There is an ‘operation on tangles’ denoted T → T − which moves the ∗-region of each of its boxes anti-clockwise by 1 and reverses the shading. There is an associated ‘operation on planar algebras’ denoted P → − P where − P is the planar algebra with spaces −

P0± = P0∓ ,

− −

Pk = Pk ,

k > 0,

and tangle action defined by ZT P = ZTP − . If P is a subfactor planar algebra, then so is Q = − P and further − Q is isomorphic to P . Now, given a subfactor planar algebra P , we define a graded, non-commutative probability space − Gr1 (P ) associated to P as follows. As a vector space − Gr1 (P ) = n1 Pn . The multiplication map • : Pm ⊗ Pn → Pm+n−1 is defined by the tangle in Fig. 10 below. The adjunction map in − Gr1 (P ) restricts to the adjunction maps in Pn for each n 1. A trace is defined in − Gr (P ) by letting τ (ξ ) for ξ ∈ P ⊆ − Gr (P ) be the sum over all Temperley–Lieb tangles T 1 n 1 of the scalar defined by Fig. 11.


2671

Fig. 10. Multiplication in − Gr1 (P ).

Fig. 11. T -component of the trace in − Gr1 (P ).

The structure maps of − Gr1 (P ) are all derived from those of Gr1 (P ) (see [5]) using the operation− . Observe that the vector space underlying both − Gr1 (P ) and Gr1 (− P ) is the same, namely, n1 Pn . A little thought now yields the following. Lemma 43. For any subfactor planar algebra P , the tracial ∗-probability spaces − Gr1 (P ) and Gr1 (− P ) are isomorphic by the identity map of the underlying vector spaces. Applying Lemma 43 with Q = − P in place of P and using that − Q ∼ = P shows that ∼ (Q) Gr (P ) as probability spaces. We now proceed towards an analogue of Proposi= 1 1 tion 40 for − Gr1 (Q). Let Γ denote the principal graph of Q. Since P is of finite depth, so is Q, and thus Γ is a finite graph. Equip Γ with its Perron–Frobenius weighting. A basis of Qn is then given by pairs of paths (ξ(+), ξ(−)) in Γ such that ξ(±) are paths of length n in Γ beginning at its ∗ and having the same end-point. Again, we identify the basis element (ξ(+), ξ(−)) with the based at ∗. loop ξ(−) ◦ ξ(+) op Observe that the 0th-graded piece of − Gr1 (Q) can be identified as an algebra with Q1 . In particular, each vertex v in Γ at distance 1 from its ∗ vertex gives a minimal central projection op op f (v, 1) in [− Gr1 (Q)]1 = Q1 and we denote by q(v, 1) any minimal projection of Q1 lying op under f (v, 1). A choice of q(v, 1) is the matrix unit (ν, ν) ∈ Q1 where ν is any path of length 1 in Γ from ∗ to v. We fix this choice. The following proposition expresses the multiplication of − Gr1 (Q) in terms of its path basis.

− Gr

Proposition 44. For paths ξ ∈ Qm ⊆ − Gr1 (Q) and η ∈ Pn ⊆ − Gr1 (Q), ξ

ξ

ξ • η = δη2m 1

η

μ(vm )μ(vn ) ξ ◦η

η

μ(vm+n−1 )μ(v1 )

ξ[0,2m−1] ◦ η[1,2n] .

2672


Fig. 12. Multiplication of − Gr1 (Q) in terms of standard tangles.

Sketch of proof. Suppose that m n. The key fact is that the tangle of Fig. 10 is expressible in terms of the inclusion, Temperley–Lieb and multiplication tangles as in Fig. 12. We omit the rest of the proof which is very similar to that of Proposition 39. 2 It follows from Proposition 44 that a basis of q(v, 1)(− Gr1 (Q))q(v, 1) is given by all paths of the form ν ◦ ξ ◦ ν where ξ ranges over all paths in Γ from v to v. This is suggestive of the following key isomorphism which is the analogue of Proposition 40 and whose proof is similar (and omitted). Proposition 45. The map θ : Gr(Γ , v) → q(v, 1)(− Gr1 (Q))q(v, 1) defined for [ξ ] ∈ P2n (Γ , v) by μ(v0ξ ) θ [ξ ] = ν ◦ ξ ◦ ν ∈ Qn+1 ξ μ(vn ) and extended linearly is an isomorphism of graded, ∗-probability spaces. We conclude with the proof of Theorem 42. Proof of Theorem 42. From Proposition 45 and the isomorphism of − Gr1 (Q) with Gr1 (P ), it follows by completing that some corner of M1 is isomorphic to M(Γ , v) – a corner of M(Γ ). Since M(Γ ) is an interpolated free group factor with finite parameter by Theorem 35, the proof is complete. 2 Acknowledgment We are deeply indebted to Ken Dykema for his constant guidance throughout the preparation of this manuscript, and even more for working overtime to identify and prove the statement needed in Section 5.


2673

References [1] K.J. Dykema, Free products of hyperfinite von Neumann algebras and free dimension, Duke Math. J. 69 (1993) 97–119. [2] K.J. Dykema, Interpolated free group factors, Pacific J. Math. 163 (1994) 123–135. [3] K.J. Dykema, Amalgamated free products of multi-matrix algebras and a construction of subfactors of a free group factor, Amer. J. Math. 117 (6) (1995) 1555–1602. [4] K.J. Dykema, A description of amalgamated free products of finite von Neumann algebras over finite dimensional subalgebras, Preprint, arXiv:0911.2052v1. [5] A. Guionnet, V.F.R. Jones, D. Shlyakhtenko, Random matrices, free probability, planar algebras and subfactors, Preprint, arXiv:0712.2904v2. [6] A. Guionnet, V.F.R. Jones, D. Shlyakhtenko, A semi-finite algebra associated to a planar algebra, Preprint, arXiv: 0911.4728v1. [7] V.F.R. Jones, D. Shlyakhtenko, K. Walker, An orthogonal approach to the subfactor of a planar algebra, Pacific J. Math. 246 (1) (2010) 187–197. [8] V.F.R. Jones, V.S. Sunder, Introduction to Subfactors, Cambridge Univ. Press, 1997. [9] Vijay Kodiyalam, V.S. Sunder, On Jones’ planar algebras, J. Knot Theory Ramifications 13 (2) (2004) 219–248. [10] Vijay Kodiyalam, V.S. Sunder, From subfactor planar algebras to subfactors, Internat. J. Math. 20 (10) (2009) 1207–1231. [11] Vijay Kodiyalam, V.S. Sunder, Guionnet–Jones–Shlyakhtenko subfactors associated to finite-dimensional Kac algebras, J. Funct. Anal. 257 (2009) 3930–3948. [12] A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability, London Math. Soc. Lecture Note Ser., vol. 335, Cambridge Univ. Press, 2006. [13] S. Popa, D. Shlyakhtenko, Universal properties of LF(∞) in subfactor theory, Acta Math. 191 (2) (2003) 225–257. [14] F. Radulescu, Random matrices, amalgamated free products and subfactors in free group factors of noninteger index, Invent. Math. 115 (1994) 347–389. [15] R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998).


Non-convex self-dual Lagrangians: New variational principles of symmetric boundary value problems Abbas Moameni a,b a Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran b Department of Mathematics and Statistics, Queen’s University, Kingston, ON, K7L 3N6 Canada

Received 31 March 2010; accepted 15 January 2011 Available online 28 January 2011 Communicated by L.C. Evans

Abstract We study the concept and the calculus of Non-convex self-dual (Nc-SD) Lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems. They indeed provide new representations and formulations for the superposition of convex functions and symmetric operators. They yield new variational resolutions for large class of Hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of Euler– Lagrange functions. These potential functions are quite flexible, and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems. Additionally, in most cases the solutions generated using this new method have greater regularity than the solutions obtained using the standard Euler–Lagrange function. Perhaps most remarkable, however, are the permanence properties of Nc-SD Lagrangians; their calculus is relatively manageable, and their applications are quite broad. © 2011 Elsevier Inc. All rights reserved. Keywords: Non-convex duality; Variational principles; Variational methods; Partial differential equations

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2675 1.1. Homogeneous boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2678


A. Moameni / Journal of Functional Analysis 260 (2011) 2674–2715

1.2. Nonlinear boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Separating duality and convex analysis . . . . . . . . . . . . . . . . . . . 2.2. Saddle functions on phase spaces . . . . . . . . . . . . . . . . . . . . . . . 2.3. Linear self-adjoint operators modulo boundary operators . . . . . . . 3. Non-convex self-dual Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Unbounded operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Symmetric operators modulo boundary operators . . . . . . . . . . . . 3.3. Characterization of non-convex self-dual Lagrangians . . . . . . . . . 4. Applications to calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Homogeneous boundary conditions . . . . . . . . . . . . . . . . . . . . . . 4.2. Nonlinear boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 5. Critical points of lower semi-continuous functionals . . . . . . . . . . . . . . . 5.1. System of transport equations . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Hamiltonian systems of PDE’s with Neumann boundary conditions Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.

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2679 2681 2681 2683 2684 2687 2690 2691 2694 2696 2696 2700 2705 2708 2711 2713 2714

1. Introduction The aim of this paper is to develop the concept of self-duality for non-convex functions well adapted to the study of certain partial differential equations that the standard Euler–Lagrange functions may not be quite manageable. Starting with an equation of the form Λu ∈ ∂ϕ(u),

(1)

it is well known that it can be formulated – and sometimes solved – whenever Λ : Dom(Λ) ⊂ V → V ∗ is a linear self-adjoint operator. Indeed, in this case it can be reduced to the inclusion 0 ∈ ∂F (u) where F is the standard Euler–Lagrange functional corresponding to this inclusion, i.e., 1 F (u) = Λu, u − ϕ(u). 2 In the case where ϕ is convex and the linear operator Λ is positive, the functional F can be written as difference of two convex functions, F (u) = ψ(Λu) − ϕ(u) where the quadratic convex function ψ on V ∗ is defined by ψ(p) = 12 Λ−1 p, p. Such problems where the objective is the difference of two convex functions has received a lot of attentions in the literature starting the works of J.F. Toland [29] and I. Singer [26]. Indeed, Toland introduced the notion of critical points of ψ ◦ Λ − ϕ that generalizes the classical definition in the case where ψ ◦ Λ and ϕ are not necessarily differentiable functions. He also established an interesting oneto-one correspondence between the critical points of ψ ◦ Λ − ϕ and ϕ ∗ ◦ Λ − ψ ∗ on V ∗ , where ϕ ∗ and ψ ∗ are Fenchel–Legendre dual of ϕ and ψ respectively. There was also established by F. Clarke and I. Ekeland [9,8,11] an interesting dual variational formulation for the case where

2676


the operator Λ is not necessarily positive and may have an infinite sequence of eigenvalues going from −∞ to ∞. In fact, similar to Toland duality they established a one-to-one correspondence between critical points of the functional F and the functional 1 FCE (u) = Λu, u − ϕ ∗ (Λu). 2 Note that even though the Clarke–Ekeland least action principle may have a somewhat related idea with Toland duality, it cannot be deduced directly from Toland’s dual principle. As seen, one can associate to an inclusion of the form (1) at least two functionals in such a way that their critical points may generate solutions for the corresponding inclusion. The question of whether these functions are all possible choices associated to a given inclusion (1) and also finding a unified source for all functions with such a property are addressed in this work. In fact, while analyzing these principles we were led to an abstract scheme that provides a unified way to obtain many more of such functions. To explain this scheme, let us start with Toland’s principle [28– 30], with a minor modification, for a class of optimization problems. Indeed, let V and V ∗ be two Banach spaces in duality and with ., . : V × V ∗ → R the corresponding bilinear form compatible with the topologies on V and V ∗ . Denote by (P ) the problem of evaluating inf J (u),

u∈V

(P )

where J : V → R is possibly a non-convex function. By embedding this problem in a family of perturbed problems a dual problem was established. In fact, by considering the perturbation Φ : V × V ∗ → R for which p → Φ(u, p) is convex and lower semi-continuous for each u ∈ V and Φ(u, 0) = −J (u), one can generate a dual problem as follows: Let LF 2 (Φ) be the Fenchel–Legendre dual of Φ with respect to the second variable, that is a function on V × V given by: LF 2 (Φ)(u, v) = sup p, v − Φ(u, p) . p∈V ∗

Denote by Φ # the Fenchel–Legendre dual of LF 2 (Φ)(., v) with respect to the first variable. Therefore Φ # is a function on phase space V × V ∗ given by Φ # (v, q) = sup q, u − LF 2 (Φ)(u, v) . u∈V

It was established that the problem inf Φ # (v, 0)

v∈V

(P ∗ )

is a dual problem for (P ) in such a way that infu∈V J (u) = infv∈V Φ # (v, 0) provided p → Φ(u, p) is bounded in a neighborhood of p = 0. There is also a one-to-one relation between minimizers of (P ) and (P ∗ ). Following this idea of obtaining a dual problem, we are led to the following notion.


2677

Definition 1.1. Say that the Lagrangian Φ on V × V ∗ is Non-convex self-dual if the following property hold. Φ # (u, p) = Φ(u, p) for all (u, p) ∈ V × V ∗ . Some basic examples of Nc-SD Lagrangians are of the form: (1) Φ1 (u, p) = ϕ ∗ (p) − ϕ(u), (2) Φ2 (u, p) = 2ϕ ∗ (p) − p, u, (3) Φ3 (u, p) = p, u − 2ϕ(u), where ϕ : V → R is convex and lower semi-continuous and ϕ ∗ its Fenchel–Legendre dual defined on V ∗ . The class of Non-convex self-dual Lagrangians is much richer though and goes well beyond saddle functions stated above, since they are naturally compatible with symmetric operators. Indeed, if Λ : V → V ∗ is self-adjoint and Φ is any Nc-SD Lagrangian on V × V ∗ then the Lagrangian Ψ (u, p) = Φ(u, Λu + p) is also Non-convex self-dual. There are also situations where the operator Λ is not purely selfadjoint provided one takes into account certain boundary terms. In fact, the operator Λ modulo the boundary operator B := (β1 , β2 ) : V → Y × Y ∗ (for some Banach spaces Y and Y ∗ that are in duality) corresponds to the “Green formula” Λu, vV ×V ∗ = u, ΛvV ×V ∗ + β1 u, β2 vY ×Y ∗ − β1 v, β2 uY ×Y ∗ . In this case if Φ : V × V ∗ → R and : Y × Y ∗ → R are Non-convex self-dual Lagrangians then the Lagrangian Ψ : V × (V ∗ × Y ∗ ) → R defined by Ψ u, (p, e) = Φ(u, Λu + p) + (β1 u, β2 u + e) is also a Non-convex self-dual Lagrangian. To connect this notion to the solutions of inclusion (1), note that u is a solution of inclusion (1) if and only if the pair (Λu, u) is a solution of one the following inclusions on the phase space V × V ∗: (1) (−Λu, u) ∈ (−∂ϕ(u), ∂ϕ ∗ (Λu)), (2) (−Λu, u) ∈ (−Λu, ∂ϕ ∗ (Λu)), (3) (−Λu, u) ∈ (−∂ϕ(u), u). Now taking into account Non-convex self-dual Lagrangians Φ1 , Φ2 and Φ3 the above inclusions can be rewritten as follows, (1) (−Λu, u) ∈ ∂Φ1 (u, Λu), (2) (−Λu, u) ∈ ∂Φ2 (u, Λu), (3) (−Λu, u) ∈ ∂Φ3 (u, Λu),

2678


respectively. Note that Φi , i = 1, 2, 3 are saddle functions and ∂Φi stands for subdifferential of saddle functions introduced by Rockafellar. As it turns out there is a close correspondence between solutions of inclusions of type (1) and critical points of Non-convex self-dual Lagrangians generated by the pair (Λ, ϕ). We shall state and summarize some particular cases of our main results in two cases: Homogeneous boundary conditions and nonlinear boundary conditions. 1.1. Homogeneous boundary conditions In this case we assume the linear operator Λ is purely self-adjoint. Here is a useful result of the variational principle we establish for homogeneous boundary conditions in Section 4. Theorem 1.2. Suppose Φ : V ×V ∗ → R∪{∞} is a saddle Nc-SD Lagrangian and Λ : Dom(Λ) ⊂ V → V ∗ is a self-adjoint linear operator that is also onto. Suppose one of the following conditions holds: (i) The operator Λ is non-negative. (ii) For each p ∈ V ∗ , the function u → Φ(u, p) is Gâteaux differentiable and ∇1 Φ(u, p) = −p. Then for every critical point u of Φ(u, Λu) there exists v ∈ V with Λu = Λv and (−Λv, v) ∈ ∂Φ(u, Λu). As a straightforward application of the above theorem the functionals Φ1 (u, Λu) := ϕ ∗ (Λu) − ϕ(u) and Φ2 (u, Λu) := 2ϕ ∗ (Λu) − Λu, u can be seen as new potentials for the inclusion (1) as follows. Corollary 1.3. Let Λ : Dom(Λ) ⊂ V → V ∗ be a non-negative self-adjoint operator that is also onto. Let ϕ : V → R be convex, lower-semi continuous and also continuous. Then every critical point of I (u) = ϕ ∗ (Λu) − ϕ(u) is a solution of the equation Λu ∈ ∂ϕ(u). This corollary was first established in [24] by the author via a direct computation (see also [22, 21] for more applications). It was then understood that all variational principles of this type fall under a unified principle as discussed in this paper. We shall use this corollary to provide an existence result for system of super-linear transport equations with a small parameter , ⎧ p−2 ⎪ v, x ∈ Ω, ⎨ a.∇u = v + |v| q−2 − a.∇v = u + |u| , x ∈ Ω, ⎪ ⎩ u = v = 0, x ∈ ∂Ω, by finding critical points of


1 I (u, v) = p

1 | a.∇u − v| dx + q p

Ω

1 | a.∇v + u| dx − p q

Ω

1 |v| dx − q

2679

|u|q dx

p

Ω

Ω

p q and q = q−1 . on W 2,q (Ω) × W 2,p (Ω) where p = p−1 Taking into account the Lagrangian Φ2 , here is another application of Theorem 1.2.

Corollary 1.4. Let Λ : Dom(Λ) ⊂ V → V ∗ be a surjective self-adjoint operator and ϕ : V → R be convex and lower-semi continuous. If u is a critical point of I (w) = 2ϕ ∗ (Λw) − Λw, w then there exists v ∈ V such that

v+u 2

is a solution of Λw ∈ ∂ϕ(w).

Note that the above corollary is nothing but the well-known Clarke–Ekeland least action principle. It is also remarkable that Φ1 and Φ2 are just two typical examples of Nc-SD Lagrangians that have already provided two different variational principles for the inclusion (1). By characterizing the class of Nc-SD Lagrangians in Section 3, we shall see one can actually obtain many more principles that fit within this theory. 1.2. Nonlinear boundary conditions Here is another useful result of the variational principle we establish for nonlinear boundary conditions in Section 4. Theorem 1.5. Let Λ : Dom(Λ) ⊂ V → V ∗ be an operator correspond to the above “Green formula” modulo the boundary operator B := (β1 , β2 ) : Dom(Λ) → Y × Y ∗ such that (Λ, β2 ) : Dom(Λ) ⊂ V → V ∗ × Y ∗ and β1 : Dom(Λ) ⊂ V → Y are onto. Let Φ : V × V ∗ → R and : Y × Y ∗ → R be saddle Non-convex self-dual Lagrangians that are Gâteaux differentiable with respect to their first variables. We also assume that Dom(Λ) ∩ Ker(β1 ) is dense in V . Suppose one of the following conditions holds: (i) For each u ∈ Dom(Λ), u, ΛuV ×V ∗ + β1 u, β2 uY ×Y ∗ 0. (ii) For each (p, e) ∈ V ∗ × Y ∗ the functions u → Φ(u, p) and l → (l, e) are Gâteaux differentiable and, ∇1 Φ(u, p) = −p and ∇1 (l, e) = −e. Suppose u is a critical point of I (u) = Ψ (u, 0) where Ψ is defined on Dom(Λ) × (V ∗ × Y ∗ ) by Ψ u, (p, e) = Φ(u, Λu + p) + (β1 u, β2 u + e). Set v ∈ ∂2 Φ(u, Λu). Then Λu = Λv, β2 u = β2 v and the pair (u, v) is a solution of the system

(−Λv, v) ∈ ∂Φ(u, Λu), (−β2 v, β1 v) ∈ ∂ (β1 u, β2 u).

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To get a better understanding and see concrete applications of this theorem we shall discuss some particular cases. Corollary 1.6. Let Λ : Dom(Λ) ⊂ V → V ∗ and B := (β1 , β2 ) : Dom(Λ) → Y × Y ∗ satisfy part (i) of Theorem 1.5. Let ϕ : V → R and ψ : Y → R be convex, lower-semi continuous and also Gâteaux differentiable. Then every critical point of I (u) = ϕ ∗ (Λu) − ϕ(u) + ψ ∗ (β2 u) − ψ(β1 u) is a solution of the inclusion

Λu = ∇ϕ(u), β2 u = ∇ψ(β1 u).

(2)

The following result can be seen as a generalization of Clarke–Ekeland duality when the operator Λ is not purely self-adjoint and one deals with boundary terms as well. Corollary 1.7. Let Λ : Dom(Λ) ⊂ V → V ∗ and B := (β1 , β2 ) : Dom(Λ) → Y × Y ∗ be as in Theorem 1.5. Let ϕ : V → R and ψ : Y → R be convex, lower-semi continuous and also Gâteaux differentiable. If u is a critical point of I (w) = 2ϕ ∗ (Λw) − Λw, wV ×V ∗ + 2ψ ∗ (β2 w) − β2 w, β1 wY ×Y ∗ then there exists v ∈ V such that

v+u 2

is a solution of (2).

As an application of this corollary we provide a new variational principle for convex Hamiltonian systems with nonlinear boundary conditions of the form: ⎧ ˙ = ∇ϕ t, u(t) , ⎨ J u(t) ⎩ u(T ) + u(0) = ∇ψ J u(T ) − J u(0) . 2 The above results are actually particular cases of a much more general Non-convex self-dual variational principle that will be stated and established in full generality in the following sections. As applications, we shall also provide many more concrete examples of this principle throughout the paper. The interested reader is referred to [10,6,7,5,2,3,25,20] for more applications of the related results to PDE’s and monotone operators. We also refer to [16–18,15] for results in convex selfduality. The paper is organized as follows. We start by reviewing in Section 2, some important definitions and results in Convex Analysis, theory of saddle functions and symmetric linear operators. In Section 3, we start by establishing some basic permanence properties of Non-convex selfdual Lagrangians and their calculus and we conclude this section by a characterization of Nc-SD Lagrangians. In Section 4, we first establish a variational principle for homogeneous boundary value problems then we deal with boundary value problems where compatible boundary Lagrangians are appropriately added to the “interior Lagrangian”, in order to solve problems with prescribed nonlinear boundary terms. In Section 5, by making use of a minimax principle for


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lower semi-continuous functionals we proceed with the proof of existence theorems stated in previous sections. 2. Preliminaries In this section we recall some important definitions and results in Convex Analysis, theory of saddle functions and linear symmetric operators used in this work. We also introduce the terminology used consistently throughout the paper for the convenience of the reader. For the proof of these results the interested reader is referred to [14,13,15,29]. 2.1. Separating duality and convex analysis Let V and V ∗ be two real Banach spaces and let , be a bilinear form on the phase space V × V ∗ . The following definition is due to J.F. Toland [29]. Definition 2.1. We say that the bilinear form puts V and V ∗ in duality. This duality is said to be separating if, (1) for 0 = u ∈ V , there exists an element p ∈ V ∗ such that u, p = 0, (2) for 0 = p ∈ V ∗ , there exists an element u ∈ V such that u, p = 0. The weak topology on V induced by , is denoted by σ (V , V ∗ ) and analogously σ (V ∗ , V ) is the weak topology on V ∗ . It is known that σ (V , V ∗ ) and σ (V ∗ , V ) are Hausdorff topologies if and only if the duality between V and V ∗ is separating. Throughout this paper we shall assume the spaces V and V ∗ are in separating duality. A function Φ : V → R is said to be lower semicontinuous if Φ(u) lim inf Φ(un ), n→∞

for each u ∈ V and any sequence un approaching u in the weak topology σ (V , V ∗ ). Let Φ : V → R ∪ {∞} be a proper convex function. The subdifferential ∂Φ of Φ is defined to be the following set-valued operator: if u ∈ Dom(Φ), set ∂Φ(u) = p ∈ V ∗ ; p, v − u + Φ(u) Φ(v) for all v ∈ V and if u ∈ / Dom(Φ), set ∂Φ(u) = ∅. If Φ is Gâteaux differentiable at u then ∂Φ(u) = {∇Φ(u)}. The Fenchel–Legendre dual of an arbitrary function Φ is denoted by Φ ∗ that is a function on V ∗ and is defined by Φ ∗ (p) = sup p, u − Φ(u); u ∈ V . Clearly Φ ∗ : V ∗ → R ∪ {∞} is convex and lower semi-continuous. Consequently Φ ∗∗ : V → R ∪ {∞} is always convex and lower semi-continuous. The following observation is crucial in the subsequent analysis.

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Proposition 2.1. Let Φ : V → R ∪ {∞} be an arbitrary function. The following statements hold: (1) Φ ∗∗ (u) Φ(u) for all u ∈ V . (2) Φ(u) + Φ ∗ (p) p, u for all u ∈ V and p ∈ V ∗ . (3) If Φ is convex and lower-semi continuous then Φ ∗∗ = Φ and the following are equivalent Φ(u) + Φ ∗ (p) = u, p

⇔

p ∈ ∂Φ(u)

⇔

u ∈ ∂Φ ∗ (p).

The following is a crucial property of convex functions. Proposition 2.2. Let V and V ∗ be in separating duality and Φ : V → R∪{∞} be a proper convex function. Suppose Φ is sub-differentiable at u, v ∈ V . If there exist p ∈ ∂Φ(u) and q ∈ ∂Φ(v) with p − q, u − v = 0

(3)

then p, q ∈ ∂Φ(u) ∩ ∂Φ(v). Proof. It follows from p ∈ ∂Φ(u) and q ∈ ∂Φ(v) that Φ(u) + Φ ∗ (p) = p, u & Φ(v) + Φ ∗ (q) = q, v. Adding up this equalities, we obtain p, u + q, v = Φ(u) + Φ ∗ (p) + Φ(v) + Φ ∗ (q). It also follows from (3) that p, u + q, v = p, v + q, u, which together with the above equation imply that p, v + q, u = Φ(u) + Φ ∗ (p) + Φ(v) + Φ ∗ (q) = Φ(v) + Φ ∗ (p) + Φ(u) + Φ ∗ (q) and therefore Φ(v) + Φ ∗ (p) − p, v + Φ(u) + Φ ∗ (q) − q, u = 0. This together with the fact that Φ(v) + Φ ∗ (p) − p, v 0,

Φ(u) + Φ ∗ (q) − q, u 0

imply that both terms are indeed zero, Φ(v) + Φ ∗ (p) − p, v = 0, Φ(u) + Φ ∗ (q) − q, u = 0, from which we have p ∈ ∂Φ(v) and q ∈ ∂Φ(u).

2

As an important and straightforward consequence of the above proposition we have the following. Theorem 2.2. Let V and V ∗ be in separating duality and Φ : V → R ∪ {∞} be a proper convex function. Suppose Φ is Gâteaux differentiable at u, v ∈ X. Then


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∇Φ(u) − ∇Φ(v), u − v = 0

if and only if ∇Φ(u) = ∇Φ(v). Proof. Since Φ is Gâteaux differentiable at u, v ∈ X, we have ∂Φ(u) = {∇Φ(u)} and ∂Φ(v) = {∇Φ(v)}. Set p = ∇Φ(u) and q = ∇Φ(v). If ∇Φ(u) − ∇Φ(v), u − v = 0, it follows from the above proposition that p, q ∈ ∂Φ(u) ∩ ∂Φ(v). This implies ∇Φ(u) = ∇Φ(v). 2 2.2. Saddle functions on phase spaces Here we summarize some of the results in the theory of saddle functions on the product space X × Y for some Banach spaces X and Y . We start with the definition of saddle functions: Definition 2.3. We call a function H : X × Y → R a saddle function if the following properties hold: (1) H (x, .) is convex and lower semi-continuous for each x ∈ X. (2) H (., y) is concave and upper semi-continuous for each y ∈ Y. Assuming the Banach spaces X and Y are in separating duality with X ∗ and Y ∗ respectively, it is easily seen that the bilinear form on (X × Y ) × (X ∗ × Y ∗ ) defined by

(x, y), (p, q) (X×Y )×(X∗ ×Y ∗ ) = x, pX×X∗ + y, qY ×Y ∗

puts X × Y and X ∗ × Y ∗ in separating duality. For a saddle function H : X × Y → R the notion of subdifferential is introduced by Rockafellar as the multivalued mapping ∂H : X × Y → X ∗ × Y ∗ defined by ∂H (x, y) = (−p, q); p is a subdifferential of the convex function −H (., y) at x and q is a subdifferential of the convex function H (x, .) at y . Thus, denoting the subdifferential with respect to the first variable by ∂1 and subdifferential with respect to the second variable by ∂2 we have ∂H (x, y) = ∂1 (−H (x, y)) × ∂2 H (x, y). For a saddle function H, the function on X ∗ × Y obtained by taking the Fenchel–Legendre dual of −H (., y) when the second argument is fixed, i.e., F (., y) = (−H (., y))∗ or F (p, y) = sup p, xX×X∗ + H (x, y) x∈X

is called the first convex parent of H. The second convex parent of H is a function on X × Y ∗ defined by G(x, .) = (H (x, .))∗ , or G(x, q) = sup q, yY ×Y ∗ − H (x, y) . y∈Y

The following is rather standard.

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Proposition 2.3. If H is a saddle function on X × Y then the following hold: (1) The first convex parent F and the second convex parent G are convex and lower semi continuous with respect to both variable and indeed F ∗ (x, q) = G(x, q)

for all (x, q) ∈ X × Y ∗ ,

G∗ (p, y) = F (p, y)

for all (p, y) ∈ X ∗ × Y,

and

where F ∗ and G∗ are Fenchel–Legendre dual of F and G with respect to both variables. (2) The following are equivalent: (−p, q) ∈ ∂H (x, y)

⇔

(x, q) ∈ ∂F (p, y)

⇔

(p, y) ∈ ∂G(x, q).

2.3. Linear self-adjoint operators modulo boundary operators For the proof of the main theorem regarding nonlinear boundary conditions and also in various applications, we are often faced with an unbounded operator Λ : Dom(Λ) ⊂ V → V ∗ which may still satisfy various aspects of symmetry. For the convenience, we now recall some standard notions on this subject. Definition 2.4. Let V and V ∗ be in separating duality. A linear operator Λ : Dom(Λ) ⊂ V → V ∗ is called symmetric if Dom(Λ) is dense in V and Λu, v = u, Λv for all elements u and v in the domain of Λ. The operator Λ is said to be non-negative if Λu, u 0 for all u ∈ Dom(Λ). We shall also deal with situations where the operator Λ is not purely symmetric provided one takes into account certain boundary terms. In fact, the operator Λ modulo the boundary operator B := (β1 , β2 ) : V → Y × Y ∗ (for some Banach spaces Y and Y ∗ that are in duality) corresponds to the “Green formula” Λu, vV ×V ∗ = u, ΛvV ×V ∗ + β1 u, β2 vY ×Y ∗ − β1 v, β2 uY ×Y ∗ . We introduce the following notion. Definition 2.5. Suppose the spaces V and V ∗ and also Y and Y ∗ are in separating duality. We say that a linear operator Λ : Dom(Λ) ⊂ V → V ∗ is symmetric modulo the linear boundary operator B = (β1 , β2 ) : Dom(Λ) → Y × Y ∗ if the following properties are satisfied: (1) The space V0 = Dom(Λ) ∩ ker(β1 ) is dense in V . (2) The operator β1 : Dom(Λ) ⊂ V → Y has a dense range. (3) For every u, v ∈ Dom(Λ) we have Λu, vV ×V ∗ = u, ΛvV ×V ∗ + β1 u, β2 vY ×Y ∗ − β1 v, β2 uY ×Y ∗ . Our definition of non-negative symmetric operators modulo the boundary operator B := (β1 , β2 ) : V → Y × Y ∗ will change accordingly. Indeed,


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Definition 2.6. We say that an operator Λ is non-negative modulo the boundary operator B = (β1 , β2 ) if the following property is satisfied: • For every u ∈ Dom(Λ) we have Λu, uV ×V ∗ + β1 u, β2 uY ×Y ∗ 0. Example 1. Let Ω be a smooth domain in RN , and ∂Ω its boundary. Note that for p > 1 and p p = p−1 Banach spaces Lp (Ω) and Lp (Ω) are in separating duality with the bilinear form

u, v =

u(x)v(x) dx. Ω

1

Consider the Laplace operator − : Dom(−) ⊂ Lp (Ω) → Lp (Ω). Let H 2 (Ω) be the 1 fractional Sobolev space of order 12 and H − 2 (Ω) its dual. Define the boundary opera1

1

tors β1 : Dom(−) ⊂ Lp (Ω) → H 2 (∂Ω) and β2 : Dom(−) ⊂ Lp (Ω) → H − 2 (Ω) by ∂u β1 u = u|∂Ω and β2 u = ∂u ∂ν where ∂ν is the normal derivative on ∂Ω. We shall show that − : Dom(−) ⊂ Lp (Ω) → Lp (Ω) is symmetric and non-negative modulo the boundary 1 1 operator B := (β1 , β2 ) : Dom(−) ⊂ Lp (Ω) → H 2 (Ω) × H − 2 (Ω). Indeed, condition (1) of Definition 2.5 holds due to the density of Cc∞ (Ω) (smooth compact supported functions) in 1 Lp (Ω). To verify condition (2), note first that Dom(−) ⊂ H 1 (Ω) and β1 : H 1 (Ω) → H 2 (Ω) is surjective by Theorem 8.3 in [23, Chapter 1]. Condition (3) of Definition 2.5 is nothing but the integration by parts in H 1 (Ω). In fact, for u, v ∈ Dom(−) we have

−u, v = −

=

u(x)v(x) dx Ω

∂u v dσ ∂ν

∇u(x).∇v(x) dx − Ω

∂Ω

=−

u(x)v(x) dx − Ω

∂u v dσ + ∂ν

∂Ω

= −v, u + β1 u, β2 v

∂v u dσ ∂ν

∂Ω 1

H 2 (∂Ω)×H

− 12

(∂Ω)

− β1 v, β2 u

1

H 2 (∂Ω)×H

− 12

(∂Ω)

.

The above computation also shows that − is non-negative modulo the boundary operator B := (β1 , β2 ). Indeed, −u, u + β1 u, β2 u 1 = Ω |∇u(x)|2 dx 0. 1 H 2 (∂Ω)×H 2 (∂Ω)

We shall use the following result in Section 4. Proposition 2.4. Suppose the spaces V and V ∗ and also Y and Y ∗ are in separating duality. The following hold: (1) Assume that the linear operator Λ : Dom(Λ) ⊂ X → X ∗ is symmetric and non-negative. If Λu − Λv, u − v = 0 for some u, v ∈ Dom(Λ) then Λu = Λv.

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(2) Assume that the linear operator Λ : Dom(Λ) ⊂ V → V ∗ is symmetric and non-negative modulo the linear boundary operator B = (β1 , β2 ) : V → Y × Y ∗ . If Λu − Λv, u − vV ×V ∗ + β1 u − β1 v, β2 u − β2 vY ×Y ∗ = 0 for some u, v ∈ Dom(Λ) then Λu = Λv and β2 u = β2 v. Proof. For the proof of part (1), note that since Λ is symmetric and non-negative, the function Φ defined on V by Φ(u) = 12 Λu, u is convex and Gâteaux differentiable on Dom(Λ). In fact ∇Φ(u) = Λu for all u ∈ Dom(Λ). Now if Λu − Λv, u − v = 0 we have

∇Φ(u) − ∇Φ(v), u − v = 0,

from which together with Theorem 2.2 one obtains ∇Φ(u) = ∇Φ(v). We now prove part (2). As in part (1) since Λ is symmetric and non-negative modulo the linear boundary operator B = (β1 , β2 ) the function Φ defined on V by Φ(u) = 12 Λu, u + 1 ∗ 2 β1 u, β2 uY ×Y is convex and Gâteaux differentiable on Dom(Λ). A straightforward computation shows that for η ∈ Dom(Λ) we have

1 1 1 1 ∇Φ(u), η = Λu, ηX×X∗ + Λη, uX×X∗ + β2 u, β1 ηY ×Y ∗ + β2 η, β1 uY ×Y ∗ . 2 2 2 2

It now follows from part (3) of Definition 2.5 that Λη, uX×X∗ + β2 η, β1 uY ×Y ∗ = Λu, ηX×X∗ + β2 u, β1 ηY ×Y ∗ and therefore

∇Φ(u), η = Λu, ηX×X∗ + β2 u, β1 ηY ×Y ∗ .

(4)

By assumption we have Λu − Λv, u − vV ×V ∗ + β1 u − β1 v, β2 u − β2 vY ×Y ∗ = 0. This together with (4) imply that

∇Φ(u) − ∇Φ(v), u − v = 0.

By Theorem 2.2 we obtain ∇Φ(u) = ∇Φ(v) from which together with (4) one has Λu − Λv, η + β2 u − β2 v, β1 ηY ×Y ∗ = 0 for all η ∈ Dom(Λ).

(5)

It follows from part (1) of Definition 2.5 that Ker(β1 ) is dense in V . This and the above equation yield that Λu − Λv, η = 0 for all η ∈ Ker(β1 ) and therefore Λu = Λv. It then follows from (5) that β2 u − β2 v, β1 ηY ×Y ∗ = 0

for all η ∈ Dom(Λ),

from which together with density of range β1 in Y, due to part (2) of Definition 2.5, we obtain β2 u = β2 v. 2


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3. Non-convex self-dual Lagrangians Let V be a Banach space that is in separating duality with the Banach space V ∗ . Functions Φ : V × V ∗ → R ∪ {∞} on phase space V × V ∗ will be called Lagrangians. We shall consider the class of Lagrangians that are convex and lower semi-continuous on the second variable. The Fenchel–Legendre dual of Φ with respect to the second variable will be denoted by LF 2 (Φ) and is a function on V × V given by: LF 2 (Φ)(u, v) = sup p, v − Φ(u, p) . p∈V ∗

We define the Non-convex dual, Φ # , of Φ by computing the Fenchel–Legendre dual of LF 2 (Φ)(., v) with respect to the first variable. Therefore Φ # is a Lagrangian on the phase space V × V ∗ given by Φ # (v, q) = sup q, u − LF 2 (Φ)(u, v) . u∈V

Definition 3.1. Suppose Φ is a Lagrangian on phase space V × V ∗ . Say that the Lagrangian Φ on V × V ∗ is Non-convex self-dual if the following property hold. Φ # (u, p) = Φ(u, p) for all (u, p) ∈ V × V ∗ . We now list some permanence properties of Nc-SD Lagrangians. Proposition 3.1. Let V be a Banach space that is in separating duality with the Banach space V ∗ . The following statements hold: (1) If ϕ : V → R is convex and lower semi-continuous and ϕ ∗ its Fenchel–Legendre dual defined on V ∗ , then the following Lagrangians are Non-convex self-dual: (i) Φ1 (u, p) := ϕ ∗ (p) − ϕ(u), (u, p) ∈ V × V ∗ , (ii) Φ2 (u, p) := ϕ ∗ (p) − p, u, (u, p) ∈ V × V ∗ , (iii) Φ3 (u, p) := p, u − ϕ(u), (u, p) ∈ V × V ∗ . (2) If Λ : V → V ∗ is symmetric and Φ is any Nc-SD Lagrangian then the Lagrangian Ψ (u, p) := Φ(u, Λu + p) is also Non-convex self-dual. (3) If Φ is an Nc-SD Lagrangian and μ > 0 then the Lagrangian μ.Φ on V × V ∗ defined by (μ.Φ)(u, p) = μ−2 Φ(μu, μp) is also Nc-SD. Proof. Fix (v, q) ∈ V × V ∗ . Note that LF 2 (Φ1 )(u, v) = ϕ(u) + ϕ(v) from which we have Φ1# (v, p) = ϕ ∗ (p) − ϕ(v) = Φ1 (v, p).

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Thus, Φ1 is an Nc-SD Lagrangian. For the Lagrangian Φ2 we have LF 2 (Φ2 )(u, v) = sup p, v − Φ2 (u, p) p∈V ∗

= sup p, v − ϕ ∗ (p) + p, u p∈V ∗

= sup p, v + u − ϕ ∗ (p) = ϕ(u + v). p∈V ∗

This implies that Φ2# (v, q) = sup q, u − ϕ(u + v) u∈V

= sup q, u + v − ϕ(u + v) − q, v u∈V ∗

= ϕ (q) − q, v = Φ2 (v, q) thereby giving that Φ2 is Nc-SD. For the Lagrangian Φ3 we have LF 2 (Φ3 )(u, v) = sup p, v − Φ3 (u, p) p∈V ∗

= sup p, v − p, u + ϕ(u) p∈V ∗

= sup p, v − u + ϕ(u) , p∈V ∗

from which we obtain LF 2 (Φ3 )(u, v) =

ϕ(u), +∞,

u = v, u = v,

and therefore Φ3# (v, q) = sup q, u − LF 2 (Φ3 )(u, v) u∈V

= q, v − ϕ(v) = Φ3 (v, q). This completes the proof of part (1). For the proof of part (2), we first compute LF 2 (Ψ )(u, v), LF 2 (Ψ )(u, v) = sup p, v − Φ(u, p + Λu) p∈V ∗

= sup p + Λu, v − Φ(u, p + Λu) − Λu, v p∈V ∗

= LF 2 (Φ)(u, v) − Λu, v.


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It follows that Ψ # (v, q) = sup q, u − LF 2 (Ψ )(u, v) u∈V

= sup q, u − LF 2 (Φ)(u, v) + Λu, v u∈V

= sup q, u + u, Λv − LF 2 (Φ)(u, v) u∈V

= sup q + Λv, u − LF 2 (Φ)(u, v) u∈V

= Φ # (v, q + Λv) = Φ(v, q + Λv) = Ψ (v, q). This proves part (2). For part (3), we have LF 2 (μ.Φ)(u, v) = sup p, v − (μ.Φ)(u, p) p∈V ∗

= sup p, v − μ−2 Φ(μu, μp) p∈V ∗

= μ−2 sup μp, μv − Φ(μu, μp) p∈V ∗

= μ−2 LF 2 (Φ)(μu, μv). Thus (μ.Φ)# (v, q) = sup q, u − μ−2 LF 2 (Φ)(μu, μv) u∈V

= μ−2 sup μq, μu − LF 2 (Φ)(μu, μv) u∈V

=μ This completes the proof of part (3).

−2

Φ(μv, μq) = (μ.Φ)(v, q).

2

It follows from Proposition 3.1 that, if Λ : X → X ∗ is a linear symmetric operator then the following Lagrangians are Nc-SD: (1) Ψ1 (u, p) := Φ1 (u, p + Λu) = ϕ ∗ (p + Λu) − ϕ(u), (2) Ψ2 (u, p) := Φ2 (u, p + Λu) = ϕ ∗ (p + Λu) − p + Λu, u, (3) Ψ3 (u, p) := Φ3 (u, p + Λu) = p + Λu, u − ϕ(u). They will be called basic Non-convex self-dual Lagrangians.

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3.1. Unbounded operators We shall see that in practice and in various applications, we are often faced with an unbounded symmetric operator Λ. As seen, in Proposition 3.1 the iteration, in an appropriate way, of NcSD Lagrangians with bounded symmetric operators are still Nc-SD. Here we extend this result to unbounded operators as well. Indeed, let Λ : Dom(Λ) ⊂ V → V ∗ be a possibly unbounded symmetric operator. If Λ is closed, consider VΛ to be the Banach space Dom(Λ) equipped with the norm: uVΛ = uV + ΛuV ∗ . Since VΛ is dense in V , it is easily seen that VΛ and V ∗ are still in separating duality with the same bilinear form that puts V and V ∗ in separating duality. Proposition 3.2. Let Λ : Dom(Λ) ⊂ V → V ∗ be a closed symmetric operator and Φ : V × V ∗ → R be a Non-convex self-dual Lagrangian. If LF 2 (Φ) is continuous on V × V then the Lagrangian Ψ : VΛ × V ∗ → R defined by Ψ (u, p) = Φ(u, Λu + p) is also a Non-convex self-dual Lagrangian. Proof. Let us first compute LF 2 (Ψ )(u, v), for u, v ∈ VΛ , LF 2 (Ψ )(u, v) = sup p, v − Φ(u, p + Λu) p∈V ∗

= sup p + Λu, v − Φ(u, p + Λu) − Λu, v p∈V ∗

= LF 2 (Φ)(u, v) − Λu, v. It follows that Ψ # (v, q) = sup q, u − LF 2 (Ψ )(u, v) u∈VΛ

= sup q, u − LF 2 (Φ)(u, v) + Λu, v u∈VΛ

= sup q + Λv, u − LF 2 (Φ)(u, v) . u∈VΛ

Since VΛ is dense in V and LF 2 (Φ) is continuous on V × V we have sup q + Λv, u − LF 2 (Φ)(u, v) = sup q + Λv, u − LF 2 (Φ)(u, v) u∈VΛ

u∈V


2691

from which we have Ψ # (v, q) = sup q + Λv, u − LF 2 (Φ)(u, v) u∈V

= Φ # (v, q + Λv) = Φ(v, q + Λv) = Ψ (v, q).

2

Example 2. Let Ω be a smooth domain in RN and ∂Ω its boundary. Consider the Laplace operator with Dirichlet boundary condition − : Dom(−) ⊂ Lp (Ω) → Lp (Ω), for p > 1 p and p = p−1 . It follows that Dom() = u ∈ Lp (Ω); u ∈ Lp (Ω) & u = 0 on ∂Ω is a Banach space when equipped with the norm: u = uLp (Ω) + uLp (Ω) . Consider also the convex function ϕ : Lp (Ω) → R defined by ϕ(u) = p1 Ω |u(x)|p dx + 1 p ∗ Ω u(x)f (x) dx where f ∈ L (Ω). An easy computation shows that ϕ (r) = p Ω |r(x) −

f (x)|p dx for all r ∈ Lp (Ω). We have that Φ(u, r) = ϕ ∗ (r) − ϕ(u) is an Nc-SD Lagrangian on Lp (Ω) × Lp (Ω). Now since 1 LF 2 (Φ)(u, v) = p

u(x)p + v(x)p dx +

Ω

u(x) + v(x) f (x) dx

Ω

is continuous on Lp (Ω) × Lp (Ω) and Dom(−) is dense in Lp (Ω) it follows from the above proposition that Ψ (u, r) :=

1 p

r(x) − u(x) − f (x)p dx − 1 p

Ω

u(x)p dx −

Ω

u(x)f (x) dx Ω

is also an Nc-SD Lagrangian on Dom(−) × Lp (Ω). 3.2. Symmetric operators modulo boundary operators For problems involving nonlinear boundary terms, we may start with an Nc-SD Lagrangian Φ, but the operator Λ : Dom(Λ) ⊂ V → V ∗ may be symmetric modulo a term involving a boundary operator B := (β1 , β2 ) : V → Y × Y ∗ for some Banach spaces Y and Y ∗ that are in separating duality. We can then try to recover Non-convex self-duality by adding a correcting term via a boundary Lagrangian on Y × Y ∗ , in such a way that a new Lagrangian Ψ u, (p, e) := Φ(u, Λu + p) + (β1 u, β2 u + e) becomes Nc-SD on VΛ × (V ∗ × Y ∗ ). Thus, we first need to define a bilinear form between VΛ and V ∗ × Y ∗ in such a way that it puts VΛ and V ∗ × Y ∗ in separating duality.

2692


Lemma 3.2. Let Λ : Dom(Λ) ⊂ V → V ∗ be a symmetric operator modulo the boundary operator B := (β1 , β2 ) : V → Y × Y ∗ . Then the bilinear form

u, (p, e) V

Λ ×(V

∗ ×Y ∗ )

:= u, pV ×V ∗ + β1 u, eY ×Y ∗

puts VΛ and V ∗ × Y ∗ in separating duality. Proof. First assume 0 = u ∈ VΛ . Since V and V ∗ are in separating duality, there exists p ∈ V ∗ such that u, pV ×V ∗ = 0 and therefore u, (p, 0)VΛ ×(V ∗ ×Y ∗ ) = 0. Now suppose 0 = (p, e) ∈ V ∗ × Y ∗ . If p = 0, since V and V ∗ are in separating duality, there exists u ∈ V such that u, pV ×V ∗ = 0. It also follows from Definition 2.5 that Ker(β1 ) is dense in V . Thus, there exists a sequence {un } ⊂ Ker(β1 ) such that un → u in V . It follows that

un , (p, e) V

Λ ×(V

∗ ×Y ∗ )

= un , pV ×V ∗ = 0,

for n large enough. If p = 0, then e must be a non-zero element of Y ∗ and the result follows from the fact that β1 has a dense range in Y. 2 We now state our result. Proposition 3.3. Let Λ : Dom(Λ) ⊂ V → V ∗ be a possibly unbounded symmetric operator modulo the boundary operator B := (β1 , β2 ) : Dom(Λ) → Y × Y ∗ . Let Φ : VΛ × V ∗ → R and : Y × Y ∗ → R be Non-convex self-dual Lagrangians. If LF 2 (Φ) and LF 2 ( ) are continuous on V × V and Y × Y respectively then the Lagrangian Ψ : VΛ × (V ∗ × Y ∗ ) → R defined by Ψ u, (p, e) = Φ(u, Λu + p) + (β1 u, β2 u + e) is also a Non-convex self-dual Lagrangian. Proof. Let us first compute LF 2 (Ψ )(u, v), for u, v ∈ VΛ , LF 2 (Ψ )(u, v) = =

sup

p, vV ×V ∗ + β1 v, eY ×Y ∗ − Φ(u, Λu + p) − (β1 u, β2 u + e)

sup

p + Λu, vV ×V ∗ + β1 v, e + β2 uY ×Y ∗

(p,e)∈V ∗ ×Y ∗ (p,e)∈V ∗ ×Y ∗

− Φ(u, p + Λu) − (β1 u, β2 u + e) − Λu, vV ×V ∗ − β1 v, β2 uY ×Y ∗

= LF 2 (Φ)(u, v) + LF 2 ( )(β1 u, β1 v) − Λu, vV ×V ∗ − β1 v, β2 uY ×Y ∗ . It follows that Ψ # (v, q) = sup q, uV ×V ∗ + β1 u, eY ×Y ∗ − LF 2 (Ψ )(u, v) u∈VΛ

= sup q, uV ×V ∗ + β1 u, eY ×Y ∗ + Λu, vV ×V ∗ + β1 v, β2 uY ×Y ∗ u∈VΛ


2693

−LF 2 (Φ)(u, v) − LF 2 ( )(β1 u, β1 v) = sup q, uV ×V ∗ + β1 u, eY ×Y ∗ + u, ΛvV ×V ∗ + β1 u, β2 vY ×Y ∗ u∈VΛ

− LF 2 (Φ)(u, v) − LF 2 ( )(β1 u, β1 v) = sup q + Λv, uV ×V ∗ + β1 u, e + β2 vY ×Y ∗ u∈VΛ

− LF 2 (Φ)(u, v) − LF 2 ( )(β1 u, β1 v)

= sup q + Λv, uV ×V ∗ + β1 (u + u0 ), e + β2 v Y ×Y ∗ u∈VΛ ,u0 ∈V0

− LF 2 (Φ)(u, v) − LF 2 ( ) β1 (u + u0 ), β1 v

where V0 = Dom(Λ) ∩ ker(β1 ). Setting w = u + u0 we have u = w − u0 and therefore Ψ # (v, q) =

sup

w∈VΛ ,u0 ∈V0

q + Λv, w − u0 V ×V ∗ + β1 (w), e + β2 v Y ×Y ∗

− LF 2 (Φ)(w − u0 , v) − LF 2 ( ) β1 (w), β1 v .

Since V0 is dense in V and LF 2 (Φ) is continuous on V × V we have sup q + Λv, w − u0 − LF 2 (Φ)(w − u0 , v) = sup q + Λv, u − LF 2 (Φ)(u, v)

u0 ∈V0

u∈V

= Φ # (v, q + Λv) from which we have Ψ # (v, q) = sup w∈VΛ

β1 (w), e + β2 v Y ×Y ∗ − LF 2 ( ) β1 (w), β1 v + Φ # (v, q + Λv).

Also taking into account that β1 : VΛ → Y has a dense range in Y and LF 2 ( ) is continuous on Y × Y we have sup w∈VΛ

β1 (w), e + β2 v Y ×Y ∗ − LF 2 ( ) β1 (w), β1 v = # (β1 v, e + β2 v).

This implies that Ψ # (v, q) = Φ # (v, q + Λv) + # (β1 v, e + β2 v) = Φ(v, q + Λv) + (β1 v, e + β2 v) = Ψ (v, q). Example 3. Let N > 4, 1 < p
0 and J : RN × RN → RN × RN be the symplectic operator defined by J (x, y) = (−y, x) and consider the following finite dimensional Hamiltonian systems in R2N , ⎧ ˙ = ∇ϕ t, u(t) , ⎨ J u(t) ⎩ u(T ) + u(0) = ∇ψ J u(T ) − J u(0) . 2

(20)

Hamiltonian systems with this type of boundary conditions are also treated in [4,16]. Here is an application of Corollary 4.10. Theorem 4.12. Let ϕ : [0, T ] × (RN × RN ) → R be differentiable, convex and lower semicontinuous in the second variable. Also let ψ : RN × RN → R be differentiable and convex. Then every critical point of the functional

T I (u) =

1 ˙ u(t) ˙ dt − J u(t), ϕ ∗ t, J u(t) 2

0

+ ψ∗

u(T ) + u(0) 1 u(T ) + u(0) − , J u(T ) − u(0) 2 2 2

is a solution of (20). Example 7 (A Hamiltonian system of PDE’s with nonlinear Neumann boundary conditions). Let N > 2 and Ω be a smooth domain in RN and ∂Ω its boundary. Consider the following infinite


2705

dimensional Hamiltonian system, ⎧ −u + u = |v|q−2 v, ⎪ ⎪ ⎪ ⎪ p−2 ⎪ u, ⎪ ⎨ −v + v = |u| ∂u = |v|q−2 v, ⎪ ⎪ ∂n ⎪ ⎪ ⎪ ⎪ ⎩ ∂v = |u|p−2 u, ∂n

x ∈ Ω, x ∈ Ω, (21)

x ∈ ∂Ω, x ∈ ∂Ω.

We have the following result. Theorem 4.13. Assume p, q > 2 and min{ p1 + q q−1 . Then the functional 1 I (u, v) = p

1 |−v + v| dx + q p

Ω

N −1 1 Nq , q

+

N −1 N −1 Np } > N .

1 |−u + u| dx + p q

Ω

q ∂u 1 + dx − 2 ∇u.∇v dx − 2 uv dx q ∂n Ω

∂Ω

Let p =

p p−1

and q =

p ∂v dx ∂n ∂Ω

Ω

has a critical point in W 2,q (Ω) × W 2,p (Ω) that is indeed a solution of the system (21). We shall prove this theorem in Section 5. As in Corollaries 4.8 and 4.10, by considering different combination of interior Nc-SD Lagrangians and boundary Nc-SD Lagrangians one can obtain different variational principles of Eq. (18). Here we state one more application of Theorem 4.7 and leave it to interested readers to generate more new principles by making use of Theorem 4.7. Corollary 4.14. Let Λ : Dom(Λ) ⊂ V → V ∗ be a symmetric operator modulo the boundary operator B := (β1 , β2 ) : V → Y × Y ∗ in such a way that (Λ, β2 ) : Dom(Λ) → V ∗ × Y ∗ is onto. Let ϕ : V → R and ψ : Y → R be convex lower semi-continuous and Gâteaux differentiable. If u is a critical point of I (w) = Λw, w − 2ϕ(w) + ψ ∗ (β2 w) − ψ(β1 w) then u is a solution of (18). 5. Critical points of lower semi-continuous functionals In this section we shall provide a minimax principle for lower semi-continuous functionals applicable for the proof of existence theorems stated in previous sections. We first recall the following minimax principle for lower semi-continuous functionals due to Szulkin [27]. Let X be a real Banach space and I : X → R ∪ {+∞} a functional on X such that I = Φ + ψ with Φ ∈ C 1 (X, R) and ψ : X → R ∪ {+∞} convex and lower semi-continuous. A point u ∈ X is said to be a critical point of I if u ∈ Dom(ψ) and if it satisfies the inequality

2706


∇Φ(u), v − u + ψ(v) − ψ(u) 0 for all v ∈ X.

We shall say that I satisfies the compactness condition of Palais–Smale type provided: (PS) If un is a sequence such that I (un ) → c ∈ R and

∇Φ(un ), v − un + ψ(v) − ψ(un ) − n v − un for all v ∈ X,

where n → 0 then un possesses a subsequent that converges strongly. The following is established by Szulkin [27]. Theorem 5.1. Suppose that I : X → R ∪ {+∞} is as above satisfying (PS) and the mountain pass geometry (in short MPG), i.e., (i) I (0) = 0 and there exist α, ρ > 0 such that I (u) α when u = ρ, (ii) I (e) 0 for some e ∈ X with e > ρ. Then I has a critical value c α which may be characterized by c = inf sup I γ (t) , γ ∈Γ t∈[0,1]

where Γ = {γ ∈ C([0, 1], X): γ (0) = 0, γ (1) = e}. By some minor changes in the proof of the above theorem one can replace (PS) condition by the following condition (PS∗ ) If un is a sequence such that I (un ) → c ∈ R and

∇Φ(un ), v − un + ψ(v) − ψ(un ) − n v − un for all v ∈ X,

where n → 0 then there exists u ∈ X such that, up to a subsequence, (i) un converges weakly to u ∈ X, (ii) Φ(un ) → Φ(u) and ∇Φ(un ), v − un → ∇Φ(u), v − u for all v ∈ X, (iii) I (un ) → I (u). We shall use the above theorem with (PS∗ ) to deal with the existence theorems stated in previous sections. To be more precise, let X and X ∗ be two real Banach spaces in separating duality and Λ : Dom(Λ) ⊂ X → X ∗ be a closed linear operator. As in Section 3 set XΛ = {u ∈ X; Λu ∈ X ∗ } that is Banach space when equipped with the norm: uVΛ = uV + ΛuV ∗ . We shall assume uX cΛuX∗ for some constant c and all u ∈ X. It then follows that u = ΛuX∗ is an equivalent norm for XΛ . We shall also assume that the embedding XΛ → X is compact. We establish the following result as a consequence of Theorem 5.1.


2707

Theorem 5.2. Suppose the operator Λ : Dom(Λ) ⊂ X → X ∗ is as above and I : XΛ → R is a functional of the form I (u) = F (Λu) − G(u) where: (1) F : X ∗ → R is convex lower semi-continuous and differentiable. (2) F is coercive, i.e., F (p) → +∞ as pX∗ → +∞, pX∗ and we also have p, ∇F (p) βF (p) for some 1 < β < 2 and all p ∈ X ∗ . (3) G ∈ C 1 (XΛ , R) and u, ∇G(u) 2G(u) for all u ∈ XΛ . (4) The functional G : XΛ → R is weakly continuous and the map ∇G : XΛ → X ∗ is weak to weak continuous. If I satisfies the mountain pass geometry then I has a critical value c α which may be characterized by c = inf sup I γ (t) , γ ∈Γ t∈[0,1]

where Γ = {γ ∈ C([0, 1], X): γ (0) = 0, γ (1) = e}. Proof. By virtue of Theorem 5.1 we just need to show that the functional I satisfies (PS∗ ). To do this, suppose un ∈ XΛ is a sequence such that I (un ) → c ∈ R and

−∇G(un ), v − un + F (Λv) − F (Λun ) − n v − un

for all v ∈ XΛ ,

(22)

where n → 0. Note that since I is differentiable the above inequality simply means ∇I (un ) → 0. We first show that un is bounded in XΛ . It follows from conditions (2) and (3) that

1 I un (t) , un (t) 2

1

1 = F (Λun ) − G(un ) − ∇F (Λun ), Λun + ∇G(un ), un 2 2

1

β ∇G(un ), un − 2G(un ) F (Λun ) − ∇F (Λun ), Λun + 2 2 2−β F (Λun ). 2

c + o(1) = I (un ) −

Since 1 < β < 2, it follows from the coercivity of F that un is bounded in XΛ . Thus, up to a subsequence, there exists u ∈ XΛ such that un → u weakly in XΛ . Due to the compact embedding XΛ → X, we have that un → u strongly in X. Fix v ∈ XΛ , it follows from condition (4) together with and convergence of un to u in a weak sense in XΛ and in a strong sense in X that

∇G(un ), v − un → ∇G(u), v − u ,

for all v ∈ XΛ . This together with (22) and lower semi-continuity of F implies that

2708


−∇G(u), v − u + F (Λv) = lim inf −∇G(un ), v − un + F (Λv) n→∞ lim inf F (Λun ) − n v − un XΛ n→∞

F (Λu). Therefore,

−∇G(u), v − u + F (Λv) F (Λu),

and since v ∈ XΛ is arbitrary, it follows that u is a critical point of I. Now we show that I (un ) → I (u). Since G : XΛ → R is weakly continuous we just need to show that limn→∞ F (Λun ) = F (Λu). Note first that since F is lower semi-continuous we have F (Λu) lim inf F (Λun ). n→∞

On the other hand by taking v = u in (22) we have

−∇G(un ), u − un + F (Λu) F (Λun ) − n u − un .

(23)

By taking lim sup from both sides we get

F (Λu) = lim sup −∇G(un ), u − un + F (Λu) n→∞

lim sup F (Λun ) − n v − un XΛ n→∞

= lim sup F (Λun ). n→∞

Thus limn→∞ F (Λun ) = F (Λu) and therefore limn→∞ I (un ) = I (u).

2

5.1. System of transport equations We now proceed with the proof of Theorem 4.3. We shall make frequent use of the following theorem while proving our existence results (see [1] for the proof). Theorem 5.3. Let Ω be a smooth domain in RN , Ω0 a bounded subdomain of Ω, and Ω0k the intersection of Ω0 with a k-dimensional plane in RN . Let j, m be integers, j 0, m 1 and let 1 r < +∞. Then the following embeddings are continuous. W j +m,r (Ω) → W j,s Ω0k if 0 < N − mr < k N and 1 s W j +m,r (Ω) → W j,s (Ω)

kr , N − mr

if mr = N, 1 k N and 1 s < ∞.

We need the following lemma to prove Theorem 4.3. Lemma 5.4. Let Ω be a smooth bounded domain of RN . If p, q > 2 and | p1 − q1 | following embeddings are continuous.

1 N

then the


2709

(1) W 2,q (Ω) → W 1,p (Ω). (2) W 2,p (Ω) → W 1,q (Ω).

Proof. By Theorem 5.3 the embedding W 2,q (Ω) → W 1,p (Ω) is continuous provided p N q 1 1 1 2,p (Ω) → W 1,q (Ω) is continuous provided N −q that is equivalent to p − q N . Similarly W 1 q

−

1 p

1 N.

2

q p ∗ q p Proof of Theorem 4.3. Set V q= L (Ω) × Lp (Ω) and V = L (Ω) × L ∗(Ω). Define 1 1 G : V → R by G(u, v) = q Ω |u| dx + p Ω |v| dx. Let Λ : Dom(Λ) ⊂ V → V be the operator Λ(u, v) = (− a.∇v − u, a.∇u − v) with

Dom(Λ) = (u, v) ∈ V ; Λ(u, v) ∈ V ∗ & u = v = 0, x ∈ ∂Ω . Note that Λ is a symmetric operator and for each (u, v) ∈ Dom(Λ) we have

Λ(u, v), (u, v) =

|∇v|2 dx +

Ω

Ω

|∇u|2 dx + 2

Ω

|∇v|2 dx + Ω

(a.∇u)v dx

|∇u|2 dx −

2 a∞ ∇uL2 (Ω;RN ) ∇vL2 (Ω;RN ) , λ1

Ω

where λ1 is the first eigenvalue of − with Dirichlet boundary condition. The above estimate λ1 1 1 1 indeed shows that Λ is non-negative provided a . Since 2 < p, q < N2N −2 and | p − q | N , ∞ it follows from Lemma 5.4 and Sobolev embeddings 2N , N −2 2N , for 2 p < N −2

for 2 q
1, from which we get the inequality 21−s |a|s − |b|s |a − b|s

for all a, b ∈ R and s > 1.

It follows from this inequality that 21−p I (u, v) p

Ω

+

2

21−q

1 |u| dx − q

Ω

Ω 21−q

1 | a.∇v| dx − q q

( a∞ ) p

|u|q dx −

q

( a∞ q

|u|q dx Ω

p

1 p

|∇u|p dx − Ω

|v|p dx Ω

Ω

1 | a.∇u| dx − p p

q

|v|p dx −

p

Ω

q

1−p

+

1 |v| dx − p p

)q

Ω

|v|p dx Ω

|∇v|q dx − Ω

1 q

|u|q dx. Ω

It follows from embeddings (24), (25) and embeddings in Lemma 5.4 that there exist c1 , c2 , c3 and c4 such that q

uLq (Ω) c1 u ∇u

p Lp (Ω)

c3 u

p Lq (Ω)

q , Lq (Ω)

and ∇v

p

vLp (Ω) c2 v q Lq (Ω)

c4 v

p , Lp (Ω)

q Lp (Ω)

for all (u, v) ∈ VΛ .

It then follows that 21−p I (u, v) p

( a∞ )p |v| dx − c3 p p

|u| dx

Ω

+

21−q q

=

p

|u|q dx − c4

v

+

p q

c2 − p

Ω

( a∞ )q q

Ω 21−p

q

p

|v| dx

p Lp (Ω)

− c3

( a∞ p

p

Ω

|v|p dx

q p

−

c1 q

u

p Lq (Ω)

|u|q dx

Ω )p

p

q q

Ω

−

c2 p v p L (Ω) p

( a∞ )q 21−q c1 q q q u − c v p − u q . 4 q (Ω) L (Ω) L L (Ω) q q q

Note that p > p and q > q . Now take ρ > 0 small enough such that for (u, v) = ρ we have

21−p p

1−q p p p q − cp2 v p 1p v p and 2 q u q Lp (Ω) L (Ω) L (Ω) L (Ω) p2 q 1 . It then follows from (u, v) = ρ that u q L (Ω) q 2q

v

−

q c1 q uLq (Ω)


2711

( a∞ )q 1 1 p q q I (u, v) p v p − c4 v p + q u q L (Ω) L (Ω) L (Ω) q p2 q2

( a∞ )p p − c3 u q . L (Ω) p a( ρ )r

Claim. Let a, b, c, ρ > 0 be four constants, 0 ρ0 ρ and r, s > 1. Let δ0 = min{ 1+c(ρ s2+( ρ )r ) , b( ρ2 )s }. 1+c(ρ r +( ρ2 )s )

2

Then for 0 < δ < δ0 we have a(ρ − ρ0 )r + bρ0s − cδ (ρ − ρ0 )s + ρ0r > δ.

The proof for this claim is elementary. Now by assuming a = q

p

1 p 2p

, b =

1 q 2q

,

c = max{c4 (aq∞ ) , c3 (ap∞ ) }, r = p , s = q , uLq (Ω) = ρ0 , vLp (Ω) = ρ − ρ0 and

δ = p +q , it follows from the above claim that for some 0 if 0 < < 0 and (u, v) = ρ then I (u, v) . The second condition of the mountain pass geometry holds for any (ru, rv) ∈ VΛ where (u, v) = (0, 0) and r ∈ R is large enough. 2 5.2. Hamiltonian systems of PDE’s with Neumann boundary conditions Here a proof to Theorem 4.13 is provided. The following lemma is a direct consequence of Theorem 5.3. Lemma 5.5. Let Ω be a smooth bounded domain in RN and ∂Ω its boundary. Let p, q > 1 and p q p = p−1 and q = q−1 . The following embeddings hold.

W 2,p (Ω) → Lq (∂Ω),

W 2,q (Ω) → Lp (∂Ω),

W 2,p (Ω) → Lq (Ω),

W 2,q (Ω) → Lp (Ω),

if

1 N −1 N −1 + > , p Nq N

if

1 N −1 N −1 + > , q Np N if

1 N −1 1 + > , q p N

if

1 N −1 1 + > . q p N

Proof of Theorem 4.13. Note first that if Ω is a smooth bounded domain in RN then for each r > 1, W 2,r (Ω) has an equivalent norm of the form r 1 r ∂u uW 2,r (Ω) := −u + urLr (Ω) + . ∂n r L (Ω)

Set V = Lp (Ω) × Lq (Ω), V ∗ = Lp (Ω) × Lq (Ω), Y = Lp (∂Ω) × Lq (∂Ω) and Y ∗ = 1 1 q p L (∂Ω) × L (∂Ω). Define Φ : V → R by Φ(u, v) = p Ω |u| dx + q Ω |v|q dx and p

2712


Ψ : Y → R by Ψ (u, v) = p1 ∂Ω |u|p dx + q1 ∂Ω |v|q dx. It follows that Φ ∗ (f, g) = 1 1 1 1 p q ∗ p q p Ω |f | dx + q Ω |g| dx and Ψ (f0 , g0 ) = p ∂Ω |f0 | dx + q ∂Ω |g0 | dx. Let Λ : Dom(Λ) ⊂ V → V ∗ , β1 : Dom(Λ) → Y and β2 : Dom(Λ) → Y ∗ be the operators ∂v Λ(u, v) = (−v + v, −u + u), β1 (u, v) = (u|∂Ω , v|∂Ω ) and β2 (u, v) = ( ∂n |∂Ω , ∂u ∂n |∂Ω ) respectively. An easy computation shows that Λ is symmetric, but not necessary non-negative, modulo the boundary operator (β1 , β2 ). Note also that due to the inequality p1 + q1 > NN−2 we

have Dom(Λ) = W 2,q (Ω) × W 2,p (Ω). It follows from Corollary 4.10 that if (u, v) is a critical point of

F (u, v) := 2Φ ∗ Λ(u, v) − Λ(u, v), (u, v) V ×V ∗ + Ψ ∗ β2 (u, v) − β2 (u, v), β1 (u, v) Y ×Y ∗ , then there exists (u, ¯ v) ¯ with Λ(u, ¯ v) ¯ = Λ(u, v) and β2 (u, ¯ v) ¯ = β2 (u, v) such that ( u+2 u¯ , v+2 v¯ ) is a solution of (21). We shall show that u = u¯ and v = v¯ and therefore (u, v) is indeed a solution of (21). In fact, if Λ(u, ¯ v) ¯ = Λ(u, v) and β2 (u, ¯ v) ¯ = β2 (u, v) we have −(u − u) ¯ + (u − u) ¯ =0 u) ¯ and ∂(u− = 0. Note that by regularity theory of Elliptic equations with Neumann boundary ∂n conditions [19], we have that u − u¯ ∈ C 1,α (Ω) for some α > 0. It then follows

0=

−(u − u) ¯ + (u − u) ¯ (u − u) ¯ dx =

Ω

2 ∇(u − u) ¯ dx +

Ω

|u − u| ¯ 2 dx, Ω

from which we obtain u = u. ¯ The same argument shows that v = v. ¯ Note also that I (u, v) = 1 F (u, v). In fact 2 F (u, v) =

2 q

|−u + u|q dx +

2 p

Ω

|−v + v|p dx +

2 q

Ω

q ∂u dx ∂n ∂Ω

p ∂v 2 + dx − (−u + u)v dx − (−v + v)u dx p ∂n

−

∂Ω

∂v u dx − ∂n

∂Ω

2 = q

Ω

Ω

∂u v dx ∂n

∂Ω

2 |−u + u| dx + p q

Ω

2 |−v + v| dx + p q

Ω

q ∂v 2 + dx − 2 ∇u.∇v dx − 2 uv dx q ∂n ∂Ω

Ω

p ∂u dx ∂n ∂Ω

Ω

= 2I (u, v). Thus, F and I have the same family of critical points. So far we have proved that if (u, v) is a critical point of I then (u, v) is a solution of (21). Now we need to show that I has at least one non-trivial critical point.


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2,p (Ω) → R by F ((f, g), Define F : V ∗ × Y ∗ → R and G : Dom(Λ) = W 2,q (Ω) ×W (f0 , g0 )) = Φ ∗ (f, g) + Ψ ∗ (f0 , g0 ) and G(u, v) = Ω ∇u.∇v dx + Ω uv dx. Define ¯ v) = (Λ(u, v), β2 (u, v)). Note that I (u, v) = Λ¯ : W 2,q (Ω) × W 2,p → V ∗ × Y ∗ by Λ(u, ¯ ¯ F (Λ(u, v)) − G(u, v), and F, G and Λ satisfy all conditions of Theorem 5.2. Therefore, to show that I has a non-trivial critical point, we just need to verify the mountain pass geometry. To do this, note that

∂v ∂u G(u, v) = (−u + u)v dx + (−v + v)u dx + u dx + v dx ∂n ∂n Ω

Ω

∂Ω

∂Ω

−u + uLq (Ω) vLq (Ω) + −v + vLp (Ω) uLp (Ω) ∂v ∂u + uLp (∂Ω) + uLq (∂Ω) ∂n q ∂n Lp (∂Ω) L (∂Ω) C v2W 2,q (Ω) + u2W 2,p (Ω) (by Lemma 5.5) for some constant C. We have 1 I (u, v) = p

1 |−v + v| dx + q p

Ω

1 |−u + u| dx + p q

Ω

∂Ω

q ∂u 1 + dx + G(u, v) q ∂n ∂Ω

1 p

|−v + v|p dx + Ω

1 q

|−u + u|q dx + Ω

p ∂v dx ∂n

1 p

q ∂u 1 + dx − C v2W 2,q (Ω) + u2W 2,p (Ω) q ∂n

p ∂v dx ∂n ∂Ω

∂Ω

=

1 1 p q v 2,p + u 2,q − C v2W 2,q (Ω) + u2W 2,p (Ω) . W (Ω) W (Ω) p q

Since p , q < 2, there exists α > 0 such that for ρ > 0 small enough we have I (u, v) > α when (u, v) = ρ. For the second condition of the mountain pass geometry take (u0 , v0 ) ∈ W 2,q (Ω) × W 2,p (Ω) with G(u0 , v0 ) > 0. It then follows that I (ru0 , rv0 ) < 0 for r ∈ R large enough. 2 Acknowledgments Some of the results on this work are related to my previous joint works with Professor Nassif Ghoussoub on self-dual Lagrangians for convex functions. I would like to express my gratitude to him for the numerous and fruitful discussions on projects in convex self-duality. I would also like to express my thanks and appreciation to Professor Ivar Ekeland for pointing me towards Toland’s duality that is indeed the main ingredient of this project.

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References [1] R.A. Adams, Sobolev Spaces, Academic Press, New York, San Francisco, London, 1975. [2] H. Attouch, H. Brézis, Duality for the sum of convex functions in general Banach spaces, in: J. Barroso (Ed.), Aspects of Mathematics and Its Applications, Elsevier Science, Amsterdam, 1986, pp. 125–133. [3] H. Attouch, M. Théra, A general duality principle for the sum of two operators, J. Convex Anal. 3 (1) (1996) 1–24. [4] J.-P. Aubin, I. Ekeland, Second-order evolution equations associated with convex Hamiltonians, Canad. Math. Bull. 23 (1) (1980) 81–94. [5] J.M. Borwein, A.S. Lewis, Practical conditions for Fenchel duality in infinite dimensions, in: J.-B. Baillon, M. Thera (Eds.), Proceedings of the International Conference on Fixed Points and Applications, in: Pitman Res. Notes Math., vol. 252, 1990, pp. 83–99. [6] H. Brézis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc. (N.S.) 8 (3) (1983) 409–426. [7] H. Brézis, J.M. Coron, L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math. 33 (1980) 667–689. [8] F. Clarke, A classical variational principle for periodic Hamiltonian trajectories, Proc. Amer. Math. Soc. 76 (1979) 186–188. [9] F. Clarke, Periodic solution to Hamiltonian inclusions, J. Differential Equations 40 (1) (1981) 1–6. [10] I. Ekeland, Legendre duality in non convex optimization and the calculus of variations, SIAM J. Control Optim. 15 (1977) 890–905. [11] I. Ekeland, Periodic solutions of Hamiltonian equations and a theorem of P. Rabinowitz, J. Differential Equations 34 (1979) 523–534. [12] I. Ekeland, A perturbation theory near convex Hamiltonian systems, J. Differential Equations 50 (3) (1983) 407– 440. [13] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, Heidelberg, New York, 1990. [14] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, American Elsevier Publishing Co., Inc., New York, 1976. [15] N. Ghoussoub, Selfdual Partial Differential Systems and Their Variational Principles, Springer Monogr. Math., Springer-Verlag, 2008, 350 pp. [16] N. Ghoussoub, A. Moameni, Selfdual variational principles for periodic solutions of Hamiltonian and other dynamical systems, Comm. Partial Differential Equations 32 (2007) 771–795. [17] N. Ghoussoub, A. Moameni, Anti-symmetric Hamiltonians (II): Variational resolutions for Navier–Stokes and other nonlinear evolutions, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (1) (2009) 223–255. [18] N. Ghoussoub, A. Moameni, Hamiltonian systems of PDEs and other evolution equations with self-dual boundary conditions, Calc. Var. Partial Differential Equations 36 (2009) 85–118. [19] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer-Verlag, Berlin, 2001, reprint of the 1998 edition. [20] J.-B. Hiriart-Urruty, Generalized differentiability, duality and optimization for problems dealing with the difference of convex functions, in: Convexity and Duality in Optimization, in: Lecture Notes in Econom. and Math. Systems, vol. 256, 1986, pp. 37–70. [21] M. Koslowsky, A. Moameni, A definite variational formulation for elliptic Hamiltonian systems with a critical nonlinearity, submitted for publication. [22] M. Lewis, A. Moameni, A new approach in convex Hamiltonian systems with nonlinear boundary conditions, submitted for publication. [23] J.-L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, New York–Heidelberg, 1972. [24] A. Moameni, A variational principle associated with a certain class of boundary value problems, Differential Integral Equations 23 (3–4) (2010) 253–264. [25] R.T. Rockafellar, Conjugate Duality and Optimisation, Regional Conf. Ser. in Appl. Math., vol. 16, SIAM Publications, Philadelphia, 1974. [26] I. Singer, A Fenchel–Rockafellar type duality theorem for maximization, Bull. Aust. Math. Soc. 20 (2) (1979) 193–198. [27] A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (2) (1986) 77–109.


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[28] J.F. Toland, Duality in nonconvex optimization, J. Math. Anal. Appl. 66 (2) (1978) 399–415. [29] J.F. Toland, A duality principle for nonconvex optimization and the calculus of variations, Arch. Ration. Mech. Anal. 60 (1979) 177–183. [30] J.F. Toland, On subdifferential calculus and duality in nonconvex optimization, Bull. Soc. Math. France Mém. 71 (1) (1979) 41–61.


Quantum double suspension and spectral triples Partha Sarathi Chakraborty ∗,1 , S. Sundar The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India Received 17 June 2010; accepted 12 January 2011 Available online 26 January 2011 Communicated by Alain Connes

Abstract In this paper we are concerned with the construction of a general principle that will allow us to produce regular spectral triples with finite and simple dimension spectrum. We introduce the notion of weak heat kernel asymptotic expansion (WHKAE) property of a spectral triple and show that the weak heat kernel asymptotic expansion allows one to conclude that the spectral triple is regular with finite simple dimension spectrum. The usual heat kernel expansion implies this property. The notion of quantum double suspension of a C ∗ -algebra was introduced by Hong and Szymanski. Here we introduce the quantum double suspension of a spectral triple and show that the WHKAE is stable under quantum double suspension. Therefore quantum double suspending compact Riemannian spin manifolds iteratively we get many examples of regular spectral triples with finite simple dimension spectrum. This covers all the odd-dimensional quantum spheres. Our methods also apply to the case of noncommutative torus. © 2011 Elsevier Inc. All rights reserved. Keywords: Local index formula; Regularity; Dimension spectrum; Heat kernel expansion; Quantum double suspension

1. Introduction Since its inception index theorems play a central role in noncommutative geometry. Here spaces are replaced by explicit K-cycles or finitely summable Fredholm modules. Through index pairing they pair naturally with K-theory. In the foundational paper [6] Alain Connes introduced cyclic cohomology as a natural recipient of a Chern character homomorphism assigning cyclic * Corresponding author.

E-mail addresses: [email protected] (P.S. Chakraborty), [email protected] (S. Sundar). 1 The author acknowledges financial support from Indian National Science Academy through its project

“Noncommutative Geometry of Quantum Groups”. 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.01.009

P.S. Chakraborty, S. Sundar / Journal of Functional Analysis 260 (2011) 2716–2741

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cocycles to finitely summable Fredholm modules. Then the index pairing is computed by the pairing of cyclic cohomology with K-theory. But finitely summable Fredholm modules occur as those associated with spectral triples and it was desirable to have cyclic cocycles given directly in terms of spectral data, that will compute the index pairing. This was achieved by Connes and Moscovici in [8]. Let us briefly recall their local index formula henceforth to be abbreviated as LIF. One begins with a spectral triple, i.e., a Hilbert space H, an involutive subalgebra A of the algebra of bounded operators on H and a self adjoint operator D with compact resolvent. It is further assumed that the commutators [D, A] give rise to bounded operators. Such a triple is finitely summable if |D|−p is trace class for some positive p. The spectral triple is said to be regular if both A and [D, A] are in the domains of δ n for all n 0, where δ is the derivation [|D|, .]. One says that the spectral triple has dimension spectrum Σ , if for every element b in the smallest algebra B containing A, [D, A] and closed under the derivation δ, the associated zeta function ζb (z) = Tr b|D|−z a priori defined on the right half plane (z) > p admits a meromorphic extension to whole of complex plane with poles contained in Σ. If a spectral triple is regular and has discrete dimension spectrum then given any n-tuple of non-negative integers k1 , k2 , . . . , kn one can consider multilinear functionals φk,n defined by φk,n (a0 , a1 , . . . , an ) = Resz=0 Tr a0 [D, a1 ](k1 ) [D, a2 ](k2 ) · · · [D, an ](kn ) |D|−n−2|k|−z , where T (r) stands for the r-fold commutator [D 2 , [D 2 , [· · · [D 2 , T ] · · ·]]] and |k| = k1 + · · · + kn . In the local index formula the components of the local Chern character in the (b, B)-bicomplex is expressed as a sum k ck,n φk,n , where the summation is over all n-tuples of non-negative integers and ck,n ’s are some universal constants independent of the particular spectral triple under consideration. Note that Remark II.1 in p. 63 of [8] says that if we consider the Dirac operator associated with a closed Riemannian spin manifold then φk,n ’s are zero for |k| = 0. Therefore most of the terms in the local Chern character are visible in truely noncommutative cases and hence should be interpreted as a signature of noncommutativity. To have a better understanding of the contribution of these terms it is desirable to have examples where these terms survive. In the foliations example the contribution of these terms becomes overwhelming and tackling them lead to new organizational principles of cyclic theory [9]. So the task of illustrating the LIF in simpler examples remained open. The first simple illustration was given by Connes in [7]. This was extended to odd-dimensional quantum spheres by Pal and Sundar in [14]. But to have a good grasp of the formula it is essential to have a systematic family of examples where one can verify the hypothesis of regularity and discreteness of the dimension spectrum. To our knowledge only Higson [10] made an attempt to this effect and gave a general scheme for verifying the meromorphic continuation. Here in this article we are also primarily concerned with the goal of developing a general procedure that will allow us to construct regular spectral triples with finite dimension spectrum from known examples. As in [10] we also draw inspiration from the classical situation. We show that a hypothesis similar to, but weaker than the heat kernel expansion which we call weak heat kernel asymptotic expansion implies regularity and discreteness of the dimension spectrum. The usual heat kernel expansion implies the weak heat kernel expansion. More importantly we show that the weak heat kernel expansion is stable under quantum double suspension, a notion introduced in [12]. Therefore by iteratively quantum double suspending compact Riemannian manifolds we get examples of noncommutative geometries which are regular with finite dimension spectrum. We show noncommutative torus satisfies the weak heat kernel expansion and there by satisfies regularity and discreteness of it’s dimension spectrum. Organization of the paper is as follows. In Section 2 we recall the basics of Mellin transform and asymptotic expansions. In the next section we introduce the weak heat kernel expansion property and show that this implies regularity and finiteness of the dimension spectrum. We also

2718


show that the usual heat kernel expansion implies the weak heat kernel expansion. In the next section we recall the notion of quantum double suspension and show that weak kernel expansion is stable under quantum double suspension. The weak heat kernel expansion of noncommutative torus is also established. In the final section we do a topological version of the theory relevant for applications in quantum homogeneous spaces. In particular we obtain the regularity and dimension spectrum of the odd-dimensional quantum spheres. This gives a conceptual explanation of the results obtained in [14]. 2. Asymptotic expansions and the Mellin transform In this section for reader’s convenience we have recalled some well-known facts about Mellin transforms [16]. We begin with a few basic facts about asymptotic expansions. Let φ : (0, ∞) → C be a continuous function. We say that φ has an asymptotic power series expansion near 0 if there exists a sequence (ar )∞ r=0 of complex numbers such that given N there exist , M > 0 such that if t ∈ (0, ) N r ar t Mt N +1 . φ(t) − r=0

We write φ(t) ∼

∞ 0

ar t r as t → 0+. Note that the coefficients (ar ) are unique. For, −1 r φ(t) − N r=0 ar t aN = lim . t→0+ tN

(2.1)

r If φ(t) ∼ ∞ r=0 ar t as t → 0+ then φ can be extended continuously to [0, ∞) simply by letting φ(0) := a0 . Let X be a topological space and F : [0, ∞) × X → C be a continuous function. Suppose that for every x ∈ X, the function t → F (t, x) has an asymptotic expansion near 0 F (t, x) ∼

∞

ar (x)t r .

(2.2)

r=0

Let x0 ∈ X. We say that expansion (2.2) is uniform at x0 if given N there exist an open set U ⊂ [0, ∞) × X containing (0, x0 ) and an M > 0 such that for (t, x) ∈ U one has N r ar (x)t Mt N +1 . F (t, x) − r=0

We say that expansion (2.2) is uniform if it is uniform at every point of X. Proposition 2.1. Let X be a topological space and F : [0, ∞)×X → C be a continuous function. Suppose that F has a uniform asymptotic power series expansion F (t, x) ∼

∞

ar (x)t r .

r=0

Then for every r 0, the function ar is continuous.


2719

Proof. It is enough to show that the function a0 is continuous. Let x0 ∈ X be given. Since the expansion of F is uniform at x0 , it follows that there exist an open set U containing x0 and δ, M > 0 such that F (t, x) − a0 (x) Mt

for t < δ and x ∈ U.

(2.3)

Let Fn (x) := F ( n1 , x). Then Eq. (2.3) says that Fn converges uniformly to a0 on U . Hence a0 is continuous on U and hence at x0 . This completes the proof. 2 The following two lemmas are easy to prove and we leave the proof to the reader. Lemma 2.2. Let X, Y be topological spaces. Let F : [0, ∞) × X → C and G : [0, ∞) × Y → C be continuous. Suppose that F and G has uniform asymptotic power series expansion. Then the function H : [0, ∞) × X × Y → C defined by H (t, x, y) := F (t, x)G(t, y) has uniform asymptotic power series expansion. Moreover if F (t, x) ∼

∞

ar (x)t r

and G(t, y) ∼

r=0

∞

br (y)t r ,

r=0

then H (t, x, y) ∼

∞

cr (x, y)t r ,

r=0

where cr (x, y) :=

am (x)bn (y).

m+n=r

Lemma 2.3. Let φ : [1, ∞) → C be a continuous function. Suppose that for every N , sup t N φ(t) < ∞. t∈[1,∞)

Then the function s →

∞ 1

φ(t)t s−1 dt is entire.

2.1. The Mellin transform In this section we recall the definition of the Mellin transform of a function defined on (0, ∞) and analyse the relationship between the asymptotic expansion of a function and the meromorphic continuation of its Mellin transform. Let us introduce some notations. We say that a function φ : (0, ∞) → C is of rapid decay near infinity if for every N > 0, supt∈[1,∞) |t N φ(t)| is finite. We let M∞ to be the set of continuous complex valued functions on (0, ∞) which has rapid decay near infinity. For p ∈ R, we let

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Mp (0, 1] := φ : (0, 1] → C: φ is continuous and sup t p φ(t) < ∞ , Mp := φ ∈ M∞ : φ|(0,1] ∈ Mp (0, 1] .

t∈(0,1]

Note that if p q then Mp ⊂ Mq and Mp ((0, 1]) ⊂ Mq ((0, 1]). Definition 2.4. Let φ : (0, ∞) → C be a continuous function. Suppose that φ ∈ Mp for some p. Then the Mellin transform of φ, denoted Mφ, is defined as follows: For Re(s) > p, ∞ Mφ(s) :=

φ(t)t s−1 dt. 0

One can show that if φ ∈ Mp then Mφ is analytic on the right half plane Re(s) > p + 2. Also 1 if φ ∈ Mp ((0, 1]) then s → 0 φ(t)t s−1 is analytic on Re(s) > p + 2. For a 0, let Ha,b,K := {σ + it: a σ b, |t| > K}. Definition 2.5. Let F be a meromorphic function on the entire complex plane with simple poles lying inside the set of integers. We say that F has decay of order r ∈ N along the vertical strips if the function s → s r F (s) is bounded on Ha,b,K for every a 0. We say that F is of rapid decay along the vertical strips if F has decay of order r for every r ∈ N. Proposition Let φ : (0, ∞) → C be a continuous function of rapid decay. Assume that 2.6. r φ(t) ∼ ∞ 0 ar t as t → 0+. Then we have the following. (1) The function φ ∈ M0 . (2) The Mellin transform Mφ of φ extends to a meromorphic function to the whole of complex plane with simple poles in the set of negative integers {0, −1, −2, −3, . . .}. (3) The residue of Mφ at s = −r is given by Ress=−r Mφ(s) = ar . (4) The meromorphic continuation of the Mellin transform Mφ has decay of order 0 along the vertical strips. Proof. By definition it follows that ∞φ ∈ M0 . Since φ has rapid decay at infinity, by Lemma 2.3, it follows that the function s → 1 φ(t)t s−1 dt is entire. Thus modulo a holomorphic function 1 r Mφ(s) ≡ 0 φ(t)t s−1 . For N ∈ N, let RN (t) := φ(t) − N r=0 ar t . Thus modulo a holomorphic function, we have ar + RN (t)t s−1 dt. Mφ(s) ≡ s +r 1

r=0

0

1 As RN ∈ M−(N +1) ((0, 1]) the function s → 0 RN (t)t s−1 dt is holomorphic on Re(s) > −N + 1. Thus on Re(s) > −N + 1, modulo a holomorphic function, one has Mφ(s) ≡

N ar . s+r r=0

(2.4)


2721

This shows that Mφ admits a meromorphic continuation to the whole of complex plane and has simple poles lying in the set of negative integers {0, −1, −2, . . .}. Also (3) follows from Eq. (2.4). Let a 0 be given. Choose N ∈ N such that N + a > 0. Then one has N ar s−1 dt + φ(t)t s−1 dt. Mφ(s) = + RN (t)t s +r ∞

1

r=0

0

1

1 As the function s → s+r is bounded for every r 0 on Ha,b,K , it is enough to show that the ∞ 1 functions ψ(s) := 0 RN (t)t s−1 dt and χ(s) := 1 φ(t)t s−1 dt are bounded on Ha,b,K . By definition of the asymptotic expansion, it follows that there exists an M > 0 such that |RN (t)| Mt N +1 . Hence for s := σ + it ∈ Ha,b,K ,

ψ(s)

M M M. σ +N +1 a+N +1

Thus ψ is bounded on Ha,b,K . Now for s := σ + it ∈ Ha,b,K , we have χ(s)

∞ ∞ σ −1 φ(t)t dt φ(t)t b−1 dt. 1

Since φ is of rapid decay, the integral This completes the proof. 2

1

∞ 1

|φ(t)|t a−1 dt is finite. Hence χ is bounded on Ha,b,K .

Corollary 2.7. Let φ : (0, ∞) → C be a smooth function. Assume that for every n, the n-th derivative φ (n) has rapid decay at infinity and admits an asymptotic power series expansion near 0. (1) For every n, the Mellin transform Mφ (n) of φ (n) extends to a meromorphic function to the whole of complex plane with simple poles in the set of negative integers {0, −1, −2, −3, . . .}. (2) The meromorphic continuation of the Mellin transform Mφ is of rapid decay along the vertical strips. Proof. (1) follows from Proposition 2.6. To prove (2), observe that Mφ (s + 1) = −sMφ(s). For Re(s) 0,

∞

Mφ (s + 1) :=

φ (t)t s dt

0

∞ =−

sφ(t)t s−1 dt 0

= −sMφ(s).

(follows from integration by parts)

2722


As Mφ and Mφ are meromorphic, it follows that Mφ (s + 1) = −sMφ(s). Now a repeated application of this equation gives Mφ(s) := (−1)n

Mφ (n) (s + n) . s(s + 1) · · · (s + n − 1)

(2.5)

Now let a < b, K > 0 and r ∈ N be given. Now (3) of Proposition 2.6 applied to φ (r) , together with Eq. (2.5), implies that the function s → s r Mφ(s) is bounded on Ha,b,K . This completes the proof. 2 The following proposition shows how to pass from the decay properties of the Mellin transform of a function to the asymptotic expansion property of the function. Proposition 2.8. Let φ ∈ Mp for some p. Assume that the Mellin transform Mφ is meromorphic on the entire complex plane with poles lying in the set of negative integers {0, −1, −2, . . .}. Suppose that the meromorphic continuation of the Mellin transform Mφ is of rapid decay along the vertical strips. Then the function φ has an asymptotic expansion near 0. r Moreover if ar := Ress=−r Mφ(s) then φ(t) ∼ ∞ a r=0 r t near 0. Proof. The proof is a simple application of the inverse Mellin transform. Let M 0. Then one has the following inversion formula. M+∞

φ(t) =

Mφ(s)t −s ds.

M−i∞

Define Ft (s) := Mφ(s)t −s . Suppose N ∈ N be given. Let σ ∈ (−N − 1, −N ) be given. For every A > 0, by Cauchy’s integral formula, we have M+iA

σ +iA

Ft (s) ds + M−iA

=

σ −iA

Ft (s) ds +

N

Ft (s) ds +

σ +iA

M+iA

M−iA

Ft (s) ds σ −iA

Ress=−r Ft (s).

(2.6)

r=0

For a fixed t, Ft has rapid decay along the vertical strips. Thus when A → ∞ the second and fourth integrals in Eq. (2.6) vanishes and we obtain the following equation

φ(t) −

N r=0

σ +i∞

Mφ(s)t −s ds.

ar t = r

(2.7)

σ −i∞

But Mφ(σ + it) has rapid decay in t. Let Mσ :=

∞

−∞ |Mφ(σ

+ it)| dt. Then Eq. (2.7) implies


2723

that N ar t r Mσ t −σ Mσ t N φ(t) −

for t 1.

r=0

r N Thus we have shown that for every N , RN (t) := φ(t) − N r=0 ar t = O(t ) as t → 0 and hence N N RN −1 (t) = RN (t) + aN t = O(t ) as t → 0. This completes the proof. 2 3. The weak heat kernel asymptotic expansion property and the dimension spectrum of spectral triples In this section we consider a property of spectral triples which we call the weak heat kernel asymptotic expansion property. We show that a spectral triple having the weak heat kernel asymptotic expansion property is regular and has finite dimension spectrum lying in the set of positive integers. Definition 3.1. Let (A, H, D) be a p+ summable spectral triple for a C ∗ -algebra A where A is a dense ∗-subalgebra of A. We say that the spectral triple (A, H, D) has the weak heat kernel asymptotic expansion property if there exists a ∗-subalgebra B ⊂ B(H) such that: (1) The algebra B contains A. (2) The unbounded derivation δ := [|D|, .] leaves B invariant. Also the unbounded derivation d := [D, .] maps A into B. (3) The algebra B is invariant under the left multiplication by F where F := sign(D). (4) For every b ∈ B, the function τp,b : (0, ∞) → C defined by τp,b (t) = t p Tr(be−t|D| ) has an asymptotic power series expansion. If the algebra A is unital and the representation of A on H is unital then (3) can be replaced by the condition F ∈ B. The next proposition proves that an odd spectral triple that has the heat kernel asymptotic expansion property is regular and has simple dimension spectrum. Theorem 3.2. Let (A, H, D) be a p+ summable spectral triple which has the weak heat kernel asymptotic expansion property. Then the spectral triple (A, H, D) is regular and has finite simple dimension spectrum. Moreover the dimension spectrum is contained in {1, 2, . . . , p}. Proof. Let B ⊂ B(H) be a ∗-algebra for which (1)–(4) of Definition 3.1 is satisfied. The fact that B satisfies (1) and (2) implies that the spectral triple (A, H, D) is regular. First we assume that D is invertible. Let b ∈ B be given. Since |D|−q is trace class for q > p, it follows that for every N > p there exists an M > 0 such that Tr(e−t|D| ) Mt −N Tr(|D|−N ). Now for 1 t < ∞ and N p one has −t|D| b Tr e−t|D| Tr be bMt −N Tr |D|−N .

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Thus the function t → Tr(be−t|D| ) is of rapid decay near infinity. Now observe that for Re(s) 0 Tr b|D|−s =

1 Γ (s)

∞

Tr be−t|D| t s−1 dt.

(3.8)

0

By assumption the function φ(t) := t p Tr(be−t|D| ) has an asymptotic power series expansion near 0. By Eq. (3.8), it follows that Mφ(s) = Γ (s + p) Tr(b|D|−s−p ). Now Proposition 2.6 implies that the function s → Γ (s) Tr(b|D|−s ) is meromorphic with simple poles lying inside {n ∈ Z: n p}. As Γ 1(s) is entire and has simple zeros at {k: k 0}, it follows that the function s → Tr(b|D|−s ) is meromorphic and has simple poles with poles lying in {1, 2, . . . , p}. Suppose D is not invertible. Let P denote the projection onto the kernel of D which is finite dimensional. Let D := D + P and b be an element in B ∞ . Now note that Tr be−t|D | = Tr(P bP )e−t + Tr be−t|D| .

Hence the function t → t p Tr(be−t|D | ) has an asymptotic power series expansion. Thus the function s → Tr(b|D |−s ) is meromorphic with simple poles lying in {1, 2, . . . , p}. Observe that for Re(s) 0, Tr(b|D |−s ) = Tr(b|D|−s ). Hence the function s → Tr(b|D|−s ) is meromorphic with simple poles lying in {1, 2, . . . , p}. This completes the proof. 2 Remark 3.3. If Tr(be−t|D| ) ∼

∞

r=−p ar (b)t

r

then (3) of Proposition 2.6 implies that

1 Resz=k Tr b|D|−z = a−k (b) k! −z Tr b|D| z=0 = a0 (b).

for 1 k p,

Remark 3.4. Let (A, H, D) be a spectral triple which has the weak heat kernel asymptotic expansion property. Then the dimension spectrum Σ is finite and lies in the set of positive integers. We call the greatest element in the dimension spectrum as the dimension of the spectral triple (A, H, D). If Σ is empty we set the dimension to be 0. Now in the next proposition we show that the usual heat kernel asymptotic expansion implies the weak heat kernel asymptotic expansion. Theorem 3.5. Let (A, H, D) be a p+ summable spectral triple for a C ∗ -algebra A. Suppose that B is a ∗-subalgebra of B(H) satisfying (1)–(4) of Definition 3.1. Assume that for every 2 2 b ∈ B, the function σp,b : (0, ∞) → C defined by σp,b (t) := t p Tr(be−t D ) has an asymptotic power series expansion. Then for every b ∈ B, the function τp,b : t → t p Tr(be−t|D| ) has an asymptotic power series expansion. Proof. It is enough to consider the case where D is invertible. Let b ∈ B be given. Let 2 2 ψ denote the Mellin transform of the function t → Tr(be−t D ) and χ denote the Mellin transform of the function t → Tr(be−t|D| ). Then a simple change of variables shows that


ψ(s) =

Γ ( 2s ) 2

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Tr(b|D|−s ). But then χ(s) = Γ (s) Tr(b|D|−s ). Thus we obtain the equation χ(s) =

2Γ (s) ψ(s). Γ ( 2s )

But we have following duplication formula for the gamma function

√ 1 = 21−2s π Γ (2s). Γ (s)Γ s + 2 Hence one has

1 s s +1 χ(s) = √ 2 Γ ψ(s). 2 π Now Proposition 2.6 implies that ψ has decay of order 0 along the vertical strips and has simple poles lying inside {n ∈ Z: n p}. Since the gamma function has rapid decay along the vertical strips, it follows that χ has rapid decay along the vertical strips and has poles lying in {n ∈ Z: n p}. If χ˜ denotes the Mellin transform of τp (., b) then χ˜ (s) = χ(s + p). Hence χ˜ has rapid decay along the vertical strips and has poles lying in the set of negative integers. Now Proposition 2.8 implies that the map t → t p Tr(be−t|D| ) has an asymptotic power series expansion near 0. This completes the proof. 2 4. Stability of the weak heat kernel expansion property and the quantum double suspension Let us recall the definition of the quantum double suspension of a unital C ∗ -algebra. The quantum double suspension is first defined in [12] and our equivalent definition is as in [13]. Let us fix some notations. We denote the left shift on 2 (N) by S which is defined on the standard orthonormal basis (en ) as Sen = en−1 and p denote the projection |e0 e0 |. The number operator on 2 (N) is denoted by N and defined as N en := nen . We denote the C ∗ -algebra generated by S in B( 2 (N)) by T which is the Toeplitz algebra. Note that SS ∗ = 1 and p = 1 − S ∗ S. Let σ : T → C(T) be the symbol map which sends S to the generating unitary z. Then one has the following exact sequence σ 0→K→T − → C(T) → 0.

Definition 4.1. Let A be a unital C ∗ -algebra. Then the quantum double suspension of A denoted Σ 2 (A) is the C ∗ -algebra generated by A ⊗ p and 1 ⊗ S in A ⊗ T . Let A be a unital C ∗ -algebra. One has the following exact sequence. ρ 0 → A ⊗ K 2 (N) → Σ 2 (A) − → C(T) → 0 where ρ is just the restriction of 1 ⊗ σ to Σ 2 (A).

2726


Remark 4.2. It can be easily shown that Σ 2 (C(T)) = C(SU q (2)) and more generally one can show that Σ 2 (C(Sq2n−1 )) = C(Sq2n+1 ). We refer to [12] or Lemma 3.2 of [14] for the proof. Thus the odd-dimensional quantum spheres can be obtained from the circle T by applying the quantum double suspension recursively. Let A be a dense ∗-subalgebra of a C ∗ -algebra A. Define

2 (A) := span a ⊗ k, 1 ⊗ S n , 1 ⊗ S ∗m : a ∈ A, k ∈ S 2 (N) , n, m 0 Σalg where S( 2 (N)) := {(amn ): m,n (1 + m + n)p |amn | < ∞ for every p}. 2 (A) is just the ∗-algebra generated by A ⊗ 2 2 Then Σalg alg S( (N)) and 1 ⊗ S. Clearly Σalg (A) is a dense subalgebra of Σ 2 (A). Definition 4.3. Let (A, H, D) be a spectral triple and denote the sign of the operator D by F . 2 (A), H ⊗ 2 (N), Σ 2 (D) := (F ⊗ 1)(|D| ⊗ 1 + 1 ⊗ N )) is called Then the spectral triple (Σalg the quantum double suspension of the spectral triple (A, H, D). 4.1. Stability of the weak heat kernel expansion We consider the stability of the weak heat kernel expansion under quantum double suspension. First observe that the following are easily verifiable. (1) The spectral triple (S( 2 (N)), 2 (N), N ) has the weak heat kernel asymptotic expansion with dimension 0. asymptotic (2) Let (Ai , Hi , Di ) be a spectral triple with the weak heat kernel expansionproperty with dimension pi for 1 i n. Then the spectral triple ( ni=1 Ai , ni=1 Hi , ni=1 Di ) has the weak heat kernel expansion property with dimension p := max{pi : 1 i n}. (3) If (A, H, D) is a spectral triple with the weak heat kernel asymptotic expansion property and has dimension p then (A, H, |D|) also has the weak heat kernel asymptotic expansion with the same dimension p. (4) Let (A, H, D) be a spectral triple with the weak heat kernel asymptotic expansion property with dimension p. Then the amplification (A ⊗ 1, H ⊗ 2 (N), |D| ⊗ 1 + 1 ⊗ N ) also has the asymptotic expansion property with dimension p + 1. We start by proving the stability of the weak heat kernel expansion under tensoring by compacts. Proposition 4.4. Let (A, H, D) be a spectral triple with the weak heat kernel asymptotic expansion property of dimension p. Then (A ⊗alg S( 2 (N)), H ⊗ 2 (N), D0 := (F ⊗ 1)(|D| ⊗ 1 + 1 ⊗ N)) also has the weak heat kernel asymptotic expansion property with dimension p. Proof. Let B ⊂ B(H) be a ∗-subalgebra for which (1)–(4) of Definition 3.1 are satisfied. We denote B ⊗alg S( 2 (N)) by B0 . We show that B0 satisfies (1)–(4) of Definition 3.1. Clearly (1) holds. We denote the unbounded derivation [|D0 |, .], [|D|, .] and [N, .] by δD0 , δD and δN respectively. By assumption δD leaves B invariant. Clearly B ⊗alg S( 2 (N)) is contained in the domain


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of δD0 and δD0 = δD ⊗ 1 + 1 ⊗ δN on B ⊗alg S( 2 (N)). Similarly one can show that the unbounded derivation [D0 , .] maps A ⊗alg S( 2 (N)) into B0 invariant. As F0 := sign(D0 ) = F ⊗ 1, (3) is clear. Now (4) follows from Lemma 2.2 and the equality t p Tr(b ⊗ k)e−t|D0 | = t p Tr(be−t|D| ) Tr(ke−tN ). This completes the proof. 2 Now we consider the stability of the heat kernel asymptotic expansion under the double suspension. Theorem 4.5. Let (A, H, D) be a spectral triple with the weak heat kernel asymptotic expansion property of dimension p. Assume that the algebra A is unital and the representation on H is unital. Then the spectral triple (Σ 2 (A), H ⊗ 2 (N), Σ 2 (D)) also has the weak heat kernel asymptotic expansion property with dimension p + 1. Proof. We denote Σ 2 (D) by D of Defini0 . Let B be a ∗-subalgebra of B(H)for which (1)–(4) tion 3.1 are satisfied. For f = n λn zn ∈ C ∞ (T), we let σ (f ) := n0 λn S n + n>0 λ−n S ∗n . 2 We denote the projection 1+F 2 by P . We let B0 to denote the algebra B ⊗alg S( (N)) as in Proposition 4.4. As in Proposition 4.4, we let δD0 , δD , δN to denote the unbounded derivations [|D0 |, .], [|D|, .] and [N, .] respectively. Define

B˜ := b + P ⊗ σ (f ) + (1 − P ) ⊗ σ (g): b ∈ B0 , f, g ∈ C ∞ (T) . Now it is clear that B˜ satisfies (1) of Definition 3.1. We have already shown in Proposition 4.4 that B0 is closed under δD0 and d0 := [D0 , .] maps A ⊗ S( 2 (N)) into B0 . Now note that δD0 P ⊗ σ (f ) = P ⊗ σ if , δD0 (1 − P ) ⊗ σ (g) = (1 − P ) ⊗ σ ig , D0 , P ⊗ σ (f ) = P ⊗ σ if , D0 , (1 − P ) ⊗ σ (g) = −(1 − P ) ⊗ σ ig . ˜ Thus it follows that δD0 leaves B˜ invariant and d0 := [D0 , .] maps Σ2 (A) into B. ˜ Now we show that B˜ Since F0 := sign(D0 ) = F ⊗ 1, it follows from definition that F0 ∈ B. satisfies (4). We have already shown in Proposition 4.4 that given b ∈ B0 , the function τp,b (t) = t p Tr(be−t|D0 | ) has an asymptotic expansion. Hence the function τp+1,b has an asymptotic expansion for every b ∈ B0 . Now note that

τp+1,P ⊗σ (f ) (t) =

τp+1,(1−P )⊗σ (g) (t) =

f (θ ) dθ t p Tr P e−t|D| t Tr e−tN , g(θ ) dθ t p Tr (1 − P )e−t|D| t Tr e−tN .

(4.9) (4.10)

Now recall that we have assumed that A is unital and hence P ∈ B. Hence t p Tr(xe−t|D| ) has an asymptotic power series expansion for x ∈ {P , 1 − P }. Thus t Tr(e−tN ) has an asymptotic

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power series expansion. From Eqs. (4.9), (4.10) and from the earlier observation that τp+1,b has ˜ the function an asymptotic power series expansion for b ∈ B0 , it follows that for every b ∈ B, τp+1,b has an asymptotic power series expansion. This completes the proof. 2 4.2. Higson’s differential pair and the heat kernel expansion Now we discuss some examples of spectral triples which satisfy the weak heat kernel asymptotic expansion property. In particular we discuss the spectral triple associated to noncommutative torus and the classical spectral triple associated to a spin manifold. Let us recall Higson’s notion of a differential pair as defined in [11]. Consider a Hilbert space H and a positive, selfadjoint and an unbounded on H. We assume k that has compact resolvent. For k ∈ N, we let Hk be the domain of the operator 2 . The vector k 2 space Hk is given a Hilbert space structure by identifying Hk with the graph of the operator . Denote the intersection k Hk by H∞ . An operator T : H∞ → H∞ is said to be of analytic order m where m ∈ Z if T extends to a bounded operator from Hk+m → Hk for every k. We say an operator T on H∞ has analytic order −∞ if T has analytic order less than −m for every m > 0. The following definition is due to Higson [11]. Definition 4.6. Let be a positive, unbounded, selfadjoint operator on a Hilbert space H with compact resolvent. Suppose that D := p0 Dq is a filtered algebra of operators on H∞ . The pair (D, ) is called a differential pair if the following conditions hold. 1. The algebra D is invariant under the derivation T → [, T ]. 2. If X ∈ Dq , then [, X] ∈ Dq+1 . 3. If X ∈ Dq , then the analytic order of X q. Now let us recall Higson’s definition of pseudodifferential operators. Definition 4.7. Let (D, ) be a differential pair. We denote the orthogonal projection onto the kernel of by P . Then P is of finite rank as has compact resolvent. Let 1 := + P . Then 1 is invertible. A linear operator T on H∞ is called a basic pseudodifferential operator of order k is for every there exist m and X ∈ Dm+k such that −m 2

T = X1

+R

where R has analytic order less than or equal to . A finite linear combinations of basic pseudodifferential operator of order k is called a pseudodifferential operator of order k. We denote the set of pseudodifferential operators of order 0 by Ψ0 (D, ). It is proved in [11] that the pseudodifferential operators of order 0 form an algebra. We need the following 1 proposition due to Higson. Denote the derivation T → [ 2 , T ] by δ. Proposition 4.8. Let (D, ) be a differential pair. The derivation δ leaves the algebra Ψ0 (D, ) invariant.


2729

r

Let (D, ) be a differential pair. Assume that − 2 is trace class for some r > 0. We say that the analytic dimension of (D, ) is p if

−r p := inf q > 0: 2 is trace class for every r > q . Let us make the following definition of the heat kernel expansion for a differential pair. Definition 4.9. Let (D, ) be a differential pair of analytic dimension p. We say that (D, ) 2 has a heat kernel expansion if for X ∈ Dm , the function t → t p+m Tr(Xe−t ) has an asymptotic expansion near 0. Now we show that if (D, ) has the heat kernel expansion then the algebra Ψ0 (D, ) has the weak heat kernel expansion. Proposition 4.10. Let (D, ) be a differential pair of analytic dimension p having the heat 1 kernel expansion. Denote the operator 2 by |D|. Then for every b ∈ Ψ0 (D, ), the function p −t|D| ) has an asymptotic power series expansion. t → t Tr(be Proof. First observe that if R : H∞ → H∞ is an operator of analytic order < −p − n − 1 then R|D|n+1 is trace class and hence by Taylor’s series n −t|D| (−1)k Tr(R|D|k ) k t + O t n+1 Tr Re = k! k=0

−m

for t near 0. Thus it is enough to show the result when b = X1 2 . For an operator T on H∞ , let ζT (s) := Tr(T |D|−s ). Then ζb (s) := ζX (s + m). As in Proposition 3.5 one can show that Γ (s)ζX (s) has rapid decay along the vertical strips. Now Γ (s)ζb (s) =

Γ (s) Γ (s + m)ζX (s + m). Γ (s + m)

Hence Γ (s)ζb (s) has rapid decay along the vertical strips. But Γ (s)ζb (s) is the Mellin transform of Tr(be−t|D| ). Hence by Proposition 2.8, it follows that t p Tr(be−t|D| ) has an asymptotic power series expansion. This completes the proof. 2 We make use of the following proposition to prove that spectral triple associated to the NC torus and that of a spin manifold posses the weak heat kernel expansion property. Proposition 4.11. Let (A, H, D) be a finitely summable spectral triple and := D 2 . Suppose that there exists an algebra of operators D := p0 Dp such that (D, ) is a differential pair of analytic dimension p. Assume that (D, ) satisfies the following: 1. the algebra D0 contains A and [D, A], 2. the differential pair (D, ) has the heat kernel expansion property, 3. the operator D ∈ D1 . Then the spectral triple (A, H, D) has the weak heat kernel asymptotic expansion property.

2730


Proof. Without loss of generality, we can assume that D is invertible. We let B be the algebra of pseudodifferential operators of order 0 associated to (D, ). Now Proposition 4.8 together with the fact that D0 ⊂ B shows that B contains A and [D, A] and is invariant under δ := [|D|, .]. −1 Since D ∈ D1 , it follows that F := D 2 ∈ B. Now (4) of Definition 3.1 follows from Proposition 4.10. This completes the proof. 2 4.3. Examples Now we discuss some examples of spectral triples which satisfy the weak heat kernel asymptotic expansion. We start with the classical example. Let M be a Riemannian spin manifold and S → M be a spinor bundle. We denote the Hilbert space of square integrable sections on L2 (M, S) by H. We represent C ∞ (M) on H by multiplication operators. Let D be the Dirac operator associated with Levi–Civita connection. Then the triple (C ∞ (M), H, D) is a spectral triple. Then the operator D 2 is then a generalised Laplacian [1]. Let D denote the usual algebra of differential operators on S. Then (D, ) is a differential pair. Moreover Proposition 2.4.6 in [1] implies that (D, ) has the heat kernel expansion. Also D ∈ D1 . Now Proposition 4.11 implies that the spectral triple (C ∞ (M), H, D) has the weak heat kernel asymptotic expansion. 4.3.1. The spectral triple associated to the NC torus Let us recall the definition of the noncommutative torus which we abbreviate as NC torus. Throughout we assume that θ ∈ [0, 2π). Definition 4.12. The C ∗ -algebra Aθ is defined as the universal C ∗ -algebra generated by two unitaries u and v such that uv = eiθ vu. Define the operators U and V on 2 (Z2 ) as follows U em,n := em+1,n , V em,n := e−inθ em,n+1 where {em,n } denotes the standard orthonormal basis on 2 (Z2 ). Then it is well known that u → U and v → V give a faithful representation of the C ∗ -algebra Aθ . Consider the positive selfadjoint operator on H := 2 (Z2 ) defined on the orthonormal basis {em,n } by (em,n ) = (m2 + n2 )em,n . For a polynomial P = p(m, n), define the operator TP on H∞ by TP (em,n ) := p(m, n)em,n . The group Z2 acts on the algebra of polynomials as follows. For x := (a, b) ∈ Z2 and P := p(m, n), define x.P := p(m − a, n − b). We denote (1, 0) by e1 and (0, 1) by e2 . k Note that if P is a polynomial of degree k, then TP − 2 is bounded on Ker()⊥ . Thus it follows that if P is a polynomial of degree k then TP has analytic order k. Also note that k

−k

12 U 12 em,n :=

k

((m + 1)2 + n2 ) 2 k

(m2 + n2 ) 2

em+1,n

if (m, n) = 0.


2731

Thus it follows that U is of analytic order 0. Similarly one can show that V is of analytic order 0. Now note the following commutation relationship U TP := Te1 .P U,

(4.11)

V TP := Te1 .P V .

(4.12)

Thus it follows that [, U α V β ] = TQ U α V β for some degree 1 polynomial Q. Let us define Dp := span{TPα,β U α V β : deg(Pα,β ) k} and let D := p Dp . The above observations can be rephrased into the following proposition. Proposition 4.13. The pair (D, ) is a differential pair of analytic dimension 2. Now we show that the differential pair (D, ) has the heat kernel expansion. Proposition 4.14. The differential pair (D, ) has the heat kernel expansion property. Proof. Let X ∈ Dq be given. It is enough to consider the case when X := TP U α V β . First note that Tr(Xe−t ) = 0 unless (α, β) = 0. Now let X := TP . Again it is enough to consider the case when P is a monomial. Let P = p(m, n) = mk1 nk2 . Now Tr TP e−t =

mk1 e−tm

2

m∈Z

nk2 e−tn

2

.

n∈Z

Now the asymptotic expansion follows from applying Proposition 2.4.6 in [1] to the standard Laplacian on the circle. This completes the proof. 2 Let Aθ be the ∗-algebra generated by U and V . We consider the direct sum representation of 0 T . Then D is selfadjoint on H ⊕ H and D 2 = 0 0 . It Aθ on H ⊕ H. Define D := Tm+in m−in 0 is well known that (Aθ , H ⊕ H, D) is a 2+ summable spectral triple. Proposition 4.15. The spectral triple (Aθ , H ⊕ H, D) has the weak heat kernel asymptotic expansion property. Proof. Let (D, ) be the differential pair considered in Proposition 4.13. Then the amplification (D := M2 (D), D 2 ) is a differential pair. Note that D ∈ D1 . Clearly Aθ ∈ D . Note the commutation relations [Tm±in , U ] = U, [Tm±in , V ] = ±iV . This implies that [D, Aθ ] ⊂ D0 . Since (D, ) has the heat kernel expansion, it follows that the differential pair (M2 (D), D 2 ) also has the heat kernel expansion. Now Proposition 4.11 implies that the spectral triple (Aθ , H ⊕ H, D) has the weak heat kernel expansion. This completes the proof. 2

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4.3.2. The torus equivariant spectral triple on the odd-dimensional quantum spheres In this section we recall the spectral triple for the odd-dimensional quantum spheres given in [5]. We begin with some known facts about odd-dimensional quantum spheres. Let q ∈ (0, 1]. The C ∗ -algebra C(Sq2 +1 ) of the quantum sphere Sq2 +1 is the universal C ∗ -algebra generated by elements z1 , z2 , . . . , z +1 satisfying the following relations (see [12]): zi zj = qzj zi ,

1 j i

+1

zi zi∗ = 1.

i=1

We will denote by A(Sq2 +1 ) the ∗-subalgebra of A generated by the zj ’s. Note that for = 0, the C ∗ -algebra C(Sq2 +1 ) is the algebra of continuous functions C(T) on the torus and for = 1, it is C(SU q (2)). There is a natural torus group T +1 action τ on C(Sq2 +1 ) as follows. For w = (w1 , . . . , w +1 ), define an automorphism τw by τw (zi ) = wi zi . Recall that N is the number operator on 2 (N) and S is the left shift on 2 (N). We also use the same notation S for the left shift on 2 (Z). We let H denote the Hilbert space 2 (N × Z). Let Yk,q be the following operators on H :

Yk,q =

⎧ ⎪ q N ⊗ · · · ⊗ q N ⊗ 1 − q 2N S ∗ ⊗ I ⊗ · · · ⊗ I if 1 k , ⎪ ⎪ ⎪ ⎨ k−1 copies

+1−k copies ⎪ qN ⊗ · · · ⊗ qN ⊗ S∗ ⎪ ⎪ ⎪ ⎩

if k = + 1.

(4.13)

copies

Then π : zk → Yk,q gives a faithful representation of C(Sq2 +1 ) on H for q ∈ (0, 1) (see Lemma 4.1 and Remark 4.5 [12]). We will denote the image π (C(Sq2 +1 )) by A (q) or by just A . Let {eγ : γ ∈ ΓΣ } be the standard orthonormal basis for H . We recall the following theorem from [5]. Theorem 4.16. (See [5].) Let D be the operator eγ → d(γ )eγ on H where the dγ ’s are given by d(γ ) =

γ1 + γ2 + · · · + γ + |γ +1 | −(γ1 + γ2 + · · · + γ + |γ +1 |)

if γ +1 0, if γ +1 < 0.

Then (A(Sq2 +1 ), H , D ) is a non-trivial ( + 1) summable spectral triple.


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But to deduce that the spectral triple (A(Sq2 +1 ), H , D ) satisfies the weak heat kernel asymptotic expansion, we need a topological version of Definition 3.1 and Proposition 4.5. We do this in the next section. 5. Smooth subalgebras and the weak heat kernel asymptotic expansion First we recall the definition of smooth subalgebras of C ∗ -algebras. For an algebra A (possibly non-unital), we denote the algebra obtained by adjoining a unit to A by A+ . Definition 5.1. Let A be a unital C ∗ -algebra. A dense unital ∗-subalgebra A∞ is called smooth in A if: 1. The algebra A∞ is a Fréchet ∗-algebra. 2. The unital inclusion A∞ ⊂ A is continuous. 3. The algebra A∞ is spectrally invariant in A i.e. if an element a ∈ A∞ is invertible in A then a −1 ∈ A∞ . Suppose A is a non-unital C ∗ -algebra. A dense Fréchet ∗-subalgebra A∞ is said to be smooth in A if (A∞ )+ is smooth in A+ . We also assume that our smooth subalgebras satisfy the condition that if A∞ ⊂ A is smooth ˆ π S( 2 (Nk )) ⊂ A ⊗ K( 2 (Nk )) is smooth. then A∞ ⊗ Let A be a unital C ∗ -algebra and A∞ be a smooth unital ∗-subalgebra of A. Assume that the topology on A∞ is given by the countable family of seminorms ( · p ). Let us denote the operator 1 ⊗ S by α. Define the smooth quantum double suspension of A∞ as follows Σ 2 A∞ :=

α ∗j (aj k ⊗ p)α k +

j,k∈N

λk α k +

k0

λ−k α ∗k : aj k ∈ A∞ ,

k>0

(1 + j + k)n aj k p < ∞, (λk ) is rapidly decreasing .

(5.14)

j,k

Now let us topologize Σ 2 (A∞ ) by defining a seminorm n,p for every n, p 0. For an element a :=

α ∗j (aj k ⊗ p)α k +

j,k∈N

k0

λk α k +

λ−k α ∗k

k>0

in Σ 2 (A∞ ) we define an,p by an,p :=

n n 1 + |j | + |k| aj k p + 1 + |k| |λk |. j,k∈N

k∈Z

It is easily verifiable that: 1. The subspace Σ 2 (A∞ ) is a dense ∗-subalgebra of Σ 2 (A).

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2. The topology on Σ 2 (A∞ ) induced by the seminorms ( n,p ) makes Σ 2 (A∞ ) a Fréchet ∗-algebra. 3. The unital inclusion Σ 2 (A∞ ) ⊂ Σ 2 (A) is continuous. The next proposition proves that the Fréchet algebra Σ 2 (A∞ ) is infact smooth in Σ 2 (A). Proposition 5.2. Let A be a unital C ∗ -algebra and let A∞ ⊂ A be a unital smooth subalgebra ˆ π S( 2 (Nk )) ⊂ A ⊗ K( 2 (Nk )) is smooth for every k ∈ N. Then the algebra such that A∞ ⊗ 2 ∞ ˆ π S( 2 (Nk )) is smooth in Σ 2 (A) ⊗ K( 2 (Nk )) for every k 0. Σ (A ) ⊗ Proof. Let us denote the restriction of 1 ⊗ σ to Σ 2 (A) by ρ. Recall the σ : T → C(T) is the symbol map sending S to the generating unitary. Then one has the following exact sequence at the C ∗ -algebra level ρ 0 → A ⊗ K 2 (N) → Σ 2 (A) − → C(T) → 0. At the subalgebra level one has the following “sub” exact sequence ρ ˆ π S 2 (N) → Σ 2 A∞ − 0 → A∞ ⊗ → C ∞ (T) → 0. ˆ π S( 2 (N)) ⊂ A ⊗ K( 2 (N)) and C ∞ (T) ⊂ C(T) are smooth, it follows from TheSince A∞ ⊗ ˆπ orem 3.2, part 2 [15] that Σ 2 (A∞ ) is smooth in Σ 2 (A). One can prove that Σ 2 (A∞ ) ⊗ S( 2 (Nk )) is smooth in Σ 2 (A) ⊗ K( 2 (Nk )) for every k > 0 along the same lines first by tensoring the C ∗ -algebra exact sequence by K( 2 (Nk )) and then by tensoring the Fréchet algebra exact sequence by S( 2 (Nk )) and appealing to Theorem 3.2, part 2 of [15]. This completes the proof. 2 5.1. The topological weak heat kernel expansion We need the following version of the weak heat kernel expansion. Definition 5.3. Let (A∞ , H, D) be a p+ summable spectral triple for a C ∗ -algebra A where A∞ is smooth in A. We say that the spectral triple (A∞ , H, D) has the topological weak heat kernel asymptotic expansion property if there exists a ∗-subalgebra B ∞ ⊂ B(H) such that: The algebra B ∞ has a Fréchet space structure and endowed with it, it is a Fréchet ∗-algebra. The algebra B ∞ contains A∞ . The inclusion B ∞ ⊂ B(H) is continuous. The unbounded derivations δ := [|D|, .] leaves B ∞ invariant and is continuous. Also the unbounded derivation d := [D, .] maps A∞ into B ∞ in a continuous fashion. (5) The left multiplication by the operator F := sign(D) denoted LF leaves B ∞ invariant and is continuous. (6) The function τp : (0, ∞) × B ∞ → C defined by τp (t, b) = t p Tr(be−t|D| ) has a uniform asymptotic power series expansion.

(1) (2) (3) (4)

We need the following analog of Proposition 4.4 and Proposition 4.5. First we need the following two lemmas.


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Lemma 5.4. Let E be a Fréchet space and F ⊂ E be a dense subspace. Let φ : (0, ∞) × E → C be a continuous function which is linear in the second variable. Suppose that φ : (0, ∞) × F → C has a uniform asymptotic power series expansion then φ : (0, ∞) × E → C has a uniform asymptotic power series expansion. r Proof. Suppose that φ(t, f ) ∼ ∞ r=0 ar (f )t . Then ar : F → C is linear and is continuous for every r ∈ N. Since F is dense in E, for every r ∈ N, the function ar admits a continuous extension to the whole of E which we still denote it by ar . Now fix N ∈ N. Then there exist a neighbourhood U of E containing 0 and , M > 0 such that N r ar (f )t Mt N +1 φ(t, f ) −

for 0 < t < , f ∈ U ∩ F.

(5.15)

r=0

Since φ(t, .) and ar (.) are continuous and as F is dense in E, Eq. (5.15) continues to hold for every f ∈ U . This completes the proof. 2 Lemma 5.5. Let E1 , E2 be Fréchet spaces and let Fi : (0, ∞) × Ei → C be continuous and ˆ π E2 → C linear in the second variable for i = 1, 2. Consider the function F : (0, ∞) × E1 ⊗ be defined by F (t, e1 ⊗ e2 ) = F1 (t, e1 )F (t, e2 ). Assume that F is continuous. If F1 and F2 have uniform asymptotic expansions then F has a uniform asymptotic power series expansion. Proof. By Lemma 5.4, it is enough to show that F : (0, ∞) × E1 ⊗alg E2 → C has a uniform asymptotic power series expansion. Let θ : E1 × E2 → E1 ⊗alg E2 be defined by θ (e1 , e2 ) = e1 ⊗ e2 . Consider the map G : (0, ∞) × E1 × E2 → C defined by G(t, e1 , e2 ) := F (t, θ ((e1 , e2 ))). By Lemma 2.2, it follows that G has a uniform asymptotic power series expansion say

G(t, e) ∼

∞

ar (e)t r .

r=0

ˆ π E2 → C be the linear The maps ar : E1 × E2 → C are continuous bilinear. We let a˜ r : E1 ⊗ maps such that a˜ r ◦ θ := ar . Let N ∈ N be given. Then there exist , M > 0 and open sets U1 , U2 containing 0 in E1 , E2 such that N r ar (e)t Mt N +1 G(t, e) −

for 0 < t < , e ∈ U1 × U2 .

(5.16)

r=0

Without loss of generality, we can assume that Ui := {x ∈ Ei : pi (x) < 1} for a seminorm pi of Ei . Now Eq. (5.16) implies that N r a˜ r θ (e) t Mt N +1 F t, θ (e) − r=0

for 0 < t < , e ∈ U1 × U2 .

(5.17)

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Hence for t ∈ (0, ) and x ∈ θ (U1 × U2 ), N r a˜ r (x)t Mt N +1 . F (t, x) −

(5.18)

r=0

Since a˜ r is linear and F is linear in the second variable, it follows that Eq. (5.18) continues to hold for x in the convex hull of θ (U1 × U2 ) which is nothing but the unit ball determined by the seminorm p1 ⊗ p2 in E1 ⊗alg E2 . This completes the proof. 2 In the next proposition, we consider the stability of the weak heat kernel asymptotic expansion property for tensoring by smooth compacts. Proposition 5.6. Let (A∞ , H, D) be a spectral triple where the algebra A∞ is a smooth subalgebra of C ∗ -algebra. Assume that (A∞ , H, D) has the topological weak heat kernel exˆ π S( 2 (N)), H ⊗ 2 (N), pansion property with dimension p. Then the spectral triple (A∞ ⊗ D0 := (F ⊗ 1)(|D| ⊗ 1 + 1 ⊗ N )) also has the weak heat kernel asymptotic expansion property with dimension p where F := sign(D). Proof. Let B ∞ ⊂ B(H) be a ∗-subalgebra for which (1)–(6) of Definition 5.3 are satisfied. We ˆ π S( 2 (N)) by B0∞ . We show that B0∞ satisfies (1)–(6) of Definition 5.3. First note denote B ∞ ⊗ that the natural representation of B0∞ in H ⊗ 2 (N) is injective. Thus (3) is clear. Also (1) and (2) are obvious. Now let us now prove (4). We denote the unbounded derivation [|D0 |, .], [|D|, .] and [N, .] by δD0 , δD and δN respectively. By assumption δD leaves B invariant and is continuous. It is also easy to see that δN leaves S( 2 (N)) invariant and is continuous. Let δ := δD ⊗ 1 + 1 ⊗ δN . Then δ : B0∞ → B0∞ is continuous. Clearly B ∞ ⊗alg S( 2 (N)) is contained in the domain of δ and δ = δ on B ∞ ⊗alg S( 2 (N)). Now let a ∈ B0∞ be given. Then there exists a sequence (an ) in B ∞ ⊗π S( 2 (N)) such that (an ) converges to a in B0∞ . Since δ is continuous on B0∞ and the inclusion B0∞ ⊂ B(H) is continuous, it follows that δD0 (an ) = δ (an ) converges to δ (a). As δD0 is a closed derivation, it follows that a ∈ Dom(δD0 ) and δD0 (a) = δ (a). Hence we have shown that δD0 leaves B0∞ invariant and is continuous. Similarly one can show that the unbounded derivation d0 := [D0 , .] ˆ π S( 2 (N)) into B0∞ invariant in a continuous manner. maps A ⊗ As F0 := sign(D0 ) = F ⊗ 1, (5) is clear. Consider the function τp : (0, ∞) × B0∞ → C defined by τp (t, b) := t p Tr(be−t|D0 | ). Then τp (t, b ⊗ k) = τp (t, b)τ0 (t, k). Hence by Lemma 5.5, it follows that τp has a uniform asymptotic power series expansion. This completes the proof. 2 Now we consider the stability of the weak heat kernel asymptotic expansion under the double suspension. Theorem 5.7. Let (A∞ , H, D) be a spectral triple with the topological weak heat kernel asymptotic expansion property of dimension p. Assume that the algebra A∞ is unital and the representation on H is unital. Then the spectral triple (Σ 2 (A∞ ), H ⊗ 2 (N), Σ 2 (D)) also has the topological weak heat kernel asymptotic expansion property with dimension p + 1. ∞ be ∗-subalgebra of B(H) for which Proof. We denote the operator Σ 2 (D) by D0 . Let B (1)–(6) of Definition 5.3 are satisfied. For f = n∈Z λn zn ∈ C(T), we let σ (f ) :=


∗n . We

denote the projection 1+F 2 by P . We assume here that P = ±1 ∞ ˆ π S( 2 (N)) as in Proposias the case P = ±1 is similar. We let B0 to denote the algebra B ∞ ⊗ tion 5.6. As in Proposition 5.6, we let δD0 , δD , δN to denote the unbounded derivations [|D0 |, .], [|D|, .] and [N, .] respectively. Define n0 λn S

n+

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n>0 λ−n S

B˜ ∞ := b + P ⊗ σ (f ) + (1 − P ) ⊗ σ (g): b ∈ B0∞ , f, g ∈ C ∞ (T) . Then B˜ ∞ is isomorphic to the direct sum B0∞ ⊕ C ∞ (T) ⊕ C ∞ (T). We give B˜ ∞ the Fréchet space structure coming from this decomposition. It is easy to see that B˜ ∞ is a Fréchet ∗-subalgebra of B(H ⊗ 2 (N)). Clearly (π ⊗ 1)(Σ 2 (A∞ )) ⊂ B˜ ∞ . Thus we have shown that (1) and (2) of Definition 5.3 are satisfied. Since B0∞ is represented injectively on H ⊗ 2 (N), it follows that B˜ satisfies (3). We have already shown in Proposition 5.6 that B0∞ is closed under δD0 and is continuous. ˆ π S( 2 (N)) into B0∞ continuously. Now note Also we have shown that d0 := [D0 , .] maps A ⊗ that δD0 P ⊗ σ (f ) = P ⊗ σ if , δD0 (1 − P ) ⊗ σ (g) = (1 − P ) ⊗ σ ig , D0 , P ⊗ σ (f ) = P ⊗ σ if , D0 , (1 − P ) ⊗ σ (g) = −(1 − P ) ⊗ σ ig . Thus it follows that δD0 leaves B˜ ∞ invariant and is continuous. Also, it follows that d0 := [D0 , .] maps Σ 2 (A∞ ) into B˜ in a continuous manner. Since F0 := sign(D0 ) = F ⊗ 1, it follows from definition that F0 ∈ B˜ ∞ . Now we show that B˜ ∞ satisfies (6). We have already shown in Proposition 5.6 that the function τp : (0, ∞) ⊗ B0∞ → C defined by τp (t, b) := t p Tr(be−t|D0 | ) has a uniform asymptotic power series expansion. Hence τp+1 restricted to B0∞ has a uniform asymptotic power series expansion. Now note that τp+1

P ⊗ σ (f ) =

τp+1 (1 − P ) ⊗ σ (g) =

f (θ ) dθ t p Tr P e−t|D| t Tr e−tN ,

(5.19)

g(θ ) dθ t p Tr (1 − P )e−t|D| t Tr e−tN .

(5.20)

Now recall that we have assumed that A∞ is unital and hence P ∈ B ∞ . Hence t p Tr(xe−t|D| ) has an asymptotic power series expansion for x ∈ {P , 1 − P }. Also t Tr(e−tN ) has an asymptotic power series expansion. Now Eqs. (5.19) and (5.20), together with the earlier observation that τp+1 restricted to B0∞ has a uniform asymptotic power series expansion, imply that the function τp+1 : (0, ∞) × B˜ ∞ → C has a uniform asymptotic power series expansion. This completes the proof. 2 d Let (C ∞ (T), L2 (T), 1i dθ ) be the canonical spectral triple on the circle. Via Fourier transform d 2 becomes the number operator. Let F be the sign of if we identify L (T) with L2 (Z) then 1i dθ

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the number operator. Then with B∞ = {f0 + Ff1 + R: f0 , f1 ∈ C ∞ (T), R infinitely smoothing} this spectral triple satisfies topological WHKAE. Hong and Szymanski proved [12] that by iteratedly quantum double suspending C(T) we get the odd-dimensional quantum spheres. It follows that the iterated quantum double suspension C ∞ (Sq2 +1 ) := Σ 2 (C ∞ (T)) is dense in C(Sq2 +1 ). d ) we get the torus Now if we quantum double suspend the spectral triple (C ∞ (T), L2 (T), 1i dθ 2 +1 equivariant spectral triple on C(Sq ) [5]. Now Theorem 5.7 implies that the torus equivariant spectral triple for the odd-dimensional quantum sphere C(Sq2 +1 ) satisfies topological weak heat kernel asymptotic expansion property with dimension + 1. Hence by Theorem 3.2 this is regular with finite dimension spectrum. This gives a conceptual proof of Proposition 3.9 in [14]. 5.2. The equivariant spectral triple on odd-dimensional quantum spheres In this section, we show that the equivariant spectral triple on Sq2 +1 constructed in [4] has the topological weak heat kernel asymptotic expansion. First let us recall that the odd-dimensional quantum spheres can be realised as the quantum homogeneous space. Throughout we assume q ∈ (0, 1). The C ∗ -algebra of the quantum group SU q (n) denoted by C(SU q (n)) is defined as the universal C ∗ -algebra generated by {uij : 1 i, j } satisfying the following conditions n

n

uik u∗j k = δij ,

k=1 n n i1 =1 i2 =1

u∗ki ukj = δij ,

(5.21)

k=1

···

n

Ei1 i2 ···in uj1 i1 · · · ujn in = Ej1 j2 ···jn

(5.22)

in =1

where Ei1 i2 ···in :=

0, (−q) (i1 ,i2 ,...,in )

if i1 , i2 , . . . , in are not distinct

where for a permutation σ on {1, 2, . . . , n}, (σ ) denotes its length. The C ∗ -algebra has the quantum group structure with the comultiplication being defined by (uij ) :=

uik ⊗ ukj .

k

Call the generators of SU q (n − 1) as vij . The map φ : C(SU q (n)) → C(SU q (n − 1)) defined by φ(uij ) :=

vi−1,j −1 δij

if 2 i, j n, otherwise

(5.23)

is a surjective unital C ∗ -algebra homomorphism such that ◦ φ = (φ ⊗ φ). In this way the quantum group SU q (n − 1) is a subgroup of the quantum group SU q (n). The C ∗ -algebra of the quotient SU q (n)/SU q (n − 1) is defined as

C SU q (n)/SU q (n − 1) := a ∈ C SU q (n) : (φ ⊗ 1)(a) = 1 ⊗ a .


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Also the C ∗ -algebra C(SU q (n)/SU q (n − 1)) is generated by {u1j : 1 j n}. Moreover the map ψ : C(Sq2n−1 ) → C(SU q (n)/SU q (n − 1)) defined by ψ(zi ) := q −i+1 u1i is an isomorphism. Let h be the Haar state on the quantum group C(SU q ( + 1)) and let L2 (SU q ( + 1)) be the corresponding GNS space. We denote the closure of C(Sq2 +1 ) in L2 (SU q ( + 1)) by L2 (Sq2 +1 ). Then L2 (Sq2 +1 ) is invariant under the regular representation of SU q ( + 1). Thus we get a covariant representation for the dynamical system (C(Sq2 +1 ), SU q ( + 1), ). We denote the representation of C(Sq2 +1 ) on L2 (Sq2 +1 ) by πeq . In [4] SU q ( + 1) equivariant spectral triples for this covariant representation were studied and a non-trivial one was constructed. It is proved in [14] that the Hilbert space L2 (Sq2 +1 ) is unitarily equivalent to 2 (N × Z × N ). Then the selfadjoint operator Deq constructed in [4] is given on the orthonormal basis {eγ : γ ∈ N × Z × N } by the formula Deq (eγ ) := dγ eγ where dγ is given by 2 +1 dγ :=

i=1 |γi | 2 +1 − i=1 |γi |

if (γ +1 , γ +2 , . . . , γ2 +1 ) = 0 and γ +1 0, else.

In [14], a smooth subalgebra C ∞ (Sq2 +1 ) ⊂ C(Sq2 +1 ) is defined and it is shown that the spectral triple (C ∞ (Sq2 +1 ), L2 (Sq2 +1 ), Deq ) is a regular spectral triple with simple dimension spectrum {1, 2, . . . , 2 + 1}. Now we show that the spectral triple (C ∞ (Sq2 +1 ), L2 (Sq2 +1 ), Deq ) has the topological weak heat kernel expansion. 2 ∞ We use the same notations as in [14]. Let A∞

:= Σ (C (T)). It follows from Corol∞ ∞ 2 +1 ∞ 2 +1 lary 4.2.3 that C (Sq ) ⊂ A . Let (C (Sq ), π , H , D ) be the torus equivariant spectral triple. Let N be the number operator on 2 (N ) defined by N eγ :=

! γi e γ .

i=1

Let us denote the Hilbert space 2 (Sq2 +1 ) by H. We identify H := 2 (N × Z) with the subspace 2 (N × Z × {0}) and we denote the orthogonal complement in H by H . Then

2 (Sq2 +1 ) = H ⊕ H . Define the unbounded operator Dtorus on H by the equation " Dtorus :=

D 0

# 0 . −|D | ⊗ 1 − 1 ⊗ N

Then in [14], it is shown that Deq = Dtorus . We denote representation π ⊕(πe ⊗1) of C(Sq2 +1 ) on H by πtorus . 2 ˆ T∞⊗ ˆ ··· ⊗ ˆ T ∞ denote the Fréchet tensor (C) and let T ∞ := T ∞ ⊗ Let T ∞ := Σsmooth product of copies. The main theorem in [14] is the following. ˆ Theorem 5.8. For every a ∈ C ∞ (Sq2 +1 ), the difference πeq (a) − πtorus (a) ∈ OP−∞ D ⊗ T . 2

Let P := 1+F 2 where F := Sign(D ). We denote the rank one projection |e0 e0 | on (N ) by P where e0 := e(0,0,...,0) .

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Proposition 5.9. The equivariant spectral triple (C ∞ (Sq2 +1 ), H, Deq ) has the topological weak heat kernel expansion. ˆ ∞ Proof. Let J := OP−∞ D ⊗ T . In [14], the following algebra is considered.

B := a1 P ⊗ P + a2 P ⊗ (1 − P ) + a3 (1 − P ) ⊗ P + a4 (1 − P ) ⊗ (1 − P ) + R: a1 , a2 , a3 , a4 ∈ A ∞

, R∈J . ∞ ∞ ∞ The algebra B is isomorphic to A∞

⊕ A ⊕ A ⊕ A ⊕ J . We give B the Fréchet space structure coming from this decomposition. In [14], it is shown that B contains C ∞ (Sq2 +1 ) and is closed under δ := [|Deq |, .] and d := [D, .]. Moreover it is shown that δ and d are continuous on B. Note that Feq := F ⊗ P − 1 ⊗ (1 − P ). Hence by definition Feq ∈ B. Now note that the torus equivariant spectral triple (A∞

, H , D ) has the topological weak heat kernel asymptotic expansion. Thus it is enough to show that the map τ2 +1 : (0, ∞) × J → C defined by τ2 +1 (t, b) := t 2 +1 Tr(be−t|Deq | ) has uniform asymptotic expansion. ∞ 2 But this follows from the fact that (OP−∞ D , H , D ) and (T , (N), N ) have the topological weak heat kernel expansion and by using Lemma 5.5. This completes the proof. 2

Remark 5.10. The method in [14] can be applied to show that the equivariant spectral triple on the quantum SU(2) constructed in [2] has the heat kernel asymptotic expansion property with dimension 3 and hence deducing the dimension spectrum computed in [7]. It has been shown in [3] that the isospectral triple studied in [17] differs from the equivariant one (with multiplicity 2) constructed in [2] only be a smooth perturbation. As a result it will follow that (since the extension B ∞ for the equivariant spectral triple satisfying Definition 5.3 contains the algebra of smoothing operators) the isospectral spectral triple also has the weak heat kernel expansion with dimension 3. References [1] Nicole Berline, Ezra Getzler, Michèle Vergne, Heat Kernels and Dirac Operators, Grundlehren Math. Wiss., Springer-Verlag, Berlin, 2004, corrected reprint of the 1992 original. [2] Partha Sarathi Chakraborty, Arupkumar Pal, Equivariant spectral triples on the quantum SU(2) group, KTheory 28 (2) (2003) 107–126. [3] Partha Sarathi Chakraborty, Arupkumar Pal, On equivariant Dirac operators for SU q (2), Proc. Indian Acad. Sci. Math. Sci. 116 (4) (2006) 531–541. [4] Partha Sarathi Chakraborty, Arupkumar Pal, Characterization of SU q ( + 1)-equivariant spectral triples for the odd-dimensional quantum spheres, J. Reine Angew. Math. 623 (2008) 25–42. [5] Partha Sarathi Chakraborty, Arupkumar Pal, Torus equivariant spectral triples for odd-dimensional quantum spheres coming from C ∗ extensions, Lett. Math. Phys. 80 (1) (April 2007) 57–68. [6] Alain Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985) 257–360. [7] Alain Connes, Cyclic cohomology, quantum group symmetries and the local index formula for SU q (2), J. Inst. Math. Jussieu 3 (1) (2004) 17–68. [8] A. Connes, H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (2) (1995) 174–243. [9] A. Connes, H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198 (1) (1998) 199–246. [10] Nigel Higson, Meromorphic continuation of zeta functions associated to elliptic operators, in: Operator Algebras, Quantization, and Noncommutative Geometry, in: Contemp. Math., vol. 365, Amer. Math. Soc., Providence, RI, 2004, pp. 129–142.


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[11] Nigel Higson, The residue index theorem of Connes and Moscovici, in: Surveys in Noncommutative Geometry, in: Clay Math. Proc., vol. 6, Amer. Math. Soc., Providence, RI, 2006, pp. 71–126. [12] Jeong Hee Hong, Wojciech Szymanski, Quantum spheres and projective spaces as graph algebra, Comm. Math. Phys. 232 (2002) 157–188. [13] Jeong Hee Hong, Wojciech Szymanski, Noncommutative balls and mirror quantum spheres, J. London Math. Soc. (March 2008). [14] Arupkumar Pal, S. Sundar, Regularity and dimension spectrum of the equivariant spectral triple for the odddimensional quantum spheres, J. Noncommut. Geom. 4 (3) (2010) 389–439. [15] Larry B. Schweitzer, Spectral invariance of dense subalgebras of operator algebras, Internat. J. Math. 4 (2) (1993) 289–317. [16] Wojciech Szpankowski, Average Case Analysis of Algorithms on Sequences, Wiley–Intersci. Ser. Discrete Math. Optim., Wiley–Interscience, New York, 2001, with a foreword by Philippe Flajolet. [17] Walter van Suijlekom, Ludwik D¸abrowski, Giovanni Landi, Andrzej Sitarz, Joseph C. Várilly, The local index formula for SU q (2), K-Theory 35 (3–4) (2006) 375–394, (2005).


Coupling of Brownian motions and Perelman’s L-functional Kazumasa Kuwada a,1 , Robert Philipowski b,∗ a Graduate School of Humanities and Sciences, Ochanomizu University, Tokyo 112-8610, Japan b Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Received 7 July 2010; accepted 28 January 2011

Communicated by C. Villani

Abstract We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that their normalized L-distance is a supermartingale. As a corollary, we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non-decreasing function and the normalized L-distance. In particular, it provides a new proof of a recent result of Topping [P. Topping, L-optimal transportation for Ricci flow, J. Reine Angew. Math. 636 (2009) 93–122]. © 2011 Elsevier Inc. All rights reserved. Keywords: Ricci flow; L-functional; Brownian motion; Coupling

1. Introduction Let M be a d-dimensional differentiable manifold, 0 τ¯1 < τ¯2 < T and (g(τ ))τ ∈[τ¯1 ,T ] a complete backwards Ricci flow on M, i.e. a smooth family of Riemannian metrics satisfying ∂g = 2 Ricg(τ ) ∂τ * Corresponding author.

E-mail addresses: [email protected] (K. Kuwada), [email protected] (R. Philipowski). 1 Partially supported by the JSPS fellowship for research abroad.


(1)

K. Kuwada, R. Philipowski / Journal of Functional Analysis 260 (2011) 2742–2766

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and such that (M, g(τ )) is complete for all τ ∈ [τ¯1 , T ]. In this situation Perelman [23, Section 7.1] (see also [6, Definition 7.5]) defined the L-functional of a smooth curve γ : [τ1 , τ2 ] → M (where τ¯1 τ1 < τ2 T ) by τ2 L(γ ) :=

2 √ τ γ˙ (τ )g(τ ) + Rg(τ ) γ (τ ) dτ,

τ1

where Rg(τ ) (x) is the scalar curvature at x with respect to the metric g(τ ). Let L(x, τ1 ; y, τ2 ) be the L-distance between (x, τ1 ) and (y, τ2 ) defined by the infimum of L(γ ) over smooth curves γ : [τ1 , τ2 ] → M satisfying γ (τ1 ) = x and γ (τ2 ) = y. The normalized L-distance Θt (x, y) (t 1) is given by Θt (x, y) := 2( τ¯2 t − τ¯1 t)L(x, τ¯1 t; y, τ¯2 t) − 2 d( τ¯2 t − τ¯1 t)2 . For a measurable function c : M × M → R, let us define the transportation cost Tc (μ, ν) between two probability measures μ and ν on M with respect to the cost function c by Tc (μ, ν) := inf π

c(x, y)π(dx, dy)

M×M

(the infimum is over all probability measures π on M × M whose marginals are μ and ν respectively). To study Perelman’s work from a different aspect, Topping [30] (see also Lott [19]) showed the following result: Theorem 1. (See Theorem 1.1 in [30].) Assume that M is compact and that τ¯1 > 0. Let p : [τ¯1 , T ] × M → R+ and q : [τ¯2 , T ] × M → R+ be two non-negative unit-mass solutions of the heat equation ∂p = g(τ ) p − Rp, ∂τ where the term Rp comes from the change in time of the volume element. Then the normalized L-transportation cost TΘt (p(τ¯1 t, ·) volg(τ¯1 t) , q(τ¯2 t, ·) volg(τ¯2 t) ) between the two solutions evaluated at times τ¯1 t respectively τ¯2 t is a non-increasing function of t ∈ [1, T /τ¯2 ]. By g(τ )-Brownian motion, we mean the time-inhomogeneous diffusion process whose generator is g(τ ) . As in the time-homogeneous case, the heat distribution p(τ, ·) volg(τ ) is expressed as the law of a g(τ )-Brownian motion at time τ . In view of this strong relation between heat equation and Brownian motion, it is natural to ask whether one can couple two Brownian motions on M in such a way that a pathwise analogue of this result holds. The main result of this paper answers it affirmatively as follows: Theorem 2. Assume that the Ricci curvature of M is bounded from below uniformly, namely there exists K 0 such that Ricg(τ ) −Kg(τ )

for any τ ∈ [τ¯1 , T ].

(2)

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Then given any points x, y ∈ M and any s ∈ [1, T /τ¯2 ], there exist two coupled g(τ )-Brownian motions (Xτ )τ ∈[τ¯1 s,T ] and (Yτ )τ ∈[τ¯2 s,T ] with initial values Xτ¯1 s = x and Yτ¯2 s = y such that the process (Θt (Xτ¯1 t , Yτ¯2 t ))t∈[s,T /τ¯2 ] is a supermartingale. In addition, we can take them so that the map (x, y) → (X, Y ) is measurable. In particular, for any ϕ : R → R being concave and non-decreasing, E[ϕ(Θt (Xτ¯1 t , Yτ¯2 t ))] is non-increasing. Obviously, (2) is satisfied if M is compact. Thus it includes the case of Theorem 1. As a result, Theorem 2 easily implies the following extension of Theorem 1 (see Section 5.3): Theorem 3. Assume that (2) holds. Let ϕ : R → R be concave and non-decreasing. Then Tϕ◦Θt (p(τ¯1 t, ·) volg(τ¯1 t) , q(τ¯2 t, ·) volg(τ¯2 t) ) is non-increasing in t ∈ [1, T /τ¯2 ] for non-negative unit-mass solutions p and q to the heat equation. We prove Theorem 2 by constructing a coupling via approximation of g(τ )-Brownian motions by geodesic random walks as studied in [14]. In the next section, we demonstrate background of the problem, review related results and compare their methods with ours. Since our method superficially looks like a detour compared with other existing coupling arguments, there we explain the reason why we choose that way. To state the idea behind our proof explicitly, we prove Theorem 2 under the assumption that there is no singularity of L-distance in Section 3. Since all technical difficulties are concentrated on the singularity of L-distance, we can study the problem there in more direct way by using stochastic calculus. Some estimates on variations of L-distance are gathered in Section 4. The proof of the full statement of Theorems 2 and 3 will be provided in Section 5. Before closing this section, we give two remarks on Theorems 2 and 3. Remark 1. As shown in [16], under backwards Ricci flow g(τ )-Brownian motion cannot explode. Hence Θt (Xt , Yt ) is well defined for all t ∈ [s, T /τ¯2 ] in Theorem 2. This fact also ensures that p(τ, ·) volg(τ ) has unit mass whenever it does at the initial time. We implicitly require this property to make Tϕ◦Θt (p(τ¯1 t, ·) volg(τ¯1 t) , q(τ¯2 t, ·) volg(τ¯2 t) ) well defined in Theorem 3. Remark 2. There are plenty of examples of backwards Ricci flow satisfying (2) even when M is non-compact. Indeed, given a metric g0 on M with bounded curvature tensor, there exists a unique solution to the Ricci flow ∂t g(t) = −2 Ricg(t) with initial condition g0 satisfying sup |Rmg(t) |g(t) (x) < ∞ x,t

for a short time (see [28] for existence and [5] for uniqueness). Then the corresponding backwards Ricci flow satisfying (2) is obtained by time-reversal. 2. Review and remarks on background of the problem The Ricci flow was introduced by Hamilton [9]. There he effectively used it to solve the Poincaré conjecture for 3-manifolds with positive Ricci curvature. By following his approach, Perelman [23–25] finally solved the Poincaré conjecture (see also [4,12,22]). There he used Lfunctional as a crucial tool. At the same stage, he also studied the heat equation in [23] in relation with the geometry of Ricci flows. It suggests that analyzing the heat equation is still an efficient


2745

way to investigate geometry of the underlying space even in the time-dependent metric case. This guiding principle has been confirmed in recent developments in this direction. For example, we refer to [36] as one of such developments. In connection with the theory of optimal transportation, McCann and Topping [21] showed the monotonicity of Tρ 2 (p(τ, ·) volg(τ ) , q(τ, ·) volg(τ ) ), g(τ ) where ρg(τ ) is the g(τ )-Riemannian distance, under backwards Ricci flow on a compact manifold. Topping’s result [30] can be regarded as an extension of it to contraction in the normalized L-transportation cost (see [19] also). By taking τ¯2 → τ¯1 , he gave a new proof of the monotonicity of Perelman’s W-entropy, which is one of fundamental ingredients in Perelman’s work. A probabilistic approach to these problems is initiated by Arnaudon, Coulibaly and Thalmaier. In [2, Section 4], they sharpened McCann and Topping’s result [21] to a pathwise contraction in the following sense: There is a coupling (Xt , Yt )t0 of two Brownian motions starting from x, y ∈ X respectively such that the g(t)-distance between Xt and Yt is non-increasing in t almost surely. In their approach, probabilistic techniques based on analysis of sample paths made it possible to establish such a pathwise estimate. As an advantage of their result, the pathwise contraction easily yields that Tϕ◦ρg(τ ) (p(τ, ·) volg(τ ) , q(τ, ·) volg(τ ) ) is non-increasing for any non-decreasing ϕ. As an application of this sharper monotonicity, we can obtain an L1 -gradient estimate of Bakry–Émery type (see [15]) for the heat semigroup. In the time-homogeneous case, this gradient estimate has been known to be very useful in geometric analysis (see e.g. [3,17]). McCann and Topping’s result only implies L2 -gradient estimate and it is weaker than the L1 estimate. (In the time-homogeneous case, it is known that L2 -estimate also implies L1 -estimate (see [3,17,26]). However, to the best of our knowledge, an extension of such equivalence in the time-inhomogeneous case is not yet established.) As another advantage of Arnaudon, Coulibaly and Thalmaier’s approach, their argument works even on non-compact M (cf. [16]). Our Theorem 2 can be regarded as an extension of their result. Indeed, our approach is the same as theirs in spirit and advantages of probabilistic approach as mentioned are also inherited to our results as we have seen in Theorem 3. We can expect that our approach makes it possible to employ several techniques in stochastic analysis to obtain more detailed behavior of Θt (Xτ¯1 t , Yτ¯2 t ), especially in the limit τ¯2 → τ¯1 , in a future development. Now we compare our method of the proof with existing arguments in coupling methods from a technical point of view. We hope that the following observation would be helpful to extend other coupling arguments than ours in this case. A common and basic idea is to couple infinitesimal motions of two Brownian particles by using a parallel transport of tangent vectors along a minimal geodesic joining them. Thus the technical difficulty arises from the singularity of (L-)distance, or the presence of (L-)cut locus. In our approach, we consider coupled geodesic random walks each of which approximates the Brownian motion. After we establish a difference inequality for time evolution of the L-distance between coupled random walks, we will obtain the result by taking a limit. Note that the convergence of our random walk in law to the Brownian motion in this time-inhomogeneous case is already established in [14]. In the time-homogeneous case, there are several arguments [8,10,32–34] to construct such a coupling by approximating it with ones which move as mentioned above if they are distant from the cut locus and move independently if they are close to the cut locus. In these cases, it will be important to estimate the size of the total time when particles are close to the cut locus. To do the same in the time-inhomogeneous case, it does not seem straightforward since the (L-)cut locus depends on time, namely it moves as time evolves. In our approach, instead of applying stochastic calculus, we only need to show a difference inequality. Thus the singularity at the L-cut locus causes less difficulties at this stage (see Remark 7 for more details).

2746


If we employ the theory of optimal transportation, we will work on coupling of heat distributions instead of coupling of Brownian motions. Once we move to the world of heat distributions, we can ignore the cut locus since they are of measure zero with respect to the Riemannian measure. However, in the derivation of the monotonicity results, the theory of optimal transportation at present covers only the case that the cost function is squared distance or L-distance. It reflects the difference of results between McCann and Topping [21] and Arnaudon, Coulibaly and Thalmaier [2]. It should be remarked that such a difference still exists between these two approaches, the one by optimal transportation and the other by stochastic analysis, even in the time-homogeneous case. Arnaudon, Coulibaly and Thalmaier [2] study the problem by developing a new method. They constructed one-parameter family of coupled particles ((Xt (u))t0 )u∈[0,1] so that Xt (u) moves as a Brownian motion for any u and (Xt (u))u∈[0,1] is a C 1 -curve whose length is non-increasing in t. Thus (Xt (0), Xt (1)) is the expected coupling. To construct it, they first consider a finite number of particles ((Xt (ui ))t0 )i which are coupled with other particles by parallel transport. Then, by increasing the number of particles, we obtain such a one-parameter family in the limit. Since they are coupled by parallel transport, the distance between two particles is of bounded variation (at least before they hit the cut locus). Thus, if adjacent particles are sufficiently close to each other at time t, we can take a deterministic δ > 0 such that they cannot hit the cut locus at least until time t + δ. Based on this observation, they succeeded in avoiding the problem coming from the cut locus by increasing the number of particles to make it constitute a one-parameter family of particles. In the case of this paper, we work on the L-distance instead of the Riemannian distance and construct a coupling by space–time parallel transport instead of a coupling by parallel transport. As a result, L-distance between coupled particle is not of bounded variation (see Remark 6 for more details). Thus, our problem differs in nature from what is studied in [1]. If we want to extend Arnaudon, Coulibaly and Thalmaier’s approach in the present case, we have to be careful and need some additional arguments. Even if we succeed in constructing a one-parameter family of particles ((Xt (u))t0 )u∈[0,1] coupled by space–time parallel transport, we cannot expect that (Xt (u))u∈[0,1] is a C 1 -curve. In our approach, such a difference causes no additional difficulty. Indeed, as studied in [13,14], we already know that it works to construct coupled particles by reflection, the distance of which is naturally regarded as a semimartingale with a non-vanishing martingale part. 3. Coupling of Brownian motions in the absence of L-cut locus Since the proof of Theorem 2 involves some technical arguments, first we study the problem in the case that the L-distance L has no singularity. More precisely, we do it here under the following assumption: Assumption 1. The L-cut locus is empty. See Section 5.1 or [6,30,35] for the definition of L-cut locus. Under Assumption 1, the following hold: τ1 τ2 of L(x, τ1 ; y, τ2 ) 1. For all x, y ∈ M and all τ¯1 τ1 < τ2 T there is a unique minimizer γxy τ1 τ2 (existence of γxy is proved in [6, Lemma 7.27], while uniqueness follows immediately from the characterization of L-cut locus, see Section 5.1). 2. The function L is globally smooth.


2747

Thus, in this case, we can freely use stochastic analysis on the frame bundle without taking any care on regularity of L. 3.1. Construction of the coupling A g(τ )-Brownian motion X˜ on M (scaled in time by the factor τ¯1 ) starting at a point x ∈ M at time s ∈ [1, T /τ¯2 ] can be constructed in the following way [1,7,16]: Let π : F (M) → M be the frame bundle and (ei )di=1 the standard basis of Rd . For each τ ∈ [τ¯1 , T ] let (Hi (τ ))di=1 be the associated g(τ )-horizontal vector fields on F (M) (i.e. Hi (τ, u) is the g(τ )-horizontal lift of uei ). Moreover let (V α,β )dα,β=1 be the canonical vertical vector fields, i.e. (V α,β f )(u) := ∂ d d ∂mαβ |m=Id (f (u(m))) (m = (mαβ )α,β=1 ∈ GLd (R)), and let (Wt )t0 be a standard R -valued Brownian motion. By Og(τ ) (M), we denote the g(τ )-orthonormal frame bundle. We first define a scaled horizontal Brownian motion as the solution U˜ = (U˜ t )t∈[s,T /τ¯1 ] of the Stratonovich SDE d U˜ t =

2τ¯1

d

Hi (τ¯1 t, U˜ t ) ◦ dWti − τ¯1

i=1

d ∂g (τ¯1 t)(U˜ t eα , U˜ t eβ )V αβ (U˜ t ) dt ∂τ

(3)

α,β=1 g(τ¯1 s)

on F (M) with initial value U˜ s = u ∈ Ox on M as

(M), and then define a scaled Brownian motion X˜

X˜ t := π U˜ t . Note that X˜ t does not move when τ¯1 = 0. The last term in (3) ensures that U˜ t ∈ Og(τ¯1 t) (M) for all t ∈ [s, T /τ¯1 ] (see [1, Proposition 1.1], [7, Proposition 1.2]), so that by Itô’s formula for all smooth f : [s, T /τ¯1 ] × M → R df (t, X˜ t ) =

∂f (t, X˜ t ) dt + 2τ¯1 (U˜ t ei )f (t, X˜ t ) dWti + τ¯1 g(τ1 t) f (t, X˜ t ) dt. ∂t d

i=1

Let us define (Xτ )τ ∈[τ¯1 s,T ] by Xτ¯1 t := X˜ t . Then Xτ becomes a g(τ )-Brownian motion when τ¯1 > 0. Remark 3. Intuitively, it might be helpful to think that Xτ lives in (M, g(τ )), or X˜ t lives in (M, g(τ¯1 t)). The same is true for Y and Y˜ which will be defined below. Similarly, for all curves γ : [τ1 , τ2 ] → M appearing in connection with L-distance, we can naturally regard γ (τ ) as in (M, g(τ )). We now want to construct a second scaled Brownian motion Y˜ on M in such a way that its infinitesimal increments d Y˜t are “space–time parallel” to those of X˜ (up to scaling effect) along the minimal L-geodesic (namely, the minimizer of L) from (X˜ t , τ¯1 t) to (Y˜t , τ¯2 t). To make this idea precise, we first define the notion of space–time parallel vector field:

2748


Definition 1 (Space–time parallel vector field). Let τ¯1 τ1 < τ2 T and γ : [τ1 , τ2 ] → M be a smooth curve. We say that a vector field Z along γ is space–time parallel if g(τ ) ∇γ˙ (τ ) Z(τ ) = − Ric#g(τ ) Z(τ )

(4)

holds for all τ ∈ [τ1 , τ2 ]. Here ∇ g(τ ) stands for the covariant derivative associated with the g(τ )Levi-Civita connection and Ric#g(τ ) is defined by regarding the g(τ )-Ricci curvature as a (1, 1)tensor via g(τ ). Since (4) is a linear first-order ODE, for any ξ ∈ Tγ (τ1 ) M there exists a unique space–time parallel vector field Z along γ with Z(τ1 ) = ξ . Remark 4. Whenever Z and Z are space–time parallel vector fields along a curve γ , their g(τ )inner product is constant in τ : g(τ )

d

∂g g(τ ) Z(τ ), Z (τ ) g(τ ) = (τ ) Z(τ ), Z (τ ) + ∇γ˙ (τ ) Z(τ ), Z (τ ) g(τ ) + Z(τ ), ∇γ˙ (τ ) Z (τ ) g(τ ) dτ ∂τ = 2 Ricg(τ ) Z(τ ), Z (τ ) − Ricg(τ ) Z(τ ), Z (τ ) − Ricg(τ ) Z(τ ), Z (τ ) = 0. Definition 2 (Space–time parallel transport). For x, y ∈ M and τ¯1 τ1 < τ2 T , we define a map mτxy1 τ2 : Tx M → Ty M as follows: mτxy1 τ2 (ξ ) := Z(τ2 ), where Z is the unique space–time τ1 τ2 parallel vector field along γxy with Z(τ1 ) = ξ . By Remark 4, mτxy1 τ2 is an isometry from (Tx M, g(τ1 )) to (Ty M, g(τ2 )). In addition, it smoothly depends on x, τ1 , y, τ2 under Assumption 1. Remark 5. The emergence of the Ricci curvature in (4) is based on the Ricci flow equation (1). Indeed, we can introduce the notion of space–time parallel transport even in the absence of (1) with keeping the property in Remark 4 by using 2−1 ∂τ g(τ )# instead of Ric#g(τ ) in (4). This would be a natural extension in the sense that it coincides with the usual parallel transport when g(τ ) is constant in τ . Similarly as in [10, Formula (6.5.1)], we now define a second scaled horizontal Brownian motion V˜ = (V˜t )t∈[s,T /τ¯2 ] on F (M) as the solution of d V˜t =

2τ¯2

d

Hi (τ¯2 t, V˜t ) ◦ dBti − τ¯2

i=1 2 dBt = V˜t−1 mτ1 ,τ U˜ π U˜ t ,π V˜t t

d ∂g (τ¯2 t)(V˜t eα , V˜t eβ )V αβ (V˜t ) dt, ∂τ

α,β=1

dWt

g(τ¯ s) ˜ let us define with initial value V˜s = v ∈ Oy 2 (M), and we set Y˜t := π V˜t . As we did for X, ˜ (Yτ )τ ∈[τ¯2 s,T ] by Yτ¯2 t := Yt to make Y a g(τ )-Brownian motion. From a theoretical point of view, it seems to be natural to work with (Xτ , Yτ ) (see Remark 3). However, for technical simplicity, we will prefer to work with (X˜ t , Y˜t ) instead in the sequel.


2749

3.2. Proof of Theorem 2 in the absence of L-cut locus Our argument in this section is based on the following Itô formula for (X˜ t , Y˜t ): Lemma 1. Let f be a smooth function on [s, T /τ¯2 ] × M × M. Then df (t, X˜ t , Y˜t ) =

∂f (t, X˜ t , Y˜t ) dt + 2τ¯1 U˜ t ei ⊕ 2τ¯2 V˜t ei∗ f (t, X˜ t , Y˜t ) dWti ∂t d

i=1

+

d

Hessg(τ¯1 t)⊕g(τ¯2 t) f |(t,X˜ t ,Y˜t )

τ¯1 U˜ t ei ⊕ τ¯2 V˜t ei∗ , τ¯1 U˜ t ei ⊕ τ¯2 V˜t ei∗ dt.

i=1

Here the Hessian of f is taken with respect to the product metric g(τ¯1 t) ⊕ g(τ¯2 t), ei∗ stands for ei∗ (U˜ t , τ¯1 t; V˜t , τ¯2 t), where 1 ,τ2 ue , ei∗ (u, τ1 ; v, τ2 ) := v −1 mτπu,πv i

and for tangent vectors ξ1 ∈ Tx M, ξ2 ∈ Ty M we write ξ1 ⊕ ξ2 := (ξ1 , ξ2 ) ∈ T(x,y) (M × M). Proof. As in [11, Formula (2.11)], Itô’s formula applied to a smooth function f˜ on [s, T /τ¯2 ] × F (M) × F (M) gives d f˜(t, U˜ t , V˜t ) =

∂ f˜ (t, U˜ t , V˜t ) dt ∂t +

d

2τ¯1 Hi,1 (τ¯1 t, ·)f˜ (t, U˜ t , V˜t ) dWti

i=1

+ +

2τ¯2 Hi,2 (τ¯2 t, ·)f˜ (t, U˜ t , V˜t ) dBti

d 2 2 τ¯1 Hi,1 (τ¯1 t, ·)f˜ (t, U˜ t , V˜t ) + τ¯2 Hi,2 (τ¯2 t, ·)f˜ (t, U˜ t , V˜t ) dt i=1

d

Hi,1 (τ¯1 t, ·)Hj,2 (τ¯2 t, ·)f˜ (t, U˜ t , V˜t ) d W i , B j t + 2 τ¯1 τ¯2 i,j =1

−

d ∂g τ¯1 (τ¯1 t)(U˜ t eα , U˜ t eβ )V αβ (U˜ t ) ∂τ

α,β=1

⊕ τ¯2

∂g (τ¯2 t)(V˜t eα , V˜t eβ )V αβ (V˜t ) f˜(t, U˜ t , V˜t ) dt, ∂τ

where Hi,1 respectively Hi,2 means Hi applied with respect to the first respectively second space variable. By the definition of B, this equals

2750


∂ f˜ (t, U˜ t , V˜t ) dt ∂t +

d

2τ¯1 Hi (τ¯1 t, U˜ t ) ⊕

2τ¯2 Hi∗ (U˜ t , τ¯1 t; V˜t , τ¯2 t) f˜(t, U˜ t , V˜t ) dWti

i=1

+

d

τ¯1 Hi (τ¯1 t, U˜ t ) ⊕

2 τ¯2 Hi∗ (U˜ t , τ¯1 t; V˜t , τ¯2 t) f˜(t, U˜ t , V˜t ) dt

i=1

−

d ∂g ∂g τ¯1 (τ¯1 t)(U˜ t eα , U˜ t eβ )V αβ (U˜ t ) ⊕ τ¯2 (τ¯2 t)(V˜t eα , V˜t eβ )V αβ (V˜t ) f˜(t, U˜ t , V˜t ) dt, ∂τ ∂τ

α,β=1

where Hi∗ (u, τ1 ; v, τ2 ) is the g(τ2 )-horizontal lift of vei∗ (u, τ1 ; v, τ2 ). The claim follows by choosing f˜(t, u, v) := f (t, πu, πv) because this f˜ is constant in the vertical direction so that the term involving V αβ f˜ vanishes. 2 Let Λ(t, x, y) := L(x, τ¯1 t; y, τ¯2 t). In order to apply Lemma 1 to the function Θ we need the following proposition, whose proof is given in the next section. Since we will use it again in Section 5, we state it without assuming Assumption 1. g(τ¯ t)

g(τ¯ t)

Proposition 1. Take x, y ∈ M, u ∈ Ox 1 (M) and v ∈ Oy 2 (M). Let γ be a √ minimizer of L(x, τ ¯ t; y, τ ¯ t). Assume that (x, τ ¯ t; y, τ ¯ t) is not in the L-cut locus. Set ξ := τ¯1 uei ⊕ 1 2 1 2 i √ τ¯2 vei∗ (u, τ¯1 t; v, τ¯2 t). Then 1 ∂Λ (t, x, y) = ∂t t

τ¯2 t 3 Rg(τ ) γ (τ ) − g(τ ) Rg(τ ) γ (τ ) − 2|Ricg(τ ) |2g(τ ) γ (τ ) τ 3/2 2τ

τ¯1 t

− d

2 1 γ˙ (τ )g(τ ) + 2 Ricg(τ ) γ˙ (τ ), γ˙ (τ ) dτ , 2τ

(5)

Hessg(τ1 )⊕g(τ2 ) Λ|(t,x,y) (ξi , ξi )

i=1

√ τ¯2 t d τ τ =τ¯2 t 1 + τ 3/2 2|Ricg(τ ) |2g(τ ) γ (τ ) + g(τ ) Rg(τ ) γ (τ ) t τ =τ¯1 t t τ¯1 t

2 − Rg(τ ) γ (τ ) − 2 Ricg(τ ) γ˙ (τ ), γ˙ (τ ) dτ τ and consequently ∂Λ (t, x, y) + Hessg(τ1 )⊕g(τ2 ) Λ|(t,x,y) (ξi , ξi ) ∂t d

i=1

(6)


2751

τ¯2 t 2 √ d 1 √ ( τ¯2 − τ¯1 ) − τ Rg(τ ) γ (τ ) + γ˙ (τ )g(τ ) dτ 2t t τ¯1 t

1 d = √ ( τ¯2 − τ¯1 ) − Λ(t, x, y). 2t t The proof of Theorem 2 is now achieved under Assumption 1 by combining Lemma 1 and Proposition 1: Proof of Theorem 2 under Assumption 1. Since Θ is bounded from below, it suffices to show that the bounded variation part of Θt (X˜ t , Y˜t ) is non-positive. By Lemma 1, dΘt (X˜ t , Y˜t ) = ∂t Θt (X˜ t , Y˜t ) +

d

Hessg(τ¯1 t)⊕g(τ¯2 t) Θt |(X˜ t ,Y˜t ) τ¯1 U˜ t ei ⊕ τ¯2 V˜t ei∗ , τ¯1 U˜ t ei ⊕ τ¯2 V˜t ei∗ dt

i=1

+

d

2τ¯1 U˜ t ei ⊕

2τ¯2 V˜t ei∗ Θt (X˜ t , Y˜t ) dWti .

i=1

For the bounded variation part we obtain ∂t Θt (X˜ t , Y˜t ) =

√

√ τ¯2 − τ¯1 ∂Λ (t, X˜ t , Y˜t ) − 2 d( τ¯2 − τ¯1 )2 Λ(t, X˜ t , Y˜t ) + 2( τ¯2 t − τ¯1 t) √ ∂t t

and d

Hessg(τ¯1 t)⊕g(τ¯2 t) Θt |(X˜ t ,Y˜t )

τ¯1 U˜ t ei ⊕ τ¯2 V˜t ei∗ , τ¯1 U˜ t ei ⊕ τ¯2 V˜t ei∗

i=1 d Hessg(τ¯1 t)⊕g(τ¯2 t) Λ|(t,X˜ t ,Y˜t ) τ¯1 U˜ t ei ⊕ τ¯2 V˜t ei∗ , τ¯1 U˜ t ei ⊕ τ¯2 V˜t ei∗ . = 2( τ¯2 t − τ¯1 t) i=1

Thus, by Proposition 1, ∂t Θt (X˜ t , Y˜t ) +

d

Hessg(τ¯1 t)⊕g(τ¯2 t) Θt |(X˜ t ,Y˜t )

i=1

τ¯1 U˜ t ei ⊕

d 1 ˜ ˜ 2( τ¯2 t − τ¯1 t) √ ( τ¯2 − τ¯1 ) − Λ(t, Xt , Yt ) 2t t √ √ τ¯2 − τ¯1 Λ(t, X˜ t , Y˜t ) − 2 d( τ¯2 − τ¯1 )2 = 0. + √ t Hence Θt (X˜ t , Y˜t ) is indeed a supermartingale.

2

τ¯2 V˜t ei∗ , τ¯1 U˜ t ei ⊕ τ¯2 V˜t ei∗

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Remark 6. Unlike the case in [2], the pathwise contraction of Θt (X˜ t , Y˜t ) is no longer true in our case. In other words, the martingale part of Θt (X˜ t , Y˜t ) does not vanish. We will see it in the τ1 τ2 of L(x, τ1 ; y, τ2 ) satisfies the L-geodesic equation following. The minimal L-geodesic γ = γxy 1 1 g(τ ) ∇γ˙ (τ ) γ˙ (τ ) = ∇ g(τ ) Rg(τ ) − 2 Ric#g(τ ) γ˙ (τ ) − γ˙ (τ ) 2 2τ

(7)

(see [6, Corollary 7.19]). Thus the first variation formula (see [6, Lemma 7.15]) yields

2τ¯1 U˜ t ei ⊕

2τ¯2 V˜t ei∗ Λ(t, X˜ t , Y˜t ) =

√

2t τ¯2 V˜t ei∗ , γ˙ (τ¯2 t) g(τ¯

2 t)

−

√

2t τ¯1 U˜ t ei , γ˙ (τ¯1 t) g(τ¯ t) . (8) 1

One obstruction √ to pathwise contraction is in the difference of time-scalings τ¯1 and τ¯2 . In addition, by (7), τ γ˙ (τ ) is not space–time parallel to γ in general (cf. Remark 4). 4. Proof of Proposition 1 In this section, we write τ1 := τ¯1 t and τ2 := τ¯2 t. We assume τ2 < T . For simplicity of notations, we abbreviate the dependency on the metric g(τ ) of several geometric quantities such as Ric, R, the inner product ·,· , the covariant derivative ∇, etc. when our choice of τ is obvious. For this abbreviation, we will think √ g(τ )). √ that γ (τ ) is in (M, g(τ )) and γ˙ (τ ) is in (Tγ (τ ) M, Note that, when τ¯1 = 0, limτ ↓τ¯1 τ γ˙ (τ ) exists while limτ ↓0 |γ˙ (τ )| = ∞. In any case, τ |γ˙ (τ )| is locally bounded (see Lemma 3). We first compute the time derivative of Λ. When τ¯1 > 0, by [30, Formulas (A.4) and (A.5)] we have 2 √ ∂L (x, τ1 ; y, τ2 ) = − τ1 Rg(τ1 ) (x) − γ˙ (τ1 ) , ∂τ1 2 ∂L √ (x, τ1 ; y, τ2 ) = τ2 Rg(τ2 ) (y) − γ˙ (τ2 ) , ∂τ2 so that ∂L ∂L ∂Λ (t, x, y) = τ¯1 (x, τ1 ; y, τ2 ) + τ¯2 (x, τ1 ; y, τ2 ) ∂t ∂τ1 ∂τ2 2 2 1 3/2 3/2 = τ2 R γ (τ2 ) − γ˙ (τ2 ) − τ1 R γ (τ1 ) − γ˙ (τ1 ) . t

(9)

Thus the integration-by-parts formula yields, 3 ∂Λ (t, x, y) = ∂t 2t

τ2

2 √ τ R γ (τ ) − γ˙ (τ ) dτ

τ1

1 + t

τ2 τ

3/2

∂R γ (τ ) + ∇γ˙ (τ ) R γ (τ ) ∂τ

τ1

− 2 ∇γ˙ (τ ) γ˙ (τ ), γ˙ (τ ) − 2 Ric γ˙ (τ ), γ˙ (τ ) dτ.

(10)


2753

Note that we have ∂R = −R − 2|Ric|2 ∂τ

(11)

(see e.g. [29, Proposition 2.5.4]). Since γ satisfies the L-geodesic equation (7), by substituting (7) and (11) into (10), we obtain (5). Note that the derivation of (9) and (10) is still valid even when τ¯1 = 0 because of the remark at the beginning of this section. Thus (5) holds when τ¯1 = 0, too. In order to estimate di=1 Hessg(τ1 )⊕g(τ2 ) Λ|(t,x,y) (ξi , ξi ) we begin with the second variation formula for the L-functional: Lemma 2 (Second variation formula). (See [6, Lemma 7.37].) Let Γ : (−ε, ε) × [τ1 , τ2 ] → M be a variation of γ , S(s, τ ) := ∂s Γ (s, τ ), and Z(τ ) := ∂s Γ (0, τ ) the variation field of Γ . Then τ =τ τ =τ √

√ d 2 L(Γs ) = 2 τ γ˙ (τ ), ∇Z(τ ) S(0, τ ) τ =τ2 − 2 τ Ric Z(τ ), Z(τ ) τ =τ2 2 1 1 ds s=0 2 τ =τ 1 + √ Z(τ ) τ =τ2 − 1 τ τ2 +

τ2

√ τ H γ˙ (τ ), Z(τ ) dτ

τ1

2 √ 1 # 2 τ ∇γ˙ (τ ) Z(τ ) + Ric Z(τ ) − Z(τ ) dτ, 2τ

(12)

τ1

where 2 ∂ Ric Z(τ ), Z(τ ) − Hess R Z(τ ), Z(τ ) + 2Ric# Z(τ ) H γ˙ (τ ), Z(τ ) := −2 ∂τ 1 − Ric Z(τ ), Z(τ ) − 2 Rm Z(τ ), γ˙ (τ ), γ˙ (τ ), Z(τ ) τ − 4(∇γ˙ (τ ) Ric) Z(τ ), Z(τ ) + 4(∇Z(τ ) Ric) γ˙ (τ ), Z(τ ) . (13) In [6] this lemma is only proved in the case τ1 = 0 and Z(τ1 ) = 0. However, the proof given there can be easily adapted to the slightly more general case needed here. Corollary 1. (See [6, Lemma 7.39] for a similar statement.) If the variation field Z is of the form Z(τ ) =

τ ∗ Z (τ ) t

with a space–time parallel field Z ∗ satisfying |Z ∗ (τ )| ≡ 1, then

(14)

2754


τ =τ τ =τ √

√ d 2 L(Γs ) = 2 τ γ˙ (τ ), ∇Z(τ ) S(0, τ ) g(τ ) τ =τ2 − 2 τ Ric Z(τ ), Z(τ ) τ =τ2 2 1 1 ds s=0 τ2 −

√

τ H γ˙ (τ ), Z(τ ) dτ +

τ1

√ τ =τ2 τ . t τ =τ1

Proof. Since Z ∗ is space–time parallel, Z satisfies 1 ∇γ˙ (τ ) Z(τ ) = − Ric# Z(τ ) + Z(τ ), 2τ so that the last term in (12) vanishes.

(15)

2

Corollary 2 (Hessian of L). (See [6, Corollary 7.40] for a similar statement.) Let Z be a vector field along γ of the form (14) and ξ := Z(τ1 ) ⊕ Z(τ2 ) ∈ T(x,y) (M × M). Then τ2 Hessg(τ1 )⊕g(τ2 ) L|(x,τ1 ;y,τ2 ) (ξ, ξ ) −

√ τ H γ˙ (τ ), Z(τ ) dτ +

τ1

√ τ =τ2 τ t τ =τ1

τ =τ √ − 2 τ Ricg(τ ) Z(τ ), Z(τ ) τ =τ2 . 1

(16)

Proof. Let Γ : (−ε, ε) × [τ1 , τ2 ] → M be any variation of γ with variation field Z and such that ∇Z(τ1 ) S(0, τ1 ) and ∇Z(τ2 ) S(0, τ2 ) vanish.

(17)

ˆ := L(Γ (s, ·)). Since l(0) ˆ = l(0) and l(s) ˆ l(s) Let l(s) := L(Γ (s, τ1 ), τ1 ; Γ (s, τ2 ), τ2 ) and l(s)

ˆ for all s ∈ (−ε, ε), we have l (0) l (0) so that, using (17), d 2 L Γ (s, τ1 ), τ1 ; Γ (s, τ2 ), τ2 2 ds s=0 d 2

ˆ = l (0) l (0) = 2 L(Γs ). ds s=0

Hessg(τ1 )⊕g(τ2 ) L|(x,τ1 ;y,τ2 ) (ξ, ξ ) =

The claim now follows from Corollary 1.

2

Let now Zi∗ (i = 1, . . . , d) be space–time parallel along γ satisfying Zi∗ (τ1 ) = uei √ fields ∗ ∗ ∗ (and consequently Zi (τ2 ) = vei ), and Zi (τ ) := τ/tZi (τ ) (so that ξi = Zi (τ1 ) ⊕ Zi (τ2 )). In order to estimate di=1 Hessg(τ1 )⊕g(τ2 ) L|(x,τ1 ;y,τ2 ) (ξi , ξi ) using Corollary 2 we will compute d i=1 H (γ˙ (τ ), Zi (τ )) in the following (see [6, Section 7.5.3] for a similar argument). Set I1 , I2 and I3 by


2755

d ∂ Ric Zi (τ ), Zi (τ ) , ∂τ

I1 := −2

i=1

I2 :=

d

2 − Hess R Zi (τ ), Zi (τ ) + 2Ric# Zi (τ )

i=1

− I3 := 4

1 Ric Zi (τ ), Zi (τ ) − 2 Rm Zi (τ ), γ˙ (τ ), γ˙ (τ ), Zi (τ ) , τ

d (∇Zi (τ ) Ric) Zi (τ ), γ˙ (τ ) − (∇γ˙ (τ ) Ric) Zi (τ ), Zi (τ ) . i=1

Then

d

i=1 H (γ˙ (τ ), Zi (τ )) = I1

+ I2 + I3 holds. By a direct computation,

1 τ 2 −R γ (τ ) + 2|Ric| γ (τ ) − R γ (τ ) + 2 Ric γ˙ (τ ), γ˙ (τ ) . I2 = t τ

(18)

The contracted Bianchi identity div Ric = 12 ∇R [18, Lemma 7.7] yields I3 =

4τ 2τ (div Ric) γ˙ (τ ) − (∇γ˙ (τ ) R) γ (τ ) = − (∇γ˙ (τ ) R) γ (τ ) . t t

(19)

For I1 , we have I1 = −2

d d Ric Zi (τ ), Zi (τ ) − (∇γ˙ (τ ) Ric) Zi (τ ), Zi (τ ) − 2 Ric ∇γ˙ (τ ) Zi (τ ), Zi (τ ) dτ i=1

d = −2 dτ =−

2τ t

d τ τ R γ (τ ) + 2 ∇γ˙ (τ ) R γ (τ ) + 4 Ric ∇γ˙ (τ ) Zi (τ ), Zi (τ ) t t i=1

∂R 1 R γ (τ ) + γ (τ ) + 4 τ ∂τ

d

Ric ∇γ˙ (τ ) Zi (τ ), Zi (τ ) .

(20)

i=1

Since Zi satisfies (15), 4

d 1 Ric ∇γ˙ (τ ) Zi (τ ), Zi (τ ) = 4 Ric − Ric# Zi (τ ) + Zi (τ ), Zi (τ ) 2τ i=1 i=1 1 2τ =− 2|Ric|2 γ (τ ) − R γ (τ ) . t τ

d

(21)

By substituting (21) into (20), I1 = −

2τ t

∂R γ (τ ) + 2|Ric|2 γ (τ ) . ∂τ

(22)

2756


Hence, by combining (22), (19) and (18), d i=1

τ ∂R −2 γ (τ ) − 2|Ric|2 γ (τ ) − R γ (τ ) H γ˙ (τ ), Zi (τ ) = t ∂τ 1 − R γ (τ ) + 2 Ric γ˙ (τ ), γ˙ (τ ) − 2(∇γ˙ (τ ) R) γ (τ ) . τ

Inserting this into (16) we obtain d

Hessg(τ1 )⊕g(τ2 ) L|(x,τ1 ;y,τ2 ) (ξi , ξi )

i=1

1 t

τ2 τ

3/2

2

∂R γ (τ ) + 2|Ric|2 γ (τ ) + R γ (τ ) ∂τ

τ1

1 + R γ (τ ) − 2 Ric γ˙ (τ ), γ˙ (τ ) + 2(∇γ˙ (τ ) R) γ (τ ) dτ τ √ τ =τ d τ τ =τ2 2τ 3/2 − + R γ (τ ) τ =τ2 1 t τ =τ1 t √ τ2 d τ τ =τ2 1 3/2 2|Ric|2 γ (τ ) + R γ (τ ) = + τ t τ =τ1 t τ1

2 − R γ (τ ) − 2 Ric γ˙ (τ ), γ˙ (τ ) dτ τ which completes the proof of Proposition 1. 5. Coupling via approximation by geodesic random walks To avoid a technical difficulty coming from singularity of L on the L-cut locus, we provide an alternative way to constructing a coupling of Brownian motions by space–time parallel transport. In this section, we first define a coupling of geodesic random walks which approximate g(τ )Brownian motion. Next, we introduce some estimates on geometric quantities in Section 5.1. Those are obtained as a small modification of existing arguments in [6,30,35]. The L-cut locus is also reviewed and studied there. We use those estimates in Section 5.2 to study the behavior of the L-distance between coupled random walks. The argument there includes a discrete analogue of the Itô formula as well as a local uniform control of error terms. Finally, we will complete the proof of Theorems 2 and 3 in Section 5.3. τ1 τ2 | τ¯1 τ1 < τ2 τ¯2 , x, y ∈ M} so that a Let us take a family of minimal L-geodesics {γxy τ1 τ2 map (x, τ1 ; y, τ2 ) → γxy is measurable. The existence of such a family of minimal L-geodesics can be shown in a similar way as discussed in the proof of [20, Proposition 2.6] since the family of minimal L-geodesics with fixed endpoints is compact (cf. [6, the proof of Lemma 7.27]). For each τ ∈ [τ¯1 , T ], take a measurable section Φ (τ ) of g(τ )-orthonormal frame bundle Og(τ ) (M)


2757

of M. For x, y ∈ M and τ1 , τ2 ∈ [τ¯1 , T ] with τ1 < τ2 , let us define Φi (x, τ1 ; y, τ2 ) ∈ F (M) for i = 1, 2 by Φ1 (x, τ1 ; y, τ2 ) := Φ (τ1 ) (x), Φ2 (x, τ1 ; y, τ2 ) := mτxy1 τ2 ◦ Φ (τ1 ) (x), where mτxy1 τ2 is as given in Definition 2. Let us take a family of Rd -valued i.i.d. random variables (λn )n∈N which are uniformly distributed on a unit ball centered at origin. We denote the (Rie(τ ) mannian) exponential map with respect to g(τ ) at x ∈ M by expx . In what follows, we define a ε ε ε coupled geodesic random walk Xt = (Xτ¯1 t , Yτ¯2 t ) with scale parameter ε > 0 and initial condition Xεs = (x1 , y1 ) inductively. First we set (Xτε¯1 s , Yτ¯ε2 s ) := (x1 , y1 ). For simplicity of notations, we set tn := (s + ε 2 n) ∧ (T /τ¯2 ). After we defined (Xεt )t∈[s,tn ] , we extend it to (Xεt )t∈[s,tn+1 ] by

Xτε¯1 t Yτ¯ε2 t

√

d + 2Φi Xτε¯1 tn , τ¯1 tn ; Yτ¯ε2 tn , τ¯2 tn λn+1 , (τ¯ t ) t − tn (1) := expXε1 n 2τ¯1 λˆ n+1 , τ¯1 tn ε (τ¯2 tn ) t − tn (2) ˆ := expY ε 2τ¯2 λn+1 τ¯2 tn ε

(i) λˆ n+1 :=

i = 1, 2,

for t ∈ [tn , tn+1 ]. We can (and we will) extend the definition of Xτε for τ ∈ [T τ¯1 /τ¯2√ , T ] in the same way. As in Section 3, Xτε¯1 t does not move when τ¯1 = 0. Note that the term d + 2 in √ the definition of λˆ (i) n+1 is a normalization factor in the sense Cov( d + 2λn ) = Id. Let us equip path spaces C([a, b] → M) or C([a, b] → M × M) with the uniform convergence topology induced from g(T ). Here the interval [a, b] will be chosen appropriately in each context. As shown in [14], (Xτε )τ ∈[τ¯1 s,T ] and (Yτε )τ ∈[τ¯2 s,T ] converge in law to g(τ )-Brownian motions (Xτ )τ ∈[τ¯1 s,T ] and (Yτ )τ ∈[τ¯2 s,T ] on M with initial conditions Xτ¯1 s = x1 , Yτ¯2 s = y1 respectively as ε → 0 (when τ¯1 > 0). As a result, Xε is tight and hence there is a convergent subsequence of Xε . We fix such a subsequence and use the same symbol (Xε )ε for simplicity of notations. We denote the limit in law of Xε as ε → 0 by Xt = (Xτ¯1 t , Yτ¯2 t ). Recall that, in this paper, g(τ )-Brownian motion means a time-inhomogeneous diffusion process associated with g(τ ) instead of g(τ ) /2. Remark 7. We explain the reason why our alternative construction works efficiently to avoid the obstruction arising from singularity of L. To make it clear, we begin with observing the essence of difficulties in the SDE approach we used in Section 3. Recall that our argument is based on the Itô formula. Hence the non-differentiability of L at the L-cut locus causes the technical difficulty. One possible strategy is to extend the Itô formula for L-distance. Since L-cut locus is sufficiently thin, we can expect that the totality of times when our coupled particles stay there has measure zero. In addition, as that of Riemannian cut locus, the presence of L-cut locus would work to decrease the L-distance between coupled particles. Thus one might think it possible to extend Itô formula for L-distance to the one involving a “local time at the L-cut locus”. If we succeed in doing so, we will obtain a differential inequality which implies the supermartingale property by neglecting this additional term since it would be non-positive. Instead of completing the above strategy, our alternative approach in this section directly provides a difference inequality without extracting the additional “local time” term. When the

2758


endpoint of minimal L-geodesic is in the L-cut locus, we divide it into two pieces. Then the pair of endpoints of each piece is not in the L-cut locus. As a result, we obtain the desired difference inequality of L-distance even in such a case (see the proof of Lemma 4 for more details). In order to follow such a procedure, it is more suitable to work with discrete time processes. 5.1. Preliminaries on properties of L-functional Recall that we assumed the uniform lower Ricci curvature bound (2). On the basis of it, we can compare Riemannian metrics at different times. That is, for τ1 < τ2 , g(τ1 ) e2K(τ2 −τ1 ) g(τ2 )

(23)

(see [29, Lemma 5.3.2], for instance). Recall that ρg(τ ) is the distance function on M at time τ . Note that a similar comparison between ρg(τ1 ) and ρg(τ2 ) follows from (2). By neglecting the term involving γ˙ in the definition of L(γ ), the condition (2) implies inf L(x, τ1 ; y, τ2 ) −

x,y∈M

2 dK 3/2 3/2 τ2 − τ1 . 3

(24)

We also obtain the following bounds for L from (2) and (23). Let γ : [τ1 , τ2 ] → M be a minimal L-geodesic. Then, for τ ∈ [τ1 , τ2 ], 2 2 3/2 e−2KT 3/2 L γ (τ1 ), τ1 ; γ (τ2 ), τ2 (25) √ √ ρg(τ¯1 ) γ (τ1 ), γ (τ ) − dK τ2 − τ1 2( τ2 − τ1 ) 3 (see [6, Lemma 7.13] or [30, Proposition B.2]). The same estimate holds for ρg(τ¯1 ) (γ (τ ), γ (τ2 ))2 instead of ρg(τ¯1 ) (γ (τ1 ), γ (τ ))2 . Taking the fact that L-functional is not invariant under reparametrization of curves into account, we will introduce a local estimate on the velocity of the minimal L-geodesic γ . Lemma 3. Let τ1 , τ2 ∈ [τ¯1 , T ] and suppose that τ2 − τ1 δ for some δ > 0. Then, for any compact set M0 ⊂ M, there exist constants C1 > 0 depending on K, M0 and δ such that, for any γ : [τ1 , τ2 ] → M with γ (τ1 ), γ (τ2 ) ∈ M0 and τ1 τ τ2 , 2 τ γ˙ (τ )g(τ ) C1 .

(26)

Proof. Though the conclusion follows by combining arguments in [6, Lemma 7.24] and [35, Proposition 2.12], we give a proof for completeness. Let o ∈ M be a reference point and take g(T ) r0 > 0 so large that Br0 /2 (o) contains M0 . Take K0 > 0 so that supτ |Rg(τ ) | K0 holds on g(T )

Br0

(o). We claim that there exists a constant C0 > 0 such that L(x, τ1 ; y, τ2 ) C0

(27)

for any x, y ∈ M0 . Take a constant speed g(T )-minimal geodesic γ0 : [τ1 , τ2 ] → M joining x g(T ) and y. Note that γ0 is contained in Br0 (o). Thus, by virtue of (23), we have


τ2 L(x, τ1 ; y, τ2 )

2759

2 √ τ γ0 (τ )g(τ ) + Rg(τ ) γ0 (τ ) dτ

τ1

3/2 3/2 2e2KT τ2 − τ1 2K0 3/2 3/2 τ2 − τ1 dg(T ) (x, y)2 + 3 3 (τ2 − τ1 )2

2K0 3/2 2T 3/2 e2KT dg(T ) (x, y)2 + T . 3 3δ 2

Thus the claim follows since x, y ∈ M0 . By combining the above claim with (25), we can show that there exists r1 > r0 which g(T ) is independent of γ such that γ (τ ) ∈ Br1 (o) for any τ ∈ [τ1 , τ2 ]. Take K1 > 0 so that g(T ) |Ricg(τ ) |g(τ ) K1 and |∇Rg(τ ) |g(τ ) K1 hold on Br1 (o) for any τ ∈ [τ¯1 , T ]. By a similar argument as in [6, Lemma 7.13(ii)], there exists τ ∗ ∈ [τ1 , τ2 ] such that 2 τ γ˙ τ ∗ g(τ ∗ ) ∗

2 dK 3/2 1 3/2 L γ (τ1 ), τ1 ; γ (τ2 ), τ2 + τ2 − τ1 . √ √ 2( τ2 − τ1 ) 3

(28)

By virtue of (2), there exist constants c1 , C1 > 0 which depends on K, K1 and T such that for all τ1 , τ2 ∈ [τ1 , τ2 ] with τ1 < τ2 , 2 2 τ2 γ˙ τ2 g(τ ) c1 τ1 γ˙ τ1 g(τ ) + C1 , 2 1 2 2 τ1 γ˙ τ1 g(τ ) c1 τ2 γ˙ τ2 g(τ ) + C1 . 1

(29) (30)

2

The first inequality in (29) can be shown similarly as [6, Lemma 7.24]. It is due to a differential inequality based on the L-geodesic equation (7) which provides an upper bound of ∂τ (τ |γ˙ (τ )|2g(τ ) ). By considering a lower bound of the same quantity instead, we obtain the second inequality (30) in a similar way. Hence the proof is completed by combining (29) and (30) with (28) and (27). 2 Let us recall that the L-cut locus, denoted by L Cut, is defined as a union of two different kinds of sets (see [35]; see [6,30] also). The first one consists of (x, τ1 ; y, τ2 ) such that there exist more than one minimal L-geodesics joining (x, τ1 ) and (y, τ2 ). The second consists of (x, τ1 ; y, τ2 ) such that (y, τ2 ) is conjugate to (x, τ1 ) along a minimal L-geodesic with respect to L-Jacobi field. Note that L is smooth on M \ L Cut (see [35, Lemma 2.9]) and that L Cut is closed (see [30]; though they assumed M to be compact, an extension to the non-compact case is straightforward). 5.2. Variations of the L-distance of coupled random walks For proving Theorem 2, our first task is to show a difference inequality of Λ(t, Xεt ) in Lemma 4. We begin with introducing some notations. Set γn := γXτ¯1εtn ,τ¯2 tn and let us define a tn √ vector field λˆ †n+1 along γn by λˆ n+1 (τ ) = τ/tn λ∗n+1 (τ ), where λ∗n+1 is a space–time parallel (1) vector field along γn with initial condition λˆ ∗n+1 (τ¯1 tn ) = λˆ n+1 . Let us define random variables ζn and Σn as follows:

2760


ζn+1 :=

√ † τ¯ t 2τ λˆ n+1 (τ ), γ˙n (τ ) g(τ ) τ2=nτ¯ t , 1 n

2 τ¯ t 1 Σn+1 := τ 3/2 Rg(τ ) γn (τ ) − γ˙n (τ )g(τ ) τ2=nτ¯ t 1 n tn √ † τ¯2 tn √ τ † ˆ ˆ + − 2 τ Ricg(τ ) λn+1 (τ ), λn+1 (τ ) tn τ =τ¯1 tn τ¯2 tn √ † − τ H γ˙ (τ ), λˆ n+1 (τ ) dτ . τ¯1 tn

Here H is as given in (13). The term ζn+1 corresponds to the martingale part of Λ(t, Xt ) and Σn does to the one dominating the bounded variation part of Λ(t, X˜ t , Y˜t ) in Section 3. As we will see in Lemma 4 below, there is a discrete analogue of the Itô formula (and the corresponding difference inequality) involving ζn and Σn . As a result of our discretization, we are no longer able to apply Proposition 1 directly to estimate Σn itself. In this case, we can do it to the conditional expectation of Σn instead. Set Gn := σ (λ1 , . . . , λn ). Then, since each Φi is isometry and (d + 2)E[ λn , ei λn , ej ] = δij , Proposition 1 yields d 1 E[Σn+1 | Gn ] √ ( τ¯2 − τ¯1 ) − Λ tn , Xεtn . tn 2tn

(31)

For M0 ⊂ M, we define σM0 : C([s, T /τ¯2 ] → M × M) → [0, ∞) by σM0 (w, w) ˜ := inf{t s | wτ¯1 t ∈ / M0 or wτ¯2 t ∈ / M0 }. ε and σ 0 respectively. As For simplicity of notations, we denote σM0 (Xε ) and σM0 (X) by σM M0 0 ε shown in [14], for any η > 0, we can take a compact set M0 ⊂ M such that limε→0 P[σM 0 T ] η holds (cf. [16]).

Lemma 4. Let M0 ⊂ M be a compact set. Then there exist a family of random variables (Qεn )n∈N, ε>0 and a family of deterministic constants (δ(ε))ε>0 with limε→0 δ(ε) = 0 satisfying

Qεn δ(ε)

(32)

ε ∧(T /τ¯ ) n; tn 0 ε ∧ (T /τ¯ ). depending on M0 such that max{|ζn |, |Λ(tn , Xεtn )|, |Σn |} C2 holds if tn σM 2 0

2762


Proof. By the definition of ζn , we have |ζn |

2(d + 2)tn−1 τ¯1 γ˙n−1 (τ¯1 tn−1 )g(τ¯

1 tn−1 )

+ τ¯2 γ˙n−1 (τ2 tn−1 )g(τ¯

2 tn−1 )

.

Thus the asserted bound for |ζn | follows from (26). Similarly, the estimate for Λ(tn , Xεtn ) follows from (24) and (27). For estimating Σn , we deal with the integral involving H in the definition of Σn . Note that every tensor field appearing in the definition of H is continuous. As in the proof of Lemma 4, take a compact set M1 ⊃ M0 such that every minimal L-geodesic joining (x, τ¯1 t) and (y, τ¯2 t) is included in M1 if x, y ∈ M0 and t ∈ [s, T /τ¯2 ]. Since Xεtn−1 ∈ M0 × M0 holds on ε ∧ (T /τ¯ )}, the upper bound (26) of √τ |γ˙ (τ )| implies that H (γ˙ (τ ), Z(τ )) the event {tn < σM 2 n 0 √ is uniformly bounded for any vector field Z(τ ) along γn of the form Z(τ ) = τ/tn Z ∗ (τ ) with a space–time parallel vector field Z ∗ (τ ) satisfying |Z ∗ (τ )|g(τ ) 1. This fact yields a required bound for the integral. For any other terms in the definition of Σn , we can estimate them as we did for ζn and Λ(tn , Xεtn ). 2 By virtue of Lemma 5, Lemma 4 yields the following: Corollary 3. Let M0 ⊂ M be a compact set. Then there exist a family of random variables ˜ ˜ = 0 satisfying (Q˜ εn )n∈N, ε>0 and a family of deterministic constants (δ(ε)) ε>0 with limε→0 δ(ε)

˜ εn δ(ε) ˜ Q

ε ∧(T /τ¯ ) n; tn 0 such that E

sup

stT /τ¯2

Θt (Xt )2 < c3 Θs (x1 , y1 )2 + C3 .

Proof. Recall that Θ is uniformly bounded from below by (24). Take b 0 so large that Θt (x, y) + b 0 for x, y ∈ M and t ∈ [s, T /τ¯2 ] and set Θˆ t (x, y) := Θt (x, y) + b. It suffices to show


E

sup

stT /τ¯2

2763

Θˆ t (Xt )2 4Θˆ s (x1 , y1 )2 + C

for some C > 0. Take a relatively compact open set M0 ⊂ M and consider Θˆ tn ∧σˆ Mε (Xεtn ∧σˆ ε ). 0

M0

ε . Thus Lemma 5 ensures that the term appearing in (34) is integrable on the event tn < σˆ M 0 Corollary 3 and (31) yield that

˜ E Θˆ tm ∧σˆ Mε Xεtm ∧σˆ ε − Θˆ tn ∧σˆ Mε Xεtn ∧σˆ ε Gn δ(ε). M0

0

M0

0

By imitating the proof of the maximal inequality (cf. [27, Chapter 2, Exercise 1.15]), we obtain P

sup ε n; stn σˆ M

0

1 ˜ Θˆ tn Xεtn r Θˆ s (x1 , y1 ) + δ(ε) r ∧(T /τ¯2 )

for r > 0. Then, by following the proof of the Doob inequality in [27], E

sup ε ∧(T /τ¯ ) stn σˆ M 2

ε 2 2 ˜ 4 Θˆ s (x1 , y1 ) + δ(ε) Θˆ tn Xt ∧ R n

(35)

0

holds for each R > 0. By (23) and the definition of X ε , there exist CM0 > 0 such that 2 2 Θˆ t∧σMε Xεt ∧ R Θˆ tn Xεtn ∧σˆ ε ∧ R + CM0 ε M0

0

for t ∈ [tn , tn+1 ]. Thus (35) yields E

sup ε ∧(T /τ¯ ) stσM 2

ε 2 2 ˜ 4 Θˆ s (x1 , y1 ) + δ(ε) Θˆ t Xt ∧ R + CM0 ε.

(36)

0

Let us turn to estimate the second moment of supt Θˆ t (Xt ). Note that we have E

sup

stT /τ¯2

2 E Θˆ t (Xt ) ∧ R

sup

stT /τ¯2

2 0 0 > t + R 2 P σM t . (37) Θˆ t (Xt ) ∧ R ; σM 0 0

Since {w | σM0 (w) > t} is open and the map w → supstT /τ¯2 (Θˆ t (wt ) ∧ R)2 is bounded and continuous on C([s, T /τ¯2 ] → M × M), (36) yields E

sup

stT /τ¯2

2 0 Θˆ t (Xt ) ∧ R ; σM > t lim inf E 0 ε→0

lim inf E ε→0

sup

stT /τ¯2

2 ε Θˆ t Xεt ∧ R ; σM > t 0

sup ε ∧(T /τ¯ ) stσM 2

ε 2 Θˆ t Xt ∧ R

0

4Θˆ s (x1 , y1 ). Thus the conclusion follows by combining the last inequality with (37) and by letting M0 ↑ M and R → ∞. 2

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5.3. Proof of Theorems 2 and 3 Proof of Theorem 2. First we remark that the map (x, y) → (Xτε¯1 · , Yτ¯ε2 · ) is obviously measurable. Thus, we obtain the same measurability for the law of (Xτ¯1 · , Yτ¯2 · ). The integrability of Θt (Xt ) follows from Lemma 6. We will show the supermartingale property in the sequel. For s s1 < · · · < sm < t < t < T and f1 , . . . , fm ∈ Cc (M ×M → R) with 0 fj 1, set F (w) := m j =1 fj (wsj ) for w ∈ C([s, T /τ¯2 ] → M × M). Take η > 0 arbitrarily and choose a relatively 0 t] η holds. Note that lim sup ε compact open set M0 ⊂ M so that P[σM ε→0 P[σM0 t] η 0 also holds since {w | σM0 (w) t} is closed. It suffices to show that there is a constant C > 0 which is independent of η and M0 such that, √ E Θt∧σ 0 (Xt∧σ 0 ) − Θt ∧σ 0 (Xt ∧σ 0 ) F (X·∧σ 0 ) C η M0

M0

M0

M0

M0

(38)

holds. In fact, once we have shown (38), then Lemma 6 yields E Θt (Xt ) − Θs (Xs ) F (X) 0 0 → ∞ almost surely as M ↑ M. since σM 0 0 Take f ∈ Cc (M × M) such that 0 f 1 and f |U ≡ 1, where U ⊂ M × M is an open set containing M¯ 0 × M¯ 0 . Then, by virtue of Lemma 6 and the choice of M0 ,

E Θt∧σ 0 (Xt∧σ 0 ) − Θt ∧σ 0 (Xt ∧σ 0 ) F (X·∧σ 0 ) M0

M0

M0

M0

M0

1/2 √ 0 E Θt (Xt ) − Θt (Xt ) f (Xt )f (Xt )F (X); σM > t + 2C4 η, 0

(39)

where C4 := c3 Θs (x1 , y1 )2 + C3 . Since {w | σM0 (w) > t} is open, 0 E Θt (Xt ) − Θt (Xt ) f (Xt )f (Xt )F (X); σM >t 0 ε lim inf E Θt Xεt − Θt Xεt f Xεt f Xεt F Xε ; σM >t 0 ε→0 ε >t . = lim inf E Θtε Xεtε − Θt ε Xεt ε f Xεtε f Xεt ε F Xε ; σM 0 ε→0

(40)

Here the last equality follows from the continuity of Θ and f . Then ε E Θtε Xεtε − Θt ε Xεt ε f Xεtε f Xεt ε F Xε ; σM >t 0 E Θtε ∧σˆ Mε Xεtε ∧σˆ ε − Θt ε ∧σˆ Mε Xεt ε ∧σˆ ε F Xε·∧σˆ ε + 2E

M0

0

sup

suT /τ¯2

0

M0

ε ε 2 1/2 ε 1/2 Θu X f X P σM0 t . u u

M0

(41)

Since a function w → sup1uT /τ¯2 |Θu (wu )f (wu )| on C([s, T /τ¯2 ] → M × M) is bounded and continuous, we have lim sup E ε→0

sup

suT /τ¯2

ε ε 2 1/2 ε 1/2 1/2 √ Θu X f X P σM0 t C4 η. u u

(42)


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Now, with the aid of Lemma 5, the iteration of (34) together with (31) yields ˜ E Θtε ∧σˆ Mε Xεtε ∧σˆ ε − Θt ε ∧σˆ Mε Xεt ε ∧σˆ ε F Xε·∧σˆ ε δ(ε). 0

M0

0

M0

M0

(43)

˜ Here δ(ε) is what appeared in Corollary 3. Hence we complete the proof by combining (40), (41), (42) and (43) with (39). 2 Proof of Theorem 3. Fix 1 s < t T /τ¯2 . We may assume Tϕ◦Θs p(τ¯1 s, ·) volg(τ¯1 t) , q(τ¯2 s, ·) volg(τ¯2 t) < ∞ without loss of generality. Let π be a minimizer of Tϕ◦Θs (p(τ¯1 s, ·) volg(τ¯1 t) , q(τ¯2 s, ·) volg(τ¯2 t) ), where the existence of π follows from [31, Theorem 4.1], using the lower bound (24). For each y (x, y) ∈ M × M, we take coupled Brownian motions (Xτx )τ ∈[τ¯1 s,T ] and (Yτ )τ ∈[τ¯2 s,T ] with iniy x x tial values Xτ¯1 s = x and Yτ¯2 s = y as in Theorem 2. Since the law of (X , Y y ) is a measurable function of (x, y), we can construct a coupling of two Brownian motions (Xτ¯1 · , Yτ¯2 · ) with initial y distribution π from ((Xτx¯1 · , Yτ¯2 · ))x,y∈M as a coordinate process on C([s, T /τ¯2 ] → M × M) by y following a usual manner. By Theorem 2, ϕ(Θt (Xτx¯1 t , Yτ¯2 t )) is a supermartingale. Hence we have E ϕ Θt (Xτ¯1 t , Yτ¯2 t ) =

y E ϕ Θt Xτx¯1 t , Yτ¯2 t π(dx, dy)

M×M

ϕ Θs (x, y) π(dx, dy)

M×M

= Tϕ◦Θs p(τ¯1 s, ·) volg(τ¯1 s) , q(τ¯2 s, ·) volg(τ¯2 s) . Since the law of (Xτ¯1 t , Yτ¯2 t ) is a coupling of p(τ¯1 t, ·) volg(τ¯1 t) and q(τ¯2 t, ·) volg(τ¯2 t) , we have Tϕ◦Θt p(τ¯1 t, ·) volg(τ¯1 t) , q(τ¯2 t, ·) volg(τ¯2 t) E ϕ Θt (Xτ¯1 t , Yτ¯2 t ) and hence the conclusion follows.

2

Acknowledgments We would like to thank Anton Thalmaier and an anonymous referee for very useful comments. References [1] M. Arnaudon, K.A. Coulibaly, A. Thalmaier, Brownian motion with respect to a metric depending on time; definition, existence and application to Ricci flow, C. R. Acad. Sci. Paris, Ser. I 346 (2008) 773–778. [2] M. Arnaudon, K.A. Coulibaly, A. Thalmaier, Horizontal diffusion in C 1 path space, in: Sém. Prob. XLIII, in: Lecture Notes in Math., vol. 2006, Springer-Verlag, Berlin, 2010, pp. 73–94. [3] D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, in: New Trends in Stochastic Analysis, Charingworth, 1994, World Sci. Publ., River Edge, NJ, 1997, pp. 43–75. [4] H.-D. Cao, X.-P. Zhu, A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton–Perelman theory of the Ricci flow, Asian J. Math. 10 (2006) 165–492.

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[5] B.-L. Chen, X.-P. Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geom. 74 (2006) 119–154. [6] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci Flow: Techniques and Applications, Part I: Geometric Aspects, American Mathematical Society, Providence, RI, 2007. [7] K.A. Coulibaly, Brownian motion with respect to time-changing Riemannian metrics, applications to Ricci flow, Ann. Inst. Henri Poincaré Probab. Stat., in press. [8] M. Cranston, Gradient estimates on manifolds using coupling, J. Funct. Anal. 99 (1991) 110–124. [9] R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255–306. [10] E.P. Hsu, Stochastic Analysis on Manifolds, American Mathematical Society, Providence, RI, 2002. [11] W.S. Kendall, Stochastic differential geometry, a coupling property, and harmonic maps, J. Lond. Math. Soc. 33 (1986) 554–566. [12] B. Kleiner, J. Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008) 2587–2855. [13] K. Kuwada, Coupling of the Brownian motion via discrete approximation under lower Ricci curvature bounds, in: Probabilistic Approach to Geometry, Kyoto, 2008, in: Adv. Stud. Pure Math., vol. 57, Math. Soc. Japan, Tokyo, 2010, pp. 273–292. [14] K. Kuwada, Coupling by reflection via discrete approximation under a backward Ricci flow, preprint, arXiv: 1007.0275v1, 2010. [15] K. Kuwada, Duality on gradient estimates and Wasserstein controls, J. Funct. Anal. 258 (11) (2010) 3758–3774. [16] K. Kuwada, R. Philipowski, Non-explosion of diffusion processes on manifolds with time-dependent metric, Math. Z., doi:10.1007/s00209-010-0704-7, in press. [17] M. Ledoux, The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse Math. (6) 9 (2) (2000) 305–366. [18] J.M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer-Verlag, New York, 1997. [19] J. Lott, Optimal transport and Perelman’s reduced volume, Calc. Var. 36 (2009) 49–84. [20] J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. Math. 169 (2009) 903–991. [21] R.J. McCann, P. Topping, Ricci flow, entropy and optimal transportation, Amer. J. Math. 132 (2010) 711–730. [22] J. Morgan, G. Tian, Ricci Flow and the Poincaré Conjecture, American Mathematical Society, Providence, RI, 2007. [23] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, arXiv:math/ 0211159v1, 2002. [24] G. Perelman, Ricci flow with surgery on three-manifolds, preprint, arXiv:math/0303109v1, 2003. [25] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint, arXiv: math/0307245v1, 2003. [26] M.-K. von Renesse, K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math. 58 (2005) 923–940. [27] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edition, Springer-Verlag, Berlin, Heidelberg, 1999. [28] W.-X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989) 223–301. [29] P. Topping, Lectures on the Ricci Flow, Cambridge University Press, 2006. [30] P. Topping, L-optimal transportation for Ricci flow, J. Reine Angew. Math. 636 (2009) 93–122. [31] C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss., vol. 338, Springer-Verlag, 2009. [32] F.-Y. Wang, Successful couplings of nondegenerate diffusion processes on compact manifolds, Acta Math. Sin. 37 (1) (1994) 116–121. [33] F.-Y. Wang, On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups, Probab. Theory Related Fields 108 (1) (1997) 87–101. [34] F.-Y. Wang, Functional Inequalities Markov Semigroups and Spectral Theory, Science Press, Beijing/New York, 2005. [35] R.G. Ye, On the l-function and the reduced volume of Perelman I, Trans. Amer. Math. Soc. 360 (1) (2008) 507–531. [36] Qi S. Zhang, Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture, CRC Press, Boca Raton, FL, 2011.


Spectral theory for commutative algebras of differential operators on Lie groups Alessio Martini Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy Received 29 July 2010; accepted 14 January 2011 Available online 1 February 2011 Communicated by P. Delorme

Abstract The joint spectral theory of a system of pairwise commuting self-adjoint left-invariant differential operators L1 , . . . , Ln on a connected Lie group G is studied, under the hypothesis that the algebra generated by them contains a “weighted subcoercive operator” of ter Elst and Robinson (1998) [52]. The joint spectrum of L1 , . . . , Ln in every unitary representation of G is characterized as the set of the eigenvalues corresponding to a particular class of (generalized) joint eigenfunctions of positive type of L1 , . . . , Ln . Connections with the theory of Gelfand pairs are established in the case L1 , . . . , Ln generate the algebra of K-invariant left-invariant differential operators on G for some compact subgroup K of Aut(G). © 2011 Elsevier Inc. All rights reserved. Keywords: Functional calculus; Differential operators; Lie groups; Joint spectrum; Eigenfunction expansions; Representation theory; Gelfand pairs

1. Introduction Let L1 , . . . , Ln be pairwise commuting smooth linear differential operators on a smooth manifold X, which are formally self-adjoint with respect to some smooth measure μ. Do these operators admit a joint functional calculus on L2 (X, μ)? In that case, what is the relationship between the joint L2 spectrum of L1 , . . . , Ln and their joint smooth (possibly non-L2 ) eigenfunctions on X? A joint functional calculus for L1 , . . . , Ln is given, via spectral integration, by a joint spectral resolution E, i.e., a resolution of the identity of L2 (X, μ) on Rn such that E-mail address: [email protected]. 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.01.008

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A. Martini / Journal of Functional Analysis 260 (2011) 2767–2814

λj dE(λ1 , . . . , λn ) Rn

is a self-adjoint extension of Lj for j = 1, . . . , n. Existence and uniqueness of E are related to the so-called “domain problems”, such as essential self-adjointness of L1 , . . . , Ln and strong commutativity of their self-adjoint extensions. Once a joint spectral resolution E is fixed, the theory of eigenfunction expansions (see, e.g., [5,39]) yields the existence, for E-almost every λ = (λ1 , . . . , λn ) in the joint L2 spectrum Σ = supp E of L1 , . . . , Ln , of a corresponding generalized joint eigenfunction φ, which (under some hypoellipticity hypothesis on L1 , . . . , Ln ) belongs to the space E(X) of smooth functions on X and satisfies Lj φ = λ j φ

for j = 1, . . . , n.

(1.1)

However, from the general theory, neither it is clear for which λ ∈ Σ there does exist a corresponding smooth eigenfunction φ, nor for which φ ∈ E(X) satisfying (1.1) the corresponding λ does belong to Σ. In this paper, we restrict to the case of X = G being a connected Lie group, with right Haar measure μ, and left-invariant differential operators L1 , . . . , Ln . In this context, the problem of existence and uniqueness of a joint spectral resolution can be stated for the operators d (L1 ), . . . , d (Ln ) in every unitary representation of G — the case of the operators L1 , . . . , Ln on L2 (G) corresponding to the (right) regular representation of G — with a possibly different joint spectrum Σ for each representation . Via techniques due to Nelson and Stinespring [45], we show in Section 3.1 that a sufficient condition for the essential self-adjointness and the existence of a joint spectral resolution in every unitary representation is that the algebra generated by L1 , . . . , Ln contains a weighted subcoercive operator. This class of hypoelliptic left-invariant differential operators, defined by ter Elst and Robinson [52] in terms of a homogeneous contraction of the Lie algebra g of G, is large enough to contain positive elliptic operators, sublaplacians and positive Rockland operators (see Section 2 for details). Under the same hypotheses on L1 , . . . , Ln , we prove that every element of the joint spectrum Σ corresponds to a joint (smooth) eigenfunction φ of L1 , . . . , Ln which is a function of positive type on G, i.e., of the form φ(x) = π(x)v, v

(1.2)

for some unitary representation π of G on a Hilbert space H and some cyclic vector v ∈ H \ {0}. More precisely, in Section 4 we show that: (a) for every unitary representation of G, Σ coincides with the set of the eigenvalues relative to the joint eigenfunctions of L1 , . . . , Ln of the form (1.2) with π (irreducible and) weakly contained in ; (b) if G is amenable, then Σ coincides with the set of the eigenvalues relative to all the joint eigenfunctions of positive type; (c) if L1 (G) is a symmetric Banach ∗-algebra, then Σ coincides with the set of the eigenvalues relative to all the bounded joint eigenfunctions.


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Recall that, if G has polynomial growth, then L1 (G) is symmetric, and this in turn implies that G is amenable (see [47]). Notice moreover that, on non-amenable groups, the previous characterization (b) of Σ cannot be expected, because of the spectral-gap phenomenon (cf. [56]). If there exists a compact group K of automorphisms of G such that the operators L1 , . . . , Ln generate the algebra of left-invariant K-invariant differential operators on G, then the theory of Gelfand pairs applies (see, e.g., [19,59]), and the joint spectral theory of L1 , . . . , Ln is related to the spectral theory of the (convolution) algebra of K-invariant L1 functions on G, i.e., to the spherical Fourier transform. The “Gelfand pair” condition, however, is quite restrictive on the groups G and the systems L1 , . . . , Ln of operators which can be considered. Under our weaker hypotheses, we develop in Section 3 a notion analogous to the spherical Fourier transform, with several similar features (Plancherel formula, Riemann–Lebesgue lemma, etc.). Finally, in Section 5 some examples are considered, involving homogeneous groups and direct products, and moreover we show how (part of) the theory of Gelfand pairs on Lie groups fits in our general setting. Some of the results presented here can be found in the literature in the case of a single operator (n = 1), particularly for a sublaplacian (see, e.g., [32,33,11,36]), often as preliminaries for spectral multiplier theorems. It appears that our setting is suited for developing a theory of joint spectral multipliers for a family of commuting left-invariant differential operators on a Lie group (cf. [37,38]). Notation For a topological space X, we denote by C(X) the space of continuous (complex-valued) functions on X, whereas C0 (X) and Cc (X) are the subspaces of continuous functions vanishing at infinity and of continuous functions with compact support respectively. If X is a smooth manifold, then E(X) and D(X) are the spaces of smooth functions and of compactly supported smooth functions on X; correspondingly, D (X) and E (X) are the spaces of distributions and of compactly supported distributions. If G is a Lie group, f is a complex-valued function on G and x, y ∈ G, then we set Rx f (y) = f (yx). Lx f (y) = f x −1 y , R : x → Rx is the (right) regular representation of G. For a fixed right Haar measure μ on G, Rx is an isometry of Lp (G) for 1 p ∞. With respect to such measure, convolution and involution of functions take the form f ∗ g(x) = f xy −1 g(y) dy, f ∗ (x) = (x)f x −1 G

(where is the modular function) and we set, for every representation π of G, π(f ) = f (x)π x −1 dx, G

so that in particular R(g)f = f ∗ g,

π(f ∗ g) = π(g)π(f ),

for every left-invariant differential operator D.

π(Df ) = dπ(D)π(f )

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2. Rockland and weighted subcoercive operators This section is devoted to summarizing and amplifying some of the results of [52], which are the basis for ours. In order to do this, however, it is useful first to recall some definitions and facts about homogeneous Lie groups; for more detailed expositions, we refer to the books [18,21,55]. 2.1. Homogeneous groups and Rockland operators A homogeneous Lie algebra is a Lie algebra g with a fixed family of automorphic dilations δt = eB log t

for t > 0,

where B is a diagonalizable derivation of g with strictly positive eigenvalues. The eigenspaces Wλ of the derivation B determine a direct-sum decomposition Wλ = Wλ1 ⊕ · · · ⊕ Wλk (2.1) g= λ∈R

(where λk > · · · > λ1 > 0 are the eigenvalues of B) such that [Wλ , Wλ ] ⊆ Wλ+λ

for all λ, λ ∈ R.

Every homogeneous Lie algebra g is nilpotent, i.e., the descending central series g[1] = g,

g[n+1] = [g, g[n] ]

is eventually null; in particular, g can be identified with the connected, simply connected Lie group G whose Lie algebra is g. Let G = g be a homogeneous Lie group, with dilations δt = eB log t . A homogeneous norm on G is a continuous function | · |δ : G → [0, +∞[ such that • |x|δ = 0 if and only if x is the identity of G; • |x −1 |δ = |x|δ ; • |δt (x)|δ = t|x|δ for all t > 0. Two homogeneous norms | · |δ , | · |δ on G are always equivalent: C −1 |x|δ |x|δ C|x|δ

for all x ∈ G,

for some constant C 1 (see [20, §3], or [21, §1.2]); moreover, there exists (see [25]) a homogeneous norm | · |δ which is smooth off the origin and subadditive: |xy|δ |x|δ + |y|δ

for all x, y ∈ G.

The quantity Qδ = tr B =

k j =1

λj dim Wj


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is called the homogeneous dimension of g; in fact, we have μ δt (U ) = t Qδ μ(U ) for every measurable U ⊆ g. Modulo rescaling (i.e., replacing t with t c for some c > 0), one can suppose that λ1 1, which shall be always understood in the rest of the paper, so that in particular Qδ dim g. The degree of polynomial growth (or dimension at infinity) of G is the unique QG ∈ N such that μ K n ∼ nQG for every compact neighborhood K = K −1 of the identity of G. This definition does not depend on the chosen dilations, and in fact it makes sense for every connected Lie group G (with polynomial growth); for a nilpotent group G, we have the following characterization, where

τK (x) = min n ∈ N: x ∈ K n . Proposition 2.1 (Guivarc’h). Suppose that G is s-step nilpotent (i.e., g[s] = 0 = g[s+1] ) and let Vj be a complement of g[j +1] in g[j ] for j = 1, . . . , s. Choose moreover norms | · |j on the Vj and set |x| =

s

1/j

|xj |j ,

(2.2)

j =1

where x = x1 + · · · + xs is the decomposition of x ∈ g = V1 ⊕ · · · ⊕ Vs . Then |x| ∼ τK (x)

for large x ∈ G,

for every compact neighborhood K = K −1 of the identity. In particular, G has polynomial growth of degree QG =

s j =1

j dim Vj =

s

dim g[j ] dim g.

j =1

Proof. See [24], particularly the proofs of Théorème II.1 and Lemme II.1.

2

A homogeneous Lie algebra g as in (2.1) is stratified if W1 generates g as a Lie algebra (this implies that λ1 , . . . , λk are integers). If G = g is stratified, then in Proposition 2.1 one can take Vj = Wj , so that (2.2) is a homogeneous norm on G and QG = Qδ . For a general homogeneous Lie group, we have the following result (cf. also [34]). Proposition 2.2. Let G be a homogeneous Lie group, with dilations δt and homogeneous dimension Qδ , and let | · |δ be a homogeneous norm on G. Let | · | be defined as in (2.2), and QG be the degree of polynomial growth of G.

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(i) One has Qδ QG , with equality if and only if G is stratified. (ii) There exist a, b, c > 0 such that c−1 |x|aδ |x| c|x|bδ

for x ∈ G large

(2.3)

(i.e., off a compact neighborhood of the identity). Moreover, we can take a = b = 1 if and only if G is stratified. Proof. (i) Decompose g as in (2.1). Notice that the subspaces g[n] composing the descending central series are characteristic ideals of g; since the dilations δt are automorphisms, the g[n] are homogeneous. A homogeneous element of g[n] , being the sum of n-fold iterated commutators of homogeneous elements of g, has a homogeneity degree which must be the sum of n of the homogeneity degrees λ1 < · · · < λk of the elements of g; since all these degrees are not less than 1, the sum is not less than n, therefore g[n] ∩ Wλ = {0} if λ < n, so that g[n] ⊆

(2.4)

Wλ .

λn

In particular, if G is s-step, QG =

s n=1

dim g[n]

s

dim Wλ

n=1 λn

k

λj dim Wλj Qδ .

(2.5)

j =1

We already know that, if G is stratified, then QG = Qδ . Conversely, if QG = Qδ , then all the inequalities in (2.5) must be equalities; this means, first of all, that the degrees λ1 , . . . , λk are integers and, secondly, that the inclusion (2.4) is an equality, so that Wn ⊆ g[n] , but then necessarily W1 generates g — i.e., G is stratified. (ii) By the definition of | · | and the equivalence of homogeneous norms, the inequalities (2.3) follow easily. If G is stratified, then also | · |δ is (modulo equivalence of homogeneous norms) of the form (2.2), with a choice of the complements Vj possibly different to the one defining | · |; therefore, by Proposition 2.1, | · |δ is equivalent in the large to | · | (both being equivalent in the large to some τK ). Conversely, since

μ x ∈ G: |x| < r ∼ r QG ,

μ x ∈ G: |x|δ < r ∼ r Qδ

for r large, if (2.3) holds with a = b = 1, then necessarily QG = Qδ , and the conclusion follows by (i). 2 The automorphic dilations δt of a homogeneous Lie algebra g extend to automorphisms δt of its complex universal enveloping algebra U(g), which is canonically isomorphic to the algebra D(G) of left-invariant differential operators on G. An element D ∈ U(g) = D(G) is said to be homogeneous of degree λ if δt (D) = t λ D

for all t > 0.


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A Rockland operator on G is a homogeneous left-invariant differential operator D ∈ D(G) such that, for every non-trivial irreducible unitary representation π of G on a Hilbert space H, dπ(D) is injective on the space H∞ of the smooth vectors of the representation. In the abelian case (G = Rn ), with isotropic dilations, the notion of Rockland operator reduces to that of constant-coefficient homogeneous elliptic operator on Rn . In the general case, by a theorem of Helffer and Nourrigat (see [27,41]), combined with a result by Miller (see [42,51]), a homogeneous L ∈ D(G) is Rockland if and only if L is hypoelliptic, i.e., for every u ∈ D (G) and every open set Ω ⊆ G, (Lu)|Ω ∈ E(Ω)

⇒

u|Ω ∈ E(Ω).

2.2. Weighted bases and contraction of a Lie algebra A weighted (algebraic) basis of a Lie algebra g is a system A1 , . . . , Ad of linearly independent elements of g which generate g as a Lie algebra, together with the assignment of a weight wj ∈ [1, +∞[ to each Aj (j = 1, . . . , d). Fix a weighted basis on g. We recall some notation from [52], analogous to the multi-index notation for partial derivatives on Rn , but taking care of the non-commutative structure. Let J (d) be the set of finite sequences of elements of {1, . . . , d}, and J+ (d) be the subset of nonempty sequences. For every α = (α1 , . . . , αk ) ∈ J (d), let |α| denote the length k of α, and set α = kj =1 wαj , as an element of U(g) , Aα = Aα1 Aα2 · · · Aαk

if α ∈ J+ (d). A[α] = . . . [Aα1 , Aα2 ], . . . , Aαk The fixed weighted basis defines an (increasing) filtration on g:

Fλ = span A[α] : α ∈ J+ (d), α λ

for λ ∈ R;

we have in fact [Fλ , Fμ ] ⊆ Fλ+μ ,

Fλ =

μ>λ

Set Fλ− =

μ 0, let Cs (ε) be the set of absolutely continuous arcs γ : [0, 1] → G such that γ (t) =

k

φj (t)Aj |γ (t)

for a.e. t ∈ [0, 1],

j =1

where φj (t)
0: ∃γ ∈ Cs (ε) with γ (0) = x, γ (1) = y . It is not difficult to show that d0 , d∞ and d∗ are left-invariant distances on G, compatible with the topology of G. In fact, d∞ is the classical “unweighted” Carnot–Carathéodory distance associated with the Hörmander system A1 , . . . , Ak (cf. [57, §III.4]), while d0 is a “weighted” Carnot–Carathéodory distance (similar to the ones studied in [44]). Moreover, for x, y ∈ G, we have d0 (x, y) 1

⇐⇒

d∞ (x, y) 1

⇐⇒

d∗ (x, y) 1,

and the same holds with strict inequalities. Finally, d (x, y) for d∗ (x, y) 1, d∗ (x, y) = 0 d∞ (x, y) for d∗ (x, y) 1. We call d∗ the control distance2 on the weighted Lie group G. The control distance d∗ induces a control modulus | · |∗ on G, given by |g|∗ = d∗ (e, g). Moreover, if Br denotes the d∗ -ball with radius r centered at the identity of G, then μ(Br ) ∼ r Q∗

for r 1,

where Q∗ is the homogeneous dimension of the contraction g∗ (see [52, Proposition 6.1]). On the other hand, the growth rate of μ(Br ) for r large coincides with the (intrinsic) volume growth of the group G (cf. [57, §III.4]); in particular, if G has polynomial growth of degree QG , then μ(Br ) ∼ r QG

for r 1.

2.4. Weighted subcoercive forms and operators Let G be a weighted Lie group, with reduced basis A1 , . . . , Ad of its Lie algebra g, and weights w1 , . . . , wd . In this context, a form is an element of the free (non-commutative associative unital) algebra over C on d indeterminates X1 , . . . , Xd ; in other words, a form is a function 2 Notice that the definition of the control distance by ter Elst and Robinson in §6 of [52] (see also [50]) is different from the one given here, and coincides with our distance d0 . Their definition has the advantage that, in the case of a homogeneous group with an adapted basis, the modulus | · |0 induced by d0 is a homogeneous norm; on the other hand, this shows (by taking, e.g., any non-stratified homogeneous Lie group, cf. Propositions 2.1 and 2.2) that in general d0 is not a “connected distance” as in [57, §III.4]. Nevertheless, in the whole papers [2,50,52] it is understood that d0 is “connected”. By a careful examination of their proofs, one sees that the specific properties of d0 are used only for small distances, whereas in the large only “connectedness” is used. Therefore, our modified definition of the control distance d fixes the problem (as it has been confirmed to us by ter Elst in a private communication). As a side-effect, since d∗ d0 everywhere, the heat kernel estimates obtained with this modification (see Theorem 2.3(e)) are stronger than the ones claimed by ter Elst and Robinson (which are therefore true a posteriori).

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C : J (d) → C null off a finite subset of J (d), which can be thought of as the non-commutative polynomial

C(α)X α .

α∈J (d)

The degree of the form C is the number

max α: α ∈ J (d), C(α) = 0 . If C is a form of degree m, then its principal part is the form P : J (d) → C which is given by the sum of the terms of C of degree m: P (α) =

if α = m, otherwise.

C(α) 0

A form is said to be homogeneous if it equals its principal part. The adjoint of a form C is the form C + defined by C + (α) = (−1)|α| C(α∗ ), where α∗ = (αk , . . . , α1 ) if α = (α1 , . . . , αk ). To each form C, we associate a differential operator dRG (C) ∈ D(G) by setting

dRG (C) =

C(α)Aα .

α∈J (d)

More generally, if π is a representation of G, we define dπ(C) = dπ dRG (C) = C(α) dπ(A)α . α∈J (d)

Notice that we have dRG C + = dRG (C)+ , where, for D ∈ D(G), D + denotes its formal adjoint (with respect to the right Haar measure μ), i.e., the element of D(G) determined by Df, g = f, D + g

for all f, g ∈ D(G),

where f, g = G f g dμ. If π is a representation of G on a Banach space V, we define seminorms and norms on (subspaces of) V by Nπ,s (x) = max dπ X α x V , α∈J (d) α=s

xπ,s = max dπ X α x V , α∈J (d) αs


2777

for s ∈ R, s 0; these quantities are certainly defined on the space V ∞ of smooth vectors of the representation. If π is the right regular representation of G on Lp (G), we use the alternative notation Np;s , · p;s for the (semi)norms, and Lp;∞ (G) for the space of smooth vectors. A form C of degree m is said to be weighted subcoercive on G if m/wi ∈ 2N for i = 1, . . . , d and if moreover the corresponding operator satisfies a local Gårding inequality: there exist μ > 0, ν ∈ R and an open neighborhood V of the identity e ∈ G such that 2 φ, dRG (C)φ μ N2;m/2 (φ) − νφ22 for all φ ∈ D(G) with supp φ ⊆ V . In this case, the operator dRG (C) is called a weighted subcoercive operator. Let G∗ be the contraction of G, with Lie algebra g∗ . Since A1 , . . . , Ad induces a reduced basis A¯ 1 , . . . , A¯ d on g∗ (with the same weights), we can associate to a form C both a differential operator dRG (C) on G and a differential operator dRG∗ (C) on G∗ : in some sense, dRG∗ (C) is the “local counterpart” of the operator dRG (C). The next theorem clarifies the relationship between the two operators. Theorem 2.3 (ter Elst and Robinson). Let C be a form of degree m, whose principal part is P , such that m/wi ∈ 2N for i = 1, . . . , d. The following are equivalent: (i) C is a weighted subcoercive form on G; (ii) dRG∗ (P + P + ) is a positive Rockland operator on G∗ ; (iii) there are constants μ > 0, ν ∈ R such that, for every unitary representation π of G on a Hilbert space H, x, dπ(C)x μx2π,m/2 − νx2H for all x ∈ H∞ ; (iv) there is a constant μ > 0 such that, for every unitary representation π of G∗ on a Hilbert space H, 2 x, dπ(P )x μ Nπ,m/2 (x) for all x ∈ H∞ . Moreover, if these conditions are satisfied, for every representation π of G on a Banach space V, we have: (a) (b) (c) (d)

the closure of dπ(C) generates a continuous semigroup {St }t0 on V; n for t > 0, St (V) ⊆ V ∞ , and moreover V ∞ = ∞ n=1 D(dπ(C) ); + ∗ if π is unitary, then dπ(C) = dπ(C ) ; there exists a representation-independent kernel kt ∈ L1;∞ ∩ C0∞ (G) (for t > 0) such that dπ X α St x = π Aα kt x =

G

for all α ∈ J (d), t > 0, x ∈ V;

α A kt (g)π g −1 x dg

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(e) the kernel satisfies the following “Gaussian” estimates: for all α ∈ J (d) there exist b, c, ω > 0 such that |g|m 1/(m−1) α Q +α A kt (g) ct − ∗ m eωt e−b t∗

for all t > 0 and g ∈ G, where Q∗ is the homogeneous dimension of g∗ and | · |∗ is the control modulus; (f) for all ρ 0, the map t → kt is continuous ]0, +∞[ → L1;∞ (G, eρ|x|∗ dx) and, for all α ∈ J (d), there exist c, ω > 0 such that α A k t

L1 (G,eρ|x|∗ dx)

ct −

α m

eωt ;

(g) the function k(t, x) =

0 kt (x)

for t 0, for t > 0

on R × G satisfies ( ∂t∂ + dRG (C))k = δ in the sense of distributions, where δ is the Dirac delta at the identity of R × G. Proof. This theorem is a summary of results contained in [52], except for (f), since in Theorem 7.2 of [52] it is only stated that the map t → kt is continuous ]0, +∞[ → L1 (G, eρ|x|∗ dx). However, the weighted L1 estimates for Aα kt in (f) are obtained by integration of the pointwise estimates (e), since the volume growth of a connected Lie group is at most exponential (cf. [24]). Moreover, by the semigroup property, we have Aα (kt+s ) = kt ∗ Aα ks

(2.8)

and, since Aα ks ∈ L1 (G, eρ|x|∗ dx), the required continuity follows from the properties of convolution. 2 Corollary 2.4. With the notation of the previous theorem, if C is a weighted subcoercive form on G, then the function k(t, x) = kt (x) is smooth off the identity of R × G, and the operator dRG (C) is hypoelliptic. Proof. From Theorem 2.3(g) we deduce that, for every r ∈ N \ {0}, the distribution r r ∂t − −dRG (C) k

(2.9)

is supported in the origin of R × G. In particular, if φ ∈ D(]0, +∞[) and ψ ∈ D(G), by applying (2.9) to φ ⊗ ψ we get ∞ ∞ r (−1) −dRG (C) kt , ψ φ(t) dt. kt , ψφ (r) (t) dt = r

0

0


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Since both t → kt and t → (−dRG (C))r kt are continuous ]0, +∞[ → L1 (G) by Theorem 2.3(f), this identity holds also for all ψ ∈ C0 (G). In other words, for all ψ ∈ C0 (G), the r-th distributional derivative of the function t → kt , ψ on ]0, +∞[ is the map r t → −dRG (C) kt , ψ ; since all these derivatives are continuous, the function t → kt , ψ is smooth on ]0, +∞[, so that also the map t → kt is smooth ]0, +∞[ → L1 (G). But then from (2.8) it follows easily that t → kt is smooth ]0, +∞[ → L1;∞ (G). By Sobolev’s embedding, we then get that t → kt is smooth ]0, +∞[ → E(G); this gives that k is smooth on ]0, +∞[ × G, and the Gaussian estimates of Theorem 2.3(e) show that k can be extended smoothly by zero to the whole R × G \ {(0, e)}. Notice that kt∗ is the kernel of dRG (C + ), which is also a weighted subcoercive operator. If we put ˜ x) = 0∗ if t 0, k(t, k−t if t 0, then k˜ is smooth on R × G \ {(0, e)} and satisfies (− ∂t∂ + dRG (C + ))k˜ = δ in the sense of distributions. By arguing analogously as in the proof of Theorem 52.1 of [54], we obtain that ∂t + dRG (C) is hypoelliptic on R × G, and the hypoellipticity of dRG (C) on G follows immediately. 2 Corollary 2.5. With the notation of Theorem 2.3, if C is a weighted subcoercive form on G, then (kt )t>0 is an approximate identity on G for t → 0+ (cf. [22, §1.2.4]), i.e., • kt ∈ L1 (G) and lim supt→0+ kt 1 < ∞; • limt→0+ G\U |kt (x)| dx = 0 for all neighborhoods U of the identity of G; • limt→0+ G kt (x) dx = 1. More generally, for every D ∈ D(G), β 0 and every neighborhood U of the identity of G, Dkt (x) dx = 0. (2.10) lim t −β t→0+

G\U

Proof. If R > 0 is such that

x ∈ G: |x|∗ < R ⊆ U,

then, by Theorem 2.3(e), for t 1 we have t

−β

Dkt (x) dx ct −γ

G\U

+∞ m 1/(m−1) σ r e−b(r /t) e dr R

for some c, b, σ, γ > 0. On the other hand, for t 1 and r R, t −γ e−b(r

m /t)1/(m−1)

eσ r e−b(r

m m m−1 −R m−1 )+σ r

e−γ log t−bR

m 1 m−1 t − m−1

,

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where the first factor on the right-hand side is integrable on ]R, +∞[ and does not depend on t, whereas the second factor is infinitesimal for t → 0+ and does not depend on r; the limit (2.10) then follows by dominated convergence. In particular, we have lim

t→0+ G\U

kt (x) dx = 0,

and moreover, by Theorem 2.3(f), the norms kt 1 are uniformly bounded for t small. Finally, if π is the trivial representation of G on C and if c = dπ(C)1, then by Theorem 2.3(d) we have

ht (x) dx = π(ht )1 = e−tc ,

G

which tends to 1 as t → 0+ .

2

In the following, we will consider connected Lie groups G with no previously fixed weighted structure; then, an operator L ∈ D(G) will be said weighted subcoercive on G if L is weighted subcoercive with respect to some weighted structure on g. In this sense, we can say that every positive Rockland operator on a homogeneous Lie group is weighted subcoercive (see [51, Lemmata 2.2 and 2.4, and Theorem 2.5]; see also [52, Example 4.4]). Moreover, it is easy to check that, for every choice of a system of linearly independent generators A1 , . . . , Ad of a Lie algebra g, the assignment of weights all equal to 1 yields a stratified contraction g∗ ; in particular, the sublaplacian L = −(A21 + · · · + A2d ) is weighted subcoercive. Further, if A1 , . . . , Ad linearly generate g, then the contraction g∗ is Euclidean (abelian and isotropic), and it is not difficult to see that positive left-invariant elliptic operators on G are weighted subcoercive with respect to this structure. 3. Algebras of differential operators Here the existence and uniqueness of a joint spectral resolution for a commuting system L1 , . . . , Ln of formally self-adjoint left-invariant differential operators on a connected Lie group G is proved, under the hypothesis that the algebra generated by L1 , . . . , Ln contains a weighted subcoercive operator. An analogue of the (inverse) spherical Fourier transform of Gelfand pairs is also defined, and its main properties are derived. In this and the following sections, results from the theory of spectral integration (as presented, e.g., in [4,48,14]) will be used without further reference. 3.1. Joint spectral resolution In the following, G will be a connected Lie group. Lemma 3.1. Let D, L ∈ D(G) and suppose that L is weighted subcoercive and formally selfadjoint. Then, for some r¯ ∈ N, we have that, for all r r¯ , Lr + D is weighted subcoercive.


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Proof. Fix a weighted structure on g with respect to which the operator L is weighted subcoercive. Then there exists a weighted subcoercive form C such that dRG (C) = L, and also a form B such that dRG (B) = D. In fact, since L+ = L, we can suppose that C + = C. Let then P be the principal part of C, so that, by Theorem 2.3, dRG∗ (P ) is Rockland. By definition, this implies that, for every r ∈ N \ {0}, dRG∗ (P r ) is Rockland too. Notice now that, if r is sufficiently large so that P r has degree greater than that of B, then the principal part of C r + B is P r and this implies, by Theorem 2.3 again, that Lr + D = dRG (C r + B) is weighted subcoercive. 2 For every D ∈ D(G) and every unitary representation π of G on a Hilbert space H, the operator dπ(D) will be considered as defined on the space H∞ of smooth vectors of π , and notions such as closure or essential self-adjointness are understood to be referred to this domain.3 Proposition 3.2. Let A be a commutative unital subalgebra of D(G) closed by formal adjunction and containing a weighted subcoercive operator. Then, for every unitary representation π of G, we have ∗ dπ(D) = dπ D + for all D ∈ A; (3.1) moreover, the operators dπ(D) for D ∈ A are normal and commute strongly pairwise. Proof. Let L ∈ A be weighted subcoercive. Since A is closed by formal adjunction, by replacing L with (L + L+ )/2, we can suppose that L is formally self-adjoint (see Theorem 2.3). Let D ∈ A. By Lemma 2.3 of [45], in order to prove (3.1) it is sufficient to show that dπ(D + D) is essentially self-adjoint. However, by Lemma 3.1, it is possible to find r ∈ N sufficiently large so that both A = L2r and C = L2r + D + D are weighted subcoercive, which implies by Theorem 2.3(c) that dπ(A) and dπ(C) are essentially self-adjoint. The conclusion that dπ(D + D) = dπ(C)−dπ(A) is essentially self-adjoint then follows as in the proof of Corollary 2.4 of [45]. From (3.1) it follows that, for every formally self-adjoint D ∈ A, dπ(D) is essentially selfadjoint. Let now

Q = D2: D = D+ ∈ A . For all A, B ∈ Q, we have that A, B, (1 + A)(1 + B) are formally self-adjoint elements of A, so that dπ(A), dπ(B), dπ((1 + A)(1 + B)) are essentially self-adjoint, and moreover dπ(A + B + AB) is positive (notice that AB ∈ Q); this implies, as in the proof of Corollary 2.4 of [45], that dπ(A) and dπ(B) commute strongly, i.e., they have commuting spectral resolutions. In order to conclude, it will be sufficient to show that every operator of the form dπ(D) for some D ∈ A is the joint function of some of the operators dπ(A) for A ∈ Q. In fact, let D = D1 + iD2 , where D1 = D + D + /2, D2 = D − D + /2i 3 For some particular representations π one may be interested in considering other domains for the operators dπ(D): for instance, for the regular representation, one could consider the space D(G) of compactly supported smooth functions. Theorem 1.1 of [45] shows that for this and other “reasonable” choices of the domain, the closure of the dπ(D) remains unvaried, thus results about essential self-adjointness do not change.

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are both formally self-adjoint elements of A. Then D12 ,

(D1 + 1/2)2 ,

D22 ,

(D2 + 1/2)2

are all elements of Q, and we can consider the joint spectral resolution E on R4 of the corresponding operators in the representation π . We then have, for j = 1, 2, dπ(Dj ) = dπ (Dj + 1/2)2 − Dj2 − 1/4 ⊆ fj dE, R4

where fj (λ1,1 , λ1,2 , λ2,1 , λ2,2 ) = λj,2 − λj,1 − 1/4, so that also (f1 + if2 ) dE,

dπ(D) ⊆

dπ D + ⊆

R4

(f1 − if2 ) dE; R4

by passing to the adjoints in the second inclusion and using (3.1), we then get dπ(D) = (f1 + if2 ) dE, R4

and we are done.

2

A system L1 , . . . , Ln ∈ D(G) will be called a weighted subcoercive system if L1 , . . . , Ln are formally self-adjoint and pairwise commuting, and if moreover the unital subalgebra of D(G) generated by L1 , . . . , Ln contains a weighted subcoercive operator. From the previous proposition and the spectral theorem we then have immediately Corollary 3.3. Let L1 , . . . , Ln ∈ D(G) be a weighted subcoercive system. For every unitary representation π of G, the operators dπ(L1 ), . . . , dπ(Ln ) admit a joint spectral resolution Eπ on Rn and, for every polynomial p ∈ C[X1 , . . . , Xn ], dπ p(L1 , . . . , Ln ) = p dEπ . (3.2) Rn

In the following, the sign of closure for operators of the form (3.2) for some weighted subcoercive system L1 , . . . , Ln will be omitted. 3.2. Kernel transform and Plancherel measure Let G be a connected Lie group. We denote by Cv 2 (G) the set of the distributions k ∈ D (G) such that the operator f → f ∗ k is bounded on L2 (G). By the Schwartz kernel theorem, there is a one-to-one correspondence between Cv 2 (G) and the set of bounded linear operators T on L2 (G) which commute with left translations: T Lx = Lx T

for all x ∈ G;


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thus we endow Cv 2 (G) with the C∗ -algebra structure of the latter. We then have the continuous embedding L1 (G) ⊆ Cv 2 (G), which is not dense [43]; the closure of L1 (G) in Cv 2 (G) (or rather the corresponding set of convolution operators) is known as the reduced C∗ -algebra of G. Let L1 , . . . , Ln be a weighted subcoercive system on G. By applying Corollary 3.3 to the (right) regular representation on L2 (G), we obtain a joint spectral resolution E of L1 , . . . , Ln . In particular, for every f ∈ L∞ (Rn , E), we can consider the operator f (L) = f (L1 , . . . , Ln ) = E[f ] =

f dE, Rn

which is a bounded left-invariant linear operator on L2 (G), so that it admits a kernel f˘ ∈ Cv 2 (G): f (L)u = u ∗ f˘ for all u ∈ D(G). In place of f˘, we use also the notation KL f . The correspondence KL : f → KL f will be called the kernel transform associated with the weighted subcoercive system L1 , . . . , Ln . The previous definitions and the properties of the spectral integral then yield immediately Lemma 3.4. (a) KL is an isometric embedding of L∞ (Rn , E) into Cv 2 (G); in particular, for every f ∈ L∞ (Rn , E), f˘Cv 2 = f L∞ (Rn ,E) ,

f˘ = (f˘)∗ .

(b) If f, g ∈ L∞ (Rn , E) and g˘ ∈ L2 (G), then (f g)˘= f (L)g, ˘ and in particular, if g˘ ∈ D(G), then (f g)˘= g˘ ∗ f˘. (c) If f, g ∈ L∞ (Rn , E), and if g(λ) = λj f (λ) for some j ∈ {1, . . . , n}, then g˘ = Lj f˘ in the sense of distributions. The resemblance of KL with an (inverse) Fourier transform goes beyond Lemma 3.4, and more refined properties of KL follow from the fact that the algebra generated by L1 , . . . , Ln contains a weighted subcoercive operator. In fact, we can find a polynomial p∗ with real coefficients

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such that p∗ (L) is weighted subcoercive; by replacing p∗ with p∗2r for some large r ∈ N, we may suppose that p∗ 0 on Rn and that moreover, if we set p0 (λ) = p∗ (λ) +

n

λ2j + 1,

j =1

pk (λ) = p0 (λ) + λk

for k = 1, . . . , n,

then p0 (L), p1 (L), . . . , pn (L) are all weighted subcoercive (see Lemma 3.1). Notice that the polynomials p0 , p1 , . . . , pn are all strictly positive on Rn and lim pk (λ) = +∞

λ→∞

for k = 0, . . . , n;

moreover, p0 (L), . . . , pn (L) generate the same subalgebra of D(G) as L1 , . . . , Ln do. Lemma 3.5. The subalgebra of C0 (Rn ) generated by the functions e−p0 , e−p1 , . . . , e−pn is a dense ∗-subalgebra of C0 (Rn ). Proof. Since the functions e−p0 , e−p1 , . . . , e−pn are real valued, the algebra generated by them is a ∗-subalgebra of C0 (Rn ). Notice that e−p0 is nowhere null. Moreover, if λ, λ ∈ Rn and λ = λ , then λk = λk for some k ∈ {1, . . . , n}, hence

either e−p0 (λ) = e−p0 (λ )

or

e−pk (λ) = e−pk (λ ) .

The conclusion then follows immediately by the Stone–Weierstrass theorem.

2

Let now JL be the subalgebra of C0 (Rn ) generated by the functions of the form e−q , where q is a non-negative polynomial on Rn such that q(L) is a weighted subcoercive operator on G and limλ→∞ q(λ) = +∞. Set moreover

C0 (L) = C0 (L1 , . . . , Ln ) = f˘: f ∈ C0 Rn . Finally, let Σ be the joint spectrum of L1 , . . . , Ln , i.e., the support of their joint spectral resolution E. Proposition 3.6. C0 (L) is a sub-C ∗ -algebra of Cv 2 (G), which is isometrically isomorphic to C0 (Σ) via the kernel transform. Moreover KL (JL ) = {f˘: f ∈ JL } is a dense ∗-subalgebra of C0 (L).


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Proof. For a function f ∈ C0 (Rn ), we have f L∞ (Rn ,E) = sup |f | = f |Σ C0 (Σ) . Σ

Since every g ∈ C0 (Σ) extends to an f ∈ C0 (Rn ) by the Tietze–Urysohn extension theorem, the first part of the conclusion follows immediately from Lemma 3.4(a). The second part follows instead from Lemma 3.5. 2 The results on weighted subcoercive operators and their heat kernels imply that the elements of KL (JL ) are particularly well-behaved. The next proposition, which shows a sort of commutativity between joint functional calculus of L1 , . . . , Ln and unitary representations of G, is a multivariate analogue of Proposition 2.1 of [36]. Proposition 3.7. For every f ∈ JL , we have f˘ ∈ L1;∞ (G) ∩ C0∞ (G) and moreover, for every unitary representation π of G, π(f˘) = f dπ(L1 ), . . . , dπ(Ln ) . If G is amenable, the last identity holds for every f ∈ C0 (Rn ) with f˘ ∈ L1 (G). Proof. Suppose first that f is one of the generators e−q of JL . Then, by Corollary 3.3 and the properties of the spectral integral, e−q dπ(L1 ), . . . , dπ(Ln ) = e−dπ(q(L)) , and, since q(L) is weighted subcoercive, we obtain from Theorem 2.3(d) that KL (e−q ) ∈ L1;∞ ∩ C0∞ (G) and e−q (dπ(L1 , . . . , Ln )) = π(KL (e−q )). The result is easily extended to every f ∈ JL by Lemma 3.4, the properties of convolution and those of the spectral integral. Suppose now that G is amenable, f ∈ C0 (Rn ) and f˘ ∈ L1 (G). By Proposition 3.6, we can find a sequence fj ∈ JL which converges uniformly to f on Rn . This implies in particular, by the properties of the spectral integral, that fj dπ(L1 ), . . . , dπ(Ln ) → f dπ(L1 ), . . . , dπ(Ln ) in the operator norm, but also that f˘j → f˘ in Cv 2 (G). Since G is amenable, the representation π is weakly contained in the regular representation (see [23, §3.5]), so that also π(f˘j ) → π(f˘) in the operator norm. But then the conclusion follows immediately from the first part of the proof. 2 We are now going to exploit the good properties of the kernels in KL (JL ) to obtain a Plancherel formula for the kernel transform KL . It should be noticed that, in the context of commutative Banach ∗-algebras, a general abstract argument yielding this kind of results is available (see [35, §26J], and also [19, Theorem 1.6.1]). However, we believe that additional insight is provided by the explicit construction presented below, which follows essentially [11], with some modifications due to our multivariate and possibly non-unimodular setting.

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Proposition 3.8. If f ∈ L∞ (Rn , E) is compactly supported, then f˘ ∈ L2;∞ ∩ C0∞ (G). Proof. Let ξt = e−tp∗ for t > 0, so that ξ˘t ∈ L1;∞ (G) ∩ C0∞ (G). Since f is compactly supported, f = g ξ1 with g = f/ξ1 ∈ L∞ (Rn , E), so that f˘ = g(L)ξ˘1 ∈ L2 (G) by Lemma 3.4. Analogously, being g compactly supported, also g˘ ∈ L2 (G), but then f˘ = ξ1 (L)g˘ = g˘ ∗ ξ˘1 ∈ L2;∞ ∩ C0∞ (G), by Lemma 3.4 and properties of convolution. 2 Thus we have plenty of kernels f˘ which are in L2 (G); as we are going to see, the L2 -norm can be interpreted as an operator norm of a convolution operator. Let · 2ˆ denote the L2 -norm with respect to the left Haar measure μ (where is the modular function), and correspondingly the operator norm from L2 (G, μ) to L∞ (G); then it is easily shown that · 2→∞ ˆ Lemma 3.9. For all f ∈ L∞ (E), we have f˘ ∈ L2 (G) if and only if f (L) ˆ < ∞, 2→∞ and in this case f˘2 = f (L)2→∞ . ˆ We are now able to obtain a Plancherel formula for the kernel transform. Theorem 3.10. The identity 2 σ (A) = E(A)2→∞ ˆ

for all Borel A ⊆ Rn

defines a regular Borel measure on Rn with support Σ , whose negligible sets coincide with those of E and such that, for all f ∈ L∞ (E), 2 |f |2 dσ = f (L)2→∞ = f˘22 . ˆ Rn

Proof. Clearly σ (∅) = 0. Moreover, σ is monotone: if A ⊆ A are Borel subsets of Rn and σ (A ) < ∞, then, by Lemma 3.9, χ˘ A ∈ L2 (G), so that, by Lemma 3.4, also χ˘ A = E(A)χ˘ A ∈ L2 (G)

and χ˘ A 2 χ˘ A 2 ,

i.e., σ (A) σ (A ). We now prove that σ is finitely additive. Let A, B ⊆ Rn be disjoint Borel sets. By monotonicity, we may suppose that σ (A), σ (B) < ∞. Then, by Lemma 3.9, both χ˘ A , χ˘ B ∈ L2 (G), but E(A ∪ B) = E(A) + E(B), so that clearly χ˘ A∪B = χ˘ A + χ˘ B ∈ L2 (G), and moreover, by Lemma 3.9, σ (A ∪ B) = χ˘ A∪B 22 = χ˘ A 22 + χ˘ B 22 = σ (A) + σ (B), since χ˘ A = E(A)χ˘ A ⊥ E(B)χ˘ B = χ˘ B in L2 (G) by Lemma 3.4.


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n Finite additivity implies that, if Aj (j ∈ N) are pairwise disjoint Borel subsets of R and A = j Aj , then

σ (Aj ) σ (A).

j

In particular, if the sum on the left-hand side diverges, then we have an equality. Suppose instead that the left-hand side sum converges. Then, as before, the χ˘ Aj are pairwise orthogonal elements of L2 (G), and their sum converges in L2 (G) to some k ∈ L2 (G) such that k22 = j σ (Aj ). But then, if u ∈ D(G), we have that, on one hand, by Lemma 3.9,

u ∗ χ˘ Aj = u ∗ k

uniformly,

j

and, on the other hand,

u ∗ χ˘ Aj =

j

E(Aj )u = E(A)u

in L2 (G),

j

which gives, by uniqueness of limits and arbitariness of u ∈ D(G), χ˘ A = k ∈ L2 (G)

and σ (A) = k22 =

σ (Aj ).

j

It is immediate from the definition that a Borel subset of Rn is σ -negligible if and only if it is E-negligible; in particular supp σ = supp E = Σ . = χ˘ A 22 is finite if A ⊆ Rn is relatively compact. By Proposition 3.8, σ (A) = χA (L)2ˆ 2→∞ We can then conclude, by Theorem 2.18 of [49], that σ is regular. Notice that, for all Borel A ⊆ Rn with σ (A) < ∞, σ coincides with the measure E(·)χ˘ A , χ˘ A on the subsets of A: in fact, for all Borel B ⊆ Rn , E(B)χ˘ A , χ˘ A = χ˘ A∩B 22 = σ (A ∩ B)

by Lemmata 3.9 and 3.4. In particular, for all f ∈ L∞ (E) with supp f ⊆ A,

|f | dσ = 2

Rn

f (λ)2 E(dλ)χ˘ A , χ˘ A = f (L)χ˘ A 2 = f˘2 = f (L)2ˆ 2 2

2→∞

Rn

by the properties of the spectral integral and Lemmata 3.9 and 3.4. Take now a countable partition of Rn made of relatively compact Borel subsets Aj (j ∈ N). Then, for every f ∈ L∞ (Rn , E), analogously as before we obtain f (L)2ˆ

2→∞

=

E(Aj )f (L)2ˆ

2→∞

j

and putting all together we get the conclusion.

=

KL (f χA )2 , j 2 j

2

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The measure σ of the previous proposition is called the Plancherel measure associated with the system L1 , . . . , Ln . Notice that L∞ Rn , E = L∞ (σ ). We now show that the estimates (for small times) on the heat kernel of weighted subcoercive operators give information on the behavior at infinity of the Plancherel measure. In the following | · |2 shall denote the Euclidean norm. Proposition 3.11. The Plancherel measure σ on Rn associated with a weighted subcoercive system L1 , . . . , Ln has (at most) polynomial growth at infinity. Proof. If ξt (λ) = e−tp∗ (λ) , then, for every r > 0, σ {p∗ r} = χ{p∗ r} 2L2 (σ ) e2 ξ1/r 2L2 (σ ) = e2 ξ˘1/r 2L2 (G) . Since ξ˘t is the heat kernel of the operator p∗ (L1 , . . . , Ln ), Theorem 2.3(e, f) gives, for large r, σ {p∗ r} Cr Q∗ /m , where m is the degree of p∗ (L1 , . . . , Ln ) with respect to a suitable weighted structure on g, and Q∗ is the homogeneous dimension of the corresponding contraction g∗ . In particular, if d is the degree of the polynomial p∗ , we get, for large a > 0, σ

λ: |λ|2 a σ p∗ C(1 + a)d C(1 + a)Q∗ d/m ,

which is the conclusion.

2

The proof of Proposition 3.11 shows that the degree of growth at infinity of the Plancherel measure σ is somehow related to the “local dimension” Q∗ of the group with respect to the control distance associated with the chosen weighted subcoercive operator (see Section 2.3). In Section 5.1 we will obtain more precise information on the behavior of σ under the hypothesis of homogeneity. By Theorem 3.10, KL |L2 ∩L∞ (σ ) extends to an isometry from L2 (σ ) onto a closed subspace of L2 (G). We give now an alternative characterization of this subspace. Namely, let ΓL2 be the closure of KL (JL ) in L2 (G). Proposition 3.12. KL |L2 ∩L∞ (σ ) extends to an isometric isomorphism L2 (σ ) → ΓL2 . In fact, this result follows immediately from Theorem 3.10 and the following Lemma 3.13. JL is dense in Lq (σ ) for 1 q < ∞.


2789

Proof. Since σ has polynomial growth at infinity (see Proposition 3.11), whereas the elements of JL decay exponentially, it is easily seen that JL is contained (modulo restriction to Σ) in L1 ∩ L∞ (σ ). Since σ is a positive regular Borel measure on Rn , in order to prove that the closure of JL in Lq (σ ) is the whole Lq (σ ), it is sufficient to show that Cc (Rn ) is contained in this closure (see [49, Theorem 3.14]). Let then m ∈ Cc (Rn ). By Lemma 3.5, we can find a sequence mk ∈ JL converging uniformly to m, so that supk mk ∞ = C < ∞. Thus, for every t > 0, mk e−tp0 converges uniformly to me−tp0 , dominated by Ce−tp0 ∈ Lq (σ ), and consequently mk e−tp0 → me−tp0 also in Lq (σ ); we then have that me−tp0 is in the closure of JL in Lq (σ ) for all t > 0, and by monotone convergence also m is in this closure. 2 −1 We now prove a sort of Riemann–Lebesgue lemma for KL .

Proposition 3.14. For every bounded Borel f : Rn → C with f˘ ∈ L1 (G), we have f L∞ (σ ) f˘1 , and moreover lim f χ{λ: |λ|2 r} L∞ (σ ) = 0.

r→+∞

Proof. The first inequality follows immediately from Lemma 3.4 and Young’s inequality. Let ξt = e−tp0 . Then, by Corollary 2.5, ξ˘t is an approximate identity for t → 0+ . In particular, if f˘ ∈ L1 (G), then KL (f ξt ) = f˘ ∗ ξ˘t → f˘ in L1 (G) for t → 0+ , which implies, by the first inequality, that lim f (1 − ξt )L∞ (σ ) = 0.

t→0+

Therefore, for every ε > 0, there exists t > 0 such that f (1 − ξt )L∞ (σ ) ε; since p0 (λ) → +∞ for λ → ∞, we may find r > 0 such that ξt χ{λ: |λ|2 r} ∞ 1/2, but then necessarily f χ{λ: |λ|2 r} ∞ 2ε.

2

An analogous (and neater) result for KL is obtained under the additional hypothesis of unimodularity. Proposition 3.15. If G is unimodular and f ∈ L1 ∩ L∞ (σ ), then f˘ ∈ C0 (G) and f˘∞ f L1 (σ ) .

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Proof. Since f ∈ L1 ∩ L∞ (σ ), for some Borel g1 , g2 : Rn → C we have f = g1 g2

and |g1 |2 = |g2 |2 = |f |;

in particular, g1 , g2 ∈ L2 ∩ L∞ (σ ). Therefore g˘ 1 , g˘ 2 ∈ L2 (G) by Theorem 3.10 and f˘ = g˘ 1 ∗ g˘ 2 by Lemma 3.4, which gives the conclusion by Young’s inequality (see [30, Theorem 20.16]).

2

3.3. Change of generators Let L1 , . . . , Ln be a weighted subcoercive system on a connected Lie group G. Let σ be the associated Plancherel measure on Rn , and Σ = supp σ . For given polynomials P1 , . . . , Pn : Rn → R, consider the operators L1 = P1 (L1 , . . . , Ln ),

...,

Ln = Pn (L1 , . . . , Ln ),

and suppose that they still form a weighted subcoercive system. Let σ be the Plancherel measure on Rn associated with the system L1 , . . . , Ln , and Σ its support. We may ask if there is a relationship between the transforms KL and KL , and between the Plancherel measures σ and σ . Let P : Rn → Rn denote the polynomial map whose j -th component is the polynomial Pj .

Lemma 3.16. The map P |Σ : Σ → Rn is a proper continuous map. Proof. Since L1 , . . . , Ln is a weighted subcoercive system, we can find a non-negative poly nomial Q : Rn → R such that Q(L ) = Q(P (L)) is a weighted subcoercive operator. By Theorem 2.3(iii), for sufficiently large C > 0 and k ∈ N, we have that k max Lj φ2 C 1 + Q P (L) φ 2 j

for φ ∈ D(G),

which means, by the spectral theorem, that k max |λj | C 1 + Q P (λ) j

for λ ∈ Σ,

since Σ is the joint spectrum of L1 , . . . , Ln . Now, if K ⊆ Rn is compact, then by continuity there exists M > 0 such that Q|K M, but then max |λj | C 1 + M k for λ ∈ Σ ∩ P −1 (K), j

thus P −1 (K) ∩ Σ is bounded in Rn , and also closed (by continuity of P ), therefore P −1 (K) is compact. 2


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Proposition 3.17. For every bounded Borel m : Rn → C, we have: m L = (m ◦ P )(L),

KL m = KL (m ◦ P ).

Moreover σ = P (σ ),

Σ = P (Σ).

Proof. The first part of the conclusion follows immediately from the spectral theorem and uniqueness of the convolution kernel. From this, the identity σ = P (σ ) is easily inferred by Theorem 3.10. In particular, σ Rn \ P −1 Σ = σ Rn \ Σ = 0, i.e., by continuity of P , P (Σ) ⊆ Σ . In order to prove the opposite inclusion, we use the fact that P |Σ is proper (see Lemma 3.16). Take λ ∈ Σ , and let Bk be a decreasing sequence of compact neighborhoods of λ in Rn such −1 that k Bk = {λ }. By definition of support, we then have σ (P (Bk )) = σ (Bk ) = 0, therefore P −1 (Bk ) ∩ Σ = ∅ for all k. Since P |Σ is proper, we have a decreasing sequence P −1 (Bk ) ∩ Σ of non-empty compacta of Rn , which therefore has a non-empty intersection. If λ belongs to this intersection, then clearly λ ∈ Σ and moreover P (λ) ∈ Bk for all k, that it, P (λ) = λ . 2 A particularly interesting case is when L1 , . . . , Ln generate the same subalgebra of D(G) as L1 , . . . , Ln . In this case, there exists also a polynomial map Q = (Q1 , . . . , Qn ) : Rn → Rn such that L1 = Q1 L ,

...,

Ln = Qn L .

Notice that in general P and Q are not the inverse one of the other: from the spectral theorem, we only deduce that (Q ◦ P )|Σ = idΣ , (P ◦ Q)|Σ = idΣ (in fact, these identities extend to the Zariski-closures of Σ and Σ ). In particular, P |Σ : Σ → Σ ,

Q|Σ : Σ → Σ

are homeomorphisms. Another way of producing new weighted subcoercive systems from a given one is via the action of automorphisms of G. Namely, if k ∈ Aut(G), then its derivative k is an automorphism of g, therefore it extends to a unique filtered ∗-algebra automorphism of D(G) ∼ = U(g) (which shall be still denoted by k ), and clearly k (L1 ),

...,

k (Ln )

(3.3)

is a weighted subcoercive system on G. Notice that, for every k ∈ Aut(G), the push-forward via k of the right Haar measure μ on G is a multiple of μ, and in fact there is a Lie group homomorphism c : Aut(G) → R+ such that k(μ) = c(k)μ.

2792


In particular, if we set Tk f = f ◦ k −1 for k ∈ Aut(G), then the properties of the spectral integral and those of convolution give immediately Proposition 3.18. For k ∈ Aut(G), Tk is a multiple of an isometry of L2 (G); more precisely Tk f 22 = c(k)−1 f 22 . Moreover, for all D ∈ D(G), k (D) = Tk DTk−1 . In particular, for every bounded Borel m : Rn → C, m k (L1 ), . . . , k (Ln ) = Tk m(L1 , . . . , Ln )Tk−1 , and consequently Kk (L) m = c(k)Tk KL m. Let O be the unital subalgebra of D(G) generated by L1 , . . . , Ln . For any automorphism k ∈ Aut(G), we say that O is k-invariant if k(O) ⊆ O, or equivalently, if k(O) = O (the equivalence is due to the fact that k is an injective linear map preserving the filtration of D(G), which is made of finitely dimensional subspaces). Let Aut(G; O) denote the (closed) subgroup of Aut(G) made of the automorphisms k such that O is k-invariant. If k ∈ Aut(G; O), then (3.3) must be a system of generators of O; therefore, we can choose a polynomial map Pk = (Pk,1 , . . . , Pk,n ) : Rn → Rn such that k (Lj ) = Pk,j (L). Hence, by putting together Propositions 3.17 and 3.18, we get Corollary 3.19. If k ∈ Aut(G; O), then, for every bounded Borel m : Rn → C, (m ◦ Pk )(L1 , . . . , Ln ) = Tk m(L1 , . . . , Ln )Tk−1 and KL (m ◦ Pk ) = c(k)Tk KL m. Moreover, Pk (σ ) = c(k)σ,

Pk (Σ) = Σ.

In particular, the restrictions Pk |Σ (which are univocally determined by k) define an action of the group Aut(G; O) on the spectrum Σ by homeomorphisms; more precisely


2793

Proposition 3.20. The map Aut(G; O) × Σ (k, λ) → Pk −1 (λ) ∈ Σ

(3.4)

is continuous, and defines a continuous (left) action of Aut(G; O) on Σ . Proof. Recall that Σ may be identified, as a topological space, with the Gelfand spectrum of the sub-C∗ -algebra C0 (L) of Cv 2 (G), where λ ∈ Σ corresponds to the multiplicative linear ˘ = m(λ). By Corollary 3.19 we then deduce functional ψλ defined by ψλ (m) ψPk (λ) = c(k)ψλ ◦ Tk , which clarifies that (3.4) defines a left action on Σ . Moreover, since C0 (L) ∩ L1 (G) is dense in C0 (L) (see Proposition 3.6), and since c(k)Tk is an isometry of Cv 2 (G), we obtain easily that k → c(k)Tk u is continuous for every u ∈ Cv 2 (G). Therefore, since the topology of the Gelfand spectrum is induced by the weak-∗ topology, we immediately obtain that (3.4) is separately continuous, and also jointly continuous since the ψλ have uniformly bounded norms. 2 In conclusion, the richer the group Aut(G; O) is, the more we may deduce about the structure of the spectrum Σ and the Plancherel measure σ . An example of this fact is illustrated in Section 5.1. 4. Spectrum and eigenfunctions Let L1 , . . . , Ln be a weighted subcoercive system on a connected Lie group G. We keep the notation of Section 3.2. Notice that every m ∈ JL is real analytic and admits a unique holomorphic extension to Cn , which we still denote by m. Proposition 4.1. Let φ ∈ D (G) be such that, for some λ = (λ1 , . . . , λn ) ∈ Cn , Lj φ = λ j φ

for j = 1, . . . , n

in the sense of distributions. Then φ ∈ E(G), and the previous equalities hold in the strong sense. Moreover, if φ ∈ L∞ (G), then, for every m ∈ JL , φ∗m ˘ = m(λ)φ

and m, ˘ φ = m(λ)φ(e).

(4.1)

Proof. From the hypothesis, we get immediately p∗ (L)φ = p∗ (λ)φ. Since p∗ (L) − p∗ (λ) is hypoelliptic by Corollary 2.4, this implies that φ ∈ E(G). Suppose now that φ is bounded. Let e−q be one of the generators of JL , and set kt = KL (e−tq ). Then, for every x ∈ G, also Lx φ is a joint eigenfunction of L1 , . . . , Ln with eigenvalue λ; therefore, by Theorem 2.3(f, g), the function t → φ ∗ kt (x) = Lx φ, kt

2794


is smooth on ]0, +∞[, with derivative t → Lx φ, −q(L)kt = −q(λ)φ ∗ kt (x). Hence we get φ ∗ kt = e−tq(λ) φ, since kt is an approximate identity for t → 0+ (see Corollary 2.5). This gives the former identity of (4.1) when m is a generator of JL , and consequently also for an arbitrary m ∈ JL ; the latter identity follows by evaluating the former in e. 2 The previous proposition shows that the joint eigenfunctions of L1 , . . . , Ln are smooth, and are also eigenfunctions of the convolution operators with kernels in KL (JL ). An analogous result holds in every unitary representation of G. Lemma 4.2. Let π be a unitary representation of G on H. The following are equivalent for v ∈ H \ {0}: (i) v ∈ H∞ and v is a joint eigenvector of dπ(L1 ), . . . , dπ(Ln ); (ii) v is a joint eigenvector of the operators π(m) ˘ for m ∈ JL . Proof. (i) ⇒ (ii) follows immediately from Proposition 3.7 and the properties of the spectral ˘ = e−pj (dπ(L)) integral. For the reverse implication, take m = e−pj for j = 0, . . . , n, so that π(m) by Proposition 3.7; by the properties of the spectral integral, ker π(m) ˘ = {0}, therefore π(m)v ˘ = cv for some c > 0. This implies that ˘ ∈ H∞ , v = c−1 π(m)v by Theorem 2.3(b), and moreover, again by the properties of the spectral integral, pj dπ(L) v = (log c)v, that is, v is an eigenvector of pj (dπ(L)) for j = 0, . . . , n. Since λj = pj (λ) − p0 (λ)

for j = 1, . . . , n,

it follows that v is a joint eigenvector of dπ(L1 ), . . . , dπ(Ln ).

2

The link between eigenfunctions on G and eigenvectors in unitary representations is given by the joint eigenfunctions of positive type. Recall that a function of positive type φ : G → C is a diagonal coefficient for some unitary representation π of G on a Hilbert space H, i.e., φ(x) = π(x)v, v (4.2) for some vector v ∈ H, which can be supposed to be cyclic for π ; in that case, the representation π is uniquely determined by φ up to equivalence (see §3.3 of [17] for details), and φ is said to be associated with π .


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Proposition 4.3. For a function of positive type φ on G, the following are equivalent: (i) φ is a joint eigenfunction of L1 , . . . , Ln and φ(e) = 1; (ii) φ has the form (4.2) for some unitary representation π of G on H and some cyclic vector v of norm 1, where v ∈ H∞ is a joint eigenvector of dπ(L1 ), . . . , dπ(Ln ); (iii) φ = 0 and, for all m ∈ JL and f ∈ L1 (G), m ˘ ∗ f, φ = f ∗ m, ˘ φ = f, φm, ˘ φ; (iv) φ = 0 and, for all m ∈ JL , m ˘ ∗m ˘ ∗ , φ = |m, ˘ φ|2 . In this case, moreover, the eigenvalue of Lj corresponding to φ is a real number and coincides with the eigenvalue of dπ(Lj ) corresponding to v. Proof. (i) ⇒ (ii). Since φ is of positive type and φ(e) = 1, then φ is of the form (4.2) for some unitary representation π of G on H and some cyclic vector v of norm 1. From (i) we have Lj φ = λj φ for some λ = (λ1 , . . . , λn ) ∈ Cn . Being L1 , . . . , Ln left-invariant, if φy (x) = Ly φ(x) = π(x)v, π(y)v , then also Lj φy = λj φy . Since v is cyclic, for all w ∈ H we can find a sequence (wn )n in span{π(y)v: y ∈ G} such that wn → w in H; if ψn (x) = π(x)v, wn ,

ψ(x) = π(x)v, w ,

then the ψn are linear combinations of the φy , so that Lj ψn = λj ψn and, passing to the limit, we also have Lj ψ = λj ψ in the sense of distributions. But then ψ ∈ E(G) by Proposition 4.1. Since w ∈ H was arbitrary, we conclude that v ∈ H∞ ; moreover λj v, w = λj ψ(e) = Lj ψ(e) = dπ(Lj )v, w , and again, from the arbitrariness of w, we get dπ(Lj )v = λj v for j = 1, . . . , n. Finally, since dπ(Lj ) is self-adjoint, we deduce that λj ∈ R. (ii) ⇒ (i). Trivial. ˘ = cv for some c ∈ C. Since v = 1, (ii) ⇒ (iii). If m ∈ JL , by Lemma 4.2, π(m) ˘ ∗ v = π(m)v we have f ∗ m, ˘ φ = π(f ∗ m)v, ˘ v = π(m)π(f ˘ )v, v = c π(f )v, v = π(f )v, v π(m)v, ˘ v = f, φm, ˘ φ. The other identity is proved analogously. (iii) ⇒ (iv). Trivial. (iv) ⇒ (ii). Being of positive type, φ has the form (4.2) for some unitary representation π of G on H and some cyclic vector v. Then (iv) can be equivalently rewritten as π(m)v ˘ v ˘ = π(m)v,

(4.3)

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for all m ∈ JL . In particular, by taking m = e−tp∗ , which is an approximate identity for t → 0+ (see Corollary 2.5), and passing to the limit, we obtain v = v2 , so that v = 1 (since φ = 0). ˘ cannot have a component orthogonal to v, Now, for an arbitrary m ∈ JL , (4.3) implies that π(m)v thus v is an eigenvector of π(m), ˘ and (ii) follows from Lemma 4.2. 2 Let PL be the set of the joint eigenfunctions φ of L1 , . . . , Ln of positive type with φ(e) = 1. For every φ ∈ PL , by Proposition 4.3 the corresponding eigenvalue λ is in Rn ; we then define ϑL : PL → Rn by setting ϑL (φ) = λ. Lemma 4.4. If PL is endowed with the topology induced by the weak-∗ topology of L∞ (G), then the map ϑL : PL → Rn is continuous. Proof. By Proposition 4.1, for j = 0, . . . , n, we have that e−pj (ϑL (φ)) = KL e−pj , φ , which is continuous in φ with respect to the weak-∗ topology of L∞ (G). In particular, if ϑL,j : PL → R is the j -th component of ϑL for j = 1, . . . , n, then e−ϑL,j (φ) = e−pj (ϑL (φ)) /e−p0 (ϑL (φ)) ; therefore the components of ϑL are continuous PL → R.

2

Proposition 4.5. The topologies on PL induced by the weak-∗ topology of L∞ (G), the compactopen topology of C(G) and the topology of E(G) coincide. Moreover, the map ϑL : PL → Rn is a continuous, proper and closed map. In particular, the image ϑL (PL ) is a closed subset of Rn and its topology as a subspace of Rn coincides with the quotient topology induced by ϑL . Proof. Since G is second-countable, the three aforementioned topologies on PL are all metrizable (cf. [40, Corollary 2.6.20]). In particular, in order to prove that they coincide, it is sufficient to show that they induce the same notion of convergence of sequences. Let (φk )k be a sequence in PL . If (φk )k converges in E(G), then a fortiori it converges in C(G). Moreover, since φk ∞ = 1 for all k, convergence in C(G) implies weak-∗ convergence in L∞ (G) by dominated convergence. Suppose now that φk → φ ∈ PL with respect to the weak-∗ topology of L∞ (G). Take m = −p e ∗ ∈ JL , so that m > 0. By Proposition 4.1, for all D ∈ D(G), we then have Dφk =

˘ φk ∗ D m , m(ϑL (φk ))

Dφ =

φ ∗ Dm ˘ ; m(ϑL (φ))

in particular, for every x ∈ G, since Rx D m ˘ ∈ L1 (G), Dφk (x) =

˘ φk ˘ φ Rx D m, Rx D m, → = Dφ(x) m(ϑL (φk )) m(ϑL (φ))

by Lemma 4.4. Moreover, again by Lemma 4.4, m(ϑL (φk )) c > 0 for some c and all k, so that ˘ 1 . This means that, for all D ∈ D(G), the family {Dφk }k is equibounded; Dφk ∞ c−1 D m


2797

but then also, for all D ∈ D(G), the family {Dφk }k is equicontinuous, so that the previously proved pointwise convergence Dφk → Dφ is in fact uniform on compacta. By arbitrariness of D ∈ D(G), we have then proved that φk → φ in E(G). Let now K ⊆ Rn be compact, and take a sequence (φk )k in PL such that ϑL (φk ) ∈ K for all k. As before, the sequence (φk )k is equibounded and equicontinuous, so that, by the Ascoli– Arzelà theorem (see [9, §X.2.5]), we can find a subsequence φkh which converges uniformly on compacta to a function φ ∈ C(G), and such that moreover ϑL (φkh ) converges to some λ ∈ K. It is now easy to show that φ is of positive type and φ(e) = 1; moreover, for all η ∈ D(G), Lj φ, η = limLj φkh , η = lim ϑL,j (φkh )φkh , η = λj φ, η, h

h

so that, by Proposition 4.1, φ is a (smooth) joint eigenfunction of L1 , . . . , Ln , hence φ ∈ PL . Since PL is metrizable, this shows that ϑL−1 (K) is compact in PL . By the arbitrariness of the compact K ⊆ Rn , we conclude that ϑL is proper and closed (see [8, Propositions I.10.1 and I.10.7]). 2 The following result, together with the Krein–Milman theorem, shows that the image of ϑL does not change if we restrict to the joint eigenfunctions associated with irreducible representations. Proposition 4.6. For λ ∈ Rn , the set ϑL−1 (λ) is a weakly-∗ compact and convex subset of L∞ (G), whose extreme points are the ones associated with irreducible representations. Proof. Clearly ϑL−1 (λ) is convex, whereas compactness follows from Proposition 4.5. In order to conclude, it will be sufficient to show that the extreme points of ϑL−1 (λ) are also extreme points of the set P1 of the functions φ of positive type on G such that φ(e) = 1 (see [17, Theorem 3.25]). Suppose then that φ ∈ ϑL−1 (λ) is not extreme in P1 , i.e., φ = θ02 φ0 + θ12 φ1 for some φ0 , φ1 ∈ P1 different from φ and some θ0 , θ1 > 0 with θ02 + θ12 = 1. For k = 0, 1, we have φk (x) = πk (x)vk , vk , where πk is a unitary representation of G on Hk and vk is a cyclic vector of norm 1. If v = (θ0 v0 , θ1 v1 ) ∈ H0 ⊕ H1 ,

H = span (π0 ⊕ π1 )(x)v: x ∈ G ,

and π is the restriction of π0 ⊕ π1 to H, then it is easy to see that v is a cyclic vector for π and that φ(x) = π(x)v, v, therefore by Proposition 4.3 it follows that v ∈ H∞ and that dπ(Lj )v = λj v for j = 1, . . . , n. If Pk : H → Hk is the restriction of the canonical projection H0 ⊕ H1 → Hk , it is immediate to check that Pk intertwines π and πk , and that Pk v = θk vk ; hence, for all w ∈ Hk and x ∈ G, πk (x)vk , w = θk−1 πk (x)Pk v, w = θk−1 π(x)v, Pk∗ w .

2798


This identity, together with the arbitrariness of w ∈ Hk , shows that also vk ∈ Hk∞ . Moreover, since Pk intertwines π(x) and πk (x) for all x ∈ G, it is easy to check that it intertwines also dπ(D) and dπk (D) for all D ∈ D(G), therefore dπk (Lj )vk = θk−1 Pk dπ(Lj )v = λj vk for j = 1, . . . , n. By Proposition 4.3, this shows that φ0 , φ1 ∈ ϑL−1 (λ), thus φ is not even extreme in ϑL−1 (λ). 2 In order to relate the joint spectrum of L1 , . . . , Ln with (some subset of) ϑL (PL ), we recall the notion of weak containment of representations. If π , are unitary representations of G, then π is said to be weakly contained in if π(f ) (f )

for all f ∈ L1 (G).

Equivalent characterizations of weak containment can be given involving functions of positive type (cf. also [23, §3.5] and [12, §3.4]): Lemma 4.7. Let be a unitary representation of G. Let moreover φ be a function of positive type, of the form (4.2) for some unitary representation π of G on the Hilbert space H and some cyclic vector v of unit norm. Then the following are equivalent: (i) π is weakly contained in ; (ii) |f, φ| (f ) for all f ∈ L1 (G); (iii) |f, φ| C (f ) for some C > 0 and all f ∈ L1 (G). Proof. (i) ⇒ (ii) ⇒ (iii). Trivial. be the Hilbert space on which acts. The hypothesis (iii) implies that φ (iii) ⇒ (i). Let H which is the closure of defines a (positive) continuous functional on the sub-C∗ -algebra of B(H) 1 (L (G)). By applying Proposition 2.1.5(ii) of [12] to this functional, one obtains, for f, g ∈ L1 (G), π(f )π(g)v 2 = g ∗ f ∗ f ∗ ∗ g ∗ , φ f ∗ f ∗ g ∗ g ∗ , φ = (f )2 π(g)v 2 . Since v is cyclic and L1 (G) contains an approximate identity, the set

π(g)v: g ∈ L1 (G)

is a dense subspace of H, therefore the previously proved inequality gives (i).

2

For a unitary representation of G, we denote by PL, the set of the functions φ ∈ PL which satisfy the equivalent conditions of Lemma 4.7. Proposition 4.8. Let be a unitary representation of G. Then PL, is a closed subset of PL . Moreover, for every λ ∈ Rn , PL, ∩ ϑL−1 (λ) is compact and convex, and its extreme points are the ones associated with irreducible representations.


2799

Proof. Condition (ii) of Lemma 4.7 is a convex and closed condition (with respect to the weak-∗ topology of L∞ (G)) for every f ∈ L1 (G). Therefore PL, is closed in PL , and moreover, for λ ∈ Rn , since ϑL−1 (λ) is compact and convex (see Proposition 4.6), PL, ∩ ϑL−1 (λ) is compact and convex too. In order to conclude, again by Proposition 4.6, it is sufficient to show that an extreme point φ of PL, ∩ ϑL−1 (λ) is also extreme in ϑL−1 (λ). Suppose then that φ = (1 − θ )φ0 + θ φ1 for some φ0 , φ1 ∈ ϑL−1 (λ) and 0 < θ < 1. For f ∈ L1 (G), we have 2 2 2 (1 − θ )f, φ0 + θ f, φ1 f ∗ f ∗ , φ (f ) by Lemma 4.7 and positivity, therefore f, φ0 (1 − θ )−1/2 (f ),

f, φ1 θ −1/2 (f ),

and again by Lemma 4.7 we obtain φ0 , φ1 ∈ PL, ∩ ϑL−1 (λ).

2

Theorem 4.9. Let be a unitary representation of G on a Hilbert space H. Then ϑL (PL, ) is the joint spectrum of d (L1 ), . . . , d (Ln ) on H. Proof. Let E be the joint spectral resolution of d (L1 ), . . . , d (Ln ). The joint spectrum of d (L1 ), . . . , d (Ln ), i.e., the support of E , can be identified with the Gelfand spectrum of the C∗ -algebra E [C0 (Rn )] (cf. the proof of Proposition 3.6), i.e., the closure in B(H) of { (m): ˘ m ∈ JL } (see Lemma 3.5 and Proposition 3.7). In particular, if φ ∈ PL, , then, by Lemma 4.7, m, ˘ for all m ∈ JL , ˘ φ (m) therefore φ defines a continuous functional on the C∗ -algebra E [C0 (Rn )], which is multiplicative by Proposition 4.3, and thus belongs to the Gelfand spectrum of E [C0 (Rn )]. Since m, ˘ φ = m ϑL (φ) for all m ∈ JL (see Proposition 4.1), the element of supp E corresponding to this functional is ϑL (φ). Conversely, if λ ∈ supp E , then we can extend the corresponding character of E [C0 (Rn )] to a positive functional ω of norm 1 on the whole B(H) (see [12, §2.10]). Since ω ◦ : L1 (G) → C is linear and continuous, there exists φ ∈ L∞ (G) such that f, φ = ω (f ) for all f ∈ L1 (G); in fact, since ω is positive, φ must be a function of positive type on G (see [17, §3.3]). Moreover, since ω extends a multiplicative functional on E [C0 (Rn )], it must be m ˘1 ∗m ˘ 2 , φ = m ˘ 1 , φm ˘ 2 , φ

for all m1 , m2 ∈ JL .

Therefore, by Proposition 4.3, φ ∈ PL , and in fact φ ∈ PL, since |f, φ| (f ) (see Lemma 4.7). Finally m ϑL (φ) = m, ˘ φ = ω (m) ˘ = m(λ) for all m ∈ JL ,

2800


by Proposition 4.1, since ω extends the character corresponding to λ, and consequently ϑL (φ) = λ (see Lemma 3.5). 2 In particular, the joint L2 spectrum Σ of L1 , . . . , Ln coincides with the set of eigenvalues ϑL (PL,R ) associated with the regular representation R of G on L2 (G). When G is amenable, every unitary representation is weakly contained in the regular representation (see [23, §3.5]), hence Corollary 4.10. We have Σ ⊆ ϑL (PL ),

(4.4)

with equality when G is amenable. Notice that, when G is not amenable, the inclusion (4.4) can be strict: for instance, if n = 1 and L1 is a sublaplacian, then 0 ∈ ϑL (PL ) \ Σ, since L1 has a spectral gap (cf. [56]). Under a more restrictive hypothesis than amenability, viz., the symmetry of the Banach ∗algebra L1 (G), we can relate the joint spectrum of L1 , . . . , Ln to the Gelfand spectrum of a closed ∗-subalgebra of L1 (G) (cf. [31–33] for the case of a single operator). Namely, let ΓL1 be the closure of KL (JL ) in L1 (G). ΓL1 is a commutative Banach ∗-subalgebra of L1 (G), and also, by Proposition 3.6, a dense ∗-subalgebra of the C∗ -algebra C0 (L). Lemma 4.11. Suppose that L1 (G) is symmetric. Then every character of ΓL1 extends to a character of C0 (L), so that the Gelfand spectra of the two Banach ∗-algebras coincide (also as topological spaces). Proof. Since G is connected and L1 (G) is symmetric, then G is also amenable (see [47, Theorem 12.5.18(e)]), so that f Cv 2 =

ρ f∗ ∗f

for all f ∈ L1 (G),

where ρ(f ) denotes the spectral radius of f in L1 (G) (see [47, Theorem 11.4.1], and also [46, p. 695]). Notice that, since ΓL1 is a closed subalgebra of L1 (G), for every f ∈ ΓL1 , the spectral radius of f in ΓL1 coincides with its spectral radius in L1 (G) (see [6, Proposition I.5.12]). Moreover, since L1 (G) is symmetric, also ΓL1 is symmetric. Hence, for every character ψ ∈ G(ΓL1 ), ψ f ∗ = ψ(f )

for all f ∈ ΓL1 ;

since ψ(f ) belongs to the spectrum of f for every f ∈ ΓL1 , we have ψ(f )2 = ψ f ∗ ∗ f ρ f ∗ ∗ f = f 2 2 . Cv This shows that every character ψ ∈ G(ΓL1 ) is continuous with respect to the norm of C0 (L), so that it extends by density to a unique character of C0 (L).


2801

Notice that, since ΓL1 is dense in C0 (L) and the elements of G(C0 (L)) have norms bounded by 1 as functionals on C0 (L), it is easy to check that the topologies of G(C0 (L)) and G(ΓL1 ) coincide. 2 Finally we obtain that, if L1 (G) is symmetric, then the joint L2 spectrum of L1 , . . . , Ln is the set of eigenvalues corresponding to all the bounded joint eigenfunctions. Proposition 4.12. If L1 (G) is symmetric, then the map Λ : PL φ → ·, φ ∈ G ΓL1 is surjective. In particular, every multiplicative linear functional on ΓL1 extends to a bounded linear functional η on L1 (G) such that η(f ∗ g) = η(f )η(g)

for all f ∈ L1 (G) and g ∈ ΓL1 .

(4.5)

Moreover

Σ = λ ∈ Cn : Lj φ = λj φ for some φ ∈ L∞ (G) \ {0} and all j = 1, . . . , n . Proof. Let ψ ∈ G(ΓL1 ). By Lemma 4.11, ψ extends to a character of C0 (L), which corresponds to some λ ∈ Σ. Now, by Corollary 4.10, there exists φ ∈ PL such that ϑL (φ) = λ, therefore, for every m ∈ JL , by Proposition 4.1, ˘ Λ(φ)(m) ˘ = m, ˘ φ = m ϑL (φ) = m(λ) = ψ(m), from which by density we deduce Λ(φ) = ψ . In particular, if η denotes the linear functional f → f, φ on L1 (G), then η extends ψ and, by Proposition 4.3, η(f ∗ m) ˘ = η(f )η(m) ˘

for all f ∈ L1 (G) and m ∈ JL ,

from which (4.5) follows by density. Finally, notice that every λ ∈ Σ is, by Corollary 4.10, the eigenvalue corresponding to some φ ∈ PL , which is a bounded function. Vice versa, if Lj φ = λj φ for some non-null φ ∈ L∞ (G) and all j = 1, . . . , n, then φ ∈ E(G) by Proposition 4.1; moreover, modulo replacing φ with Lx −1 φ/φ(x) for some x ∈ G with φ(x) = 0, we may suppose that φ(e) = 1. This means, again by Proposition 4.1, that ·, φ is a multiplicative linear functional on ΓL1 , hence ·, φ = ·, ψ on ΓL1 for some ψ ∈ PL , by surjectivity of Λ. Then necessarily λ = ϑL (ψ) ∈ Σ by Proposition 4.1 and Corollary 4.10, since G is amenable. 2 5. Examples 5.1. Homogeneous groups Let G be a homogeneous Lie group, with automorphic dilations δt and homogeneous dimension Qδ . A weighted subcoercive system L1 , . . . , Ln on G will be called homogeneous if each Lj is δt -homogeneous.

2802


In the following, L1 , . . . , Ln will be a homogeneous weighted subcoercive system, with associated Plancherel measure σ , and rj will denote the degree of homogeneity of Lj , i.e., δt (Lj ) = t rj Lj . The unital subalgebra of D(G) generated by L1 , . . . , Ln is δt -invariant for every t > 0. Therefore, if we set Dt f = f ◦ δt −1 , and if we denote by t the dilations on Rn given by t (λ) = t r1 λ1 , . . . , t rn λn ,

(5.1)

then from Corollary 3.19 we immediately deduce Proposition 5.1. For every bounded Borel m : Rn → C, we have (m ◦ t )(L) = Dt m(L)Dt −1 ,

(m ◦ t )˘= t −Qδ m ˘ ◦ δt −1 .

Moreover, the support Σ of σ is t -invariant, and σ t (A) = t Qδ σ (A) for all Borel A ⊆ Rn . In particular, the Plancherel measure σ admits a “polar decomposition”: if S = {λ ∈ Rn : |λ| = 1} for some t -homogeneous norm | · | , then there exists a regular Borel measure τ on S such that +∞

f dσ = Rn

f t (ω) dτ (ω) t Qδ −1 dt.

0 S

In the context of homogeneous groups, an equivalent characterization of homogeneous weighted subcoercive systems can be given, which is analogous to the definition of Rockland operator. Theorem 5.2. Let L1 , . . . , Ln ∈ D(G) be homogeneous, pairwise commuting and formally selfadjoint. (i) If L1 , . . . , Ln is a weighted subcoercive system, then the algebra generated by L1 , . . . , Ln contains a Rockland operator if and only if the degrees of homogeneity of L1 , . . . , Ln have a common multiple. (ii) L1 , . . . , Ln is a weighted subcoercive system if and only if, for every non-trivial irreducible unitary representation π of G on a Hilbert space H, the operators dπ(L1 ), . . . , dπ(Ln ) are jointly injective on H∞ , i.e., dπ(L1 )v = · · · = dπ(Ln )v = 0 for all v ∈ H∞ .

⇒

v=0


2803

Proof. Suppose that L1 , . . . , Ln is a weighted subcoercive system. Let p be a real polynomial such that p(L) = p(L1 , . . . , Ln ) is a weighted subcoercive operator. Choose moreover a system X1 , . . . , Xd of generators of g made of δt -homogeneous elements, so that δt (Xk ) = t νk Xk for some νk > 0. From Theorem 2.3(iii) we deduce that, possibly by replacing p with some power p m , there exists a constant C > 0 such that, for every unitary representation π of G on a Hilbert space H, dπ(Xk )v 2 C v2 + dπ p(L) v 2

(5.2)

for v ∈ H∞ , k = 1, . . . , d. Fix a non-trivial irreducible unitary representation π of G on a Hilbert space H, and let v ∈ H∞ be such that dπ(L1 )v = · · · = dπ(Ln )v = 0. For t > 0, since δt ∈ Aut(G), πt = π ◦ δt is also a unitary representation of G; moreover, it is easily checked that smooth vectors for πt coincide with smooth vectors for π , and that dπt (D) = dπ δt (D)

for every D ∈ D(G).

In particular, dπt p(L) v = dπ (p ◦ t )(L) v = p(0)v, thus from (5.2) applied to the representation πt we get dπ(Xk )v 2 t −2νk C 1 + p(0)2 v2 , and, for t → +∞, we obtain dπ(X1 )v = · · · = dπ(Xd )v = 0. Since X1 , . . . , Xd generate g, this means that the function x → π(x)v is constant, i.e., π(x)v = v

for all x ∈ G,

but π is irreducible and non-trivial, thus v = 0. Suppose now conversely that dπ(L1 ), . . . , dπ(Ln ) are jointly injective on H∞ for every nontrivial irreducible representation π on a Hilbert space H, and that moreover the degrees r1 , . . . , rn of homogeneity of L1 , . . . , Ln have a common multiple M. Then 2M/r1

= L1

2M/rn

+ · · · + Ln

is homogeneous of degree 2M and belongs to the subalgebra of D(G) generated by L1 , . . . , Ln . Moreover, for every irreducible unitary representation π of G on H, and for every v ∈ H∞ , we have 2 2 dπ()v, v = dπ(L1 )M/r1 v H + · · · + dπ(Ln )M/rn v H ,

2804


so that, if dπ()v = 0, then also dπ(Lj )v = 0 for j = 1, . . . , n, therefore v = 0. This proves that is a (positive) Rockland operator, and in particular it is weighted subcoercive, so that L1 , . . . , Ln is a weighted subcoercive system. If instead dπ(L1 ), . . . , dπ(Ln ) are jointly injective for every non-trivial irreducible representation π , but the degrees of homogeneity of L1 , . . . , Ln do not have a common multiple, by the results of [42] (see in particular Proposition 1.1 and its proof), we can find another homogeneous structure on G with integral degrees, with respect to which the operators L1 , . . . , Ln are still homogeneous. In particular the degrees of homogeneity of L1 , . . . , Ln in this new structure must have a common multiple, so that, by the previous part of the proof, L1 , . . . , Ln is again a weighted subcoercive system, and this last notion is independent of the homogeneous structure. Finally, if the algebra generated by L1 , . . . , Ln contains a Rockland operator, then (see [42, Proposition 1.3]; see also [51]) the homogeneity degrees of the elements of g must have a common multiple, and a fortiori this is true also for the degrees of L1 , . . . , Ln . 2 Notice that, while the existence of a Rockland operator on G forces the homogeneity degrees of g to have a common multiple, this is not the case for the existence of a homogeneous weighted subcoercive system. For instance, the system of the partial derivatives −i∂1 , . . . , −i∂n on Rn is a homogeneous weighted subcoercive system with respect to any family of dilations of the form δt (x1 , . . . , xn ) = t λ1 x1 , . . . , t λn xn for λ1 , . . . , λn ∈ [1, +∞[. 5.2. Direct products In order to have a system of commuting operators, the simplest way is to start from operators living on different Lie groups, and then to consider them as operators on the direct product of the groups. Here we show that the notion of weighted subcoercive system is compatible with this construction, in the sense that weighted subcoercive systems on different groups can be put together in a single weighted subcoercive system on the direct product. For l = 1, . . . , , let Gl be a connected Lie group, and set G× = G1 × · · · × G . We then have the identification g× = g1 ⊕ · · · ⊕ g . Moreover, for l = 1, . . . , , if D ∈ D(Gl ) and D × is the image of D via the derivative of the canonical inclusion Gl → G× , then D × (f1 ⊗ · · · ⊗ f ) = f1 ⊗ · · · ⊗ fl−1 ⊗ (Dfl ) ⊗ fl+1 ⊗ · · · ⊗ f ; in this case, we say that D × is the differential operator along the l-th factor of G× corresponding to D ∈ D(Gl ).


2805

Lemma 5.3. For l = 1, . . . , , suppose that Al,1 , . . . , Al,dl is a reduced basis of gl , with weights wl,1 , . . . , wl,dl . Then A1,1 , . . . , A1,d1 , . . . , A,1 , . . . , A,d

(5.3)

is a reduced basis of g× , with weights w1,1 , . . . , w1,d1 , . . . , w,1 , . . . , w,d . Moreover, if (Vl,λ )λ is the filtration on gl corresponding to the chosen reduced basis for l = 1, . . . , , then Vλ× = V1,λ ⊕ · · · ⊕ V,λ gives the filtration on g× corresponding to the reduced basis (5.3); therefore, by passing to the quotients, we obtain for the contractions × g ∗ = (g1 )∗ ⊕ · · · ⊕ (g )∗ . Proof. An iterated commutator A[α] of the elements of (5.3) is not null only if it coincides with an iterated commutator (Al )[α ] of Al,1 , . . . , Al,nl for some l ∈ {1, . . . , } (this can be checked by induction on the length |α| of the commutator). The identities involving the filtrations then follow immediately, from which we get easily the conclusion. 2 Theorem 5.4. Suppose that Dl ∈ D(Gl ) is a self-adjoint weighted subcoercive operator on Gl , for l = 1, . . . , , and let Dl× ∈ D(G× ) be the differential operator on G× along the l-th factor corresponding to Dl . Then 2 2 D = D1× + · · · + D× is a positive weighted subcoercive operator on G× . Proof. For l = 1, . . . , , let Al,1 , . . . , Al,dl be a reduced basis of gl , such that, for some selfadjoint weighted subcoercive form Cl , we have Dl = dRGl (Cl ); let moreover Pl be the principal part of Cl . Clearly, modulo rescaling the weights of the reduced bases, we may suppose that the forms C1 , . . . , C have the same degree m. By Lemma 5.3, the concatenation of the bases of g1 , . . . , gl gives a reduced basis (5.3) of g× . We can then consider, for l = 1, . . . , , the forms Cl× , Pl× corresponding to Cl , Pl but re-indexed on the basis (5.3). In particular, if 2 2 C = C1× + · · · + C× ,

2 2 P = P1× + · · · + P× ,

then P = P + is the principal part of C, and moreover 2 2 dRG× (C) = dRG1 (C1 )× + · · · + dRG (C )× = D.

2806


On the other hand, again by Lemma 5.3, we have the identification × G ∗ = (G1 )∗ × · · · × (G )∗ , so that 2 2 dR(G× )∗ (P ) = dR(G1 )∗ (P1 )× + · · · + dR(G )∗ (P )× . By Theorem 2.3, we have that dR(Gl )∗ (Pl ) is Rockland on (Gl )∗ for l = 1, . . . , ; in order to conclude, it is sufficient to show that dR(G× )∗ (P ) is Rockland on (G× )∗ . If π is a non-trivial irreducible unitary representation of G× on a Hilbert space H, then (see [17, Theorem 7.25]) we may suppose that π = π1 ⊗ · · · ⊗ π , where πl is an irreducible unitary ˆ ··· ⊗ ˆ H and at representation of Gl on a Hilbert space Hl for l = 1, . . . , , so that H = H1 ⊗ least one of π1 , . . . , π is non-trivial. Let (wl,νl )νl be a complete orthonormal system for Hl , for l = 1, . . . , , so that(w1,ν1 ⊗ · · · ⊗ w,ν )ν is a complete orthonormal system for H. Then, for every element v = ν1 ,...,ν aν1 ,...,ν w1,ν1 ⊗ · · · ⊗ w,ν of H, we have

dπ dR(G× )∗ (P ) v, v H =

l=1 ν1 ,...,νl−1 ,νl+1 ,ν

2 ; dπl dR(G ) (Pl ) a w ν ,...,ν l,ν l l ∗ 1 νl

Hl

since at least one of the dπl (dR(Gl )∗ (Pl )) is injective (being dR(Gl )∗ (Pl ) Rockland and πl nontrivial), this formula gives easily that v = 0 i.e., dπ(dR(G× )∗ (P )) is injective.

⇒

dπ dR(G× )∗ (P ) v = 0,

2

Theorems 5.4 and 3.10, together with the properties of the spectral integral, yield easily Corollary 5.5. For l = 1, . . . , , let Ll,1 , . . . , Ll,nl ∈ D(Gl ) be a weighted subcoercive system. × Let moreover L× l,j be the differential operator on G along the l-th factor corresponding to Ll,j . Then × × × L× 1,1 , . . . , L1,n1 , . . . , L,1 , . . . , L,n

(5.4)

is a weighted subcoercive system on G× . Further: (a) if ml is a bounded Borel function on Rnl for l = 1, . . . , , then KL× (m1 ⊗ · · · ⊗ m ) = KL1 m1 ⊗ · · · ⊗ KL m ; (b) if σl is the Plancherel measure associated with the system Ll,1 , . . . , Ll,nl for l = 1, . . . , , and if moreover σ × is the Plancherel measure associated with the system (5.4), then σ × = σ1 × · · · × σ .


2807

5.3. Gelfand pairs Let G be a connected Lie group. In this paragraph, we describe a particular way of obtaining weighted subcoercive systems on G, which has been extensively studied in the literature. Let K be a compact subgroup of Aut(G). A function (or distribution) f on G is said to be K-invariant if Tk f = f

for all k ∈ K.

We add a subscript K to the symbol representing a particular space of functions or distributions p in order to denote the corresponding subspace of K-invariant elements; for instance, LK (G) p denotes the Banach space of K-invariant L functions on G. Since Tk f ∗ = (Tk f )∗ , Tk (f ∗ g) = (Tk f ) ∗ (Tk g), it is immediately proved that L1K (G) is a Banach ∗-subalgebra of L1 (G). We also define the projection onto K-invariant elements: PK : f → Tk f dk, K

where the integration is with respect to the Haar measure on K with mass 1. This projection satisfies PK f ∗ (PK g) = PK (PK f ) ∗ g = (PK f ) ∗ (PK g), PK f ∗ = (PK f )∗ . Among the left-invariant differential operators on G, we can consider those which are Kinvariant, i.e., which commute with Tk for all k ∈ K. The set DK (G) of left-invariant K-invariant differential operators on G is a ∗-subalgebra of D(G), which is finitely generated since K is compact (cf. [28, Corollary X.2.8 and Theorem X.5.6]). Moreover, DK (G) contains an elliptic operator (e.g., the Laplace–Beltrami operator associated with a left-invariant K-invariant metric on G, cf. [29, proof of Proposition IV.2.2]). Therefore, if one chooses a finite system of formally self-adjoint generators of DK (G), the only property which is missing in order to have a weighted subcoercive system is commutativity of DK (G). In fact, under these hypotheses, the following properties are equivalent (cf. [53], or [59, §8.3]): • DK (G) is a commutative ∗-subalgebra of D(G); • L1K (G) is a commutative Banach ∗-subalgebra of L1 (G). The latter condition corresponds to the fact that (G K, K) is a Gelfand pair.4 We now summarize in our context some of the main notions and results from the general theory of Gelfand pairs, 4 If S is a locally compact group, and K a compact subgroup of S, then (S, K) is said to be a Gelfand pair if the (convolution) algebra L1 (K; S; K) of bi-K-invariant integrable functions on S is commutative. The study of a Gelfand pair (S, K) involves the K-homogeneous space S/K. In the case S = G K, the space S/K can be identified with G, and most of the notions and results about Gelfand pairs can be rephrased in terms of the algebraic structure of G (see, e.g., [10,3]); this has to be kept in mind when comparing the results presented in the literature with the ones mentioned

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for which we refer mainly to [13,59,28,29]. In the following, we always suppose that L1K (G) is commutative; consequently, G must be unimodular (cf. [29, Theorem IV.3.1]). The K-invariant joint eigenfunctions φ of the operators in DK (G) with φ(e) = 1 are called K-spherical functions. The set GK of bounded K-spherical functions, with the topology induced by the weak-∗ topology of L∞ (G), is identified with the Gelfand spectrum G(L1K (G)) of the commutative Banach ∗-algebra L1K (G), via the correspondence which associates to a bounded K-spherical function φ the (multiplicative) linear functional f → f, φ on L1K (G). According to this identification, the Gelfand transform — which is also called the K-spherical Fourier transform — of an element f ∈ L1K (G) is the function GK f : GK φ → f, φ ∈ C. Let PK denote the set of K-invariant functions φ of positive type on G with φ(e) = 1. Then PK is a closed and convex subset of P1 , whose extreme points are the elements of G+ K = GK ∩ PK , i.e., the K-spherical functions of positive type; in particular, by the Krein–Milman theorem, + the convex hull of G+ K is weakly-∗ dense in PK . By restricting K-spherical transforms to GK , one obtains that ∗ GK f G+ = (GK f )|G+ , K

K

therefore the map f → (GK f )|G+ is a ∗-homomorphism L1K (G) → C0 (G+ K ) with unit norm K

and dense image. Moreover, there exists a unique positive regular Borel measure σK on G+ K, which is called the Plancherel measure of the Gelfand pair (G K, K), such that f (x)2 dx = GK f (φ)2 dσK (φ) G+ K

G

for all f ∈ L1K ∩ L2K (G); further, the map f → (GK f )|G+ extends to an isomorphism L2K (G) → K

L2 (G+ K , σK ). Choose now a finite system L1 , . . . , Ln of formally self-adjoint generators of DK (G). As we have seen before, the system L1 , . . . , Ln is a weighted subcoercive system on G. If the map ϑL of Section 4 is extended to all the joint eigenfunctions of L1 , . . . , Ln , then it is known (see [15]) that ϑL |GK : GK → Cn is a homeomorphism with its image ϑL (GK ), which is a closed subset of Cn . Notice that G+ K ⊆ PL ,

ϑL G+ K = ϑL (PL );

consequently, for every λ ∈ ϑL (PL ), there exists a unique element of ϑL−1 (λ) ∩ PL which is a K-spherical function (cf. [29, Proposition IV.2.4]). here. Notice that, according to Vinberg’s reduction theorem (see [58]), Gelfand pairs in “semidirect-product form” are one of the two structural constituents of general Gelfand pairs.


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The embedding ϑL allows us to compare the notions of K-spherical transform GK and Plancherel measure σK of the Gelfand pair (G K, K) with the notions of kernel transform KL and Plancherel measure σ associated with the weighted subcoercive system L1 , . . . , Ln . Notice that, in the case of nilpotent G and Schwartz multipliers, results similar to the following are proved in [1,16] (cf. also §1.7 of [19]). As a preliminary remark, notice that from Proposition 3.18 it follows that, for every bounded Borel m : Rn → C, the corresponding kernel KL m is K-invariant. Proposition 5.6. Let f ∈ L1K (G). Then there exists m ∈ C0 (Rn ) such that GK f (φ) = m ϑL (φ)

for φ ∈ G+ K.

For any of such m, and for every unitary representation π of G, we have π(f ) = m dπ(L1 ), . . . , dπ(Ln ) , and in particular f = KL m. Proof. Since GK f |G+ ∈ C0 (G+ is a homeomorphism with its image, which K ), and since ϑL |G+ K K n is a closed subset of R , then by the Tietze–Urysohn extension theorem we can find m ∈ C0 (Rn ) extending (GK f ) ◦ (ϑL |G+ )−1 . K By Proposition 3.7, for every u ∈ JL and every unitary representation π of G, we have π(u) ˘ = u dπ(L1 ), . . . , dπ(Ln ) ; therefore the map JL u → u˘ ∈ L1 (G) extends by density (see Proposition 3.6) to a ∗-homomorphism Φ : C0 Rn → C ∗ (G), and we have π Φ(u) = u dπ(L1 ), . . . , dπ(Ln ) for all u ∈ C0 (Rn ) and all unitary representations π of G. The conclusion will then follow if we prove that f = Φ(m) as elements of C ∗ (G). Recall that every φ ∈ P1 defines a positive continuous functional ωφ on C ∗ (G) with unit norm, extending L1 (G) h → h, φ ∈ C.

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In fact, the norm of an arbitrary g ∈ C ∗ (G) is given by g∗ = sup ωφ g ∗ g ∗ φ∈P1

(see [17, Proposition 7.1]); therefore, in order to conclude, it will be sufficient to show that the set A of the φ ∈ P1 such that ∗ ωφ f − Φ(m) ∗ f − Φ(m) = 0 coincides with the whole P1 . ∗ (G) of L1 (G) in C ∗ (G), and it is Notice that both f and Φ(m) belong to the closure CK K ∗ easily checked that, for φ ∈ P1 and g ∈ CK (G), ωφ (g) = ωPK φ (g); consequently, we are reduced to prove that PK ⊆ A. In fact, since A is a closed convex subset of P1 , it is sufficient to prove the inclusion G+ K ⊆ A. 1 On the other hand, the functionals ωφ for φ ∈ G+ K are multiplicative on LK (G), thus they are ∗ also multiplicative on CK (G) by continuity, therefore ∗ 2 2 ωφ f − Φ(m) ∗ f − Φ(m) = ωφ f − Φ(m) = GK f (φ) − m ϑL (φ) = 0 for every φ ∈ G+ K , and we are done.

2

Thus, by applying first GK and then KL , we are back at the beginning. The composition of the transforms in reverse order is considered in the following statement, which gives also an improvement of Proposition 3.14 in this particular context. ˘ ∈ L1 (G). Then m ˘ ∈ Corollary 5.7. Let m : Rn → C be a bounded Borel function such that m 1 LK (G) and GK (KL m)(φ) = m ϑL (φ) for all φ ∈ G+ K with ϑL (φ) ∈ Σ. In particular m|Σ ∈ C0 (Σ). Proof. We already know that m ˘ is K-invariant, so that m ˘ ∈ L1K (G). Therefore, by Proposin tion 5.6, we can find u ∈ C0 (R ) such that GK m(φ) ˘ = u ϑL (φ) for all φ ∈ G+ ˘ = u, ˘ i.e., K , and we have m m(L1 , . . . , Ln ) = u(L1 , . . . , Ln ), which means that m and u must coincide on the joint spectrum Σ of L1 , . . . , Ln , and we are done. 2 Finally, we compare the Plancherel measures σ and σK .


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Corollary 5.8. We have σ = ϑL |G+ (σK ), K

σK = (ϑL |G+ )−1 (σ ). K

Proof. Recall that ϑL |G+ is a homeomorphism with its image, which is a closed subset of Rn K containing the support Σ of σ , thus the two equalities to be proved are equivalent. Set σ˜ = (ϑL |G+ )−1 (σ ). Then σ˜ is a positive regular Borel measure on G+ K . Moreover, if K

f ∈ L1K ∩ L2K (G), then by Proposition 5.6 there is m ∈ C0 (Rn ) such that GK f (φ) = m ϑL (φ)

for all φ ∈ G+ K

and f = m. ˘ Since f ∈ L2 (G), by Theorem 3.10 we also have m ∈ L2 (σ ), and

f (x)2 dx =

G

|m| dσ = 2

Rn

|GK f |2 d σ˜

G+ K

by the change-of-variable formula for push-forward measures. By the arbitrariness of f ∈ L1K ∩ L2K (G) and the uniqueness of the Plancherel measure of a Gelfand pair, we obtain that σK = σ˜ , and we are done. 2 We have thus shown that the study of the algebra DK (G) of differential operators associated with a Gelfand pair (G K, K) fits into the more general setting of weighted subcoercive systems, where in general there is no compact group K of automorphisms which determines the algebra of operators. It should be noticed that the hypothesis of Gelfand pair is quite restrictive. We have already mentioned that, if L1K (G) is commutative, then G must be unimodular. Moreover, the algebra DK (G) always contains an elliptic operator, while a general weighted subcoercive operator is not even analytic hypoelliptic (see, e.g., [26]). Further, if G is solvable, then G must have polynomial growth, and, if G is nilpotent, then G is at most 2-step (see [3]). In this last case, notice that it is always possible to find a family of automorphic dilations on G which commute with the elements of K, and any system L1 , . . . , Ln of homogeneous formally self-adjoint generators of DK (G) is a homogeneous weighted subcoercive system. On the other hand, the results of this paper can be applied to homogeneous groups which are 3-step or more, and which therefore do not belong to the realm of Gelfand pairs. Take for instance the free 3-step nilpotent group N2,3 with 2 generators, defined by the relations [X1 , X2 ] = Y,

[X1 , Y ] = T1 ,

[X2 , Y ] = T2 ,

where X1 , X2 , Y, T1 , T2 is a basis of its Lie algebra, and notice that the group SO2 acts on N2,3 by automorphisms given by simultaneous rotations of RX1 + RX2 and RT1 + RT2 . Although

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the whole algebra of SO2 -invariant left-invariant differential operators on N2,3 cannot be commutative, the operators − X12 + X22 ,

2X2 T1 − 2X1 T2 − Y 2 ,

− T12 + T22

generate a non-trivial homogeneous commutative subalgebra to which our results apply, as well as they apply to the larger algebra generated by − X12 + X22 ,

2X2 T1 − 2X1 T2 − Y 2 ,

−iT1 ,

−iT2

(which is no longer made of SO2 -invariant operators). Acknowledgments I thank Fulvio Ricci for introducing me to the theory of Gelfand pairs, and for his continuous support and encouragement. I also thank A.F.M. ter Elst for the exchange of comments about his work. References [1] F. Astengo, B. Di Blasio, F. Ricci, Gelfand pairs on the Heisenberg group and Schwartz functions, J. Funct. Anal. 256 (2009) 1565–1587. [2] P. Auscher, A.F.M. ter Elst, D.W. Robinson, On positive Rockland operators, Colloq. Math. 67 (1994) 197–216. [3] C. Benson, J. Jenkins, G. Ratcliff, On Gel’fand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc. 321 (1990) 85–116. [4] S.K. Berberian, Notes on Spectral Theory, Van Nostrand Math. Stud., vol. 5, Van Nostrand Co., Inc., Princeton, NJ, 1966. Corrected second edition (2009) available on the web at www.ma.utexas.edu/mp_arc. [5] J.M. Berezans’ki˘ı, Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monogr., vol. 17, American Mathematical Society, Providence, RI, 1968 (translated from the Russian by R. Bolstein, J.M. Danskin, J. Rovnyak, L. Shulman). [6] F.F. Bonsall, J. Duncan, Complete Normed Algebras, Ergeb. Math. Grenzgeb., Band 80, Springer-Verlag, New York, 1973. [7] N. Bourbaki, Lie Groups and Lie Algebras, Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989 (translated from the French; reprint of the 1975 edition). [8] N. Bourbaki, General Topology, Chapters 1–4, Elements of Mathematics, Springer-Verlag, Berlin, 1998 (translated from the French; reprint of the 1989 English translation). [9] N. Bourbaki, General Topology, Chapters 5–10, Elements of Mathematics, Springer-Verlag, Berlin, 1998 (translated from the French; reprint of the 1989 English translation). [10] G. Carcano, A commutativity condition for algebras of invariant functions, Boll. Unione Mat. Ital. B (7) 1 (1987) 1091–1105. [11] M. Christ, Lp bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. 328 (1991) 73–81. [12] J. Dixmier, C ∗ -Algebras, revised ed., North-Holland Math. Library, vol. 15, North-Holland Publishing Co., Amsterdam, 1982 (translated from the French by Francis Jellett). [13] J. Faraut, Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques, in: Analyse Harmonique, Les Cours du C.I.M.P.A., CIMPA/ICPAM, Nice, 1982, pp. 315–446. [14] J.M.G. Fell, R.S. Doran, Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, vol. 1, Pure Appl. Math., vol. 125, Academic Press Inc., Boston, MA, 1988. [15] F. Ferrari Ruffino, The topology of the spectrum for Gelfand pairs on Lie groups, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10 (2007) 569–579. [16] V. Fischer, F. Ricci, Gelfand transforms of SO(3)-invariant Schwartz functions on the free group N3,2 , Ann. Inst. Fourier (Grenoble) 59 (2009) 2143–2168. [17] G.B. Folland, A Course in Abstract Harmonic Analysis, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1995.


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[18] G.B. Folland, E.M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes, vol. 28, Princeton University Press, Princeton, NJ, 1982. [19] R. Gangolli, V.S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, Ergeb. Math. Grenzgeb. (Results in Mathematics and Related Areas), vol. 10, Springer-Verlag, Berlin, 1988. [20] R. Goodman, Filtrations and asymptotic automorphisms on nilpotent Lie groups, J. Differential Geom. 12 (1977) 183–196. [21] R.W. Goodman, Nilpotent Lie Groups: Structure and Applications to Analysis, Lecture Notes in Math., vol. 562, Springer-Verlag, Berlin, 1976. [22] L. Grafakos, Classical Fourier Analysis, second ed., Grad. Texts in Math., vol. 249, Springer, New York, 2008. [23] F.P. Greenleaf, Invariant Means on Topological Groups and Their Applications, Van Nostrand Math. Stud., vol. 16, Van Nostrand Reinhold Co., New York, 1969. [24] Y. Guivarc’h, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973) 333–379. [25] W. Hebisch, A. Sikora, A smooth subadditive homogeneous norm on a homogeneous group, Studia Math. 96 (1990) 231–236. [26] B. Helffer, Conditions nécessaires d’hypoanalyticité pour des opérateurs invariants à gauche homogènes sur un groupe nilpotent gradué, J. Differential Equations 44 (1982) 460–481. [27] B. Helffer, J. Nourrigat, Caracterisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué, Comm. Partial Differential Equations 4 (1979) 899–958. [28] S. Helgason, Differential Geometry and Symmetric Spaces, Pure Appl. Math., vol. XII, Academic Press, New York, 1962. [29] S. Helgason, Groups and Geometric Analysis, Pure Appl. Math., vol. 113, Academic Press Inc., Orlando, FL, 1984. [30] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vol. I, second ed., Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences), vol. 115, Springer-Verlag, Berlin, 1979. [31] A. Hulanicki, On the spectrum of convolution operators on groups with polynomial growth, Invent. Math. 17 (1972) 135–142. [32] A. Hulanicki, Subalgebra of L1 (G) associated with Laplacian on a Lie group, Colloq. Math. 31 (1974) 259–287. [33] A. Hulanicki, Commutative subalgebra of L1 (G) associated with a subelliptic operator on a Lie group G, Bull. Amer. Math. Soc. 81 (1975) 121–124. [34] J.W. Jenkins, Dilations and gauges on nilpotent Lie groups, Colloq. Math. 41 (1979) 95–101. [35] L.H. Loomis, An Introduction to Abstract Harmonic Analysis, D. Van Nostrand Company, Inc., Toronto, 1953. [36] J. Ludwig, D. Müller, Sub-laplacians of holomorphic Lp -type on rank one AN-groups and related solvable groups, J. Funct. Anal. 170 (2000) 366–427. [37] A. Martini, Algebras of differential operators on Lie groups and spectral multipliers, Tesi di perfezionamento (PhD thesis), Scuola Normale Superiore, Pisa, arXiv:1007.1119, 2010. [38] A. Martini, Analysis of joint spectral multipliers on Lie groups of polynomial growth, arXiv:1010.1186, 2010. [39] K. Maurin, General Eigenfunction Expansions and Unitary Representations of Topological Groups, Monografie Matematyczne, Tom 48, PWN – Polish Scientific Publishers, Warsaw, 1968. [40] R.E. Megginson, An Introduction to Banach Space Theory, Grad. Texts in Math., vol. 183, Springer-Verlag, New York, 1998. [41] A. Melin, Parametrix constructions for right invariant differential operators on nilpotent groups, Ann. Global Anal. Geom. 1 (1983) 79–130. [42] K.G. Miller, Parametrices for hypoelliptic operators on step two nilpotent Lie groups, Comm. Partial Differential Equations 5 (1980) 1153–1184. [43] P. Milnes, Identities of group algebras, Proc. Amer. Math. Soc. 29 (1971) 421–422. [44] A. Nagel, E.M. Stein, S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985) 103–147. [45] E. Nelson, W.F. Stinespring, Representation of elliptic operators in an enveloping algebra, Amer. J. Math. 81 (1959) 547–560. [46] T.W. Palmer, Classes of nonabelian, noncompact, locally compact groups, Rocky Mountain J. Math. 8 (1978) 683– 741. [47] T.W. Palmer, Banach Algebras and the General Theory of *-Algebras, vol. 2, Encyclopedia Math. Appl., vol. 79, Cambridge University Press, Cambridge, 2001. [48] W. Rudin, Functional Analysis, McGraw–Hill Ser. Higher Math., McGraw–Hill Book Co., New York, 1973. [49] W. Rudin, Real and Complex Analysis, second ed., McGraw–Hill Ser. Higher Math., McGraw–Hill Book Co., New York, 1974.

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[50] A.F.M. ter Elst, D.W. Robinson, Weighted strongly elliptic operators on Lie groups, J. Funct. Anal. 125 (1994) 548–603. [51] A.F.M. ter Elst, D.W. Robinson, Spectral estimates for positive Rockland operators, in: Algebraic Groups and Lie Groups, Austral. Math. Soc. Lect. Ser., vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 195–213. [52] A.F.M. ter Elst, D.W. Robinson, Weighted subcoercive operators on Lie groups, J. Funct. Anal. 157 (1998) 88–163. [53] E.G.F. Thomas, An infinitesimal characterization of Gel’fand pairs, in: Conference in Modern Analysis and Probability, New Haven, Conn., 1982, Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 379–385. [54] F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967. [55] V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Prentice-Hall Ser. Modern Anal., PrenticeHall Inc., Englewood Cliffs, NJ, 1974. [56] N.T. Varopoulos, Hardy–Littlewood theory on unimodular groups, Ann. Inst. Henri Poincare Probab. Stat. 31 (1995) 669–688. [57] N.T. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math., vol. 100, Cambridge University Press, Cambridge, 1992. [58] È.B. Vinberg, Commutative homogeneous spaces and co-isotropic symplectic actions, Uspekhi Mat. Nauk 56 (2001) 3–62. [59] J.A. Wolf, Harmonic Analysis on Commutative Spaces, Math. Surveys Monogr., vol. 142, American Mathematical Society, Providence, RI, 2007.


Classification of a class of crossed product C ∗ -algebras associated with residually finite groups José R. Carrión Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States Received 7 August 2010; accepted 1 February 2011

Communicated by D. Voiculescu

Abstract A residually finite group acts on a profinite completion by left translation. We consider the corresponding crossed product C ∗ -algebra for discrete countable groups that are central extensions of finitely generated abelian groups by finitely generated abelian groups (these are automatically residually finite). We prove that all such crossed products are classifiable by K-theoretic invariants using techniques from the classification theory for nuclear C ∗ -algebras. © 2011 Elsevier Inc. All rights reserved. Keywords: Classification of C ∗ -algebras; Crossed product; Residually finite group; C(X)-algebra; Decomposition rank; Generalized Bunce–Deddens algebra

1. Introduction In [16] Orfanos introduced a class of C ∗ -algebras generalizing the classical Bunce–Deddens algebras [4]. These generalized Bunce–Deddens algebras can be constructed starting with any discrete, countable, amenable and residually finite group—the construction yields the classical ˜ G, Bunce–Deddens algebras when starting with Z. They are C ∗ -algebras of the form C(G) ˜ where G acts on a profinite completion G by left translation. From the point of view of the classification theory for nuclear C ∗ -algebras [10,23], these C ∗ -algebras enjoy many desirable properties: they are simple, separable, nuclear and quasidiagonal; they have real rank zero, stable rank one, unique trace and comparability of projections (see [16]). E-mail address: [email protected]. 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.02.002

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J.R. Carrión / Journal of Functional Analysis 260 (2011) 2815–2825

In this note we prove that the generalized Bunce–Deddens algebras associated with a discrete, countable and residually finite group G that is a central extension of a finitely generated abelian group by a finitely generated abelian group have finite decomposition rank (Proposition 2.1). This result covers all the profinite completions of G, see Section 2. For this, we will use that the decomposition rank of a C(X)-algebra is finite as long as X has finite covering dimension and the decomposition rank of the fibers is uniformly bounded (Lemma 3.1). We are then free to use powerful theorems of Lin and Winter [14,25] to observe that the generalized Bunce–Deddens algebras associated with such groups G are classified by K-theory. The above class of groups contains some examples of interest, such as the integer Heisenberg group (of any dimension). It is also shown that the decomposition rank of C ∗ (G) is finite for all such groups (Theorem 2.2). The classical Bunce–Deddens algebras were classified by Bunce and Deddens along the same lines as the Uniformly Hyperfinite (UHF) algebras. Tensor products of Bunce–Deddens algebras were classified by Pasnicu in [18]. In both cases the algebras in question are inductive limits of circle algebras—in fact, of algebras of the form C(T) ⊗ Mr . The trivial observation that these algebras have slow dimension growth (owing to the fact that T has covering dimension equal to 1) can be used to subsume these results under subsequent and very general classification theorems ([8,5], see also [23, Theorem 3.3.1]). In the present case, the algebras in question are inductive limits of C ∗ -algebras of the form ∗ C (L) ⊗ Mr for certain (usually non-abelian) subgroups L ⊂ G. In the place of covering dimension we use decomposition rank, a noncommutative analog of covering dimension introduced by Kirchberg and Winter [13]. Let C be the class of all unital, separable and simple C ∗ -algebras A with real rank zero and finite decomposition rank that satisfy the Universal Coefficient Theorem, and such that ∂e T (A), the extreme boundary of the tracial state space, is compact and zero-dimensional. The Elliott invariant for such algebras is their ordered K-theory: Ell(A) = K0 (A), K0 (A)+ , [1A ]0 , K1 (A) . (See [10] for the general version of the Elliott invariant.) A combination of results of Lin and Winter gives: Theorem 1.1. (See [25, Corollary 6.5].) Let A, B ∈ C. Then A ∼ = B if and only if Ell(A) ∼ = Ell(B). In the early 1990s Elliott [7] initiated a program to classify nuclear C ∗ -algebras and conjectured such a result (for a larger class and with a refined invariant, including for example the space of tracial states). One aim of this paper is to show that the class of generalized Bunce–Deddens algebras associated with a large collection of groups is a natural example where the program succeeds. This paper has three additional sections. In Section 2 we recall the relevant definitions, state our main result (Theorem 2.3) and the key technical result behind its proof (Proposition 2.1). In Section 3 we recall the definition of decomposition rank and prove a lemma concerned with the decomposition rank of C(X)-algebras. Section 4 contains the proof of Proposition 2.1. The proof uses a continuous field structure of certain group C ∗ -algebras due to Packer and Raeburn [17]. The decomposition rank such a group C ∗ -algebra is estimated in terms of the ranks of the normal subgroup and its quotient (Theorem 2.2). For this, we use a result of Poguntke [20] concerning the structure of twisted group algebras of abelian groups. The decomposition rank of these twisted


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group algebras is then estimated using the decomposition rank of noncommutative tori, for which we provide an estimate in Lemma 4.4. 2. Main results We first recall Orfanos’ definition of a generalized Bunce–Deddens algebra. Let G be a discrete, countable and amenable group. Assume also that G is residually finite, so that there is a nested sequence of finite index, normal subgroups of G with trivial intersection. Let G ⊃ L1 ⊃ L2 ⊃ · · · be such a sequence. The profinite completion of G with respect to this sequence is the inverse limit ˜ = lim G/Li G ←− with connecting maps xLi+1 → xLi . This is a compact, Hausdorff and totally disconnected group. The group G acts by left multiplication. The corresponding crossed product ˜ G C(G) is called a generalized Bunce–Deddens algebra associated with G. For example, let q = (qi ) be a sequence of positive integers with qi |qi+1 for every i. With G = Z and Li = qi Z, the above construction yields the usual Bunce–Deddens algebra of type q. Let G be the collection of all (discrete) groups G with the following property: there exist finitely generated abelian groups N and Q such that G is a central extension 1→N →G→Q→1

(1)

(where by “central” we mean that the image of N lies in the center of G). A celebrated theorem of Hall [22, 15.4.1] states that a finitely generated group that is an extension of an abelian group by a nilpotent group is residually finite. The groups in G are therefore residually finite. They are also amenable, as an extension of amenable groups is itself amenable. As noted in the introduction, G contains the integer Heisenberg groups of all dimensions (see Example 2.4 below). ˜ G associated Proposition 2.1. If G ∈ G, then any generalized Bunce–Deddens algebra C(G) ˜ with G has finite decomposition rank. (This covers all profinite completions G of G.) See Section 3 for the definition of decomposition rank and Section 4 for the proof. One ingredient of its proof is the following result, also proved in Section 4. Theorem 2.2. Let G be a countable, discrete group that is a central extension 1→N →G→Q→1 where N and Q are finitely generated abelian groups of ranks n and m, respectively. Then dr C ∗ (G) (n + 1) 2m2 + 4m + 1 − 1. Our main result can now be stated as:

2818


˜ G associated with groups Theorem 2.3. The generalized Bunce–Deddens algebras C(G) G ∈ G are classified by their Elliott invariant. Proof. Corollary 7 and Theorem 12 of [16] show that all generalized Bunce–Deddens algebras are unital, separable, simple, and that they have real rank zero and unique trace. By Proposition 2.1 the ones associated with groups in G have finite decomposition rank, so they satisfy the hypothesis of the classification Theorem 1.1. 2 Example 2.4. Fix a positive integer m. Define H2m+1 as the group of all square matrices of size m + 2 of the form ⎞ ⎛ 1 xt z ⎝ 0 1m y ⎠ 0 0 1 where x, y ∈ Zm , z ∈ Z, and 1m is the identity matrix in Mm . This is a discrete, countable and residually finite group. Indeed, let (qi ) be a sequence of positive integers such that qi |qi+1 for all i. For each i let Li be the subgroup of H2m+1 consisting of those matrices with x, y ∈ (qi Z)m and z ∈ qi Z. This provides a nested sequence of finite index, normal subgroups with trivial intersection. Moreover, H2m+1 is a central extension 1 → Z → H2m+1 → Z2m → 1, as one can check (notice that the center of H2m+1 is generated by the matrix with x = y = 0 and z = 1). Thus H2m+1 ∈ G. It is well known that C ∗ (H3 ) may be regarded as the C ∗ -algebra of sections of a continuous field of (irrational and rational) rotation algebras over T [1]. Similarly, one can prove that the C ∗ -algebra of H2m+1 may be regarded as the C ∗ -algebra of a continuous field of noncommutative tori (in fact, of m-th tensor powers of rotation algebras) over T. A theorem of Packer and Raeburn [17] shows that this is an instance of a more general phenomenon, a fact we use in Section 4. 3. The decomposition rank of C(X)-algebras For any two C ∗ -algebras A and B, we have dr(A ⊗ B) (dr A − 1)(dr B − 1) + 1, see Remark 3.2 of [13]. In this section we prove an analogous estimate for C(X)-algebras. Let us first recall some of the relevant definitions. Let A and B be C ∗ -algebras. A completely positive (c.p.) map φ : F → A is said to have order zero if φ(a)φ(b) = 0 for all a, b ∈ A+ with ab = 0. We refer the reader to [26] for a detailed treatment of order zero maps. A c.p. map φ : si=1 Mri → A is k-decomposable if there is a partition kj =0 Ij = {1, . . . , s} such that φ restricted to i∈Ij Mri has order zero for all j ∈ {0, . . . , k}. In [13], Kirchberg and Winter introduced a noncommutative analog of topological covering dimension called decomposition rank. A separable C ∗ -algebra A has decomposition rank k,


2819

dr A = k for short, if k is the least integer such that the following holds: for any finite subset F of A and any > 0, there are a finite dimensional C ∗ -algebra F and c.p. contractions ψ : A → F , φ : F → A, with φ k-decomposable, satisfying

a − φψ(a) < for all a in F . (That is, (F, ψ, φ) is a c.p. approximation of F to within .) We have, for example, that dr C(X) = dim X when X is a compact metric space (see [13, Proposition 3.3]). A C(X)-algebra is a C ∗ -algebra A endowed with a nondegenerate ∗-homomorphism of C(X) to the center of the multiplier algebra of A. (Nondegeneracy means that C(X)A is dense in A.) Consult for example [3, §2.1] for an introduction to C(X)-algebras. Such algebras were introduced by Kasparov in [12] and generalize continuous fields of C ∗ -algebras; they may be regarded as “upper semicontinuous” fields of C ∗ -algebras. Let A be a C(X)-algebra. For a closed subset F ⊂ A we write A|F for the quotient of A by the (closed) ideal C0 (X \ F )A. If F consists of a single point x we write A(x) instead of A|{x} and call A(x) the fiber of A at x. The image of an element a ∈ A under the quotient map of A to A(x) is written a(x). Lemma 3.1. Let X be a compact metric space and A a (separable) C(X)-algebra. If dim X l and dr A(x) k for every x ∈ X, then dr A (l + 1)(k + 1) − 1. Proof. Let a finite subset F of A and > 0 be given. We claim that for every x ∈ X there are a finite dimensional C ∗ -algebra Fx and c.p. contractions ψx

φx

A −→ Fx −→ A, with φx k-decomposable, such that

(φx ψx − idA )(a)(x) <

(2)

for all a in F . Fix x ∈ X. Because dr A(x) k, there is a c.p. approximation (Fx , ψ x , φ x ) of {a(x): a ∈ F } to within /2, with φ x k-decomposable. Let ψx be the composition of ψ x with the quotient map A → A(x). By Remark 2.4 of [13], φ x lifts to a c.p. map Φ x : Fx → A that is k-decomposable, but not necessarily contractive. We may assume that Φ x (1)(x) = φ x (1) = φ x = 0 (we abbreviate 1Fx to 1). Recall from [3] that for every a ∈ A the function N (a) : y → a(y)A(y) is upper semicontinuous on X and satisfies N(f a) = |f |N (a) for every f ∈ Cb (X). Then there is a δ > 0 such that d(x, y) < δ implies N(Φ x (1))(y) 1 + /2. Let h ∈ C(X)+ be of norm 1/(1 + /2), taking on its maximum value at x and vanishing on {y: d(x, y) δ}. Then

x

hΦ (1) = sup N hΦ x (1) (y) = y∈X

sup

{y: d(y,x) 0 such that * Corresponding author.

E-mail addresses: [email protected] (A. Haraux), [email protected] (M.A. Jendoubi). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.01.012

A. Haraux, M.A. Jendoubi / Journal of Functional Analysis 260 (2011) 2826–2842 ∞

2827

φn Ln (E,F ) r n < ∞

1

and ∀h ∈ E,

h < r

⇒

f (a + h) − f (a) =

∞

φn (h, . . . , h).

1

It is clear that analyticity is a local notion and it is known that analytic maps between Banach spaces enjoy many properties in common with usual (numerical) analytic functions. In particular the composition g ◦ f of two analytic maps f : U → F and g : V → G is analytic: U → G whenever f (U ) ⊂ V open subset of G, cf. for instance [21] and the references therein. In addition if E = RN , F = R, the above definition reduces to the usual notion of real analytic functions: RN → R. Let V be a real Hilbert space and U an open subset of V . It is a natural question to ask whether an analytic function F : U → R satisfies the Łojasiewicz gradient inequality which means that for any a ∈ U there exist θ ∈ (0, 1/2), a V -neighborhood W of a in U and c > 0 for which 1−θ ∀u ∈ W, DF(u)V cF (u) − F (a) where V is the topological dual of V . After the pioneering works of S. Łojasiewicz [17,18] in the finite-dimensional case, many results of this type have been proved in the literature in various contexts with main applications to partial differential equations, the main objectives being convergence results of bounded solutions of equations such as u + DF(u) = 0 (or more general problems such as second order damped systems) to stationary ones or decay estimates of the difference between the solution and its limiting equilibrium, cf. for instance [1–16,20]. Usually in the PDE framework one makes use of a compactness hypothesis of the resolvent of the linearization of DF(u) around an arbitrary equilibrium. It is however reasonable to wonder what would be a minimal framework to extend the Łojasiewicz theory to analytic functionals in infinite dimensions. This paper gives rather simple answers to this question, first in the case where F is quadratic or equivalently DF is a linear operator independent of u where the gradient inequality can already fail without additional assumptions, and secondly in the socalled semilinear case (the gradient being a lower order nonlinear perturbation of a coercive linear operator) where the situation turns out to be slightly more complicated. At the end of the paper we recover the usual Łojasiewicz–Simon results in the PDE framework, but only in the simplest cases where the basic Sobolev space imbeds in the space of bounded functions. After understanding these easier cases the reader should be able to consider the classical PDE literature, the last step reducing to technical regularity considerations which are always present in PDE problems and are not specific of the subject matter of this paper. 2. Quadratic forms and the linear case Throughout this section we consider a real Hilbert space H and a linear operator A such that A ∈ L(H );

A∗ = A

(1)

2828

A. Haraux, M.A. Jendoubi / Journal of Functional Analysis 260 (2011) 2826–2842

and the associated quadratic form Φ : H → R defined by ∀u ∈ H,

1 Φ(u) = Au, u . 2

(2)

We denote by |u| the norm of a vector u ∈ H . Our main result is the following. Theorem 2.1. The following properties are equivalent: (i) 0 is not an accumulation point of sp(A). (ii) For some ρ > 0 we have ∀u ∈ ker(A)⊥ ,

|Au| ρ|u|.

(iii) Φ satisfies the Łojasiewicz gradient inequality at the origin for some θ > 0. (iv) Φ satisfies the Łojasiewicz gradient inequality at any point for θ = 12 . Proof. We establish [(i) ⇒ (ii) ⇒ (iv)] and the contraposition of [(iii) ⇒ (i)]. Since [(vi) ⇒ (iii)] is obvious, the result follows. Step 1. Assuming that 0 is an accumulation point of sp(A) we prove that the Łojasiewicz gradient inequality at 0 fails. We state first an easy Lemma 2.2. Assume that for some η > 0 we have ∀u ∈ H,

|Au − λu| η|u|.

Then λ ∈ / sp(A). Proof. Indeed since A is bounded, A − λI has closed graph and consequently (A − λI )−1 , which is well defined on H has also closed graph and is therefore bounded. 2 As a consequence of this lemma, we can find a sequence λn of positive numbers tending to 0 and a sequence of vectors un ∈ H for which ∀n ∈ N,

|Aun − λn un |
0 can be satisfied in a neighborhood of 0. Step 2. (i) ⇒ (ii). As a consequence of Theorem VIII.4, p. 260 from [19], up to an isometric isomorphism we may assume H = L2 (Ω, dμ) where (Ω, dμ) is some positively measured space and ∀u ∈ H,

(Au)(x) = a(x)u(x),

μ-a.e. in Ω.

We define Ω+ = x ∈ Ω, a(x) > 0 ;

Ω− = x ∈ Ω, a(x) < 0 ;

Ω0 = x ∈ Ω, a(x) = 0 .

First if 0 ∈ / sp(A), then A is an isomorphism and then the result is obvious. Indeed in that case Φ(u) |Au||u| C|Au|2 . On the other hand if 0 ∈ sp(A) and 0 is isolated in sp(A), it means that for some ρ > 0 we have [−ρ, ρ] ∩ sp(A) = {0}. We claim that μ a −1 (0, ρ) ∩ Ω+ = 0. Indeed assuming μ(a −1 (0, ρ) ∩ Ω+ ) > 0, there is first of all η ∈ (0, ρ) for which μ a −1 (η, ρ) ∩ Ω+ > 0. Then we have either

ρ+η ∩ Ω+ > 0 μ a −1 η, 2 or

(3)

2830


ρ +η , ρ ∩ Ω+ > 0 μ a −1 2 and by inductive dichotomy we find a sequence of integers kn ∈ [0, 2n − 1] for which, setting

ρ−η ρ −η In = η + kn n , η + (kn + 1) n 2 2 the following properties hold In ⊂ In−1 ⊂ · · · ⊂ I1 and μ a −1 (In ) ∩ Ω+ > 0.

∀n ∈ N, Let

ρ∗ :=

In .

n1

It is clear that μ a −1 B(ρ∗ , ε) ∩ Ω+ > 0.

∀ε > 0, Letting

ωε = a −1 B(ρ∗ , ε) ∩ Ω+ ;

φε = 1ωε

we find ∀ε > 0,

(A − ρ∗ )φε 2 ε 2 |φε |2 .

Hence ρ∗ ∈ sp(A), a contradiction. Similarly we have μ a −1 (−ρ, 0) ∩ Ω− = 0.

(4)

Finally given u ∈ H , we have |Au| = 2

a(x)u(x)2 dμ(x) ρ 2

Ω

u (x) dμ(x) + 2

Ω+

u (x) dμ(x) 2

Ω−

and the result is now obvious since ⊥

∀u ∈ ker(A) ,

u (x) dμ(x) + 2

Ω+

Ω−

u2 (x) dμ(x) = |u|2 .


2831

Step 3. (ii) ⇒ (iv). Given u ∈ H , let v = Qu be the orthogonal projection of u on ker(A)⊥ . We have clearly A A 2Φ(u) = 2Φ(v) A|v|2 2 |Av|2 = 2 |Au|2 ρ ρ which is precisely the gradient inequality at 0 with θ = 12 . The gradient inequality for any 1 θ > 0 is trivially satisfied at any point x with Ax = 0 and if Ax = 0 we have |Φ(u)−Φ(x)| = |Φ(u)|, so that at such a point x the gradient inequality reduces to the same inequality at 0. 2 Remark 2.3. Condition (ii) is in fact equivalent to R(A) is closed in H. Indeed (ii) easily implies that R(A) is closed and conversely if R(A) is closed, (ii) follows from the closed graph theorem. However in practice, in the examples (ii) usually follows from a combination of elementary constructive steps which do not require the (non-constructive) closed graph theorem, and this is the reason why the formulation (ii) will be preferred to a closed range condition. 3. What happens in the nonlinear case? A natural question is whether the necessary and sufficient condition of Theorem 2.1 gives the right condition for the second derivative at a critical point a in order for an analytic functional F : H → R to fulfill a Łojasiewicz inequality near a. Since in the quadratic case the Łojasiewicz inequality is either false, or satisfied with the best possible value θ = 12 , it is clear that some additional difficulties will appear. The next result shows that if the second derivative is “bad”, at least the functional cannot satisfy the Łojasiewicz inequality with θ = 12 . Proposition 3.1. Let F : U → R be an analytic functional where U ⊂ H is an open neighborhood of 0 and assume F (0) = DF(0) = 0. If 0 is an accumulation point of sp(D 2 F (0)), then an inequality ∀u ∈ U,

∇F (u) cF (u)1−θ

for some c > 0 implies θ 13 . Proof. As a consequence of the hypothesis, as in the proof of Theorem 2.1 we can find a sequence λn of positive numbers tending to 0 and a sequence of vectors un ∈ H with |un | = ρ which can be taken arbitrarily small independently of λn , and for which ∀n ∈ N,

3 |Aun | < λn ρ; 2

Φ(un ) >

λn 2 ρ 4

2832


where A = D 2 F (0) and Φ is the quadratic part of F . Then by taking the next (second order) approximation we find for some fixed constants C, D 0 ∇F (un ) 2λn ρ + Cρ 2 ;

F (un )

λn 2 ρ − Dρ 3 . 4

Choosing ρ = ελn with ε > 0 small enough we find for some M > 0 ε2

λn 3 8

1−θ Mλn 2 .

Therefore λn 1−3θ is bounded and by letting n go to infinity we conclude that θ 13 .

2

Remark 3.2. We have been unable to construct an example of the above situation in which θ = 13 . The next example shows that θ = 14 can actually happen. Proposition 3.3. Let H = L2 (0, 1) and F : H → R be the analytic functional given by 1 F (u) := 4

2

1 2

u (x) dx

1 + 2

0

1 xu2 (x) dx. 0

Then ∀u ∈ H,

3 (∇F )(u) F (u) 4 .

However [0, 1] ⊂ sp(D 2 F (0)). Proof. It is easily verified that F is an analytic (actually polynomial) functional with

1 (∇F )(u) = xu +

2

u dx u. 0

In particular (∇F )(u)2 =

2 1 3 1 1 1 2 2 2 2 u dx u dx x u dx + u dx . x+ 0

0

0

On the other hand 1

2 2

u dx 0

1 =

3 4 2

2

u dx 0

3

4 (∇F )(u) 3 .

0


2833

In addition 3

1

1

2

2

xu (x) dx

3 1

0

0

x 2 u2 dx + 0

x u dx + 2 2

0

1

1 =

u dx

1

1

4

2

x u dx

0

3

4

2 2

3 4 4

2

u dx 0

3 u2 dx

0

therefore

1

1

xu (x) dx 0

1 x u dx +

2

2 2

3 2 3

2

u dx

0

2× 2 4 (∇F )(u) 3 = (∇F )(u) 3

0

and we end up with 1 F (u) := 4

2

1 2

u (x) dx

1 + 2

0

1

4 3 xu2 (x) dx (∇F )(u) 3 4

0

which clearly implies the result.

2

Remark 3.4. In the opposite direction it is natural to wonder whether the condition on the spectrum of D 2 F (0) is enough to ensure the existence of a Łojasiewicz gradient inequality. The following example where ker D 2 F (0) = H shows that it is not the case. Proposition 3.5. Let H = l 2 (N) and F : H → R be the analytic functional given by F (u1 , u2 , . . . , un , . . .) :=

∞ |uk |2k+2 . (2k + 2)! k=2

Then F satisfies no Łojasiewicz gradient inequality. Proof. First we note that D 2 F (0) = 0, hence sp(D 2 F (0)) = {0} and in particular 0 is isolated in sp(D 2 F (0)). Defining (ei )j = δij , an immediate calculation shows that ∀t > 0,

F (tek ) =

t 2k+2 ; (2k + 2)!

∇F (tek ) =

In particular for each θ > 0 we have F (tek )1−θ = c(θ, k)t 1−(2k+2)θ . |∇F (tek )| Choosing k large enough gives a contradiction for t small.

2

t 2k+1 . (2k + 1)!

2834


4. A basically optimal nonlinear result In the application, in particular to PDE problems, the basic Hilbert space will not be identified with its dual since the gradient operators we are dealing with are semilinear perturbations of an unbounded self-adjoint linear operator. For the statement of the main result of this section we consider two real Hilbert spaces V , H where V ⊂ H with continuous and dense imbedding and H , the topological dual of H is identified with H , therefore V ⊂ H = H ⊂ V with continuous and dense imbeddings. The duality product of φ ∈ V with v ∈ V is denoted as φ, v . The examples of Section 3 suggest that the following result is essentially optimal. Theorem 4.1. Let F : U → R be analytic in the sense of [22] where U ⊂ V is an open neighborhood of 0 and assume F (0) = 0;

DF(0) = 0.

We assume the two following conditions: (i) N := ker D 2 F (0) is finite-dimensional. (ii) There is ρ > 0 for which ∀u ∈ V ∩ N ⊥ ,

2 D F (0)u

V

ρuV ,

where N ⊥ stands for the orthogonal in the sense of H . Then there exist θ ∈ (0, 1/2), a neighborhood W of 0 and c > 0 for which ∀u ∈ W,

DF(u) cF (u)1−θ . V

Proof. We set A = D 2 F (0) ∈ L(V , V ) and we introduce the orthogonal projection Π in H on N = ker(A). As a consequence of (ii), it is clear that A has closed range in V . Indeed let un ∈ V be such that Aun − f V → 0 for some f ∈ V . Since A(un − Πun ) = Aun , A(un − Πun ) is a Cauchy sequence in V , and by (ii), un − Πun = vn is Cauchy in V . Setting v = limV vn we clearly find f = Av ∈ R(A), proving the claim. From now on we follow the method introduced by Simon in [20]. First we show that the linear operator L := Π + A restricted to V is one-to-one and onto. Actually we shall see that for some η>0 ∀u ∈ V ,

LuV ηuV .

Thanks to the fact that A is symmetric: V → V we have R(A) ∩ N = {0}.

(5)


2835

Indeed if v ∈ R(A) ∩ N we have v = Au for some u ∈ V and then 0 = Av, u = Au, v = |v|2H = 0. Then (5) becomes an immediate consequence of the next lemma. Lemma 4.2. Let W be a real Hilbert space endowed with the norm · W and N , F two closed subspaces with N finite-dimensional. Then, assuming F ∩ N = {0} there is a constant σ > 0 such that ∀(n, f ) ∈ N × F,

n + f W σ nW + f W .

Proof. First we denote by Q the projection onto F ⊥ in the sense of W . We observe that the function n ∈ N → p(n) = QnW is a norm on N and since N is finite-dimensional we find immediately the existence of ν > 0 for which ∀n ∈ N,

QnW νnW .

Now we have n + f = Qn + (I − Q)n + f and (I − Q)n + f ∈ F , therefore by orthogonality in W we deduce n + f W QnW νnW . Then it suffices to observe that f W = n + f − nW n + f W + nW 1 + ν −1 n + f W and the result follows with σ=

ν . ν+2

2

In order to prove (5) it suffices to apply Lemma 4.2 with W = V , F = R(A), N = ker A ⊂ V ⊂ V , n = Πu, f = Au, n + f = Lu. Indeed we have ∀u ∈ V ,

uV u − ΠuV + ΠuV ρ −1 A(u − Πu)V + KΠuV

2836


by using (ii) and the equivalence of the norms in V and V in the finite-dimensional space N . Then ∀u ∈ V ,

uV max ρ −1 , K AuV + ΠuV

and the lemma gives ∀u ∈ V ,

uV σ −1 max ρ −1 , K LuV

that is (5) with η = σ min{ρ, K −1 }. Now we observe that (5) implies the one-to-one character of L. In addition since L is symmetric we have R(L) = [ker L]⊥ = V . Then from (5) it follows that L is onto. Indeed given any ϕ ∈ V , there exists a sequence ϕn = Lun with ϕn − ϕV → 0. In particular (ϕn ) is a Cauchy sequence in V , and by (5) (un ) is Cauchy in V . Setting u = limV un we clearly conclude that ϕ = Lu ∈ R(L). Finally, (5) gives immediately L−1 ∈ L V , V . Let now N : V → V , u → Πu + DF(u). By using the hypotheses, we deduce that N is analytic in the neighborhood of 0 and DN (0) = L. Applying the local inversion theorem (analytic version cf. [22, Corollary 4.37, p. 172]), we can find a neighborhood of 0, W1 (0) in V , a neighborhood of 0, W2 (0) in V and an analytic map Ψ : W2 (0) → W1 (0) which satisfies N Ψ (f ) = f, ∀f ∈ W2 (0), Ψ N (u) = u, ∀u ∈ W1 (0), Ψ (f ) − Ψ (g) C1 f − gV , ∀f, g ∈ W2 (0), C1 > 0. V

(6)

Since F is C 1 , we also have DF(u) − DF(v)

V

C2 u − vV ,

∀(u, v) ∈ W1 (0).

(7)

Let {ϕ }m be a basis of the finite-dimensional space N . For ξ ∈ Rm small enough to achieve m j j =1 j =1 ξj ϕj ∈ W2 (0), we define the map Γ by Γ (ξ ) = F Ψ

m j =1

ξj ϕj

.


2837

By using the chain rule, since F : U → R is analytic, the function Γ is real analytic in some neighborhood of 0 in Rm . Let u ∈ W1 (0) such that Π(u) = m j =1 ξj ϕj ∈ W2 (0). For any k ∈ {1, . . . , m} we have the formula ∂Γ d = F Ψ ξj ϕj + (ξk + s)ϕk ∂ξk ds s=0

= DF Ψ

j =k

m

ξj ϕj

,Ψ

j =1

m

ξj ϕj ϕk .

j =1

By (6), it is clear that for each k ∈ {1, . . . , m}, Ψ ( m j =1 ξj ϕj )ϕk is bounded in V . Then by using (6) and (7) we obtain ∇Γ (ξ )

Rm

m C3 DF Ψ ξj ϕj j =1

V

= C3 DF Ψ Π(u) V = C3 DF Ψ Π(u) − DF(u) + DF(u)V C3 DF(u)V + C4 Ψ Π(u) − uV = C3 DF(u)V + C5 Ψ Π(u) − Ψ Πu + DF(u) V C3 DF(u) + C5 DF(u) V

V

hence ∇Γ (ξ )

Rm

C6 DF(u)V .

(8)

On the other hand F (u) − Γ (ξ ) = F (u) − F Ψ Π(u) 1 d = F u + t Ψ Π(u) − u dt dt 0

1 = DF u + t Ψ Π(u) − u , Ψ Π(u) − u dt 0

Ψ Π(u) − u

1

DF u + t Ψ Π(u) − u

V

V 0

dt

2838


1

DF(u)

+ tC7 Ψ Π(u) − u V dt Ψ Π(u) − uV V

0

C8 DF(u)V Ψ Π(u) − Ψ Π(u) + DF(u) V hence F (u) − Γ (ξ ) C9 DF(u)2 . V

(9)

Applying the classical Łojasiewicz inequality to the scalar analytic function Γ defined on some neighborhood of 0 in Rm , we now obtain F (u)1−θ Γ (ξ )1−θ + Γ (ξ ) − F (u)1−θ ∇Γ (ξ ) m + Γ (ξ ) − F (u)1−θ . R

(10)

By combining (8), (9), (10) we obtain ) F (u)1−θ C6 DF(u) + C 1−θ DF(u)2(1−θ . 9 V V Then since 2(1 − θ ) 1, there exist σ > 0, c > 0 such that DF(u)

V

1−θ cF (u)

Theorem 4.1 is completely proved.

for all u ∈ V such that uV < σ.

2

5. Remarks and application Theorem 4.1 suggests a few observations. Remark 5.1. Theorem 4.1 is of course applicable near any equilibrium point a ∈ E = v ∈ V , DF(v) = 0 and gives, assuming that A = D 2 F (a) satisfies the relevant hypothesis, the existence of a neighborhood W of a in V such that ∀u ∈ W,

DF(u)

V

1−θ cF (u) − F (a) .

Remark 5.2. Theorem 4.1 is essentially optimal, since in order for such a general result to be true we at least need it to apply to quadratic forms. The finite dimensionality hypothesis on N is motivated by the example of Proposition 3.5. Remark 5.3. If the imbedding V → H is compact and for some m0 ∈ R the operator D 2 F (0) + m0 I is invertible, then Lemma 6.1 from [12] shows that the condition is automatically fulfilled. Actually we have the following more general result.


2839

Proposition 5.4. Let V , W be two reflexive Banach spaces and L ∈ L(V , W ) be such that for some compact operator K ∈ L(V , W ), L + K is invertible. Let X be a closed linear subspace of V such that X ∩ ker L = {0}. Then there exists c > 0 such that ∀v ∈ X,

LvW cvV .

Proof. If X = {0} there is nothing to prove. Otherwise, assuming that the result is not true, for each integer n 1, let wn ∈ X be such that wn ∈ X,

wn V = 1,

1 Lwn W . n

Then we can replace wn by a subsequence (still denoted wn ) such that wn → w

weakly in V

and Lwn → 0 strongly in W.

Since L is continuous from (V , weak) to (W , weak) we have Lw = 0. Since X is closed, hence weakly sequentially closed in V , we also have w ∈ X, hence w = 0. In particular, we have (L + K)wn = Lwn + Kwn → 0 strongly in W therefore wn → 0 strongly in V , which contradicts wn V = 1.

2

Corollary 5.5. Let V , H be as in the statement of Theorem 4.1, let F : U → R be an analytic functional where U ⊂ V is an open neighborhood of a and assume DF(a) = 0. We assume the two following conditions: (i) N := ker D 2 F (0) is finite-dimensional. (ii) For some compact operator K ∈ L(V , V ), L + K is invertible. Then there exist θ ∈ (0, 1/2), a neighborhood W of a and c > 0 for which ∀u ∈ W,

DF(u)

V

1−θ cF (u) − F (a) .

Proof. This result is an immediate consequence of Theorem 4.1 and Proposition 5.4.

2

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Corollary 5.6. Let Ω be a bounded open interval of R, let G : R → R be an analytic function and g = G . The functional Φ defined by ∀u ∈ H01 (Ω),

Φ(u) =

1 2 ux + G(u) dx 2

Ω

is such that for any solution ϕ of ϕ ∈ H01 (Ω),

−ϕ + g(ϕ) = 0

there is θ ∈ (0, 12 ) and ε > 0, C > 0 for which ∀u ∈ H01 (Ω),

u − ϕH 1 (Ω) ε 0

⇒

Φ(u) − Φ(ϕ)1−θ C −u + g(u) −1 . H (Ω)

Proof. The functional Φ is clearly analytic since it is the sum of a continuous quadratic functional on H01 (Ω) and a Nemytskii operator which is analytic on the Banach algebra H01 (Ω) ⊂ L∞ (Ω). Moreover we have ∀u ∈ H01 (Ω),

DΦ(u) = −u + g(u)

and for any ϕ as above ∀u ∈ H01 (Ω),

D 2 Φ(ϕ)(u) = −u + g (ϕ)(u).

In particular for m > 0 large enough, the operator Λ = D 2 Φ(ϕ) + mI where I stands for the identity operator is coercive, thus invertible as an operator from H01 (Ω) to H −1 (Ω). In addition the kernel of D 2 Φ(ϕ) is clearly finite-dimensional. Since I : H01 (Ω) → H −1 (Ω) is compact, the result follows from Corollary 5.5 applied with V = H01 (Ω), H = L2 (Ω). 2 Corollary 5.7. Let Ω be a bounded open domain of RN with N 3, let G : R → R be an analytic function and g = G . The functional Φ defined by ∀u ∈ H02 (Ω),

Φ(u) =

1 2 |u| + G(u) dx 2

Ω

is such that for any solution ϕ of ϕ ∈ H02 (Ω),

2 ϕ + g(ϕ) = 0


2841

there is θ ∈ (0, 12 ) and ε > 0, C > 0 for which ∀u ∈ H02 (Ω),

u − ϕH 2 (Ω) ε 0

⇒

Φ(u) − Φ(ϕ)1−θ C 2 u + g(u) −2 . H (Ω)

Proof. The functional Φ is clearly analytic since it is the sum of a continuous quadratic functional on H02 (Ω) and a Nemytskii operator which is analytic on the Banach algebra H02 (Ω) ⊂ L∞ (Ω). Moreover we have ∀u ∈ H02 (Ω),

DΦ(u) = −u + g(u)

and for any ϕ as above we have ∀u ∈ H02 (Ω),

D 2 Φ(ϕ)(u) = 2 u + g (ϕ)(u).

In particular for m > 0 large enough, the operator Λ = D 2 Φ(ϕ) + mI is coercive, thus invertible as an operator from H02 (Ω) to H −2 (Ω). In addition the kernel of D 2 Φ(ϕ) is clearly finite-dimensional. Since I : H02 (Ω) → H −2 (Ω) is compact, the result follows from Corollary 5.5 applied with V = H02 (Ω), H = L2 (Ω). 2 Remark 5.8. In higher dimensions, Theorem 4.1 cannot be applied directly since the nonlinear perturbation is no longer analytic in the topology of V = H01 (Ω) for the first example when N > 1, and the topology of V = H02 (Ω) when N > 3 in the second example. In higher dimensions one makes use of the fact that the equilibria are smoother and the analyticity of the functional is used on a smaller Banach space in order to be able to treat the finite-dimensional term at the end, cf. [20,16,10,8] for precise statements. Acknowledgments A major part of this work was done when the second author was invited by the Mathematics Department of Versailles University. He wishes to express here his thanks for their kind hospitality. The authors are grateful to the referees for pointing out some pedagogical weaknesses of the initial version of this paper, thus contributing to make it easier to read. References [1] I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations, Asymptot. Anal. 69 (1–2) (2010) 31–44. [2] I. Ben Hassen, L. Chergui, Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation, forthcoming. [3] L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dynam. Differential Equations 20 (2008) 643–652. [4] L. Chergui, Convergence of global and bounded solutions of the wave equation with nonlinear dissipation and analytic nonlinearity, J. Evol. Equ. 9 (2009) 405–418. [5] R. Chill, On the Łojasiewicz–Simon gradient inequality, J. Funct. Anal. 201 (2003) 572–601.

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[6] R. Chill, A. Haraux, M.A. Jendoubi, Applications of the Łojasiewicz–Simon gradient inequality to gradient-like evolution equations, Anal. Appl. 7 (2009) 351–372. [7] R. Chill, M.A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal. 53 (2000) 1017–1039. [8] E. Feireisl, F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Differential Equations 12 (2000) 647–673. [9] A. Haraux, M.A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differential Equations 144 (1998) 313–320. [10] A. Haraux, M.A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations 9 (1999) 95–124. [11] A. Haraux, M.A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Anal. 26 (2001) 21–36. [12] A. Haraux, M.A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations, J. Evol. Equ. 7 (2007) 449–470. [13] A. Haraux, M.A. Jendoubi, O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evol. Equ. 3 (2003) 463–484. [14] S.J. Huang, Gradient Inequalities, with Applications to Asymptotic Behavior and Stability of Gradient-like Systems, Math. Surveys Monogr., vol. 126, Amer. Math. Soc., Providence, RI, 2006. [15] S.J. Huang, P. Takác, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal. 46 (2001) 675–698. [16] M.A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal. 153 (1998) 187–202. [17] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, in: Colloques internationaux du CNRS: Les équations aux dérivées partielles, Paris, 1962, Editions du CNRS, Paris, 1963, pp. 87–89. [18] S. Łojasiewicz, Ensembles Semi-Analytiques, Publ. Inst. Hautes Etudes Sci., Bures-sur-Yvette, 1965, preprint. [19] M. Reed, B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, second ed., Academic Press, New York, 1980. [20] L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math. 118 (1983) 525–571. [21] E.F. Whittlesey, Analytic functions in Banach spaces, Proc. Amer. Math. Soc. 16 (1965) 1077–1083. [22] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Tome 1, Springer-Verlag, 1985.

Journal of Functional Analysis 260 (2011) 2843 www.elsevier.com/locate/jfa

Editorial

Nigel J. Kalton (1946–2010). Obituary Nigel Kalton was born on 20 June 1946 in Bromley (Great Britain). He was educated at Dulwich College, before studying mathematics at Trinity College Cambridge. An exceptionally talented student in mathematics, he was also an outstanding chess player, among the 20 best in England. He received his PhD in 1970 under the supervision of D.J.H. Garling, and his thesis was awarded the Rayleigh Prize for research excellence. After holding positions in Lehigh University (Pennsylvania), Warwick and Swansea, Nigel Kalton joined the University of Missouri–Columbia in 1979, where he spent the rest of his career. Among other honors bestowed on Nigel Kalton is the prestigious Stefan Banach Medal, which was awarded to him in 2004 by the Polish Academy of Sciences. He was also honored at a 2006 conference, organized by Beata and Narcisse Randrianantoanina, held in Miami University (Ohio) on the occasion of his sixtieth birthday. This conference gathered more than 160 participants from all over the world. At the time of his death from a stroke on August 31, 2010, Nigel served on the editorial boards of several journals, including that of the Journal of Functional Analysis. Nigel Kalton was an amazingly prolific researcher, having (according to MathSciNet) published 255 papers and 6 books, not counting those which were about to be submitted when he died. Several of these references are full Memoirs, and although Nigel’s articles were always concisely written, his published work fills thousands of pages. His research encompassed (but was not limited to) every branch of functional analysis. He also supervised 14 students, who now hold positions on 4 continents. In addition to his great achievements, Nigel was an engaging man who was excellent company. He was also incredibly generous and helpful, qualities of which his colleagues, students, and coauthors were all beneficiaries. Nigel’s attitude was that everyone around him should be given the chances to improve and progress, and his list of co-authors reflects this attitude. Nigel Kalton is survived by Jennifer Kalton, his wife of 41 years, their children Neil Kalton and Helen Kurtz, and four grandchildren. There is no doubt that Nigel’s colleagues feel, that, on August 31, 2010, they too lost a much loved family member.

0022-1236/$ – see front matter © 2011 Published by Elsevier Inc. doi:10.1016/j.jfa.2011.02.008


Pullbacks, C(X)-algebras, and their Cuntz semigroup Ramon Antoine, Francesc Perera ∗ , Luis Santiago Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain Received 25 January 2011; accepted 17 February 2011 Available online 24 February 2011 Communicated by D. Voiculescu

Abstract In this paper we analyse the structure of the Cuntz semigroup of certain C(X)-algebras, for compact spaces of low dimension, that have no K1 -obstruction in their fibres in a strong sense. The techniques developed yield computations of the Cuntz semigroup of some surjective pullbacks of C∗ -algebras. As a consequence, this allows us to give a complete description, in terms of semigroup valued lower semicontinuous functions, of the Cuntz semigroup of C(X, A), where A is a not necessarily simple C∗ -algebra of stable rank one and vanishing K1 for each closed, two-sided ideal. We apply our results to study a variety of examples. © 2011 Elsevier Inc. All rights reserved. Keywords: C∗ -algebras; Cuntz semigroup; Classification

0. Introduction To any C∗ -algebra A, one can attach an ordered semigroup Cu(A), the Cuntz semigroup of A. It was originally devised by Cuntz in [10], and can be constructed using suitable equivalence classes of positive elements in the stabilization of A, in a way similar to how the projection semigroup is constructed from Murray–von Neumann equivalence. Coward, Elliott and Ivanescu proved in [9] that the order relation in Cu(A) has additional properties and a new category Cu was defined for ordered semigroups with this structure. This category was shown to be closed un* Corresponding author.

E-mail addresses: [email protected] (R. Antoine), [email protected] (F. Perera), [email protected] (L. Santiago). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.02.016

R. Antoine et al. / Journal of Functional Analysis 260 (2011) 2844–2880

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der sequential inductive limits, and it was furthermore proved that the assignment A → Cu(A), from the category of C∗ -algebras to the category Cu, is functorial and (sequentially) continuous. The study of the Cuntz semigroup has had a resurgence in recent years, mainly due to its impact in Elliott’s classification program. Notably, one of its order properties is the key to distinguishing two non-isomorphic C∗ -algebras that agree on the Elliott invariant and several possible extensions of it; see [31]. For well behaved simple algebras, this semigroup can be recovered from the classical Elliott invariant (see [5,6]), and in the non-simple case it has already been used to prove actual classification results (see, e.g., [7,8,24,25,29]). However, because of its complexity, this semigroup becomes a very difficult object to describe. Already in the commutative setting, it is necessary to understand the isomorphism classes of fibre bundles over the spectrum X of the algebra. In cases where there are no cohomological obstructions, a description via point evaluations has been obtained in terms of (extended) integer valued lower semicontinuous functions on X (see [23,26]). A natural class to consider consists of the algebras C0 (X, A) where X is a locally compact Hausdorff space. When A is a unital, simple, non-type I ASH-algebra with slow dimension growth, the case C0 (X, A) has been studied in [30]. The description of Cu(C0 (X, A)) is given in terms of pairs of certain projection valued functions and semigroup valued lower semicontinuous functions (see below). In this paper we study the Cuntz semigroup of C(X)-algebras A when X is a second countable, compact Hausdorff space of dimension at most one. We also assume that each fibre of A has stable rank one and vanishing K1 for every closed, two-sided ideal. Our approach is based on describing Cuntz equivalence classes by means of the corresponding classes in the fibres, thus seeking to recover global information from local data. We do so by analysing the natural map Cu(A) →

Cu A(x) ,

[a] →

a(x) x∈X .

x∈X

Of particular interest will be the algebras of the form C0 (X, A) for a not necessarily simple algebra A, as in this case the range of the previous map can be completely identified. This is achieved in Theorem 3.4. The strategy combines a number of ingredients, each of which may be of independent interest. We discuss them below. In Section 2, we are exclusively concerned with the case X = [0, 1] and prove in Theorem 2.1 that the map above is an order-embedding for a C([0, 1])-algebra whose fibres have the said conditions. If, further, the algebra has the form C([0, 1], A), we show that the range of the map can be described as the semigroup of lower semicontinuous functions on X with values in Cu(A), denoted by Lsc(X, Cu(A)) (Corollary 2.7). To do this, we need to show that this semigroup belongs to the category Cu, and this requires a deeper analysis of general semigroups of the form Lsc(X, M) with M a semigroup in Cu. In order not to interrupt the flow of the paper, we postpone this discussion until Section 5, where in fact we prove that, for any finite-dimensional second countable, compact Hausdorff space X and any separable C∗ -algebra A, the semigroup Lsc(X, Cu(A)) belongs to Cu (see Theorem 5.15). Pullbacks are the main theme in Section 3 as they provide us with a way to deal with more general spaces. We consider surjective pullbacks of the form B ⊕A(Y ) A, where A is a C(X)-algebra, Y is a closed subset of X and B is any C∗ -algebra (where the maps are given by the natural projection A → A(Y ) and a ∗-homomorphism B → A(Y )). Any such pullback gives rise to a diagram of the corresponding Cuntz semigroups. We relate, in Theorems 3.1, 3.2 and 3.3, the pullback in the category of ordered semigroups with the Cuntz semigroup of the C∗ -algebra pullback.

2846


The methods developed allow us to prove, in Theorem 3.4, that the Cuntz semigroup of C(X, A) is order-isomorphic to Lsc(X, Cu(A)), where X has dimension at most one, A has stable rank one and vanishing K1 for each closed, two-sided ideal. If, further, A is an AF-algebra, the same ˇ result is available for spaces of dimension at most 2 and vanishing second Cech cohomology group (Corollary 3.6). A key ingredient in the proofs is the continuity of the functors Lsc(X, −) and Lsc(−, Cu(A)), also proved to hold in Section 5 (Proposition 5.18). This description yields, as a consequence, a cancellation result, namely that Cu(C(X, A)) is order-cancellative with respect to the relation (see below). In Section 4 we discuss applications of the previous results in some computations of Cu(A). Combining these results, we can give a description of the Cuntz semigroup for algebras obtained by a successive pullback construction. This includes one-dimensional rsh-algebras and, more specifically, one-dimensional non-commutative CW-complexes. We also offer a computation of the Cuntz semigroup of dimension drop algebras over the interval and over certain two-dimensional spaces, as well as a description of the Cuntz semigroup of the mapping torus of an algebra A. 1. Preliminary results and definitions 1.1. Cuntz semigroup Let A be a C∗ -algebra. Recall that a positive element a is said to be Cuntz subequivalent to b ∈ A+ , written a b, if there exists a sequence (xn ) in A such that xn bxn∗ → a. This defines a preorder in A+ and we say that a is Cuntz equivalent to b, a ∼ b, if a b and b a. Proposition 1.1. (See [27, Proposition 2.4], [16, Proposition 2.6].) Let A be a C∗ -algebra, and a, b ∈ A+ . The following are equivalent: (i) a b. (ii) For all > 0, (a − )+ b. (iii) For all > 0, there exists δ > 0 such that (a − )+ (b − δ)+ . Furthermore, if A is stable, this conditions are equivalent to (iv) For every > 0 there is a unitary u ∈ U(A∼ ) such that u(a − )+ u∗ ∈ Her(b). Proof. The proof of the equivalence between (i) and (iv) is essentially that of [27, Proposition 2.4(v)], where it is stated for algebras with stable rank one. We briefly sketch the necessary modifications to extend it to the case of stable algebras. Given > 0, write (a − /2)+ = zz∗ with z∗ z ∈ Her(b). By [1, Lemma 4.8], we know that A ⊂ GL(A∼ ). Therefore dist(z∗ , GL(A∼ )) = 0, so [19, Corollary 8] applies to find u ∈ U(A∼ ) with u(a − )+ u∗ = v(a − )+ v ∗ = z∗ z − /2 + ∈ Her(b), where v is the partial isometry in the polar decomposition of z∗ .

2


2847

The Cuntz semigroup of A is defined as the set of Cuntz equivalence classes in the stabilized algebra, (A ⊗ K)+ / ∼ and is denoted by Cu(A). Equip Cu(A) with the order induced by Cuntz subequivalence and addition given by [a] + [b] =

a 0

0 b

,

so that it becomes an ordered Abelian semigroup with [0] as its least element (the positive element inside the brackets in the right side of the equation above is identified with its image in A ⊗ K by any isomorphism of M2 (A ⊗ K) with A ⊗ K induced by an isomorphism of K and M2 (K)). Recall that, in an ordered semigroup M, a is said to be compactly contained in b, denoted a b, if whenever b supn cn for some increasing sequence (cn ) with supremum in M, implies there exists n0 such that a cn0 . A sequence (an ) such that an an+1 is said to be rapidly increasing. The following theorem summarizes the structure of Cu(A). Theorem 1.2. (See [9].) Let A be a C∗ -algebra. Then: (i) Suprema of increasing sequences exist in Cu(A). (ii) Any element in Cu(A) is the supremum of a rapidly increasing sequence. (iii) The operation of taking suprema and are compatible with addition. The conditions (i)–(iii) of the last theorem define a category of ordered semigroups of positive elements, denoted by Cu, which is closed under countable inductive limits, and such that Cu(− ⊗ K) defines a sequentially continuous functor from the category of C∗ -algebras to the category Cu (see [9]). It can be shown that the rapidly increasing sequence in (ii) for a positive element a ∈ A ⊗ K, can be chosen as ([(a − n1 )+ ]). Let M be a semigroup in the category Cu. Endow M with the ω-Scott topology, that is, the topology generated by the open sets a := {c ∈ M | a c} where a ∈ M (see [15]). Adopting the terminology of [15], we will say that an object M in Cu is countably based if there is a countable subset X in M such that every element of M is the supremum of a rapidly increasing sequence of elements coming from X. If M is countably based, then M satisfies the second axiom of countability as a topological space equipped with the Scott topology (see, e.g., [15, Theorem III-4.5]). Lemma 1.3. Let A be a separable C∗ -algebra. Then Cu(A) is countably based. Proof. We first claim that, if M is a semigroup in Cu and X ⊆ M is a countable subset such that, given a ∈ M and a a, there is b ∈ X with a b a, then M is countably based. Indeed, given a ∈ M write a = supn an , where (an ) is a rapidly increasing sequence in M. Then, by our assumption there exists a sequence (bn ) in X such that a1 b1 a2 b2 a3 · · · . This implies that (bn ) is rapidly increasing and that supn bn = supn an = a.

2848


Let now A be a separable C∗ -algebra, which we may take to be stable. Let F be a countable dense subset of A+ , and consider the set X=

(a − 1/m)+ a ∈ F, m ∈ N ⊆ Cu(A).

Given b ∈ A+ and x [b], find m such that x [(b − 1/m)+ ]. For this m, there is a ∈ F such that b − a < 1/4m. From this, we first obtain (a − 1/2m)+ (a − 1/4m)+ [b]. Observe also that (a − 1/2m)+ − b < 1/2m + 1/4m = 3/4m, whence x (b − 1/m)+ (b − 3/4m)+ (a − 1/2m)+ [b]. Thus the result follows from the first part of the proof and the fact that Cu(A) is a semigroup in Cu. 2 The following lemma is a modification of Lemma 2 of [8]: Lemma 1.4. Let A be a C∗ -algebra, and let B be a hereditary subalgebra such that B ⊆ GL(B ∼ ). If a is a positive contraction, and x0 , x1 ∈ A are such that x0 x0∗ , x1 x1∗ ∈ B, and a − x ∗ x0 < , 0

a − x ∗ x1 < , 1

then there exists a unitary u in B ∼ such that

x0 − ux1 < 9. Proof. By Proposition 1 of [25], applied to the elements a and x0 , there exists y0 ∈ A such that (a − )+ = y0∗ y0 ,

y0 − x0 < 4,

y0 y0∗ ∈ B.

(A straightforward computation shows that Proposition 1 of [25] holds for C = 4.) Similarly, there exists y1 ∈ A such that (a − )+ = y1∗ y1 ,

y1 − x1 < 4,

y1 y1∗ ∈ B.

It follows now from Lemma 2 of [8], applied to the elements y0 and y1 , that there exists a unitary u ∈ B ∼ such that

y0 − uy1 < . (Note that Lemma 2 of [8] still holds if the assumption of B having stable rank one is replaced by B ⊆ GL(B ∼ ).) Therefore,

x0 − ux1 < x0 − y0 + y0 − uy1 + uy1 − ux1 < 4 + + 4 = 9.

2


2849

1.2. C(X)-algebras Let X be a compact Hausdorff space. Recall that a C(X)-algebra is a C∗ -algebra A endowed with a unital ∗-homomorphism θ from C(X) to the center Z(M(A)) of the multiplier algebra M(A) of A (see, e.g., [11]). We shall refer to the map θ as the structure map. For each closed subset Y of X, we define A(Y ) to be the quotient of A by the closed twosided ideal C0 (X\Y )A, which is a C(X)-algebra in the natural way. The quotient map is denoted by πY : A → A(Y ). If, further, Z ⊆ Y is closed then πZ = πZY ◦ πY , where πZY : A(Y ) → A(Z) denotes the quotient map. In the case that Y = {x} the C∗ -algebra A(x) := A({x}) is called the fibre of A at x, and we write πx for π{x} . The image by πx of an element a ∈ A is denoted by a(x). It is well known that, for all a ∈ A, the map x → a(x) is upper semicontinuous. Moreover, if a ∈ A one has that a = supx∈X a(x) and the supremum is attained (see, e.g., [2, Proposition 2.8]). The following lemmas will be used in a number of instances, and are quite possibly well known. We include their proofs for completeness. Lemma 1.5. Let A be a C(X)-algebra. Then, A ⊗ K is a C(X)-algebra and for any closed set Y of X there exists a ∗-isomorphism ϕY : (A ⊗ K)(Y ) → A(Y ) ⊗ K, such that ϕY ◦ πY = πY ⊗ 1K , where πY : A → A(Y ) and πY : A ⊗ K → (A ⊗ K)(Y ) denote the quotient maps. In particular, for any x ∈ X, we have (A ⊗ K)(x) ∼ = A(x) ⊗ K with (a ⊗ k)(x) → a(x) ⊗ k. Proof. Abusing the language, define θ : C(X) → Z(M(A ⊗ K)) on elementary tensors by θ (f )(a ⊗ k) = (θ (f )a) ⊗ k, whenever a ∈ A and k ∈ K, and where θ is the structure map for A. This endows A ⊗ K with a structure map. Now, if Y is closed in X, we have an exact sequence 0 → C0 (X\Y )A → A → A/C0 (X\Y )A → 0. Tensoring by the compacts we obtain another exact sequence 0 → C0 (X\Y )A ⊗ K → A ⊗ K → A/C0 (X\Y )A ⊗ K → 0. Since (C0 (X\Y )A) ⊗ K = C0 (X\Y )(A ⊗ K), it follows that (A ⊗ K)(Y ) = (A ⊗ K)/C0 (X\Y )(A ⊗ K) ∼ = A(Y ) ⊗ K.

2

Remark 1.6. Let A be a C(X)-algebra. Then, Lemma 1.5 implies that the natural map πx : A → A(x) induces, at the level of the Cuntz semigroup, a map Cu(A) → Cu(A(x)) that can be viewed as [a] → [a(x)]. In turn, these maps define a map α : Cu(A) →

x∈X

Cu A(x) .

2850


Similarly, if Y is closed in X the map πY induces a map Cu(πY ) : Cu(A) → Cu(A(Y )) such that Cu(πY )[a] = [a|Y ]. Thus, by the previous lemma, when computing the Cuntz semigroup of A we may assume that A, A(x) and A(Y ) are stable. Lemma 1.7. Let A be a C(X)-algebra and let B := A + C(X) · 1M(A) . Then: (i) B is a C(X)-algebra that contains A as a closed two-sided ideal. In particular, A is C(X)subalgebra of B. (ii) The restriction of πx : B → B(x) to A induces an isomorphism A(x) ∼ = πx (A) for all x ∈ X. Proof. (i) Since the quotient of B by the C∗ -algebra A is a C∗ -algebra, B itself is a C∗ -algebra. It is clear that B is a C(X)-algebra and that A is a closed two-sided ideal of B. The second part of the lemma follows from part (v) of [11, Lemma 2.1]. 2 2. The Cuntz semigroup of C([0, 1], A) Theorem 2.1. Let A be a C[0, 1]-algebra such that for t in a dense subset of [0, 1] the fibre C∗ -algebra A(t) is separable, has stable rank one, and K1 (I ) = 0 for any ideal I of A(t). Then, the map α : Cu(A) →

Cu A(t) ,

t∈[0,1]

given by α[a] = ([a(t)])t∈[0,1] is an order-embedding. Proof. By Remark 1.6 and since our assumptions on A and its fibres are stable, we may assume that A and its fibres are stable at the outset. Let 0 < < 1 be fixed, and let us suppose that a, b ∈ A are positive contractions such that a(t) b(t), for all t ∈ [0, 1]. We need to show that a b. Let B := A + C[0, 1] · 1M(A) . Then B is a C∗ -algebra that contains A as a closed two-sided ideal by (i) of Lemma 1.7. In addition, by (ii) of Lemma 1.7 we have that a(t) b(t) in B(t), for all t ∈ [0, 1]. By the definition of the Cuntz order and since B(t) is a quotient of B for each t ∈ [0, 1] there exists d ∈ B such that a(t) − d(t)∗ b(t)d(t) < . By the upper semicontinuity of the norm the inequality above also holds in a neighbourhood of t. Hence, since [0, 1] is a compact set, there exist a finite covering of [0, 1] consisting of open intervals Ui , i = 1, 2, . . . , n, and elements (di )ni=1 ⊂ B such that a(t) − di (t)∗ b(t)di (t) < , for all t ∈ Ui and all 1 i n. Moreover, we may choose the open intervals (Ui )ni=1 such that t < t if t ∈ Ui and t ∈ Ui+2 , for i = 1, 2, . . . , n − 2. 1 For each 1 i n, set b 2 di = xi . Then, xi xi∗ ∈ Her(b), and a(t) − xi (t)∗ xi (t) < ,


2851

for all t ∈ Ui . By assumption, there exists ti ∈ Ui ∩ Ui+1 such that the stable rank of πti (A) ∼ = A(ti ) is one, where πti : B → B(ti ) denotes the quotient map. Therefore, since Her(b(ti )) = b(ti )B(ti )b(ti ) is also a hereditary subalgebra of πti (A), the stable rank of Her(b(ti )) is one. We now have xi (ti )xi (ti )∗ , xi+1 (ti )xi+1 (ti )∗ ∈ Her b(ti ) , a(ti ) − xi+1 (ti )∗ xi+1 (ti ) < . a(ti ) − xi (ti )∗ xi (ti ) < , Hence, by Lemma 1.4 there exists a unitary ui in the unitization of Her(b(ti )) such that xi (ti ) − ui xi+1 (ti ) < 9. Note that since πti (A) ∼ / πti (A), whence 1M(A) (ti ) ∈ / Her(b(ti )). = A(ti ) is stable 1M(A) (ti ) ∈ This implies that the unitization of Her(b(ti )) is isomorphic to the C∗ -algebra Her(b(ti )) + C · 1M(A) (ti ). Therefore, we may assume that the unitary ui belongs to this algebra. Using [3, Theorem 2.8] and that A(ti ) is separable we conclude that Her(b(ti )) is stably isomorphic to an ideal of πti (A), which is in turn isomorphic to an ideal of A(ti ). Hence, it follows from our assumptions that K1 (Her(b(ti ))) = 0. Since sr(Her(b(ti ))) = 1, we know from [22, Theorem 2.10] that U(Her(b(ti ))) is connected. Therefore, ui can be connected to 1M(A) (ti ) in Her(b(ti )) + C · 1M(A) (ti ). Since this C∗ -algebra is the image by πti of the C∗ algebra Her(b) + C · 1M(A) , and unitaries in the connected component of the identity lift (see, e.g., [32, Corollary 4.3.3]), there exists a unitary vi in Her(b) + C · 1M(A) such that vi (ti ) = ui . Let (yi )ni=1 be the elements defined by y1 = x1 , and yi = v1 v2 · · · vi−1 xi , for i = 2, . . . , n. Since xi ∈ b1/2 A, it follows that yi ∈ A for all i. Also, yi yi∗ ∈ Her(b), and a(t) − yi (t)∗ yi (t) < ,

(1)

for all t ∈ Ui . Moreover, (yi − yi+1 )(ti ) = v1 · · · vi−1 (xi − vi xi+1 )(ti ) = (xi − vi xi+1 )(ti ) = xi (ti ) − ui xi+1 (ti ). Thus, yi (ti ) − yi+1 (ti ) < 9. Since the norm is upper semicontinuous there exists δ > 0 such that yi (t) − yi+1 (t) < 9, for all t ∈ (ti − δ, ti + δ) and for all 1 i n − 1. Let us consider the open intervals (Vi )ni=1 defined by

2852


⎧ ⎨ [0, t1 + δ) Vi = (ti−1 − δ, ti + δ) ⎩ (tn−1 − δ, 1]

if i = 1; if 2 i n − 1; if i = n.

n

Then, i=1 Vi = [0, 1], and Vi ⊆ Ui for all 1 i n. Let (λi )ni=1 be a partition of unity associated to the open covering (Vi )ni=1 . Let us consider the element y=

n

λi yi .

i=1

Then, y ∈ A, yy ∗ ∈ Her(b), and ⎧ y1 (t) ⎪ ⎪ ⎨ yi (t) y(t) = ⎪ ⎪ ⎩ yn (t) λi (t)yi (t) + λi+1 (t)yi+1 (t)

if 0 t t1 − δ; if ti−1 + δ t ti − δ, and 2 i n − 1; if tn−1 + δ t 1; if ti − δ < t < ti + δ, and 1 i n − 1.

(2)

Let us show that a − y ∗ y < 28. By (1) and (2) it is enough to show that a(t) − y(t)∗ y(t) < 28 for t ∈ (ti − δ, ti + δ). We have a(t) − y(t)∗ y(t) ∗ = a(t) − λi (t)yi (t) + λi+1 (t)yi+1 (t) λi (t)yi (t) + λi+1 (t)yi+1 (t) a(t) − yi (t)∗ yi (t) ∗ + yi (t)∗ yi (t) − λi (t)yi (t) + λi+1 (t)yi+1 (t) λi (t)yi (t) + λi+1 (t)yi+1 (t) < + 27 = 28. We have found an element y ∈ A such that a − y ∗ y < 28, and yy ∗ ∈ Her(b). This implies by Lemma 2.2 of [17] that (a − 28)+ b in A. Therefore, [a] = sup (a − 28)+ [b]. >0

This concludes the proof of the theorem.

2

In the particular case of C(X, A), for a given C∗ -algebra A, all fibres are naturally isomorphic to A, and hence the image of the map α in the theorem above can be viewed as functions from X to Cu(A), that are lower semicontinuous in a certain topology. Proposition 2.2. (See [30, Proposition 3.1].) Let A be a C∗ -algebra, X a compact Hausdorff space and f ∈ C(X, A). Then, for any a ∈ A, the set {x ∈ X | [a] [f (x)]} is open. Given a separable C∗ -algebra A, the sets {[b] ∈ Cu(A) | [a] [b]} define a basis of a topology in Cu(A) named the Scott Topology (see, e.g., [15]).


2853

Definition 2.3. Let X be a topological space, M a semigroup in Cu, and f : X → M a function. We say that f is lower semicontinuous if, for all a ∈ M, the set f −1 (a ) := {t ∈ X | a f (t)} is open in X. We shall denote the set of all lower semicontinuous functions from X to M by Lsc(X, M). In Section 5 we prove that in the general case of a second countable finite-dimensional topological space X and a countably based semigroup M in Cu, Lsc(X, M), equipped with the pointwise order and addition, is a semigroup in the category Cu. The key step in the argument is to show that any function f ∈ Lsc(X, M) is a supremum of a rapidly increasing sequence of functions that have a special form. We describe these functions in the particular case of the interval X = [0, 1], and refer the reader to Section 5 for the general case. Definition 2.4. Let A be a C∗ -algebra. Given the following data: (i) A partition 0 = t0 < t1 < · · · < tn−1 < tn = 1 of [0, 1] with n = 2r + 1 for some r 1; (ii) Elements x0 , . . . , xn−1 in M, with x2i , x2i+2 x2i+1 for 0 i r − 1, a piecewise characteristic function is a map g : [0, 1] → Cu(A) such that g(s) =

x2i x2i+1

if s ∈ [t2i , t2i+1 ], if s ∈ (t2i+1 , t2i+2 ).

If moreover g f for some f ∈ Lsc([0, 1], Cu(A)), we then say that g is a piecewise characteristic function for f . We denote the set of all such functions by χ(f ). It is easily verified that a piecewise characteristic function as above is lower semicontinuous. Lemma 2.5. Let A be a separable stable C∗ -algebra, f ∈ Lsc([0, 1], Cu(A)), and f1 f2 be piecewise characteristic functions for f . Then, there exists a continuous function g2 ∈ C([0, 1], A) such that f1 α([g2 ]) f2 and α([g2 ]) ∈ χ(f ). Proof. Suppose that f2 is described as in Definition 2.4 with xi = [ai ] for some ai ∈ A. Let f2, be the function with the same form as f2 but with [a2i ] replaced by [(a2i − )+ ] for 0 i r. Note that f2, ∈ χ(f ) and that f2 = sup f2, in Lsc([0, 1], Cu(A)). Hence, since f1 f2 , there exists > 0 such that f1 f2, f2 . Since [a2i ] [a2i−1 ], [a2i+1 ], by condition (iv) in Proposition 1.1 there exist unitaries ui and vi in A∼ such that ui (a2i − )+ u∗i ∈ Her(a2i−1 ),

vi (a2i − )+ vi∗ ∈ Her(a2i+1 ).

(3)

Since A is stable, the unitary group of the multiplier algebra M(A) is connected in the norm topology (see, e.g., [32, Corollary 16.7]). Therefore, for each i = 0, 1, . . . , r there exists a continuous path wi : [0, 1] → U(M(A)) such that wi (t) = ui if t ∈ [0, t2i ], and wi (t) = vi for t ∈ [t2i+1 , 1]. Let (λi )ri=0 be sequence of continuous positive real-value functions on [0, 1] that are supported in the open sets

2854


[0, t2 ),

(t2i−1 , t2i+2 )r−1 i=1 ,

(t2r−1 , 1],

respectively (i.e., they are non-zero in each point of the corresponding interval and zero elsewhere). Let us define g2 ∈ C([0, 1], A) by g2 (t) =

r λi (t)wi (t)(a2i − )+ wi∗ (t) + λi (t)λi+1 (t)a2i+1 . i=0

(In the equation above we are taking λr+1 = 0 and ar+1 = 0.) If t ∈ [t2j , t2j +1 ] with 0 j r, then g2 (t) = λi (t)wj (t)(a2j − )+ wj (t) ∼ (a2j − )+ . Hence, α([g2 ])(t) = [(a2j − )+ ]. If t ∈ (t2j +1 , t2j +2 ) with 0 j r − 1, then g2 (t) = λj (t)wj (t)(a2j − )+ wj (t)∗ + λj +1 (t)wj +1 (t)(a2j +2 − )+ wj +1 (t)∗ + λj (t)λj +1 (t)a2j +1 = λj (t)vj (a2j − )+ vj∗ + λj +1 (t)uj +1 (a2j +2 − )+ u∗j +1 + λj (t)λj +1 (t)a2j +1 . By (3) the element g2 (t) belongs to Her(a2j +1 ), whence g2 (t) a2j +1 . Also, we have g2 (t) λj (t)λj +1 (t)a2j +1 ∼ a2j +1 . Therefore, α([g2 ])(t) = [a2j +1 ]. It follows that α([g2 ]) = f2, , which proves the result.

2

Theorem 2.6. Let A be a separable C∗ -algebra. If the natural map α : Cu C [0, 1], A → Lsc [0, 1], Cu(A) is an order-embedding, then it is an isomorphism in the category Cu. Proof. Without loss of generality we may assume that A is stable. In addition, we only need to prove that α is surjective since by our assumptions this will imply that it is an order-isomorphism, whence an isomorphism in the category Cu. Let f ∈ Lsc([0, 1], Cu(A)). We know from Proposition 5.14 combined with Lemma 1.3 that there is a rapidly increasing sequence of functions (fn ) in χ(f ) such that f = sup fn . By Lemma 2.5, we may suppose that there exists gn ∈ C([0, 1], A)+ with α([gn ]) = fn . As α is an order-embedding by assumption, the sequence ([gn ]) is increasing. Let [g] = sup[gn ]. Then α [g] = sup α [gn ] = sup fn = f, n

as desired.

n

2

From Theorems 2.1 and 2.6, we immediately obtain the corollary below. Much more is true, as will be shown in the next section.


2855

Corollary 2.7. Let A be a separable C∗ -algebra with stable rank one such that K1 (I ) = 0 for every closed two-sided ideal I of A. Then, the map α : Cu C [0, 1], A → Lsc [0, 1], Cu(A) , given by α([a])(t) = [a(t)] is an isomorphism in the category Cu. 3. Pullbacks In this section we extend the previous results to general spaces of dimension at most one. En route to our result, we analyse the behavior of the functor Cu under the formation of certain pullbacks, which we describe below. Let A, B, and C be C∗ -algebras. Let π : A → C and φ : B → C be ∗-homomorphisms. We can form the pullback

B ⊕C A = (b, a) ∈ B ⊕ A φ(b) = π(a) . By applying the Cuntz functor Cu(·) to the ∗-homomorphisms π and φ we obtain Cuntz semigroup morphisms (in the category Cu) Cu(π) : Cu(A) → Cu(C),

and Cu(φ) : Cu(B) → Cu(C).

Let us consider the pullback (in the category of ordered semigroups) Cu(B) ⊕Cu(C) Cu(A) =

[b], [a] ∈ Cu(B) ⊕ Cu(A) Cu(φ)[b] = Cu(π)[a] .

Then, we have a natural order-preserving map β : Cu(B ⊕C A) → Cu(B) ⊕Cu(C) Cu(A),

(4)

defined by β([(b, a)]) = ([b], [a]). Observe that since Cu(π) and Cu(φ) are maps in Cu, the pullback semigroup Cu(B) ⊕Cu(C) Cu(A) is closed under suprema of increasing sequences. Note also that the map β preserves suprema. Theorem 3.1. Let A, B, and C be C∗ -algebras such that C is separable, has stable rank one, and K1 (I ) = 0 for every closed two-sided ideal I of C. Let φ : B → C and π : A → C be ∗-homomorphisms such that π is surjective. Then, the map β : Cu(B ⊕C A) → Cu(B) ⊕Cu(C) Cu(A), given by β[(b, a)] = ([b], [a]) is an order-embedding. Proof. By [20, Theorem 3.9] applied to Y = K we may assume that A, B, and C are stable. Let (b1 , a1 ) and (b2 , a2 ) be positive contractions of B ⊕C A such that a1 a2 and b1 b2 . Let 0 < < 1. Then, by the definition of the Cuntz relation there are x ∈ A and y ∈ B such that

2856


a1 − x ∗ x < , b1 − y ∗ y < ,

xx ∗ ∈ Her(a2 ), yy ∗ ∈ Her(b2 ).

Since π(a1 ) = φ(b1 ) and π(a2 ) = φ(b2 ), the equations above imply that π(a1 ) − φ(y)∗ φ(y) < , π(x)π(x)∗ , φ(y)φ(y)∗ ∈ Her π(a2 ) .

π(a1 ) − π(x)∗ π(x) < ,

(5)

By Lemma 1.4 there is a unitary u ∈ Her(π(a2 ))∼ such that uπ(x) − φ(y) < 9. Using [3, Theorem 2.8] and that C is separable it follows that Her(a2 ) is stable isomorphic to an ideal of C. Hence, by our assumptions K1 (Her(π(a2 ))) = 0. Since sr(A) = 1, we have by [22, Theorem 2.10] that U(Her(π(a2 ))) = U0 (Her(π(a2 ))). Therefore, u is in the connected component of the identity. By the surjectivity of the map π there exists a unitary v ∈ Her(a2 )∼ such that π (v) = u, where π : A∼ → C ∼ is the extension of π to the unitization of the algebras A and C. In addition, there exists y ∈ Her(a2 ) such that π(y ) = φ(y). Hence, we have π vx − y = uπ(x) − φ(y) < 9. Since vx − y ∈ a2 A there exists z ∈ a2 A ∩ Ker(π) such that vx − y − z < 9. Set y + z = z. Then, π(z) = φ(y),

zz∗ ∈ Her(a2 ),

vx − z < 9.

Also, a1 − z∗ z a1 − x ∗ x + x ∗ x − z∗ z < + (vx)∗ (vx) − z∗ z + (vx)∗ (vx − z) + (vx − z)∗ z + vx

vx − z + z

vx − z + 2 vx − z + 11 vx − z < 118. Since π(z) = φ(y) the element (y, z) belongs to B ⊕C A, and by the previous computation (b1 , a1 ) − (y, z)∗ (y, z) < 118.


2857

In addition, since yy ∗ ∈ Her(b2 ) and zz∗ ∈ Her(a2 ) we have 1 1 (y, z)(y, z)∗ = lim (b2 , a2 ) n (y, z)(y, z)∗ (b2 , a2 ) n ∈ Her (b2 , a2 ) . n→∞

By [17, Lemma 2.2] we have (b1 , a1 ) − 118 + (y, z)∗ (y, z) ∼ (y, z)(y, z)∗ (b2 , a2 ). Therefore, (b1 , a1 ) = sup (b1 , a1 ) − 118 + (b2 , a2 ) .

2

>0

Theorem 3.2. Let X be a one-dimensional compact Hausdorff space and let Y be a closed subset of X. Let A be a C(X)-algebra and let πY : A → A(Y ) be the quotient map. Let B be a C∗ -algebra and let φ : B → A(Y ) be a ∗-homomorphism. Suppose that, for every x ∈ X, the C∗ -algebra A(x) is separable, has stable rank one, and K1 (I ) = 0 for every two-sided ideal I of A(x). Then, the map β : Cu(B ⊕A(Y ) A) → Cu(B) ⊕Cu(A(Y )) Cu(A), given by β[(b, a)] = ([b], [a]) is an order-embedding. Proof. By Remark 1.6 and [20, Theorem 3.9] we may assume that A, A(Y ) and B are stable. Let (b1 , a1 ) and (b2 , a2 ) be positive elements of B ⊕A(Y ) A such that a1 a2 and b1 b2 . Let > 0. By the definition of the Cuntz order, there exist d ∈ B and c ∈ A such that a1 − c∗ a2 c < ,

b1 − d ∗ b2 d < .

(6)

Since φ(b1 ) = πY (a1 ) and φ(b2 ) = πY (a2 ), the second inequality in the equation above implies that πY (a1 ) − φ(d)∗ πY (a2 )φ(d) < . Choose an element h ∈ A such that πY (h) = φ(d). Then, we have a1 (x) − h(x)∗ a2 (x)h(x) < ,

(7)

for all x ∈ Y . By upper semicontinuity, there exists an open neighbourhood U of Y such that the inequality above holds for all x ∈ U . Since Y ⊆ X is compact and X is normal there exists an open subset V such that Y ⊆ V ⊆ V ⊆ U . Moreover, by Theorem [14, 4.2.2] we may assume that V has an empty or zero-dimensional boundary. Let D := A + C(X) · 1M(A) . Then, D is a C(X)-algebra that contains A as a closed two1

1

sided ideal by the first part of Lemma 1.7. Consider the elements y1 = a22 c and y2 = a22 h. Then y1 y1∗ , y2 y2∗ ∈ Her(a2 ). Moreover, rewriting the first inequality in (6) and the inequality (7), we have

2858


a1 − y ∗ y1 < , 1

a1 (x) − y2 (x)∗ y2 (x) < ,

for all x ∈ V , where the second inequality holds in D(x) (here we are using that A(x) ∼ = πx (A) by the second part of Lemma 1.7, where πx : D → D(x) denotes the quotient map). In particular, if x ∈ bd(V ) we have y1 (x)y1 (x)∗ , y2 (x)y2 (x)∗ ∈ Her(a2 (x)), and a1 (x) − y1 (x)∗ y1 (x) < ,

a1 (x) − y2 (x)∗ y2 (x) < .

(8)

Since by assumption A(x) has stable rank one, the hereditary algebra Her(a2 (x)) has stable / πx (A), rank one. In addition, since A(x) is stable and A ∼ = πx (A) we have that 1M(A) (x) ∈ ∼ . By / Her(a2 (x)). This implies that Her(a2 (x)) + C · 1M(A) ∼ (x)) whence 1M(A) (x) ∈ Her(a = 2 Lemma 1.4 applied to (8), there exists a unitary ux ∈ Her(a2 (x)) + C · 1M(A) (x) such that ux y1 (x) − y2 (x) < 9.

(9)

Since the K1 -group of every ideal of A(x) is trivial, it follows that K1 (Her(a2 (x))) = 0 by [3, Theorem 2.8] and the separability of A(x). Hence, by [22, Theorem 2.10] every unitary in Her(a2 (x)) + C · 1M(A) (x) is connected to the identity, and in particular ux . Since Her(a2 (x)) + C · 1M(A) (x) is the image of Her(a2 ) + C · 1M(A) by πx and unitaries in the connected component of the identity lift, there exists a unitary v x ∈ U0 (Her(a2 ) + C · 1M(A) ) such that v x (x) = ux . Suppose first that bd(V ) = ∅. Note that, by (9), x v y1 − y2 (x) = ux y1 (x) − y2 (x) < 9, for all x ∈ X. Hence by the upper semicontinuity of the norm and since bd(V ) is zerodimensional and compact, there are points x1 , . . . , xn ∈ bd(V ) and an open cover of bd(V ) consisting of pairwise disjoint neighbourhoods (Vi )ni=1 with xi ∈ Vi such that x v i y1 − y2 (x) < 9,

(10)

for all i and all x ∈ Vi . Since the sets (Vi )ni=1 are open, closed, pairwise disjoint,and form a n ∗ cover of bd(V ), the C∗ -algebra D(bd(V n )) canx be identified with the C -algebra i=1 D(Vi ). i Let us consider the element v = i=1 πVi (v ) ∈ D(bd(V )) (here we are using the previous identification). Then, v is a unitary in πbd(V ) (Her(a2 ) + C · 1M(A) ) that is connected to the identity πbd(V ) (1M(A) ). Hence, there is a unitary u ∈ Her(a2 ) + C · 1M(A) such that πbd(V ) (u) = v. By (10) we have u(x)y1 (x) − y2 (x) < 9, for all x ∈ bd(V ). Set uy1 = y1 . Since y1 ∈ a2 A. It follows that y1 ∈ a2 A. Further, y1 y1 ∗ ∈ Her(a2 ), and a1 − y ∗ y < , 1

1

y (x) − y2 (x) < 9, 1

for all x ∈ bd(V ). By upper semicontinuity of the norm there exists an open neighbourhood W of bd(V ) such that W ∩ Y = ∅, and


y (x) − y2 (x) < 9,

2859

(11)

1

for all x ∈ W . Let f1 , f2 ∈ C(X) be a partition of unity associated to the covering of X given by the open sets V c ∪ W and V ∪ W . In case bd(V ) = ∅ we proceed analogously with u = 1M(A) and W = ∅, since now V , V c are both open sets. Consider the element z = f1 y1 + f2 y2 . Then, z ∈ A and zz∗ ∈ Her(a2 ). Next, using (11), a computation analogous to the one carried out in the proof of Theorem 2.1 shows that ∗ z z − a1 < 28. 1

1

By construction πY (z) = φ(b22 d), whence the element (z, b22 d) ∈ B ⊕A(Y ) A, and 1 1 z, b 2 d ∗ z, b 2 d − (a1 , b1 ) < 28.

2

2

1

1

In addition, since zz∗ ∈ Her(a2 ) and b22 d(b22 d)∗ ∈ Her(b2 ), we have 1 1 1 1 1 1 ∗ ∗ z, b22 d z, b22 d = lim (a2 , b2 ) n z, b22 d z, b22 d (a2 , b2 ) n ∈ Her (a2 , b2 ) .

n→∞

Hence, by [17, Lemma 2.2] (a1 , b1 ) − 28 + (a2 , b2 ). Therefore, (a1 , b1 ) = sup (a1 , b1 ) − 28 + (a2 , b2 ) .

2

Theorem 3.3. Let X be a compact Hausdorff space and let Y be a closed subset of X. Let A be a C(X)-algebra, and let B be any C∗ -algebra. Suppose that the map α : Cu(A) →

Cu A(x) ,

x∈X

given by α([a])(x) = [a(x)] is an order-embedding. Then: (i) The map β : Cu(B ⊕A(Y ) A) → Cu(B) ⊕Cu(A(Y )) Cu(A), defined by β([(b, a)]) = ([b], [a]) is surjective. (ii) The pullback semigroup Cu(B) ⊕Cu(A(Y )) Cu(A) is in the category Cu. Proof. (i) By Remark 1.6 and [20, Theorem 3.9] we may assume that A, A(Y ), and B are stable. Let a ∈ A and b ∈ B be positive elements such that πY (a) ∼ φ(b). Choose a positive element c ∈ A such that πY (c) = φ(b). Then we have πY (a) ∼ πY (c).

2860


Let > 0. Since πY (a) πY (c), by (iii) of Proposition 1.1 there exists 0 < δ < such that πY ((a − )+ ) πY ((c − δ)+ ). Therefore, by the definition of the Cuntz order and since πY is surjective there exists d ∈ A such that πY (a − )+ − πY (d)∗ πY (c − δ)+ πY (d) < . In particular, in the fibre algebras A(x), with x ∈ Y , we have (a − )+ (x) − d(x)∗ (c − δ)+ (x)d(x) < . By upper semicontinuity of the norm, there exists an open neighbourhood U of Y such that the inequality above holds for all x ∈ U . Since X is normal there exists an open set W such that Y ⊆ W ⊆ W ⊆ U . Without loss of generality we may assume that U = W and that (a − )+ (x) − d(x)∗ (c − δ)+ (x)d(x) < , holds for all x ∈ U . It follows now that π (a − )+ − d ∗ (c − δ)+ d < . U By [17, Lemma 2.2] and since πU is surjective, there exists f ∈ A such that πU ((a − 2)+ ) = πU (f ∗ (c − δ)+ f ). This implies that πU (a − 2)+ πU (c − δ)+ , πU (a − 3)+ = πU f ∗ (c − δ)+ f − + .

(12) (13)

Since f ∗ (c − δ)+ f (c − δ)+ , by (iv) of Proposition 1.1 there exists a unitary u ∈ A∼ such that u∗ f ∗ (c − δ)+ f − + u ∈ Her (c − δ)+ .

(14)

Let us consider the element a = u∗ au. Then, Eqs. (13) and (14) imply that πU a − 3 + ∈ πU Her (c − δ)+ = Her πU (c − δ)+ . Hence, passing to the fibres we have a − 3 + (x) ∈ Her (c − δ)+ (x) ,

(15)

for all x ∈ U . In addition, by (12) we have a − 2 + (x) (c − δ)+ (x),

(16)

for all x ∈ U . Now let us use that πY (c) πY (a) ∼ πY (a ). Arguing as above, there exists g ∈ A such that c(x) − g(x)∗ a (x)g(x) < δ < δ,


2861

for all x in a closed neighbourhood V of Y . Without loss of generality, we may assume that U = V . Therefore, π c − g ∗ a g < δ < δ. U Hence, by [17, Lemma 2.2] we have πU ((c − δ )+ ) πU (a ). This implies by (iii) of Proposition 1.1 that πU ((c − δ)+ ) πU ((a − 3 )+ ), for some < . Now passing to the fibres we have that (c − δ)+ (x) a − 3 + (x),

(17)

for all x ∈ U . Let f1 , f2 ∈ C(X) be a partition of unity associated to the open sets U and X\Y . Let us consider the element z = f1 (c − δ)+ + f2 a − 3 + . Then, z(x) = c(x) − δ + if x ∈ Y, z(x) = a (x) − 3 + if x ∈ X\U, z(x) (c − δ)+ (x) or z(x) a − 3 + (x) z(x) ∈ Her c(x) − δ + if x ∈ U,

if x ∈ U, (18)

where the last equation follows by (15). By the choice of c and the first equation in (18) we have z(x) = (c − δ)+ (x) = (φ(b) − δ)(x), for all x ∈ Y . Hence, πY (z) = φ((b − δ)+ ), and so ((b − δ)+ , z) is an element of the pullback. We also obtain by the first and third equation of (18) and by (16) that (a − 3)+ (x) z(x), for all x ∈ X. In addition, by the first and last equation of (18) and by (17) we obtain that z(x) (a − 3 )+ (x), for all x ∈ X. Since by assumption the map α is an order-embedding, this yields a − 3 + z a − 3 + . Therefore, we can choose sequences δn < n decreasing to zero and elements zn in A+ such that ((b − δn )+ , zn ) ∈ B ⊕A(Y ) A, (b − δn )+ , zn (b − δn+1 )+ , zn+1 , for all n, supn [(b − δn )+ ] = [b], and supn [zn ] = [a ] = [a]. Moreover, ([(b − δn )+ ], [zn ]) is by construction rapidly increasing and supn ([(b − δn )+ ], [zn ]) = ([b], [a]). It also follows that β sup (b − δn )+ , zn = sup β (b − δn )+ , zn = sup (b − δn )+ , [zn ] = [b], [a] , n

n

which proves that β is surjective.

n

2862


(ii) We need to show that Cu(B) ⊕Cu(A(Y )) Cu(A) satisfies the axioms of the category Cu, that is to say, (i), (ii), and (iii) of Theorem 1.2. It was previously shown in the proof of the first part of the theorem that the pullback satisfies (ii). That the pullback satisfies the rest of the axiom follows easily using this fact and that Cu(πY ) and Cu(φ) are morphisms in the category Cu. 2 We now turn our attention to the C(X)-algebras of the form C(X, A). In order to deal with general one-dimensional spaces, we will first analyse the case where the underlying space is a graph. These algebras can be conveniently described in pullback form, as follows (see, e.g., [13, Section 3.1] combined with [20, Theorem 3.8]). As a directed graph, write X = (V , E, r, s), where V = {v1 , . . . , vn } is the set of vertices, E = {e1 , . . . , em } is the set of edges, and r, s : E → V are the range and source maps. For 1 k m, denote by ik : A → Am ⊕ Am the inclusion in the kth component of the first summand. Likewise, we may define jk : A → Am ⊕ Am for the second summand. Next, define φ : C(V , A) → Am ⊕ Am by φ(g) =

n l=1

k∈s −1 (vl )

ik g(vl ) +

jk g(vl ) .

k∈r −1 (vl )

Finally, let π{0,1} : C([0, 1], A) → C({0, 1}, A) denote the quotient map. Then C(X, A) ∼ = C [0, 1], Am ⊕Am ⊕Am C(V , A) (where Am ⊕ Am is identified with C({0, 1}, Am ) in the obvious manner). Theorem 3.4. Let X be a locally compact Hausdorff space that is second countable and onedimensional. Let A be a separable C∗ -algebra with stable rank one such that K1 (I ) = 0 for every closed two-sided ideal I of A. Then, the map α : Cu(C0 (X, A)) → Lsc(X, Cu(A)) given by α([a])(x) = [a(x)], for all a ∈ C0 (X, A) and x ∈ X, is an isomorphism in the category Cu. Proof. By Corollary 2.7, the result holds when X = [0, 1]. Now, let X be a finite graph. By Theorems 3.3 and 3.2 and the comments previous to this theorem Cu C(X, A) ∼ = Cu C [0, 1], Am ⊕Cu(Am ⊕Am ) Cu C(V , A) ∼ = Lsc [0, 1], Cu Am ⊕Cu(Am ⊕Am ) Lsc V , Cu(A) ∼ = Lsc X, Cu(A) . Note that the isomorphism between Cu(C(X, A)) and Lsc(X, Cu(A)) obtained above is given by the map [a] → (x ∈ X → [a(x)]). Next, any compact Hausdorff space X that is second countable and one-dimensional can be written as a projective limit X = lim ←−(Xi , μi,j )i,j ∈N , where Xi are finite graphs and μi,j : Xj → Xi , with i j , are surjective maps (see [14, p. 153]). By [9, Theorem 2] this implies that


2863

Cu C(X, A) = lim −→ Cu C(Xi , A) , Cu(ρi,j ) i,j ∈N , where ρi,j : C(Xi ) → C(Xj ), with i j , is the ∗-homomorphism induced by μi,j , i.e. ρ(f ) = f ◦ μi,j . In addition, by (i) of Proposition 5.18 we have Lsc X, Cu(A) = lim −→ Lsc Xi , Cu(A) , Lsc(μi,j ) i,j ∈N , where the maps Lsc(μi,j ) : Lsc(Xi , Cu(A)) → Lsc(Xj , Cu(A)) are given by Lsc(f ) = f ◦ μi,j . Consider the following diagram:

Cu(C(X1 , A))

Cu(ρ1,2 )

α

Cu(C(X2 , A))

Cu(ρ2,3 )

···

Cu(C(X, A))

α

Lsc(X1 , Cu(A))

Lsc(μ1,2 )

α

Lsc(X2 , Cu(A))

Lsc(μ2,3 )

···

Lsc(X, Cu(A)).

This diagram is clearly commutative. Hence, since by the argument above the vertical arrows in the finite stages are isomorphisms the map between the limit semigroups is an isomorphism. This proves the theorem in the case that X is compact. Let X is an arbitrary locally compact space. Then, applying the first part of the proof to = X ∪ {∞} we conclude that the map α : Cu(C(X, A)) → its one-point compactification X Lsc(X, Cu(A)) is an isomorphism in the category Cu. It is easy to check that the image by α A)) is {f ∈ Lsc(X, Cu(A)) | f (∞) = 0}. The latter of the order-ideal Cu(C0 (X, A)) of Cu(C(X, is in turn order-isomorphic to Lsc(X, Cu(A)) (via restriction). Thus, the result follows. 2 Corollary 3.5. Let X be a compact Hausdorff space that is second countable and onedimensional. Let A be a separable C∗ -algebra with stable rank one such that K1 (I ) = 0 for every closed two-sided ideal I of A. Let B be any C∗ -algebra and suppose φ : B → C(Y, A) is a ∗-homomorphism, where Y ⊆ X is a closed subset of X. Then Cu B ⊕C(Y,A) C(X, A) ∼ = Cu(B) ⊕Lsc(Y,Cu(A)) Lsc X, Cu(A) , in the category Cu. Proof. Combine Theorems 3.2, 3.3 and 3.4.

2

Corollary 3.6. Let X be a second countable, compact Hausdorff space of dimension at most ˇ ˇ 2 (X, Z) vanishes. Let Y be a closed subtwo such that its second Cech cohomology group H space of X of dimension zero, let A be an AF-algebra and let B be an arbitrary C∗ -algebra. If π : C(X, A) → C(Y, A) is the quotient map and φ : B → C(Y, A) is a ∗-homomorphism, then Cu B ⊕C(Y,A) C(X, A) ∼ = Cu(B) ⊕Lsc(Y,Cu(A)) Lsc X, Cu(A) , in the category Cu.

2864


Proof. Since A is an AF-algebra, A admits an inductive limit decomposition A1 → A2 → · · · → A, where the C∗ -algebras Ai , i = 1, 2, . . . , are finite-dimensional. By [9, Theorem 2] and by (ii) of Proposition 5.18 we have Cu(A) = lim −→ Cu(Ai ) and Lsc(X, Cu(A)) = lim −→ Lsc(X, Cu(Ai )). Consider the following commutative diagram: Cu(C(X, A1 ))

Cu(C(X, A2 ))

...

Cu(C(X, A))

Lsc(X, Cu(A1 ))

Lsc(X, Cu(A2 ))

...

Lsc(X, Cu(A))

where the vertical arrows are given by the Cuntz semigroup morphisms induced by the rank function. These maps are isomorphisms by [23, Theorem 1]. Hence, they induce an isomorphism between the limit semigroups, that is, Cu(C(X, A)) ∼ = Lsc(X, Cu(A)). Since Y is compact and zero-dimensional, and A is AF, we see that C(Y, A) is AF. It thus follows that K1 (I ) = 0 for any ideal I of C(Y, A). The corollary now follows from Theorems 3.1 and 3.3. 2 Corollary 3.7. Let X be a locally compact Hausdorff space that is second countable and onedimensional. Let A be a separable C∗ -algebra with stable rank one such that K1 (I ) = 0 for every closed two-sided ideal I of A. Then, the semigroup Cu(C0 (X, A)) is order-cancellative with respect to . Proof. By Theorem 3.4, α : Cu(C0 (X, A)) → Lsc(X, Cu(A)) is an order-isomorphism. Let [a], [b], [c] ∈ Cu(C0 (X, A)) be such that [a] + [b] [a] + [c]. There exists then > 0 with [a] + [b] [(a − )+ ] + [(c − )+ ]. Applying α we obtain a(x) + b(x) a(x) − + + c(x) − + for all x ∈ X. Using now [28, Theorem 4.3], we conclude that [b(x)] [(c(x) − )+ ] for all x ∈ X. Since α is an order-isomorphism, we get [b] [(c − )+ ], so that [b] [c], as desired. 2 Recall that an element a in an ordered semigroup is compact if a a. Compact elements in Cu(A) are strongly related to equivalence classes of projections (see, e.g., [9,4]). Corollary 3.8. Let X be a locally compact Hausdorff space that is connected, second countable, and one-dimensional. Let A be a separable C∗ -algebra with stable rank one such that K1 (I ) = 0 for every closed two-sided ideal I of A. Then, an element [f ] ∈ Cu(C0 (X, A)) is compact if and only if there exists a compact element a ∈ Cu(A) such that [f (t)] = a, for all t ∈ X.


2865

Proof. Upon identifying Cu(C0 (X, A)) with Lsc(X, Cu(A)), which we may by Theorem 3.4, we assume that f ∈ Lsc(X, Cu(A)) is compact. Let t ∈ X, and write f (t) = supn fn (t) as in Proposition 5.5 where the functions fn are constant in a neighbourhood Vt of t. Then f has a constant value at in a neighbourhood of t. Since f f implies f (t) f (t) for all t ∈ X, we have at at . Further, since X is compact, we can find a finite cover (Vti )ki=1 with associated compact elements ati . It is clear that Vti ∩ Vtj = ∅ if ati = atj , so using the connectedness of X, we find a unique value ati , and so f is constant. 2 4. Examples We now give some examples of the computation of Cu(A) for certain C∗ -algebras. 4.1. Recursive sub-homogeneous algebras The class of Recursive Subhomogeneous Algebras (rsh-algebras) defined in [21] is the smallest class R of C∗ -algebras which contains C(X, Mn ) for all compact Hausdorff spaces X and n 1, and which is closed under isomorphisms and pullbacks of the type A ϕ

C(X, Mn )

ρ

(19)

C(Y, Mn )

where A is in R, ϕ is a unital ∗-homomorphism, Y ⊆ X is a closed subspace of X and ρ is the restriction map. If we restrict to the class of rsh-algebras R constructed using compact Hausdorff spaces of dimension at most one, we can describe their Cuntz semigroup by an iterated use of Corollary 3.5 as Cu(R) ∼ = . . . Lsc(X0 , N) ⊕Lsc(Y1 ,N) Lsc(X1 , N) ⊕Lsc(Y2 ,N) Lsc(X2 , N) . . . ⊕Lsc(Yk ,N) Lsc(Xk , N), where Xi are second countable compact Hausdorff spaces of dimension at most one and Yi ⊆ Xi are closed subsets. Note that N = N ∪ {∞} can be naturally identified with the Cuntz semigroup of Mn . 4.2. Non-commutative CW-complexes Non-commutative CW-complexes introduced by Eilers, Loring and Pedersen in [13] are a particular case of rsh-algebras. A one-dimensional NCCW-complex is the resulting C∗ -algebra pullback of the following diagram

2866


E ϕ

C([0, 1], F )

ρ

(20)

F ⊕F

where E, F are finite-dimensional C∗ -algebras, ϕ is an arbitrary ∗-homomorphism, and ρ is given by evaluation at 0 and 1. One-dimensional NCCW-complexes cover a large amount of C∗ algebras, including dimension drop algebras, spitting interval algebras, and the building blocks used in the classification of one-parameter continuous fields of AF-algebras (see [12]). The classification of inductive limits of one-dimensional NCCW-complexes with trivial K1 -group was carried out in [24] using the functor Cu∼ which is related to the functor Cu. Using Corollary 3.5, the Cuntz semigroup of a one-dimensional NCCW-complex can be computed as the induced pullback of ordered semigroups in Cu. We identify the Cuntz semigroup of a finite-dimensional C∗ -algebra with Nk for some k. Since ϕ : E → F ⊕ F is any C∗ -algebra map, we obtain a semigroup map Cu(ϕ) : Nr → N2s which is thus described by a matrix A ∈ M2s,r (N). Now the map Cu(ρ) : Cu(C([0, 1], F )) → Cu(F ⊕ F ) is given by evaluation at 0 and 1 of lower semicontinuous functions f : [0, 1] → N2s . Therefore, the ordered semigroup pullback is isomorphic to t

(f, b) ∈ Lsc [0, 1], Ns ⊕ Nr f (0), f (1) = Ab , which is thus completely determined by the matrix A. 4.3. Dimension drop algebras over the interval Dimension drop algebras are a particular case of non-commutative CW-complexes. In fact we will consider a slightly more general case since we need not restrict to finite-dimensional algebras. Given two positive integers p, q the dimension drop algebra is defined as

Zp,q = f ∈ C [0, 1], Mp (C) ⊗ Mq (C) f (0) ∈ Mp (C) ⊗ Iq , f (1) ∈ Ip ⊗ Mq (C) , and can be described as the pullback of the following diagram Mp (C) ⊕ Mq (C) φ

C([0, 1], Mp (C) ⊗ Mq (C))

(λ0 ,λ1 )

(Mp (C) ⊗ Mq (C))2

where λi (f ) = f (i) and φ(A, B) = (A ⊗ Iq , Ip ⊗ B). Identifying Cu(Mr (C)) with 1r N we obtain by Corollary 3.5 1 1 1 Cu(Zpq ) ∼ N f (0) ∈ N, f (1) ∈ N . = f ∈ Lsc [0, 1], pq p q


2867

In case p, q are coprime, Zpq is called a prime dimension drop algebra. The Jiang-Su algebra Z is constructed as an inductive limit of different prime dimension drop algebras Zpn qn which is simple and has a unique trace. This construction can be slightly simplified using a unique dimension drop algebra Zpq and a unique morphism γ : Zpq → Zpq but allowing p, q to be supernatural numbers of infinite type, that is to say, pi∞ = pi (see [28]). The construction for Zpq can also be done using a pullback as before but now Mp (C) should be replaced by the corresponding UHF-algebra. The Cuntz semigroup of these UHF-algebras can be computed using the description given in e.g. [5] as 1 N {∞}. := R++ Cp := Cu lim −→ Mn (C); n | p n

(21)

n|p

Hence, observing that Cp , Cq ⊆ Cpq , we have

Cu(Zpq ) ∼ = f ∈ Lsc [0, 1], Cpq f (0) ∈ Cp , f (1) ∈ Cq . 4.4. Dimension drop algebras over a two-dimensional space ˇ 2 (X, Z) = 0, and let Let X be compact Hausdorff space of dimension two such that H x1 , x2 , . . . , xn ∈ X. Given supernatural numbers p1 , p2 , . . . , pn of infinite type, let us consider the dimension drop algebra Zp1 ,p2 ,...,pn =

f ∈ C X,

n

Mpi

f (xi ) ∈ Ip1 ⊗ · · · ⊗ Ipi−1 ⊗ Mpi ⊗ Ipi+1 ⊗ · · · ⊗ Ipn ,

i=1

i = 1, 2, . . . , n . This algebra can be described as the pullback of the following diagram: n

i=1 Mpi φ

C(X,

!n

λ

i=1 Mpi )

Lsc(Y,

!n

i=1 Mpi )

where Y = {x1 , x2 , . . . , xn }, λ(f ) = f |Y , and φ(A1 , A2 , . . . , An ) i = Ip1 ⊗ · · · ⊗ Ipi−1 ⊗ Ai ⊗ Ipi+1 ⊗ · · · ⊗ Ipn , for i = 1, 2, . . . , n. By Corollary 3.6 Cu(Zp1 ,p2 ,...,pn ) ∼ =

n " i=1

Hence, we have

Cpi

⊕Lsc(Y,Cp1 p2 ···pn ) Lsc(X, Cp1 p2 ···pn ).

2868


Cu(Zp1 ,p2 ,...,pn ) ∼ = f ∈ Lsc(X, Cp1 p2 ···pn ) f (xi ) ∈ Cpi , i = 1, 2, . . . , n , where Cp is as in (21). 4.5. Mapping torus of A Let A be a C∗ -algebra and φ : A → A an automorphism. The mapping torus of the pair (A, φ) is defined by Tφ (A) = f ∈ C [0, 1], A f (1) = φ f (0) . Observe that Tφ (A) can be obtained as the pullback in the following diagram Tφ (A)

A (id,φ)

C([0, 1], A)

ρ{0,1}

A ⊕ A.

Therefore, if A has stable rank one and K1 (I ) = 0 for every ideal in A, using Corollary 3.5 we obtain Cu Tφ (A) ∼ = f ∈ Lsc [0, 1], Cu(A) f (1) = Cu(φ) f (0) . 5. Semigroups of lower semicontinuous functions Our aim in this section is to prove that the set Lsc(X, M) of lower semicontinuous functions from a compact Hausdorff space X with finite covering dimension to a countably based semigroup M in Cu, equipped with the pointwise order and addition, is also a semigroup in Cu. Throughout this section, X will always denote a topological space that is second countable, compact, and Hausdorff, whence metrizable. The following lemmas are easy to prove and hence we omit the details. Lemma 5.1. If M is a semigroup in Cu, then f ∈ Lsc(X, M) if and only if, given t ∈ X and a f (t), there exists an open neighbourhood Ut of t such that a f (s), for all s ∈ Ut . Lemma 5.2. Let f ∈ Lsc(X, M) and Y ⊆ X be a closed set. Then: (i) The restriction f |Y of f to Y is a function in Lsc(Y, M). (ii) If g ∈ Lsc(Y, M) and g f |Y , then f↓g (t) := is a function in Lsc(X, M)

g(t) f (t)

if t ∈ Y, otherwise


2869

A source of examples of these functions is obtained by the action of the characteristic functions of open subsets of X in Lsc(X, M). This action is described as follows: given an open set U ⊆ X and a function f ∈ Lsc(X, M), f · χU is defined by (f · χU )(x) :=

f (x) 0

if x ∈ U, otherwise.

This function belongs to Lsc(X, M) by Lemma 5.2. Remark 5.3. If M is a semigroup in Cu, and (an ) is an increasing sequence with a supn an , then there exists m such that a am . Indeed, write supn an = supk bk , for a rapidly increasing sequence (bk ) in M, and find k such that a bk . As bk+1 am for some m, this yields a bk bk+1 am , so a am . Lemma 5.4. Let M be a semigroup in Cu. Then: (i) Lsc(X, M) endowed with the pointwise addition and order is an ordered semigroup. (ii) Lsc(X, M) is closed under (pointwise) suprema of increasing sequences. Proof. (i) Let a ∈ M and t ∈ (f + g)−1 (a ). Let us write f (t) = supn fn and g(t) = supn gn where (fn ), (gn ) are rapidly increasing sequences. Then (f + g)(t) = f (t) + g(t) = supn (fn + gn ). Since a (f + g)(t) = f (t) + g(t) = supn (fn + gn ), there exists N 0 such that a fN + gN . Next, since fN f (t), gN g(t) and f, g ∈ Lsc(X, M), there are open neighbourhoods Ut and Vt of t such that fN f (s) if s ∈ Ut and gN g(s) if s ∈ Vt . Hence, Wt = Ut ∩ Vt is an open neighbourhood of t, and clearly a fN + gN f (s) + g(s) = (f + g)(s), for all s ∈ Wt . Therefore Wt ⊆ (f + g)−1 (a ), which proves that (f + g)−1 (a ) is open, whence (f + g) ∈ Lsc(X, M). (ii) Let (fn ) be an increasing sequence in Lsc(X, M), and put f (t) := supn fn (t). Since M is closed under suprema of increasing sequences, f exists. For any t ∈ X, and a f (t) = supn fn (t), there exists by Remark 5.3 a number N such that a fN (t). Lower semicontinuity of fN now provides a neighbourhood Ut such that a fN (s) f (s) for all s ∈ Ut . Therefore f ∈ Lsc(X, M) and clearly f = supn fn . 2 The following proposition provides a characterization of compact containment in these function spaces. Proposition 5.5. Let M be a semigroup in Cu. Given f, g ∈ Lsc(X, M) we have g f if and only if for every t ∈ X there are an open neighbourhood Ut of t, and ct ∈ M such that g(s) ct f (s) for all s ∈ Ut . Proof. Suppose g f . Given t ∈ X let us write f (t) = supm am where (am ) is a rapidly increasing sequence. Since f ∈ Lsc(X, M) and X is metrizable there exists, for all m, an open using that X is metrizneighbourhood Vm of t such that am f (s) for all s ∈ V m . Moreover, # able we may choose these neighbourhoods to be such that {t} = m Vm .

2870


Consider the following functions in Lsc(X, M), fm (s) :=

am

if s ∈ Vm ,

f (s)

otherwise.

Using Lemma 5.2 we see that fm ∈ Lsc(X, M), and it is clear that f = supm fm . Hence, there exists m0 such that g fm0 , and this inequality proves the result by taking Ut = Um0 and ct = a m 0 . Now suppose the condition holds, and consider an increasing sequence hn ∈ Lsc(X, M) such that f supn hn . For each t ∈ X there are a neighbourhood Vt of t and ct ∈ M such that g(s) ct f (s), for all s ∈ Vt . Thus, g(t) ct f (t) supn hn (t). Hence, there exists nt ∈ N such that ct hnt (t). Lower semicontinuity of hnt now provides a neighbourhood Ut such that ct hnt (s) for all s ∈ Ut . Put Wt = Ut ∩ Vt . Then, g(s) ct hnt (s), whenever s ∈ Wt . By compactness of X, there is a finite cover Wt1 , . . . , Wtk such that g(s) hnti (s) for every s ∈ Wti . Since the sequence (hn ) is increasing, there exists N such that hnti hN for all i and thus g(s) hN (s) for all s ∈ X. We thus have g hN . This implies that g f , as desired. 2 With this characterization at hand, we can now prove that addition in Lsc(X, M) is compatible with compact containment. Corollary 5.6. Let M be a semigroup in Cu. Let f1 , f2 , g1 , g2 ∈ Lsc(X, M) such that f1 g1 and f2 g2 . Then, f1 + f2 g1 + g2 . Proof. Let t ∈ X. As f1 g1 and f2 g2 , we obtain by Proposition 5.5 neighbourhoods Ut and Vt of t, and elements ct , dt ∈ M such that f1 (s) ct g1 (s) for all s ∈ Ut and f2 (s) dt g2 (s) for all s ∈ Vt . Hence, Wt = Ut ∩ Vt is an open neighbourhood of t such that for all s ∈ Wt , (f1 + f2 )(s) = f1 (s) + f2 (s) ct + dt g1 (s) + g2 (s) = (g1 + g2 )(s). A second usage of Proposition 5.5 yields f1 + f2 g1 + g2 .

2

Corollary 5.7. Let M be a semigroup in Cu, f ∈ Lsc(X, M) and Y ⊆ X a closed set. Then, we have: (i) If a f (s) for all s ∈ Y , there exists an open neighbourhood Y ⊆ U such that a f (s) for all s ∈ U . Furthermore a · χV f for all open sets V ⊆ Y . (ii) If g ∈ Lsc(X, M) and g f , then g|Y f |Y . Proof. The first assertion in (i) is a straightforward application of Lemma 5.1. Then, since Y is compact, this can be used to find the open sets required by Proposition 5.5. Finally, (ii) is a consequence of Proposition 5.5. 2


2871

Recall that in a topological space X, the covering dimension is defined as the least n such that any open cover has an open refinement of multiplicity n + 1, or infinity in case this n does not exist. Here, a cover U = {Uλ }λ∈Λ has multiplicity k if every x ∈ X belongs to at most k subsets in U . We proceed to show that, in case that X is finite-dimensional, every function f ∈ Lsc(X, M) can be written as the supremum of a directed set of functions. In relevant situations such set can be taken to be a sequence and that will show that Lsc(X, M) is an object in Cu. We first generalize the step functions defined in the case of X = [0, 1] to an arbitrary space X (cf. Definition 2.4). Notation 5.8. Given a family of open sets U = {Ui }i∈Λ , we write FU ,t , FU ,t , and AU for the sets: FU ,t := {i ∈ Λ | t ∈ Ui },

FU ,t := {i ∈ Λ | t ∈ Ui },

AU := FU ,t , FU ,t t ∈ X .

When clear we will omit U in the notation. Observe that AU is a subset of the power set P(Λ) hence we order it by inclusion. Definition 5.9. Let X be an n-dimensional topological space, M a semigroup in Cu and f ∈ Lsc(X, M). A function g : X → M will be termed a piecewise characteristic function for f if there are: m (i) A family of open sets U = {Ui }m i=1 of X such that {U i }i=1 has multiplicity n + 1. (ii) An ordered map ϕ : AU → M with ϕ(∅) = 0 satisfying, for all t ∈ X:

g(t) = ϕ(FU ,t ) ϕ FU ,t f (t). We will use the notation g := χ(U, ϕ) to refer to such a function. The set of all piecewise characteristic functions for f will be denoted by χ(f ). Lemma 5.10. If f ∈ Lsc(X, M) and g is a piecewise characteristic function for f , then g ∈ Lsc(X, M) and g f . Proof. Consider g = χ({Ui }m 5.9 with g(t) = ϕ(Ft ) ϕ(Ft ) f (t). i=1 , ϕ) as in Definition # Given a g(t), observe that t ∈ Vt := i∈Ft Ui which is an open neighbourhood of t. Then, for all s ∈ Vt we have Fs ⊇ Ft and therefore a g(t) = ϕ(Ft ) ϕ(Fs ) = g(s) for all s ∈ Vt . By Proposition 5.1, this proves that g ∈ Lsc(X, M). Now observe that given t ∈ X, there is a neighbourhood Vt of t that only intersects the sets Ui with i ∈ Ft . Hence Fs ⊆ Ft for all s ∈ Vt . Therefore, g(s) = ϕ(Fs ) ϕ(Ft ) f (t) for all s ∈ Vt . By lower semicontinuity of f we can choose this neighbourhood in such a way that g(s) ϕ(Ft ) f (s) for all s ∈ Vt . This implies by Proposition 5.5 that g f . 2 Lemma 5.11. Let M be a semigroup in Cu and f ∈ Lsc(X, M). Then

f = sup g g ∈ χ(f ) .

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Proof. Given t ∈ X, let us write f (t) = supn an where an an+1 . Given n, lower semicontinuity of f provides us with a neighbourhood Un such that an f (s) for all s ∈ Un . Since X is normal, we can find an open neighbourhood Un such that U n ⊆ Un . Hence, gn = an · χUn = χ(Un , ϕn ), with Un = {Un }, ϕ(∅) = 0 and ϕ({n}) = an , is a piecewise characteristic function for f with gn (t) = an . If now h ∈ Lsc(X, M) is such that h g for all g ∈ χ(f ), then in particular h(t) gn (t). Thus h(t) an for each n, that is, h(t) f (t). Since the supremum in Lsc(X, M) is the pointwise supremum, we obtain that f is the supremum of its piecewise characteristic functions. 2 The previous lemma describes each element f in Lsc(X, M) as the supremum of piecewise characteristic functions that are compactly contained in f . But to prove that Lsc(X, M) is an object in Cu, we need to write f as the supremum of a sequence. In order to prove this, we will first show that the set of piecewise characteristic functions form a directed set, and that it can be chosen to be countable if furthermore M is countably based. To this end, we shall use induction on the dimension of the space. The key of the inductive step is encoded in the following lemma. Lemma 5.12. Let Y ⊂ X be a closed set, let f ∈ Lsc(X, M), and let g ∈ Lsc(Y, M) be a piecewise characteristic function for f |Y . Then, there exists a piecewise characteristic function h for f such that g h|Y . Moreover, if g = χ(W, φ), then for every δ > 0, h can be constructed as h = χ(W , φ ), where 0 < < δ and W consists of the -neighbourhoods of the elements of W, and φ is a restriction of φ. Proof. Suppose g = χ(W, φ) where W = {Wi }ki=1 . For every > 0 and i = 1, . . . , k, we denote by Wi the -neighbourhood of Wi . That is, Wi = {x ∈ X | d(x, Wi ) < }. We claim that can be chosen in such a way that AW ⊇ AW . Observe that for all > 0 and t ∈ X, we have FW ,t ⊇ FW ,t ⊇ FW ,t . For each j such that t ∈ / Wj , there exists δj > 0, such that t ∈ / Wjδ for all δ δj . Hence, for each t there exists t > 0 such that FW t ,t = FW t ,t = FW ,t and hence there is a neighbourhood Vt of t such that FW t ,s = FW t ,s = FW ,t for all s ∈ Vt . Since X is compact, we can find a finite number of such neighbourhoods Vti covering X and hence there is 0 < < ti for which we will have AW ⊇ AW . In this situation, {Wi } has the same multiplicity as {Wi }. # Now, for each F ∈ AW and t ∈ i∈F Wi we have F ⊆ Ft , and therefore φ(F ) # φ(Ft ) f (t). Hence φ(F ) f (t) for all t ∈ i∈F Wi = WF . Since WF is closed in X, by Corollary 5.7 there is an open neighbourhood U# F of WF (in X) such that φ(F ) f (t). Thus we can make > 0 smaller if necessary so that i∈F Wi ⊂ UF , and in such a way that this is true for all F ∈ AW . Then, φ(F ) f (t)

for all t ∈

$

Wi .

i∈F

Now, since AW ⊇ AW , we consider h = χ(W , φ ) where φ = φ|AW .

(22)


Given t ∈ X we have t ∈

#

i∈FW ,t

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Wi . Thus, by (22) we have

h(t) = φ (FW ,t ) φ FW ,t f (t), proving that h is a piecewise characteristic function for f . Finally, it is clear that given t ∈ Y , FW ,t ⊆ FW ,t ∈ AW ⊆ AW . Hence, g(t) = φ(FW ,t ) φ(FW ,t ) = φ (FW ,t ) = h(t), implying h|Y g. The last assertion is clear by the construction of h. 2 Proposition 5.13. Let M be a semigroup in Cu. Let f , g1 and g2 ∈ Lsc(X, M) and suppose that g1 , g2 f . Then, there exists h ∈ χ(f ) such that g1 , g2 h. In particular, χ(f ) is an upwards directed set. Proof. Let η > 0. We will prove, by induction on the dimension of X, that there exists an open cover U of X and h = χ(U, ϕ) such that g1 , g2 h f and each open set Ui ∈ U is contained in an η-ball. By Proposition 5.5, for any t ∈ X we can find an open neighbourhood Vt of t and elements at , bt ∈ M such that g1 (s) at f (t) and g2 (s) bt f (t) for all s ∈ Vt . We may further assume that each Vt is a δt -ball with center in t for some 0 < δt < η. As elements compactly contained in f (t) form a directed set, there exists ct ∈ M such that at , bt ct f (t). Now, since f is also lower semicontinuous, we can choose the previous neighbourhoods in such a way that ct f (s) for all s ∈ Vt , and therefore g1 (s), g2 (s) ct f (s) for all s ∈ Vt . Now let Vt be a δt /2-ball with center t, so that g1 (s), g2 (s) ct f (s)

for all s ∈ Vt .

By compactness, there exists a finite cover for X of the form V = {Vti }ki=1 . In case X has dimension 0 this cover has a finite disjoint refinement, which means we can assume V is a finite cover of disjoint clopen sets. Hence, AV = {{1}, . . . , {k}}, and we can consider the piecewise characteristic function h := χ(V, ϕ), where ϕ({i}) = cti . By construction each Vti is contained in an η-ball, and it is not difficult to check that g1 , g2 h. Now suppose dim X = n 1 and that the result holds true for spaces of smaller dimension. Retain the construction of Vti , Vt i , cti as before. Using [14, 4.2.2], we may assume without loss of generality that the boundary, Y = ki=1 bd(Vti ) has dimension at most n − 1. Let δ = min{δti /3} and so we have Vtδi Vt i for all i = 1, . . . , k, where Vtδi is a δ-neighbourhood of Vti . Since Y ⊆ X is a closed set, f |Y ∈ Lsc(Y, M) (by Lemma 5.2). For all i = 1, . . . , k, put VtYi = Y ∩ Vtδi and we have g1 (s), g2 (s) cti f (s) for all s ∈ VtYi . Hence we have cti χV Y f |Y for all i by Corollary 5.7(i). By induction, there exists an open ti

cover W = {Wj }rj =1 of Y , with each Wj contained in a δ/3-ball, and a piecewise characteristic function gY = χ(W, φ) for f |Y such that cti χV Y gY f |Y ti

for all i = 1, . . . , k.

(23)

Observe that whenever Wj ∩ Vti = ∅, we have Wj ⊆ Vtδi ⊆ Vt i . Now we use Lemma 5.12 to obtain a piecewise characteristic function h = χ(W , φ) for f such that gY h |Y . Decreasing

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if necessary, we can further assume that each Wj is contained in a δ/2-ball, and hence Wj ⊆ Vtδi ⊆ Vt i whenever Wj ∩ Vti = ∅. Let Y = rj =1 Wj . Put Ui = Wi for each Fj ∈ AV = {F1 , . . . , Fr }, let#Ur+j be an /3# i = 1, . . . , r. For neighbourhood of i∈Fj Vti \(Y ∪ ( k ∈F / j Vtk )). Observe that Ur+j ⊆ i∈Fj Vti , and that Ur+l ∩ Ur+l = ∅ for all l = l . Next consider the cover U = {U1 , . . . , Ur , Ur+1 , . . . , Ur+r }. Observe that if F ∈ AU , then either F ∈ AW or F = F ∪ {r + l} for some F ∈ AW , or else F = {r + l}. Since W has multiplicity at least n, we see that U has multiplicity at least n + 1. Let ϕ : AU → M be defined by ⎧ ⎪ ⎨ φ(F ) ϕ(F ) = φ(F ) ⎪ ⎩c ti1

if F ∈ AW , if F = F ∪ {r + l} for some l 1, and F ∈ AW , if F = {r + l}, and Fl = {i1 < · · · < ikl }

and let h = χ(U, ϕ). We claim that h is a piecewise characteristic function for f such that g1 , g2 h. (i) ϕ is an ordered map. By definition of AU , we only have to consider the following cases: F ⊆ F ,

F ∪ {r + l} ⊆ F ∪ {r + l},

F ⊆ F ∪ {r + l},

{r + l} ⊆ F ∪ {r + l},

where F ⊆ F ∈ AW . By definition of ϕ and since φ is an ordered map, the only non-trivial case is {r + l} ⊆ F ∪ {r + l}. # Then, suppose Fl = {i1 < · · · < ikl }, and FU ,t = F ∪ {r + l}. This means t ∈ ( j ∈F Wj ) ∩ # ( kjl=1 Vtij ). There exists tY ∈ Y such that F = FW ,tY . Given j ∈ F , since t ∈ Wj ∩ Vti1 = ∅,

we have Wj ⊆ Vt i , whence tY ∈ VtYi . Therefore, using (23), we have 1

1

ϕ {r + l} = cti1 = cti1 · χV Y (tY ) ti 1

gY (tY ) h (tY ) = φ(FW ,tY ) φ FW = φ F = ϕ F ∪ {r + l} , ,t Y proving that ϕ is an ordered map. (ii) h is a piecewise characteristic function for f . Let t ∈ X. If t ∈ Y , then FU ,t equals FW ,t or FW ,t ∪ {r + l}. Hence ϕ(FU ,t ) = φ(FW ,t ) f (t) since h is a characteristic function for f . Otherwise, if t ∈ / Y , then t ∈ Ur+l for some l 1, and FU ,t = FU ,t = {r + l}, where Fl = {i1 < · · · < il }. In particular, t ∈ Vt1 , therefore ct1 f (t), and we have ϕ FU ,t = ϕ {r + l} = ct1 f (t). This proves that h ∈ χ(f ).


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(iii) g1 h and g2 h. Let t ∈ X. If t ∈ Wj for some j , since Wj ⊆ Vt i for some i, we have g1 (s), g2 (s) cti gY (s) h(s) / Y , we have t ∈ Ur+l ⊆ Vti1 , for some l and i1 the first element for all s ∈ Wj . Otherwise if t ∈ in Fl . Hence for all s ∈ Ur+l , g1 (s), g2 (s) ct1 = ϕ {r + l} ϕ(FU ,s ) = h(s), proving that g1 , g2 h.

2

Proposition 5.14. Let X be a finite-dimensional topological space, let M be an object in Cu, and let f ∈ Lsc(X, M). If M is countably based, then f is the supremum of a rapidly increasing sequence of elements from χ(f ). Proof. For any function h : X → M, put

Uh = (t, a) ∈ X × M a h(t) , which is an open set when h is lower semicontinuous. We know from Lemma 5.11 that f = sup{g | g ∈ χ(f )}, whence Uf = g∈χ(f ) Ug . Since by assumption M is countably based, X × M has a countable basis, and so does U f . As these spaces have the Lindelöf property, there is a sequence (gn ) in χ(f ) such that Uf = Ugn . The sequence (gn ) may be taken to be rapidly increasing by virtue of Proposition 5.13. This implies that f = sup gn , as was to be shown. 2 Assembling the results above, we obtain the following: Theorem 5.15. Let X be a second countable finite-dimensional compact Hausdorff topological space, and let M be an object in Cu. If M is countably based, then Lsc(X, M) (with the pointwise order and addition) is also a semigroup in Cu. Proof. Combine Lemma 5.4, Corollary 5.6 and Proposition 5.14.

2

We now proceed to study some functorial properties of Lsc. Lemma 5.16. Let X, Y be finite-dimensional second countable compact Hausdorff spaces and M, N be countably based semigroups in Cu: (i) If f : X → Y is a (proper) continuous map then, Lsc(f, M) : Lsc(Y, M) → Lsc(X, M), g → g ◦ f is a map in Cu.

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(ii) If α : M → N is a map in Cu, then Lsc(X, α) : Lsc(X, M) → Lsc(X, N ), g → α ◦ g, is also a map in Cu. Proof. (i) It is easy to see that the map is well defined since f is continuous. It also preserves order, addition and suprema, since those are defined pointwise. To prove preservation of compact containment, we will use Proposition 5.5. Assume g1 g2 . For each t ∈ X, since f (t) ∈ Y and g1 g2 , there exist cf (t) ∈ M and Vf (t) an open neighbourhood of f (t) such that g1 (s) cf (t) g2 (s) for all s ∈ Vf (t) . Hence, (g1 ◦ f )(s) cf (t) (g2 ◦ f )(s) for all s ∈ f −1 (Vf (t) ) which is an open neighbourhood of t. Therefore g1 ◦ f g2 ◦ f . (ii) Let us first see that Lsc(X, α) is well defined, which is to say, Lsc(X, α)(g) ∈ Lsc(X, N ) for all g ∈ Lsc(X, M). Let g ∈ Lsc(X, M) be fixed, and let x ∈ X and a ∈ N be such that a Lsc(X, α)(g)(x). Choose a rapidly increasing sequence (bn )n∈N in M such that supn bn = f (x). Since a Lsc(X, α)(g)(x) = supn α(bn ) there exists n0 1 such that a α(bn0 ). Since g is lower semicontinuous and bn0 g(x) there exists a neighbourhood U of x such that bn0 g(y), for all y ∈ U . Hence, it follows that a α(bn0 ) α g(y) = Lsc(X, α)(g)(y), for all y ∈ U . Since x and a are arbitrary this implies that Lsc(X, α)(g) ∈ Lsc(X, N ). It is clear that Lsc(X, α) preserves the zero element, the order, and suprema of increasing sequences since α does. To complete the proof let us show that Lsc(X, α) preserves compact containment. Let g, h ∈ Lsc(X, M) be such that g h. By Proposition 5.5 for all x ∈ X there exist cx ∈ M and a neighbourhood U of x such that g(y) cx h(y), for all y ∈ U . It follows that Lsc(X, α)(g)(y) = α g(y) α(cx ) α h(y) = Lsc(X, α)(h)(y), for all y ∈ U . Therefore, by applying Proposition 5.5 again we conclude that Lsc(X, α)(g) Lsc(X, α)(h). 2 As a consequence of Lemma 5.16 and Theorem 5.15 we obtain Theorem 5.17. Let X be a finite-dimensional second countable compact Hausdorff space and let M be a countably based semigroup in Cu. Then Lsc(·, M) defines a contravariant functor from the category of finite-dimensional, second countable, compact Hausdorff topological spaces to Cu, and Lsc(X, −) defines a covariant functor from the category of countably based semigroups in Cu to Cu. This functor is seen to be sequentially continuous in the following cases:


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Proposition 5.18. (i) Let (Xi , μi,j )i,j ∈N be an inverse system of compact Hausdorff one-dimensional spaces with surjective maps μi,j : Xj → Xi for j i. If M is a countably based semigroup in Cu, then Lsc lim ←−(Xi , μi,j ), M = lim −→ Lsc(Xi , M), Lsc(μi,j , M) . (ii) Let (Mi , αi,j )i,j ∈N be a directed system of countably based semigroups in Cu. If X is a second countable, compact, Hausdorff space, then, Lsc X, lim −→(Mi , αi,j ) = lim −→ Lsc(X, Mi ), Lsc(X, αi,j ) . Proof. Recall that the category Cu has limits of inductive sequences (see, e.g., [9]), and that given a directed set (Si , γi,j ), a semigroup S with maps γi : Si → S is the directed limit lim −→(Si , γi,j ) if and only if the following two conditions are satisfied: (a) For all s ∈ S, s = supi γi (si ) for some si ∈ Si . (b) If si ∈ Si and sj ∈ Sj are such that γi (si ) γj (sj ), then, for all x si there exists k ∈ N such that γi,k (x) γj,k (sj ). (i) Let X = lim ←−(Xi , μi,j )i,j ∈N with μi : X → Xi the canonical maps. Note that X is also a one-dimensional compact Hausdorff space and that the canonical maps μi are all surjective. The open sets in X can be described as i∈N μ−1 i (Ui ) where each Ui is an open set in Xi . −1 Furthermore, the Ui ’s can be chosen in such a way that μ−1 i (Ui ) ⊆ μj (Uj ) if i j (see, e.g., [18, Propositions 1-7.1 and 1-7.5]). By Theorem 5.15, both Lsc(Xi , M) and Lsc(X, M) are objects in Cu. Therefore, using Theorem 5.17, we obtain a directed system (Lsc(Xi , M), ρi,j )i,j ∈N in Cu, with maps ρi : Lsc(Xi , M) → Lsc(X, M) given by ρi (f ) = f μi , and ρi,j : Lsc(Xi , M) → Lsc(Xj , M) given by ρi,j (g) = gμi,j whenever i j . To prove condition (a) above, since f ∈ Lsc(X, M) can be described as the supremum of a sequence in χ(f ), we may assume that f itself is a piecewise characteristic function. Hence suppose f = χ(U, φ) where U = {Uj }rj =1 is a family of open sets such that U = {U j }rj =1 has multiplicity at most one, and φ : AU → M is an ordered map. Let us write each Uj as −1 −1 −1 i∈N μi (Uj,i ) for some open sets Uj,i in Xi and such that μi (Uj,i1 ) ⊆ μi (Uj,i2 ) if i1 i2 . −1 r Now, for each k 1 we consider the family of open sets Uk = {μk (Uj,k )}j =1 . We observe that both Uk and Uk have multiplicity at most one. For each k 1 we consider the map φk : {FUk ,t | t ∈ X} → M, φ(F ) F → 0

if F ∈ AU , otherwise.

and fk : X → M defined by fk (t) = φk (FUk ,t ). Following the proof of Lemma 5.10, if φk is an ordered map then fk is lower semicontinuous. But since φ is already an ordered map we only need to check the case F1 ⊂ F2 with F2 ∈ / AU and F1 ∈ AU . In this case, since F2 = FUk ,t for

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some t, there exists some F3 ∈ AU such that F2 ⊂ F3 . But U has multiplicity at most one and therefore subsets in AU have at most two elements. Therefore F3 = {i, j }, F2 = {i} and F1 = ∅. Thus we obtain φk (F1 ) = φk (∅) = 0 φk (F2 ). For all t ∈ X, there exists k ∈ N such that FUk ,t = FU ,t . Hence, for all t ∈ X there exists k such that f (t) = φ(FU ,t ) = φk (FUk ,t ) = fk (t), and since supremum in Lsc(X, M) is the pointwise supremum, we obtain f = supk∈N fk . Now we consider Vk = {Uj,k }rj =1 . Then

t ∈ μ−1 k (Uj,k ) if and only if μk (t) ∈ Uj,k . It follows that FVk ,μk (t) = FUk ,t , and hence, taking gk (s) = φk (FVk ,s ), we have gk ∈ Lsc(Xk , M) and fk = ρk (gk ) since fk (t) = φk (FUk ,t ) = φk (FVk ,μk (t) ) = gk μk (t). We now prove condition (b). Suppose that for some i j , gi ∈ Lsc(Xi , M) and gj ∈ Lsc(Xj , M) are such that ρi (gi ) ρj (gj ). Let h gi and hence ρi (h) ρi (gi ) ρj (gj ) which implies ρi (h) ρj (gj ). Using Proposition 5.5, there is for each t ∈ X an open neighbourhood Vt of t and ct ∈ M such that ρi (h)(s) ct ρj (gj )(s)

for all s ∈ Vt .

We may assume Vt = μ−1 it (Ut ) for some open set Ut ⊆ Xit , where it i, j . Then, for all s ∈ Vt , ρi (h)(s) ct ρj (gj )(s) ⇔

⇔

hμi (s) ct gj μj (s)

hμit ,i μit (s) ct gj μit ,j μit (s),

and hence, for all s ∈ Ut , hμit ,i s ct gj μit ,j s

⇔

ρit ,i h s ct ρit ,j gj s .

(24)

rUsing compactness of X, we obtain a finite number of open sets Vti such that X = i=1 Vti , and we can moreover choose it1 = · · · = itr = k for some k ∈ N. Since μk is surjective then Xk = Ut1 ∪ · · · ∪ Utr which, together with (24) and Proposition 5.5 proves ρk,i (h) ρk,j (gj ). (ii) Let M = lim −→(Mi , αi,j ) with αi : Mi → M the canonical maps. Let f, g ∈ Lsc(X, Mi ) be such that Lsc(X, αi )(f ) Lsc(X, αi )(g), and let h f . By Proposition 5.5 for all x ∈ X there exist cx ∈ M and a neighbourhood U of x such that h(y) cx f (y), for all y ∈ U . We have αi f (x) = Lsc(X, αi )(f )(x) Lsc(X, αi )(g)(x) = αi g(x) . Hence, since cx f (x) by the characterization of inductive limits in the category Cu that there exists j i such that αi,j (cx ) αi,j g(x) = Lsc(X, αi,j )(g)(x). By the lower semicontinuity of the function Lsc(X, αi,j )(g) there exists a neighbourhood V of x such that αi,j (cx ) Lsc(X, αi,j )(g)(y), for all y ∈ V . Since h(y) cx for all y ∈ U it follows that


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Lsc(X, αi,j )(h)(y) = αi,j h(y) αi,j (cx ) Lsc(X, αi,j )(g)(y), for all y ∈ U ∩ V . We have shown that for each x ∈ X there exist j i and a neighbourhood W of X such that Lsc(X, αi,j )(h)(y) Lsc(X, αi,j )(g)(y), for all y ∈ W . Therefore, by the compactness of X we may choose j i such that Lsc(X, αi,j )(h)(y) Lsc(X, αi,j )(g)(y), for all y ∈ X. It follows now that Lsc(X, αi,j )(h) Lsc(X, αi,j )(g). This proves condition (b). Observe that condition (a) above is equivalent to saying that i Lsc(X, αi )(Lsc(X, Mi )) forms a dense subset in Lsc(X, M). Let g1 , g2 ∈ Lsc(X, M) be such that g1 g2 . By Proposition 5.13 (and its proof), there exists a piecewise characteristic function h = χ(U, ϕ) such that g1 h g2 whose range can be chosen in a dense subset of M. Since i1 αi (Mi ) forms a dense subset of M, and a piecewise characteristic function takes a finite number of values in M, we can find k 1 such that furthermore ϕ(AU ) ⊆ αk (Mk ). For each F ∈ AU let us write ϕ(F ) = αk (cF ) for some cF ∈ Mk . Recall that ϕ is an ordered map and, in fact, by the proof of Proposition 5.13, we actually have αk (cF ) αk (cF ) whenever F F . Let us write each cF as the supremum of a rapidly increasing sequence in Mk , cF = supi cFi . Hence, i for each N 0 there exists iN such that iN N and αk (cF ) αk (cFN ) whenever F F . Now, using that M = lim −→i Mi and since AU has a finite number of inequalities, we can choose lN k such that αk,lN (cFN ) αk,lN (cFN ) for all F F . Therefore ϕN (F ) := αk,lN (cFN ) defines an ordered map ϕN : AU → MlN , and hN := χ(U, ϕN ) is a piecewise characteristic function in Lsc(X, MlN ). It follows that h = supN Lsc(X, αlN )(hN ). Since g1 h g2 there exists N0 such that g1 Lsc(X, αlN0 )(hN0 ) g2 , proving that the images Lsc(X, αi )(Lsc(X, Mi )) form a dense subset in Lsc(X, M). 2 i

i

i

Acknowledgments This work has been partially supported by an MEC-DGESIC grant (Spain) through Project MTM2008-06201-C02-01/MTM, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. References [1] B. Blackadar, A. Tikuisis, A.S. Toms, L. Robert, W. Winter, An algebraic approach to the radius of comparison, Trans. Amer. Math. Soc. (2011), in press. [2] E. Blanchard, Déformations de C∗ -algèbres de Hopf, Bull. Soc. Math. France 124 (1) (1996) 141–215. [3] L.G. Brown, Stable isomorphism of hereditary subalgebras of C∗ -algebras, Pacific J. Math. 71 (2) (1977) 335–348. [4] N.P. Brown, A. Ciuperca, Isomorphism of Hilbert modules over stably finite C∗ -algebras, J. Funct. Anal. 257 (1) (2009) 332–339. [5] N.P. Brown, F. Perera, A.S. Toms, The Cuntz semigroup, the Elliott conjecture, and dimension functions on C∗ algebras, J. Reine Angew. Math. 621 (2008) 191–211. [6] N.P. Brown, A.S. Toms, Three applications of the Cuntz semigroup, Int. Math. Res. Not. IMRN 2007 (19) (2007), Art. ID rnm068, 14 pp. [7] A. Ciuperca, G.A. Elliott, A remark on invariants for C∗ -algebras of stable rank one, Int. Math. Res. Not. IMRN 2008 (5) (2008), Art. ID rnm 158, 33 pp. [8] A. Ciuperca, G.A. Elliott, L. Santiago, On inductive limits of type I C∗ -algebras with one-dimensional spectrum, Int. Math. Res. Not. IMRN (2010), doi:10.1093/imrn/rnq157. [9] K.T. Coward, G.A. Elliott, C. Ivanescu, The Cuntz semigroup as an invariant for C∗ -algebras, J. Reine Angew. Math. 623 (2008) 161–193. [10] J. Cuntz, Dimension functions on simple C∗ -algebras, Math. Ann. 233 (2) (1978) 145–153.

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[11] M. Dadarlat, Continuous fields of C∗ -algebras over finite-dimensional spaces, Adv. Math. 222 (5) (2009) 1850– 1881. [12] M. Dadarlat, G.A. Elliott, Z. Niu, One-parameter continuous fields of Kirchberg algebras II, Canad. J. Math. (2011) doi:10.4153/CJM-2011-001-6. [13] S. Eilers, T.A. Loring, G.K. Pedersen, Stability of anticommutation relations: an application of noncommutative CW-complexes, J. Reine Angew. Math. 499 (1998) 101–143. [14] R. Engelking, Dimension Theory, North-Holland Math. Library, vol. 19, North-Holland Publishing Co., Amsterdam, 1978, translated from the Polish and revised by the author. [15] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, Continuous Lattices and Domains, Encyclopedia Math. Appl., vol. 93, Cambridge University Press, Cambridge, 2003. [16] E. Kirchberg, M. Rørdam, Non-simple purely infinite C ∗ -algebras, Amer. J. Math. 122 (3) (2000) 637–666. [17] E. Kirchberg, M. Rørdam, Infinite non-simple C∗ -algebras: absorbing the Cuntz algebras O∞ , Adv. Math. 167 (2) (2002) 195–264. [18] A.R. Pears, Dimension Theory of General Spaces, Cambridge University Press, Cambridge/England, 1975. [19] G.K. Pedersen, Unitary extensions and polar decompositions in a C ∗ -algebra, J. Operator Theory 17 (2) (1987) 357–364. [20] G.K. Pedersen, Pullback and pushout constructions in C∗ -algebra theory, J. Funct. Anal. 167 (2) (1999) 243–344. [21] N.C. Phillips, Recursive subhomogeneous algebras, Trans. Amer. Math. Soc. 359 (10) (2007) 4595–4623 (electronic). [22] M.A. Rieffel, The homotopy groups of the unitary groups of noncommutative tori, J. Operator Theory 17 (2) (1987) 237–254. [23] L. Robert, The Cuntz semigroup of some spaces of dimension at most 2, preprint, arXiv:0711.4396, 2007. [24] L. Robert, Classification of inductive limits of 1-dimensional NCCW complexes, preprint, arXiv:1007.1964v1, 2010. [25] L. Robert, L. Santiago, Classification of C ∗ -homomorphisms from C0 (0, 1] to a C ∗ -algebra, J. Funct. Anal. 258 (3) (2010) 869–892. [26] L. Robert, A. Tikuisis, Hilbert C∗ -modules over a commutative C∗ -algebra, Proc. London Math. Soc. 102 (2) (2011) 229–256. [27] M. Rørdam, On the structure of simple C∗ -algebras tensored with a UHF-algebra. II, J. Funct. Anal. 107 (2) (1992) 255–269. [28] M. Rørdam, W. Winter, The Jiang-Su algebra revisited, J. Reine Angew. Math. 642 (2010) 129–155. [29] L. Santiago, A classification of inductive limits of splitting interval algebras, preprint, arXiv:1011.6559v1, 2010. [30] A. Tikuisis, The Cuntz semigroup of continuous functions into certain simple C∗ -algebras, Int. J. Math. (2011), in press. [31] A.S. Toms, On the classification problem for nuclear C ∗ -algebras, Ann. of Math. (2) 167 (3) (2008) 1029–1044. [32] N.E. Wegge-Olsen, K-Theory and C∗ -Algebras, Oxford Sci. Publ., The Clarendon Press/Oxford University Press, New York, 1993.


Sharp Gårding inequality on compact Lie groups Michael Ruzhansky a,∗,1 , Ville Turunen b a Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom b Aalto University, Institute of Mathematics, P.O. Box 1100, FI-00076 Aalto, Finland

Received 3 September 2010; accepted 16 February 2011 Available online 23 February 2011 Communicated by P. Delorme

Abstract We establish the sharp Gårding inequality on compact Lie groups. The positivity condition is expressed in the non-commutative phase space in terms of the full matrix symbol, which is defined using the representations of the group. Applications are given to the L2 and Sobolev boundedness of pseudo-differential operators. © 2011 Elsevier Inc. All rights reserved. Keywords: Pseudo-differential operators; Compact Lie groups; Microlocal analysis; Gårding inequality

1. Introduction The sharp Gårding inequality on Rn is one of the most important tools of the microlocal analysis with numerous applications in the theory of partial differential equations. Improving on the m (Rn ) and p(x, ξ ) 0, original Gårding inequality in [6], Hörmander [7] showed that if p ∈ S1,0 then Re p(x, D)u, u L2 −Cu2H (m−1)/2

(1)

holds for all u ∈ C0∞ (Rn ). The scalar case was also later extended to matrix-valued operators by Lax and Nirenberg [11], Friedrichs [5] and Vaillancourt [22]. Further improvements on the * Corresponding author.

E-mail addresses: [email protected] (M. Ruzhansky), [email protected] (V. Turunen). 1 The first author was supported in part by the EPSRC Leadership Fellowship EP/G007233/1.


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lower bound in the scalar case were also obtained by Beals and Fefferman [1] and Fefferman and Phong [4]. Notably, the sharp Gårding inequality (1) requires the condition p(x, ξ ) 0 imposed on the full symbol. This is different from the original Gårding inequality for elliptic operators which can be readily extended to manifolds. The main difficulty in obtaining (1) in the setting of manifolds is that the full symbol of an operator cannot be invariantly defined via its localisations. While the standard localisation approach still yields the principal symbol and thus the standard Gårding inequality, it cannot be extended to produce an improvement of the type in (1). Nevertheless, for pseudo-differential operators P ∈ Ψ 2 (M) on a compact manifold M, under certain geometric restrictions on the characteristic variety of the principal symbol p2 0 and certain hypothesis on p1 , Melin [13] and Hörmander [8] obtained a lower bound known as the Hörmander–Melin inequality. See also Taylor [20]. The aim of the present paper is to establish the lower bound (1) on any compact Lie group G, with the statement given in Theorem 2.1. On compact Lie groups, the non-commutative analogue where G is the unitary dual of G. We use a global quantization of of the phase space is G × G, operators on G consistently developed by the authors in [18] and [16]. For a continuous linear operator A : C ∞ (G) → D (G) it produces a full matrix-valued symbol σA (x, ξ ) defined for Thus, in Theorem 2.1 we will show the lower bound (1) under the assumption (x, [ξ ]) ∈ G × G. that the full symbol satisfies σA 0, i.e. when the matrices σA (x, ξ ) are positive for all (x, [ξ ]) ∈ In general, if a full symbol is positive in the phase space, the corresponding pseudoG × G. differential operator does not have to be positive in the operator sense. However, it still has lower bounds like the one in (1). An important example is the group SU(2) ∼ = S3 , with the group operation (matrix product) in SU(2) corresponding to the quaternionic product in S3 . Details of the global quantization have been worked out in [16,18]. We note that the standard Gårding inequality on compact Lie groups was derived in [2] using Langlands’ results for semigroups on Lie groups [10], but no quantization yielding full symbols is required in this case because of the ellipticity assumed on the operator. The global quantization used in [18] and [16] will be briefly reviewed in Section 3. We note that it is different from the one considered by Taylor [21] because we work directly on the group without referring to the exponential mapping and the symbol classes on the Lie algebra. We note that one of the assumptions for the Hörmander–Melin inequality to hold is the vanishing of the principal symbol p2 0 on the set {p2 = 0} to exactly second order. Thus, for example, it does not apply to operators of the form −∂X2 plus lower order terms, where ∂X is the derivative with respect to a vector field X, unless dim G = 1. For higher order operators, again, the operator ∂X4 − LG , with the bi-invariant Laplace operator LG , gives an example when the Hörmander–Melin inequality does not work while the full matrix-valued symbol is positive definite, so that Theorem 2.1 applies. The relaxation of the transversal ellipticity has been analysed recently by Mughetti, Parenti and Parmeggiani, and we refer to [14] for further details on this subject. A usual proof of (1) in Rn relies on the Friedrichs symmetrisation of an operator done in the frequency variables (see [7,11,5,9,20]). This does not readily work in the setting of Lie groups forms only a lattice which does not behave well enough for this type of because the unitary dual G arguments. Thereby our construction uses mollification in x-space, more resembling those used by Calderón [3] or Nagase [15] for the proof of the sharp Gårding inequality in Rn . Other proofs, e.g. using the anti-Wick quantisation, are also available on Rn , see [12] and references therein. We would also like to point out that the proof of Nagase [15] can be extended to prove (1) on the torus Tn under the assumption that the toroidal symbol p(x, ξ ) of the operator P ∈ Ψ m (Tn )


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satisfies p(x, ξ ) 0 for all x ∈ Tn and ξ ∈ Zn . The toroidal quantization necessary for this proof was developed by the authors in [17] but we will not give such a proof here because such result is now included as a special case of Theorem 2.1 which covers the non-commutative groups as well. The system as well as (ρ, δ) versions of the sharp Gårding inequality will appear elsewhere. The proof of Theorem 2.1 consists of approximating the operator A with non-negative symbol σA by a positive operator P . Although this approximation has a symbol of type (1, 1/2) and not of type (1, 0), it is enough to prove Theorem 2.1 due to additional cancellations in the error terms, ensured by the construction. We note that working with symbol classes of type (1, 1/2) is a genuine global feature of the proof and of our construction because the operators of such type cannot be defined in local coordinates. As usual, for a compact Lie group G we denote by Ψ m (G) the Hörmander pseudo-differential operators on G, i.e. the class of operators which in all local coordinate charts give operators in Ψ m (Rn ). Operators in Ψ m (Rn ) are characterised by the symbols satisfying α β ∂ ∂ a(x, ξ ) C 1 + |ξ | m−|α| ξ x

for all multi-indices α, β and all x, ξ ∈ Rn . An operator in Ψ m (G) is called elliptic if all of its localisations are locally elliptic. Here and in the sequel we use the standard notation for the μ multi-indices α = (α1 , . . . , αμ ) ∈ N0 , where μ may vary throughout the paper depending on the context. The paper is organised as follows. In Section 2 we introduce the full matrix-valued symbols and state the sharp Gårding inequality in Theorem 2.1. We apply it in Corollaries 2.2 and 2.3 to the L2 boundedness of pseudo-differential operators. In Section 3 we collect facts necessary for the proof, and develop an expansion of amplitudes of type (ρ, δ) required for our analysis. In Section 4 we approximate operators with positive symbols by positive operators and derive the error estimates. The authors would like to thank Jens Wirth for discussions and a referee for useful remarks. 2. Sharp Gårding inequality Let G be a compact Lie group of dimension n with the neutral element e. Its Lie algebra will denote the unitary dual of G, i.e. set of be denoted by g. We now fix the necessary notation. Let G all equivalence classes of (continuous) irreducible unitary representations of G and let Rep(G) be the set of all such representations of G. For f ∈ C ∞ (G) and ξ ∈ Rep(G), let f(ξ ) =

f (x)ξ(x)∗ dx

G

be the (global) Fourier transform of f , where integration is with respect to the normalised Haar measure on G. For an irreducible unitary representation ξ : G → U(Hξ ) we have the linear operator f(ξ ) : Hξ → Hξ . Denote by dim(ξ ) the dimension of ξ , dim(ξ ) = dim Hξ . If ξ is a is discrete and all matrix representation, we have f(ξ ) ∈ Cdim(ξ )×dim(ξ ) . Since G is compact, G

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of its elements are finite dimensional. Consequently, by the Peter–Weyl theorem we have the Fourier inversion formula f (x) =

dim(ξ ) Tr ξ(x)f(ξ ) .

[ξ ]∈G

The Parseval identity takes the form f 2L2 (G) =

[ξ ]∈G

2 dim(ξ )f(ξ )H S ,

For a linear continuwhere f(ξ )2H S = Tr(f(ξ )f(ξ )∗ ), which gives the norm on 2 (G). ∞ ous operator from C (G) to D (G) we introduce its full matrix-valued symbol σA (x, ξ ) ∈ Cdim(ξ )×dim(ξ ) by σA (x, ξ ) = ξ(x)∗ (Aξ )(x). Then it was shown in [18] and [16] that Af (x) =

dim(ξ ) Tr ξ(x)σA (x, ξ )f(ξ )

(2)

[ξ ]∈G

holds in the sense of distributions, and the sum is independent of the choice of a representation Moreover, we have ξ from each class [ξ ] ∈ G. σA (x, ξ ) =

RA (x, y)ξ(y)∗ dy

G

in the sense of distributions, where RA is the right-convolution kernel of A: Af (x) =

K(x, y)f (y) dy =

G

f (y)RA x, y −1 x dy.

G

the symbol of a continuous linear operator Symbols σA can be viewed as mappings on G × G: A : C ∞ (G) → C ∞ (G) is a mapping σA : G × Rep(G) →

End(Hξ ),

ξ ∈Rep(G)

where σA (x, ξ ) : Hξ → Hξ is linear for every x ∈ G and ξ ∈ Rep(G), see [16, Rem. 10.4.9], and End(Hξ ) is the space of all linear mappings from Hξ to Hξ . If η ∈ [ξ ], i.e. there is an intertwining isomorphism U : Hη → Hξ such that η(x) = U −1 ξ(x)U , then σA (x, η) = U −1 σA (x, ξ )U . instead of G × Rep(G). In this sense we may think that the symbol σA is defined on G × G For further details of these constructions and their properties we refer to [16].


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A (possibly unbounded) linear operator P on a Hilbert space H is called positive if P v, v H 0 for every v ∈ V for a dense subset V ⊂ H. A matrix P ∈ Cn×n is called positive if the natural corresponding linear operator Cn → Cn is positive, where Cn has the standard inner product. A matrix pseudo-differential symbol σA is called positive if the matrix σA (x, ξ ) ∈ Cdim(ξ )×dim(ξ ) is positive for every x ∈ G and ξ ∈ Rep(G). In this case we write σ (x, ξ ) 0. We note that for each ξ ∈ Rep(G), the condition σA (x, ξ ) 0 implies σA (x, η) 0 for all η ∈ [ξ ]. We can also note that this symbol positivity does not change if we move from left symbols to right symbols: σA (x, ξ ) := ξ ∗ (x)(Aξ )(x) = ξ(x)∗ ρA (x, ξ )ξ(x), ρA (x, ξ ) := (Aξ )(x)ξ ∗ (x) = ξ(x)σA (x, ξ )ξ(x)∗ ; that is, σA is positive if and only if ρA is positive. Moreover, this positivity concept is natural in the sense that a left- or right-invariant operator is positive if and only if its symbol is positive, as it can be seen from the equalities a ∗ f, f L2 (G) =

[ξ ]∈G

f ∗ a, f L2 (G) =

dim(ξ ) Tr f(ξ ) a (ξ )f(ξ )∗ ,

(3)

dim(ξ ) Tr f(ξ )∗ a (ξ )f(ξ ) ,

(4)

[ξ ]∈G

which can be shown by a simple calculation which we give in Proposition 3.6. At the same time, the operator Mf of multiplication by a smooth function f ∈ C ∞ (G) is positive if and only if the function satisfies f (x) 0 for every x ∈ G. The symbol of such multiplication operator is σMf (x, ξ ) = f (x)Idim(ξ ) , so that this means the positivity of the matrix symbol again. Now we can formulate the main result of this paper: Theorem 2.1. Let A ∈ Ψ m (G) be such that its full matrix symbol σA satisfies σA (x, ξ ) 0 for Then there exists C < ∞ such that all (x, [ξ ]) ∈ G × G. Re(Au, u)L2 (G) −Cu2H (m−1)/2 (G) for every u ∈ C ∞ (G). As a corollary of Theorem 2.1 we obtain the following statement on compact Lie groups, analogous to the corresponding result on Rn , which is often necessary in the proofs of pseudodifferential inequalities (see e.g. Theorem 3.1 in [20]). Corollary 2.2. Let A ∈ Ψ 1 (G) be such that its matrix symbol σA satisfies σA (x, ξ )

op

C

Then A is bounded from L2 (G) to L2 (G). for all (x, [ξ ]) ∈ G × G.

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Here · op denotes the 2 → 2 operator norm of the linear finite dimensional mapping (matrix multiplication by) σA (x, ξ ), i.e. σA (x, ξ )

op

= sup σA (x, ξ )v 2 : v ∈ Cdim(ξ ) , v 2 1 .

The weights for measuring the orders of symbols are expressed in terms of the eigenvalues span of the bi-invariant Laplacian LG . Matrix elements of every representation class [ξ ] ∈ G an eigenspace of the bi-invariant Laplace–Beltrami operator LG on G with the corresponding eigenvalue −λ2ξ . Based on these eigenvalues we define 1/2 ξ = 1 + λ2ξ . For further details and properties of these constructions we refer to [16]. In particular, for the To fix the norm on usual Sobolev spaces, we have f ∈ H s (G) if and only if ξ s f(ξ ) ∈ 2 (G). s H (G) for the following statement, we can then set

f H s (G) :=

1/2 dim(ξ ) ξ 2s Tr f(ξ )∗ f(ξ ) ,

[ξ ]∈G

and we can write this also as ξ s f(ξ ) 2 (G) . Also, we note that by [16, Lemma 10.9.1] (or by Theorem 3.1 below), if A ∈ Ψ m (G), then there is a constant 0 < M < ∞ such that σA (x, ξ )op M ξ m holds for all x ∈ G and [ξ ] ∈ G. As another corollary of Theorem 2.1 we can get a norm-estimate for pseudo-differential operators on compact Lie groups: Corollary 2.3. Let A ∈ Ψ m (G) and let M=

sup

−m ξ σA (x, ξ )op .

(x,[ξ ])∈G×G

Then for every s ∈ R there exists a constant C > 0 such that Au2H s (G) M 2 u2H s+m (G) + Cu2H s+m−1/2 (G) for all u ∈ C ∞ (G). 3. Preliminary constructions In this section we collect and develop several ideas which will be used in the proof of Theorem 2.1. These include characterisations of the class Ψ m (G), the Leibniz formula, the amplitude operators on G, and some properties of even and odd functions.


2887

3.1. On symbols and operators First we collect several facts and definitions required for our proof. We now introduce the notation for the symbol classes on the group G and give a characterisation of classes Ψ m (G) in terms of the matrix-valued symbols. In this, we follow the notation of [19]. We say that Qξ is a difference operator of order k if it is given by Qξ f(ξ ) = q Q f (ξ ), for a function q = qQ ∈ C ∞ (G) vanishing of order k at the identity e ∈ G, i.e., (Px qQ )(e) = 0 for all left-invariant differential operators Px ∈ Diffk−1 (G) of order k − 1. We denote the set of all difference operators of order k as diffk (G). is called admisA collection of μ n first order difference operators 1 , . . . , μ ∈ diff1 (G) sible, if the corresponding functions q1 , . . . , qμ ∈ C ∞ (G) satisfy qj (e) = 0, dqj (e) = 0 for all j = 1, . . . , μ, and if rank(dq1 (e), . . . , dqμ (e)) = n. An admissible collection is called strongly μ admissible if j =1 {x ∈ G: qj (x) = 0} = {e}. For a given admissible selection of difference operators on a compact Lie group G we use α multi-index notation αξ = α1 1 · · · μμ and q α (x) = q1 (x)α1 · · · qμ (x)αμ . Furthermore, there exist corresponding differential operators ∂x ∈ Diff|α| (G) such that Taylor’s formula (α)

f (x) =

|α|N −1

1 α q (x)∂x(α) f (e) + O dist(x, e)N α!

(5)

holds true for any smooth function f ∈ C ∞ (G) and with dist(x, e) the geodesic distance from (α) x to the identity element e. An explicit construction of operators ∂x in terms of q α (x) can be (α) found in [16, Section 10.6]. In addition to these differential operators ∂x ∈ Diff|α| (G) we introduce operators ∂xα as follows. Let {∂xj }nj=1 ⊂ Diff1 (G) be a collection of left-invariant first order differential operators corresponding to some linearly independent family of the left-invariant vector fields on G. We denote ∂xα = ∂xα11 · · · ∂xαnn . We note that in most estimates we can freely replace (α) operators ∂x by ∂xα and in the other way around since they can be expressed in terms of each other. For further details and properties of the introduced constructions we refer to [16]. We now record the characterisation of Hörmander’s classes as it appeared in [19]: Theorem 3.1. Let A be a linear continuous operator from C ∞ (G) to D (G), and let m ∈ R. Then the following statements are equivalent: (A) A ∈ Ψ m (G). (B) For every left-invariant differential operator Px ∈ Diffk (G) of order k and every difference of order there is the symbol estimate operator Qξ ∈ diff (G) Qξ Px σA (x, ξ )

op

CQξ Px ξ m− .

we have (C) For an admissible collection 1 , . . . , μ ∈ diff1 (G) α β ∂ σA (x, ξ ) ξ x

op

Cαβ ξ m−|α|

for all multi-indices α, β. Moreover, sing supp RA (x, ·) ⊆ {e}.

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we have (D) For a strongly admissible collection 1 , . . . , μ ∈ diff1 (G) α β ∂ σA (x, ξ ) ξ x

op

Cαβ ξ m−|α|

for all multi-indices α, β. m (G) = The set of symbols σA satisfying either of conditions (B)–(D) will be denoted by S1,0 m S (G). We note that if conditions (C) or (D) hold for one admissible (strongly admissible, resp.)

collection of first order difference operators, they automatically hold for all admissible (strongly admissible, resp.) collections. For the purposes of this paper, we will also need larger classes of symbols which we now m (G) if it is smooth introduce. We will say that a matrix-valued symbol σA (x, ξ ) belongs to Sρ,δ we have in x and if for a strongly admissible collection 1 , . . . , μ ∈ diff1 (G) α β ∂ σA (x, ξ ) ξ x

op

Cαβ ξ m−ρ|α|+δ|β|

(6)

for all multi-indices α, β, uniformly in x ∈ G and ξ ∈ Rep(G). Remark 3.2. As it was pointed out in [19], in Theorem 3.1 we still have the equivalence of conditions (B), (C), (D), also if we replace symbolic inequalities in Theorem 3.1 by inequalities of the form (6). Also in this setting, if conditions (C) or (D) hold for one admissible (strongly admissible, resp.) collection of first order difference operators, they automatically hold for all admissible (strongly admissible, resp.) collections. m (G) if for every multi-index β and for every x ∈ G we have We will also write a ∈ Sρ,δ# 0 β

m+δ|β|

μ

∂x a(x0 , ·) ∈ Sρ# (G), where for a multiplier b = b(ξ ) we write b ∈ Sρ# (G) if for every multi-index α there is a constant Cα such that α b(ξ ) ξ

op

Cα ξ μ−ρ|α|

We record the following straightforward lemma that follows from the holds for all [ξ ] ∈ G. smoothness of symbols in x and the compactness of G: m (G) if and only if a ∈ S m (G). Lemma 3.3. We have a ∈ Sρ,δ ρ,δ#

Another tool which will be required for the proof is the finite version of the Leibniz formula which appeared in [19]. Given a continuous unitary matrix representation ξ 0 = [ξij0 ]1i,j : G → C × , = dim(ξ 0 ), let q(x) = ξ 0 (x) − I (i.e. qij = ξij0 − δij with Kronecker’s deltas δij ), and define Dij f(ξ ) := q ij f (ξ ). 2

In the previous notation, we could also write Dij = qij . For a multi-index γ ∈ N 0 , we write γ γ |γ | = i,j =1 |γij |, and for higher order difference operators we write Dγ = D1111 D1212 · · · γ , −1 γ

D , −1 D

. In contrast to the asymptotic Leibniz rule [16, Thm. 10.7.12] for arbitrary difference operators, operators D satisfy the finite Leibniz formula:


2889

2

Proposition 3.4. For all γ ∈ N 0 we have

Dγ (ab) =

Cγ εδ Dε a Dδ b ,

|ε|,|δ||γ ||ε|+|δ| 2

with the summation taken over all ε, δ ∈ N 0 satisfying |ε|, |δ| |γ | |ε| + |δ|. In particular, for |γ | = 1, we have Dij (ab) = (Dij a)b + a(Dij b) +

(Dik a)(Dkj b).

(7)

k=1

Difference operators D lead to strongly admissible collections (see [19]): Lemma 3.5. The family of difference operators associated to the family of functions {qij = ξij − δij }[ξ ]∈G, 1i,j dim(ξ ) is strongly admissible. Moreover, this family has a finite subfamily associated to finitely many representations which is still strongly admissible. We now give a simple proof of the equalities (3) and (4). Proposition 3.6. We have a ∗ f, f L2 (G) =

[ξ ]∈G

f ∗ a, f L2 (G) =

dim(ξ ) Tr f(ξ ) a (ξ )f(ξ )∗ , dim(ξ ) Tr f(ξ )∗ a (ξ )f(ξ ) .

[ξ ]∈G

Proof. The second claimed equality follows from the following calculation: f ∗ a, f L2 (G) = (f ∗ a)(x)f (x) dx G

=

dim(ξ ) Tr ξ(x) a (ξ )f(ξ ) dim(η) Tr η(x)f(η) dx

G [ξ ]∈G

=

[η]∈G

dim(ξ )

G [ξ ]∈G

=

[ξ ]∈G

=

[ξ ]∈G

dim(ξ )

dim(ξ )

ξ(x)kl a (ξ )lm f(ξ )mk

k,l,m=1 dim(ξ )

a (ξ )lm f(ξ )mk f(ξ )lk

k,l,m=1

dim(ξ ) Tr a (ξ )f(ξ )f(ξ )∗ ,

[η]∈G

dim(η)

dim(η) p,q=1

η(x)pq f(η)qp dx

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where we used the orthogonality of the matrix elements of the representations. The first claimed equality can be proven in an analogous way. 2 We also record the Sobolev boundedness result that was Theorem 10.8.1 in [16]: Theorem 3.7. Let G be a compact Lie group. Let A be a continuous linear operator from C ∞ (G) to C ∞ (G) and let σA be its symbol. Assume that there exist constants m, Cα ∈ R such that α ∂ σA (x, ξ ) x

op

Cα ξ m

holds for all x ∈ G, ξ ∈ Rep(G), and all multi-indices α. Then A extends to a bounded operator from H s (G) to H s−m (G) for all s ∈ R. 3.2. Amplitudes on G Let 0 δ, ρ 1. An amplitude a ∈ Am ρ,δ (G) is a mapping defined on G × G × Rep(G), smooth in x and y, such that for an irreducible unitary representation ξ : G → U(Hξ ) we have2 linear operators a(x, y, ξ ) : Hξ → Hξ , and for a strongly admissible collection of difference operators αξ the amplitude satisfies the amplitude inequalities α β γ ∂ ∂y a(x, y, ξ ) ξ x

op

Cαβγ ξ m−ρ|α|+δ|β+γ | ,

For an amplitude a, the amplitude for all multi-indices α, β, γ and for all (x, y, [ξ ]) ∈ G×G× G. ∞ operator Op(a) : C (G) → D (G) is defined by Op(a)u(x) :=

[η]∈G

dim(η) Tr η(x) a(x, y, η)u(y)η(y)∗ dy .

(8)

G

Notice that if here a(x, y, η) = σA (x, η) then Op(a) = A as in (2). This definition can be justified as follows: Proposition 3.8. Let 0 δ < 1 and 0 ρ 1, and let a ∈ Am ρ,δ (G). Then Op(a) is a continuous linear operator from C ∞ (G) to C ∞ (G). Proof. By the definition of η we have (1 − LG )η(y) = η 2 η(y). On the other hand, the Weyl spectral asymptotics formula for the Laplace operator LG implies that η −1 C dim(η)−2/ dim(G) (see Proposition 10.3.19 in [16]). Consequently, integrating by parts in the dy-integral in (8) with operator η −2 (I − LG ) arbitrarily many times, we see that the η-series in (8) converges, so that Op(a)u ∈ C ∞ (G) provided that u ∈ C ∞ (G). The continuity of Op(a) on C ∞ (G) follows by a similar argument. 2 2 Especially, if ξ is a unitary matrix representation of dimension d, then a(x, y, ξ ) ∈ Cd×d .


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Remark 3.9. In the proof we used the inequality dim(η) C η n/2 , n = dim G, which easily follows from the Weyl spectral asymptotic formula (see Proposition 10.3.19 in [16]), and which is enough for the purposes of the proof. However, a stronger inequality dim(η) C η (n−l)/2 can be obtained from the Weyl character formula, with l = rank G. For the details of this, see e.g. [23, (11), (12)]. Proposition 3.10. Let 0 δ < ρ 1 and let a ∈ Am ρ,δ (G). Then A = Op(a) is a pseudom (G). Moreover, σ has the asymptotic differential operator on G with a matrix symbol σA ∈ Sρ,δ A expansion σA (x, ξ ) ∼

1 ∂ (α) αξ a(x, y, ξ )|y=x . α! y

α0

Proof. If σA is the matrix symbol of the continuous linear operator A = Op(a) : C ∞ (G) → C ∞ (G), we can find it from the formula σA (x, ξ ) = ξ(x)∗ (Aξ )(x). By fixing some basis in the representation spaces, we have dim(ξ )

σA (x, ξ )mn =

ξ x −1 ml (Aξln )(x)

l=1 dim(ξ )

=

ξ x −1 ml

l=1

=

=

G [η]∈G

[η]∈G

dim(η) ξ x −1 y mn dim(η) η y −1 x j k a(x, y, η)kj dy [η]∈G

G

dim(η) Tr η(x)a(x, y, η)ξ(y)ln η(y)∗ dy

ξ x −1 y mn dim(η) Tr η y −1 x a(x, y, η) dy

G

=

j,k=1

dim(η) −1 ξ z mn dim(η) η(z)j k a x, xz−1 , η kj dz [η]∈G

G

j,k=1

dim(η) 1 ∂u(α) dim(η) a(x, u, η)kj |u=x ξ z−1 mn η(z)j k qα (z) dz, α!

∼

α0

[η]∈G

j,k=1

G

by the Taylor expansion (5). Using difference operators αξ s(ξ ) := q α s(ξ ), we find

dim(η)

[η]∈G

= G

dim(η)

a(x, u, η)kj

j,k=1

ξ(z)∗ qα (z)

ξ z−1 η(z)j k qα (z) dz

G

[η]∈G

dim(η) Tr η(z)a(x, u, η) dz = αξ a(x, u, ξ ).

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Thus σA (x, ξ ) ∼

1 dim(η) Tr η(z)a(x, u, η) dzu=x ∂u(α) ξ(z)∗ qα (z) α!

α0

=

[η]∈G

G

1 ∂ (α) αξ a(x, u, ξ )|u=x . α! u

α0

The remainder in this asymptotic expansion can be dealt with in a way similar to the argument for the composition formulae (see [16]), so we omit the proof. 2 3.3. Properties of even and odd functions On a group G, function f : G → C is called even if it is inversion-invariant, i.e. if f (x −1 ) = f (x) for every x ∈ G. Function f : G → C is called odd if f (x −1 ) = −f (x) for every x ∈ G. Recall that f : G → C is central if f (xy) = f (yx) for all x, y ∈ G. Linear combinations of characters χξ = (x → Tr(ξ(x))) of irreducible unitary representations ξ of a compact group G are central, and such linear combinations are dense among the central functions of C(G). When G is a compact Lie group, for Y ∈ g and f ∈ C ∞ (G) we define LY f (x) :=

d f x exp(tY ) t=0 , dt

RY f (x) :=

d f exp(tY )x t=0 , dt

so that LY , RY are the first order differential operators, LY being left-invariant and RY rightinvariant. For a central function f we have LY f = RY f , which would not be true for an arbitrary smooth function f . Moreover, if f is even and central then LY f x −1 = −LY f (x), i.e. LY f is odd in this case. Similarly LY f is even for odd central functions f , but LY f does not have to be central. More precisely, for central f ∈ C ∞ (G) we obtain LY f u−1 xu = LuY u−1 f (x), where u ∈ G. For higher order derivatives of even and odd functions, taking the differential of f x exp(t1 X1 ) · · · exp(tk Xk ) = ±f x −1 exp (−tk )Xk · · · exp (−t1 )X1 at t1 = · · · = tk = 0, we obtain: Proposition 3.11. Let f ∈ C ∞ (G) be even and central, and X1 , . . . , Xk ∈ g. Then LX1 LX2 · · · LXk−1 LXk f x −1 = (−1)k LXk LXk−1 · · · LX2 LX1 f (x). Similarly, if f ∈ C ∞ (G) is an odd central function, then we have the equality LX1 LX2 · · · LXk−1 LXk f x −1 = (−1)k+1 LXk LXk−1 · · · LX2 LX1 f (x).


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4. Proof of the sharp Gårding inequality We notice that if a linear operator Q : H (m−1)/2 (G) → H −(m−1)/2 (G) is bounded then Re(Qu, u)L2 −(Qu, u)L2 −QuH −(m−1)/2 uH (m−1)/2 −QL(H (m−1)/2 ,H −(m−1)/2 ) u2H (m−1)/2 . Hence Theorem 2.1 would follow if we could show that A = P + Q, where P is positive (on C ∞ (G) ⊂ L2 (G)) and Q : H (m−1)/2 (G) → H −(m−1)/2 (G) is bounded. The proof of this decomposition will be done in several steps. 4.1. Construction of wξ First, we construct an auxiliary function wξ which will play a crucial role for our proof. We can treat G as a closed subgroup of GL(N, R) ⊂ RN ×N for some N ∈ N. Then its Lie algebra g ⊂ RN ×N is an n-dimensional vector subspace (hence identifiable with Rn ) such that [A, B] := AB − BA ∈ g for every A, B ∈ g. Let U ⊂ G be a neighbourhood of the neutral element e ∈ G, and let V ⊂ g be a neighbourhood of 0 ∈ g ∼ = Rn , such that the matrix exponential mapping is a diffeomorphism exp : V → U . For the construction and for the notation only in Section 4.1, we define the central norm | · | on g as follows.3 Take the Euclidean norm | · |0 on g and define (9) |X| = uXu−1 0 du, G

where we may view the product under the integral as the product of matrices in RN ×N . Then by definition the norm (9) is invariant by the adjoint representation, and we have, in particular | exp−1 (xy)| = | exp−1 (yx)|, etc. We may assume that V is the open ball V = B(0, r) = {Z ∈ Rn : |Z| < r} of radius r > 0. Let φ : [0, r) → [0, ∞) be a smooth function such that (Z → φ(|Z|)) : g → R is supported in V and φ(s) = 1 for small s > 0. For every ξ ∈ Rep(G) we define wξ (x) := φ exp−1 (x) ξ 1/2 ψ exp−1 (x) ξ n/4 ,

(10)

where −1/2 f (Y )−1/2 , ψ(Y ) = C0 det D exp(Y ) D exp is the Jacobi matrix of exp, f (Y ) is the density with respect to the Lebesgue measure of the Haar pulled back to g ∼ = Rn by the exponential mapping, and with constant measure2 on G−1/2 . By Idim(ξ ) we denote the identity mapping on Cdim(ξ ) . For x, y ∈ G C0 = ( Rn φ(|Z|) dZ) close to each other, dist(x, y) is the geodesic distance between x and y. 3 In fact, any central norm | · | on g will work.

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Lemma 4.1. We have wξ ∈ C ∞ (G), wξ (e) = C0 ξ n/4 , wξ is central and inversion-invariant, i.e. wξ (xy) = wξ (yx) and wξ (x −1 ) = wξ (x) for every x, y ∈ G. Also, dist(x, e) ≈ | exp−1 (x)| r ξ −1/2 on the support of wξ . Moreover, wξ L2 (G) = 1 for all ξ ∈ Rep(G). Finally, we have n/4

((x, ξ ) → wξ (x)Idim(ξ ) ) ∈ S1,1/2 (G). Proof. It is easy to see that wξ ∈ C ∞ (G), wξ (e) = C0 ξ n/4 , and that wξ is inversion-invariant. Clearly dist(x, e) ≈ | exp−1 (x)| r ξ −1/2 on the support of wξ in view of properties of the function φ. In particular, (10) is well defined and supp wξ ⊂ U . From (9) it also follows that wξ is central since f is invariant under adjoint representation as a density of two bi-invariant measures. Let us now show that wξ L2 (G) = 1 for all ξ ∈ Rep(G). Indeed,

wξ (x)2 dx = ξ n/2

G

= C02

2 2 φ |Y | ξ 1/2 ψ(Y ) det D exp(Y )f (Y ) dY

Rn

2 φ |Z| dZ,

Rn

so that wξ L2 (G) = 1 in view of the choice of the constant C0 . Thus, the main thing is to check n/4

that wξ Idim(ξ ) ∈ S1,1/2 (G). By Lemma 3.3, we need to check that for every multi-index β and β

n/4+|β|/2

every x0 ∈ G we have ∂x wξ (x0 ) ∈ S1# sums of terms of the form

(G). We observe that the x-derivatives of wξ are

−1 1/2 n/4+l/2 exp (x) ξ

ξ

Idim(ξ ) , χ exp−1 (x) φ

(11)

∈ C ∞ (R), φ is constant near the origin, and l is an integer such that where χ ∈ C0∞ (V ), φ 0 n/4+l/2 0 l |β|. We note that ξ

Idim(ξ ) is the symbol of the pseudo-differential operator n/4+l/2 n/4+|β|/2 n/8+l/4 n/4+l/2 , and hence ξ

Idim(ξ ) ∈ S1# ⊂ S1# . Moreover, we can elim(1 − LG ) inate it from the formulae by the composition formulae for the matrix-valued symbols (see [16, Thm. 10.7.9]). Thus we have to check that for every x0 ∈ G, the other terms in (11) fixed at 0 (G), i.e. that x = x0 are in S1# 0 exp−1 (x0 ) ξ 1/2 Idim(ξ ) ∈ S1# φ (G).

(12)

If exp−1 (x0 ) = 0, then this symbol is a constant times the identity Idim(ξ ) and hence it is in 0 (G). On the other hand, if exp−1 (x ) = 0, then the symbol (12) is compactly supported S1# 0 in ξ , and hence defines a smoothing operator. Indeed, in this case it has decay of any order in ξ , together with all difference operators applied to it, with constants depending on x0 , so it is smoothing by Theorem 3.1. Let us also give an alternative argument relating this operator to a corresponding operators (|v|t) and using the characterisation of pseudo-differential operators in on g. Writing ϕv (t) := φ Theorem 3.1, we notice that (12) holds if for all x0 ∈ G, the operators ϕexp−1 (x0 ) ((I − LG )1/4 ) belong to Ψ 0 (G). Looking at these operators locally near every point x ∈ G and introducing


2895

θ ∈ C0∞ (Rn ) such that θ ◦ exp−1 x is supported in a small neighbourhood near x, with expx := (Z → x exp(Z)) : g → G the exponential mapping centred at x, we have to show that θ (y)ϕv (B) ∈ Ψ 0 Rn

(13)

holds locally on the support of θ , for all v = exp−1 (x0 ), where operator B is the pullback by 1/2 expx of the operator (I − LG )1/4 near x. In particular, we have B ∈ Ψ1,0 (Rn ), B is elliptic on the support of θ , and its symbol is real-valued. We now observe that if v = 0, then the operator in (13) is the multiplication operator by a smooth function, so that (13) is true in this case. If v = 0, we can show that the operator in (13) is actually a smoothing operator, so that (13) is also true. Here ϕv ∈ C0∞ (R) since v = 0. We denote 1 ∂t . Let f ∈ L2 (Rn ) be compactly supported, and let u = u(t, x) be the solution to the Dt = i2π Cauchy problem Dt u = Bu,

u(0, ·) = f.

We can write u(t, ·) = ei2πtB f and we have u(t, ·) ∈ L2 (Rn ). Consequently, ϕv (B)f =

i2πtB e f ϕv (t) dt =

R

B −k u(t, ·)Dtk ϕv (t) dt,

R

where we integrated by parts k times using the relation u = B −1 Dt u, and where we can localise k/2 to a neighbourhood of a point x at each step. Consequently, we obtain that ϕv (B)f ∈ Hloc (Rn ) + ∞ n for all k ∈ Z , so that actually ϕv (B)f ∈ C (R ). Thus, the operator ϕv (B) is smoothing and (13) holds also for v = 0. 2 4.2. Auxiliary positive operator P We now introduce a positive operator P which will be important for the proof of the sharp Gårding inequality. This operator P will give a positive approximation to our operator A. m (G). Let us define an amplitude p by Proposition 4.2. Let σA ∈ S1,0

p(x, y, ξ ) :=

wξ xz−1 wξ yz−1 σA (z, ξ ) dz,

G

where wξ ∈ C ∞ (G) is as in (10). Let the amplitude operator P = Op(p) be given by P u(x) =

dim(ξ ) Tr ξ y −1 x p(x, y, ξ ) u(y) dy.

G [ξ ]∈G

Then p ∈ Am 1,1/2 (G) and the operator P is positive.

(14)

2896


Proof. We observe that p(x, y, ξ )

op

−1 −1 wξ xz wξ yz dz supσA (z, ξ ) C ξ m op z∈G

G

because wξ 2L2 (G) = 1 by Lemma 4.1. Then p ∈ Am 1,1/2 (G) follows from Lemma 4.1 and the Leibniz formula in Proposition 3.4 by an argument similar to the one which will be given in the proof of Lemma 4.4, so we omit it. Let (ek ) k=1 be an orthonormal basis for C . For matrices M, Q ∈ C × , where Q is positive, we have

∗ M QMek , ek C = QMek , Mek C 0. Tr M ∗ QM = k=1

(15)

k=1

Let us denote M(z, ξ ) :=

∗ wξ yz−1 ξ yz−1 u(y) dy.

G

We can now show that the operator P is positive: P u, u L2 (G) = P u(x)u(x) dx G

=

dim(ξ ) Tr ξ(x)p(x, y, ξ )u(y)ξ(y)∗ dy u(x) dx

G G [ξ ]∈G

=

G [ξ ]∈G

=

dim(ξ ) G

Tr ξ(x) wξ xz−1 wξ yz−1 σA (z, ξ ) dz u(y)ξ(y)∗ dy u(x) dx G

dim(ξ ) Tr M(z, ξ )∗ σA (z, ξ )M(z, ξ ) dz,

G [ξ ]∈G

which is non-negative because of (15).

2

4.3. The difference p(x, x, ξ ) − σA (x, ξ ) In the earlier notation, we show here that p(x, x, ξ ) − σA (x, ξ ) is a symbol of a bounded operator from H s (G) to H s−(m−1) (G). Lemma 4.3. Let s ∈ R. Then the pseudo-differential operator with the symbol p(x, x, ξ ) − σA (x, ξ ) is bounded from H s (G) to H s−(m−1) (G).


2897

Proof. By Theorem 3.7 it is enough to show that β ∂ p(x, x, ξ ) − σA (x, ξ ) Cβ ξ m−1 x op holds for every multi-index β. By Lemma 4.1 we have ∂xβ p(x, x, ξ ) − σA (x, ξ ) =

wξ (z)2 ∂xβ σA xz−1 , ξ − ∂xβ σA (x, ξ ) dz.

G

We notice that dist(z, e) C ξ −1/2 on the support of wξ , and we can use the Taylor expansion β of ∂x σA (xz−1 , ξ ) at x to get (γ ) ∂xβ σA xz−1 , ξ = ∂xβ σA (x, ξ ) + ∂x ∂xβ σA (x, ξ )qγ (z) + O dist(z, e)2 .

(16)

|γ |=1

Taking the Taylor polynomials qγ to be odd, qγ (z) = −qγ (z−1 ), and using the evenness of wξ from Lemma 4.1, we can conclude that G wξ (z)2 qγ (z) dz = 0. Since for all β and γ we have (γ ) β

∂x ∂x σA (x, ξ )op C ξ m , we can estimate β ∂ p(x, x, ξ ) − σA (x, ξ ) x

C ξ m op

wξ (z)2 qγ (z) dz C ξ m−1

|γ |=2 G

because |qγ (z)| C ξ −1 on the support of wξ , for |γ | = 2.

2

4.4. The difference σP (x, ξ ) − p(x, x, ξ ) Let σP be the matrix symbol of the operator P from Proposition 4.2. Lemma 4.4. Let s ∈ R. Then the pseudo-differential operator with the symbol σP (x, ξ ) − p(x, x, ξ ) is bounded from H s (G) to H s−(m−1) (G). Proof. Observe that for a fixed s ∈ R, it is enough to take sufficiently many derivatives (and not infinitely many) for the Sobolev boundedness in Theorem 3.7. Thus it is enough to prove that for sufficiently many β ∈ Nn0 it holds that β ∂ σP (x, ξ ) − p(x, x, ξ ) x

op

Cβ ξ m−1 .

By an argument in the proof of Proposition 3.10 we have the expansion σP (x, ξ ) ∼

1

α ∂ (α) p(x, y, ξ )|y=x , α! ξ y

α0

whose asymptotic properties we will discuss below. Instead of studying the terms β β (α) ∂x ( αξ ∂y p(x, y, ξ )|y=x ), we may study ∂x ( αξ ∂yα p(x, y, ξ )|y=x ) as well. Moreover, abusing

2898


the notation slightly, without loss of generality we can look only at the right-invariant derivatives β ∂yα and left-invariant derivatives ∂x . Recalling that p(x, y, ξ ) =

wξ xz−1 wξ yz−1 σA (z, ξ ) dz,

G

we notice that ∂xβ

α α

ξ ∂y p(x, y, ξ )|y=x = αξ

wξ (z) ∂zα wξ (z)∂xβ σA z−1 x, ξ dx.

(17)

G

We notice also that by Remark 3.2 we can replace differences ξ by Dξ with a suitable correction for multi-indices. The application of Dα here introduces (due to the Leibniz formula in Proposition 3.4) a finite sum of terms of the type

κ μ Dξ wξ (z) Dλξ ∂zα wξ (z) Dξ σA z−1 x, ξ dz,

(18)

G n/4

where |κ + λ + μ| |α|. Recalling that wξ ∈ S1,1/2 by Lemma 4.1, we get that κ D wξ (z) Dλ ∂ α wξ (z) Dμ σA z−1 x, ξ C ξ m+n/2−|α|/2 . ξ z

ξ

ξ

Taking into account that the support of z → wξ (z) is contained in the set of measure C ξ −n/2 by Lemma 4.1, and that taking differences in ξ does not increase the support in z, we get that the integral in (18) can be estimated by C ξ m−|α|/2 . Thus, we get β α α ∂ ∂ p(x, y, ξ )|y=x x

ξ y

op

C ξ m−|α|/2 .

(19)

For |α| 2 this implies the desired bound by C ξ m−1 for the Sobolev boundedness of the corresponding operator. Now, assume that |α| = 1. Taking the Taylor expansion of σA (z−1 x, ξ ) at x similar to the one in (16) we see that the first term vanishes:

wξ (z) ∂zα wξ (z) dz = 0

G

for |α| = 1 because functions wξ and ∂zα wξ are even and odd, respectively, see Proposition 3.11. Consequently, for |γ | 1, we can estimate wξ (z) ∂ α wξ (z)qγ (z) C ξ n/2+|α|/2−|γ |/2 , z

which together with (17) gives β α α ∂ ∂ p(x, y, ξ )|y=x x

ξ y

op

C ξ m−|α|/2−|γ |/2 C ξ m−1


2899

because |α| = 1 and |γ | 1. Finally, let us look at the remainder σRN (x, ξ ) = σP (x, ξ ) −

1

α ∂ (α) p(x, y, ξ )|y=x . α! ξ y

|α| 0 there exist a diagonal operator Dε and a compact operator Kε with Kε ε such that A = Dε + Kε . Our main result is Theorem 2.12, which shows that a bounded operator A belongs to a certain set associated with its self-commutator whenever A − λI can be approximated by invertible operators for all λ ∈ C. Theorem 2.12 implies both the BDF and Huaxin Lin’s theorems. Moreover, it allows us to refine the former and to extend the latter to operators of infinite rank and other norms (see Sections 3.1 and 3.3, Corollary 2.14 and Remark 2.11). In particular, we obtain (i) a quantitative version of Theorem 0.1, which links the BDF and Weyl–von Neumann–Berg theorems for bounded operators, and (ii) an analogue of Huaxin Lin’s theorem for the Schatten norms of finite matrices. Theorem 2.12 holds for any unital C ∗ -algebra L of real rank zero, but in the general case we need a slightly stronger condition on A. Namely, we assume that A − λI belong to the closure of the connected component of unity in the set of invertible elements in L for all λ ∈ C. Our proof of Theorem 2.12 uses the C ∗ -algebra technique developed in [15] and [16] and cannot be significantly simplified by assuming that L is the C ∗ -algebra of all bounded operators. One of the main ingredients in the proof is the part of Corollary 2.5 which says that a normal operator satisfying the above condition can be approximated by normal operators with finite spectra. This statement is contained in [16, Theorem 3.2]. The authors indicated how it could be proved but did not present complete arguments. Therefore, for reader’s convenience, we give

2904

N. Filonov, Y. Safarov / Journal of Functional Analysis 260 (2011) 2902–2932

operator-theoretic proofs of this and relevant results. More precisely, we deduce Corollary 2.5 from Theorem 2.1, which seems to be new and may be of independent interest. 1. Notation and auxiliary results 1.1. Notation and definitions Let H be a complex Hilbert space (not necessarily separable). Throughout the paper, • B(H ) is the C ∗ -algebra of all bounded operators in H ; • σ (A) denotes the spectrum of A ∈ B(H ); • L is a unital C ∗ -algebra represented on the Hilbert space H , so that L ⊆ B(H ). Recall that, by the Gelfand–Naimark theorem, such a representation exists for any unital C ∗ algebra L. Every operator √ A ∈ B(H ) admits the polar decomposition A = V |A|, where |A| is the selfadjoint operator A∗ A and V is an isometric operator such that V H = AH and ker V = ker A = ker |A|. If A is normal then V |A| = |A|V . Remark 1.1. If A ∈ L then f (|A|) ∈ L for any continuous function f . If, in addition, A is invertible then V = A|A|−1 is a unitary element of L because |A|−1 can be approximated by continuous functions of |A|. In particular, this implies that A−1 ∈ L. If A ∈ L is not invertible then the isometric operator V in its polar representation does not have to belong to L. Remark 1.2. The spectral projections of a normal operator A ∈ L may not lie in L. However, the spectral projection corresponding to a connected component of σ (A) belongs to L, since it can be written as a continuous function of A. Further on • • • • •

L−1 is the set of invertible operators in L; −1 containing the identity operator; L−1 0 denotes the connected component of L Ln , Lu , and Ls are the sets of normal, unitary and self-adjoint operators in L respectively; Lf is the set of operators A ∈ L with finite spectra; if M ⊂ L then M denotes the norm closure of the set M in L.

Clearly, the sets L−1 and L−1 0 are open in L, and the sets Ln , Lu and Ls are closed. One says that • L has real rank zero if L−1 ∩ Ls = Ls . The concept of real rank of a C ∗ -algebra was introduced in [6]. A unital C ∗ -algebra L has real rank zero if and only if any self-adjoint operator A ∈ L is the norm limit of a sequence of selfadjoint operators from L with finite spectra (see Corollary 2.4 and Section 5.1). Remark 1.3. Note that any self-adjoint operator A ∈ B(H ) is approximated in the norm topology by invertible self-adjoint operators of the form f (A), where f are suitable real-valued Borel


2905

functions. Therefore, all von Neumann algebras (in particular, B(H ) and the algebra of finite m × m-matrices) have real rank zero. Example 1.4. The minimal unital C ∗ -algebra LA containing a given bounded self-adjoint operator A consists of normal operators f (A), where f are continuous complex-valued functions on / L−1 ∩ Ls and, consequently, σ (A). If there is an open interval (a, b) ⊂ σ (A) then A − a+b 2 I ∈ LA is not of real rank zero. Our main results hold for C ∗ -algebras of real rank zero and operators A ∈ L satisfying the following condition (C) A − λI ∈ L−1 0 for all λ ∈ C. 1.2. Auxiliary lemmas We shall need the following simple lemmas. be the subset of the direct product C × L which consists of all pairs (λ, A) Lemma 1.5. Let L −1 such that λ ∈ / σ (A). If A0 − λ0 I ∈ L−1 0 for some (λ0 , A0 ) ∈ L then A − λI ∈ L0 for all (λ, A) that contains (λ0 , A0 ). lying in the closure of the connected component of L containing (λ0 , A0 ). Since the set L is open, 0 be the connected component of L Proof. Let L 0 is path-connected. If (λ, A) ∈ L 0 and (λt , At ) ⊂ L 0 is a path in L 0 from (λ0 , A0 ) to (λ, A) L then At − λt I ∈ L−1 for all t, which implies that A − λI ∈ L−1 0 . If (λ, A) belongs to the closure of L0 then A − λI can be approximated by operators An − λn I ∈ L−1 0 with (λn , An ) ∈ L0 . 2 −1 Remark 1.6. If |μ| > A then A − μI ∈ L−1 0 because [0, 1] t → tA − μI is a path in L −1 from −μI ∈ L0 to A − μI . Therefore, Lemma 1.5 implies that A satisfies the condition (C) whenever C \ σ (A) is a dense connected subset of C. In particular, (C) is fulfilled for all compact operators A ∈ L, all A ∈ Ls ∪ Lf , and all unitary operators A ∈ Lu whose spectra do not coincide with the whole unit circle. −1 Lemma 1.7. Let A ∈ L−1 and A = U |A|. Then A ∈ L−1 0 if and only if U ∈ L0 ∩ Lu .

Proof. [0, 1] t → U (tI + (1 − t)|A|) is a path in L−1 from A to U .

2

Lemma 1.8. Assume that L has real rank zero. Then U ∈ Lu ∩ L−1 0 if and only if for every ε ∈ (0, 1) there exist unitary operators Uε , Wε ∈ Lu such that U = Wε Uε , −1 ∈ / σ (Uε ) and Wε − I ε. Proof. Recall that the point −1 does not belong to the spectrum of U ∈ Lu if and only if U is the Cayley transform of a self-adjoint operator X, that is, U = (iI − X)−1 (iI + X) where X = i(U + I )−1 (U − I ) ∈ Ls . For every such an operator U , the principal branch of the argument Arg is continuous in a neighbourhood of σ (U ), so that Arg U ∈ Ls and exp(it Arg U ) is a path in Lu ∩ L−1 0 from I to U .

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Assume first that U = Wε Uε where Uε and Wε satisfy the above conditions. Then U ∈ Lu , −1 ∈ / σ (Wε ) and exp(it Arg Wε ) exp(it Arg Uε ) is a path in Lu ∩ L−1 0 from I to U . Thus U ∈ −1 Lu ∩ L 0 . −1 Assume now that U ∈ Lu ∩ L−1 0 . Then there exists a path Z(t) ⊂ L0 from I to U . The ˜ = Z(t)|Z(t)|−1 lies in Lu ∩ L−1 and also joins I and U . Let us choose “normalized” path Z(t) 0 a finite collection of points 0 = t0 < t 1 < t 2 < · · · < t m = 1 ˜ j ) − Z(t ˜ j −1 ) < 1 and define Vj := Z(t ˜ j )Z˜ −1 (tj −1 ). Then U = Vm Vm−1 . . . V1 such that Z(t and Vj − I < 1, so that −1 ∈ / σ (Vj ) for all j . Let V = (iI − Y )−1 (iI + Y ) and V˜ = (iI − Y˜ )−1 (iI + Y˜ ) where Y, Y˜ ∈ Ls . Since L has real rank zero, for each δ > 0 we can find Yδ ∈ Ls ∩ L−1 such that Y − Yδ < δ, and then Y˜δ ∈ Ls such that Y˜ − Y˜δ < δ and Y˜δ − Yδ−1 ∈ L−1 . If Vδ = (iI − Yδ )−1 (iI + Yδ ) and V˜δ = (iI − Y˜δ )−1 (iI + Y˜δ ) then Vδ V˜δ − V V˜ → 0 as δ → 0 because the function t → (iI − t)−1 (iI + t) is continuous. Also, −1 ∈ / σ (Vδ V˜δ ) because (iI − Yδ )(Vδ V˜δ + I )(iI − Y˜δ ) = (iI + Yδ )(iI + Y˜δ ) + (iI − Yδ )(iI − Y˜δ ) = 2(Yδ Y˜δ − I ) = 2Yδ Y˜δ − Yδ−1 ∈ L−1 . Thus we see that the composition of two unitary operators whose spectra do not contain −1 can be approximated by unitary operators with the same property. By induction, the same is true for the composition of any finite collection of unitary operators. In particular, there exists Uε ∈ Lu ∩ L−1 / σ (Uε ) and U − Uε < ε. Taking Wε := U Uε−1 , we obtain the 0 such that −1 ∈ required representation of U . 2 Lemma 1.9. Assume that L has real rank zero. Then −1 −1 −1 (1) for every A ∈ L−1 0 and every δ > 0 there exists an operator Sδ ∈ L0 such that Sδ δ and A − Sδ 2δ. (2) If A ∈ Ls then one can find a self-adjoint operator Sδ ∈ L−1 satisfying the above conditions. −1 (3) For every S ∈ L−1 0 there exists a continuous path Z : [0, 1] → L0 such that Z(0) = I , Z(1) = S, Z(t)−1 max{1, S −1 } for all t ∈ [0, 1] and

Z(t) − Z(r) |t − r|(1 + 2π) max 1, S ,

∀t, r ∈ [0, 1].

(1.1)

Proof. (1) Let us choose an arbitrary operator B ∈ L−1 0 such that A − B δ, and let B = V |B| be its polar decomposition. Then, by Lemma 1.7, we have V ∈ L−1 0 ∩ Lu and Sδ := V ((|B| − −1 −1 −1 δI )+ + δI ) ∈ L0 . Obviously, Sδ = ((|B| − δI )+ + δI ) δ −1 and B − Sδ = (|B| − δI )+ + δI − |B| δ, so that A − Sδ 2δ. (2) If A ∈ Ls then there exists an operator B = V |B| ∈ L−1 0 ∩ Ls such that A − B δ. As in (1), we can take Sδ := V ((|B| − δI )+ + δI ) ∈ Ls . (3) Let S := U |S| be the polar representation of S. By Lemma 1.7, U ∈ L−1 0 ∩ Lu . Therefore A := Wε Uε |S|, where Wε and Uε are unitary operators satisfying the conditions of Lemma 1.8.


2907

Let us define Z1 (t) := exp(it Arg Wε ), Z2 (t) := exp(it Arg Uε ) and Z3 (t) := t|Aδ | + (1 − t)I , where t ∈ [0, 1]. Each Zj (t) is a path in L−1 0 , and so is Z(t) := Z1 (t)Z2 (t)Z3 (t). Obviously, Z(0) = I , Z(1) = S and Z(t)−1 = t|S| + (1 − t)I −1 = t S −1 −1 + (1 − t) −1 max 1, S −1 . One can easily see that Z1 (t) = Z2 (t) = 1, Z3 (t) max{1, S} and Z3 (t) − Z3 (r) |t − r| max 1, S . Since |eitθ − eirθ | π|t − r| for all r, t ∈ R and θ ∈ (−π, π), we also have Zj (t) − Zj (r) π|t − r| for j = 1, 2. These inequalities imply (1.1). 2 2. Main results 2.1. Resolution of the identity The following theorem will be proved in Section 4. Roughly speaking, it says that a normal operator A ∈ Ln satisfying the condition (C) has a resolution of the identity in L associated with any finite open cover of σ (A). Theorem 2.1. Assume that L has real rank zero. Let A ∈ Ln , and let {Ωj }m j =1 be a finite open cover of σ (A). If A satisfies the condition (C) then there exists a family of mutually orthogonal projections Pj ∈ L such that m

Pj = I

and Pj H ⊂ ΠΩj H

for all j = 1, . . . , m,

(2.1)

j =1

where ΠΩj are the spectral projections of A corresponding to the sets Ωj . Remark 2.2. The operators Pj can be thought of as approximate spectral projections of A. If L is a von Neumann algebra then the spectral projections of A belong to L and one can simply take Pj = ΠΩj , where {Ωj }m j =1 is an arbitrary collection of mutually disjoint subsets Ωj ⊂ Ωj covering σ (A). However, even in this situation Theorem 2.1 may be useful, since the projections Pj constructed in the proof continuously depend on A in the norm topology. The following simple lemma shows how Theorem 2.1 can be applied for approximation purposes. Lemma 2.3. Let A ∈ Ln , and let {Ωj }m j =1 be a finite open cover of σ (A) whose multiplicity does not exceed k. If there exist mutually orthogonal projections Pj satisfying (2.1) then m √ zj Pj k max(diam Ωj ) A − j j =1

for any collection of points zj ∈ Ωj .

(2.2)

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Proof. If zj ∈ Ωj then 2 m m m ΠΩ (Au − zj u)2 zj Pj u = Pj Au − zj Pj u2 Au − j j =1

j =1

=

j =1

m m (A − zj I )ΠΩ u2 ΠΩj u2 (diam Ωj )2 j j =1

j =1

ku max(diam Ωj ) 2

2

j

for all u ∈ H . Taking the supremum over u, we obtain (2.2).

2

Theorem 2.2 and Lemma 2.3 imply the following corollaries. Corollary 2.4. The following statements are equivalent. (1) A C ∗ -algebra L has real rank zero. (2) Every self-adjoint operator A ∈ Ls has approximate spectral projections in the sense of Theorem 2.1, associated with any finite open cover of its spectrum. (3) Ls = Lf ∩ Ls . Corollary 2.5. Assume that L has real rank zero. Then for every normal operator A ∈ Ln the following statements are equivalent. (1) The operator A satisfies the condition (C). (2) The operator A has approximate spectral projections in the sense of Theorem 2.1, associated with any finite open cover of its spectrum. (3) A ∈ Lf ∩ Ln . Proof. The corollaries are proved in the same manner. By Remark 1.6, every self-adjoint operator A ∈ Ls satisfies the condition (C). Therefore the implications (1) ⇒ (2) follow from Theorem 2.1. Any subset of C admits a cover {Ωj }m j =1 of multiplicity four by open squares Ωj of arbitrarily small size. If A ∈ Ln has approximate spectral projections Pj associated with all such covers of its spectrum then, in view of (2.2), the operator A can be approximated by operators m of the z P ∈ L ∩ L . Moreover, if A ∈ L then we can take z ∈ R, so that form m j j f n s j j =1 j =1 zj Pj ∈ Lf ∩ Ls . Thus (2) ⇒ (3). mFinally, in view of Remark 1.2, every operator T ∈ Lf ∩ Ln can be written in the form j =1 zj Πj , where zj ∈ R whenever T ∈ Ls and Πj are mutually orthogonal projections lying in L. If A is approximated by a sequence of such operators then it can also be approximated by a sequence of operators m j =1 z˜ j Πj , where Πj are the same projections, z˜ j = 0 and Im z˜ j = Im zj . This shows that (3) ⇒ (1). 2 Remark 2.6. The implications (1) ⇔ (3) in the above corollaries are known results (see Section 5.1 and [16, Theorem 3.2]). In [16], the authors explained that the part (1) ⇒ (3) of Corollary 2.5 would follow from the existence of projections ‘that approximately commute with


2909

A and approximately divide σ (A) into disjoint components’. The implications (1) ⇒ (2) ⇒ (3) in Corollary 2.5 give a precise meaning to their statement. 2.2. The main theorem In this subsection • B(r) := {T ∈ L: T r} is the closed ball about the origin in L of radius r; • MT denotes the convex hull of the set S1 ,S2 ∈B(1) S1 T S2 , where T ∈ L; • JT is the two-sided ideal in L generated by the operator T ∈ L. Remark 2.7. The ideal JT consists of all finite linear combinations

of operators of the form S1 T S2 where S1 , S2 ∈ L. Therefore, MT ⊂ JT ∩ B(T ) and JT = t0 tMT . (see, for examRemark 2.8. The unit ball B(1) coincides with the closed convex hull of Lu

ple, [26]). This implies that MT is a subset of the closed convex hull of the set U,V ∈Lu U T V .

Moreover, if L−1 = L then B(1) coincides with the convex hull of Lu (see [25]) and, consequently, every element of MT is a finite convex combination of operators of the form U T V with U, V ∈ Lu . We shall say that a continuous real-valued function f satisfies the condition (Cε,r ) for some ε, r > 0 if f is defined on the interval [−r − ε, r + ε] and (Cε,r ) there exists ε ∈ (0, ε) such that the set {x ∈ R: f (x) = y} is an ε -net in {x ∈ R: x 2 + y 2 (r + ε)2 } for each y ∈ [−r − ε, r + ε].

The condition (Cε,r ) is fulfilled whenever the function f sufficiently rapidly oscillates between −r − ε and r + ε. In particular, it holds for the function f (x) = (r + ε) cos(πx/ε). Lemma 2.9. Let L have real rank zero, and let X, Y ∈ Ls be self-adjoint operators such that the operator A := X + iY satisfies the condition (C). Then, for every function f satisfying the condition (Cε,r ) with r = A, the operator Y belongs to the closure of the set f (X + B(ε) ∩ Ls ) + J[X,Y ] ∩ Ls . Proof. Assume first that J[X,Y ] = {0}, so that A is normal. By Corollary 2.5, for each δ ∈ (0, ε] there exists an operator Aδ ∈ Ln ∩ Lf with finite spectrum σ (Aδ ) = {z1 , . . . , zm } such that A − Aδ δ and, consequently, |zj | r + δ for all j . In view of (Cε,r ), one can find real numbers εk ∈ [−ε , ε ] such that zk + εk lie on the graph of f for all k = 1, . . . , m. Let Aδ be the operator with eigenvalues zk + εk and the same spectral projections as Aδ . Then Im Aδ = f (Re Aδ ), Y − Im Aδ δ and X − Re Aδ δ + ε . This implies that Y + B(δ) ∩ f X + B(ε) ∩ Ls = ∅,

∀δ ∈ 0, ε − ε .

Letting δ → 0, we see that Y ∈ f (X + B(ε) ∩ Ls ). Assume now that J[X,Y ] = {0} and denote L := J[X,Y ] . Let us consider the quotient C ∗ algebra L/L and the corresponding quotient map π : L → L/L . Since the map π is continuous

2910


and πS = π(S + S ∗ )/2 for all self-adjoint elements πS ∈ L/L , the quotient algebra also has −1 real rank zero and πS ∈ (L/L )−1 0 whenever S ∈ L0 . The latter implies that the normal element πA of the quotient algebra L/L also satisfies the condition (C). Applying the previous result with ε replaced by an arbitrary ε0 ∈ (ε , ε) to πA = πX + iπY , we can find a sequence of operators Xn ∈ Ls such that f (πXn ) → πY as n → ∞ and πX − πXn ε0 for all n. Since T Re T for all T ∈ L, we have πT := inf T − R = R∈L

inf

R∈L ∩Ls

T − R,

∀T ∈ Ls .

(2.3)

Therefore, there exist operators Rn ∈ L ∩ Ls such that Xn + Rn ∈ X + B(ε) ∩ Ls . Since f can be uniformly approximated by polynomials on any compact subset of R and Q(πXn ) = πQ(Xn ) for any polynomial Q, we have f (πXn ) = f π(Xn + Rn ) = πf (Xn + Rn ) and, consequently, π(f (Xn + Rn ) − Y ) → 0. In view of (2.3), there exist operators R˜ n ∈ L ∩ Ls such that f (Xn + Rn ) + R˜ n → Y as n → ∞. This implies that Y belongs to the closure of the set f (X +B(ε)∩Ls )+L ∩Ls which coincides with f (X + B(ε) ∩ Ls ) + J[X,Y ] ∩ Ls . 2 Corollary 2.10. Assume that L has real rank zero. If A ∈ L satisfies (C) then A ∈ B A ∩ Ln + J[A∗ ,A] ∩ Ls .

(2.4)

Proof. Let r := A. Given ε > 0, let us choose a function fε satisfying the condition (Cε,r ) whose graph lies in the disc {x 2 + y 2 (r + ε)2 }. Applying Lemma 2.9, we can find an operator Xε ∈ Re A + B(ε) ∩ Ls such that Im A ∈ fε (Xε ) + J[A∗ ,A] ∩ Ls + B(ε) ∩ Ls . The operator A˜ ε := Xε + ifε (Xε ) is normal, and A − A˜ ε ∈ J[A∗ ,A] ∩ Ls + B(2ε) ∩ Ls . Therefore, there exist operators Rε ∈ J[A∗ ,A] ∩ Ls such that A˜ ε + Rε → A as ε → 0. Since x 2 + (f (x))2 (r + ε)2 , we have 2 A˜ ε 2 = Xε 2 + fε (Xε ) (r + ε)2 . If Aε := r(r + ε)−1 A˜ ε then, by the above, Aε ∈ B(r) ∩ Ln and Aε + Rε → A as ε → 0.

2

Remark 2.11. By Corollary 2.10, Ln + L = Ln + L ∩ Ls for any two-sided ideal L ⊂ L in a C ∗ -algebra L of real rank zero. Indeed, if A ∈ Ln + L then J[A∗ ,A] ⊂ L and, in view of (2.4), A can be approximated by operators from Ln + L ∩ Ls . The following refinement of Corollary 2.10 is the main result of the paper. Theorem 2.12. There is a nonincreasing function h : (0, ∞) → [0, ∞) such that h(ε) = 0 for all ε 1 and A ∈ B A ∩ Ln + h(ε)M[A∗ ,A] ∩ Ls + B(ε)

(2.5)

for all ε ∈ (0, ∞), all C ∗ -algebras L of real rank zero and all operators A ∈ B(1) satisfying the condition (C).


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Remark 2.13. In other words, the inclusion (2.5) means that for each ε > 0 there exist a normal operator T (ε) ∈ Ln and a finite linear combination S(ε) =

c(j, ε)S1 (j, ε) A∗ , A S2 (j, ε)

(2.6)

j

with Sk (j, ε) ∈ L and c(j, ε) ∈ (0, 1] such that T (ε) A, S(ε) ∈ Ls , Sk (j, ε) 1, j c(j, ε) = 1 and A − T (ε) − h(ε)S(ε) ε. Note that (2.6) can be written as a linear combination of self-adjoint operators, S(ε) =

∗ ∗ c(j, ε) S+ (j, ε) A∗ , A S+ (j, ε) − S− (j, ε) A∗ , A S− (j, ε)

(2.7)

j

where S± (j, ε) 1. Indeed, if S± (j, ε) := 12 (S1∗ (j, ε) ± S2 (j, ε)) then the real part of each term in the right-hand side of (2.6) coincides with corresponding term in (2.7). Proof. Let us consider a family of C ∗ -algebras Lξ of real rank zero parameterised by ξ ∈ Ξ , where Ξ is an arbitrary index set, and let L be their direct product. By definition, the C ∗ -algebra L consists of families S = {Sξ } with Sξ ∈ Lξ such that SL := supξ ∈Ξ Sξ < ∞, S ∗ := {Sξ∗ } and S S˜ = {Sξ S˜ξ }. Let BL (r) and Bξ (r) be the balls of radius r about the origin in L and Lξ respectively. In view of Lemma 1.9(2), L has real rank zero. Lemma 1.9(3) implies that {Sξ } ∈ L−1 0 −1 whenever {Sξ } ∈ L, Sξ ∈ (Lξ )−1 0 for each ξ ∈ Ξ and supξ ∈Ξ Sξ < ∞. From here and Lemma 1.9(1) it follows that A = {Aξ } ∈ L satisfies the condition (C) whenever all the operators Aξ satisfy (C). Let us fix ε ∈ (0, 1) and consider an arbitrary family A = {Aξ } ∈ L of operators Aξ ∈ Lξ satisfying (C). Applying Corollary 2.10 to A, we can find families of operators Tε = {Tξ,ε } ∈ BL (AL ) ∩ Ln and Rε = {Rξ,ε } ∈ J[A∗ ,A] ∩ Ls such that A − Tε − Rε L ε. The estimate for the norm holds if and only if Aξ − Tξ,ε − Rξ,ε ∈ Bξ (ε) for all ξ ∈ Ξ . The inclusion Tε ∈ BL (AL ) ∩ Ln means that Tξ,ε ∈ Bξ (AL ) ∩ (Lξ )n for all ξ ∈ Ξ . Finally, by Remark 2.7, J[A∗ ,A] = t0 tM[A∗ ,A] . This identity and the inclusion Rε ∈ J[A∗ ,A] ∩ Ls imply that Rξ,ε ∈ tM[A∗ξ ,Aξ ] ∩ (Lξ )s for all ξ ∈ Ξ and some t independent of ξ . Thus we obtain Aξ ∈ Bξ AL ∩ (Lξ )n + tM[A∗ξ ,Aξ ] ∩ (Lξ )s + Bξ (ε),

∀ξ ∈ Ξ,

(2.8)

where t is a nonnegative number which does not depend on ξ . If (2.5) were not true for any h(ε) ∈ [0, ∞) then there would exist families of C ∗ -algebras Lξ and operators Aξ ∈ Lξ satisfying the condition (C), for which (2.8) would not hold with any t independent of ξ . However, by the above, it is not possible. Thus we have (2.5) with some function h for all ε ∈ (0, 1). Since A ∈ B(1), we can extend h(ε) by zero for ε 1. It remains to notice that the function h can be chosen nonincreasing because the same inclusion holds for ˜ h(ε) := suptε h(t). 2

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If t ∈ [0, ∞), let F (t) := inf h(ε)t + ε ,

(2.9)

ε>0

where h is the function introduced in Theorem 2.12. The function F is nondecreasing, F (0) = 0 and 0 < F (t) 1 for all t > 0. Since the subgraph of F coincides with an intersection of halfplanes, F is concave and, consequently, continuous. Corollary 2.14. Let L be a C ∗ -algebra of real rank zero, and let · be a continuous seminorm on L such that U SV S

and S C S for all S ∈ L and all U, V ∈ Lu ,

(2.10)

where C is a positive constant. Then inf

T ∈Ln : T A

A − T C F C −1 A∗ , A

(2.11)

for all operators A ∈ B(1) satisfying the condition (C). Proof. In view of Remark 2.8, from the inequalities (2.10) it follows that S1 SS2 S1 S S2 for all S, S1 , S2 ∈ L

(2.12)

and, consequently, S [A∗ , A] for all S ∈ M[A∗ ,A] . Since R C R εC for all R ∈ B(ε), the inclusion (2.5) implies that inf

T ∈Ln : T A

A − T h(ε) A∗ , A + εC ,

Taking the infimum over ε > 0, we obtain (2.11).

∀ε > 0.

2

Remark 2.15. It is clear from the proof that (2.11) can be extended to general functions · : L → R+ satisfying (2.10) and suitable quasiconvexity conditions. Example 2.16. Let J be a two-sided ideal in L. Then the seminorm A := dist(A, J ) satisfies the conditions (2.10) with C = 1. Corollary 2.14 implies that dist A, J + B A ∩ Ln F dist A∗ , A , J for all A ∈ B(1) satisfying the condition (C).

(2.13)


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3. Applications Throughout this section • • • • •

C(H ) is the C ∗ -algebra of compact operators in H ; Sp are the Schatten classes of compact operators; and · p are the corresponding norms (we shall always be assuming that p 1); Aess := infK∈C (H ) A − K is the distance from A to C(H ); F is the function defined by (2.9).

3.1. Matrices Let L be the linear space of all complex m × m matrices. Then the Schatten norms · p on L satisfy (2.10) with C = m1/p . Corollary 2.14 implies that inf A − T

T ∈Ln

A − T F A∗ , A

(3.1)

A − T p m1/p F m−1/p A∗ , A p

(3.2)

inf

T ∈Ln : T A

and inf A − T p

T ∈Ln

inf

T ∈Ln : T A

for all p ∈ [1, ∞), all m = 1, 2, . . . and all A ∈ L such that A 1. Note that the S2 -distance from a given m × m-matrix A to the set of normal matrices admits the following simple description. Lemma 3.1. Let A be an m × m-matrix, and let Σm (A) be the set of all complex vectors z ∈ Cm of the form z = (Au1 , u1 ), (Au2 , u2 ), . . . , (Aum , um ) where {u1 , u2 , . . . , um } is an orthonormal basis. Then inf A − T 22 =

T ∈Ln

inf

T ∈Ln : T A

A − T 22 = A22 −

sup |z|2 .

(3.3)

z∈Σm (A)

Proof. If T is an arbitrary normal matrix and {u1 , u2 , . . . , um } is the basis formed by its eigenvectors then A − T 22

m

(Auj , uk ) 2 = A2 −

(Auj , uj ) 2 A2 − 2

j =k

2

j =1

sup |z|2 .

(3.4)

z∈Σm (A)

Therefore infT ∈Ln A − T 22 A22 − supz∈Σm (A) |z|2 . On the other hand, since the set Σm (A) is compact, supz∈Σm (A) |z|2 = |z0 |2 for some z0 ∈ Σm (A). Let us write down the matrix A in a corresponding orthonormal basis {v1 , . . . , vm } and denote by T0 the normal matrix obtained by removing the off-diagonal elements. Then T0 A and A − T0 22 = j =k |(Avj , vk )|2 = A22 − |z0 |2 . 2

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The following example shows that for p = 2 the estimate (3.2) is order sharp as m → ∞. m Example 3.2. Let m be even, and let {ej }m j =1 be the standard Euclidean basis in C . Consider the m × m-matrix ⎞ ⎛ 0 1 0 0 ... 0 0 ⎜0 0 0 0 ... 0 0⎟ ⎟ ⎜ ⎜0 0 0 1 ... 0 0⎟ ⎟ ⎜ A = ⎜0 0 0 0 ... 0 0⎟ ⎟ ⎜ ⎜ . . . . ... . . ⎟ ⎠ ⎝ 0 0 0 0 ... 0 1 0 0 0 0 ... 0 0

defined by the identities Ae2i = e2i−1 and Ae2i−1 = 0, where i = 1, . . . , m/2. By direct calculation, A = 1 and [A, A∗ ] = diag(1, −1, . . . , 1, −1, ), so that [A, A∗ ]2 = m1/2 . For this matrix A, inf A − T 2 = (m/4)1/2 = m1/2 2−1 m−1/2 A, A∗ 2 .

T ∈Ln

m Indeed, let {uj }m j =1 be an orthonormal basis in C . Then

uj =

m/2 (αi,j e2i−1 + βi,j e2i ), i=1

where αi,j and βi,j are complex numbers such that (Auj , uj ) =

m/2

m/2

2 i=1 (|αi,j |

(βi,j e2i−1 , uj ) =

i=1

m/2

+ |βi,j |2 ) = 1. Clearly,

βi,j αi,j .

i=1

Therefore m/2

|αi,j |2 + |βi,j |2 = 1 2 (Auj , uj ) i=1

m/2 and, consequently, j =1 |(Auj , uj )|2 m/4. Thus we have |z|2 m/4 for all z ∈ Σm (A). Since A22 = m/2, Lemma 3.1 implies that A − T 22 m/4 for all normal matrices T . On the other hand, if T0 = Re A then A − T0 22 = Im A22 = m/4. Remark 3.3. If T is a normal matrix and A is an arbitrary matrix of the same size then ∗ ∗ ∗ A , A = A , T + A , A − T = (A − T )∗ , T + A∗ , (A − T ) . Estimating the Schatten norms of the right- and left-hand sides, we obtain ∗ A , A 2 A + T A − T p . p


2915

This implies that inf

T ∈Ln : T A

A − T p

[A∗ , A]p 4A

(3.5)

for all finite matrices A and all p ∈ [1, ∞]. Substituting the matrix A from Example 3.2 into (3.5), we see that the second estimate (3.2) is order sharp as m → ∞ for all p ∈ [1, ∞). 3.2. Bounded and compact operators If L is the C ∗ -algebra obtained from C(H ) by adjoining the unity then, by Remark 1.6, = L−1 0 and all A ∈ L satisfy the condition (C). Thus our main results hold for all compact operators A. If L = B(H ) then L−1 = L−1 0 because every unitary operator can be joined with I by the path exp(it Arg U ) (see Lemma 1.7). However, in the infinite dimensional case L−1 = B(H ). The following result was obtained in [13] (it also follows from [9, Theorem 4.1] or [3, Theorem 3]). L−1

Lemma 3.4. Let H be separable, and let L = B(H ). Then an operator A satisfies the condition (C) if and only if for each λ ∈ C either the range of (A − λI ) is not closed or dim ker(A − λI ) = dim ker(A∗ − λI ). In other words, Lemma 3.4 states that in the separable case (C) is equivalent to the condition on the index function in the BDF theorem. In particular, this implies that normal operators and their compact perturbations satisfy the condition (C). Remark 3.5. An explicit description of the closure of the set of invertible operators in a nonseparable Hilbert space was obtained in [4]. 3.3. The BDF theorem In this subsection we are always assuming that H is separable and L = B(H ). Recall that an operator A ∈ B(H ) is called quasidiagonal if it can be represented as the sum of a block diagonal and a compact operator, that is, if there exist mutually orthogonal finite dimensional subspaces Hk and operators Sk : Hk → Hk such that H = ∞ k=1 Hk and A = diag{S1 , S2 , . . .} + K, where K ∈ C(H ). We shall need the following well-known result. Lemma 3.6. The set of compact perturbations of normal operators on a separable Hilbert space is norm closed and coincides with the set of quasidiagonal operators S ∈ B(H ) such that [S ∗ , S] ∈ C(H ). Lemma 3.6 follows from the BDF theorem but it also admits a simple independent proof based on Theorem 0.2 (see [16, Proposition 2.8]). Obviously, the BDF theorem is an immediate consequence of Corollary 2.10 and Lemma 3.6. One obtains a slightly better result by applying

2916


the following lemma, which shows that the BDF theorem holds with a normal operator T such that T A. Lemma 3.7. Let H be separable, and let L = B(H ). Then, for each fixed r > 0, the set B(r) ∩ Ln + C(H ) is closed and coincides with the set of quasidiagonal operators of the form diag{S1 , S2 , . . .} + K such that K ∈ C(H ), Sk are normal and Sk r for all k. Proof. Obviously, if A = diag{S1 , S2 , . . .} + K then A ∈ B(r) ∩ Ln + C(H ) whenever Sk and K satisfy the conditions of the lemma. Assume now that A ∈ B(r) ∩ Ln + C(H ). Then Aess r, [A∗ , A] ∈ C(H ) and, by acting in Lemma 3.6, A = diag{S1 , S2 , . . .} + K , where K ∈ C(H ) and Sk are operators some mutually orthogonal finite dimensional subspaces Hk such that H = k Hk . Since the self-commutator [A∗ , A] is compact, we have [(Sk )∗ , Sk ] → 0 as k → ∞. By (3.1), there are normal operators Sk : Hk → Hk such that Sk − Sk → 0 as k → ∞. The operator diag{S1 − S1 , S2 − S2 , . . .} is compact, so that A = diag{S1 , S2 , . . .} + K where K ∈ C(H ). Since Aess r, we have lim supk→∞ Sk r. Define Sk :=

if Sk r, Sk , −1 rSk Sk , if Sk > r.

Clearly, Sk are normal and Sk r. The estimate for the upper limit implies that lim supSk − Sk = lim sup Sk − r + = 0. k→∞

k→∞

It follows that the operator diag{S1 − S1 , S2 − S2 , . . .} is compact and, consequently, A = diag{S1 , S2 , . . .} + K where K ∈ C(H ). 2 Theorem 2.12 and Lemma 3.7 also imply the following quantitative version of the BDF theorem. Theorem 3.8. Let H be separable, and let A ∈ B(H ) be an operator with A 1 satisfying the condition (C). (1) If [A∗ , A] ∈ C(H ) then for each ε > 0 there exists a diagonal operator Tε ∈ B(H ) such that A − Tε ∈ C(H ), Tε A and A − Tε F A∗ , A + ε. (2) If [A∗ , A] ∈ / C(H ) then for each ε > 0 there exists a diagonal operator Tε ∈ B(H ) such that A − Tε ess 2F ([A∗ , A]ess ) and A − Tε 5F A∗ , A + 3F 2F A∗ , A ess + ε. Proof. Since a block diagonal normal operator is represented by a diagonal matrix in the orthonormal basis formed by its eigenvectors, it is sufficient to construct a block diagonal normal Tε satisfying the above conditions.


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Assume first that [A∗ , A] ∈ C(H ). Then, by Corollary 2.10 and Lemma 3.7, we have A = diag{S1 , S2 , . . .} + K, where K ∈ C(H ) and Sk are normal operators in finite dimensional subsuch that Sk A. Let us denote by En the orthogonal projections onto the spaces Hk subspaces nk=1 Hk and define δn := K − En KEn . Since (En AEn )∗ , En AEn = En A∗ En A − AEn A∗ En = En A∗ [En , A] + A∗ , A + A A∗ , En En , [En , A]En = [En , K]En = (En KEn − K)En and [A∗ , En ]En = (K ∗ − En K ∗ En )En , we have (En AEn )∗ , En AEn A∗ , A + 2δn A A∗ , A + 2δn . Applying (3.1) to the finite rank operators En AEn , we can find normal operators An acting in En H such that An En AEn A and En AEn − An F A∗ , A + 2δn + δn . The block diagonal operators Tñ := An ⊕ diag{Sn+1 , Sn+2 , . . .} are normal, Tñ A and A − Tñ = (En AEn − An ) + (K − En KEn ),

∀n = 1, 2, . . . .

The above identity implies that A − Tñ ∈ C(H ) and A − Tñ F A∗ , A + 2δn + 2δn ,

∀n = 1, 2, . . . .

Since K is compact, limn→∞ δn = 0 and, consequently, limn→∞ A − Tñ = F ([A∗ , A]). Thus we can take Tε := Tñ with a sufficiently large n. Assume now that [A∗ , A]ess > 0. From (2.13) with J = C(H ) it follows that A = S + K + R, where S is a bounded normal operator, K ∈ C(H ) and R is a bounded operator with R 2F ([A∗ , A]ess ). Since |F | 1, we have S + K 3. Let A := 13 (S + K). Then [(A )∗ , A ] ∈ C(H ), A 1 and, in view of Lemma 3.4, A satisfies the condition (C). Applying (1) to A , we can find a block diagonal normal operator Tε and a compact operators Kε such that A = Tε + Kε and Kε F ([(A )∗ , A ]) + ε/3. The identities 3A = S + K = A − R and the above estimates for A, R and S + K imply that ∗ 1 ∗ A , A A , A + 2RS + K + 2AR 9 A∗ , A + 2F A∗ , A ess . Obviously, A = 3Tε + 3Kε + R and A − 3Tε ess = Ress 2F ([A∗ , A]ess ). Since [A∗ , A]ess [A∗ , A] and the function F is nondecreasing and concave, from the above estimates it follows that

2918


ε K F A∗ , A + 2F A∗ , A + ε ess 3 ∗ ε ∗ F A , A + F 2F A , A ess + 3 and, consequently, A − 3T R + 3K 5F A∗ , A + 3F 2F A∗ , A + ε. ε ε ess Thus we can take Tε := 3Tε .

2

Remark 3.9. Since ε can be chosen arbitrarily small, the distance from an operator A satisfying the conditions of Theorem 3.8 to the set of diagonal operators does not exceed 5F ([A∗ , A]) + 3F (2F ([A∗ , A]ess )). If A is normal then this sum is equal to zero and Theorem 3.8 turns into the Weyl–von Neumann–Berg theorem for bounded operators. 3.4. Truncations of normal operators Let G be a positive unbounded self-adjoint operator in a separable Hilbert space H whose spectrum consists of eigenvalues of finite multiplicity accumulating to ∞. Denote its spectral projections corresponding to the intervals (0, λ) by Pλ , and let N(λ) := rank Pλ

and N1 (λ) := sup N (μ) − N (μ − 1) . μλ

If B ∈ B(H ) and [G, B] ∈ B(H ) then, according to [19, Theorem 1.3], (I − Pλ )BPλ 2 (I − Pλ )B(Pλ − Pλ−1 )2 + (I − Pλ )[G, B](G − λI )−1 Pλ−1 2 . 2 2 2

(3.6)

2

A direct calculation shows that (G − λI )−1 Pλ−1 22 π6 N1 (λ) (see [19] for details). This estimate, (3.6) and the obvious inequality Pλ − Pλ−1 22 N1 (λ) imply that 2 (I − Pλ )BPλ 2 B2 + π [G, B]2 N1 (λ). 2 6

(3.7)

Let A ∈ B(H ) be a normal operator such that [G, A] ∈ B(H ), and let Aλ := Pλ APλ be its truncation to the subspace Pλ H . Then

A∗λ , Aλ = Pλ A∗ (I − Pλ )APλ − Pλ A(I − Pλ )A∗ Pλ

and, consequently, ∗ A , Aλ Pλ A∗ (I − Pλ )APλ + Pλ A(I − Pλ )A∗ Pλ . λ 1 1 1 Since Pλ B ∗ (I − Pλ )BPλ 1 = Tr(Pλ B ∗ (I − Pλ )BPλ ) = (I − Pλ )BPλ 22 , applying (3.7) with B = A and B = A∗ , we obtain


∗ A , Aλ CA N1 (λ), λ 1 where CA := 2A2 +

π2 2 3 [G, A] .

2919

(3.8)

The inequalities (3.2) and (3.8) imply that

inf Aλ − Tλ 1 N (λ)F CA N1 (λ)/N (λ) , Tλ

(3.9)

where the infimum is taken over all normal operators Tλ acting in the finite dimensional subspace Pλ H . Assume that there exist positive constants c and such that N (λ) = cλ + o(λ ) as λ → ∞ (that is, we have a Weyl type asymptotic formula for the counting function N (λ)). Then N1 (λ)/N (λ) → 0 and, consequently, F (CA N1 (λ)/N (λ)) → 0 as λ → ∞. Therefore, in view of (3.9), there exist normal operators T˜λ acting in the subspaces Pλ H such that λ− Aλ − T˜λ 1 → 0 as λ → ∞. Roughly speaking, this means that, under the above conditions on A and N (λ), the normalized truncations λ− Aλ are asymptotically close to normal matrices with respect to the S1 -norm. Remark 3.10. The Weyl asymptotic formula holds for elliptic self-adjoint pseudodifferential operators on closed compact manifolds and differential operators on domains with appropriate boundary conditions. If G is a pseudodifferential operator of order 1 and A is the multiplication by a smooth function, as in the Szegö limit theorem [28], then A and [G, A] are bounded in the corresponding space L2 and we have (3.9). Remark 3.11. In [19], the classical Szegö limit theorem was extended to wide classes of self-adjoint (pseudo)differential operators G and A. More precisely, the authors proved that Tr f (Aλ ) ∼ Tr Pλ f (A)Pλ as λ → ∞ for all sufficiently smooth functions f : R → R and all self-adjoint operators G and A satisfying the above conditions. If f : C → C and the operator A is normal then the right-hand side of the above asymptotic formula is well defined. However, generally speaking, the truncations Aλ are not normal matrices and the left-hand side does not make sense. The results of this subsection suggest that similar limit theorems can be obtained for (almost) normal operators A, provided that Tr f (Aλ ) is understood in an appropriate sense. For instance, it is plausible that the asymptotic formula holds for all sufficiently smooth functions f : C → C if one defines Tr f (Aλ ) := j f (μj ), where μj are the eigenvalues of Aλ (see [27]). 4. Resolution of the identity The proof of Theorem 2.1 is based on successive reductions of the operator A ∈ Ln to normal operators whose spectra do not contain certain subsets of the complex plane. One can think of this process as removing subsets from σ (A). After each step we obtain a new normal operator lying in Ln . The main problem is that, in order to carry on the reduction procedure, one has to ensure that the removal of a subset Ω from the spectrum does not change the spectral projection corresponding to C \ Ω, and that the new operator still satisfies the condition (C). In our scheme this is guaranteed by the equality A(I − ΠΩ ) = AΩ (I − ΠΩ ) and the condition (a4 ). Further on • Dr (λ) is the open disc of radius r centred at λ ∈ C, and ∂Dr (λ) is its boundary; • S := ∂D1 (0) is the unit circle about the origin.

2920


We shall need the following lemmas which will be proved in the next two subsections. Lemma 4.1. Let A ∈ Ln , and let ΠΩ be the spectral projection of A corresponding to an open set Ω ⊂ C. If Ω is homeomorphic to the disc D1 (0) and A − μI ∈ L−1 0 for some μ ∈ Ω then there exists a normal operator RΩ in the subspace ΠΩ H such that (a1 ) (a2 ) (a3 ) (a4 )

(A − RΩ )ΠΩ ∈ L and, consequently, AΩ := A(I − ΠΩ ) ⊕ RΩ ∈ Ln ; σ (RΩ ) ⊂ ∂Ω, so that σ (AΩ ) ⊂ (σ (A) \ Ω) ∪ ∂Ω; AΩ − λI ∈ L−1 0 for all λ ∈ Ω; if A satisfies the condition (C) then so does the operator AΩ .

In other words, we can remove the set Ω from σ (A) by adding a perturbation which does not change A(I − ΠΩ ). Moreover, (a5 ) A − AΩ = (A − RΩ )ΠΩ 2r, where r is the radius of the minimal disc containing Ω. This shows that the perturbation is small whenever Ω is a subset of a small disc. However, the new operator AΩ may have additional spectrum lying on ∂Ω. In view of the above, Lemma 4.1 is not sufficient for the study of operators with onedimensional spectra, as does not allow one to split the one-dimensional spectrum into disjoint components. This problem is resolved by Lemma 4.2. Let the conditions of Lemma 4.1 be fulfilled. Assume, in addition, that (i) L has real rank zero; (ii) σ (A) ∩ Ω is a subset of a simple contour γ which intersects ∂Ω at two points; (iii) A − μI ∈ L−1 0 for all μ ∈ Ω \ γ . Then there exists a normal operator RΩ : ΠΩ H → ΠΩ H satisfying the conditions (a1 ), (a3 ), (a4 ) and (a2 ) σ (RΩ ) ⊂ γ ∩ ∂Ω, so that σ (AΩ ) ⊂ (σ (A) \ Ω) ∪ (γ ∩ ∂Ω). 4.1. Proof of Lemma 4.1 The proof proceeds in three steps. 4.1.1. Assume first that Ω = Dε (0) and μ = 0, that is, A ∈ L−1 0 . For the sake of brevity, we shall denote Πε := ΠDε (0) , Rε := RDε (0) and Aε := ADε (0) . Let A = V |A| be the polar decomposition of A. Let us consider a sequence of operators Bn ∈ L−1 0 such that Bn → A as n → ∞, and let Bn = Vn |Bn | be their polar decompositions. Then |Bn | → |A| and Vn |A| → V |A| as n → ∞. Since Vn |Bn | = |Bn∗ |Vn and |Bn∗ | → |A∗ | = |A|, we also have |A|Vn → |A|V as n → ∞. It follows that Vn ρ(|A|) → Vρ(|A|) and ρ(|A|)Vn → ρ(|A|)V as n → ∞ for every continuous function ρ : R+ → R vanishing near the origin.


2921

Let us fix arbitrary continuous nonnegative functions ρ1 and ρ2 of R+ such that ρ1 ≡ 1 and ρ2 ≡ 0 on the interval [ε, ∞), ρ1 ≡ 0 near the origin, and ρ12 + ρ22 ≡ 1. Let Sn := Vρ12 |A| + ρ2 |A| Vn ρ2 |A| . The operators Sn belong to L because Vn ∈ L (see Remark 1.1), ρ(|A|) ∈ L for all continuous functions ρ, and Vρ1 (|A|) = Aρ˜1 (|A|) where ρ˜1 (τ ) := τ −1 ρ1 (τ ) is a continuous function. Since V commutes with |A|, we have Sn − Vn = (V − Vn )ρ12 |A| − (V − Vn ) I − ρ2 |A| ρ2 |A| − I − ρ2 |A| (Vn − V )ρ2 |A| . By the above, the norm of the right-hand side converges to zero as n → ∞. Since Vn ∈ L−1 0 ∩ Lu −1 (see Lemma 1.7), this implies that Sn ∈ L0 for all sufficiently large n. Let us fix n such that Sn ∈ L−1 0 and consider the polar decomposition Sn = Un |Sn |. By −1 Lemma 1.7, Un ∈ L0 ∩ Lu . Since Sn (I − Πε ) = V (I − Πε ), the operator Sn coincides with the orthogonal sum V (I − Πε ) ⊕ Sn Πε . The unitary operator Un has the same block structure, Un = V (I − Πε ) ⊕ Un Πε . Let Rε be the restriction of εUn to the subspace Πε H . Obviously, σ (Rε ) is a subset of ∂Dε (0). Since (A − Rε )Πε = A − fε (|A|)Un where fε (t) := ε + (t − ε)+ , the operator Rε satisfies the condition (a1 ). We have Aε = A(I − Πε ) ⊕ Rε = fε (|A|)Un , where fε ε > 0 and Un ∈ L−1 0 . Therefore

−1 Aε ∈ L−1 0 (see Lemma 1.7). Since σ (Aε ) ∩ Dε (0) = ∅, Lemma 1.5 implies that Aε − λI ∈ L0 for all λ ∈ Dε (0). It remains to prove that Aε − λI ∈ L−1 / Dε (0) whenever A satisfies the condition 0 for λ ∈ (C). Let Πδ,λ be the spectral projection of A corresponding to the open disc Dδ (λ) of radius δ < |λ| − ε. Applying the above arguments to the operator A − λI , we can find an operator Rδ,λ acting in Πδ,λ H such that σ (Rδ,λ ) ⊂ ∂Dδ (λ), (A − Rδ,λ )Πδ,λ ∈ L and

A − (A − Rδ,λ )Πδ,λ − λI = A(I − Πδ,λ ) ⊕ Rδ,λ − λI ∈ L−1 0 .

(4.1)

Denote Bt,δ := (1 − t)A + tAε (I − Πδ,λ ) ⊕ Rδ,λ . Since Πε Πδ,λ = Πδ,λ Πε = 0 and Aε = A − (A − Rε )Πε , we have (1)

(2)

Bt,δ = A(I − Πδ,λ ) ⊕ Rδ,λ − t (A − Rε )Πε = Bδ ⊕ Bt

⊕ Rδ,λ ,

(4.2)

where Bδ(1) := A(I − Πδ,λ )(I − Πε ) and Bt(2) := ((1 − t)A + tRε )Πε . Obviously, σ (Bδ(1) ) ⊂ (2) (2) C \ Dδ (λ) and σ (Bt ) ⊂ Dε (0) for all t ∈ [0, 1] (because Bt,δ ε). Thus the spectra of all the operators in the orthogonal sum on the right-hand side of (4.2) do not contain the point λ. Therefore the operator Bt,δ − λI is invertible. The first equality (4.2) implies that Bt,δ ∈ L,

2922


so we have Bt,δ − λI ∈ L−1 for all t ∈ [0, 1]. By (4.1), B0,δ − λI ∈ L−1 0 and, consequently, −1 B1,δ − λI ∈ L0 . Now, letting δ → 0, we obtain lim (B1,δ − λI ) = lim Aε (I − Πδ,λ ) ⊕ Rδ,λ − λI = Aε − λI ∈ L−1 0 .

δ→0

δ→0

4.1.2. Let B ∈ Ln , and let ϕ : C → C be a homeomorphism isotopic to the identity. The results obtained in Section 4.1.1 imply that ϕ(B) satisfies the condition (C) whenever so does the operator B. Indeed, let us fix μ ∈ C and consider the isomorphism ψ : z → ϕ(z + ϕ −1 (μ)). Denote A := B − ϕ −1 (μ)I , and let Aε be the operators constructed in Section 4.1.1. We have μ ∈ / σ (ψ(Aε )) for all ε > 0 because ψ −1 (μ) = 0 ∈ / σ (Aε ). Moreover, since ϕ is isotopic to the identity, the same is true for ψ and, by Lemma 1.5, ψ(Aε ) − μI ∈ L−1 0 for all ε > 0. This implies that ϕ(B) − μI = ψ(A) − μI = lim ψ(Aε ) − μI ∈ L−1 0 . ε→0

4.1.3. Assume now that Ω is an arbitrary domain and μ ∈ Ω is an arbitrary point satisfying the conditions of the lemma. Let us fix a homeomorphism ψ : C → C isotopic to the identity such that ψ : Ω → D1 (0) and ψ(μ) = 0. Denote A˜ := ψ(A). Then ΠΩ coincides with the spectral projection of A˜ corresponding to the open disc D1 (0). By 4.1.1, there exists an operator R˜ 1 acting in the subspace ΠΩ H such that (A˜ − R˜ 1 )ΠΩ ∈ L ˜ − ΠΩ ) ⊕ R˜ 1 and RΩ := ψ −1 (R˜ 1 ). Obviously, the inand σ (R˜ 1 ) ⊂ ∂D1 (0). Let A˜ 1 := A(I verse image RΩ satisfies (a1 ) and (a2 ), and AΩ = ψ −1 (A˜ 1 ). Since ψ is isotopic to the identity, Lemma 1.5 implies (a3 ). Finally, by 4.1.2, if A satisfies the condition (C) then the same is true ˜ A˜ 1 (as was shown in 4.1.1) and AΩ . for the operators A, 4.2. Proof of Lemma 4.2 It is sufficient to prove the lemma in the case where Ω = D1 (0) and γ ∩ Ω = (−1, 1). After that, the general result is obtained by choosing a homeomorphism ψ isotopic to the identity such that ψ : Ω → D1 (0) and ψ : γ ∩ Ω → (−1, 1) and repeating the same arguments as in 4.1.3. Further on we always assume that Ω, γ and σ (A) are as above and write Π1 , R1 and A1 instead of ΠΩ , RΩ and AΩ . 4.2.1. Suppose first that σ (A) lies on a simple closed contour γ homeomorphic to S. Let ϕ : C → C be a homeomorphism isotopic to the identity such that ϕ : γ → S and ϕ(0) = −1. The operator ϕ(A) belongs to Lu because its spectrum lies in S. The condition (iii) and Lemma 1.5 imply that ϕ(A) ∈ L−1 0 . Therefore, by Lemma 1.8, there exist operators Wn ∈ Lu such that Wn → ϕ(A) as n → ∞ and −1 ∈ / σ (Wn ). Let Bn := ϕ −1 (Wn ) be their inverse images. Then Bn belong to Ln , σ (Bn ) ⊂ γ \ {0} for all n, and Bn → A as n → ∞. The rest of this subsection is similar to Section 4.1.1. Let us fix continuous nonnegative functions ρ1 and ρ2 of R+ such that ρ1 ≡ 1 and ρ2 ≡ 0 on the interval [1, ∞), ρ1 ≡ 0 near the origin, and ρ12 + ρ22 ≡ 1. Define Sñ := Vρ12 |A| + ρ2 |A| (Re Vn )ρ2 |A|


2923

where V and Vn are the isometric operators in the polar representations A = V |A| and Bn = Vn |Bn |. Note that ρ2 |Bn | (Re Vn )ρ2 |Bn | = (Re Vn )ρ22 |Bn | = Vn ρ22 |Bn |

(4.3)

because σ (Bn ) ∩ D1 (0) ⊂ (−1, 1) and ρ2 ≡ 0 outside the interval [0, 1). We have Sñ − Vn = (V − Vn )ρ12 |A| + ρ2 |A| (Re Vn )ρ2 |A| − Vn ρ22 |A| . Since ρ1 ≡ 0 in a neighbourhood of the origin, the first term in the right-hand side converges to zero. Since ρ2 (|Bn |) → ρ2 (|A|), the identity (4.3) implies that the second term also converges to zero. Thus Sñ − Vn → 0 as n → ∞ and, consequently, Sñ ∈ L−1 for all sufficiently large n. Let us fix n such that Sñ ∈ L−1 and consider the polar decomposition Sñ = U˜ n |Sñ |. The unitary operator U˜ n has the same block structure as Un in the proof of Lemma 4.1 but now, in addition, its restrictions to the subspace Π1 H is self-adjoint. Let R1 = U˜ n |Π1 H . Then R1 satisfies (a1 ) and its spectrum can contain only the points ±1, so we have (a2 ) instead of (a2 ). By Remark 1.6, A1 satisfies the condition (C), which implies (a3 ) and (a4 ). 4.2.2. Suppose now that σ (A) \ (−1, 1) is an arbitrary subset of C \ D1 (0). In the process of proof we shall introduce auxiliary operators A(1) and A(2) lying in Ln , such that ( ) the spectral projection of A(j ) corresponding to D1 (0) coincides with Π1 , and A(j ) Π1 = AΠ1 . Every next operator will have a simpler spectrum, and R1 will be defined in terms of A(2) . Let us consider the homotopy ψt : C → C defined by z ψt (z) = (1 − t)z + t z |z|

if z ∈ D1 (0), if z ∈ / D1 (0),

where t ∈ [0, 1],

and let A(1) := ψ1 (A). Since ψ1 : C \ D1 (0) → S and ψ1 (z) = z for all z ∈ D1 (0), the operator A(1) satisfies the condition ( ) and σ (A(1) ) ⊂ (−1, 1) ∪ S. In view of Lemma 1.5, A(1) also satisfies (iii). Denote by Π˜ the spectral projection of A(1) corresponding to the open lower semicircle S− := {z ∈ S: Im z < 0}. Now let us consider the homotopy ϕt : D1 (0) → D1 (0) defined by ϕt =

z − it Im z + it 1 − (Re z)2 z + it 1 − (Re z)2

if Im z 0, if Im z 0,

where t ∈ [0, 1],

and let A˜ := ϕ1 (A(1) ). Since ϕ1 : S− → (−1, 1), ϕ1 : (−1, 1) → S+ and ϕ1 : S+ → S+ , the ˜ lies on the contour γ formed by the interval [−1, 1] and the upper semicircle spectrum σ (A) S+ := {z ∈ S: Im z > 0}. By Lemma 1.5, the operator A˜ satisfies (iii).

2924


Note that ϕ1 |S− is a homeomorphism between S− and (−1, 1). Therefore, the spectral projection of A˜ corresponding to the interval (−1, 1) coincides with Π˜ . Applying 4.2.1 to the opera˜ we can find a self-adjoint operator R˜ acting in the subspace Π˜ H such that (A˜ − R) ˜ Π˜ ∈ L, tor A, ˜ ˜ ˜ ˜ ˜ σ (R) ⊂ {−1, 1} and A(I − Π ) ⊕ R satisfies the condition (iii). Since the restriction of A˜ to ΠH ˜ ˜ ˜ is self-adjoint, we have (A − R)Π ∈ Ls . ˜ Π˜ ), where ϕ˜ −1 : (−1, 1) → S− is the inverse mapping Let A(2) := A(1) − ϕ˜1−1 ((A˜ − R) 1 −1 (2) ˜ This implies that σ (A(2) ) ⊂ γ and A(2) ˜ ⊕ R. (ϕ1 |S− ) . Then we have A = A(1) (I − Π) satisfies the condition ( ). Moreover, A(2) satisfies (iii) because ˜ Π˜ − μI [0, 1] t → A(1) − t ϕ˜ 1−1 (A˜ − R) is a path in L−1 from A(1) − μI to A(2) − μI for each μ lying in the open domain bounded by γ . Finally, applying 4.2.1 to A(2) , we obtain an operator R1 in the subspace Π1 H such that (2) (A − R1 )Π1 = (A − R1 )Π1 ∈ Ls and σ (R1 ) ⊂ {−1, 1}. The latter inclusion and (ii) imply that σ ((A − tA + tR1 )Π1 ) ⊂ [−1, 1] for all t ∈ [0, 1]. Thus we have A − t (A − R1 )Π1 − μI ∈ L−1 ,

∀μ ∈ D1 (0) \ (−1, 1), ∀t ∈ [0, 1].

Since the operator A satisfies (iii), it follows that A1 − μI ∈ L−1 0 for all μ ∈ D1 (0), where A1 = A(I − Π1 ) ⊕ R1 . Now (a3 ) and (a4 ) are proved in the same way as in Lemma 4.1. 4.3. Proof of Theorem 2.1 Every open set Ωj coincides with the union of a collection of open discs. Since the spectrum σ (A) is compact, it is sufficient to prove the theorem assuming that Ωj is the union of a finite projections Pj,k such collection of open disks Dj,k . If there exist mutually orthogonal that j,k Pj,k = I and Pj,k H ⊂ ΠDj,k H then we can take Pj := k Pj,k . Thus we only need to prove the theorem for open discs Ωj . In the rest of the proof we shall be assuming that Ωj = Drj (zj ). The proof is by induction on m. If m = 1 then the result is obvious. Suppose that the theorem holds for m − 1 and consider a family of m open discs {Ωj }m j =k Ωj j =1 covering σ (A). If Ωk ⊂ for some k then we can take P = 0 and apply the induction assumption. Further on we shall be k

assuming that Ωk ⊂ j =k Ωj for all k = 1, . . . , m. 4.3.1. If r > t > 0, let Dr := Dr (zm )

and Dt,r := z ∈ C: t < |z − zm | < r .

Note that

σ (A) \ Dt ⊂

m−1 j =1

Ωj

(4.4)


2925

whenever rm − t is small enough. Indeed, if this were not true then there would exist a sequence of points μn ∈ σ (A) \ ( m−1 j =1 Ωj ) converging to ∂Ωm , and the limit point would not belong to

m j =1 Ωj . In the rest of the proof t ∈ (0, rm ) is assumed to be so close to rm that (4.4) holds. 4.3.2. In this subsection we are going to construct auxiliary operators A(i) ∈ Ln satisfying (C) and the following condition (i) (i) ( ) ΠΩ H ⊂ ΠΩj H for all j = 1, . . . , m, where ΠΩ are the spectral projections of A(i) corj j responding to Ωj .

Assume first that ∂Ωm ∩ Ωjk = ∅ for some indices jk m − 1. Let us fix an arbitrary r ∈ (t, rm ) and consider the open annulus Dt,r . The circles ∂Ωjk split Dt,r into a finite collection

of connected disjoint open sets Λα such that Dt,r = α Λα . Each set Λα is a circular polygon whose edges are arcs of the circles ∂Ωjk , ∂Dt and ∂Dr . Since Ωk ⊂ Dt,r for all k = 1, . . . , m, the boundaries ∂Λα are connected and, consequently, each polygon Λα is homeomorphic to a disc. Let us remove from σ (A) the open sets Λα , repeatedly applying Lemma 4.1. Then we obtain an operator A(1) ∈ Ln satisfying the condition (C), such that σ A(1) ⊂ Dt ∪

∂Λα ∪ (C \ Dr ) and σ A(1) \ Dr = σ (A) \ Dr .

α

Note that Λα ⊂ Ωj whenever ∂Λα ∩ Ωj = ∅. In view of (a2 ), this implies that the removal of Λα from the spectrum can only reduce the eigenspace corresponding to Ωj . Therefore A(1) satisfies the condition ( ). Now, repeatedly applying Lemma 4.2, let us remove from σ (A(1) ) ∩ Dt,r the interiors of all edges of the polygons ∂Λα lying in the open annulus Dt,r . Then we obtain an operator A(2) ∈ Ln satisfying the condition (C), such that σ A(2) ⊂ Dt ∪ Σ ∪ ∂Dr ∪ σ (A) \ Dr and σ A(2) \ Dr = σ (A) \ Dr ,

(4.5)

where Σ is the set of vertices of the polygons Λα . If at least one point of a closed edge of Λα belongs to Ωj , then the interior part of this edge also lies in Ωj . In view of (a2 ), this implies that the removal of open arcs does not increase the eigenspaces corresponding to Ωj . Therefore A(2) satisfies ( ). By (4.5), the set of z ∈ σ (A(2) ) \ Dr which do not belong to σ (A) \ Dr consists of a countable collection of arcs γβ of the circle ∂Dr , whose end points belong either to Σ ∩ ∂Dr or to σ (A) ∩ ∂Dr . Each interior point of γβ is separated from σ (A(2) ) \ Dr (otherwise it would belong to

σ (A)). The set Σ is finite and, by (4.4), the intersection σ (A) ∩ ∂Dr is a subset of m−1 j =1 Ωj .

This implies that (σ (A(2) ) ∩ ∂Dr ) \ ( m−1 Ω ) is covered by a finite subcollection of arcs γβ j

m−1 j =1 whose end points belong to Σ ∪ ( j =1 Ωj ). Repeatedly applying Lemma 4.2, let us remove the interior parts of the arcs γβ from σ (A(2) ). Then we obtain an operator A(3) ∈ Ln satisfying the

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condition (C), such that σ A(3) ⊂ Dt ∪ Σ ∪

m−1

Ωj \ Dr .

(4.6)

j =1

For the same reason as before, A(3) also satisfies the condition ( ). If ∂Ωm ∩ Ωj = ∅ for all j = 1, . . . , m − 1 then σ (A) is separated from the boundary ∂Ωm , and we define A(3) = A. Obviously, in this case A(3) also satisfies ( ) and (4.6) with Σ = ∅ and some t ∈ (0, rm ) and r ∈ (t, rm ). 4.3.3. Let Pm be the spectral projection of the operator A(3) corresponding to the set Dt ∪ Σ. Since t < r and Σ is finite, this set is separated from σ (A(3) ) \ Dr and, consequently, Pm ∈ L. Since Dt ∪ Σ ⊂ Ωm , the condition ( ) implies that Pm H ⊂ ΠΩm H . Given z ∈ C, let us consider the operator Az := zPm + (I − Pm )A(3) . From (4.6) it follows that σ (Az ) ⊂ {z} ∪

m−1

Ωj \ Dr ,

∀z ∈ C.

(4.7)

j =1

If z ∈ Dt then for each sufficiently small δ > 0 there is a homeomorphism ϕz,δ : C → C isotopic to the identity, which maps a neighbourhood of Dt ∪ Σ onto Dδ (z) and coincides with the identity on a neighbourhood of σ (A(3) ) \ Dr . By 4.1.2, all the operators ϕz,δ (A(3) ) satisfy the condition (C). Since ϕz,δ (A(3) ) converge to Az as δ → 0, this implies that Az also satisfy the condition (C) for all z ∈ Dt . If z ∈ / Dt and λ = z, let us fix a point z˜ ∈ Dt and a path μ(s) from z˜ to z which does not go through λ. Assume that ε > 0 is so small that z˜ ∈ / Dε (λ). Then, applying Lemma 4.1 with Ω = Dε (λ) to Az˜ , we can find an operator Az˜ ,ε := Az˜ ,Ω ∈ Ln such that Pm Az˜ ,ε = Az˜ ,ε Pm = z˜ Pm , −1 Az˜ ,ε − λI ∈ L−1 0 and limε→0 Az˜ ,ε = Az˜ . Since μ(s)Pm + Az˜ ,ε (I − Pm ) − λI is a path in L from Az˜ ,ε − λI to zPm + Az˜ ,ε (I − Pm ) − λI , the latter operator also belongs to L−1 0 . Therefore Az − λI = lim zPm + Az˜ ,ε (I − Pm ) − λI ∈ L−1 0 , ε→0

∀λ = z.

Obviously, the same inclusion holds for λ = z. Thus the operators Az satisfy the condition (C) for all z ∈ C.

4.3.4. Let us fix an arbitrary point z ∈ Ω1 \ ( m j =2 Ωj ) and denote A := Az . In view of

(4.7), we have σ (A ) ⊂ m−1 j =1 Ωj . Applying the induction assumption to the operator A , we can m−1 find mutually orthogonal projections P1 , P2 , . . . , Pm−1 such that P1 + j =2 Pj = I , P1 H ⊂ H and P H ⊂ Π H for all j = 2, . . . , m − 1, where Π are the spectral projections of ΠΩ j Ωj Ωj 1

, . . . , Π A corresponding to Ωj . Since z ∈ / m−1 Ω , the projections ΠΩ j j =2 Ωm−1 coincide with 2


2927

= 0 for all j = the spectral projections of the truncation A(3) |(I −Pm )H . Thus we have Pm ΠΩ j 2, . . . , m − 1 and, consequently, Pm H ⊂ P1 H . Let 1 := P1 − Pm . Then, by the above, P1 , . . . , Pm are mutually orthogonal projections such P m that j =1 Pj = I . It remains to notice that, in view of ( ),

(3) P1 − Pm H ⊂ ΠΩ1 H ⊂ ΠΩ1 H H ⊂ Π H ⊂ Π H for all j = 2, . . . , m − 1. and ΠΩ Ωj Ωj j (3)

5. Remarks and references ∗ 5.1. One can easily show that Lf ∩ Ls ⊂ L−1 ∩ Ls and Lf ∩ Ln ⊂ L−1 0 ∩ Ln in any C algebra L (see the proof of Corollaries 2.4 and 2.5). If L has real rank zero then

(1) Ls = Lf ∩ Ls (this is the implication (1) ⇒ (3) in Corollary 2.4) and (2) L−1 0 ∩ Lu = Lf ∩ Lu . The first result is well known and elementary (see, for example, [10, Theorem V.7.3] or [6, Theorem 2.6]). The second is due to Huaxin Lin [20]. Note that (2) is an immediate consequence of (1) and Lemma 1.8. Theorem 2.1 and Lemma 2.3 imply the following “quantitative” versions of (1) and (2). If L has real rank zero then √

(1 ) for every A ∈ Ls there exist self-adjoint operators A1 , A2 , . . . ∈ L such that A − An n2 and σ (An ) contains at most 2nA + 1 points, (2 ) for every A ∈ L−1 0 ∩ Lu there exist unitary operators A1 , A2 , . . . ∈ L such that A − An √

2 n

and σ (An ) contains at most 2πn + 1 points.

Indeed, in both cases A is a normal operator satisfying the condition (C) and σ (A) has an open 1 cover {Ωj }m j =1 of multiplicity 2 by open discs Ωj of diameter n , where m = 2nA + 1 in the first case and m = 2πn + 1 in the second case. 5.2. Let L be an arbitrary unital C ∗ -algebra. If an operator A ∈ L satisfies the condition (C) then (C0 ) A − λI ∈ L−1 / σ (A). 0 for all λ ∈ If the complement C \ σ (A) is dense in C then (C0 ) implies (C), but in the general case it is not true. One says that a C ∗ -algebra L has weak (FN) if every normal operator satisfying (C0 ) can be approximated by normal operators with finite spectra. In view of Corollary 2.5, a C ∗ -algebra of real rank zero has weak (FN) if and only if the conditions (C) and (C0 ) are equivalent for all normal operators.

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In [21] and [22], Huaxin Lin has described some classes of simple real rank zero C ∗ -algebras with weak (FN). The following modification of the example from [21] shows that the simplicity condition is important for the weak (FN) property. Let L be a unital C ∗ -algebra such that Ln ⊂ L−1 0 . Consider the direct sum L = L ⊕ L, and

let A = A1 ⊕ A2 where A1 , A2 are normal elements of L such that A1 ∈ / L−1 0 , A1 1 and σ (A2 ) = {z ∈ C: |z| 1}. Then A ∈ Ln and σ (A) coincides with the unit disc about the origin. −1 −1 / L−1 In view of Remark 1.6, A satisfies (C0 ). However, A ∈ 0 = L0 ⊕ L0 . 5.3. Theorem 2.1 remains valid for self-adjoint operators A in a general C ∗ -algebra L satisfying the condition (Cs ) A − λI ∈ L−1 ∩ Ls for all λ ∈ R.

Indeed, if we take Bn ∈ L−1 ∩ Ls in Section 4.1.1 then the operators Vn , Sn and Un are selfadjoint, and so is the operator Aε . The same arguments show that Aε still satisfies the condition (Cs ). Therefore, iterating this procedure, we can remove from σ (A) an arbitrary finite collection of open intervals without changing the spectral projections corresponding to the complements of their closures. This allows us to construct approximate spectral projections in the same manner as in Section 4.3, with obvious simplifications due to the fact that σ (A) ⊂ R. Using this observation, one can refine Corollary 2.4 as follows. 5.4. In an arbitrary C ∗ -algebra L, the following statements about a self-adjoint operator A ∈ Ls are equivalent. (1) The operator A satisfies the condition (Cs ). (2) The operator A has approximate spectral projections in the sense of Theorem 2.1, associated with any finite open cover of its spectrum. (3) A ∈ Lf ∩ Ls . As explained in Section 5.3, (2) follows from (1), and the other two implications are proved in the same way as in Section 2.1. −1 5.5. It is clear from the proof that Lemma 4.1 remains valid if we replace L−1 0 with L . However, in Lemma 4.2 the assumption (iii) is of crucial importance.

5.6. For a disc Ω = Dε (0), Lemma 4.1 without the condition (a4 ) can easily be deduced from [24, Theorem 5] (see also [25, Theorem 2.2]). In the both papers the theorem was proved for A ∈ L−1 , but in [16, Section 3] the authors explained that the approximating operator belongs −1 L−1 0 whenever A ∈ L0 . [24, Theorem 5] holds for a general operator A satisfying the condition dist(A, L−1 ) < ε, whereas we assumed that dist(A, L−1 0 ) = 0 and, in addition, that A is normal. Our proof slightly differs from those in [24,25,16]. It gives a weaker result in the general case but is better suited for the study of operators with one-dimensional spectra. It also shows that one can choose approximating operators satisfying the condition (C).


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5.7. Lemma 4.2 seems to be new. Possibly, one could deduce the (1) ⇒ (3) part of Corollary 2.5 from [21, Theorem 5.4], but our approach gives more information about the approximating operators. In particular, Theorem 2.1 implies a quantitative (in the same sense as in Section 5.1) version of [21, Theorem 5.4]. 5.8. One can further refine Theorem 2.12 by introducing subsets MTn ⊂ MT , which consist of convex combinations of operators of the form S1 T S2 containing at most n terms. The same n(ε) proof shows that M[A∗ ,A] in (2.5) can be replaced with M[A∗ ,A] where n(ε) is an integer-valued nonincreasing function of ε ∈ (0, ∞). 5.9. A review of results on almost commuting operators and matrices can be found in [11]. The authors listed several dimension-dependent results and discussed the following known example by D. Voiculescu [29]. Let {e0 , e1 , . . . , em } be an orthonormal basis in Cm+1 , and let Am and Bm be (m + 1) × (m + 1)-matrices defined by the identities 2j ej for all j = 0, . . . , m, Am ej = 1 − m 2 Bm ej = (j + 1)(m − j )ej +1 for all j = 0, . . . , m − 1, m+1 Bm em = 0. ∗ , B ] 4 and [A , B ] 2 , so that the HerThen Am = 1, Bm 1, Am = A∗m , [Bm m m m m m mitian matrices Am , Re Bm and Im Bm are almost commuting for large values of m. However, the distance between the pair {Am , Bm } and any pair of commuting (m + 1) × (m + 1)-matrices is estimated from below by a constant independent of m (see [8]). This example shows that, without additional assumptions, B(ε) in (2.5) cannot be replaced by B(ε) ∩ Ls (or, in other words, it is not sufficient to adjust only one operator in a pair of almost commuting self-adjoint operators to obtain a pair of commuting self-adjoint operators). Indeed, if (2.5) held with B(ε) ∩ Ls then, applying Theorem 2.12 to the matrices Re Bm + iAm and Im Bm + iAm , we could find Hermitian (m + 1) × (m + 1)-matrices Xm and Ym such that [Am , Xm ] = [Am , Ym ] = 0 and Xm + iYm − Bm → 0 as m → ∞.

5.10. Theorem 2.12 allows one to obtain approximation results for bounded operators with compact self-commutators. For instance, if A ∈ B(H ) satisfies the condition (C), A 1, [A∗ , A] ∈ Sp and [A∗ , A]p c then the number of eigenvalues of each operator from h(ε)M[A∗ ,A] lying outside the interval (−ε, ε) does not exceed (cε −1 h(ε))p . In view of (2.5), this implies that for each ε > 0 there exist a normal operator Tε and a self-adjoint operator Rε of finite rank such that A − Tε − Rε 2ε and rank Rε (cε −1 h(ε))p . Moreover, if the operator A is compact then one can take Tε ∈ C(H ). Since Theorem 2.12 does not give an explicit estimate for h(ε), the above observation is of limited interest. However, it shows that rank Rε is bounded by a constant depending only on ε and p. 5.11. If A ∈ B(H ) and ε > 0, let us define Specε (A) := σ (A) ∪ z ∈ C: (A − zI )−1 > ε −1 .

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The set Specε (A) is called the ε-pseudospectrum of A. It is known that Specε (A) =

σ (A + R)

Rε

(see, for instance, [12, Theorem 9.2.13] and [7, Lemma 2]). Let dA (ε) :=

dist λ, σess (A)

sup

and dA (0) := sup dist λ, σess (A) ,

λ∈Specε (A)

λ∈σ (A)

where σess (A) is the spectrum of the corresponding element of the Calkin algebra. In [2] the authors proved the following statement. If [A∗ , A] c2 and (A − λI )−1 dist λ, σess (A) − c −1 ,

∀λ: dist λ, σess (A) > c,

then the normal operator T in the BDF theorem can be chosen in such a way that σ (T ) = σess (A) and A − T f (c), where f : [0, ∞) → [0, ∞) is some (unknown) continuous function vanishing at the origin that depends only on σess (A). Note that, under the above condition on (A − λI )−1 , we have dA (ε) c + ε for all ε > 0. Theorem 3.8(1) implies the following more precise result which holds without any a priori assumptions about the resolvent. 5.12. Under the conditions of the BDF theorem, there exists a normal operator T such that σ (T ) = σess (A), A − T ∈ C(H ) and A − T 2AF A−2 A∗ , A + dA 2AF A−2 A∗ , A , where F : [0, ∞) → [0, 1] is a nondecreasing concave function vanishing at the origin, which does not depend on A. Indeed, applying Theorem 3.8(1) to the operator A−1 A, we can find a normal operator T such that A − T ∈ C(H ) and A − T 2AF A−2 A∗ , A . By the above, σ (T ) ⊂ Specδ (A) for all δ > A − T and, consequently, dist λ, σess T = dist λ, σess (A) dA A − T ,

∀λ ∈ σ T .

Now, using the spectral theorem, one can easily construct a normal operator T such that T − T ∈ C(H ), σ (T ) = σess (T ) and T − T dA (A − T ). This operator satisfies the required conditions.


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5.13. Theorem 2.12 states that (2.5) holds for all C ∗ -algebras L of real rank zero with some universal function h. The function F is determined only by h and, therefore, (2.11) is true for all C ∗ -algebras L of real rank zero and all seminorms satisfying the conditions (2.10). Our proof is by contradiction and does not give explicit estimates for h and F . For a particular C ∗ -algebra L and a seminorm · ∗ on L, it may be possible to optimise the choice of functions h and F or to obtain additional information about their behaviour. Note that (i) if (2.5) holds with some function h then we also have (2.11) with F defined by (2.9) for all seminorms · ∗ satisfying (2.10); (ii) lim infε→0 (εh(ε)) > 0 for any function h satisfying (2.5); and (iii) lim inft→0 (t −1/2 F (t)) > 0 for any function F satisfying (2.11) (otherwise we obtain a contradiction by substituting an operator δA and letting δ → 0). 5.14. In [11] the authors conjectured that the estimate (3.1) holds with a function F such that F (t) ∼ t 1/2 as t → 0. In the recent paper [18], Hastings announced (3.1) with F (t) = t 1/5 F˜ (t), where F˜ is a function growing slower than any negative power of t as t → 0. Note that the proof of Theorem 3.8(1) uses only (3.1). Therefore, if Hastings’ result is correct, we obtain the following corollary. If A satisfies the conditions of the BDF theorem and A 1 then for each ε, δ > 0 there exists a diagonal operator Tε,δ such that A − Tε,δ ∈ C(H ) and 1/5−δ A − Tε,δ Cδ A∗ , A + ε, where Cδ is a constant depending only on δ. 5.15. The estimate (3.2) with a function F depending on p follows from [17, Theorem 4.2]. In [14] the authors have shown that for p = 2 one can take F (t) = ct −1/4 in (3.2), where c is some constant independent of the dimension. 5.16. In most statements, for the sake of simplicity, we assumed that A 1. One can easily get rid of this condition by applying the corresponding result to the operator A−1 A (as was done in Section 5.12). Acknowledgment The research was supported by the EPSRC grant GR/T25552/01. References [1] I.D. Berg, An extension of the Weyl–von Neumann theorem to normal operators, Trans. Amer. Math. Soc. 160 (1971) 365–371. [2] I.D. Berg, K.R. Davidson, Almost commuting matrices and a quantitative version of the Brown–Douglas–Fillmore theorem, Acta Math. 166 (1991) 121–161. [3] R. Bouldin, The essential minimum modulus, Indiana Univ. Math. J. 30 (1981) 513–517. [4] R. Bouldin, Closure of invertible operators on a Hilbert space, Proc. Amer. Math. Soc. 108 (1990) 721–726.

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[5] L.G. Brown, R.G. Douglas, P.A. Fillmore, Unitary equivalence modulo the compact operators and extensions of C ∗ algebras, in: Proceedings of a Conference on Operator Theory, in: Lecture Notes in Math., vol. 345, 1973, pp. 58–128. [6] L.G. Brown, G.K. Pedersen, C ∗ -algebras of real rank zero, J. Funct. Anal. 99 (1991) 131–149. [7] F. Chaitin-Chatelin, A. Harrabi, About definitions of pseudospectra of closed operators in Banach spaces, Tech. Rep. TR/PA/98/08, CERFACS. [8] M.D. Choi, Almost commuting matrices need not be nearly commuting, Proc. Amer. Math. Soc. 102 (1988) 529– 533. [9] L.A. Coburn, A. Lebow, Algebraic theory of Fredholm operators, J. Math. Mech. (Indiana Univ. Math. J.) 15 (4) (1966) 577–584. [10] K.R. Davidson, C ∗ -Algebras by Example, Amer. Math. Soc., 1996. [11] K.R. Davidson, S.J. Szarek, Local operator theory, random matrices and Banach spaces, in: W.B. Johnson, J. Lindenstrauss (Eds.), Handbook of the Geometry of Banach Spaces, vol. I, North-Holland, Amsterdam, 2001, pp. 317–366. [12] E.B. Davies, Linear Operators and Their Spectra, Cambridge University Press, 2007. [13] J. Feldman, R.V. Kadison, The closure of the regular operators in a ring of operators, Proc. Amer. Math. Soc. 5 (1954) 909–916. [14] N. Filonov, I. Kachkovskiy, A Hilbert–Schmidt analog of Huaxin Lin’s theorem, preprint, 2010, http://arxiv.org/ abs/1008.4002v2. [15] P. Friis, M. Rørdam, Almost commuting self-adjoint matrices – a short proof of Huaxin Lin’s theorem, J. Reine Angew. Math. 479 (1996) 121–131. [16] P. Friis, M. Rørdam, Approximation with normal operators with finite spectrum, and an elementary proof of a Brown–Douglas–Fillmore theorem, Pacific J. Math. 199 (2001) 347–366. [17] D. Hadwin, W. Li, A note on approximate liftings, Oper. Matrices 3 (1) (2009) 125–143. [18] M.B. Hastings, Making almost commuting matrices commute, preprint, 2010, http://arxiv.org/abs/0808.2474v3. [19] A. Laptev, Y. Safarov, Szegö type limit theorems, J. Funct. Anal. 138 (1996) 544–559. [20] H. Lin, Exponential rank of C ∗ -algebras with real rank zero and Brown–Pedersen’s conjecture, J. Funct. Anal. 114 (1993) 1–11. [21] H. Lin, Approximation by normal elements with finite spectra in C ∗ -algebra of real rank zero, Pacific J. Math. 173 (1996) 443–489. [22] H. Lin, C ∗ -algebras with weak (FN), J. Funct. Anal. 150 (1997) 65–74. [23] H. Lin, Almost commuting selfadjoint matrices and applications, in: Operator Algebras and Their Applications, in: Fields Inst. Commun., vol. 13, 1997, pp. 193–233. [24] G.K. Pedersen, Unitary extensions and polar decompositions in a C ∗ -algebra, J. Operator Theory 17 (1987) 357– 364. [25] M. Rørdam, Advances in the theory of unitary rank and regular approximation, Ann. of Math. 128 (1988) 153–172. [26] B. Russo, H.A. Dye, A note on unitary operators in C ∗ -algebras, Duke Math. J. 33 (1966) 413–416. [27] Y. Safarov, Berezin and Gårding inequalities, Funktsional. Anal. i Prilozhen. 39 (4) (2005) 69–77 (in Russian); English translation in: Funct. Anal. Appl. 39 (2005) 301–307. [28] G. Szegö, Beiträge zur theorie der Toeplizschen formen, Math. Z. 6 (1920) 167–202. [29] D. Voiculescu, Asymptotically commuting finite rank unitary operators without commuting approximants, Acta Sci. Math. (Szeged) 45 (1983) 429–431.


Convergence and decay estimates for a class of second order dissipative equations involving a non-negative potential energy Imen Ben Hassen a , Alain Haraux b,∗ a Département de Mathématiques, Faculté des Sciences de Bizerte, 7021 Jarzouna, Bizerte, Tunisia b Laboratoire Jacques-Louis Lions, boite courrier 187, Université Pierre et Marie Curie, 4 place Jussieu,

75252 Paris Cedex 05, France Received 26 August 2010; accepted 9 February 2011 Available online 22 February 2011 Communicated by J. Coron

Abstract We estimate the rate of decay of the difference between a solution and its limiting equilibrium for the following abstract second order problem u(t) ¨ + g u(t) ˙ + M u(t) = 0,

t ∈ R+ ,

where M is the gradient operator of a non-negative functional and g is a non-linear damping operator, under some conditions relating the Łojasiewicz exponent of the functional and the growth of the damping around the origin. The main result is applied to non-linear wave or plate equations, in some cases direct constructive proofs of the Łojasiewicz gradient inequality are given, applicable to some non-analytic functionals in presence of multiple critical points. At the end similar results are obtained when a fast decaying source term is added in the right-hand side. © 2011 Elsevier Inc. All rights reserved. Keywords: Convergence; Decay estimates; Second order equation; Łojasiewicz exponent


E-mail addresses: [email protected] (I. Ben Hassen), [email protected] (A. Haraux). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.02.010

2934

I. Ben Hassen, A. Haraux / Journal of Functional Analysis 260 (2011) 2933–2963

1. Introduction The convergence problem for bounded solutions of semilinear dissipative wave equation has been the object of many specialized works in the last 30 years. Assuming Ω to be a bounded open connected domain of RN , a convergence result has been shown in [9] for the problem

utt + cut − u + f (u) = 0, u(t, x) = 0, on R+ × ∂Ω

in R+ × Ω,

(1)

when the function s → f (s) + λ1 s is non-decreasing, relying on the fact that the set of equilibria is then one-dimensional. This result was established under a growth assumption on f and assuming precompactness of the solution curve (u(t, .), ut (t, .)) in the energy space. A first generalization allowing some types of non-linear dampings was done by E. Zuazua in [22]. A much more general theory dealing with one-dimensional sets of equilibria was developed later by J. Hale and G. Raugel [8]. The hypothesis on the dimension of the equilibrium set can be relaxed if f is analytic, and general convergence results as well as rates of convergence were proved in this direction by M.A. Jendoubi and the second author, cf. [18,13,14] by using the Łojasiewicz gradient inequality, cf. [20,21]. The case of a genuinely non-linear damping seems to be more delicate. As a model example, a convergence result for bounded solutions of the problem ⎧ α ⎪ ⎨ utt + |ut | ut − u + f (u) = 0, in R+ × Ω, u(t, x) = 0, on R+ × ∂Ω, ⎪ ⎩ u(0, ·) = u0 ∈ H01 (Ω), ut (0, ·) = u1 ∈ L2 (Ω)

(2)

has been proved by L. Chergui in [6] assuming 0 < α < 1 and the following conditions f : R → R is analytic,

(3)

there exist C 0 and η > 0 with (N − 2)η < 2 such that: ∀s ∈ R,

f (s) C 1 + |s|η .

(4)

α More precisely L. Chergui had to assume the existence of θ ∈ ] α+1 , 12 ] and c > 0 such that for all ϕ in the equilibrium set

Σ = ψ ∈ H 2 (Ω) ∩ H01 (Ω) and ψ = f (ψ) there is σϕ such that for every u ∈ H01 (Ω) one has u − ϕH 1 (Ω) < σϕ 0

⇒

u + f (u)

H −1 (Ω)

1−θ c E(u) − E(ϕ) .

(5)

However [6] does not contain any estimate of the rate of convergence. This comes, among other things, from the method relying on an abstract, non-constructive, compactness result as well as the nature of the Liapunov functional used by Chergui which does not allow us to control the decay of the solution. At the present time it is not known whether it is possible to devise another approach providing decay rates in any function space.


2935

The situation appears better for the finite-dimensional system α ˙ u(t) u(t) ¨ + u(t) ˙ + ∇F u(t) = 0, t ∈ R+ , studied by L. Chergui [5] who, by using the same kind of Liapunov function, was able to compute a decay rate. In this paper we shall obtain such an estimate for a class of equations which contains (2) when f satisfies some additional assumptions which ensure a reinforced positivity condition on the potential energy. Since the energy is defined up to an additive constant, in practice this condition means some kind of minimization property of the potential energy on the elements of the weak omega-limit set but does not require a global convexity assumption on the potential, cf. Remark 2.3. The plan of the paper is as follows: in Section 2 we state and prove our main result. In Section 3 we give simple applications of this result. In Section 4 we show how to check the hypotheses for a more general class of non-linear operators, in particular the non-linearity does not need to be analytic and in the case of Neumann boundary conditions we can handle multiple equilibria. Section 5 contains the convergence and decay results in the more elaborate examples related to the results of Section 4. Section 6 is devoted to the case where a rapidly decaying source term appears in the right-hand side of the equation, in the spirit of [2,3,7,17]. 2. Main result 2.1. Functional setting Throughout this article we let H and V be two Hilbert spaces. We assume that V is densely and continuously embedded into H . Identifying H with its dual H , we obtain V → H = H → V . We denote by ·, · scalar products and duality relations; the spaces in question will be specified by subscripts. The notation f, u without any subscript will be used sometimes to denote f, u V ,V . Throughout the text, we let C1 0 be such that vV C1 vH C12 vV ,

v ∈ V.

(6)

Other constants in the calculations will be denoted by Ci (i 2). Let E ∈ C 2 (V , R), and denote by M ∈ C 1 (V , V ) the first derivative of E. Throughout the text we shall assume that E and M are bounded on bounded sets of V with values in V and V respec1,1 2,1 (R+ , V ) ∩ Wloc (R+ , H ) tively. We study the asymptotic behaviour of some solutions u ∈ Wloc of the following abstract Cauchy problem: ⎧ ⎪ ¨ + g u(t) ˙ + M u(t) = 0, t 0, ⎨ u(t) (7) u0 ∈ V , u(0) = u0 , ⎪ ⎩ u(0) ˙ = u1 , u1 ∈ H, under the following assumptions on g and E: 1) g : H → V is such that there exist α ∈ ]0, 1[, ρ1 > 0 and ρ2 > 0 for which

∀v ∈ V , g(v), v V ,V ρ1 vα+2 H , α+1 ∀v ∈ H, g(v) V ρ2 vH .

(8) (9)

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2) There exist a real number θ such that θ∈

1 α , , α+1 2

(10)

and some constants c1 > 0, c2 > 0 and a bounded subset B of V such that the function E is non-negative on B and satisfies the following assumption ∀u ∈ B, c1 E(u)γ M(u) V c2 E(u)1−θ (11) where γ > 0 is such that: 1 − α(1 − θ ) γ 1 − θ. 2

(12)

Remark 2.1. 1) The left inequality in (11) implies in particular that whenever u ∈ B and E(u) = 0, we have M(u) = 0. 2) If V is finite-dimensional, under the above condition and assuming E analytic, the standard Łojasiewicz inequality implies the existence of a positive γ for which (11) is fulfilled. It is not clear, however, to decide whether γ can be taken satisfying (12). 3) In all the examples considered in this paper, (11) will be fulfilled with γ = 12 . It would be interesting to find sufficient conditions under which an analytic E defined on a finite-dimensional space V fulfills (11) with γ = 12 . 1,1 2,1 4) The local regularity condition u ∈ Wloc (R+ , V ) ∩ Wloc (R+ , H ) corresponds to the socalled strong solutions and is usually fulfilled in the application if the initial state (u0 , u1 ) is smooth enough. It is used to justify the differentiations in the proof of Theorem 2.2 below. On the other hand, we do not require any global higher order bound for the solution, as a consequence in the examples the result of Theorem 2.2 will be applicable also to weak solutions which are usually obtained by density for any (u0 , u1 ) ∈ V × H , cf. e.g. [10]. 2.2. The result The main result of this paper is 1,1 2,1 Theorem 2.2. Let u ∈ Wloc (R+ , V ) ∩ Wloc (R+ , H ) be a solution of (7) such that u(t) ∈ B for t large where B denotes a closed bounded subset of V . Assume that the hypotheses (8), (9) and (11) are satisfied. Assume in addition that there exists a constant C 0 such that for u ∈ B, the following inequality holds: (13) ∀v ∈ V , M (u)v, v V Cv2H .

Let θ be as in (10) and (11). Let us introduce ξ=

1 − (α + 1)(1 − θ ) ; (α + 2)(1 − θ ) − 1

λ=

1 . (α + 2)(1 − θ ) − 1

Then there exist a ∈ B and a constant C > 0 such that ∀t T , u(t) − a H Ct −ξ .

(14)


2937

Moreover there are positive constants M1 , M2 such that λ u(t) ˙ H M1 t − 2 , ∀t T , E u(t) M2 t −λ .

∀t T ,

(15) (16)

Finally, if either u has precompact range in V , or M : B → V is weakly continuous for the topology of V , we have

a ∈ E := y ∈ V , M(y) = 0 . Remark 2.3. 1) Let us observe that in the case V = H = RN , g(v) = v ˙ α v˙ and M(u(t)) = ∇F (u(t)), L. Chergui [5] studied the differential system α ˙ u(t) ˙ + ∇F u(t) = 0, u(t) ¨ + u(t)

t ∈ R+ .

(17)

He proved a convergence result for bounded solutions of (17) and he obtained the rate of decay given by formula (14). In his case (15) and (16) are irrelevant. 2) The hypothesis (11) implies that any equilibrium point a ∈ B satisfies E(a) = 0. Hence the minimum of E on B is achieved on equilibrium points. In particular, if either u has precompact range in V , or M : B → V is weakly continuous for the topology of V , the existence of a region B with the above mentioned properties implies the not so trivial conclusion that the minimum of E on B is achieved and equal to 0. 3) In contrast with the result of [6], no compactness assumption on the trajectory associated to u needs to be made or proved. This assumption is replaced by assuming that u(t) ∈ B for t large and the uniform property (11). In practice, in the examples, B will be an arbitrary closed ball in V , so that the uniform bound of the solution, following easily from the energy dissipation property, will be enough to get the convergence result with an estimate of the convergence rate in H . We shall see that in many examples, the energy decay will finally provide an estimate of the convergence rate in V . 4) Our result is a convergence result to a local minimum of the potential energy but does not require convexity of the potential, although in some of the examples we limited ourselves to a convex situation with a simple explicit formula for the non-linear conservative term. Nonconvex situations appear in the examples of Subsection 3.1 and Theorem 5.6. Similar variants are also possible with Dirichlet boundary conditions, but for the moment we cannot handle multiple equilibria in that case and for this reason we just mentioned the simplest examples. 5) The question naturally arises of comparing our result to those of [1,15,19] which deal with an extension of the fundamental parabolic result of [4] in presence of a convex potential. The situation is in fact quite different from ours, since our dissipation is non-linear, a situation which cannot be handled by Alvarez’s method consisting in generalizing Bruck’s approach. When the dissipation is made non-linear, the analog parabolic problem is badly degenerate and convexity is no longer enough, some kind of reinforced convexity assumption might provide results but such conditions still have to be found. Moreover, the above quoted results do not encompass the case of unbounded gradient operators, as a result they do not apply to PDE problems. Finally the results of these papers are weak convergence results without estimate of the convergence rate in any norm at all, while the main objective of the present work is to obtain a convergence rate.

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2.3. Proof of Theorem 2.2 1,1 2,1 (R+ , V ) ∩ Wloc (R+ , H ) and let T > 0 be Let u be a solution of Eq. (7) such that u ∈ Wloc such that u(t) ∈ B for t T . Let us define the non-negative function

2 1 ˙ H + E u(t) . E(t) = u(t) 2 Then we have

E (t) = − u(t), ˙ g u(t) ˙ 0. V ,V Therefore E, being non-increasing and non-negative, remains bounded. In particular u(t) ˙ is bounded in H. Now let 0 < ε 1 be a real constant. By analogy with [11, formula (3.53), p. 47] we introduce the function

˙ V , H (t) = E(t) + εE(t)β M u(t) , u(t) where β = α(1 − θ ), θ is the Łojasiewicz exponent defined in (11). (Note that when M = − : H01 (Ω) → H −1 (Ω) we have θ = 12 and therefore β = α2 , which reduces our formula to (3.53) from [11].) Our bounded solution u being fixed, we can choose ε small enough in order to achieve E(t) H (t) 2E(t), 2

for all t T .

(18)

In fact, due to the definition of H together with assumption (11) we have for all t T

˙ V H (t) = E(t) + εE(t)β M u(t) , u(t) ˙ V E(t) + εE(t)β M u(t) V u(t) √ 1 E(t) + εC1 c1 2E(t)α(1−θ)+γ + 2 . 1

Now, since E is bounded, it follows from (12) that E(t)α(1−θ)+γ + 2 KE(t), for all t T , where K is a positive constant. Then by choosing ε small enough we get H (t) 2E(t). The reverse inequality follows similarly. We now compute

˙ V + εE(t)β M u(t) u(t), ˙ u(t) ˙ V H (t) = E (t) + εβE (t)E(t)β−1 M u(t) , u(t) 2

˙ − εE(t)β M u(t) , g u(t) − εE(t)β M u(t) V . V Then we find

H (t) = − u(t), ˙ V ˙ g u(t) ˙ − εβ u(t), ˙ g u(t) ˙ E(t)β−1 M u(t) , u(t) V ,V V ,V

+ εE(t)β M u(t) u(t), ˙ u(t) ˙ V − εE(t)β M u(t) , g u(t) ˙ V 2 − εE(t)β M u(t) V .


2939

By using (8), u(t), ˙ g(u(t)) ˙ V ,V is non-negative and then we get thanks to the definition of E together with assumption (11)

1 ˙ V C2 u(t), ˙ g u(t) ˙ E(t)β−1 M u(t) , u(t) E(t)β−1 E(u)γ E(t) 2 − u(t), ˙ g u(t) ˙ V ,V V ,V

1 ˙ g u(t) ˙ C2 u(t), E(t)β+γ − 2 V ,V

1 ˙ g u(t) ˙ = C2 u(t), E(t)α(1−θ)+γ − 2 . V ,V From the last inequality together with assumption (12) and since E is bounded we deduce

− u(t), ˙ g u(t) ˙ ˙ V C3 u(t), ˙ g u(t) ˙ E(t)β−1 M u(t) , u(t) . V ,V V ,V

(19)

By assumption (13) and since E(t) 0 for all t T , by applying Young’s inequality we get for any δ > 0 given in advance 2 α+2

(α+2) δε ˙ H E(t)β α + εC5 u(t) ˙ H , εE(t)β M u(t) u(t), ˙ u(t) ˙ V εC4 E(t)β u(t) 2 where C5 depends on δ < 1 to be chosen small enough later on. Now by using the definition of E and β we obtain, since (1 − θ )(α + 2) < 2 α+2

δε ˙ u(t) ˙ V E(t)(1−θ)(α+2) + εC5 u(t) ˙ H εE(t)β M u(t) u(t), 2 2(1−θ)(α+2) α+2 ˙ ˙ . δε u(t) + E(u)(1−θ)(α+2) + 2εC5 u(t) H

H

Since u˙ is bounded in H we get α+2

εE(t)β M u(t) u(t), ˙ H + δεE(u)α(1−θ) E(u)2(1−θ) . ˙ u(t) ˙ V εC6 u(t) By using assumption (11) we deduce by a suitable choice of δ > 0 which will remain fixed from now on: α+2 ε 2

˙ H + E(u)α(1−θ) M u(t) V εE(t)β M u(t) u(t), ˙ u(t) ˙ V εC6 u(t) 4 α+2 ε 2 ˙ εC6 u(t) + E(t)α(1−θ) M u(t) V . H 4 Therefore we have for all t T α+2 ε 2

˙ H + E(t)β M u(t) V . εE(t)β M u(t) u(t), ˙ u(t) ˙ V εC6 u(t) 4

(20)

Using the Cauchy–Schwarz inequality together with assumption (9), we obtain α+1

˙ H . ˙ ερ2 E(t)β M u(t) V u(t) εE(t)β M u(t) , g u(t) V Let C7 = 1 + supR+ M(u(t))V , by using Young’s inequality there exists C8 0 such that

2940


α+1 ˙ H ε ε M u(t) V u(t)

α+2 1−α M u(t) α+2 ˙ H . + εC8 u(t) α V 4(1 + α)C7

Then since E is bounded we obtain for all t T α+2

2 ε ˙ E(t)β M u(t) V + εC9 u(t) εE(t)β M u(t) , g u(t) ˙ H . V 4

(21)

Thanks to assumptions (8), by combining (19), (20) and (21) and by choosing ε small enough it follows that for all t T α+2 2 ˙ H + E(t)β M u(t) V . H (t) −C10 u(t)

(22)

Now by using the last inequality together with assumption (11) and the definition of E we get α+2 ˙ H + E(u)(α+2)(1−θ) −H (t) C10 u(t) C10 E(u)(α+2)(1−θ) 2 (α+2)(1−θ) 1 ˙ C10 E(t) − u(t) H 2 2(α+2)(1−θ) ˙ C11 E(t)(α+2)(1−θ) − C12 u(t) H

C11 E(t)

(α+2)(1−θ)

α+2 ˙ − C13 u(t) H

C11 E(t)

(α+2)(1−θ)

+ C14 H (t).

Then we obtain −C15 H (t) E(t)(α+2)(1−θ) .

(23)

By combining (23) and (18) we obtain the next differential inequality for all t T H (t)(α+2)(1−θ) + C16 H (t) 0. It follows by applying Lemma 2.8 from [2] that for all t T H (t) C17 t −λ , where λ =

1 (α+2)(1−θ)−1 .

2t t

Since we have

(24)

By using (22) together with (24) we obtain for all t T

α+2 1 u(s) ˙ H ds − C10

2t t

H (s) ds

1 H (t) C18 t −λ . C10


2t

α+1 u(s) ˙ H ds t α+2

t

2t

α+2 u(s) ˙ ds

2941

1 α+2

H

.

t

Therefore we get for all t T 2t

λ α+1 u(s) ˙ H ds C19 t − α+2 t α+2 = C19 t −ξ ,

t

where ξ = 1−(α+1)(1−θ) (α+2)(1−θ)−1 . Then we obtain for all t T k+1 ∞ ∞ 2 t u(s) u(s) ˙ H ds ˙ H ds

t

k=0

C20

2k t ∞ k −ξ 2 t k=0

C20 t −ξ . In particular, u˙ ∈ L1 (T , ∞, H ). Hence, u(t) has a limit a in H as t → ∞ and u(t) − a Ct −ξ . H The other estimates follow rather easily from (24) and (18). Then by (11), we see that M(u(t)) tends to 0 as t → ∞ and the last conclusion follows immediately. 3. Direct applications In this section we apply our main result to various simple example in order to test the sharpness of the estimates given by that theorem. 3.1. A second order ODE As a first application of the abstract Theorem 2.2 let us consider the following second order ODE: α u + u u + f (u) = 0, where α ∈ (0, 1) and f is such that for some reals a, b with a b f (s) = m(s) (s − b)+ − (s − a)− where

(25)

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I. Ben Hassen, A. Haraux / Journal of Functional Analysis 260 (2011) 2933–2963 1,∞ m ∈ Wloc (R)

and

inf m(s) > 0.

s∈R

(26)

For this example it suffices to choose V = R, then conditions (11) and (12) are verified with θ = γ = 12 for any compact interval B of R. The assumptions (8) and (9) are clearly true for Eq. (25). As a consequence of (26) it is easy to see that all solutions of (25) are bounded and globally Lipschitz on R+ . By applying Theorem 2.2 we prove that there exists c ∈ R such that 1 |u(t) − c| Ct − α +1 , where C is a positive constant. Clearly c ∈ [a, b]. This result is optimal, since when a < b, the solutions starting in (a, b) with a sufficiently small non-zero initial velocity remain in (a, b) for all times and since the velocity u then satisfies the equation u + |u |α u = 0, 1 the solution u tends to a limit c with |u(t) − c| equivalent to Ct − α +1 for some C > 0. 3.2. A critical semilinear wave equation In what follows, Ω is a bounded connected open subset of RN . As a second application of the abstract Theorem 2.2, we let V = H01 (Ω), H := L2 (Ω), M(u) = −u − λ1 u + |u|p−1 u, where λ1 is the first eigenvalue of − and p > 1 satisfies (N − 2)p < N + 2, 1 E(u) = 2

|∇u|2 − λ1 |u|2 dx +

Ω

1 p+1

|u|p+1 dx Ω

and we consider the following system

utt + g(ut ) − u − λ1 u + |u|p−1 u = 0, u(t, x) = 0, on R+ × ∂Ω,

in R+ × Ω,

(27)

where g : H → V satisfies (8)–(9) with 0 < α < p1 . It has been established in [13] that under the above conditions, E ∈ C 2 (V , V ) and (13) is fulfilled. In order to apply our main result to this example the main assumption remaining to be checked is therefore assumption (11). Now it has been proved in [16] that for u small enough in V M(u)

H −1

1−θ c1 E(u) ,

where θ =

1 . p+1

This result suffices to study the asymptotics of solutions knowing in advance that they converge to 0 in V. Actually a refinement of the method of [16] allows us to verify that for any R > 0, there is c1 (R) > 0 for which ∀u ∈ V ,

uV R

⇒

M(u)

H −1

p c1 (R) E(u) p+1 .

For the proof, see Section 4, Corollary 4.2 and Remark 4.3, 2). On the other hand it is not difficult to check that (11) holds true on any bounded subset of V with γ = 12 . Indeed M(u)

H −1

We claim

−u − λ1 uH −1 + |u|p−1 u H −1 .


2943

1 −u − λ1 uH −1 2E(u) 2 . Indeed ∀v ∈ V ,

−u − λ1 u, v =

(∇u.∇v − λ1 uv) dx. Ω

By the Cauchy–Schwarz inequality we deduce ∀v ∈ V ,

−u − λ1 u, v

∇u2 − λ1 |u|2 dx

1 2

Ω

∇v2 − λ1 |v|2 dx

1 2

Ω

∇u2 − λ1 |u|2 dx

1 2

vV

Ω

and the result follows. Now, if N 2 the estimate of the non-linear part is obvious, while if p +2 1 1 1 N > 2 since p < N N −2 then we have p+1 < 2 + N = (2∗ ) . So we obtain p−1 |u| u

∗ L(2 )

c2 |u|p−1 u

p+1 L p

p c3 E(u) p+1 .

Then we get p−1 |u| u

H −1

p c4 E(u) p+1 .

In order to apply Theorem 2.2 we first observe that 1 1 1 ∀u ∈ V , E(u) = |∇u|2 − λ1 |u|2 dx + |u|p+1 dx |∇u|2 dx − M 2 p+1 2 Ω

Ω

Ω

for some positive constant M (the proof for p = 1 is just slightly more delicate), and as a consequence of the fact that E(t) is non-increasing we deduce that any solution u is such that u(t) remains bounded in V . Then we compute ξ=

1 − αp 1 − (α + 1)(1 − θ ) = ; (α + 2)(1 − θ ) − 1 (α + 1)p − 1

λ=

1 p+1 = . (α + 2)(1 − θ ) − 1 (α + 1)p − 1

Applying (14) we obtain 1−αp u(t) − a Ct − (α+1)p−1

for some a which will turn out to be 0 by the last part of the theorem. On the other hand, since p+1 (cf. Proposition 4.1, formula (34)) for some η > 0 we have E(u) ηuV , applying (16) we find

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1 u(t) Mt − (α+1)p−1 V which is always sharper. This shows that it is sometimes preferable to apply directly the energy estimate rather than (14). 3.3. A semilinear wave equation with Neumann boundary conditions As a third application of the abstract Theorem 2.2, we let V = H 1 (Ω), H = L2 (Ω), M(u) = −u + |u|p−1 u, where p > 1 satisfies (N − 2)p < N + 2, E(u) =

1 2

|∇u|2 dx + Ω

1 p+1

|u|p+1 dx Ω

and we consider the following system ⎧ ⎨ utt + g(ut ) − u + |u|p−1 u = 0, ∂u ⎩ = 0, on R+ × ∂Ω, ∂n

in R+ × Ω, (28)

where g : H → V satisfies (8)–(9) with 0 < α < p1 . As in the previous example the results of [13] show that under the above conditions, E ∈ C 2 (V , V ) and (13) is fulfilled. The main assumption to be checked is again assumption (11). It is rather easy to verify that for any R > 0, there is c1 (R) > 0 for which ∀u ∈ V ,

uV R

⇒

M(u)

H −1

p c1 (R) E(u) p+1 .

For the proof, see Section 4, Corollary 4.4 and Remark 4.5, 2). Moreover it is not difficult to check that (11) holds true on any bounded subset of V with γ = 12 . Indeed M(u)

V

−uV + |u|p−1 u V .

We claim 1 −uV 2E(u) 2 . Indeed ∀v ∈ V ,

−u, v =

∇u.∇v dx. Ω

By the Cauchy–Schwarz inequality we deduce


1

∀v ∈ V ,

−u, v

2

∇u dx 2

Ω

1 ∇v dx 2

2

Ω

2945

1 ∇u2 dx

2

vV .

Ω

And the result follows. Then by the same method as in the Dirichlet case we obtain easily p−1 |u| u

V

p c4 E(u) p+1 .

In order to apply Theorem 2.2 we first observe that ∀u ∈ V ,

1 E(u) = 2

1 |∇u| dx + p+1

Ω

|u|

2

p+1

dx δ

Ω

|∇u|2 + u2 dx − M

Ω

for some constants M 0, δ > 0, and as a consequence of the fact that E(t) is non-increasing we deduce that any solution u is such that u(t) remains bounded in V . By applying Theorem 2.2 and using Proposition 4.1, formula (34), we obtain exactly the same estimates as in the previous example. In particular we find 1 u(t) Mt − (α+1)p−1 . V The degree of sharpness of this estimate is not clear, cf. Remark 3.4. 3.4. Examples of damping operators and convergence of weak solutions In the examples of both Sections 3.2 and 3.3, the initial value problem can be solved for any initial state in V × H under relevant conditions on the damping term. In this subsection we shall consider 2 basic examples. Example 3.1. Let γ : R → R be a locally Lipschitz function such that ∃K > 0,

γ (s) −K,

a.e. on R.

Assume that γ satisfies the following conditions ∀s ∈ R, ∀s ∈ R,

γ (s)s ρ1 |s|α+2 , γ (s) ρ2 |s|α+1 .

The typical case is α+1 α+1 γ (s) = γ1 s + − γ2 s − where γ1 , γ2 are some positive constants. By setting g(v)(x) = γ v(x)

a.e. on Ω

(C1 ) (C2 )

2946


we define an operator g : H → V with V = H 1 (Ω) (resp. V = H01 (Ω)) for any α ∈ [0, 1] if N 2 and for any α ∈ [0, N2 ] if N 3. In addition in such a case g satisfies automatically (8)–(9). Example 3.2. Let m be a locally Lipschitz function such that ∀s ∈ R+ ,

σ1 s α m(s) σ2 s α

(M)

and ∃K > 0,

m (s) −K,

a.e. on R.

By setting g(v)(x) = m vH v(x)

a.e. on Ω

we define an operator g : H → H ⊂ V with V = H 1 (Ω) (resp. V = H01 (Ω)) for any α > 0. In addition in such a case g satisfies automatically (8)–(9). By applying the results of [10], it is rather straightforward to see that for any g satisfying the conditions of Example 3.1, the initial value problems associated to both equations (27) and (28) are well posed for any initial state in V × H . In addition, for initial data in D(A) × V , the solution has the regularity required for the applicability of Theorem 2.2. In addition the solution depends continuously on the initial state as a map from V × H to C([0, T , V ]) ∩ C 1 ([0, T , H ]) for any T > 0. A careful inspection of the results from [10] shows that exactly the same property can be deduced for the same equation when g has the non-local form described in Example 3.2. By combining these properties with the results previously obtained, we obtain Corollary 3.3. For any g satisfying the conditions of Example 3.1 or Example 3.2, for any (u0 , u1 ) ∈ V × H there is a unique weak global solution u ∈ C(R+ , V ) ∩ C 1 (R+ , H ) of (27) (resp. (28)) in the sense of [10] which satisfies u(0, .) = u0 and ut (0, .) = u1 . In addition if we assume α < p1 and in the case of Example 3.1, for N > 2, the additional condition α N2 , then we have, for some constants M, M > 0 ∀t 1,

1 u(t, .) Mt − (α+1)p−1 V

and ∀t 1,

p+1 ut (t, .) M t − 2[(α+1)p−1] . H

Proof. For a strong solution, the result is a direct consequence of the fact that g satisfies (8)–(9). Moreover it follows obviously from our method of the proof that the estimates on u and ut are uniform when the initial data (u0 , u1 ) remain bounded in V × H . Then the result in the general case follows by density. 2


2947

Remark 3.4. The Neumann case contains in particular the case of the ODE α u + c u u + |u|p−1 u = 0 which was studied in [12]. It is not difficult to see that the rate of decay given by Corollary 3.3 is optimal for space-independent solutions only if p = 1. 4. A gradient inequality for some non-analytic functionals In this section we shall find the optimal Łojasiewicz exponent for a class of non-negative potentials associated to semi-linear PDE problems. In what follows, Ω is a bounded connected open subset of RN . 4.1. A general class of possibly non-analytic functionals. Application to various operators and boundary conditions In this subsection, V is a Hilbert space continuously imbedded in H = L2 (Ω), A ∈ L(V , V ) is symmetric, such that ∀u ∈ V ,

Au, u 0

(29)

and we set M(u) = Au + f (u) where f : V → V is the gradient of a functional F ∈ C 1 (V ). The energy functional is 1 E(u) = Au, u + F (u). 2 We assume that N = ker A is finite-dimensional and we denote by P : H → N the orthogonal projection on N in H . Proposition 4.1. Under the following hypotheses ∃η > 0, ∀v ∈ V ∩ N ⊥ , Av, v ηv2V ,

∃μ > 0, ∀u ∈ V , f (u), u μF (u),

(30) (31)

for any R > 0 there is a constant M(R) such that ∀u ∈ V ,

uV R

⇒

ur+1 H M(R)F (u).

(32)

Then for any R > 0 1) there is a constant C(R) such that ∀u ∈ V ,

uV R

⇒

r E(u) r+1 C(R) M(u) V ,

(33)

2948


2) there is a constant P (R) such that ur+1 V P (R)E(u).

∀u ∈ V ,

uV R R

∀u ∈ V ,

Mu, u = A(u − P u), u − P u + f (u), u

⇒

(34)

Proof. We have

ηu − P u2V + μF (u). On the other hand ∀u ∈ V ,

uV u − P uV + P uV u − P uV + C1 P uH ,

∀u ∈ V ,

uV R

⇒

1

uV u − P uV + C2 (R)F (u) r+1 .

Hence ∀u ∈ V ,

uV R

⇒

2 1 uV C3 (R) u − P uVr+1 + F (u) r+1

which implies by using the inequality a + b 2(a q + bq )1/q applied with q = r + 1 ∀u ∈ V ,

uV R

⇒

1 uV 2C3 (R) u − P u2V + F (u) r+1 .

Therefore (34) is proved and moreover ∀u ∈ V ,

uV R

⇒

ηu − P u2V + μF (u)

1 Mu, u 2C3 (R) u − P u2V + F (u) r+1 MuV .

Then (33) becomes an immediate consequence of the simple inequality

Au, u = A(u − P u), u − P u AL(V ,V ) u − P u2V .

2

We now state 3 simple applications of Proposition 4.1 1) The Dirichlet case. Let V = H01 (Ω), H := L2 (Ω). We consider p q M(u) = −u − λ1 u + c1 u+ − c2 u− where λ1 is the first eigenvalue of − and c1 , c2 > 0, 1 inf{p, q} and either N 2, or +2 sup{p, q} < N N −2 . The energy is given by 1 E(u) = 2

Ω

|∇u|2 − λ1 |u|2 dx +

c1 p+1

Ω

+ p+1 c2 dx + u q +1

− q+1 dx. u

Ω

Corollary 4.2. Under the above conditions, we have (33) with r = sup{p, q}.


2949

Proof. We set p q f (u) = c1 u+ − c2 u−

Au = −u − λ1 u; and F (u) :=

c1 p+1

Ω

+ p+1 c2 u dx + q +1

− q+1 u dx.

Ω

Then under the assumptions on p, q, f is a continuous map from V to V and it is in fact the gradient of F . In addition (31) is fulfilled with μ = 1 + inf{p, q} 2 and we also have clearly, using the positive character of both c1 and c2 + p+1 u K1 F (u) H

and − q+1 u K2 F (u). H

By addition we find + r+1 − r+1 r−p r−q r−p r−q u + u H K1 F (u)uH + K2 F (u)uH KF (u) uH + uH H and (32) follows easily. Finally we observe that since Ω is connected, N is one-dimensional and to check (30), introducing the second eigenvalue λ2 of − on V we find ∀v ∈ V ∩ N ⊥ ,

Av, v λ2 v2H .

In addition by definition of the norm in V we have ∀v ∈ V ,

v2V = Av, v + λ1 v2H .

Hence ∀v ∈ V ∩ N ⊥ , which gives (30) with η = result follows. 2

v2V Av, v +

λ2 λ1 +λ2 .

λ1 λ1 Av, v Av, v = 1 + λ2 λ2

Therefore all conditions of Proposition 4.1 are fulfilled and the

Remark 4.3. 1) If p = q or c1 = c2 , the functional is not analytic since even its restriction to the subspace of multiples of the first eigenfunction is not analytic. 2) If p = q and c1 = c2 we recover the example of Section 3.2.

2950


2) The Neumann case. Let V = H 1 (Ω), H := L2 (Ω). We consider p q M(u) = −u + c1 u+ − c2 u− where c1 , c2 > 0, 1 inf{p, q} and either N 2, or sup{p, q} < 1 E(u) = 2

c1 |∇u| dx + p+1

2

Ω

Ω

N +2 N −2 .

+ p+1 c2 u dx + q +1

The energy is given by

− q+1 u dx.

Ω

Corollary 4.4. Under the above conditions, we have (33) with r = sup{p, q}. Proof. We set Au = −u;

p q f (u) = c1 u+ − c2 u−

and F (u) :=

c1 p+1

Ω

+ p+1 c2 u dx + q +1

− q+1 u dx.

Ω

The properties of f and F are similar to the previous example. Since Ω is connected, N is one-dimensional, being reduced to constant functions and to check (30), introducing the second eigenvalue λ2 of − on V we find ∀v ∈ V ∩ N ⊥ ,

Av, v λ2 v2H .

In addition by definition of the norm in V we have here ∀v ∈ V ,

v2V = Av, v + v2H .

Hence ⊥

∀v ∈ V ∩ N , which gives (30) with η = result follows. 2

v2V

λ2 1+λ2 .

1 1 Av, v Av, v + Av, v = 1 + λ2 λ2

Therefore all conditions of Proposition 4.1 are fulfilled and the

Remark 4.5. 1) If p = q or c1 = c2 , the functional is not analytic since even its restriction to the subspace of multiples of the first eigenfunction is not analytic. 2) If p = q and c1 = c2 we recover the example of Section 3.3. 3) A fourth order operator. Let V = H02 (Ω), H = L2 (Ω) and let λ1 denote here the first eigenvalue of 2 on H02 (Ω). We assume c1 > 0, c2 > 0 and p, q > 1 with (N − 4) sup{p, q} < N + 4.


2951

We consider p q M(u) = 2 u − λ1 u + c1 u+ − c2 u− . The energy is given by 1 E(u) = 2

|u|2 − λ1 |u|2 dx +

Ω

c1 p+1

Ω

+ p+1 c2 u dx + q +1

− q+1 u dx.

Ω

Corollary 4.6. Under the above conditions, we have (33) with r = sup{p, q}. Proof. We set Au = 2 u − λ1 u;

p q f (u) = c1 u+ − c2 u−

and c1 F (u) := p+1

Ω

+ p+1 c2 u dx + q +1

− q+1 u dx.

Ω

Then under the assumptions on p, q, f is a continuous map from V to V and it is in fact the gradient of F . The rest of the proof is identical to the proof of Corollary 4.4. 2 4.2. Multiple equilibria under Neumann boundary conditions In this subsection we let V = H 1 (Ω), H = L2 (Ω), and we set M(u) = −u + f (u) 1 2 |∇u| dx + F u(x) dx. E(u) = 2 Ω

Ω

The hypotheses that we shall make on f will imply that the energy functional is bounded from below since it will be the case for any primitive of f . We shall choose for F the primitive with minimum equal to 0. More specifically, let us define for some reals a, b with a b ∀s ∈ R,

ρ(s) = (s − b)+ − (s − a)− .

Let p > 1 be such that (N − 2)p < N + 2 in order that V ⊂ Lp+1 (Ω).

2952


1,∞ We assume that f ∈ Wloc (R) with f = 0 on [a, b] and for some c > 0

∀s ∈ R,

p+1 f (s)ρ(s) c ρ(s) .

(35)

We define then s F (s) :=

f (u) du. a

In particular F = 0 on [a, b] and it follows clearly from (35) that F satisfies ∀s ∈ R,

p+1 c ρ(s) . p+1

F (s)

We assume that F satisfies for some C > 0 the upper bound p+1 . F (s) C ρ(s)

∀s ∈ R,

(36)

Proposition 4.7. Under the conditions (35), (36), for any R > 0 there is a constant C(R) such that ∀u ∈ V ,

uV R

⇒

p E(u) p+1 C(R) M(u) V .

(37)

Proof. Setting P v :=

1 |Ω|

v(x) dx Ω

we have since f is non-decreasing ∀u ∈ H 2 (Ω),

Mu, u − P u =

|∇u|2 + f (u)(u − P u) dx Ω

|∇u|2 + f (P u)(u − P u) dx =

Ω

|∇u|2 dx Ω

similarly ∀u ∈ H 2 (Ω),

Mu, ρ(u) =

Ω

ρ (u)|∇u|2 + f (u)ρ(u) dx

c

p+1

Ω

hence by addition

ρ(u) p+1 dx = c ρ(u) p+1


∀u ∈ H 2 (Ω),

2953

p+1 Mu, u − P u + ρ(u) ∇u22 + c ρ(u) p+1 .

As a consequence of Poincaré’s inequality, we have u − P uV K1 ∇u2 . In addition ρ(u) ∇ ρ(u) + ρ(u) ∇u2 + K2 ρ(u) V 2 2 p+1 hence u − P u + ρ(u) K3 ∇u2 + ρ(u) . V p+1 As a consequence we find for some fixed δ > 0 p+1

∇u22 + ρ(u)p+1

Mu∗ δ

∇u2 + ρ(u)p+1

.

By writing 2 1− p+1

∇u2 ∇u2

2

2 1− p+1

∇u2p+1 uV

2

∇u2p+1

we deduce p+1

∇u22 + ρ(u)p+1

Mu∗ δ

2 1− p+1

(1 + uV

.

2

)(∇u2p+1 + ρ(u)p+1 )

By using the inequality a + b 2(a q + bq )1/q applied with q = p + 1 we find p+1

Mu∗ δ

p

(∇u22 + ρ(u)p+1 ) p+1 2 1− p+1

2(1 + uV

)

and finally for some M > 0 p 1− 2 E(u) p+1 M 1 + uV p+1 Mu∗ .

2

5. More elaborate convergence results In this section we state and prove the convergence results corresponding to the specific cases of Section 4. These convergence results will contain as special cases all the examples given in Section 3.

2954


5.1. Preliminary results In order to apply Theorem 2.2, we shall need among other things to verify the left-hand side of (11). For this we shall rely in all cases on the following simple preliminary result. Proposition 5.1. Let Ω be a bounded open subset of RN , V a Hilbert space continuously imbedded in H = L2 (Ω), and A ∈ L(V , V ) symmetric, satisfying (29) and (30). Let f : V → V be the gradient of a non-negative functional F ∈ C 1 (V ). Assume that for any R > 0 there is a constant K(R) such that ∀u ∈ V ,

uV R

⇒

f (u)

V

1 K(R) F (u) 2 .

(38)

Then for any R > 0, there is a constant M(R) such that ∀u ∈ V ,

uV R

⇒

Au + f (u)

V

1 M(R) Au, u + 2F (u) 2 .

(39)

Proof. For any u ∈ V we have 1 A AuV = A(u − P u) V Au − P uV 1 Au, u 2 . η2 The result follows by combining this inequality with (38).

2

In practice, to treat the non-linear part f (u), we shall use repeatedly the following simple lemma. Lemma 5.2. Let Ω, V , H be as above and p 1 be such that V ⊂ Lp+1 (Ω) with continuous and dense imbedding. Then for any c ∈ R and any R > 0, there is a constant P (R) such that ∀u ∈ V ,

uV R

Proof. By duality we have L we have

(u − c)+ p

p+1 p

V

⇒

(u − c)+ p

V

p+1 2 P (R) (u − c)+ p+1 .

(40)

(Ω) ⊂ V with continuous imbedding. Therefore for any u ∈ V

p K1 (u − c)+

p+1 L p

p = K1 (u − c)+ p+1 .

Since p p+1 2 , the result follows easily by combining this inequality with the imbedding V ⊂ Lp+1 (Ω). 2 Finally, boundedness of the solutions in V will follow in all examples as a consequence of the following lemmas.


2955

Lemma 5.3. Let V , H be as above and A ∈ L(V , V ) be symmetric, satisfying (29). Assume that there is λ0 > 0, γ0 > 0 such that ∀u ∈ V ,

Au, u + λ0 u2H γ0 u2V .

In addition, assume that there is σ0 > 0, K0 0 such that ∀u ∈ V ,

F (u) σ0 u2H − K0 .

(41)

E(u) σ u2V − K0 .

(42)

Then there exists σ > 0 such that ∀u ∈ V , Proof. If σ0

λ0 2 ,

the result is obvious since then

1 1 1 1 2 E(u) = Au, u + F (u) γ0 uV + σ0 − λ0 u2H − K0 γ0 u2V − K0 . 2 2 2 2 If σ0
1, c > 0, A 0 Lemma 5.4. Let f ∈ Wloc

∀s ∈ R,

|s| A

⇒

f (s)s c|s|p+1 .

(43)

Then for any primitive F of f there is δ > 0 and C > 0 for which ∀s ∈ R,

F (s) δ|s|2 − C.

In particular in this case, the functional F (u) :=

F u(x) dx

Ω

satisfies (41). Proof. The first inequality is an obvious consequence of a trivial lower estimate on F and the second one follows by integration since Ω is bounded. Of course without additional conditions on f the functional F is not necessarily defined on all u ∈ V , it may take infinite values. In the examples we always assume that the functional is defined on V as a consequence of growth conditions on F . 2

2956


5.2. Convergence to 0 and rate of decay Again, in what follows, Ω is a bounded connected open subset of RN . We consider the following problems

p q utt + g(ut ) − u − λ1 u + c1 u+ − c2 u− = 0, u(t, x) = 0, on R+ × ∂Ω,

in R+ × Ω,

(44)

and ⎧ p q ⎨ utt + g(ut ) − u + c1 u+ − c2 u− = 0, ⎩ ∂u = 0, on R+ × ∂Ω. ∂n

in R+ × Ω, (45)

The main result of this subsection is the following: Theorem 5.5. Define V , H as in Section 4.1. Assuming that p, q, c1 , c2 fulfill the conditions of 1,1 2,1 Section 4.1 and g satisfies (8)–(9), any solution u ∈ Wloc (R+ , V ) ∩ Wloc (R+ , H ) of one of these systems converges to 0 in V as t → ∞ and we have 1 u(t, .) Mt − (α+1)r−1 ; V

r+1 ut (t, .) M t − 2[(α+1)r−1] H

where r = sup{p, q} and M, M are some positive constants depending on the solution. Proof. The main assumptions to be checked are (11) and boundedness of u(t) in V . The relevant Łojasiewicz inequality is uniform on any bounded subset of V and has been proved in Corollary 4.2 as a consequence of Proposition 4.1. We now check that the left-hand side of (11) holds true on any bounded subset of V with γ = 12 . This is in fact a simple consequence of Lemma 5.2 applied with c = 0. Indeed under the hypothesis of Theorem 5.5, we have ∀u ∈ V ,

uV R

⇒

+ p u

V

p+1 1 1 2 P (R) u+ p+1 p 2 P (R) F (u) 2 .

Changing u to −u and replacing p by q we find ∀u ∈ V ,

uV R

⇒

− q u

V

1 1 q 2 Q(R) F (u) 2 .

Then by an obvious combination we find than f (u) = c1 (u+ )p − c2 (u− )q satisfies (38). By applying Proposition 5.1 we conclude that the left-hand side of (11) holds true on any bounded subset of V with γ = 12 . Now (11) holds true on any bounded subset of V . A simple application of Theorem 2.2 will conclude the proof as soon as boundedness of u(t) in V is established. But it is clear that f satisfies (43) with A = 0. Then Lemma 5.4 and Lemma 5.3 applied with λ0 = λ1 in the Dirichlet case, λ0 = 1 in the Neumann case conclude the proof, since the non-increasing character of the energy provides the required V -bound. 2


2957

5.3. Multiple equilibria under Neumann boundary conditions We consider the system ⎧ ⎨ utt + g(ut ) − u + f (u) = 0, ∂u ⎩ = 0, on R+ × ∂Ω. ∂n

in R+ × Ω, (46)

The main result of this subsection is the following: Theorem 5.6. Define V , H as in Section 4.2. Assume that in addition to the conditions of Proposition 4.6, f ∈ C 1 (R) satisfies (35) and ∀s ∈ R,

f (s) C ρ(s) p−1

with (N − 2)p < N + 2. Assume that g satisfies (8)–(9). 1,1 2,1 (R+ , V ) ∩ Wloc (R+ , H ) of one of these systems converges in V Then any solution u ∈ Wloc as t → ∞ to a constant c ∈ [a, b] and we have 1−αp u(t, .) − c Mt − (α+1)p−1 ; V

p+1 ut (t, .) + ∇u(t, .) M t − 2[(α+1)p−1] . H H

Proof. The main assumptions to be checked are again (11) and boundedness of u(t) in V . Boundedness in V is similar to the previous case since f satisfies (43) with A = 2 max{|a|,|b|} +1 for instance. The relevant Łojasiewicz inequality is uniform on any bounded subset of V and has been proved in Proposition 4.7. We now check that the left-hand side of (11) holds true on any bounded subset of V with γ = 12 . This is in fact a simple consequence of Lemma 5.2. Indeed under the hypothesis of Theorem 5.5, we have ∀u ∈ V ,

uV R

⇒

(u − b)+ p

V

p+1 1 2 P (R) (u − b)+ p+1 KP (R) F (u) 2 .

Similarly we find ∀u ∈ V ,

uV R

⇒

(u − a)− q

V

1 K Q(R) F (u) 2 .

Then we find that f satisfies (38). By applying Proposition 5.1 we conclude that the left-hand side of (11) holds true on any bounded subset of V with γ = 12 . Since (11) holds true on any bounded subset of V , a simple application of Theorem 2.2 concludes the proof. 2 5.4. An example with a fourth order operator in space Again, in what follows, Ω is a bounded connected open subset of RN . We consider the following problem ⎧ α ⎪ ⎨ utt − c |∇ut |2 dx 2 ut + 2 u − λ1 u + c1 u+ p − c2 u− q = 0, ⎪ ⎩

Ω

u(t, x) = |∇u| = 0,

on R+ × ∂Ω,

in R+ × Ω,

(47)

2958


where λ1 denotes here the first eigenvalue of 2 on H02 (Ω). We define V = H02 (Ω) and H = L2 (Ω). We assume c1 > 0, c2 > 0 and p, q > 1 with (N − 4) sup{p, q} < N + 4. The main result of this subsection is the following: Theorem 5.7. Assuming that p, q, c1 , c2 fulfill the conditions above and 0 < α < 1r , with r := 1,1 2,1 sup{p, q}, any solution u ∈ Wloc (R+ , V ) ∩ Wloc (R+ , H ) of (47) converges to 0 in V as t → ∞ and we have 1 r+1 ut (t, .) M t − 2[(α+1)r−1] u(t, .) Mt − (α+1)r−1 ; V H where M, M are some positive constants depending on the solution. Proof. The proof of the V -bound of u(t) and (11) are quite similar to the analogous steps in the proof of Theorem 5.5. The relevant Łojasiewicz inequality is uniform on any bounded subset of V and has been proved in Proposition 4.6. The proof that the left-hand side of (11) holds true on any bounded subset of V with γ = 12 is already done in the proof of Theorem 5.5. Hence (11) holds true on any bounded subset of V . However here Theorem 2.2 cannot be applied as it stands because g is not defined on H , but from H01 to H −1 . A thorough inspection of the proof of Theorem 2.2 allows us, mutatis mutandis, to obtain the necessary extension, and this concludes the proof. 2 5.5. The case of weak solutions When g satisfies the conditions of either Example 3.1 or Example 3.2, Theorems 5.5 and 5.6 are applicable to any weak solution u ∈ C(R+ , V ) ∩ C 1 (R+ , H ) in the sense of [10]. The same remark applies to Theorem 5.7 for weak solutions obtained by the standard monotonicityperturbation method. The general idea is that local (in time) approximation by strong solutions is enough to justify convergence of the estimates since all constants depend boundedly on the natural energy norm. We skip the details which are an easy adaptation from the methods of [10]. 6. Convergence and rate of convergence in the non-autonomous case In this section we assume that the hypothesis of Theorem 2.2 is satisfied and we consider the following abstract system ⎧ ⎪ ¨ + g u(t) ˙ + M u(t) = h(t), t 0, ⎨ u(t) (48) u0 ∈ V , u(0) = u0 , ⎪ ⎩ u(0) ˙ = u1 , u1 ∈ H, where h : R+ → H is such that ∃c 0, ∃δ > 0,

h(t) H

c , (1 + t)1+δ+α

The main result of this section is the following:

for all t ∈ R+ .

(49)


2959

1,1 2,1 Theorem 6.1. Let u ∈ Wloc (R+ , V ) ∩ Wloc (R+ , H ) be a solution of (48) such that u(t) ∈ B for t large where B denotes a closed subset of V . Assume that hypotheses (8), (9), (11), (13) and (49) are satisfied. Let θ be as in (10) and (11). Let us introduce

δ 1 − (α + 1)(1 − θ ) , ; μ = inf (α + 2)(1 − θ ) − 1 α + 1

ν = inf

1 α+2 ,α + 1 + δ . (α + 2)(1 − θ ) − 1 α+1

Then there exist T > 0, a ∈ B and some constants C, M > 0 such that u(t) − a Ct −μ , H ∀t T , E u(t) Mt −ν .

∀t T ,

(50) (51)

6.1. Proof of Theorem 6.1 1,1 2,1 Let u be a solution of Eq. (48) such that u ∈ Wloc (R+ , V ) ∩ Wloc (R+ , H ) and u(t) ∈ B for t large. Let us define the non-negative function

2 1 ˙ H + E(u). E(t) = u(t) 2 We have by (8)

E (t) = − g(u), ˙ u˙ V ,V + h, u ˙ H −ρ1 u ˙ α+2 ˙ H. H + hH u Therefore we have by Young’s inequality E (t) −

α+2 ρ1 α+1 u ˙ α+2 H + KhH . 2

In particular α+2

E (t) KhHα+1 ∈ L1 (R+ )

(52)

hence E(t) is bounded on R+ . Now let 0 < ε 1 be a real constant. We define the function

H (t) = E(t) + εE(t)β M u(t) , u(t) ˙ V +

∞

∞

t

t

h(s), u(s) ˙ ds + H

2 E(s)β h(s) H ds,

where β = α(1 − θ ), θ is the Łojasiewicz exponent defined in (11). We claim that for some constants c8 , c9 to be defined later H (t) c8 E(t) +

c9 , (1 + t)λ

where λ = α + 1 + δ( α+2 α+1 ). In fact, we have

for all t T ,

(53)

2960


M(u) V u ˙ V +

β

H (t) E(t) + εE(t)

∞

∞ hH u ˙ H ds +

t

2 E(s)β h(s) H ds.

t

By integrating (52) we obtain ∞

∞ u ˙ α+2 H ds

c4 E(t) + c3

t

α+2

hHα+1 ds. t

And since E is bounded this implies ∞

2 h(s) H ds c5

β

E(s) t

∞ h(s) 2 ds. H t

On the other hand, by the above inequalities together with (11) we get α(1−θ)+γ + 12

H (t) E(t) + εc6 E(t)

∞ + c3

α+2 α+1

hH

∞ 2 + c4 E(t) + c5 h(s) H ds.

t

t

1

Since E(t)α(1−θ)+γ + 2 c7 E(t), then by observing that 2 > we get H (t) c8 E(t) +

α+2 α+1

and using the assumption (49)

c9 , (1 + t)λ

where λ = α + 1 + δ( α+2 α+1 ), as claimed. Now we have

H (t) = E (t) − h, u ˙ V ˙ H + εβ u(t), ˙ h(t) − g u(t) ˙ E(t)β−1 M u(t) , u(t) V ,V

˙ u(t) ˙ V + εE(t)β M u(t) , h(t) − g u(t) ˙ + εE(t)β M u(t) u(t), V 2 2 β β − εE(t) M u(t) − E(t) h(t) . V

(54)

H

By the Cauchy–Schwarz inequality

εβ u(t), ˙ h(t) V ,V E(t)β−1 M u(t) , u(t) ˙ V εβc10 u ˙ 2H M u(t) V hH E(t)β−1 . Then we obtain since u ˙ 2H 2E(t)

εβ u(t), ˙ h(t) V ,V E(t)β−1 M u(t) , u(t) ˙ V εβc11 E(t)β hH M u(t) V . By the Cauchy–Schwarz inequality together with the last inequality we get


2961

˙ V εE(t)β M u(t) , h(t) V ,V + εβ u(t), ˙ h(t) V E(t)β−1 M u(t) , u(t) εE(t)β hH M u(t) V + εβc11 E(t)β hH M u(t) V . Once again by applying the Cauchy–Schwarz inequality we get

εE(t)β M u(t) , h(t) V ,V + εβ u(t), ˙ V ˙ h(t) V E(t)β−1 M u(t) , u(t) 2 2 ε E(t)β M u(t) V + εc13 E(t)β h(t) H . 4 On the other hand we have by the calculations of Section 1

˙ V c14 u(t), ˙ g u(t) ˙ E(t)β−1 M u(t) , u(t) , − u(t), ˙ g u(t) ˙ V ,V V ,V α+2 ε 2

˙ H + E(t)β M u(t) V , εE(t)β M u(t) u(t), ˙ u(t) ˙ V εc15 u(t) 4 α+2 2

ε β β ˙ H . εE(t) M u(t) , g u(t) ˙ E(t) M u(t) V + εc16 u(t) V 4

(55) (56) (57)

Then by combining the last 4 inequalities we obtain for a fixed ε small enough α+2 2 ˙ H + E(t)β M u(t) V . H (t) −c17 u(t)

(58)

Then by following the steps of the proof of Theorem 2.2 and by using (53) we get the following differential inequality c18 H (t)(α+2)(1−θ) + c19 H (t)

c20 . (1 + t)λ(1−θ)(α+2)

Therefore by applying Lemma 2.1 from [3] we obtain H (t) c21 (1 + t)−ν ,

(59)

1 where ν = inf{ (α+2)(1−θ)−1 , λ}. Now, by using (58) together with the last inequality we have for all t T

2t

−ν u ˙ α+2 H ds c22 (1 + t) .

t

Since we have 2t u ˙ H ds t t

we obtain

α+1 α+2

2t u ˙ α+2 H ds t

1 α+2

,

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2t u ˙ H ds t

c23 , (1 + t)−μ

δ where μ = inf{ 1−(α+1)(1−θ) (α+2)(1−θ)−1 , α+1 }. Then the conclusion follows as in the proof of Theorem 2.2 and we have u˙ ∈ L1 (T , ∞, H ) for T > 0 large enough. Hence, u(t) has a limit a in H as t → ∞ and

u(t) − a Ct −μ . H On the other hand we have by the calculation leading to (54)

α(1−θ)+γ − 12

H (t) E(t) − εc6 E(t)

∞ − c3

α+2 α+1

hH

∞ 2 − c4 E(t) − c5 h(s) H ds.

t

t

Then by choosing ε small enough we get H (t)

c24 E(t) − . 2 (1 + t)λ

Therefore we have E(t) 2H (t) +

c24 , (1 + t)λ

and then thanks to (59) we obtain, since λ ν E(t)

c25 . (1 + t)ν

It follows that there exists a constant M such that E u(t) M(1 + t)−ν . 6.2. Applications Theorem 6.1 is applicable to perturbations of any of the particular systems considered in Sections 3 and 5 by a sufficiently fast decaying forcing term. To avoid heavy repetitions, the details of application to those examples are left to the reader. Acknowledgments The authors are grateful to the referees for interesting observations which contributed to make their work easier to read.


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References [1] F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert spaces, SIAM J. Control Optim. 38 (4) (2000) 1102–1119. [2] I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations, Asymptot. Anal. 69 (2010) 31–44. [3] I. Ben Hassen, L. Chergui, Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation, J. Dynam. Differential Equations, in press. [4] R.E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal. 18 (1975) 15–26. [5] L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dynam. Differential Equations 20 (2008) 643–652. [6] L. Chergui, Convergence of global and bounded solutions of the wave equation with nonlinear dissipation and analytic nonlinearity, J. Evol. Equ. 9 (2009) 405–418. [7] R. Chill, M.A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal. 53 (7–8) (2000) 1017–1039. [8] J.K. Hale, G. Raugel, Convergence in gradient-like systems with applications to PDE, Z. Angew. Math. Phys. 43 (1) (1992) 63–124. [9] A. Haraux, Asymptotics for some nonlinear hyperbolic equations with a one-dimensional set of rest points, Bol. Soc. Brasil. Mat. 17 (2) (1986) 51–65. [10] A. Haraux, Semi-linear hyperbolic problems in bounded domains, in: J. Dieudonne (Ed.), in: Math. Rep., vol. 3, Harwood Academic Publishers, Gordon & Breach, 1987 (Part 1). [11] A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Res. Notes Math., vol. 17, Masson, Paris, 1991. [12] A. Haraux, Sharp decay estimates of the solutions to a class of nonlinear second order ODE, Anal. Appl. (2010), in press. [13] A. Haraux, M.A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations 9 (2) (1999) 95–124. [14] A. Haraux, M.A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Anal. 26 (1) (2001) 21–36. [15] A. Haraux, M.A. Jendoubi, On a second order dissipative ODE in Hilbert space with an integrable source term, in press. [16] A. Haraux, M.A. Jendoubi, O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evol. Equ. 3 (3) (2003) 463–484. [17] S.Z. Huang, P. Takac, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal. 46 (5) (2001) 675–698. [18] M.A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Differential Equations 144 (2) (1998) 302–312. [19] M.A. Jendoubi, R. May, On an asymptotically autonomous system with Tikhonov type regularizing term, Arch. Math. 95 (2010) 389–399. [20] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, in: Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles, Paris, 1962, Editions du C.N.R.S., Paris, 1963, pp. 87–89. [21] S. Łojasiewicz, Ensembles semi-analytiques, preprint, I.H.E.S. Bures-sur-Yvette, 1965. [22] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems, Asymptot. Anal. 1 (2) (1988) 161–185.


A generalization of the cellular indecomposable property via fiber dimension Guozheng Cheng a,∗ , Xiang Fang b,1 a School of Mathematics, Wenzhou University, Wenzhou, Zhejiang 325035, China b Department of Mathematics, Kansas State University, Manhattan, KS 66502, United States

Received 24 August 2010; accepted 9 February 2011 Available online 22 February 2011 Communicated by D. Voiculescu

Abstract The cellular indecomposable property, introduced by Olin and Thomson in 1984 [11], is well known for the Dirichlet space, but it fails trivially for the vector-valued case. The purpose of this paper is to use the fiber dimension to reformulate the property such that it naturally extends the scalar-valued case, yet fix the vector-valued case in a meaningful way. Using the new formulation, we are able to generalize several previous results to the vector-valued setting. In particular, we extend a theorem of Bourdon relating the cellular indecomposable property and the codimension-one property to codimension-N . Several of our results appear to be new even for the Hardy space over the unit disc. © 2011 Elsevier Inc. All rights reserved. Keywords: Cellular indecomposable property; Fiber dimension; Codimension; Invariant subspace

1. Introduction The cellular indecomposable property (CIP), introduced by R. Olin and J. Thomson in [11], states that any two nontrivial invariant subspaces M1 , M2 ⊂ H of a Hilbert space H , with respect to an operator T ∈ B(H ), have a nontrivial intersection M1 ∩ M2 = {0}. It is well known that (CIP) holds for the Dirichlet space D over the unit disk D, see Richter and Shields [14]. * Corresponding author.

E-mail addresses: [email protected] (G. Cheng), [email protected] (X. Fang). 1 Partially supported by the National Science Foundation Grant DMS 0801174 and the Laboratory of Mathematics for

Nonlinear Science, Fudan University. 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.02.009

G. Cheng, X. Fang / Journal of Functional Analysis 260 (2011) 2964–2985

2965

Theorem. (See [14].) Any two nonzero invariant subspaces M1 , M2 of the Dirichlet space D with respect to the Dirichlet shift Mz have a nontrivial intersection M1 ∩ M2 = {0}. This important property has many applications in operator theory, even to the transitive algebra problem [3,13]. But it fails trivially for the vector-valued case: just consider invariant subspaces M1 = D ⊕ {0} and M2 = {0} ⊕ D of H = D ⊕ D. It is desirable to extend (CIP) to the vector-valued case in a meaningful way and the purpose of this paper is to present such an extension. Theorem 1. For any two invariant subspaces M1 , M2 ⊂ D ⊗ CN , N ∈ N, if f d(M1 ) + f d(M2 ) > N, then M1 ∩ M2 = {0}. Here the fiber dimension f d(M) of an invariant subspace M is defined as f d(M) = sup dim M(λ) λ∈D

with M(λ) = f (λ): f ∈ M ⊂ CN . Theorem 1 clearly generalizes the above result of Richter and Shields [14], since f d(M1 ) = f d(M2 ) = 1 when M1 , M2 are nonzero invariant subspaces of D. Instead of proving Theorem 1 directly, we will show that a quantitative result is indeed true: under the condition of Theorem 1, we have f d(M1 ∩ M2 ) f d(M1 ) + f d(M2 ) − N.

(1)

Further, we are able to establish a relative version of the above (1); namely, the two invariant subspaces M1 , M2 can be chosen relative to another invariant subspace M ⊂ D ⊗ CN . We also point out that it is not hard to extend the result to spaces with complete Nevanlinna– Pick (NP for short) kernels, which certainly cover the Dirichlet space. We state the next theorem in this more general setting. Let H(k) denote a Hilbert space of analytic functions over a domain Ω ⊂ C containing the origin, determined by a reproducing kernel k with the complete NP kernel property. Theorem 2. Let k be a complete NP kernel. For any multiplier invariant subspace M ⊂ H(k) ⊗ CN , N ∈ N, and two multiplier invariant subspaces M1 , M2 ⊂ M, f d(M1 ∩ M2 ) f d(M1 ) + f d(M2 ) − f d(M). In particular, one has that (†) if f d(M1 ) + f d(M2 ) > f d(M), then M1 ∩ M2 = {0}.

(2)

2966


Here an analytic function ϕ on Ω is a multiplier of H(k) if ϕf ∈ H(k) for every f ∈ H(k). The closed graph theorem implies that each multiplier ϕ induces a bounded multiplication operator Mϕ : f → ϕf on H(k). A subspace M of H(k) ⊗ CN is called multiplier invariant if it is invariant for each Mϕ . When k is a complete NP kernel, any multiplier invariant subspace of H(k) ⊗ CN has the following nice property, which follows from Theorem 0.7 in [10]. Lemma 3. Suppose that k is a complete NP kernel and M is a multiplier invariant subspace of H(k) ⊗ CN , N ∈ N, then the subset f ∈ H(k) ⊗ CN : f has multiplier entries ∩ M is dense in M. Here f = (f1 , . . . , fN ) ∈ H(k) ⊗ CN has multiplier entries if each fi is a multiplier. The rest of the paper is organized as follows: In Section 2 we gather preliminary facts on the fiber dimension and a notion called “occupy invariant” which is introduced in [8] and will be needed in the proof of Theorem 2 in Section 3. Section 3 is devoted to the proof of Theorem 2. Section 4 contains a direct application of Theorem 2 which yields a subadditivity result (Theorem 8) for Samuel multiplicities on coinvariant subspaces. In Section 5, we introduce the “complete cellular indecomposable property (CCIP)”, which extends the familiar cellular indecomposable property (CIP) of Olin and Thomson [11]. We also introduce a weaker version (CCIP ) and show that it implies the stronger (CCIP) under a natural complementary condition (C), see Theorem 12. Sections 6 and 7 are devoted to generalizations of two results of Bourdon [2]: First, Bourdon showed that the cellular indecomposable property implies the wellknown codimension-one property. On the other hand, it is a folklore that for the vector-valued case, the codimension-one property is replaced by the codimension-N property. In Section 6 we show that one can indeed establish a parallel result for codimension-N (Theorem 13) if using the complete cellular indecomposable property introduced in Section 5. The second result of Bourdon which we will generalize in Section 7 is a partial converse of the cellular indecomposable property, see Theorem 18. It is not hard to see that the converse of Bourdon’s result is not true. 2. Preliminaries on fiber dimension and occupy invariant In this section we gather some basic facts on the fiber dimension and a notion called “occupy invariant” which provides a way to describe the structure of certain invariant subspaces and will be used in the proof of Theorem 2. Definition 4. For any subspace M ⊂ H(k) ⊗ CN , N ∈ N, define the occupy invariant of M, denoted by lM , to be the maximal dimension of a subspace E of CN with the following property: there exists a basis (not necessarily orthonormal) e1 , . . . , el (l = lM ) of E and h1 , . . . , hl ∈ M such that PH(k)⊗E hi (= 0) ∈ H(k) ⊗ ei ,

i = 1, . . . , l.

When E has the above property we say that M occupies H(k) ⊗ E in H(k) ⊗ CN . The following is probably the most useful fact about lM for our purpose.


2967

Lemma 5. Let k be a complete NP kernel. If M is a multiplier invariant subspace of H(k) ⊗ CN , N ∈ N, then lM = f d(M). For a proof see Proposition 3.3 of [3], which essentially follows from Lemma 23 of [8]. A subspace M ⊂ H(k) ⊗ CN , N ∈ N, is called a d-graph subspace (d N ) [3] if there exists a basis of CN such that with respect to this basis, M has the form M = (f1 , . . . , fd , T1 f, . . . , TN −d f ): f = (f1 , . . . , fd ) ∈ L , where L is the linear manifold of the first d entries of elements in M and we assume f d(L) = d. Moreover, each Ti is a linear transform from L to H(k). When M is multiplier invariant, one has that Ti Mϕ = Mϕ Ti for any multiplier ϕ. If k is a complete NP kernel and M occupies H(k) ⊗ E for some E ⊂ CN with dim(E) = f d(M), then we can extend the basis of E (as in Definition 4) to a basis of CN . With respect to this basis, it is easy to check that M is a d-graph subspace with d = f d(M). For the details, see Theorem 3.6 of [3]. Lemma 6. Let k be a complete NP kernel. Suppose that M ⊂ H(k) ⊗ CN , N ∈ N, is a multiplier invariant subspace with f d(M) = d, then M is a d-graph subspace. The following lemma plays a key role in the proof of Theorem 2 and is of independent interests. Lemma 7. Let k be a complete NP kernel. If M ⊂ H(k) ⊗ CN , N ∈ N, is a multiplier invariant subspace, then it occupies H(k) ⊗ M(λ) for any point λ ∈ Ω. Proof. Assume dim M(λ) = d. Then we take f1 , . . . , fd ∈ M such that f1 (λ), . . . , fd (λ) form a basis for M(λ). Moreover, by Lemma 3 we can require that each fi has multiplier entries. Extend {fi (λ)}di=1 to a basis of CN and with respect to this basis, we write fi = (fi1 , . . . , f1N ),

1 i d.

By our choice of fi , the determinant of matrix Θ = (fij )di,j =1 , denoted by det(Θ), is a nonzero analytic function and is nonzero at λ in particular. Moreover, note that det(Θ) is a multiplier on H(k). Recall that the inverse matrix of Θ is given by det1Θ (Aij )di,j =1 , where Aij is the (d − 1) × (d − 1) minor of Θ associated with fj i . The useful fact here is that Aij is still a multiplier on H(k). It follows that

2968


⎛

(Aij )d×d · (fij )d×N

det(Θ) ⎜ 0 ⎜ =⎜ ⎝ ··· 0

0 det(Θ) ··· 0

··· 0 ··· 0 ··· ··· · · · det(Θ)

g11 g21 ··· gd1

⎞ · · · g1,N −d · · · g2,N −d ⎟ ⎟ ⎟. ··· ··· ⎠ · · · gd,N −d

So the following vectors are in M because Aij are multipliers, i -th

gi = 0, . . . , 0, det(Θ), 0, . . . , 0, gi1 , . . . , gi,N −d , They show that M occupies H(k) ⊗ M(λ).

1 i d.

2

Lastly we observe that the fiber dimension f d(M) of a subspace M ⊂ H(k)⊗CN is achieved at almost all points λ ∈ Ω; namely, f d(M) = dim M(λ),

a.e. λ ∈ Ω.

(3)

Here M(λ) = {f (λ): f ∈ M} ⊂ CN . We say that λ is a maximal fiber point if (3) holds for λ. Any point μ ∈ Ω in the complement of maximal fiber points is called a degenerate point. The set of degenerate points is denoted by Zdg (M). Let λ be a maximal fiber point of M, and let {e1 , . . . , ed } be an orthonormal basis for M(λ) ⊂ CN , and extend it to an orthonormal basis {e1 , . . . , eN } for CN . Let f1 , . . . , fd be such that {f1 (λ), . . . , fd (λ)} form a basis for M(λ). Write fi = (fi1 , . . . , fiN ) according to the orthonormal basis {e1 , . . . , eN }. Then the determinant F (z) = det(fij )di,j =1 is a nonzero function and is nonzero at λ in particular. It follows that

Zdg (M) ⊂ Z F (z) , the zero set of F (z). So, in general, Zdg (M) is a discrete subset of the domain Ω. 3. Proof of Theorem 2 Proof. Without loss of generality, we assume that M = M1 ∨ M2 , the closed subspace spanned by M1 and M2 . Let λ0 be a maximal fiber point for M, M1 and M2 . By Lemma 7, M and Mi occupy H(k) ⊗ M(λ0 ) and H(k) ⊗ Mi (λ0 ), respectively, i = 1, 2. Let E = M1 (λ0 ) ∩ M2 (λ0 ), E1 = M1 (λ0 ) E , Then assume

and E2 = M2 (λ0 ) E .


dim(Ei ) = di ,

2969

i = 1, 2,

and

dim E = d . We take bases, not necessarily orthonormal, {e1 , . . . , ed1 },

{ed1 +1 , . . . , ed1 +d2 },

{ed1 +d2 +1 , . . . , ed1 +d2 +d }

for E1 , E2 , E , respectively. Obviously, E = {e1 , . . . , ed1 , ed1 +1 , . . . , ed1 +d2 , ed1 +d2 +1 , . . . , ed1 +d2 +d } is a basis for M(λ0 ). Let d = d1 + d2 + d = f d(M), the fiber dimension of M. Extend E to a basis of CN , denoted by E . Under the basis E , by Lemma 6, M is a d-graph subspace. As in Lemma 6, let L be the linear manifold of the first d-components of elements in M, so there are N − d linear transformations Tj : L → H(k) such that M is of the form M = (f1 , . . . , fd , T1 f, . . . , TN −d f ): f = (f1 , . . . , fd ) ∈ L . The rest of the proof is divided into three steps. Step I: In this step, we will choose functions of a particular form in M1 and M2 to represent the fiber dimensions in a way suitable for considering the fiber dimension of M1 ∩ M2 . Since by Lemma 7, M1 occupies H(k) ⊗ (E1 + E ), we can find the following d1 + d elements, all with multiplier entries, in M1 : Fi = (Fi , T1 Fi , . . . , TN −d Fi ),

1 i d1 + d ,

(4)

where for i = 1, . . . , d1 , d1 ’s

d ’s

d2 ’s

Fi = 0, . . . , 0, fi , 0, . . . , 0, hi1 , . . . , hi,d2 , 0, . . . , 0 , and for i = d1 + 1, . . . , d1 + d , d1 ’s

d2 ’s

d ’s

Fi = 0, . . . , 0, ki1 , . . . , ki,d2 , 0, . . . , 0, fi , 0, . . . , 0 . Here each fi is a nonzero function. In particular, since we assume that λ0 is a maximal fiber point for M1 and by the proof of Lemma 7, fi (λ0 ) = 0.

(5)

2970


For simplifying symbols, let Ai (1 i d1 ) be a d1 × d2 matrix with only one nonzero row which is the i-th row. Similarly, let Bi (d1 + 1 i d1 + d ) be a d × d2 matrix

hi,d ( hfi1i , . . . , fi 2 ),

with only one nonzero row ( kfi1i , . . . , for i = 1, . . . , d1 ,

ki,d2 fi ),

which is the i-th row. Then Fi can be rewritten as: d ’s

d2 ’s

d1 ’s

Fi = 0, . . . , 0, fi , 0, . . . , 0, (0, . . . , 0, fi , 0, . . . , 0)Ai , 0, . . . , 0 , and for i = d1 + 1, . . . , d1 + d , d ’s

d2 ’s

d1 ’s

Fi = 0, . . . , 0, (0, . . . , 0, fi , 0, . . . , 0)Bi , 0, . . . , 0, fi , 0, . . . , 0 . Similarly, we can find the following d2 + d elements in M2 Gi = (Gi , T1 Gi , . . . , TN −d Gi ),

1 i d2 + d ,

(6)

where for i = 1, . . . , d2 , d1 ’s

d ’s

d2 ’s

Gi = (0, . . . , 0, gi , 0, . . . , 0)Ci , 0, . . . , 0, gi , 0, . . . , 0, 0, . . . , 0 ,

(7)

and for i = d2 + 1, . . . , d2 + d , d1 ’s

d2 ’s

d ’s

Gi = (0, . . . , 0, gi , 0, . . . , 0)Di , 0, . . . , 0, 0, . . . , 0, gi , 0, . . . , 0 ,

(8)

with gi = 0, and each Ci or Di is a d2 × d1 or d × d1 matrix respectively. Similarly, we have gi (λ0 ) = 0.

(9)

In particular, it follows from (5) and (9) that

dim span F1 (λ0 ), . . . , Fd1 (λ0 ), G1 (λ0 ), . . . , Gd2 (λ0 ) = d1 + d2 .

(10)

Step II: In this step we are mainly concerned with solving Eq. (12) by analyzing its coefficient matrix. Our previous choices of vectors Fi and Gi in Step I make explicit analysis of the coefficient matrix of (12) possible. In order to consider the fiber dimension of M1 ∩ M2 , we consider those (d + d )-tuples (˜r1 , . . . , r˜d+d ), with multiplier entries and not all being zeros, such that r˜1 F1 + · · · + r˜d1 +d Fd1 +d = r˜d1 +d +1 G1 + · · · + r˜d+d Gd2 +d

(= 0).

(11)

Next we need the reduction of Eq. (11) from Fi and Gi to Fi and Gi . To do this, observe that for a vector f in M1 ∨ M2 , if the first d entries of f are all zero, then f must be zero since M = M1 ∨ M2 is a d-graph subspace. Now because the linear transformations Tj are


2971

commuting with multiplications induced by multipliers, we only need to consider the tuples (˜r1 , . . . , r˜d+d ) such that r˜1 F1 + · · · + r˜d1 +d Fd1 +d = r˜d1 +d +1 G1 + · · · + r˜d+d Gd2 +d

(= 0).

Now we rearrange the above equation as (r1 F1 + · · · + rd1 Fd1 ) + (rd1 +1 G1 + · · · rd1 +d2 Gd2 ) + (rd1 +d2 +1 Fd1 +1 + · · · + rd1 +d2 +d Fd1 +d ) + (rd1 +d2 +d +1 Gd2 +1 + · · · + rd1 +d2 +2d Gd2 +d ) = 0.

(12)

Therefore, we need to consider the following d × (d + d ) coefficient matrix of Eq. (12) ⎛

W1 ⎝ = AW1 0

CV1 V1 0

⎞ DV2 0 ⎠, V2

0 BW2 W2

where the columns are FiT and GT i with T denoting transposition; namely, W1 is the d1 × d1 diagonal matrix W1 = diag(f1 , . . . , fd1 ) and AW1 is a d2 × d1 matrix with columns T

(0, . . . , 0, fi , 0, . . . , 0)Ai ,

1 i d1 .

Similarly, W2 = diag(fd1 +1 , . . . , fd1 +d ), V1 = diag(g1 , . . . , gd2 ), V2 = diag(gd2 +1 , . . . , gd2 +d ). Moreover, BW2 , CV1 and DV2 are understood in the same way as AW1 . Let Θ be the first 3 × 3 block matrix of , that is, ⎛

W1

Θ = ⎝ AW1 0

CV1 V1 0

0

⎞

BW2 ⎠ . W2

Claim. The determinant of Θ is a nonzero analytic function.

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Proof. It is sufficient to show that det

W1 AW1

CV1 V1

= 0

since W2 is a diagonal matrix with nonzero diagonal entries.

(13) 2

Note that the column vectors in (13) are the (E1 + E2 )-components of F1 , . . . , Fd1 , G1 , . . . , Gd2 .

(14)

Let us recall three facts here: (1) E -components of all vectors in (14) are zero. (2) For any vector in M, if its (E1 + E2 + E )-component is zero, then the vector is itself zero, because M is a d-graph subspace. (3) Vectors in (14), if evaluated at λ0 , are independent. See (10). Now (13) follows from the above three facts, because otherwise it implies that vectors in (14) are always dependent when evaluated at any point λ. Step III: In this step, we (explicitly) solve Eq. (12) and observe that the degree of freedom in the solution is d , hence completing the proof of Theorem 2. First, recall that the inverse matrix of Θ is given by Θ −1 =

1 (Aij )di,j =1 , det Θ

where Aij are the (d − 1) × (d − 1) minors of Θ, and they are all multipliers of D. If we write as = (Θ, Θ1 ), then

(Aij )di,j =1 · = det(Θ) · Id , det(Θ) · Θ −1 · Θ1 at the level of matrix multiplication, where Id is the identity matrix with size d. Note that det(Θ) · Θ −1 · Θ1 is a d × d matrix and we write it as det(Θ) · Θ −1 · Θ1 = Θ = (hij )i=1,...,d, j =1,...,d . Then the following equation (15) is obtained by multiplying Eq. (12) with (Aij )di,j =1 , ⎛ ⎞ r1

. det(Θ) · Id , Θ ⎝ .. ⎠ = 0. rd+d

(15)

Hence any solution of (15) is also a solution of (12). Now it is not hard to see that the solutions of (12) have d many free variables. To be more precise, we write down explicitly the following d tuples of R = (r1 , . . . , rd+d ) which are the solutions of Eq. (15), hence of Eq. (12),


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(d+i)-th

Ri = hi1 , . . . , hid , 0, . . . , 0, − det(Θ), 0, . . . , 0 ,

1 i d .

(16)

So we have the following d vectors in M1 ∩ M2 : hi,d1 +1 G1 + · · · + hi,d1 +d2 Gd2 − det(Θ)Gd2 +i ∈ M1 ∩ M2 ,

i = 1, . . . , d .

(17)

Moreover, by the particular forms of Gi in (7), (8), we know that vectors in (17) show that M1 ∩ M2 occupies at least H(k) ⊗ E . This completes the proof of the theorem since

f d(M1 ) + f d(M2 ) − f d(M) = d1 + d + d2 + d − d1 + d2 + d = d .

2

4. An application: subadditivity of Samuel multiplicity In commutative algebra, the additivity of Samuel multiplicity ([5, p. 273, p. 279], [9, p. 52]) is of fundamental importance for applications in algebraic geometry and a parallel version in operator theory is proved, say, for the Hardy space H 2 (D) and the Dirichlet space D over the unit disk [7]. The purpose of this section is to show that Theorem 2 can lead to a subadditivity result (18) for Samuel multiplicities. Although the proof is short, this type of results seems to be new in operator theory literature, so we record it here. For an operator T ∈ B(H ) acting on a Hilbert space H such that dim(H /T H ) < ∞, the Samuel multiplicity is defined by [6] dim(H /T k H ) , k→∞ k

e(T , H ) = lim

which is well defined and is indeed a finite integer. For an invariant subspace M ⊂ D ⊗ CN , we define e(M⊥ ) to be e(M⊥ , Sz ). Here Sz is the compression of Mz to M⊥ , the orthogonal complement of M in D ⊗ CN . This section concerns ⊥ ⊥ ⊥ subadditivity of e(M⊥ ); namely, the relationship between e(M⊥ 1 ∨ M2 ) and e(M1 ) + e(M2 ). In this paper ∨ denotes the closed span of two subspaces. Theorem 8. For any two invariant subspaces M1 , M2 ⊂ D ⊗ CN , N ∈ N, we have

⊥

⊥

⊥ e M⊥ 1 ∨ M2 e M1 + e M2 .

(18)

⊥

⊥

⊥ ⊥ ⊥ e M⊥ 1 ∨ M2 + e M1 ∩ M2 e M1 + e M2 .

(19)

Indeed, we have

Proof. In [7], the second author obtained

f d(M) + e M⊥ = N for any invariant subspace M ⊂ D ⊗ CN . Meanwhile, by Theorem 2

(20)

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f d(M1 ∩ M2 ) f d(M1 ) + f d(M2 ) − f d(M1 ∨ M2 ). Combining (20) and (21), one gets (19).

(21)

2

Remark. One can also extend the above result to the complete NP case, but this will require one to extend the corresponding result from [7], which will cause unnecessary complexity for this paper. 5. Complete cellular indecomposable property Next we introduce a stronger version of (CIP), which is clearly motivated by Theorems 1 and 2. In this section, H denotes a Hilbert space of analytic functions over a domain Ω ⊂ C. Moreover, Mz , the multiplication by the coordinate function, is assumed to be bounded and all invariant subspaces are with respect to Mz . Definition 9. H has the complete cellular indecomposable property (CCIP) if for any invariant subspace M ⊂ H ⊗ CN , N ∈ N, two invariant subspaces M1 , M2 ⊂ M such that f d(M1 ) + f d(M2 ) > f d(M) have a nontrivial intersection M1 ∩ M2 = {0}. It is also natural to consider the following weaker definition, replacing M by the whole space H ⊗ CN . Definition 10. H has (CCIP ) if any two invariant subspaces M1 , M2 ⊂ H ⊗ CN , N ∈ N, such that f d(M1 ) + f d(M2 ) > N have a nontrivial intersection M1 ∩ M2 = {0}. The purpose of this section is to show that under a natural complementary condition (C) the weaker property (CCIP ) implies the stronger version (CCIP). Definition 11. H is said to satisfy the complementary condition (C) if for any invariant subspace M ⊂ H ⊗ CN , N ∈ N, there is another invariant subspace M such that


2975

1. f d(M ) + f d(M) = N ; 2. f d(M ∨ M ) = N ; 3. M and M have a positive angle. The condition (C) appears to be a fairly general property, and we conjecture that it holds for the Hardy space, the Dirichlet space, and even Bergman space. Yet even the Hardy space case is not previously known, and we intend to pursue these problems in a forthcoming work. Here the angle between M1 , M2 , denoted by angle(M1 , M2 ), is defined to be θ ∈ [0, π2 ] such that cos(θ ) = sup f, g: f = g = 1, f ∈ M1 , g ∈ M2 .

(22)

The invariant subspace M satisfying the above conditions is called a complementary space of M. Theorem 12. If H satisfies the complementary condition (C), then the weaker (CCIP , Definition 10) implies the stronger (CCIP, Definition 9). Proof. We argue by contradiction. Assume that H satisfies (CCIP ) and there is an invariant subspace M ⊂ H ⊗ CN such that it has two invariant subspaces M1 , M2 ⊂ M satisfying f d(M1 ) + f d(M2 ) > f d(M) and M1 ∩ M2 = {0}. Consider M , a complementary space of M, and M2 + M , which is automatically closed since M and M , hence M2 and M , have a positive angle, by condition 3. First, we show

f d M2 + M = f d(M2 ) + f d M .

(23)

Choose a λ ∈ Ω such that it is a maximal fiber point for M, M , and M ∨ M . In particular, dim M(λ) = f d(M)

and

dim M (λ) = f d M .

By condition 2,

dim M ∨ M (λ) = dim M(λ) + M (λ) = N, which implies M(λ) + M (λ) = CN . On the other hand, M(λ) ∩ M (λ) = {0} since dim M(λ) + dim M (λ) = N by condition 1.

(24)

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It follows from (24) that M2 (λ) ∩ M (λ) = {0}. If we further assume that λ is a maximal fiber point for M2 and M2 + M , then

f d M2 + M = dim M2 + M (λ) which is equal to dim M2 (λ) + dim M (λ) since the latter two spaces have a trivial intersection. So Eq. (23) is proved. Now by the assumption of (CCIP ),

f d(M1 ) + f d M2 + M = f d(M1 ) + f d(M2 ) + f d M

> f d(M) + f d M = N,

(25)

hence there exists a nonzero intersection element

m1 = m2 + m ∈ M1 ∩ M2 + M , where m ∈ M and mi ∈ Mi , i = 1, 2. So m1 − m2 = m ∈ M ∩ M = {0}. So m = 0 and m1 = m2 ∈ M1 ∩ M2 . Contradiction.

2

6. A generalization of Bourdon’s result on codimension-one property Recall that Bourdon’s result says that the cellular indecomposable property (CIP) implies the well-known codimension-one property. For a subspace M in a Hilbert space H of analytic functions over a domain Ω ⊂ C, we say that M has the division property at λ ∈ Ω if (z − λ)g ∈ M for some g ∈ H implies g ∈ M. Theorem. (See Bourdon [2].) Suppose that H is a Hilbert space of analytic functions over the unit disk D satisfying that


2977

1. the polynomials are dense in H ; 2. H has the division property at zero; 3. the linear functional of evaluation at each point of D is continuous. If H has the cellular indecomposable property (CIP), then for any nonzero invariant subspace M ⊂ H,

dim M (z − λ)M = 1,

|λ| < r1 (Mz ),

where r1 (Mz ) is a positive constant. Note that when considering vector-valued spaces, it is by now customary in operator theory to replace codimension-one by codimension-N . In this section we show that one can indeed obtain a codimension-N version of Bourdon’s result using (CCIP), see Theorem 13. Moreover, we will prove a stronger result (Theorem 14) which is the main result of this section. Assumptions. In this and the next sections, Ω denotes a domain in C containing the origin and H is a Hilbert space of analytic functions over Ω satisfying that (1) Mz , the multiplication by the coordinate function z, is bounded on H ; (2) the linear functional of evaluation at each point of Ω is continuous; (3) H has the division property at each point λ ∈ Ω. It is known that the above condition (3) implies that for λ ∈ Ω, Mz − λ has a closed range, which is just the kernel of the evaluation at λ. So Mz − λ is bounded below. For more details, say, see [12]. This in turn implies that Mz − λ is semi-Fredholm. Since Mz − λ has a trivial kernel, for any invariant M,

dim(M zM) = dim M (z − λ)M

(26)

by general Fredholm theory. Note that this co-dimension can be infinite. Theorem 13. If H has the complete cellular indecomposable property (CCIP), then any invariant subspace M ⊂ H ⊗ CN , N ∈ N, satisfies dim(M zM) N.

(27)

Recall that (27) is proved for the Dirichlet space by Richter in [13]. Then it is improved to be an equality by the second author in [7], dim(M zM) = f d(M).

(28)

Note that f d(M) is, by definition, at most N . Next we show that this equality (28) indeed holds for any space with (CCIP). So Theorem 13 will follow from Theorem 14.

2978


Theorem 14. If H has the complete cellular indecomposable property (CCIP), then any invariant subspace M ⊂ H ⊗ CN , N ∈ N, satisfies dim(M zM) = f d(M).

(29)

This result extends the one variable case of Corollary 4.6 in [3]. Before proving Theorem 14, the following two lemmas are needed. Lemma 15. If an invariant subspace M ⊂ H ⊗ CN , N ∈ N, satisfies (29), then M has the division property at any maximal fiber point. Proof. Let f d(M) = d. Choose a λ ∈ Ω to be a maximal fiber point for M and take f1 , . . . , fd ∈ M such that f1 (λ), . . . , fd (λ) form a basis for M(λ). Let Pλ be the projection onto M (z − λ)M and hi = Pλ (fi ),

1 i d.

Obviously, fi (λ) = hi (λ), hence h1 (λ), . . . , hd (λ) are linearly independent. So are h1 , . . . , hd , which implies that h1 , . . . , hd form a basis for M (z − λ)M since

dim M (z − λ)M = dim(M zM) = f d(M) = d. If (z − λ)g ∈ M, write (z − λ)g = c1 h1 + · · · + cd hd + (z − λ)g for some g ∈ M. Let z = λ in the above identity, one has that c1 = · · · = cd = 0 since h1 (λ), . . . , hd (λ) are linearly independent. Hence g = g ∈ M, as desired.

2

Lemma 16. If g ∈ H ⊗ CN , N ∈ N, and g(λ) = 0, then [g] has the division property at λ.


2979

Proof. We first observe that both f d([g]) and dim([g] z[g]) are one. Then note that λ is a maximal fiber point for [g]. So the proof follows from Lemma 15. 2 Now we are ready to prove Theorem 14. Proof of Theorem 14. Given (CCIP), we first show the following claim. Claim. If λ ∈ Ω is a maximal fiber point, namely, dim M(λ) = f d(M), then M has the division property at λ. Proof. Assume that (z − λ)g ∈ M, and we need to show g ∈ M. We first deal with the case g(λ) = 0. Let f d(M) = d and pick f1 , . . . , fd ∈ M such that

dim span f1 (λ), . . . , fd (λ) = d.

(30)

Now consider M = [g, f1 , . . . , fd ], the invariant subspace generated by g, f1 , . . . , fd . Observe that

f d [g, f1 , . . . , fd ] = f d (z − λ)g, f1 , . . . , fd and since (z − λ)g ∈ M, we have

f d M = d. So,

f d [g] + f d [f1 , . . . , fd ] > f d M and by (CCIP), we have [g] ∩ [f1 , . . . , fd ] = {0}. Subclaim. dim([f1 , . . . , fd ] (z − λ)[f1 , . . . , fd ]) = d. Proof. Denote [f1 , . . . , fd ] by M1 and it is easy to see

dim M1 (z − λ)M1 d since M1 is generated by d elements. Next decompose fi = fi1 + fi2 ,

1 i d,

(31)

2980


with fi1 ∈ M1 (z − λ)M1 and fi2 ∈ (z − λ)M1 . To show the equality in (31) it is sufficient to show that 1 (32) f1 , . . . , fd1 are linearly independent in M1 . To show (32) it is sufficient to show that 1 f1 (λ), . . . , fd1 (λ)

(33)

are linearly independent in CN . Now (33) follows from fi (λ) = fi1 (λ) and the fact that {fi (λ)} are linearly independent (30). So the subclaim is proved.

2

Let us continue with the proof of the claim. By Lemma 15, Lemma 16, and the subclaim, both invariant subspaces [g] and [f1 , . . . , fd ] have the division property at λ. It is easy to see that if two invariant subspaces have the division property at λ, then so does their intersection, if nontrivial. Moreover, if an invariant subspace has the division property at λ, then it contains a function which is nonvanishing at λ. So we can pick h ∈ [g] ∩ [f1 , . . . , fd ] such that h(λ) = 0. Meanwhile, there are polynomials pn , qn1 , . . . , qnd such that pn g → h

d

and

qni fi → h,

as n → ∞.

(34)

i=1

Because the evaluation at λ is continuous, pn (λ)g(λ) → h(λ),

as n → ∞.

. h(λ) , pn (λ) → c = g(λ)

as n → ∞.

So (35)

Note that c is a nonzero constant. Hence, (34) and (35) can be rewritten such that d p (z) − p (λ) n n (z − λ)g − qni fi + cg z−λ i=1

→ 0,

as n → ∞.

H ⊗CN

Since (z − λ)g ∈ M and fi ∈ M, we have g ∈ M. The claim is proved when g(λ) = 0.

(36)


2981

For general g such that (z − λ)g ∈ M, we still need to show g ∈ M. Write g = (z − λ)c g1 for some positive integer c and g1 (λ) = 0. Note that g1 ∈ H since we assume that H has the division property at λ. Then, similar to the above arguments, by picking an h, h ∈ [g1 ] ∩ [f1 , . . . , fd ] such that h(λ) = 0, one has a similar statement as (36) which will show that g1 ∈ (z − λ)g1 , f1 , . . . , fd . So g = (z − λ)c g1 ∈ (z − λ)c+1 g1 , (z − λ)c f1 , . . . , (z − λ)c fd ⊂ M. The claim is proved.

2

To continue with the proof of Theorem 14, it is an easy general fact that

dim(M zM) = dim M (z − λ)M f d(M).

(37)

If the above inequality (37) is strictly greater, then we can find d + 1 linearly independent functions g1 , . . . , gd , gd+1 ∈ M (z − λ)M. On the other hand, g1 (λ), . . . , gd (λ), gd+1 (λ) must be linearly dependent in CN since f d(M) = d, so there are not all zero constants c1 , . . . , cd+1 such that d+1

ci gi (λ) =

i=1

d+1

ci gi (λ) = 0,

i=1

which implies that, by the division property of M at λ, or by the claim, d+1

ci gi ∈ (z − λ)M.

i=1

Contradiction. This completes the proof of Theorem 14.

2

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7. A partial converse for CIP Recall that Bourdon’s result states that (CIP) implies the codimension-one property. Moreover, Bourdon proves a partial converse: If Mz on H is such that each nontrivial invariant subspace has codimension one, . cod(M) = dim(M zM) = 1, then any two nontrivial invariant subspaces M1 , M2 have a zero angle [2]. Note that the converse of Bourdon’s result is not true as illustrated by the following example. Let dμ = dA + χS |dz|, where A is the (normalized) area measure on the disk, |dz| the (normalized) Lebesgue measure on the unit circle and χS the characteristic function of the upper semicircle S ⊂ T. Furthermore, let H = P 2 (μ), the closure of polynomials in L2 (dμ). Then it is well known to experts that, just like the Bergman space, one can find two zero sequences of H such that their union is not a zero sequence. This can also be shown directly by imitating the proof of Horowitz’s theorem, see Theorem 3 of Chapter 4 in [4]. Then for these two zero sequences, their corresponding invariant subspaces have a trivial intersection. On the other hand, by [1], any nontrivial invariant subspace of H has codimension one. In Section 6 we showed that (CCIP) implies the codimension-N property; indeed, we proved a stronger result (Theorem 14); namely, given (CCIP), one has cod(M) = dim(M zM) = f d(M) for any invariant subspace M ⊂ H ⊗ CN , N ∈ N. Definition 17. We say that an invariant subspace M ⊂ H ⊗ CN , N ∈ N, satisfies the cod-fd condition if its codimension is equal to the fiber dimension; namely, cod(M) = f d(M). The purpose of this section is to give a partial converse for Theorem 14. Recall that H in this section satisfies the assumptions in Section 6. Theorem 18. Suppose that each invariant subspace of H ⊗ CN , N ∈ N, satisfies the cod-fd condition. If two invariant subspaces M1 , M2 ⊂ H ⊗ CN satisfy f d(M1 ) + f d(M2 ) > N,

(38)

then angle(M1 , M2 ) = 0. Observe that if M1 ∩ M2 = {0}, then angle(M1 , M2 ) = 0. Also observe that it is not enough to just assume that M1 and M2 have the cod-fd condition. Proof of Theorem 18. Let f d(M2 ) = t and take g1 , . . . , gt ∈ M2 such that for some point, hence for almost every point, λ ∈ Ω,

dim span g1 (λ), . . . , gt (λ) = t. Define


2983

N0 = M 1 and Ni = [M1 , g1 , . . . , gi ],

i = 1, . . . , t.

Then the following finite, increasing sequence t f d(Ni ) i=0 has to stabilize at some stage, due to the fact that f d(Ni ) N and the assumption (38). That is, there exists an r < t such that f d(Nr ) = f d(Nr+1 ) = d for the first time. Choose λ0 ∈ Ω such that

dim Nr (λ0 ) = dim Nr+1 (λ0 ) = d and (39)

g1 (λ0 ), . . . , gr+1 (λ0 ) are linearly independent. Observe that for any λ ∈ Ω,

f d(Nr+1 ) = f d M1 , g1 , . . . , gr , (z − λ)gr+1 . Then λ0 is also a maximal fiber point for M = M1 , g1 , . . . , gr , (z − λ0 )gr+1 since

dim M (λ0 ) = dim Nr (λ0 ) = dim Nr+1 (λ0 )

= f d(Nr+1 ) = f d M .

By the assumption of the theorem, M has the cod-fd condition. Hence by Lemma 15, M has the division property at λ0 . It follows that gr+1 ∈ M1 , g1 , . . . , gr , (z − λ0 )gr+1 . So there exist functions mn ∈ M1 and polynomials pn1 , . . . , pnr+1 such that mn + p 1 g1 + · · · + p r gr + (z − λ0 )p r+1 gr+1 − gr+1 → 0, n n n H ⊗CN

as n → ∞.

(40)

2984


Let Rn = pn1 g1 + · · · + pnr gr + (z − λ0 )pnr+1 gr+1 − gr+1 , and we claim inf Rn H ⊗CN > 0.

(41)

n

Otherwise, there is a subsequence Rnj H ⊗CN → 0,

as j → ∞.

Meanwhile, the evaluation at λ0 is continuous, so Rnj (λ0 ) → 0,

as j → ∞,

which implies that gr+1 (λ0 ) ∈ span g1 (λ0 ), . . . , gr (λ0 ) . This contradicts the fact that g1 (λ0 ), . . . , gr+1 (λ0 ) are linearly independent, or (39). By (40) and (41), we also have the fact that inf mn H ⊗CN > 0. n

Hence (40) leads to mn Rn + → 0, m R n n H ⊗CN This implies that angle(M1 , M2 ) = 0.

as n → ∞.

2

Acknowledgments The authors thank the referee for several valuable suggestions which greatly improve the presentation of this paper. References [1] A. Aleman, S. Richter, C. Sundberg, Nontangential limits in P t (μ)-spaces and the index of invariant subspaces, Ann. of Math. 169 (2009) 449–490. [2] P. Bourdon, Cellular-indecomposable operators and Beurling’s theorem, Michigan Math. J. 33 (1986) 187–193. [3] G. Cheng, K. Guo, K. Wang, Transitive algebras and reductive algebras on reproducing analytic Hilbert spaces, J. Funct. Anal. 258 (2010) 4229–4250. [4] P. Duren, A. Schuster, Bergman Spaces, Math. Surveys Monogr., vol. 100, American Mathematical Society, 2004. [5] D. Eisenbud, Commutative Algebra. With a View toward Algebraic Geometry, Grad. Texts in Math., vol. 150, Springer-Verlag, New York, 1995. [6] X. Fang, Samuel multiplicity and the structure of semi-Fredholm operators, Adv. Math. 186 (2004) 411–437.


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[7] X. Fang, Invariant subspaces of the Dirichlet space and commutative algebra, J. Reine Angew. Math. 569 (2004) 189–211. [8] X. Fang, The Fredholm index of a pair of commuting operators, Geom. Funct. Anal. 16 (2006) 367–402. [9] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math., vol. 52, Springer-Verlag, New York, 1977. [10] S. McCullough, T. Trent, Invariant subspaces and Nevanlinna–Pick kernels, J. Funct. Anal. 178 (2000) 226–249. [11] R. Olin, J. Thomson, Cellular-indecomposable subnormal operators, Integral Equations Operator Theory 7 (1984) 392–430. [12] S. Richter, Invariant subspaces in Banach spaces of analytic functions, Trans. Amer. Math. Soc. 304 (1987) 585– 616. [13] S. Richter, Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 386 (1988) 205–220. [14] S. Richter, A. Shields, Bounded analytic functions in the Dirichlet space, Math. Z. 198 (1988) 151–159.


On quantification of weak sequential completeness ✩ O.F.K. Kalenda a , H. Pfitzner b , J. Spurný a,∗ a Department of Mathematical Analysis, Faculty of Mathematics and Physic, Charles University,

Sokolovská 83, 186 75, Praha 8, Czech Republic b Université d’Orléans, BP 6759, F-45067 Orléans Cedex 2, France

Received 30 November 2010; accepted 3 February 2011 Available online 16 February 2011 Communicated by G. Schechtman

Abstract We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in L-embedded Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton and D. Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space X with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G. Godefroy. © 2011 Elsevier Inc. All rights reserved. Keywords: Weakly sequentially complete Banach space; L-embedded Banach space; Quantitative versions of weak sequential completeness

1. Introduction and statement of the results If X is a Banach space, we recall that it is weakly sequentially complete if any weakly Cauchy sequence in X is weakly convergent. In the present paper we investigate quantitative versions of this property. To this end we use several quantities related to a given bounded sequence (xk ) in X. ✩ The first and third authors were supported in part by the grant GAAV IAA 100190901 and in part by the Research Project MSM 0021620839 from the Czech Ministry of Education. * Corresponding author. E-mail addresses: [email protected] (O.F.K. Kalenda), [email protected] (H. Pfitzner), [email protected] (J. Spurný).


O.F.K. Kalenda et al. / Journal of Functional Analysis 260 (2011) 2986–2996

2987

Let clustX∗∗ (xk ) denote the set of all weak∗ cluster points of (xk ) in X ∗∗ . By δ(xk ) we will denote the diameter of clustX∗∗ (xk ) (see also (4) below). Further, if A, B are nonempty subsets of a Banach space X, then d(A, B) = inf a − b: a ∈ A, b ∈ B denotes the usual distance between A and B and the Hausdorff non-symmetrized distance from A to B is defined by d(A, B) = sup d(a, B): a ∈ A . Note that a space X is weakly sequentially complete if for each bounded sequence (xk ) in X satisfying δ(xk ) = 0 (this just means that the sequence is weakly Cauchy) we have d(clustX∗∗ (xk ), X) = 0 (i.e., all the weak∗ cluster points are contained in X, which for a weakly Cauchy sequence means that it is weakly convergent). It is thus natural to ask which Banach spaces satisfy a quantitative version of weak sequential completeness, i.e., the inequality d clustX∗∗ (xk ), X C · δ(xk )

(1)

for all bounded sequences (xk ) in X and for some C > 0. The starting point of our investigation was the following remark made by G. Godefroy in [3, p. 829]: If X is complemented in X ∗∗ by a projection P satisfying ∗∗ ∗∗ ∗∗ x = P x + x − P x ∗∗ ,

x ∗∗ ∈ X ∗∗ ,

(2)

then X is weakly sequentially complete and d clustX∗∗ (xk ), X δ(xk )

(3)

for any sequence (xk ) in X. It can be easily seen that δ(xk ) = sup

x ∗ ∈BX∗

= sup

lim sup x ∗ (xk ) − lim inf x ∗ (xk ) k→∞

k→∞

lim sup x ∗ (xl ) − x ∗ (xj ) : l, j n .

x ∗ ∈BX∗ n→∞

(4)

The first formula of (4) is used in [1, Section 2.1], the second one in [3, p. 829]. Banach spaces satisfying assumption (2) above are called L-embedded, see [6, Section III.1]. The proof of (3) can be found in [4, Lemma IV.7]. By what has been said above, inequality (3) is a quantitative form of weak sequential completeness. In [3, p. 829] G. Godefroy mentions that it is not clear which weakly sequentially complete spaces can be renormed to have such a quantitative form of weak sequential completeness. The aim of our paper is twofold. On the one hand we show that the answer to G. Godefroy’s question cannot be positive for all weakly sequentially complete Banach spaces, more precisely

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we construct a weakly sequentially complete space that cannot be renormed in such a way that (3) holds, see Example 4 below. On the other hand we put inequality (3) into context by studying some modifications and possible converses, see the following theorem. In particular, we slightly improve inequality (3) – see (6) in the theorem – but such that now the additional factor 2 is optimal. We will use one more quantity (cf. [8] but appearing implicitly in [1]) which in some situations is more natural than the quantity δ, namely δ (xk ) = inf δ(xkj ): (xkj ) is a subsequence of (xk ) . Theorem 1. Let X be a Banach space and (xk ) be a bounded sequence in X. Then δ (xk ) 2 d clustX∗∗ (xk ), X .

(5)

If the space X is L-embedded, then also the following inequalities hold: 2 d clustX∗∗ (xk ), X δ(xk ), 2d clustX∗∗ (xk ), X δ (xk ).

(6) (7)

Since we have trivially that δ δ and d d it is natural to ask whether one of these quantities can be replaced by a sharper one in the inequalities of the theorem. The following remark and Example 3 show that this cannot be done in any of the inequalities (5)–(7). Remark 2. (a) In (6), δ cannot be replaced by δ and in (7) d cannot be replaced by d. This is witnessed by the sequence (xk ) in X = 1 such that x2k−1 = 0 and x2k = ek for all k ∈ N. Then δ (xk ) = 0, d(clustX∗∗ (xk ), X) = 1 and δ(xk ) = 2. d(clustX∗∗ (xk ), X) = (b) Inequality (5) is a kind of converse of (3) and holds in all Banach spaces. We note that δ cannot be replaced by δ in (5), in other words, inequality (3) cannot be reversed as it is, neither in L-embedded spaces. Indeed, let X = 1 . We consider the elements xk = 0 and yk = e1 , k ∈ N. d(clust∗∗ (zk ), 1 ) = 0 because all weak∗ cluster Let (zk ) be the sequence x1 , y1 , x2 , y2 , . . . . Then 1 points of (zk ) are contained in 1 , but δ(zk ) lim sup e1 (zk ) − lim inf e1 (zk ) = 1. k→∞

k→∞

(c) We further remark that in all inequalities in Theorem 1 the factor 2 is optimal, as witnessed by the sequence (ek ) in X = 1 . Indeed, then d clustX∗∗ (ek ), X = d clustX∗∗ (ek ), X = 1

and δ (ek ) = δ(ek ) = 2.

It is also natural to ask whether d can be replaced by d in the inequality (5), i.e., whether the inequality (7) can be reversed (at least for L-embedded spaces). This is not the case by the following example. Example 3. There is an L-embedded space X and a bounded sequence (xk ) in X such that δ (xk ) = 2 and d(clustX∗∗ (xk ), X) = 0.


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The negative partial answer to the mentioned question of G. Godefroy is given by the following example. In fact, we obtain a slightly stronger result. Not only there is a weakly sequentially complete Banach space not satisfying (1) for all bounded sequences and some C > 0, but we get even a weakly sequentially complete space not satisfying a weaker form of (1) – with d in place of d. Example 4. There exists a separable Banach space X with the Schur property – in particular, X is weakly sequentially complete – which is 1-complemented in its bidual, such that there is no constant C > 0 satisfying d clustX∗∗ (xk ), X C · δ(xk ) for every bounded sequence (xk ) in X. We remark that a separable space with the Schur property belongs to the class of so-called strongly weakly compactly generated spaces (see [9, Examples 2.3]) and thus such a quantitative form of weak sequential completeness does not hold even for this class of spaces. 2. Proof of Theorem 1 The proof relies on two simple properties of 1 -sequences which are formulated in the following lemma. Lemma 5. Let X be a Banach space and (xn ) be a bounded sequence in X. Suppose that c > 0 is such that n n c α x |αj | j j j =1

j =1

whenever n ∈ N and α1 , . . . , αn are real numbers. Then (i) δ(xn ) 2c, (ii) d(clustX∗∗ (xk ), X) c. Proof. (i) It is clear that the sequence (xn ) is linearly independent. Hence there is a unique linear functional defined on its linear span whose value is c at x2k−1 and −c at x2k for each k ∈ N. By the assumption, the norm of this functional is at most 1. Let x ∗ ∈ BX∗ be its Hahn–Banach extension. Then x ∗ witnesses that δ(xn ) 2c. (ii) Let x ∗∗ be any weak∗ cluster point of the sequence (xn ) in X ∗∗ and x ∈ X be arbitrary. It follows from [7, Proposition 4.2] that there is an index m ∈ N such that ∞ ∞ αj (xj − x) c |αj | j =m

j =m

for every sequence (αj )∞ j =m with finitely many nonzero elements. In particular, it follows that the vectors xj − x, j m, are linearly independent. So, there is a unique linear functional on their linear span whose value at each xj − x is equal to c. By the above inequality, the norm of

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this functional is at most one. Let x ∗ ∈ X ∗ be its Hahn–Banach extension. Then we have ∗∗ x − x x ∗∗ − x x ∗ lim inf x ∗ (xj − x) = c. j →∞

2

This completes the proof of the lemma.

Now we are ready to prove Theorem 1. We start by proving (5): Let (xk ) be a bounded sequence in X. We assume that δ (xk ) > 0 because otherwise (5) holds trivially. Let c ∈ (0, δ (xk )) be arbitrary. The key ingredient is provided by a result of E. Behrends (see [1, Theorem 3.2]) that yields a subsequence (xnk ) such that k k c αi xni |αi | 2 i=1

i=1

whenever k ∈ N and α1 , . . . , αk ∈ R. By Lemma 5(ii) we get d(clustX∗∗ (xnk ), X) 2c , hence δ (xk )) is arbitrary, (5) follows. d(clustX∗∗ (xk ), X) 2c . As c ∈ (0, We continue by proving (6): We set c = d(clustX∗∗ (xk ), X) and assume that c > 0 because otherwise (6) holds trivially. Let ε ∈ (0, c) be arbitrary and let x ∗∗ be a weak∗ cluster point of the sequence (xk ) in X ∗∗ such that d(x ∗∗ , X) > c − 2ε . Set x = P x ∗∗ and xs = x ∗∗ − x where P denotes the projection on X as in (2). Then d(x ∗∗ , X) = xs . We claim that there is a subsequence (xkn ) such that n n αi (xki − x) c − 1 − 2−n ε |αi | (8) i=1

i=1

for all n ∈ N and all (αi )ni=1 in Rn . This will be proved by G. Godefroy’s ‘ace of 3 argument’ [6, p. 170], cf. the proof of [6, Proposition IV.2.5]. Since xs is a weak∗ cluster point of the sequence (xk − x), there is k1 such that xk1 − x > c − 2ε which settles the first induction step. Suppose we have constructed xk1 , . . . , xkn . Let (α l )L l=1 be a finite sequence of elements of the n+1 l unit sphere of 1 such that αn+1 = 0 for all l ∈ {1, . . . , L} and such that for each α in the unit there is an element α l such that sphere of n+1 1 α − α l

n+1 1

c − 1 − 2−n ε

n+1

l

α = c − 1 − 2−n ε. i

i=1


2991

It follows that there is kn+1 > kn such that n+1 l αi (xki − x) > c − 1 − 2−n ε i=1

for all l ∈ {1, . . . , L}. By a straightforward calculation using the choice of the α l and the triangle inequality we get that inequality (8), with n + 1 instead of n, holds for all α in the unit sphere of and hence for all elements of Rn+1 . n+1 1 This finishes the construction. By Lemma 5(i) we get δ(xkn − x) 2(c − ε), hence clearly δ(xk ) δ(xkn ) = δ(xkn − x) 2(c − ε). As ε ∈ (0, c) is arbitrary, we get (6). Finally, let us prove (7): We take any subsequence (xkn ) and observe that 2d clustX∗∗ (xk ), X 2 d clustX∗∗ (xkn ), X δ(xkn ) by (6). Then we can pass to the infimum over all (xkn ). This finishes the proof of the theorem. 3. Proof of Example 3 For n ∈ N set Xn = n∞ and let X be the 1 -sum of all the spaces Xn , n ∈ N. Then X is L-embedded by [6, Proposition IV.1.5]. Further, let e1n , . . . , enn be the canonical basic vectors of Xn and let (xk ) be the sequence in X containing subsequently these basic vectors, i.e., the sequence e11 , e12 , e22 , e13 , e23 , e33 , e14 , . . . , e44 , . . . . Then we have δ (xk ) = 2 as each subsequence of (xk ) contains a further subsequence isometrically equivalent to the canonical basis of 1 . It remains to show that d(clustX∗∗ (xk ), X) = 0. To do so, it is enough to prove that 0 is a weak cluster point of the sequence (xk ). To verify this, we fix g 1 , . . . , g m ∈ X ∗ and ε > 0. Let K = max{g 1 , . . . , g m }. The dual X ∗ can be canonically identified with the ∞ -sum of the spaces Xn∗ , n ∈ N. Moreover, Xn∗ is canonically isometric to n1 . Thus each g ∈ X ∗ can be viewed as a bounded sequence (gn )n∈N , where gn = (gn,j )nj=1 ∈ n1 for each n ∈ N. We find N ∈ N such that K be such that n > mN . Let k ∈ {1, . . . , m} be N < ε and let n ∈ N k |, the set k k k arbitrary. We have gn g K. As gn = nj=1 |gn,j

k K

j ∈ {1, . . . , n}: gn,j N

2992


has at most N elements. It follows that the set

k K

j ∈ {1, . . . , n}: ∃k ∈ {1, . . . , m}, gn,j N k | < K < ε for has at most mN elements. As n > mN , there is some j ∈ {1, . . . , n} such that |gn,j N n k each k ∈ {1, . . . , m}. It means that |g (ej )| < ε for each k ∈ {1, . . . , m}. Since ejn is an element of the sequence (xk ), this completes the proof that 0 is in the weak closure of the sequence, hence 0 is a weak cluster point (as the sequence (xk ) does not contain 0).

4. Proof of Example 4 ˇ We recall that βN is the Cech–Stone compactification of N and M(βN) is the space of all signed Radon measures on βN considered as the dual of ∞ . Let us fix α > 0 and consider the space Yα = 1 , α · 1 ⊕1 C[1, ω], · ∞ . Here · 1 denotes the usual norm on 1 , ω is the first infinite ordinal, C[1, ω] stands for the space of all continuous functions on the ordinal interval [1, ω] and · ∞ is the standard supremum norm. Note that we have the following canonical identifications: 1 Yα∗ = ∞ , · ∞ ⊕∞ 1 [1, ω], · 1 , and α ∗∗ Yα = M(βN), α · M(βN) ⊕1 ∞ [1, ω], · ∞ . For k ∈ N, let xk = (ek , χ[k,ω] ) ∈ Yα , where ek denotes the k-th canonical basic vector in 1 and χ[k,ω] is the characteristic function of the interval [k, ω]. Let Xα be the closed linear span of the set {xk : k ∈ N}. We observe that

n ηk for all n ∈ N . Xα = (ηk ), f ∈ Yα : f (n) =

(9)

k=1

Indeed, the set on the right-hand side is a closed linear subspace of Yα containing xk for each k ∈ N. This proves the inclusion ‘⊂’. To prove the converse one, let us take any point ((ηk ), f ) in the set on the right-hand side. Since (ηk ) ∈ 1 , we get ∞ ηk x k ∈ X α (ηk ), f = k=1

as the series is absolutely convergent. It follows that for each ((ηk ), f ) ∈ Xα we have α (ηk )1 (ηk ), f (α + 1)(ηk )1 ,


2993

hence Xα is isomorphic to 1 . More precisely, the projection on the first coordinate is an isomorphism onto 1 . In particular, Xα has the Schur property (and thus it is weakly sequentially complete). We further observe that Xα∗∗ is canonically identified with the weak∗ closure of Xα in Yα∗∗ , thus Xα∗∗ = (μ, f ) ∈ M(βN) × ∞ [1, ω]: ∀n ∈ N: f (n) = μ{1, . . . , n} and f (ω) = μ(βN) .

(10)

Indeed, the set on the right-hand side is a weak∗ closed linear subspace of Yα∗∗ containing Xα , which proves the inclusion ‘⊂’. To prove the converse one let us fix (μ, f ) in the set on the right-hand side. Take a bounded net (uτ ) in 1 which weak∗ converges to μ. For each τ there is a unique fτ ∈ C[1, ω] such that (uτ , fτ ) ∈ Xα . Then (fτ ) is clearly a bounded net in ∞ [1, ω]. Moreover, we will show that (fτ ) weak∗ converges to f . Since the weak∗ topology on bounded sets coincides with the topology of pointwise convergence, it suffices to show that fτ pointwise converge to f . Indeed, fτ (n) =

n

uτ (k) → μ{1, . . . , n} = f (n),

for each n ∈ N,

k=1

fτ (ω) =

∞

uτ (k) → μ(βN) = f (ω).

k=1

It follows that Xα is 1-complemented in its bidual. To show that we set P (μ, f ) =

μ{k} , f − μ(βN \ N) · χ{ω} ,

(μ, f ) ∈ Xα∗∗ .

Then P is a projection of Xα∗∗ onto Xα of norm one. Indeed, if (μ, f ) ∈ Xα , then μ(βN \ N) = 0 and hence P (μ, f ) = (μ, f ). Further, by (9) and (10) we get that P (μ, f ) ∈ Xα for each (μ, f ) ∈ Xα∗∗ . Thus P is a projection onto Xα . To show it has norm one, it is enough to observe that, given (μ, f ) ∈ Xα∗∗ , we have (μ{k})1 μ, and that f − μ(βN \ N) · χ{ω} is a continuous function on [1, ω] coinciding on [1, ω) with f and so f − μ(βN \ N) · χ{ω} ∞ f ∞ . Further, for the sequence (xk ), its weak∗ cluster points in Xα∗∗ are equal to (εt , χ{ω} ): t ∈ βN \ N , where εt denotes the Dirac measure at a point t ∈ βN. We claim that, for our sequence (xk ), we have 1 d clustXα∗∗ (xk ), Xα 2

and δ(xk ) = 2α.

(11)

To see the first inequality, we use the fact that the distance of any weak∗ cluster point of (xk ) from Xα is at least d(χ{ω} , C[1, ω]) = 12 . On the other hand, if t, t ∈ βN \ N are distinct, then (εt , χ{ω} ) − (εt , χ{ω} )

Xα∗∗

= (εt − εt , 0)X∗∗ = αεt − εt M(βN) = 2α. α

2994


This verifies (11). Now we use the described procedure to construct the desired space X. For n ∈ N, let αn = and let X 1 be the space constructed for αn . Let

1 n

n

X=

∞ n=1

X1

n

1

be the 1 -sum of the spaces X 1 . We claim that X is the required space. n First, since each X 1 has the Schur property, X, as their 1 -sum, possesses this property as n well (this follows by a straightforward modification of the proof that 1 has the Schur property, see [2, Theorem 5.19]). Hence X is weakly sequentially complete. Further, observe that ∗

X =

∞ n=1

∗

and X

X1 n

∗∗

⊃

∞ n=1

∞

∗∗

X1

n

. 1

Note that the latter space is not equal to X ∗∗ but it is 1-complemented in X ∗∗ (cf. the proof of [6, Proposition IV.1.5]). Now it follows that X is 1-complemented in X ∗∗ . n-th

Finally, fix n ∈ N. We consider a sequence xk = (0, . . . , 0, xk , 0, . . .), where the elements n-th = (0, . . . , 0, (εt , χ{ω} ), 0, . . .),

xk ∈ X 1 , k ∈ N, are defined above. Let y where t ∈ βN \ N, be a n xk ) in X ∗∗ . Then, for any z = (z(1), z(2), . . .) ∈ X, weak∗ cluster point of ( 1 y − zX∗∗ (εt , χ{ω} ) − z(n)X∗∗ 1 2 n by (11). Hence 1 d clustX∗∗ ( xk ), X . 2 On the other hand, 2 δ( xk ) = δ(xk ) = , n again by (11). From this observation the conclusion follows. 5. Final remarks Even though the second part of Theorem 1 is formulated for L-embedded spaces, using results of A.S. Granero and M. Sánchez we can prove the following variant of Theorem 1. Let X be a subspace of an L-embedded Banach space Y and (xk ) be a bounded sequence in X. Then d clustX∗∗ (xk ), X δ(xk )

and

δ (xk ). d clustX∗∗ (xk ), X

(12)


2995

To verify the first inequality, we consider x ∗∗ ∈ clustX∗∗ (xk ). Since clustX∗∗ (xk ) = clustY ∗∗ (xk ), using [5, Lemma 2.2] (with X, Y for D, X) and Theorem 1, we obtain d clustY ∗∗ (xk ), Y δ(xk ). d x ∗∗ , X 2d x ∗∗ , Y 2 This proves the first statement because x ∗∗ ∈ clustX∗∗ (xk ) was arbitrary. The second one can be deduced from the first one just as in the proof of Theorem 1. However, we do not know whether it is possible to obtain not only (12) but (6) and (7) of Theorem 1 for subspaces of L-embedded spaces. Up to now we have tacitly assumed that we are dealing with real Banach spaces. In fact, our proofs work for real spaces but all the results can be easily transferred to complex spaces as well. Let us indicate how to see this. Let X be a complex Banach space. Denote by XR the same space considered over the field of real numbers (i.e., we just forget multiplication by imaginary numbers). Let φ : X ∗ → (XR )∗ be defined by φ x ∗ (x) = Re x ∗ (x),

x ∗ ∈ X ∗ , x ∈ X.

It is well known that φ is a real-linear isometry of X ∗ onto (XR )∗ . Let us define a mapping ψ : X ∗∗ → (XR )∗∗ by the formula ψ x ∗∗ y ∗ = Re x ∗∗ φ −1 y ∗ ,

x ∗∗ ∈ X ∗∗ , y ∗ ∈ (XR )∗ .

It is easy to check that the mapping ψ satisfies the following properties: (i) ψ is a real-linear isometry of X ∗∗ onto (XR )∗∗ . (ii) ψ is a weak∗ -to-weak∗ homeomorphism. (iii) ψ(X) = XR . It follows that for any sequence in X all the quantities in question (i.e., δ, δ , d and d) are the same with respect to X and with respect to XR . (Recall that δ is defined as the diameter of weak∗ cluster points, which has good sense in a complex space as well, even though in the complex case only the second formula of (4) works.) If, moreover, we observe that XR is L-embedded whenever X is L-embedded, we conclude that Theorem 1 is valid for complex spaces as well. As for Examples 3 and 4, it is clear that they work also in the complex setting – we can just consider complex versions of the respective spaces. We finish by recalling that G. Godefroy’s question, for which Banach spaces (3) holds, remains open. In particular, the following question seems to be open. Question. Let X be a Banach space which is a u-summand in its bidual, i.e., there is a projection P : X ∗∗ → X with I − 2P = 1. Does (1) hold for X for some C > 0? We conjecture that the space from Example 4, although it is 1-complemented in its bidual, is not a u-summand. At least the projection we have constructed does not work. Acknowledgment The authors would like to thank the referee for several interesting comments and suggestions.

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References [1] E. Behrends, New proofs of Rosenthal’s l 1 -theorem and the Josefson–Nissenzweig theorem, Bull. Pol. Acad. Sci. Math. 43 (4) (1996) 283–295, 1995. [2] M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucía, J. Pelant, V. Zizler, Functional Analysis and InfiniteDimensional Geometry, CMS Books Math./Ouvrages Math. SMC, vol. 8, Springer-Verlag, New York, 2001. [3] G. Godefroy, Renormings of Banach spaces, in: Handbook of the Geometry of Banach Spaces, vol. I, North-Holland, Amsterdam, 2001, pp. 781–835. [4] G. Godefroy, N.J. Kalton, D. Li, Operators between subspaces and quotients of L1 , Indiana Univ. Math. J. 49 (1) (2000) 245–286. [5] A.S. Granero, M. Sánchez, Distances to convex sets, Studia Math. 182 (2) (2007) 165–181. [6] P. Harmand, D. Werner, W. Werner, M-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Math., vol. 1547, Springer-Verlag, Berlin, 1993. [7] H. Knaust, E. Odell, On c0 sequences in Banach spaces, Israel J. Math. 67 (2) (1989) 153–169. [8] H. Pfitzner, Boundaries of Banach spaces determine weak compactness, Invent. Math. 182 (3) (2010) 585–604. [9] G. Schlüchtermann, R.F. Wheeler, On strongly WCG Banach spaces, Math. Z. 199 (3) (1988) 387–398.


Composition operators with closed range for smooth injective symbols R → Rd Nicolas Kenessey, Jochen Wengenroth ∗ Universität Trier, FB IV – Mathematik, 54286 Trier, Germany Received 22 September 2010; accepted 29 January 2011 Available online 18 February 2011 Communicated by G. Schechtman

Abstract In 1998, Allan, Kakiko, O’Farrell, and Watson proved a description of the closure (with respect to the uniform convergence of all derivatives on compact sets) of A (ψ) = {F ◦ ψ: F ∈ E (Rd )} for a smooth injective symbol ψ : R → Rd in terms of formal Taylor series. In that article it was conjectured that A (ψ) is closed if ψ is proper and has only critical points of finite order. In the present paper we first give a simple counterexample and then rectify the conjecture by adding a geometrical property for the curve ψ(R). This yields a characterization of A (ψ) = A (ψ). © 2011 Elsevier Inc. All rights reserved. Keywords: Composition operator; Composite function problem; Algebras of smooth functions

1. Introduction and main result Composition operators F → F ◦ ψ on the space E (Rd ) of C ∞ -functions are well understood if ψ is an analytic symbol, we only mention Glaeser’s article [4], the books of Malgrange [6] and Tougeron [9], and the work of Bierstone and Milman [2,3]. However, if the symbol is only smooth relatively little is known and even for an injective ψ : R → Rd the behavior of the algebra A (ψ) = F ◦ ψ: F ∈ E Rd * Corresponding author. Fax: +49 651 2014182.

E-mail addresses: [email protected] (N. Kenessey), [email protected] (J. Wengenroth). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.01.019

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N. Kenessey, J. Wengenroth / Journal of Functional Analysis 260 (2011) 2997–3006

is not completely understood although, as the reviewer S.J. Sidney of [1] in the Mathematical Reviews writes, “this is the sort of thing one imagines was tended to years ago”. In 1998, Allan, Kakiko, O’Farrell, and Watson [1] described the closure of this algebra in E (R) (endowed with the topology of uniform convergence of all derivatives on compact sets) in terms of formal Taylor series. We will need the following version of their result: Theorem (Allan, Kakiko, O’Farrell, and Watson). For a smooth injective ψ : R → Rd and f ∈ E (R) one has f ∈ A (ψ) if and only if for every critical point a of ψ (i.e., ψ (a) = 0) there is F ∈ E (Rd ) such that f (n) (a) = (F ◦ ψ)(n) (a) for all n ∈ N0 . The original result of Allan, Kakiko, O’Farrell, and Watson is somewhat stronger requiring only that the formal Taylor series of f in a is a certain composition with the Taylor series of ψ – this is immediately implied by the condition in the formulation above. The aim of this article is to characterize when A (ψ) is closed. For the sake of precision let us quote from [1] where A = A (ψ) and D is its closure: The referee of this paper conjectured that A = D is probably true for those ψ : R → Rd that are proper, injective, and have only critical points of finite order. This is a reasonable conjecture, and could probably be approached by using the methods that work for analytic functions. Recall that a map is proper if preimages of compact sets are always compact, and having finite order means, of course, that ψ (n) (a) = 0 for some n ∈ N, the minimal n with this property being the order of a. Here is a very simple example disproving this conjecture. Example. Consider ϕ(t) = exp(−1/t) for t > 0, ϕ(t) = 0 for t 0, and ψ(t) = (t 2 , ϕ(t)). Then ψ : R → R2 is smooth, injective, and proper with only one critical point of order 2 but A (ψ) is not closed. Indeed, as a consequence of the above cited theorem of Allan, Kakiko, O’Farrell, and Watson every f ∈ E (R) which is flat on the critical set (which means that √ the function and all its derivatives vanish there) belongs to A (ψ). Hence, in particular, f (t) = ϕ(t) is in the closure of A (ψ). Assume that there is F ∈ E (R2 ) with f = F ◦ ψ . For all positive t we thus obtain 1 exp − = f (t) − f (−t) = F t 2 , ϕ(t) − F t 2 , ϕ(−t) . 2t As a continuously differentiable function F is locally Lipschitz continuous and there is thus a constant C > 0 such that for all 0 < t 1 2 2 1 1 = F t , ϕ(t) − F t , ϕ(−t) C ϕ(t) − ϕ(−t) = C exp − . exp − 2t t For t → 0 we get a contradiction. As our theorem below will show, the reason for A (ψ) = A (ψ) in this example is the very sharp cusp of ψ(R) at ψ(0). Following the book of L. Schwartz [8, III.9] we call an arcwise connected set M ⊆ Rd Whitney regular if for every ξ ∈ M there are C, ε > 0 and an exponent


2999

γ ∈ (0, 1] such that the geodesic distance of any two points x, y ∈ M ∩ B(ξ, ε) is less than C|x − y|γ (i.e., they can be joined by a rectifiable curve in M of that length). Our main result reads as follows: Theorem. Let ψ : R → Rd be a smooth injective curve. Then A (ψ) = {F ◦ ψ: F ∈ E (Rd )} is closed in E (R) if and only if ψ is proper, has only critical points of finite order, and has a Whitney regular image ψ(R). Since ψ is injective the geodesic distance between two points ψ(a) and ψ(b) with a < b is b of course a |ψ (t)| dt. Using this we obtain that the composition operator has closed range if and only if ψ is not only a homeomorphism between R and ψ(R) (which always holds if ψ is proper) but even a local Hölder isomorphism: Corollary. A smooth injective symbol ψ : R → Rd induces a composition operator E (Rd ) → E (R), F → F ◦ ψ with closed range if and only if the inverse of ψ is locally Hölder continuous, i.e., for all z ∈ ψ(R) there are ε, C, α > 0 such that |ψ −1 (u) − ψ −1 (v)| C|u − v|α for all u, v ∈ ψ(R) ∩ B(z, ε). Compared to Whitney regularity the condition in this corollary might look more appropriate for a direct proof that A (ψ) is closed which would require concrete estimates of all derivatives of F in terms of those of F ◦ ψ in order to show that the composition operator is open onto its range A (ψ). We do not claim that this approach is impossible but in view of Faà di Bruno’s formula for the n-th derivative of a composition one would probably run into very technical estimates. We are going to use instead the closed range theorem for operators T : X → Y between Fréchet spaces which says that T has closed range if and only if every continuous linear functional u ∈ X which vanishes on the kernel of T has a representation u = v ◦ T for some v ∈ Y . In our case, it will quite easily turn out that u is a distribution on Rd with compact support in ψ(R). If this set is Whitney regular one can get continuity estimates for |u(F )| only in terms of derivatives of F on the set ψ(R) which then can be used to construct v. (Recall that, in general, one would need either derivatives of F on some neighborhood of the support of u or the far less handy Whitney norms of F on the support.) Before starting with the proofs we would like to mention that it is not the flatness of the second component in the example which causes A (ψ) = A (ψ). This is shown by the example ˜ ˜ = A (ψ). On the other hand, the ψ(t) = (t 2 , t 2 + ϕ(t)) which generates the same algebra A (ψ) 3 symbol (t , ϕ(t)) generates a composition operator with closed range (which follows from the corollary) although one component is flat at the critical point. 2. Sufficiency Throughout, ψ : R → Rd is a smooth and injective function. Let us start with two elementary consequences of the conditions from our theorem. Lemma 1. (a) If ψ is proper then it is a homeomorphism R → ψ(R) with closed image. (b) If every critical point is of finite order then E = {a ∈ R: ψ (a) = 0} is discrete.

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Proof. The first statement follows from the fact that ψ|K is a homeomorphism for each compact set K ⊆ R which is enough since ψ is proper and Rd is locally compact. To prove the second part assume that E is not discrete. Then there is a monotone convergent sequence an → a in E. Applying Rolle’s theorem to each component ψj of ψ we find monotone sequences bj,n → a such that ψj (bj,n ) = 0 which implies ψ (a) = 0 by continuity. Repeating this argument we get the contradiction ψ (n) (a) = 0 for all n ∈ N. 2 The next lemma is already a very particular (local and one-dimensional) case of our result. For a compact set M ⊆ Rd let D(M) be the closed subspace of E (Rd ) of smooth functions with support in M. Lemma 2. Let φ : [a, b] → [c, d] be a smooth bijection whose only critical point is a which is of finite order. Then its inverse θ = φ −1 : [c, d] → [a, b] induces a continuous composition operator D([a, b]) → D([c, d]), f → f ◦ θ . Proof. We may assume φ(a) = c. The continuity of the operator will follow from the closed graph theorem once we have shown that it maps indeed D([a, b]) into D([c, d]). For f ∈ D([a, b]) the composition f ◦ θ is continuous on [c, d] and C ∞ in (c, d] since θ is C ∞ there. We thus have to show (f ◦ θ )(n) (y) → 0 for y → c and all n ∈ N0 which we will do by induction. The case n = 0 is clear since θ is continuous, and the induction step will follow from the chain rule

(f ◦ θ )

(n+1)

(n) = f ◦ θ · θ =

f ◦θ φ ◦ θ

(n)

once we have shown f /φ ∈ D([a, b]). This however follows again by induction using l’Hospital’s rule and the hypothesis that a is of finite order. 2 From now on we will assume that ψ satisfies all three hypotheses of the theorem and has the (discrete) critical set E = {a ∈ R: ψ (a) = 0}. The first step in the proof is contained in [1, factorization lemma]. We will give an explicit construction which will be useful later on. Step 1. Every f ∈ E (R) with support in R \ E belongs to A (ψ). Proof. For each x ∈ supp(f ) there is a coordinate ψj (x) of ψ whose derivative has no zero in some open interval Ix x so that this coordinate is a diffeomorphism there. This will easily give a local factorization: We set Jx = ψj (x) (Ix ) and we choose an open set Mx ⊆ Rd such that ψ(Ix ) = ψ(R) ∩ Mx and πj (x) (Mx ) ⊆ Jx where πj : Rd → R denotes the canonical projection (there is such an Mx because ψ is a homeomorphism by Lemma 1). Then Fx = f ◦ ψj−1 (x) ◦ πj (x) is a smooth function on Mx satisfying Fx ◦ ψ = f on the interval Ix .


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To obtain a global factorization f = F ◦ ψ we choose a locally finite smooth partition of unity (φx )x∈supp(f ) subordinate to the covering (Mx )x∈supp(f ) of ψ(supp(f )) and set F=

φx Fx .

2

x∈supp(f )

Step 2. Let I be a compact interval containing only one critical point a. Then every f ∈ D(I ) which is flat at a belongs to A (ψ). Proof. Since we know how to factorize in the noncritical case and since we can paste local factorizations as in the proof of Step 1 we may decrease the interval in such a way that there is a coordinate ψk having only the critical point a in I which is of finite order. Since we may write f ∈ D(I ) as f = f + + f − with f ± having support to the right respectively left of a we only need to consider f ∈ D([a, b]) for some b > a such that [a, b] ∩ E = {a}. Put in a functional analytic form it is enough to prove that the Fréchet space D([a, b]) is contained in the range of the restricted composition operator T : X → D [a, b] ,

F → F ◦ ψ

where X = {F ∈ E (Rd ): F flat on ψ(R \[a, b])} is also a Fréchet space (being closed in E (Rd )). Note that the space D((a, b)) of functions having support in the open interval (a, b) is dense in D([a, b]) and contained in the range of T by Step 1. It is therefore enough to show that the range of T is closed which we will prove using the closed range theorem, see e.g. [7, 26.3]. We thus have to show kern(T )⊥ ⊆ range T t where T t : D([a, b]) → X , v → v ◦ T is the transposed operator between the continuous duals and kern(T )⊥ = {u ∈ X : u(F ) = 0 for all F ∈ X with F ◦ ψ = 0} is the annihilator of the kernel of T . Let us thus fix u ∈ kern(T )⊥ which, by the Hahn–Banach theorem, can be extended to a distribution with compact support u˜ ∈ E (Rd ). Then u˜ has support in ψ(R) since every φ ∈ ˜ = u(φ) = 0. D(Rd ) with supp(φ) ∩ ψ(R) = ∅ satisfies φ ∈ X, T (φ) = φ ◦ ψ = 0, and thus u(φ) We are looking for v ∈ D([a, b]) with v ◦ T = u and we will first define v on the dense subspace D((a, b)) of D([a, b]) using the construction in Step 1. For f ∈ D((a, b)) we choose an open set M ⊆ Rd such that ψ((a, b)) = M ∩ ψ(R) and πk (M) ⊆ ψk ((a, b)) as well as a cutoff function ϕ ∈ D(M) which is equal to 1 in a neighborhood of ψ(supp(f )). Then we define F = ϕ · (f ◦ ψk−1 ◦ πk ) on M and extend it by 0 outside M so that F ∈ X and F ◦ ψ = f . We may thus define v(f ) = u(F ). Note that v : D((a, b)) → C is a well-defined linear map since u ∈ kern(T )⊥ . In order to extend v to all of D([a, b]) (either by the Hahn–Banach theorem or by uniform continuity) we have to prove its continuity and this is the point where the Whitney regularity of ψ(R) is needed (remembering the counterexample with the very sharp cusp one sees that, without further assumptions,

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M had to be very thin near ψ(a) and thus ϕ would be probably too steep to allow an estimate of |v(f )| only in terms of the derivatives of f ). The crucial property of Whitney regular sets (for connected sets it is even a characterization) is described in [5, Theorem 2.3.11]: Every distribution with compact support in ψ(R) has a continuity estimate involving only finitely many derivatives on ψ(R). There are thus n ∈ N and C > 0 such that for all φ ∈ D(Rd ) u(φ) ˜ Cφn,ψ(R) = C sup ∂ α φ(z): |α| n, z ∈ ψ(R) . Now, all derivatives of F = ϕ · (f ◦ ψk−1 ◦ πk ) and f ◦ ψk−1 ◦ πk coincide in a neighborhood of ψ(supp(f )) ⊆ ψ([a, b]). Since, moreover, ϕ ∈ D(M) we therefore obtain v(f ) = u ϕ · f ◦ ψ −1 ◦ πk C ϕ · f ◦ ψ −1 ◦ πk k k n,ψ(R)∩M −1 −1 = C f ◦ ψk ◦ πk n,ψ(R)∩M = C f ◦ ψk ◦ πk n,ψ([a,b]) . This can be further estimated by some norm f m,[a,b] because the projection and, in view of Lemma 2, also ψk−1 induce continuous composition operators. We have shown that v : D((a, b)) → C is continuous with respect to the Fréchet space topology of D([a, b]) and hence we can extend it to v˜ ∈ D([a, b]) . We still have to show v˜ ◦ T = u which is the case on the subspace {F ∈ X: F ◦ψ ∈ D((a, b))} of X since its elements are mapped to D((a, b)). It thus remains to notice that this subspace is dense in X which follows, e.g., from [5, Theorem 2.3.3] and the theorem of bipolars (or, without duality theory, by multiplying with cut-off functions as in the proof of that theorem from Hörmander’s book). 2 Step 3. A (ψ) is closed. Proof. Fix f ∈ A (ψ). Since the critical set E is discrete and ψ is proper we may choose open and pairwise disjoint intervals Ia a as well as open disjoint Ma ⊆ Rd with ψ(Ia ) = Ma ∩ ψ(R) for all a ∈ E. Moreover, we fix ϕa ∈ D(Ma ) which are equal to 1 in some neighborhood of ψ(a). The theorem of Allan et al. gives Fa ∈ E (Rd ) such that f − Fa ◦ ψ are flat at a, and hence ga = (f − Fa ◦ ψ) · (ϕa ◦ ψ) are flat at a with support in the interval Ia whose only critical point is a. Because of Step 2 each ga is thus a composition ga = Ga ◦ ψ where we can assume that Ga has support in Ma (by multiplying with a cut-off function in D(Ma ) which is equal to 1 near ψ(supp(ga ))). We set G = a∈E Ga + Fa ϕa and h = a∈E ϕa ◦ ψ and obtain fh=

a∈E

f · (ϕa ◦ ψ) =

ga + (Fa ◦ ψ) · (ϕa ◦ ψ) = G ◦ ψ.

a∈E

Moreover, f (1 − h) has support outside the critical set and is thus a composition by Step 1, hence f = hf + (1 − h)f is a composition. 2 3. Necessity Throughout this section let ψ : R → Rd be an injective and smooth symbol such that A (ψ) is closed in E (R). The open mapping theorem then implies that the composition operator T : E (Rd ) → A (ψ) is open. Using that sets of the form {F ∈ E (Rd ): F m,K ε} with m ∈ N,


3003

K ⊆ Rd compact, and ε > 0 form a basis of the 0-neighborhoods we obtain that for all K ⊆ Rd compact there are n ∈ N, I ⊆ R compact, and c > 0 such that

f ∈ A (ψ): f n,I 1 ⊆ T F ∈ E Rd : F 1,K c .

For all f ∈ A (ψ) there is thus F ∈ E (Rd ) with f =F ◦ψ

and F 1,K cf n,I

if f n,I = 0 and F 1,K 1/2 otherwise.

Step 1. ψ is proper. Proof. For a compact set K ⊆ Rd we take n, I , and c as above and we claim that K ∩ ψ(R) ⊆ ψ(I ). Otherwise there are y = ψ(x) ∈ K and G ∈ D(Rd \ ψ(I )) with G(y) = 1. Then G ◦ ψ ∈ A (ψ) and there is thus F ∈ E (Rd ) with G ◦ ψ = F ◦ ψ and F 1,K 1/2. This yields the contradiction 1 = |G(ψ(x))| = |F (ψ(x))| = |F (y)| 1/2. 2 To continue we use the chain rule f (x) = inequality to obtain f (x) cf n,I ψ (x)

d

j =1 ∂j F (ψ(x))ψj (x)

and the Cauchy–Schwarz

whenever ψ(x) ∈ K and f n,I = 0.

In order to evaluate this condition we fix ϕ ∈ D((−1, 1)) with ϕ (0) = 1 and we set cn = ϕn,R . For every y ∈ / E with ψ(y) ∈ K we define r = r(y) = dist(y, E) = min{|a − y|: a ∈ E} and f (x) = ϕ((x − y)/r). Then supp(f ) ⊆ (y − r, y + r) ⊆ R \ E and hence f ∈ A (ψ) by Step 1 in the proof of sufficiency. Since f (y) = 1/r and f n,I cn /r n we obtain (with εn = 1/ccn ) (∗)

ψ (y) εn r n−1

whenever ψ(y) ∈ K

because ψ(y) ∈ K implies y ∈ I and hence f n,I = 0. Step 2. Every critical point is of finite order and, in particular, E is discrete. Proof. Assume that there is a critical point a of infinite order. We apply the above to a fixed compact neighborhood K of ψ(a) to obtain n and εn . Next we choose δ > 0 such that L = [a − δ, a + δ] satisfies ψ(L) ⊆ K and (n) ψ (x) < εn /d(2n)n−1

for all x ∈ L.

Since dist(·, E) is continuous on L there is y ∈ L such that r = dist(y, E) = max dist(x, E): x ∈ L . Then every closed subinterval of L of length 2r contains a critical point, and hence every subinterval of length 2nr contains n critical points. If y is contained in a subinterval J of length 2nr we repeatedly apply Rolle’s theorem to each component ψj of ψ and obtain points z1 , . . . , zn−1 ∈ J (k) with ψj (zk ) = 0. If y is not contained in a subinterval of L of length 2nr we have |y − a| 2nr

3004


and we put z1 = · · · = zn−1 = a and J = L. In any case we then have |zk − w| 2nr for all w ∈ J . We now apply the mean value theorem n − 1 times to obtain ξ2 , . . . , ξn ∈ J such that ψ (y) = ψ (y) − ψ (z1 ) = ψ (ξ2 )|y − z1 | ψ (ξ2 ) − ψ (z2 )2nr j

j

j

j

j

j

(3) (3) (3) = ψ (ξ3 )|ξ2 − z2 |2nr ψ (ξ3 ) − ψ (z3 )(2nr)2 j

j

j

.. .

(n) ψj (ξn )(2nr)n−1 . This last product, however, is strictly less than εn r n−1 /d, and hence we conclude |ψ (y)| < εn r n−1 in contradiction with (∗). 2 Step 3. The inverse ψ −1 : ψ(R) → R is locally Hölder continuous. Proof. Since ψ is a diffeomorphism outside the critical set E it is enough to prove the Hölder continuity of ψ −1 at every ψ(a) for a critical point a which, for notational convenience, we may assume to be 0. Since we already know that E is discrete and ψ is proper, ψ(0) is an isolated point of ψ(E) and we may choose a compact, convex neighborhood K of ψ(0) with ψ(E) ∩ K = {ψ(0)}. As before we take n, I , c, and εn from above according to K. (m) Let m be the order of the critical point 0 and choose a coordinate ψk with ψk (0) = 0. Then there is an interval J = [−δ, δ] such that ψk has no further zero in J . Applying l’Hospital’s rule m − 1 times we see that all quotients ψj (x)/ψk (x) have a limit at 0 and we thus obtain ψ (x) c˜ψ (x) k

for all x ∈ J and some constant c˜ 1. Decreasing δ we may additionally assume ψ(J ) ⊆ K and dist(x, E) = |x| for all x ∈ J . We are going to prove an estimate 1/n |x − y| C ψ(x) − ψ(y) in J , and to do so we distinguish the two cases where x, y have equal or different signs. In the first case we may assume 0 < x < y so that ψk has constant sign on [x, y]. Using (∗) from above we obtain y y ψ(y) − ψ(x) ψk (y) − ψk (x) = ψ (t) dt = ψ (t) dt k k x

1 c˜

y x

ψ (t) dt εn c˜

εn (y − x)n , nc˜

x

y t n−1 dt = x

εn n y − xn nc˜


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where we used for z = y/x 1 the (very rough) inequality zn − 1 = (z − 1)

n−1

z (z − 1)zn−1 (z − 1)n .

=0

Multiplying with nc/ε ˜ n and taking the n-th root gives the desired estimate in this first case. In the second case we have x < 0 < y and we apply the estimates from the open mapping theorem to different functions. We consider a fixed χ ∈ D((−1, 1)) with χ(0) = 1 and kn = χn,R . Since for x ∈ J we have dist(x, E) = |x|, the function f (t) = χ((t − x)/|x|) has support in (2x, 0) ⊆ R \ E. As before, f ∈ A (ψ) and there is thus F ∈ E (Rd ) with f = F ◦ ψ and F 1,K cf n,I ckn /|x|n . Since x < 0 < y we have f (y) = 0 and we obtain ckn 1 = f (x) − f (y) = F ψ(x) − F ψ(y) F 1,K ψ(x) − ψ(y) n ψ(x) − ψ(y), |x| which implies |x|n ckn ψ(x) − ψ(y). Changing the roles of x and y we also have |y|n ckn |ψ(x) − ψ(y)| which finally gives n ckn 2n ψ(x) − ψ(y). |x − y|n 2 max |x|, |y|

2

To finish the proof of the theorem as well as the corollary it remains to note: Step 4. If ψ −1 : ψ(R) → R is locally Hölder continuous then ψ is proper, has only critical points of finite order, and has a Whitney regular image ψ(R). Proof. That ψ is proper is implied by the mere continuity of ψ −1 , and that all critical points are of finite order follows from Taylor’s theorem. To show Whitney regularity we note that for x < y y the geodesic distance in ψ(R) is just x |ψ (t)| dt. Combining the Hölder continuity |x − y| y C|ψ(x) − ψ(y)|γ with the trivial estimate x |ψ (t)| dt sup{|ψ (t)|: t ∈ [x, y]}|x − y| ends the proof. 2 Acknowledgment We thank Leonhard Frerick for several useful discussions about the subject of this article. In particular, he suggested to us a version of the corollary. References [1] Graham Allan, Grayson Kakiko, Anthony G. O’Farrell, Richard O. Watson, Finitely-generated algebras of smooth functions, in one dimension, J. Funct. Anal. 158 (2) (1998) 458–474, MR 1648487 (99m:46127). [2] Edward Bierstone, Pierre D. Milman, Geometric and differential properties of subanalytic sets, Bull. Amer. Math. Soc. (N.S.) 25 (2) (1991) 385–393, MR 1102751 (92h:32008). [3] Edward Bierstone, Pierre D. Milman, Geometric and differential properties of subanalytic sets, Ann. of Math. (2) 147 (3) (1998) 731–785, MR 1637671 (2000c:32027).

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[4] Georges Glaeser, Fonctions composées différentiables, Ann. of Math. (2) 77 (1963) 193–209, MR 0143058 (26 #624). [5] Lars Hörmander, The Analysis of Linear Partial Differential Operators. I: Distribution Theory and Fourier Analysis, second ed., Grundlehren Math. Wiss., vol. 256, Springer-Verlag, Berlin, 1990, MR 1065993 (91m:35001a). [6] Bernard Malgrange, Ideals of Differentiable Functions, Tata Inst. Fund. Res. Stud. Math., vol. 3, Tata Institute of Fundamental Research, Bombay, 1967, MR 0212575 (35 #3446). [7] Reinhold Meise, Dietmar Vogt, Introduction to Functional Analysis, Oxf. Grad. Texts Math., vol. 2, Clarendon Press, Oxford University Press, New York, 1997, translated from the German by M.S. Ramanujan and revised by the authors, MR 1483073 (98g:46001). [8] Laurent Schwartz, Théorie des distributions, Nouvelle édition, entiérement corrigée, refondue et augmentée, Publ. Inst. Math. Univ. Strasbourg, vols. IX–X, Hermann, Paris, 1966, MR 0209834 (35 #730). [9] Jean-Claude Tougeron, Idéaux de Fonctions Différentiables, Ergeb. Math. Grenzgeb., vol. 71, Springer-Verlag, Berlin, 1972, MR 0440598 (55 #13472).


Asymptotic integration of Navier–Stokes equations with potential forces. II. An explicit Poincaré–Dulac normal form Ciprian Foias a , Luan Hoang b,∗ , Jean-Claude Saut c a Department of Mathematics, 3368 TAMU, Texas A&M University, College Station, TX 77843-3368, USA b Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX 79409-1042, USA c Laboratoire de Mathematiques, Universite Paris Sud, Batiment 425, 91405 Orsay Cedex, France

Received 4 January 2011; accepted 2 February 2011 Available online 22 February 2011 Communicated by H. Brezis

Abstract We study the incompressible Navier–Stokes equations with potential body forces on the threedimensional torus. We show that the normalization introduced in the paper [C. Foias, J.-C. Saut, Linearization and normal form of the Navier–Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1) (1987) 1–47], produces a Poincaré–Dulac normal form which is obtained by an explicit change of variable. This change is the formal power series expansion of the inverse of the normalization map. Each homogeneous term of a finite degree in the series is proved to be well-defined in appropriate Sobolev spaces and is estimated recursively by using a family of homogeneous gauges which is suitable for estimating homogeneous polynomials in infinite dimensional spaces. © 2011 Elsevier Inc. All rights reserved. Keywords: Navier–Stokes equations; Poincaré–Dulac normal form; Nonlinear dynamics; Homogeneous gauge

Contents 1. 2.

Introduction and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3008 Homogeneous gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012


E-mail addresses: [email protected] (L. Hoang), [email protected] (J.-C. Saut). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.02.005

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C. Foias et al. / Journal of Functional Analysis 260 (2011) 3007–3035

3. Homogeneous polynomials in the normal form 4. A Poincaré–Dulac normal form . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3017 3025 3034 3035

1. Introduction and preliminaries The initial value problem for the incompressible Navier–Stokes equations in the threedimensional space R3 with a potential body force is ⎧ ∂u ⎪ + (u · ∇)u − νu = −∇p − ∇φ, ⎨ ∂t ⎪ ⎩ div u = 0, u(x, 0) = u0 (x),

(1.1)

where ν > 0 is the kinematic viscosity, u = u(x, t) is the unknown velocity field, p is the unknown pressure, (−∇φ) is the body force specified by a given function φ and u0 (x) is the known initial velocity field. We consider only solutions u(x, t) such that for any t 0, u(x) = u(x, t) satisfies u(x + Lej ) = u(x) for all x ∈ R3 , j = 1, 2, 3

(1.2)

(where {e1 , e2 , e3 } is the canonical orthonormal basis in R3 ), and u(x) dx = 0

(1.3)

Ω

where L > 0 is fixed and Ω = (−L/2, L/2)3 . We call the functions satisfying (1.2) L-periodic functions. Throughout this paper we take L = 2π and ν = 1. The general case is easily recovered by a change of scale. For the theory of Navier–Stokes equations, the reader is referred to the pioneering works by J. Leray [12–14] as well as the books [11,3,17,19]. For the dynamical point of view of Navier–Stokes equations, see [16,18]. Let V be the set of all L-periodic trigonometric polynomials on Ω with values in R3 which are divergence-free as well as satisfy the condition (1.3). We define H , resp. V , the closure of V in L2 (Ω)3 , resp. H 1 (Ω)3 , where H l (Ω), l = 0, 1, 2, . . . , denote the Sobolev spaces W l,2 (Ω). √ Let a · b denote the standard scalar product of vectors a, b in R3 and |a| = a · a. The inner product and norm in L2 (Ω)3 are given by u, v =

u(x) · v(x) dx,

|u| = u, u1/2 ,

u = u(·), v = v(·) ∈ L2 (Ω)3 .

Ω

Though the notation |·| denotes the length of vectors in R3 as well as the L2 -norm of vector fields in L2 (Ω)3 , its meaning will be clear in the context.


3009

On V we consider the norm · defined by u =

1/2 3 ∂uj (x) 2 , ∂x dx k

for u = u(·) = (u1 , u2 , u3 ) ∈ V .

j,k=1 Ω

Define the Stokes operator A by Au = −u

for all u ∈ D(A),

where D(A) is the closure of V in H 2 (Ω)3 . The condition (1.3) implies that on D(A), the norm |Aw|, w ∈ D(A), is equivalent to the usual Sobolev norm of H 2 (Ω)3 . We recall that the spectrum σ (A) of the Stokes operator A consists of the eigenvalues λ1 < λ2 < λ3 < · · · of the form λj = |k|2 for some k ∈ Z3 \{0} where j = 1, 2, 3, . . . . Note that the additive semi-group generated by σ (A) coincides with the set N = {1, 2, 3, . . .} of all natural numbers. We denote by Rn the orthogonal projection of H onto the eigenspace of A associated to n if n is an eigenvalue of A, otherwise Rn = 0. Define Pn = R1 + R2 + · · · + Rn . Fractional powers of A: For α 0, define Aα u =

k =0

ik·x |k|2α u(k)e ˆ ,

for u =

ik·x u(k)e ˆ ∈ H.

k =0

The domain of Aα is D(Aα ) = {u ∈ H : |Aα u| < ∞}. Note that D(A0 ) = H , D(A1/2 ) = V and u = |A1/2 u| for u ∈ V . Also, |Aα ξ | |Aβ ξ | and hence D(Aβ ) ⊂ D(Aα ) for all β α 0. We also define the bilinear mapping associated with the nonlinear term in the Navier–Stokes equations by B(u, v) = PL (u · ∇v) for all u, v ∈ D(A),

(1.4)

where PL denotes the orthogonal projection in L2 (Ω)3 onto H . We denote by R the set of all initial values u0 ∈ V such that there is a (unique) solution u(t), t > 0, satisfying the functional form of (1.1): du(t) + Au(t) + B u(t), u(t) = 0, dt

t > 0,

(1.5)

with the initial data u(0) = u0 , where the equation holds in H , and u(t) is continuous from [0, ∞) into V . In other words, R is the set of all initial data u0 ∈ V such that the solution u(t) of the Navier–Stokes equations (1.5) is regular on [0, ∞). Note that R is an open set in V that contains infinitely many unbounded closed linear manifolds of infinite dimension, see Remark 7 of [7] (and also Proposition 6.4 of [4]). Let us recall some known results on the asymptotic expansions and the normal form of the regular solutions to the Navier–Stokes equations (see [8,9] for more details). First, for any u0 ∈ R the solution u(t) has the asymptotic expansion u(t) ∼ q1 (t)e−t + q2 (t)e−2t + q3 (t)e−3t + · · · ,

(1.6)

3010


where qj (t), j 1, is a polynomial in t with values in V. This means that for any N ∈ N the

−j t satisfies correction term u˜ N +1 (t) = u(t) − N j =1 qj (t)e u˜ N +1 (t) = O e−(N +ε)t

as t → ∞ for some ε = εN > 0.

(1.7)

In fact, u˜ N +1 (t) belongs to C 1 ([0, ∞), V ) ∩ C ∞ ((0, ∞), C ∞ (R3 )), and for each m ∈ N relation (1.7) holds for the Sobolev norm u˜ N +1 (t)H m (Ω) and ε = εN,m > 0. Define the normalization map W by W (u0 ) = W1 (u0 ) ⊕ W2 (u0 ) ⊕ · · ·, where Wj (u0 ) = Rj qj (0) for j ∈ N. Then W is a one-to-one analytic mapping from R to the Fréchet space SA = R1 H ⊕ R2 H ⊕ · · · endowed with the topology induced by the family of semi-norms |Rj u| (u ∈ SA ), j ∈ N. If u0 ∈ R then the polynomials qj (t) are the unique polynomial solutions to the following equations qj (t) + (A − j )qj (t) + βj (t) = 0,

t ∈ R,

(1.8)

with Rj qj (0) = Wj (u0 ), where the terms βj (t) are defined by β1 (t) = 0 and βj (t) =

B qk (t), ql (t)

for j > 1.

(1.9)

k+l=j

Given arbitrary ξ = (ξn )∞ n=1 ∈ SA , the polynomial solutions qj (t, ξ ) of (1.8) satisfying the initial condition Rj qj (0) = ξj , are explicitly given by the recursive formula t qj (t, ξ ) = ξj −

Rj βj (τ ) dτ +

(−1)n+1 (A − j )−n−1

n0

0

dn (I − Rj )βj , dt n

(1.10)

for j ∈ N. Here (A − j )−n−1 is used to denote [(A − j )|(I −Rj )H ]−n−1 and is defined by (A − j )−n−1 u =

|k|2 =j

ak eik·x (|k|2 − j )n+1

for u = |k|2 =j ak eik·x ∈ V. Above I denotes the identity map on H . ∞ The SA -valued function ξ(t) = (ξj (t))∞ j =1 = (Wj (u(t)))j =1 = W (u(t)) satisfies the following system of differential equations ⎧ dξ1 (t) ⎪ ⎪ ⎪ ⎨ dt + Aξ1 (t) = 0, dξj (t) ⎪ Rj B qk 0, ξ(t) , ql 0, ξ(t) = 0, + Aξj (t) + ⎪ ⎪ ⎩ dt

j > 1.

(1.11)

k+l=j

For ξ ∈ SA , denote Pj (ξ ) = qj (0, ξ ),

j 1,

(1.12)


3011

and B = (Bj )∞ j =1 where B1 (ξ ) = 0,

Bj (ξ ) =

Rj B Pk (ξ ), Pl (ξ ) = 0,

j > 1.

(1.13)

k+l=j

Note that Pj (ξ ) is a polynomial in ξ ; here and throughout, concerning the polynomials between vector spaces, we refer to Chapter IV in [10] (with the caveat that we consider the zero map a homogeneous polynomial of any degree). In fact, Pj (ξ ), ξ = (ξn )∞ n=1 , is a V-valued polynomial in the variables ξ1 , ξ2 , . . . , ξj , each belonging to a finite dimensional space. For example, P1 (ξ ) = ξ1 , P2 (ξ ) = ξ2 − (A − 2)−1 (I − R2 )B(ξ1 , ξ1 ). Regarding the notation, hereafter, for any polynomial Q in ξ regardless if it depends on t, we denote Q[d] , for d 0, the sum of all its monomials of degree d, i.e., the homogeneous part of degree d of Q. The system (1.11) written in the vector form in SA is dξ + Aξ + B(ξ ) = 0. dt

(1.14)

This system is the normal form in SA of the Navier–Stokes equations (1.5) associated with the asymptotic expansion (1.6). It is easy to check that the solution of (1.11) with initial data 0 −j t )∞ . Thus, formula (1.10) yields the normal ξ 0 = (ξj0 )∞ j =1 ∈ SA is precisely (Rj qj (t, ξ )e j =1 form and its solutions. It was proved by G. Minea in [15] that this type of normalization for ordinary differential equations in the finite dimensional case “coincides with the distinguished normalization in the sense of A.D. Brjuno” [15]. However, whether this is true for Navier–Stokes equations is still an open question. In previous works [5,6] the normalization map W and the normal form (1.11) are studied and well understood in a Banach space of the following type: SA

= ξ

= (ξn )∞ n=1

∞ def

∈ SA : ξ =

ρn ξn < ∞ ,

(1.15)

n=1

where the sequence (ρn )∞ n=1 of positive weights decays suitably fast to zero. However these spaces are much too large to easily connect to the concrete approach of the classical Poincaré– Dulac theory (see e.g. [1], and also our relevant summary of the subject in Section 4 below). In this paper we will show that the Poincaré–Dulac theory can be extended to the Navier–Stokes equations (1.5) by using the normal form (1.14) restricted to the space E ∞ = C ∞ (R3 , R3 ) ∩ V which is continuously embedded in SA . In fact we prove that if Pj[d] (ξ ) and Bj[d] (ξ ) denote the sum of all homogeneous monomials of degree d of Pj (ξ ) and Bj (ξ ), respectively, then the series

[d]

[d] ∞ to continuous polynomials P [d] (ξ ) and B [d] (ξ ), j Pj (ξ ) and j Bj (ξ ) converge in E respectively, such that ∞

dξ + Aξ + B [d] (ξ ) = 0 dt d=2

(1.16)

3012


is a Poincaré–Dulac normal form for the Navier–Stokes equations obtained by the formal change of variable u=ξ +

∞

P [d] (ξ ).

(1.17)

d=2

In contrast to [5,6] where we relied on the recursive relations between the polynomials qj (t, ξ ) (j ∈ N), here we rely on the recursive formulas of the homogeneous terms in the normal form. Our main tool in estimating their Sobolev norms is a family of homogeneous gauges [[ξ ]]d,n introduced in Section 2 (see (2.8)). These gauges have useful multiplicative properties (Lemma 2.1) as well as estimates in terms of Sobolev norms (Lemma 2.2). Also in Section 2, we recall some necessary estimates for the bilinear operator B(u, v) (Lemma 2.3). In Section 3, by using those

∞ [d]

[d] gauges we establish the absolute convergence of the series ∞ j =1 Pj (ξ ) and j =1 Bj (ξ ) in α α+3d/2 D(A ), for ξ ∈ D(A ) and α 1/2. The main result of the paper (Theorem 4.9), the rigorous definition of its framework, and its full proof (necessitating in several steps) are given in Section 4. 2. Homogeneous gauges In this section, we give several estimates which are of independent interest as well as needed in our next sections.

Hereafter, we identify ξ ∈ SA with u ∼ j ξj , and hence can define fractional power Aα , for α 0, on SA as well as its domain D(Aα ) as a subspace of SA . In working with polynomials ininfinite dimensional spaces, it is convenient to introduce the set of general multi-indices GI = ∞ n=1 GI(n) where for n 1, GI(n) = α¯ = (αk )∞ / σ (A) . k=1 , αk ∈ {0, 1, 2, . . .}, αk = 0 for k > n or k ∈ For α¯ ∈ GI, define |α| ¯ =

∞

αk

and α ¯ =

k=1

∞

(2.1)

kαk .

k=1

For d, n 1, define the set of special multi-indices (see Lemma 3.1 for its motivation): SI(d, n) = α¯

= (αk )∞ =1

∈ GI, |α| ¯ =

∞

αk = d, α ¯ =

k=1

∞

kαk = n ;

(2.2)

k=1

note 1 d n hence SI(d, n) ⊂ GI(n). Also, for n d 1 and n d 1 we have SI(d, n) + SI d , n ⊂ SI d + d , n + n . Let ξ = (ξk )∞ ¯ = (αk )∞ k=1 ∈ SA and α k=1 ∈ GI, define [ξ ]α¯ =

αk >0

|ξk |αk .

(2.3)


3013

We have the following properties

¯ α¯ [ξ ]α¯ [ξ ]α¯ = [ξ ]α+ ,

r [ξ ]r α¯ = [ξ ]α¯

(2.4)

for r = 0, 1, 2, . . . ,

(2.5)

[ξ ]2α¯ = |ξ |2d .

(2.6)

|α|=d ¯

For n d 1, we define the (d, n)-gauge of ξ ∈ SA by [[ξ ]]d,n =

[ξ ]2α¯

1/2

=

[ξ ]2α¯

1/2 .

(2.7)

|α|=d, ¯ α=n ¯

α∈SI(d,n) ¯

The family of gauges referred to in the abstract and introduction is {[[·]]d,n : n d 1}. Three useful properties of these gauges are given below. First, note that

[[ξ ]]d,n

2α¯

1/2

[ξ ]

|Pn ξ |d .

(2.8)

α∈GI(n), ¯ |α|=d ¯

Next is a multiplicative inequality: Lemma 2.1. Let ξ ∈ SA , n d 1 and n d 1. Then

[[ξ ]]d,n · [[ξ ]]d ,n ed+d [[ξ ]]d+d ,n+n .

(2.9)

Proof. By (2.4), [[ξ ]]2d,n · [[ξ ]]2d ,n =

α∈SI(d,n) ¯

=

[ξ ]2α¯

[ξ ]2α¯

α¯ ∈SI(d ,n )

¯ α¯ ) [ξ ]2(α+

(2.10)

α∈SI(d,n) ¯ α¯ ∈SI(d ,n ) By (2.3) the index γ¯ = α¯ + α¯ belongs n ), where α, ¯ α¯ are as in (2.10). We

to SI(d + d , n2γ+ ¯ need to compare the above sum to γ¯ ∈SI(d+d ,n+n ) [ξ ] . For that we estimate the number of

¯ α¯ ) in (2.10). times [ξ ]2γ¯ is summed up as [ξ ]2(α+ ∞ Fix γ¯ = (γk )k=1 ∈ SI(d, n) + SI(d , n ). We count the number of ways to write each γ¯ as the sum α¯ + α¯ . If k > n or k > n then αk = 0, αk = γk or αk = 0, αk = γk , hence one way. Let k min{n, n }. Counting via αk : the set of possible values for αk is {0, 1, 2, . . . , γk }, hence at most γk + 1 values. Thus the number of repetition of γ¯ as the sum α¯ + α¯ is at most

N(γ¯ ) = (γ1 + 1)(γ2 + 1) · · · (γn + 1) (γ1 + 1)(γ2 + 1) · · · (γn+n + 1).

3014


By generalized Young’s inequality:

(γ1 + 1) + (γ2 + 1) + · · · + (γn+n + 1) n+n N (γ¯ ) n + n d + d + n + n n+n d + d n+n = = 1 + . n + n n + n

Since the function f (x) = (1 + a/x)x is increasing for x a > 0 we have N (γ¯ ) ed+d . It follows that α∈SI(d,n) ¯ α¯ ∈SI(d ,n )

¯ α¯ ) [ξ ]2(α+

N (γ¯ )[ξ ]2γ¯ ed+d

γ¯ ∈SI(d+d ,n+n )

[ξ ]2γ¯ .

γ¯ ∈SI(d+d ,n+n )

2

Combining with (2.10), we obtain (2.9).

It is noteworthy that the factor ed+d on the right-hand side of (2.9) depends on neither n nor n . Lemma 2.2. For any ξ ∈ SA , any numbers α, s 0 and n d 1, one has α A ξ d,n

s s d d α+s d A ξ d,n Pn Aα+s ξ . n n

(2.11)

Proof. For |α| ¯ = d and α ¯ = n we have [ξ ]2α¯ =

|ξk |2αk =

αk >0

αk >0

=

|k s ξk |2αk k 2αk s

s 2αk [As ξ ]2α¯ α >0 |k ξk | k = . ( αk >0 k αk )2s ( αk >0 k αk )2s

Let k0 = max{k: αk = 0}. Then n =

kαk k0 ( αk ) = k0 d. Hence k0 n/d and αk0

k αk k0

k0 n/d.

(2.12)

αk >0

Therefore 2α¯ [ξ ]2α¯ (d/n)2s As ξ .

(2.13)

Summing over α¯ ∈ SI(n, d) one obtains [[ξ ]]d,n (d/n)s As ξ d,n .

(2.14)


3015

Replacing ξ by Aα ξ in (2.14) yields the first inequality of (2.11). The second one results from (2.8). 2 We now turn to the estimates of the bilinear mapping B(u, v). We will use the following simple inequalities. Lemma 2.3. For α 0 one has α A B(u, v) 4α C A1/2 u 1/2 |Au|1/2 Aα+1/2 v + Aα u 1/2 Aα+1/2 u 1/2 |Av| , (2.15) α 1/2 1/2 A B(u, v) 4α C A1/2 u |Au|1/2 Aα+1/2 v + Aα+1/2 u A1/2 v |Av|1/2 , (2.16) where C > 0 is an absolute constant defined by (2.25). In particular, when α 1/2 one has α A B(u, v) K α Aα+1/2 u Aα+1/2 v ,

(2.17)

for all u, v ∈ D(Aα+1/2 ), where K = 4(max{2C, 1})2 . Proof. Suppose u, v, w ∈ H with u=

−ik·x ˆ u(k)e ,

v=

k =0

vˆ (k)e−ik·x ,

w=

k =0

−ik·x ˆ w(k)e .

(2.18)

k =0

Let u∗ =

u(k) ˆ e−ik·x , k =0

v∗ =

vˆ (k) e−ik·x ,

w∗ =

k =0

w(k) ˆ e−ik·x .

(2.19)

k =0

The relation between the Sobolev norms of u and u∗ is: α A u = (−)α u∗ for all α 0. We have Aα B(u, v) · w dx = 8π 3 Ω

. ˆ ˆ · l vˆ (l) · w(m) |m|2α u(k)

k+l+m=0

Using the inequality |m|α 2α (|k|α + |l|α ) we estimate Aα B(u, v) · w dx 8π 3 4α Ω

ˆ ˆ |l| vˆ (l) w(m) |k|2α u(k)

k+l+m=0

+ 8π 3 4α

u(k) . ˆ |l|2α+1 vˆ (l) w(m) ˆ

k+l+m=0

Rewriting the last two sums in terms of u∗ , v∗ and w∗ we have

(2.20)

3016


Aα B(u, v) · w dx 8π 3 4α (−)α u∗ (−)1/2 v∗ w∗ dx Ω

Ω

+ 8π 3 4α u∗ (−)α+1/2 v∗ w∗ dx .

(2.21)

Ω

For the first integral on the right-hand side of (2.21), we have two options for Hölder’s inequality using powers 3, 6, 2 or 6, 3, 2, and then use interpolation and Sobolev inequalities and the relation (2.20): (−)α u∗ (−)1/2 v∗ w∗ dx Ω

1/2 1/2 3/2 C1 (−)α u∗ (−)α+1/2 u∗ (−)v∗ |w∗ | 1/2 1/2 3/2 C1 Aα u Aα+1/2 u |Av||w|, (−)α u∗ (−)1/2 v∗ w∗ dx

(2.22)

Ω

1/2 1/2 3/2 C1 (−)α+1/2 u∗ (−)1/2 v∗ (−)v∗ |w∗ | 1/2 3/2 C1 Aα+1/2 u A1/2 v |Av|1/2 |w|,

(2.23)

where C1 > 0 is the Sobolev constant for the embedding of V into L6 (Ω). For the second integral on the right-hand side of (2.21), applying the Hölder inequality and then using the Agmon inequality for the embedding of D(A) into L∞ (Ω), we obtain u∗ (−)α+1/2 v∗ w∗ dx Ω

u∗ L∞ (Ω) (−)α+1/2 v∗ |w∗ | 1/2 1/2 C2 (−)1/2 u∗ (−)u∗ (−)α+1/2 v∗ |w∗ | 1/2 C2 A1/2 u |Au|1/2 Aα+1/2 v |w|.

(2.24)

Combining (2.21) with (2.24) and (2.22), resp. (2.23), yields (2.15), resp. (2.16) with 3/2 C = 8π 3 max C1 , C2 . For α 1/2, by either (2.15) or (2.16): α A B(u, v) 2C4α Aα+1/2 u Aα+1/2 v , and hence (2.17) follows.

2

(2.25)


3017

3. Homogeneous polynomials in the normal form In this section we show that the homogeneous polynomials P [d] (ξ ) in (1.17) and B [d] (ξ ) in (1.16) are well-defined. Let ξ = (ξk )∞ k=1 ∈ SA and qj (t, ξ ) be the polynomial solutions given by (1.10). Using the explicit formula (1.10), one can easily verify the following properties by induction: (a) Each qj (t, ξ ) is a polynomial in t and in ξ1 , ξ2 , . . . , ξj , hence is a V-valued polynomial on the finite dimensional space R × Pj H . (b) The degree in t: degt qj (t, ξ ) j − 1. (c) The degree in ξ : degξ qj (t, ξ ) j . Thus we can write qj (t, ξ ) =

j −1

qj,m (ξ )t m =

m=0

j j −1

[d] qj,m (ξ )t m =

m=0 d=1

j

qj[d] (t, ξ ),

(3.1)

d=1

[d] where qj,m (ξ ) is a polynomial in ξ , and qj,m (ξ ) and qj[d] (t, ξ ) are homogeneous polynomials in ξ of degree d.

[d],(α) ¯ [d],(α) ¯ qj (t, ξ ), where α¯ = (αk )∞ (t, ξ ) is We also write qj[d] (t, ξ ) = |α|=d ¯ k=1 ∈ GI and qj

the sum of all monomials (in ξ ) of qj[d] (t, ξ ) having degree αk in ξk for all k 1. Similarly, we

[d],(α) ¯ [d] (ξ ) = |α|=d qj,m (ξ ). We also have write qj,m ¯ βj (t, ξ ) =

j −2

βj,m (ξ )t m =

m=0

j j −2

[d] βj,m (ξ )t m =

m=0 d=1

j

βj[d] (t, ξ ),

d=1

[d] where β1,m (ξ ) = β1,m (ξ ) = β1[d] (t, ξ ) = 0 for all m, d, t and ξ ,

βj,m (ξ ) =

l+l =j r+r =m [d] βj,m (ξ ) =

B ql,r (ξ ), ql ,r (ξ ) ,

l+l =j r+r =m s+s =d

[s] B ql,r (ξ ), ql[s ,r] (ξ ) ,

(3.2) (3.3)

for j 2 and 0 m j − 2. ¯ We need the following properties of the degrees in t and in ξ of qj[d] (t, ξ ) and qj[d],(α) (t, ξ ). Lemma 3.1. (i) The degree in t of the polynomial qj[d] (t, ξ ) is less than or equal to d − 1, i.e.,

degt qj[d] (t, ξ ) d − 1.

¯ = 0 then α¯ ∈ SI(d, j ). (ii) If qj[d],(α) (iii) Consequently, for each (nonzero) monomial of Pj (ξ ), j 1, having degree αk in ξk , ¯ Also, for each (nonzero) monok 1, one has α¯ = (αk )∞ k=1 belongs to SI(d, j ) where d = |α|. mial of B(Pm (ξ ), Pn (ξ )), having degree αk in ξk , k 1, one has α¯ = (αk )∞ k=1 belongs to SI(d, m + n) where d = |α|. ¯

3018


Proof. We prove (i) and (ii) by induction in j and d. Since q1 (t, ξ ) = ξ1 and qj[1] (t, ξ ) = ξj , the statements (i) and (ii) hold for j = 1 and all d’s, as well as for d = 1 and all j ’s. Let j 2, d 2 and assume that for all 1 j < j and 1 d < d, we have degt qj[d ] (t, ξ ) d − 1 and [d ],(α) ¯

= 0. α¯ ∈ SI(d , j ) whenever qj Observe from (3.3) and the induction hypothesis that m = r + r (s − 1) + (s − 1) = d − 2 [s] [s ] whenever B(ql,r , ql ,r ) = 0, hence degt βj[d] d − 2. From the integral in the formula (1.10), degt qj[d] degt βj[d] + 1 d − 1, hence (i) is verified for j and d.

[d],(α) ¯ βj (t, ξ ), where For (ii), we also have that βj[d] (t, ξ ) = |α|=d ¯ [d],(α) ¯

βj

(t, ξ ) =

l+l =j

k+k =d

γ¯ +γ¯ =α¯

[k],(γ¯ ) [k ],(γ¯ ) B ql (t, ξ ), ql (t, ξ ) .

(3.4)

[k],(γ¯ )

By the induction hypothesis γ¯ ∈ SI(k, l) and γ¯ ∈ SI(k , l ) whenever the summand B(ql [k ],(γ¯ ) ql )

k, l

,

l)

in (3.4) is not identically zero, hence by (2.3) α¯ ∈ SI(k + + = SI(d, j ). This completes the proof of (ii). The first statement of (iii) is a direct consequence of (ii) since Pj (ξ ) = qj (0, ξ ). Each mono[d],(α) ¯ mial in the second statement of (iii) is of the form βm+n (0, ξ ) as in (3.4), hence the argument in (ii) readily shows α¯ ∈ SI(d, m + n). 2 In the following calculations we will use the convention 0/0 = 0 as well as the shorthand notation j |d = min{j, d − 1}

for all j, d.

[d] [d] = 0 for m > (j − 1)|d , and βj,m = 0 for m > (j − 2)|d−1 . It is clear from Lemma 3.1 that qj,m Applying the projection Rk to (1.10) we have

Rk qj (t, ξ ) = Rk ξj −

t j −2

Rk Rj βj,m τ m dτ

0 m=0

+

j −2 j −2 (−1)n+1 d n Rk (I − Rj )βj,m t m n n+1 dt (k − j ) n=0

= Rk ξj −

m=0

j −1 Rk Rj βj,m−1 m t m

m=1

+

j −2 j −2 (−1)n+1 m! Rk (I − Rj )βj,m t m−n . n+1 (m − n)! (k − j ) m=n n=0

By a suitable relabelling we obtain


Rk qj (t, ξ ) = Rk ξj −

3019


m=1

+

j −2−n j −2 (−1)n+1 (m + n)! Rk (I − Rj )βj,m+n t m n+1 m! (k − j ) n=0

m=0

= Rk ξj −


m=1

j −2 j −2−m (−1)n+1 (m + n)! m Rk (I − Rj )βj,m+n t . + m! (k − j )n+1 m=0

n=0

Hence for m = 0: Rk qj,0 = Rk ξj +

j −2 (−1)n+1 n! n=0

(k − j )n+1

Rk (I − Rj )βj,n ;

(3.5)

for m = 1, . . . , j − 2: j −2−m (−1)n+1 (m + n)! Rk Rj βj,m−1 Rk qj,m = − + Rk (I − Rj )βj,m+n ; m m! (k − j )n+1

(3.6)

n=0

and for m = j − 1: Rk qj,j −1 = −

j −1 Rk Rj βj,j −2 . j −1

(3.7)

m=1

Collecting the homogeneous components of degree d 1 in ξ of the identities (3.5), (3.6) and (3.7) yields [d] Rk qj,0

= Rk ξj[d]

+

j −2 (−1)n+1 n! n=0

= Rk ξj[d]

(k − j )n+1

(j −2)|d−1

+

n=0

[d] Rk qj,m =−

=−

[d] Rk Rj βj,m−1

m [d] Rk Rj βj,m−1

m

+

Rk (I

(−1)n+1 n! [d] Rk (I − Rj )βj,n , (k − j )n+1

j −2−m n=0

n=m

(3.8)

(−1)n+1 (m + n)! [d] Rk (I − Rj )βj,m+n m! (k − j )n+1

(j −2)|d−1

+

[d] − Rj )βj,n

(−1)n−m+1 n! [d] Rk (I − Rj )βj,n (k − j )n−m+1 m!

(3.9)

3020


for m = 1, . . . , (j − 2)|d , and [d] Rk qj,j −1 = −

[d] Rk Rj βj,j −2

j −1

(3.10)

.

[d] After this preparation, we will establish recursive estimates for the norms of qj,m .

Lemma 3.2. For j 2, d 1, α 0 and ξ ∈ SA , one has

(j −2)|d−1 α [d] 2 Aα (I − Rj )β [d] (ξ ) 2 ; A q (ξ ) 2(d!)(d − 1)! Aα ξ [d] 2 + j j,n j,0

(3.11)

n=0

α [d] |A Rj βj,m−1 (ξ )|2 α [d] A q (ξ ) 2 (d!)(d − 1)! j,m m2 1 + 2 m!

(j −2)|d−1

α A (I − Rj )β [d] (ξ ) 2 j,n

(3.12)

n=0

for m = 1, . . . , (j − 2)|d ; and [d] 2 |Aα Rj βj,j −2 (ξ )|2 . j,j −1 (ξ ) = (j − 1)2

α [d] A q

(3.13)

Proof. For d = 1, the inequality (3.11) is trivially true, because qj[1] (t, ξ ) = ξj for all j , and the sum on the right-hand side of (3.11) is missing; also (3.12) and (3.13) trivially hold since all polynomials involved are zero. Let d 2. For m = 0 we apply the Cauchy–Schwarz inequality to the sum on the right-hand side of (3.8), which consists of at most d terms, to obtain (j −2)|d−1 α 2 [d] A Rk q d Aα Rk ξ [d] 2 + j j,0 n=0

α n!2 A Rk (I − Rj )β [d] 2 j,n 2(n+1) |k − j |

d−1 α 2 (j −2)| 2 [d] [d] 2 α d A Rk ξ j + n! A Rk (I − Rj )βj,n n=0

(j −2)|d−1 2 2 Aα Rk (I − Rj )β [d] . (d − 1)!2 d Aα Rk ξj[d] + j,n n=0

Then summing this estimate over k yields (3.11). Similarly, for m = 1, . . . , (j − 2)|d , we have from (3.9) that


3021

[d] α 2 2 |Aα Rk Rj βj,m−1 |2 [d] A Rk q (d − 1)! d j,m m2 1 + 2 m!

α A Rk (I − Rj )β [d] 2 , j,n

(j −2)|d−1

n=0

which gives (3.12) after summing over k. Finally, the formula (3.10) yields (3.13) directly.

2

Our next aim is to prove that the following two formal series P

[d]

(ξ ) =

∞

Pj[d] (ξ ) =

j =d

B

[d]

(ξ ) =

∞ j =1

∞

[d] qj,0 (ξ ) =

j =d

Bj[d] (ξ ) =

∞ j =1

∞

qj[d] (0, ξ ),

d 1,

(3.14)

j =d

[m] [n] Rj B Pk (ξ ), Pl (ξ ) ,

d 2,

(3.15)

k+l=j m+n=d

are in fact convergent in Sobolev spaces and define homogeneous polynomials acting between appropriate Sobolev spaces. We note that when d = 1, Pj[1] (ξ ) = ξj = Rj ξ

for all j,

hence P [1] (ξ ) = ξ

for ξ ∈ H.

(3.16)

[d] (ξ )| and |Aα Pj[d] (ξ )| by using the (d, j )-gauge. First, we estimate the norms |Aα qj,m

Proposition 3.3. For j d 1 and 0 m (j − 1)|d , one has α [d] A q (ξ ) c(α, d) Aα+ 32 (d−1) ξ j,m

d,j

(3.17)

,

for all ξ ∈ SA and α 1/2, where the positive number c(α, d) is c(α, d) = (Md )(α+τd )(d−1) ,

(3.18)

with Md = K 2 + d 6 e2d (d!)2

and τd = (d − 1)/2.

(3.19)

In particular, when m = 0 one has α [d] A P (ξ ) c(α, d) Aα+ 32 (d−1) ξ . j d,j

(3.20)

Proof. For this proof, we rewrite (3.17) in the following convenient form α [d] A q (ξ ) c(α, d) Aα+hδ(d−1) ξ j,m d,j

with δ = 1/2, h =

δ+1 = 3. δ

(3.21)

3022


We will prove (3.21) by induction in j . For j = 1, we have q1 (t, ξ ) = ξ1 ; it is clear that d = 1, m = 0 and (3.21) holds with c(α, 1) = 1 for all α 0. Let j 2. Suppose α [d ] A q (ξ ) c α , d Aα +hδ(d −1) ξ

(3.22)

d ,j

j ,m

for all 1 j < j , 1 d j , and 0 m (j − 1)|d −1 , all ξ ∈ SA and α 1/2. [1] For d = 1, qj,m (ξ ) is ξj for m = 0 and is 0 otherwise; hence (3.17) holds for all α 0. Consider d 2. We will use the recursive estimates in Lemma 3.2. First, we estimate [d] |Aα βj,n |. Note that Md > 1 and both Md and τd are increasing in d. Using formula (3.3), inequality (2.17) and the induction hypothesis (3.22) we have α [d] A β (ξ ) j,m


[s] K α Aα+δ ql,r (ξ ) Aα+δ ql[s ,r] (ξ ) K α c(α + δ, s)c α + δ, s


× Aα+δ+hδ(s−1) ξ s,l Aα+δ+hδ(s −1) ξ s ,l .

Applying the multiplicative inequality (2.9) yields α [d] A β (ξ ) j,m


(α+δ+τs )(s −1)

K α Ms(α+δ+τs )(s−1) Ms

× es+s Aα+δ+hδ(s+s −2) ξ s+s ,l+l (α+δ+τd )[(s−1)+(s −1)] d α+δ+hδ(d−2) K α Md e A ξ d,j l+l =j r+r =m s+s =d


d 2K α

l+l =j

(α+δ+τd )(d−2) d α+δ+hδ(d−2)

K α Md

e

A

(α+δ+τd )(d−2) d α+δ+hδ(d−2)

Md

e

A

ξ

d,j

ξ

d,j

,

thus α [d] A β (ξ ) K α d 2 ed M (α+δ+τd )(d−2) · j Aα+δ+hδ(d−2) ξ j,m

d

d,j

.

Applying the inequality (2.11) with s = 1, we obtain α [d] α+δ+hδ(d−2)+1 A β (ξ ) K α d 2 ed M (α+δ+τd )(d−2) · j · d A ξ d,j d j,m j (α+δ+τd )(d−2) α+δ+hδ(d−2)+1 A K α d 3 e d Md ξ d,j .


3023

Note that h and δ satisfy δ + hδ(d − 2) + 1 = hδ(d − 1). Setting N = N (α, d) = K α d 3 ed × (α+δ+τd )(d−2) Md , we have α [d] A β (ξ ) N Aα+hδ(d−1) ξ j,m

d,j

(3.23)

.

For m = 1, 2, . . . , (j − 2)|d , it follows from (3.12) and (3.23) that

(j −2)|d−1 α [d] 2 α+hδ(d−1) 2 α+hδ(d−1) 2 2 2 A q (d − 1)!d! N A ξ d,j + N A ξ d,j j,m n=0

2 d! N Aα+hδ(d−1) ξ d,j 2

2

2(α+δ+τd )(d−2) α+hδ(d−1)

= K 2α (d!)2 d 6 e2d Md

A

ξ

2 d,j

.

Since the numbers Md and τd in (3.19) satisfy K 2 , (d!)2 d 6 e2d Md

and α + 1 + 2(α + δ + τd )(d − 2) 2(α + τd )(d − 1),

it follows that α [d] 2 A q M α+1+2(α+δ+τd )(d−2) Aα+hδ(d−1) ξ 2 j,m

d

2(α+τd )(d−1) α+hδ(d−1) 2 A Md ξ d,j .

d,j

Therefore (3.21) is readily obtained for j , d and m = 0, j − 1. Similarly, we obtain (3.21) for [d] [d] (ξ )| and |Aα qj,j |Aα qj,0 −1 |, resp., by using (3.11) and (3.13), resp., together with (3.23). This concludes the induction step and completes our proof. 2 Theorem 3.4. Let α 1/2, d 1 and ξ ∈ D(Aα+3d/2 ). Then P [d] (ξ ) defined in (3.14) converges absolutely in D(Aα ) and satisfies ∞ α [d] α [d] A P (ξ ) A P (ξ ) M(α, d) Aα+3d/2 ξ d , j

(3.24)

j =d

where M(α, d) = (1 + 2d)c(α, d) with c(α, d) given by (3.18). Moreover, P [d] (ξ ) is a continuous homogeneous polynomial of degree d from D(Aα+3d/2 ) to D(Aα ). Proof. Let r > 1. From Proposition 3.3 and inequality (2.11) we have ∞ ∞ α [d] A q (ξ ) c(α, d) Aα+hδ(d−1) ξ j,m

j =d

j =1

∞ j =d

d,j

r d α+hδ(d−1)+r d A c(α, d) ξ j

d = c0 (d, r)c(α, d) Aα+hδ(d−1)+r ξ ,

3024


where c0 (d, r) = d r

∞

j =d

1/j r . Particularly, when m = 0 we have

∞ α [d] α [d] A P (ξ ) A P (ξ ) c0 (d, r)c(α, d) Aα+hδ(d−1)+r ξ d . j

(3.25)

j =1

The constant c0 (d, r) can be bounded by c0 (d, r) d

r

1 + dr

∞ d

1 d . dt = 1 + tr r −1

Taking r = 3/2 and recalling that δh = 3/2, we have α + hδ(d − 1) + r = α + 3d/2 and c0 (d, 3/2) 1 + 2d.

(3.26)

Hence (3.24) follows (3.25) and (3.26). By the uniform estimate (3.24) and Theorem 4.2.9 in [10], the limit function P [d] (ξ ) is a continuous homogeneous polynomial of degree d. 2 Theorem 3.5. Let α 1/2, d 2 and ξ ∈ D(Aα+3d/2 ). Then one has for all m 0 that ∞ α [d] A β (ξ ) C(α, d) Aα+3d/2 ξ d , j,m

(3.27)

j =2

where C(α, d) = (2d + 1)N(α, d) with N (α, d) being the same number as in (3.23). Consequently, B [d] (ξ ), d 2, defined in (3.15) is a continuous homogeneous polynomial of degree d from D(Aα+3d/2 ) to D(Aα ) for all α 1/2, and satisfies ∞ α [d] α [d] A B (ξ ) A B (ξ ) C(α, d) Aα+3d/2 ξ d . n

(3.28)

n=1

Proof. By (3.23), and with r > 1 we have ∞ ∞ d α [d] A β (ξ ) N(α, d)(d/j )r Aα+hδ(d−1)+r ξ j,m

j =1

j =1

d c0 (d, r)N(α, d) Aα+hδ(d−1)+r ξ .

Taking r = 3/2 and using (3.26), we obtain (3.27). The relation (3.28) is obtained by taking m = 0 in (3.27). The same argument as in Theorem 3.4 shows that B [d] is a continuous homogeneous polynomial of degree d from D(Aα+3d/2 ) to D(Aα ). 2


3025

4. A Poincaré–Dulac normal form The normal form (1.14), originally constructed in SA , now can be rewritten formally as (1.16). Concerning the framework for (1.16) where B [d] (ξ ) is well-defined for all d 2, Theorem 3.5 with the estimate (3.28) of |Aα B [d] (ξ )| in terms of |Aα+3d/2 ξ | for all α 1/2 and d 1 suggests an appropriate framework, namely, the Fréchet space E ∞ = C ∞ (R3 , R3 ) ∩ V . We recall that the classical L. Schwartz’s topology on E ∞ is the one given by the family of norms |Aα u| (u ∈ E ∞ ), α 0. Theorems 3.4 and 3.5 directly imply

∞ [d] [d] [d] Lemma 4.1. The series P [d] (ξ ) = ∞ j =d Pj (ξ ) and B (ξ ) = j =d Bj (ξ ), for d 2, converge in E ∞ . Moreover P [d] (ξ ) and B [d] (ξ ), for d 2, are continuous homogeneous polynomials of degree d from E ∞ to E ∞ . We denote by H[d] (E ∞ ), d 1, the linear space of continuous homogeneous polynomials of degree d from E ∞ to E ∞ . For Q ∈ H[d] (E ∞ ) we denote by Qˆ the continuous symmetric ˆ d-linear map from (E ∞ )d to E ∞ representing Q, that is, Q(ξ ) = Q(ξ, ξ, . . . , ξ ) for ξ ∈ E ∞ . We recall from [9] that the asymptotic expansion (1.6) can be written formally as

u(t) =

∞

∞ Pj W u(t) = Pj ξ(t) .

j =1

(4.1)

j =1

Therefore it is reasonable to consider that u = ∞ j =1 Pj (ξ ) is the formal inverse of the normalization map ξ = W (u).

∞ j [d] We write u = ∞ j =1 Pj (ξ ) = j =1 d=1 Pj (ξ ), then by formally interchanging the order of summations, we have the formal relation

u=

∞ ∞

Pj[d] (ξ ) =

j =1 d=1

∞ ∞

Pj[d] (ξ ) =

d=1 j =1

∞

P [d] (ξ ).

d=1

Therefore we formally write ∞ def

u = P(ξ ) =

P [d] (ξ ) = ξ +

d=1

∞

P [d] (ξ ).

(4.2)

d=2

Similarly, for the nonlinear term in (1.14), we have

B(ξ ) =

∞ j =1

Bj (ξ ) =

∞ ∞ j =1 d=2

Bj[d] (ξ ) =

∞ ∞ d=2 j =1

Bj[d] (ξ ) =

∞

B [d] (ξ ).

(4.3)

d=2

We aim to prove that (1.16) is a Poincaré–Dulac normal form in E ∞ which can be obtained from the Navier–Stokes equations (1.5) by using the explicit formal power series (4.2).

3026


The power series (4.2) has its formal right inverse of the form def ˜ = ξ = P(u)

∞

P˜ [d] (u) = u +

d=1

∞

P˜ [d] (u),

(4.4)

d=2

where each P˜ [d] (u), d 1, is a homogeneous polynomial of degree d, particularly, P˜ [1] (u) = P [1] (u) = u. To compute this inverse we introduce (4.4) into (4.2) and formally obtain u=u+

∞

P˜ [d] (u) +

d=2

∞

P

[d]

d=2

∞

[k] ˜ P (u) ,

k=1

here P [d] (v) = Pˆ [d] (v, v, . . . , v), where Pˆ [d] is the continuous symmetric d-linear mapping from (E ∞ )d into E ∞ that represents P [d] , hence ∞

˜ [d]

P

(u) +

d=2

∞

∞

ˆ [d]

P

˜ [k1 ]

P

u, . . . ,

k1 =1

d=2

∞

˜ [kd ]

P

u = 0.

kd =1

Then collecting the homogeneous terms of the same degree d, we have

P˜ [d] (u) = −

d m=2

Pˆ [m] P˜ [k1 ] u, . . . , P˜ [km ] u

k1 +···+km =d

= −P [d] (u) −

d−1 m=2

Pˆ [m] P˜ [k1 ] u, . . . , P˜ [km ] u

(4.5)

k1 +···+km =d

for d > 2, and P˜ [2] (u) = −P [2] (u) when d = 2. Using formula (4.5), we define recursively all homogeneous polynomials P˜ [d] (u) which are ˜ continuous from E ∞ to E ∞ . Obviously (P ◦ P)(u) = u for all u ∈ E ∞ . Similarly, one can find the formal right inverse P˜˜ of (4.4) and can verify that P˜˜ = P, hence (P˜ ◦ P)(ξ ) = ξ for all ξ ∈ E∞. We make the formal change of variable (4.2) in the Navier–Stokes equations and obtain the following formal relation du = −Au − B(u, u) dt = −Aξ −

∞ d=2

AP

[d]

(ξ ) −

∞

B P [k] (ξ ), P [l] (ξ ) .

k,l=1

Letting u = u(t) and ξ = ξ(t) in (4.4) and formally taking the derivative in t, we obtain

(4.6)


3027

∞ du ˜ [d] du dξ = + D P (u) dt dt dt d=2

= −Aξ −

∞

AP [d] (ξ ) −

d=2

∞

B P [k] (ξ ), P [l] (ξ )

k,l=1

∞

∞ [d] [l] ˜

D P (u) u= ∞ P [k] (ξ ) − AP (ξ ) k=1 d=2

−

l=1

∞

∞

d=2

k,l=1

D P˜ [d] (u) u= ∞ P [m] (ξ ) m=1

[k] [l] B P (ξ ), P (ξ ) .

(4.7)

Above, D applied to a polynomial Q ∈ H[d] (E ∞ ) indicates the (Fréchet) derivation of Q, that is ˆ DQ(ξ ) v = d Q(v, ξ, . . . , ξ ),

ξ ∈ E∞, v ∈ E∞.

(4.8)

Using (4.8), we rewrite the sums in (4.7). For instance, ∞

∞ [k] [d] [l] D P˜ (u) u= ∞ P [m] (ξ ) B P (ξ ), P (ξ ) m=1 d=2

∞ ˜ [d] = dP

=

k,l=1

∞

d=2

k,l=1

∞ ∞

∞

∞ ∞ [k] [l] [m2 ] [md ] B P (ξ ), P (ξ ) , P (ξ ), . . . , P (ξ ) m2 =1

md =1

˜ [d] B P [k] (ξ ), P [l] (ξ ) , P [m2 ] (ξ ), . . . , P [md ] (ξ ) . dP

(4.9)

d=2 k,l=1 m2 ,...,md =1

Note that each summand in the last expression has degree k + l + m2 + · · · + md . Hence each homogeneous component of a finite degree in (4.9) is only a finite sum of those summands. Similar calculations apply to other simpler sums in (4.7). By collecting the homogeneous terms of the same degree in (4.7), we then derive ∞

dξ [d] + Q (ξ ) = 0, dt

(4.10)

d=1

where Q[d] (ξ ) ∈ H[d] (E ∞ ), d 1; however the series in (4.10) is still formal. Although Q[d] (ξ ) can be computed recursively in terms of P [m] (ξ ) and P˜ [m] (ξ ), we will express it in terms of P [m] (ξ ) only. This will be helpful in our later considerations. Obviously, we have Q[1] (ξ ) = Aξ . Up to degree d 2 in ξ we formally calculate from (4.2), knowing that (4.10) already holds, dξ du dξ = + DP [m] (ξ ) dt dt dt m2

3028


= −Aξ −

Q[d] (ξ ) −

d2

DP [k] (ξ ) Aξ + Q[l] (ξ ) . k2

l2

We substitute this into the left-hand side of (4.6) and collect the homogeneous polynomials of the same degree to derive Q[d] (ξ ). Note that either P [m] = 0 or deg((DP [m] (ξ ))Q(ξ )) = m + n − 1 for any m 1 and Q ∈ H[n] (E ∞ ). Hence Q[d] (ξ ) = AP [d] (ξ ) − DP [d] (ξ )Aξ + −

B P [k] (ξ ), P [l] (ξ )

k+l=d

DP [k] (ξ ) Q[l] (ξ ).

k,l2 k+l=d+1

Therefore we obtain the recursive formula for d 2:

Q[d] (ξ ) = HA(d) P [d] (ξ ) +

k+l=d

−

B P [k] (ξ ), P [l] (ξ )

DP [k] (ξ ) Q[l] (ξ );

(4.11)

2k,ld−1 k+l=d+1

above HA(d) : Q(ξ ) → AQ(ξ ) − (DQ(ξ ))Aξ , where Q ∈ H[d] (E ∞ ), is the analogue of Poincaré homology operator of order d (cf. see e.g. [1]) in the present framework. Clearly (d) HA (H[d] (E ∞ )) ⊂ H[d] (E ∞ ), for all d ∈ N. Now Eq. (4.10) can be rewritten as ∞

dξ + Aξ + Q[d] (ξ ) = 0, dt

(4.12)

d=2

where the polynomials Q[d] (ξ ), d 2, are given in more explicit way by (4.11). We will prove that Q[d] (ξ ) = B [d] (ξ ) for all d 2 and ξ ∈ E ∞ . The Poincaré homology operators have the following property. (d)

Lemma 4.2. For d 1 the homology operator HA satisfies HA P [d] (ξ ) = (d)

∞ (A − j )Pj[d] (ξ )

for all ξ ∈ E ∞ .

(4.13)

j =1

For the proof we need the following preliminaries, which will also be useful later. Definition 4.3. Let Q ∈ H[d] (E ∞ ). Then Q(ξ ) (ξ ∈ E ∞ and ξj = Rj ξ , j ∈ N), is a monomial of degree αki > 0 in ξki where i = 1, 2, . . . , m, αk1 + · · · + αkm = d and k1 < k2 < · · · < km , if it can be represented as


3029

˜ k1 , . . . , ξk1 , ξk2 , . . . , ξk2 , . . . , ξkm , . . . , ξkm ), Q(ξ ) = Q(ξ αk1

(4.14)

αkm

αk2

˜ (1) , ξ (2) , . . . , ξ (d) ) is a continuous d-linear map from (E ∞ )d to E ∞ . where Q(ξ

The monomial Q(ξ ) defined by (4.14), with degree d 2, is called resonant if m i=1 αki ki = j and Q = Rj Q = 0. In (4.14), by the symmetrization of Q˜ in each group of variables, specifically, αk1 variables of ξ (1) , ξ (2) , . . . , ξ αk1 , αk2 variables of ξ (αk1 +1) , ξ (αk1 +2) , . . . , ξ (αk1 +αk2 ) , . . . , and αkm variables of ξ (d−αkm +1) , ξ (d−αkm +2) , . . . , ξ (d) , we will always assume without loss of generality that Q˜ is symmetric in each of these groups of variables. In later calculations, it will be convenient to have αk defined for Q(ξ ) for all k ∈ N, by defining / {k1 , . . . , km }. the degree αk in ξk to be zero if k ∈ For j ∈ N, let Nj = dim Rj H and {ϕj,1 , ϕj,2 , . . . , ϕj,Nj } be a fixed orthonormal basis of Rj H . Let J (A) = {(j, i): j ∈ σ (A), 1 i Nj }. For any ξ ∈ E ∞ we have the unique expansion

ξ=

where xj,i = ξ, ϕj,i ∈ R,

xj,i ϕj,i ,

(4.15)

(j,i)∈J (A)

and the series is absolutely convergent in D(Aα ) for all α 0 hence converges in E ∞ . Here xj,i , with (j, i) ∈ J (A), are the coordinates of ξ with respect to the orthonormal basis {ϕj,i : (j, i) ∈ J (A)} of H . Then any function of ξ can be viewed as a function of x = (xj,i ). Let Q(ξ ) be a monomial with representation (4.14). Using the preceding notation, we have ˜ k1 , . . . , ξk1 , ξk2 , . . . , ξk2 , . . . , ξkm , . . . , ξkm ) Q(ξ ) = Q(ξ αk1

αkm

αk2

Nk Nk1 Nk2 Nk2 1 ˜ =Q xk1 ,i ϕk1 ,i , . . . , xk1 ,i ϕk1 ,i , xk2 ,i ϕk2 ,i , . . . , xk2 ,i ϕk2 ,i , i=1

i=1

i=1

αk1 Nkm

...,

xkm ,i ϕkm ,i , . . . ,

i=1

i=1

αk2

Nkm

xkm ,i ϕkm ,i .

i=1

αkm

˜ By the multi-linearity of Q: N

N

k1

Q(ξ ) = (1)

(αk ) 1 =1

i1 ,...,i1

xk

Nkm

k2

(1) 1 ,i1

˜ × Q(ϕ k

(αk ) 2 =1

(1)

(αk ) 1

k1 ,i1

,...,ϕ

xk

(1) 2 ,i2

(αk ) k1 ,i1 1

(α

(1)

)

im ,...,im km =1

i2 ,...,i2

···x

(1) 1 ,i1

··· ···x

, ϕk

(αk ) 2

k2 ,i2

(1) 2 ,i2

· · · xk

(1) m ,im

,...,ϕ

(αk ) k2 ,i2 2

···x

(α

km ,im km

, . . . , ϕk

(1) m ,im

)

,...,ϕ

(α ) km ,im km

) .

3030


˜ . .) of the last expression, we have Applying the expansion (4.15) to the term Q(. N

Q(ξ ) =

N

k1

(j ∗ ,h∗ )∈J (A)

xk

(1) 1 ,i1

˜ × Q(ϕ k

(1) m ,im

(αk ) (α ) 1 =1 i (1) ,...,i k2 =1 2 2

(1)

···x

(αk ) 1

k1 ,i1

,...,ϕ

xk

(1) 2 ,i2

(αk ) 1

k1 ,i1

,...,ϕ

(α ) km ,im km

···x

, ϕk

···

i1 ,...,i1

(1) 1 ,i1

ϕk

Nkm

k2

(αk ) 2

k2 ,i2

(1) 2 ,i2

(α

(1)

)

im ,...,im km =1

· · · xk

(1) m ,im

,...,ϕ

), ϕj ∗ ,h∗ ϕj ∗ ,h∗ .

(αk ) 2

k2 ,i2

···x

(α

km ,im km

)

,..., (4.16)

Each summand in (4.16) above has the form cξ, ϕj1 ,h1 αj1 ,h1 ξ, ϕj2 ,h2 αj2 ,h2 . . . ξ, ϕjn ,hn αjn ,hn ϕj ∗ ,h∗ ,

ξ ∈ E∞,

(4.17)

where c ∈ R, (j ∗ , h∗ ) ∈ J (A), (ji , hi ) with i = 1, 2, . . . , n, are distinct elements in J (A), and αji ,hi > 0 for i = 1, . . . , n. Note that the degree αk of Q(ξ ) in ξk satisfies αk =

αji ,hi

for k ∈ {k1 , k2 , . . . , km } = {j1 , j2 , . . . , jn }.

(4.18)

{i: ji =k}

Let q(ξ ) be the function of ξ defined by (4.17). Obviously q(ξ ) as a function of x is a monomial having degree αji ,hi in xji ,hi , for i = 1, 2, . . . , n. Again we define the degree αj,h in xj,h ((j, h) ∈ J (A)) to be zero if (j, h) ∈ / {(ji , hi ): i = 1, . . . , n}. Since this type of monomials is used in the classical theory of normal form (see again [1]), we call the monomial in (4.17) classical monomial. Also, q(ξ ) can be written as q(ξ ) = cξj1 , ϕj1 ,h1 αj1 ,h1 ξj2 , ϕj2 ,h2 αj2 ,h2 . . . ξjn , ϕjn ,hn αjn ,hn ϕj ∗ ,h∗ ,

(4.19)

hence q(ξ ) is a monomial of degree αk =

αji ,hi

in ξk for k ∈ {j1 , . . . , jn }.

(4.20)

{i: ji =k}

Therefore it is easy to prove that: Lemma 4.4. The classical monomial (4.17) is resonant if and only if c = 0 and j ∗ = αj1 ,h1 j1 + αj2 ,h2 j2 + · · · + αjn ,hn jn .

(4.21)

Remark 4.5. Lemma 4.4 implies that for classical monomials, being resonant (in the sense of Definition 4.3) is the same as being resonant in the classical theory [1].


3031

We will use the lexicographical order in J (A), namely, (j, h) < (j , h ) if (j < j ) or (j = j and h < h ). Denote (J (A)n , 0

hence each such monomial is resonant. Therefore (1.16) is a Poincaré–Dulac normal form for the Navier–Stokes equations (1.5). 2 Remark 4.10. The classical Poincaré–Dulac theory has a second part concerning the convergence of the series which give the change of variable, and the convergence of the series in the normal form (cf. [1]). We have not yet been able to extend this part of the theory to the Navier– Stokes equations. Remark 4.11. Our particular Poincaré–Dulac normal form (1.16) is uniquely determined by the asymptotic expansion (1.6) and its associated normal form (1.14) in SA . (Indeed, each summand B [d] in (1.16) is obtained by collecting the homogeneous terms of degree d in (1.14), see (4.33).) It is still open whether (1.16) is Brjuno’s distinguished normal form [2]. Remark 4.12. The methods developed in this paper can be applied to other dissipative partial differential equations with quadratic nonlinearity and a singleton global attractor. In particular, they apply directly to the 2D Navier–Stokes equations, since the latter are the restriction of the 3D ones to an appropriate invariant linear manifold. Acknowledgment L.H. acknowledges the support by NSF Grant DMS-0908177.


3035

References [1] V.I. Arnol’d, Geometrical Methods in the Theory of Ordinary Differential Equations, second edition, Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences), vol. 250, Springer-Verlag, New York, 1988, translated from the Russian by Joseph Szücs [József M. Sz˝ucs]. [2] A.D. Brjuno, Analytic form of differential equations. I, II, Tr. Mosk. Mat. Obs. 25 (1971) 119–262; Tr. Mosk. Mat. Obs. 26 (1972) 199–239. [3] P. Constantin, C. Foias, Navier–Stokes Equations, Chicago Lectures Math., University of Chicago Press, Chicago, IL, 1988. [4] C. Foias, L. Hoang, B. Nicolaenko, On the helicity in 3D-periodic Navier–Stokes equations. I. The non-statistical case, Proc. Lond. Math. Soc. (3) 94 (1) (2007) 53–90. [5] C. Foias, L. Hoang, E. Olson, M. Ziane, On the solutions to the normal form of the Navier–Stokes equations, Indiana Univ. Math. J. 55 (2) (2006) 631–686. [6] C. Foias, L. Hoang, E. Olson, M. Ziane, The normal form of the Navier–Stokes equations in suitable normed spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (5) (2009) 1635–1673. [7] C. Foias, J.-C. Saut, Asymptotic behavior, as t → +∞, of solutions of Navier–Stokes equations and nonlinear spectral manifolds, Indiana Univ. Math. J. 33 (3) (1984) 459–477. [8] C. Foias, J.-C. Saut, Linearization and normal form of the Navier–Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1) (1987) 1–47. [9] C. Foias, J.-C. Saut, Asymptotic integration of Navier–Stokes equations with potential forces. I, Indiana Univ. Math. J. 40 (1) (1991) 305–320. [10] E. Hille, Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloq. Publ., vol. 31, American Mathematical Society, New York, 1948. [11] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, translated from the Russian by Richard A. Silverman and John Chu, second English edition, revised and enlarged, Math. Appl., vol. 2, Gordon and Breach Science Publishers, New York, 1969. [12] J. Leray, Etude de diverse equations integrales non lineares et de quelques problemes que pose l’hydrodynamique, J. Math. Pures Appl. 12 (1933) 1–82. [13] J. Leray, Essai sur les mouvements plans d’un liquide visqueux que limitent des parois, J. Math. Pures Appl. 13 (1934) 331–418. [14] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1) (1934) 193–248. [15] G. Minea, Investigation of the Foias–Saut normalization in the finite-dimensional case, J. Dynam. Differential Equations 10 (1) (1998) 189–207. [16] G.R. Sell, Y. You, Dynamics of Evolutionary Equations, Appl. Math. Sci., vol. 143, Springer-Verlag, New York, 2002. [17] R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, second edition, CBMS–NSF Regional Conf. Ser. in Appl. Math., vol. 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. [18] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, second edition, Appl. Math. Sci., vol. 68, Springer-Verlag, New York, 1997. [19] R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001, reprint of the 1984 edition.


Asymptotic behaviors of solutions to evolution equations in the presence of translation and scaling invariance Yoshiyuki Kagei a , Yasunori Maekawa b,∗ a Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan b Department of Mathematics, Graduate School of Science, Kobe University, 1-1, Rokkodai, Nada-ku,

Kobe, 657-8501, Japan Received 20 August 2009; accepted 2 February 2011 Available online 24 February 2011 Communicated by I. Rodnianski

Abstract There are wide classes of nonlinear evolution equations which possess invariant properties with respect to a scaling and translations. If a solution is invariant under the scaling then it is called a self-similar solution, which is a candidate for the asymptotic profile of general solutions at large time. In this paper we establish an abstract framework to find more precise asymptotic profiles by shifting self-similar solutions suitably. © 2011 Elsevier Inc. All rights reserved. Keywords: Evolution equations; Large time behaviors of solutions; Self-similar solutions

1. Introduction In [8] Escobedo and Zuazua studied large time behaviors of solutions to convection–diffusion equations such as ∂t Ω − Ω + a · ∇ |Ω|p Ω = 0,

t > 0, x ∈ Rn ,


E-mail addresses: [email protected] (Y. Kagei), [email protected] (Y. Maekawa). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.02.004

(n-B)

Y. Kagei, Y. Maekawa / Journal of Functional Analysis 260 (2011) 3036–3096

3037

where n 1; a ∈ Rn is a given constant vector; p > 0 is a given number; = ni=1 ∂x2i ; and ∇ = (∂x1 , . . . , ∂xn ) . When p > n1 in (n-B), which is called the weakly nonlinear case, the nonlinear term can be regarded as a perturbation from the scaling point of view. It is proved in [8] that solutions asymptotically behaves like the Gauss kernel which is the self-similar solution to the linear heat equations ∂t Ω − Ω = 0,

t > 0, x ∈ Rn .

(H)

On the other hand, when p = n1 , which is called the critical case, the nonlinear term is not a perturbation and the Gauss kernel no longer describes the large time behaviors of solutions to (n-B). Instead, in this case the self-similar solutions to the nonlinear equation (n-B) itself describe large time behaviors of solutions to (n-B). The existence and the uniqueness of selfsimilar solutions are shown in [1], and it is proved in [8] that these self-similar solutions give the large time asymptotics of solutions in the following sense: Ω(t) − t − n2 Uδ √· t

− n (1− 1 ) p , =o t 2

t → ∞, 1 p ∞.

(1)

Lp (Rn )

Here Ω(t) is a solution to(n-B) with initial data Ω0 and Uδ is the profile function of the selfsimilar solution satisfying Rn Uδ (x) dx = δ := Rn Ω0 (x) dx. Eq. (n-B) is considered as a generalization of the well-known viscous Burgers equations 1 ∂t Ω − ∂x2 Ω + ∂x Ω 2 = 0, 2

t > 0, x ∈ R.

(1-B)

For (1-B) a more precise asymptotic profile was given by [21]; see also [4]. Indeed, when the initial data has a suitable decay at spatial infinity the next estimate holds with some y ∗ , t ∗ ∈ R: ∗ 1 Ω(t) − t + t ∗ − 2 Uδ √· + y t + t∗

−2+ 1 2p , =o t

t → ∞, 1 p ∞.

(2)

Lp (R)

This result is extended by [27], which established further improvements of the rate of convergence for solutions to (1-B). Roughly speaking, the above result implies that large time behaviors of solutions are described more precisely by suitably shifted self-similar solutions. The key idea in [21] and [27] is to reduce (1-B) to the linear heat equations by using the Hopf–Cole transformation. Hence, if we try to obtain analogous results with (2) for (n-B) or other nonlinear equations, we cannot use the arguments in [21,27]. Recently analogous observations with (2) were achieved in [24] for a one-dimensional parabolic system modeling chemotaxis, usually called a Keller–Segel system. In [24] it is proved that a suitably shifted Gaussian gives a more precise asymptotic profile of solutions than the Gaussian itself, which improves the results of [22,17]. The shift is determined by solving some ODEs. The one-dimensional Keller–Segel system treated in [24] is classified in the category of the weakly nonlinear case. As is remarked in [24], it is expected that an analogous result to [24] also holds for the multi-dimensional Keller–Segel systems when they are in the category of the weakly nonlinear case. On the other hand, there is a two-dimensional Keller–Segel system which is classified in the critical case. For such a system it is known [2,23] that self-similar solutions

3038


describe large time behavior as in (1), while an estimate like (2) seems to have not yet been achieved for this case. In [26] Witelski and Bernoff presented the idea to improve the asymptotic profile in terms of shifted self-similar solutions as in (2) for the one-dimensional heat equations and the onedimensional porous medium equations. Their argument is based on a relation between decay rates and the symmetries such as translation invariances in space and time and a scaling invariance. The idea in [26] is then used in [10] to obtain the second and third order asymptotics of solutions to the two-dimensional vorticity equations. But the way how to determine the shifts in [26] or [10] seems to rely on explicit formulas of self-similar solutions and of eigenprojections related to the linearized operators. So it will be worth establishing a general method based on the abstract functional analysis for wider applications, where such explicit representations are not always available. The aim of this paper is thus to provide an abstract framework which clarifies the structure leading to the estimates like (2). This enables us to capture more precise asymptotic behaviors of solutions at time infinity for equations other than (1-B) in a systematic way. Our approach is based on two symmetries of the equations as in [26]; translation invariance and scaling invariance. If an evolution equation is invariant with respect to some scaling, we can expect the existence of self-similar solutions and they are candidates for the asymptotic profiles of general solutions. It is well known that, by introducing the self-similar variables, this problem can be reduced to the stability problem of stationary solutions for the equations in the new variables. The key observation here is that if translations and the scaling are suitably connected, then the linearized operators around the stationary solutions are shown to possess some point spectra which reflect the symmetries but also correspond to the second order asymptotic profiles of solutions. Then we determine the (time-dependent) shifts of stationary solutions so that the difference of solutions and shifted stationary solutions always belongs to a complement subspace of the eigenspace for these eigenvalues, by which we obtain more precise asymptotic profiles of solutions. In our arguments the time dependent shifts are constructed by solving suitable ODEs which are derived from the condition, stated as (T1) in Section 2.2.1, between infinitesimal generators associated with translations and scaling. The condition (T1) seems to be a new observation due to the abstract treatment, which plays important roles overall our arguments. We also note that in the abstract setting a one-parameter family of translations is introduced as well as a simple translation. This enables us to treat equations, such as the Vlasov–Poisson–Fokker–Planck equations without friction term, that are not invariant with respect to usual translations. As an example of the application, we will consider the convection–diffusion equation (n-B). By applying our method we will show the new estimate · + y˜ ∗ Ω(t) − (1 + t)− n2 Uδ √ 1+t

− n2 (1− p1 )−1+

= O(1 + t)

,

Lp

with small > 0 and some y˜ ∗ ∈ Rn , which gives an improvement of the estimate (1) along the direction of the estimate for (1-B) given in (2). To illustrate that our method systematically leads to the convergence of solutions to a shifted self-similar solutions as in (2), we will also present the applications to two-dimensional vorticity equation for viscous incompressible flows and twodimensional Keller–Segel systems. In the case of the Keller–Segel system, the result shows that the convergence rate to a shifted self-similar solution depends on the initial data. Since rather complicated computations are required, a verification of the conditions to apply our method to the Keller–Segel system will be given in a separate paper [15]. In [15] an interesting spectral


3039

behavior of the linearized operator is also described. As for the vorticity equation we note that in [10] Gallay and Wayne obtained the second and third order asymptotic expansion by proving the convergence of solutions to a shifted profile function of the self-similar solution. So our result on the vorticity equation is a restatement of their results from our abstract framework. We will further present that our method is applicable to the Vlasov–Poisson–Fokker–Planck equations without friction term, which is a degenerate parabolic equation and is not invariant with respect to a usual translation of a spatial variable. We will show that the one-dimensional Vlasov–Poisson– Fokker–Planck equation possesses a self-similar solution; and an application of our method will lead to the convergence of solutions to a shifted self-similar solution for a restricted class of initial data. This paper is organized as follows. From Section 2 to Section 6 we discuss in rather abstract settings, by regarding translations and scaling as the actions of the additive groups and the multiplicative group, respectively. In Section 2.1 we introduce the idea of translation and scaling in the Banach spaces and define the self-similarity of functions. We also fix the idea of solutions for the abstract Cauchy problems in this section. Especially, we always deal with mild solutions which are solutions to the integral equations defined through strongly continuous semigroups. In Section 2.2 we give several assumptions for operators. The main results in this paper are stated in Section 2.3. In Section 3 we derive the equations in self-similar variables in the abstract settings. Then the existence and the stability of self-similar solutions to the original equations are shown to be equivalent with those of stationary solutions to the new equations. The existence and the uniqueness of stationary solutions are proved in Section 4. In Section 5 we establish the local stability of the stationary solutions with a “rough” rate of convergence. Section 6 is the main contribution of this paper, in which we determine a suitable shift of self-similar solutions so as to get the more precise asymptotic profiles of solutions. For this purpose in Section 6.1 we study the spectrum of the linearized operator around stationary solutions by applying the general perturbation theory and semigroup theory of linear operators. In Section 6.2 we construct a time dependent shift by solving the nonlinear ODEs. Finally, in Section 6.3 we calculate the rate of convergence to shifted stationary solutions (or equivalently, shifted self-similar solutions), which completes the proofs of the main results in Section 2.3. In Section 7 we apply our abstract results obtained in Sections 2–6 to several nonlinear PDEs. 2. Preliminaries and main results 2.1. Scaling and translations in abstract settings We consider the evolution equations in a Banach space X: d Ω − AΩ + N (Ω) = 0, dt

t > 0.

(E)

Here A is a closed linear operator in X and N is a nonlinear operator. We are interested in large time behaviors of solutions to (E) in the presence of several invariant properties with respect to actions of groups, which are defined as scaling or translation in Definition 2.1 below. These definitions might be rather general to call them “scaling” or “translation”, but we use this naming in this paper because of typical examples in Section 7 and for simplicity of notations. We denote by R× the multiplicative group {λ ∈ R | λ > 0} and by R+ the additive group R. Both groups are endowed with the usual Eucledian topology. Let B(X) be the Banach space of all bounded linear operators in X.

3040


Definition 2.1. (1) We say R = {Rλ }λ∈R× ⊂ B(X) a scaling if R is a strongly continuous action of R× on X, i.e., Rλ1 λ2 = Rλ1 Rλ2 ,

λ1 , λ2 ∈ R × ,

(3)

R1 = I,

(4)

Rλ u → Rλ u in X when λ → λ for each u ∈ X.

(5)

(2) We say T = {τa }a∈R+ ⊂ B(X) a translation if T is a strongly continuous group acting on X. For one-parameter family of translations {Tθ }θ∈R with Tθ = {τa,θ }a∈R+ , we say that it is strongly continuous if τa,θ (f ) → τa,θ (f ) in X as θ → θ for each a ∈ R+ and f ∈ X. When (j ) there are n one-parameter families of translations {Tθ }θ∈R , j = 1, . . . , n, we say that they are

+ independent if for all a, a , θ ∈ R it follows that (i) (j )

(j )

(i)

τa,θ τa ,θ = τa ,θ τa,θ ,

1 i, j n.

(6)

The generator of {Rλ }λ∈R× is the operator B given by

R1+h f − f

Dom(B) = f ∈ X lim exists , h→0 h R1+h f − f , h→0 h

Bf = lim

f ∈ Dom(B).

(7)

Note that if f ∈ Dom(B) then Rλ f ∈ Dom(B) and BRλ f = Rλ Bf . Moreover, if f ∈ d Dom(B) then Rλ f is differentiable in X at each λ ∈ R× , and we have dλ Rλ f |λ=λ0 = λ10 Rλ0 Bf . A scaling R = {Rλ }λ∈R× naturally induces an action on C((0, ∞); X) as follows. For f ∈ C((0, ∞); X) we set Θλ (f )(t) = Rλ f (λt) ,

λ ∈ R× .

(8)

Then it is not difficult to see Proposition 2.1. (1) {Θλ }λ∈R× is an action of R× on C((0, ∞); X) and Θλ is linear for each λ ∈ R× . (2) Θλ (f )(t) → Θλ (f )(t) in X as λ → λ for each t > 0 and f ∈ C((0, ∞); X). We call {Θλ }λ∈R× the scaling induced by R. Let {Tθ }θ∈R be a strongly continuous oneparameter family of translations. The generator of Tθ for each θ ∈ R is the operator Dθ given by

τa,θ (f ) − f

Dom(Dθ ) = f ∈ X lim exists , a→0 a Dθ f = lim

a→0

τa,θ (f ) − f , a

f ∈ Dom(Dθ ).

(9)


3041

We can also consider a linear operator Γa,θ which is a derivative of τa,θ with respect to θ :

τa,θ+h (f ) − τa,θ (f )

exists , Dom(Γa,θ ) = f ∈ X lim h→0 h τa,θ+h (f ) − τa,θ (f ) , h→0 h

Γa,θ (f ) = lim

f ∈ Dom(Γa,θ ).

(10)

Definition 2.2. Let {Θλ }λ∈R× be the scaling induced by R = {Rλ }λ∈R× . We say that f ∈ C((0, ∞); X) is self-similar with respect to {Θλ }λ∈R× iff λ ∈ R× .

Θλ (f ) = f,

(11)

Then we easily see that Proposition 2.2. The function f ∈ C((0, ∞); X) is self-similar with respect to {Θλ }λ∈R× if and only if there is a function h ∈ X such that f (t) = R 1 (h).

(12)

t

Next let us fix the idea of solutions to (E). Throughout of this paper we consider only mild solutions, so we assume that A generates a strongly continuous (C0 ) semigroup et A in X, which gives a mild solution to the linear equation d Ω − AΩ = 0, dt

t > 0.

(E0 )

Definition 2.3. We say that Ω(t) ∈ C((0, ∞); X) is a mild solution to (E) if t

(t−s)A e N Ω(s) X ds < ∞

0

for any t > 0 and Ω(t) satisfies the equality

Ω(t) = e

(t−s)A

t Ω(s) −

e(t−τ )A N Ω(τ ) dτ,

for all t > s > 0.

(13)

s

Moreover, if Ω(t) satisfies in addition t lim Ω(t) = Ω0 ∈ X,

t→0

lim

t→0

(t−s)A e N Ω(s) X ds = 0,

0

then we say that Ω(t) is a mild solution to (E) with initial data Ω0 .

(14)

3042


Remark 2.1. If Ω(t) is a mild solution to (E) with initial data Ω0 , then Ω ∈ C([0, ∞); X) and it satisfies Ω(t) = e

tA

t

e(t−s)A N Ω(s) ds,

Ω0 −

for all t 0.

(15)

0

Definition 2.4. We call Ω(t) ∈ C((0, ∞); X) a self-similar solution to (E) with respect to {Θλ }λ∈R× if Ω(t) is a mild solution to (E) and is self-similar with respect to {Θλ }λ∈R× . The existence of self-similar solutions and its asymptotic stability are related with the invariant properties of (E) for the associated scaling. Moreover, if (E) also possesses a translation invariance, we can expect more precise informations on the asymptotic profiles of general solutions. The main purpose of this paper is to verify this idea under some assumptions on the operators. Definition 2.5. Let R = {Rλ }λ∈R× be a scaling and let {Tθ }θ∈R be a strongly continuous oneparameter family of translations. (i) We say that (E) is invariant with respect to {Θλ }λ∈R× if Θλ (Ω)(t) is a mild solution to (E) with initial data Rλ (Ω0 ) for each λ ∈ R× whenever Ω(t) is a mild solution to (E) with initial data Ω0 . (ii) We say that (E) is invariant with respect to {Tθ }θ∈R if τa,t+θ (Ω(t)) is a mild solution to (E) with initial data τa,θ (Ω0 ) for each a ∈ R+ and θ 0 whenever Ω(t) is a mild solution to (E) with initial data Ω0 . Let (E0 ) be invariant with respect to {Θλ }λ∈R× and {Tθ }θ∈R . Then the above definition is expressed by the equalities Rλ (eλt A Ω0 ) = Θλ (et A Ω0 ) = Θλ (Ω)(t) = et A Rλ (Ω0 ) and τa,t+θ et A Ω0 = et A τa,θ Ω0 , which holds for any Ω0 ∈ X, λ ∈ R× , a ∈ R+ , and t, θ 0. Hence we have key relations, which are equivalent representations of the above definitions, such as Rλ eλt A = et A Rλ ,

(16)

τa,t+θ et A = et A τa,θ .

(17)

Next we introduce the “similarity transform”. When (E0 ) is invariant with respect to {Θλ }λ∈R× induced by R = {Rλ }λ∈R× we set Θ(t) = Ret e(e −1)A = e(1−e t

−t )A

Re t ,

t 0.

(18)

Lemma 2.1. The one parameter family {Θ(t)}t0 defined by (18) is a strongly continuous semigroup in X. Proof. From (18) and (16) we have Θ(t)Θ(s) = e(1−e = e(1−e

−t )A −t )A

Ret Res e(e Re(t+s) ee

s −1)A

t+s (e−t −e−t−s )A


= e(1−e = e(1−e

−t )A

e(e

−t−s )A

−t −e−t−s )A

3043

Re(t+s)

Re(t+s)

= Θ(t + s). It is easy to see that Θ(t) is strongly continuous and Θ(0) = I . This completes the proof.

2

Let A be the generator of Θ(t), i.e.,

Θ(t)f − f

exists , Dom(A) = f ∈ X lim t→0 t Af = lim

t→0

Θ(t)f − f , t

f ∈ Dom(A).

(19)

Then it follows that Dom(A) ∩ Dom(B) ⊂ Dom(A) and Af = Af + Bf,

for f ∈ Dom(A) ∩ Dom(B).

(20)

In general, we cannot expect that B is relatively bounded with respect to A. Especially, the spectral properties of A can quite differ from those of A. Typical examples of A, R, and T (j ) are n 1 (j ) , (Rλ f )(x) = λ 2 f (λ 2 x), and (τa f )(x) = f (x1 , . . . , xj + a, . . . , xn ). Then the generators of x (j ) are given by B = 2 · ∇ + n2 and D (j ) = ∂xj . R and T 2.2. Several assumptions In this section we collect several assumptions on (E) and operators which we deal with. 2.2.1. Assumptions on (E0 ) We first state the assumptions on (E0 ). As stated in the previous section, the operator A is assumed to generate a strongly continuous semigroup et A in X. (E1) There is a scaling R = {Rλ }λ∈R× such that (E0 ) is invariant with respect to {Θλ }λ∈R× . (E2) There are finite numbers of strongly continuous one-parameter families of translations (j ) {Tθ }θ∈R , 1 j n, such that they are independent and (E0 ) is invariant with respect to (j ) {Tθ }θ∈R for each j . (j )

(j )

(j )

In other words, we have (16) and (17) for each Rλ and τa,θ . Let B, Dθ , and Γa,θ be (j )

(j )

the generator of Rλ , the generator of Tθ , and the derivative of τa,θ with respect to θ defined by (10), respectively. For a pair of linear operators L1 , L2 its commutator is defined by [L1 , L2 ] = L1 L2 − L2 L1 . The next assumption represents the relation between the scaling R (j ) and translations {Tθ }θ∈R . (T1) For all a, θ ∈ R and j = 1, . . . , n the inclusion (j ) (j ) (j ) τa,θ Dom(A) ∩ Dom(B) ∩ Dom Dθ ∩ Dom Γa,θ ⊂ Dom(B)

3044


holds, and there is a μj > 0 such that

(j ) (j ) (j ) (j ) B, τa,θ f + θ Γa,θ f = −aμj Dθ τa,θ f (j )

(j )

holds for f ∈ Dom(A) ∩ Dom(B) ∩ Dom(Γa,θ ) ∩ Dom(Dθ ). (T2) For any nontrivial f belonging to Dom(A) ∩ Dom(B) ∩ (1) (n) {Bf, D1 f, . . . , D1 f }

(21)

n

(j ) j =1 Dom(D1 )

the functions

are linearly independent. (j )

(j )

(j ) (j )

The relation (T1) is formally expressed as [B, τa,θ ] + θ Γa,θ = −aμj Dθ τa,θ . Especially, (j )

(j )

(j )

(j )

when τa,θ = τa for all θ ∈ R it is just [B, τa ] = −aμj D (j ) τa . As an example, if X = L2 (R), 1

1

(Rλ f )(x) = λ 2 f (λ 2 x), and (τa f )(x) = f (x + a), then (T1) formally holds with μ = 12 , i.e., we have [B, τa ] = − a2 ∂x τa . Note that Γa,θ = 0 in this case. The values μj in (T1) are related with the eigenvalues of A and they play important roles in our arguments. We set μ∗ = max{μ1 , . . . , μn , 1},

μ∗ = min{μ1 , . . . , μn , 1}.

(22)

2.2.2. Assumptions on A t We recall that A is the generator of the strongly continuous semigroup Θ(t) = Ret e(e −1)A . In our arguments it is more convenient to put assumptions on A than on A. So the additional assumptions are given for A instead of A as follows. Let σ (A) be the spectrum of A and let ress (etA ) be the radius of the essential spectrum of etA ; see [6, Chapter IV] for definitions. (A1) There is a positive number such that σ (A) ⊂ {0} ∪ {μ ∈ C | Re(μ) − }. Moreover, 0 is a simple eigenvalue of A in X. (A2) There is a positive number ζ such that ζ > max{ , μ∗ } and ress (etA ) e−ζ t . Let w0 be the eigenfunction to the eigenvalue 0 of A normalized to be 1 in X. We will see (j ) that (T1) and (T2) are sufficient conditions for D1 w0 to be an eigenfunction to the eigenvalue −μj of A if w0 possesses a suitable regularity. Thus in this case μ∗ holds by (A1). Let us introduce the eigenprojections P0,0 and Q0,0 , which are defined by P0,0 f = f, w0∗ w0 ,

Q0,0 f = f − P0,0 f

(23)

where , is a dual coupling of X and its dual space X ∗ , and w0∗ is the eigenfunction to the eigenvalue 0 of the adjoint operator A∗ in X ∗ with w0 , w0∗ = 1. From (A2) the set {μ ∈ σ (A) | Re(μ) > −ζ } consists of isolated eigenvalues with finite algebraic multiplicities; see [6, Corollary IV-2-11]. From the spectral mapping theorem we have Proposition 2.3. Assume that (A1) and (A2) hold. Then we have tA e Q0,0

B(X)

for any > 0.

C e−( −)t ,

t > 0,

(24)


3045

2.2.3. Assumptions on N Finally we give the assumptions on the nonlinear operator N . For a linear operator T we denote by · Dom(T ) the graph norm of T . 1+q

(N1) N maps Dom(A) into Q0,0 X and there is q > 0 such that N (f )X Cf Dom(A) holds for any f Dom(A) 1. (N2) There are α ∈ (0, 1], β ∈ [0, 1), and 0 ∈ [0, ) such that for each t > 0 the operator N (t, ·) = etA N (·)

(25)

is a C 1+α map from Dom(A) into Q0,0 X satisfying the estimate β

N (t, f )h − N (t, g)h C 1 + t e−( −0 )t f − gα hX , X X t

(26)

for all f, g, h ∈ Dom(A) with a constant C > 0 depending only on α, β, , 0 , and M > 0 when f X + gX + hX M. Here N (t, f ) is a Fréchet derivative of N (t, ·) at f . (j ) (j ) (N3) There is a dense set W in X such that λRλ N = N Rλ and τa,θ N = N τa,θ hold in W × + for any λ ∈ R , a ∈ R , θ ∈ R, and j . Note that from (N1) and (N2), N (t, 0) = 0 and N (t, 0) = 0 follow. Moreover, for each t > 0, N(t, ·) is extended as a C 1+α map from X into Q0,0 X. Let A−1 be the generator of the semigroup etA in the negative order space X−1 : · X−1 = (−A + I )−1 ·X .

X−1 = Dom (−A + I )−1 ,

(27)

Note that the domain of A−1 is just X; see [6, II-5]. Then the Laplace formula −1+θ

(−A + I )

1 = Γ (1 − θ )

∞

t −θ e−t e−tA dt,

θ ∈ [0, 1),

(28)

0

with the Euler Γ function Γ (1 − θ ) and the estimates in (N2) lead to the following lemma. Lemma 2.2. The nonlinear operator N (·) can be extended as a C 1+α map from X into X−1 and its Fréchet derivative N is given by 1 (−A−1 + I )1−γ N (f ) = Γ (1 − γ )

∞

t −γ e−t N (t, f ) dt,

(29)

t > 0,

(30)

0

for any γ ∈ [0, 1 − β). Moreover, the equality etA−1 N (f ) = N (t, f ),

holds for any f ∈ X. Especially, etA−1 N (f ) is extended as a bounded linear operator from X into Q0,0 X and satisfies the estimate

3046


β tA e −1 N (f )h − etA−1 N (g)h C 1 + t e−( −)t f − gα hX , X X t

(31)

for any > 0. Here the constant C is independent of f X , gX , hX , and t > 0. Proof. We first note that if f ∈ X−1 then (−A−1 + I )θ f = (−A + I )θ f , and hence (−A−1 + I )−θ f = (−A + I )−θ f for any f ∈ X. Moreover, if f ∈ X then etA−1 f = etA f . From N (t, f ) ∈ X and (N2) the right-hand side of (29) makes sense for any f ∈ X. By the 1 density argument, it suffices to prove (29) for any f ∈ Dom(A). Set Cγ = Γ (1−γ ) . For any f, h ∈ Dom(A) we have ∞ 1−γ −γ −t

t e N (t, f )h dt N (f + h) − N (f ) − Cγ (−A−1 + I )

X−1

0

∞ −γ −γ −t tA

= Cγ (−A + I ) t e e N (f + h) − N (f ) − N (t, f )h dt

X

0

∞ −γ −γ −t

= Cγ (−A + I ) t e N (t, f + h) − N (t, f ) − N (t, f )h dt

X

0

∞ C

t −γ e−t

1+t t

β 1+α dt hX

0 1+α ChX .

So the Fréchet derivative N (f ) : X → X−1 is given by the right-hand side of (29). To prove (30) we note that etA−1 N (s, f ) = etA N (s, f ) = N (t + s, f ) = esA N (t, f ) for t, s > 0 which can be seen from the density argument and the semigroup property of etA . Then we observe from (2.2) that

e

tA−1

∞

N (f ) = Cγ (−A−1 + I )

1−γ

s −γ e−s etA−1 N (s, f ) ds

0

∞ = Cγ (−A + I )

1−γ

s −γ e−s esA N (t, f ) ds

0

= N (t, f ). Hence (30) holds. The estimate (31) follows from the semigroup property e(t+t0 )A−1 N (f ) = etA−1 et0 A−1 N (f ), Proposition 2.3, (30), and (N2). We omit the details here. This completes the proof of the lemma. 2


3047

2.3. Main results Let us state the main results in this paper. Due to the nonlinearity, we only deal with sufficiently small initial data and solutions. The first result gives the existence of self-similar solutions to (E). Theorem 2.1. Assume that (E1), (A1), (N1), (N2), and (N3) hold. Let {Θλ }λ∈R× be the scaling induced by R in (E1), and q, α be the numbers in (N1), (N2). Then there is a number δ0 > 0 such that the following statement holds. There is a family of self-similar solutions {R 1 Uδ }|δ|δ0 t

to (E) with respect to {Θλ }λ∈R× such that Uδ is C 1+α in X with respect to δ and written in the form Uδ = δw0 + vδ for some vδ ∈ Q0,0 X with vδ Dom(A) C|δ|1+q . The second result is on the existence of time global solutions to (E) and their self-similar asymptotics at time infinity. Theorem 2.2. Assume that (E1), (A1), (A2), (N1), (N2), and (N3) hold. Let be the number in (A1). If Ω0 X is sufficiently small, then there is a unique mild solution Ω(t) ∈ C([0, ∞); X) to (E) with initial data Ω0 such that

R1+t Ω(t) − Uδ C(1 + t)− 2 Ω0 − Uδ X , X

t > 0.

(32)

Here δ = Ω0 , w0∗ and Uδ is the function in Theorem 2.1. Remark 2.2. If there is a Banach space Y such that X is continuously embedded in Y and if Rλ f Y = K(λ)f Y holds with a constant K(λ) satisfying supλ1 K(λ) < ∞, then from (32) we have Ω(t) − R Since R

1 1+t

1 1+t

1 1 Uδ Y = K( 1+t )R1+t Ω(t) − Uδ Y CK( 1+t )(1 + t)− 2 Ω0 − Uδ X .

1 Uδ Y = K( 1+t )Uδ Y , R

1 1+t

Uδ gives an asymptotic profile of Ω(t) at t 1.

The estimate (32) in Theorem 2.2 implies that solutions are approximated by the self-similar

solution in large time with accuracy up to O(t − 2 ). In view of Proposition 2.3, the rate O(t − 2 ) could be improved but in general at most up to O(t − + ) for any > 0. Our aim is to present an abstract method to capture more precise asymptotic profiles of solutions by making use of symmetries of equations, translation and scaling invariances. Especially, in many applications our method gives a suitable shift of the self-similar solution as an asymptotic approximation with accuracy beyond O(t − ). For y = (y1 , . . . , yn+1 ) ∈ Rn+1 we define the shift operator (1)

(n)

S(y; f ) = τy1 ,1+yn+1 · · · τyn ,1+yn+1 R

1 1+yn+1

f.

(33)

Note that if O ⊂ Rn+1 is a sufficiently small open ball centered at the origin, then S(y; f ) is a continuous map from O to X. The following lemma represents the relations between symmetries of (E0 ) and the operator A. Lemma 2.3. Assume that (E1), (E2), (T1), (T2), and (A1) hold. Let w0 be the eigenfunction for the eigenvalue 0 of A in (A1) with w0 X = 1. Suppose that S(·; w0 ) : O → X is C 1 . Then Bw0

3048

Y. Kagei, Y. Maekawa / Journal of Functional Analysis 260 (2011) 3036–3096 (j )

and D1 w0 are eigenfunctions of A for the eigenvalues −1 and −μj , respectively. Moreover, (j ) w0 , Bw0 and D1 w0 , j = 1, . . . , n, are linearly independent. If in addition (A2) holds, then the set {μ ∈ σ (A) | Re(μ) −μ∗ } with μ∗ = max{1, μ1 , . . . , μn } consists of finite numbers of eigenvalues with finite algebraic multiplicities (note that the relation μ∗ holds by (A1) and Lemma 2.3). Let E0 be the total eigenprojection to the eigenvalues {μ ∈ σ (A) | Re(μ) −μ∗ }, that is, 1 E0 = (λ − A)−1 dλ, (34) 2πi γ˜

where γ˜ is a suitable curve around {μ ∈ σ (A) | Re(μ) −μ∗ }. Set e0,0 = w0 ,

e0,n+1 = c0,n+1 Bw0 ,

(j )

e0,j = c0,j D1 w0 ,

j = 1, . . . , n.

(35)

Here c0,j is taken as e0,j X = 1. Then {e0,j }n+1 j =0 forms a part of the basis of the generalized

∗ }n+1 ⊂ X ∗ which forms a part of the basis eigenspace E0 X = {E0 f | f ∈ X}. So there are {e0,j j =0 of the generalized eigenspace associated with the eigenvalues {μ ∈ σ (A∗ ) | Re(μ) −μ∗ } to ∗ = δ , where , is a dual coupling the adjoint operator A∗ and satisfies the relations e0,j , e0,k jk ∗ (= w ∗ ) is the of X and its dual space X ∗ , and δj k is kronecker’s delta. By (A1) at least e0,0 0 eigenfunction for the simple eigenvalue 0 of A∗ . We set the projections as

∗ P0,j f = f, e0,j e0,j , P0 f =

n+1

Q0,j f = f − P0,j f, P0,j f,

0 j n + 1,

Q0 f = f − P0 f.

(36) (37)

j =0

Note that P0 X is a subset of E0 X, and, in general, P0 X does not coincide with E0 X. Let −ν0 be the growth bound of etQ0 AQ0 , that is,

−ν0 = inf μ ∈ R ∃Cμ > 0 s.t. etQ0 AQ0 f X Cμ eμt f X , ∀f ∈ Q0 X .

(38)

Then we always have

ν0 ζ,

(39)

where and ζ are the numbers in (A1) and (A2). Next we set H (y0 , y; Uδ ) = S(y; Uδ+y0 ).

(40)

Then H (y0 , y; Uδ ) is continuous from (−δ0 + δ, δ0 − δ) × O ⊂ Rn+2 to X for each δ ∈ (−δ0 , δ0 ). Set μ0 = 0 and μn+1 = 1. We will show that each −μj is an eigenvalue of the linearized operator A − N (Uδ ); see Section 6 for the precise realization of A − N (Uδ ). Especially, 0(= μ0 ) is shown to be a simple eigenvalue by the general perturbation theory. If ν0 μ∗ and each −μj is also a semisimple eigenvalue then we can derive the analogous estimate with (2) in the abstract settings. The main contribution of this paper is as follows.


3049

Theorem 2.3. Set μn+1 = 1. Assume that (E1), (E2), (T1), (T2), (A1), (A2), and (N1)–(N3) hold. Suppose that S(y; w0 ) is C 1 near y = 0 and H (y0 , y; Uδ ) is C 1+γ near (y0 , y) = (0, 0) for some γ > 0. Let Ω(t) be the mild solution in Theorem 2.2 with δ = 0 and let ν0 be the number

in (38). Assume that ν0 μ∗ and {−μj }n+1 j =1 are semisimple eigenvalues of A − N (Uδ ). Then ∗ n+1 such that there exist η(δ) ∈ R and y ∈ R ∗ yn+1 y1∗ yn∗ −ν0 +η(δ)+ R1+t Ω(t) − S ,..., , ; Uδ C (1 + t) μ μ n 1 (1 + t) (1 + t) 1+t

(41)

X

holds for all > 0 and t 1. Here η(δ) satisfies limδ→0 η(δ) = 0 and C is independent of t 1. Especially, if ν0 > μ∗ and |δ| is sufficiently small then {−μj }n+1 j =1 are semisimple eigen

values of A − N (Uδ ), and thus (41) holds in this case. Remark 2.3. The distribution of the spectrum of A assumed in Theorem 2.3 is visualized in Fig. 1. When ν0 μ∗ we can take = μ∗ := min1j n+1 {μj } by (A1) and the definitions of ν0 . So ν0 μ∗ μ∗ = > 0 holds, and the asymptotic profile is improved by (41) if ν0 > μ∗ . We will give some examples in Section 7 such that ν0 > μ∗ holds. Remark 2.4. The value of η(δ) in Theorem 2.3 is determined by the spectrum of A − N (Uδ ). Indeed, we will show that there exists an η(δ) ∈ R with limδ→0 η(δ) = 0 such that

σ A − N (Uδ ) ⊂ {−μj }n+1 j =0 ∪ μ ∈ C Re(μ) −ν0 + η(δ) .

(42)

The number η(δ) in Theorem 2.3 is nothing but η(δ) in (42). Moreover, in Theorem 2.3 if in addition ζ > ν0 and {μ ∈ σ (A − N (Uδ )) | Re(μ) −ν0 + η(δ)} consists of semisimple eigenvalues, then we can take = 0 in (41); see Remark 6.3 for details. In Theorem 2.3 we consider the shifts of Uδ with respect to both translations and scaling. In fact, we can also consider the shifts of Uδ with respect to only translations under weaker assumptions on A. Set μ˜ ∗ = max{μ1 , . . . , μn },

(43)

and P˜ 0 f =

n

˜ 0 f = f − P˜ 0 f. Q

P0,j f,

(44)

j =0 ˜

˜

Let −˜ν0 be the the growth bound of et Q0 AQ0 , that is,

˜ ˜ ˜ 0X . −˜ν0 = inf μ ∈ R ∃Cμ > 0 s.t. et Q0 AQ0 f X Cμ eμt f X , ∀f ∈ Q Instead of (A2) we consider the case (A2)’ There is ζ > max{ , μ˜ ∗ } such that ress (etA ) e−ζ t .

(45)

3050


Fig. 1. The distribution of the spectrum σ (A) assumed in Theorem 2.3. By (A2) the continuous spectrum is included in the gray region, and there are only isolated eigenvalues with finite multiplicity in {μ ∈ C | Re(μ) > −ζ } (black dots in the figure). Each −μj is shown to be an eigenvalue of A by Lemma 2.3. The definition of ν0 and the assumption ν0 μ∗ imply {μ ∈ σ (A) | Re(μ) > −ν0 } = {−μj | μj < ν0 , 0 j n + 1}.

Since μ∗ μ˜ ∗ , (A2)’ is weaker than (A2) in general. For y˜ = (y1 , . . . , yn ) ∈ Rn let us define ˜ y; the shift operator S( ˜ f ) by (1) (n) ˜ y; S( ˜ f ) = S(y, ˜ 0; f ) = τy1 ,1 · · · τyn ,1 f.

(46)

Then we have Theorem 2.4. Assume that (E1), (E2), (T1), (T2), (A1), (A2)’, and (N1)–(N3) hold. Suppose that S(y; w0 ) is C 1 near y = 0 and H (y0 , y; Uδ ) is C 1+γ near (y0 , y) = (0, 0) for some γ > 0. Let Ω(t) be the mild solution in Theorem 2.2 with δ = 0 and let ν˜ 0 be the number in (45). Assume ˜ ∈R that ν˜ 0 μ˜ ∗ and {−μj }nj=1 are semisimple eigenvalues of A−N (Uδ ). Then there exist η(δ) n and y˜ ∈ R such that y1∗ yn∗ −˜ν0 +η(δ)+ ˜ R1+t Ω(t) − S˜ , . . . , ; U δ C (1 + t) μ μ (1 + t) 1 (1 + t) n

(47)

X

holds for all > 0 and t 1. Here η(δ) ˜ satisfies limδ→0 η(δ) ˜ = 0, C is independent of t 1. Especially, if ν˜ 0 > μ˜ ∗ and |δ| is sufficiently small then {−μj }nj=1 are semisimple eigenvalues of A − N (Uδ ), and thus (47) holds in this case. Remark 2.5. From (38), (45), and Lemma 2.3 we have ν0 ν˜ 0 and 1 ν˜ 0 . Hence, in general, the shift of self-similar solutions with respect to both translations and scaling, (41), can give more precise asymptotic profile than the shift with respect to only translations, (47), if (A2) is satisfied. A typical example such that ν0 > ν˜ 0 follows is the viscous Burgers equations (1-B); see Section 7.1. Even when ν0 = ν˜ 0 , the rate in (41) is better than the one in (47) if η(δ) is negative


3051

∗ and yn+1 = 0. Indeed, we first note that ν0 = ν˜ 0 = 1 in this case, for ν0 μ∗ is assumed in Theorem 2.3 and 1 ν˜ 0 . Then if η(δ) < 0 we have from (41),

R1+t Ω(t) − S˜

y1∗ yn∗ ,..., ; Uδ (1 + t)μ1 (1 + t)μn X ∗ ∗ ∗ yn+1 y1 y1∗ yn yn∗ ; Uδ S ,..., , 0; Uδ − S ,..., , (1 + t)μ1 (1 + t)μn (1 + t)μ1 (1 + t)μn 1 + t X ∗ ∗ ∗ yn+1 y1 yn − R1+t Ω(t) − S (1 + t)μ1 , . . . , (1 + t)μn , 1 + t ; Uδ X

∗ | |yn+1

1+t

BUδ X − C(1 + t)−1−c

for some c > 0. Here we used the C 1+γ regularity of S(y; Uδ ) near y = 0 and the fact that ∂yn+1 S(y; Uδ )|y=0 = BUδ . Hence η(δ) ˜ + 0 must hold in (47), which proves the above assertion. The examples such that ν0 = ν˜ 0 = 1 and η(δ) < 0 hold are given in Section 7.2 and Section 7.3. The proof of Theorem 2.4 is just a simple modification of the one of Theorem 2.3, so the details will be omitted; see Section 6.4. Applications of main results to concrete nonlinear PDEs will be given in Section 7. 3. Invariant property and reduction by similarity transform In this section we prove that (E) is invariant with respect to the scaling R and the translations j = 1, . . . , n under the assumptions stated in the previous section. We also derive an equation by the “similarity transform” which enables us to analyze large time behaviors of mild solutions to (E) in terms of a stability problem of stationary solutions to the new equation. s We first consider the relation between et A N and etA N . Since esA = Res e(e −1)A we have for f ∈ Dom(A), (j ) {Tθ }θ∈R ,

N (t, f ) := et A N (f ) = R

1 1+t

elog(1+t)A N (f ) = R

1 1+t

N log(1 + t), f .

(48)

Since there is a sequence {fn } ⊂ X such that fn ∈ Dom(A) and fn → f in X, N (t, ·) can be extended as a C 1+α map on X and the above equality holds for any f ∈ X. Moreover, by density arguments we have Lemma 3.1. Assume that (E1), (E2), (A1), (N2), and (N3) hold. Then it follows that λRλ N (λt, f ) = N (t, Rλ f ), t > 0, (j ) (j ) τa,t+θ N (t, f ) = N t, τa,θ f , t > 0, for any f ∈ X, λ ∈ R× , a ∈ R+ , θ ∈ R, and j .

(49) (50)

3052


Proof. Here we give a proof for the first equality only, since the second one is shown in the same way. From (16) and (N3) it follows that λRλ N (λt, f ) = limn→∞ λRλ eλt A N (fn ) = limn→∞ et A λRλ N (fn ) = limn→∞ et A N (Rλ fn ) = N (t, Rλ f ). This completes the proof. 2 t From (N2) and (48) we observe that the term s e(t−τ )A N (Ω(τ )) dτ in (13) makes sense t for all Ω(t) ∈ C((0, ∞); X) by rewriting it as s N (t − τ, Ω(τ )) dτ . Moreover, if Ω(t) ∈ t C([0, ∞); X) then we can show limt→0 0 N (t − s, Ω(s))X ds = 0. Hence as a corollary of Lemma 3.1 we have Corollary 3.1. Assume that (E1), (E2), (A1), (N2), and (N3) hold. Then (E) is invariant with (j ) respect to {Θλ }λ∈R× and {Tθ }θ∈R , j = 1, . . . , n. Proof. Let Ω(t) ∈ C([0, ∞); X) be a mild solution to (E) with initial data Ω0 . Then Θλ (Ω)(t) = Rλ Ω(λt) ∈ C([0, ∞); X) and we have

Θλ (Ω)(t) = Rλ e

λt A

λt Ω 0 − Rλ

N λt − s, Ω(s) ds

0

= e t A Rλ Ω 0 −

t

λRλ N λ t − s , Ω λs ds

0

=e

tA

t Rλ Ω 0 −

N t − s , Θλ (Ω) s ds .

0 (j )

This implies that (E) is invariant with respect to {Θλ }λ∈R× . The invariance for {Tθ }θ∈R is proved in a similar manner, so we omit the details. This completes the proof. 2 Next we consider the following integral equation: t u(t) = e u0 − tA

N t − s, u(s) ds.

(51)

0

The main result in this section is as follows. Lemma 3.2. Assume that (E1), (A1), (N2), and (N3) hold. If Ω(t) ∈ C([0, ∞); X) is a mild solution to (E) with initial data Ω0 , then u(t) = Ret Ω(et − 1) is a mild solution to (51) with initial data Ω0 . Conversely, if u(t) ∈ C([0, ∞); X) is a mild solution to (51) with initial data u0 , then Ω(t) = R 1 u(log(1 + t)) is a mild solution to (E) with initial data u0 . Moreover, u ∈ X is 1+t a stationary solution to (51) if and only if R 1 u is a self-similar solution to (E). t


3053

Proof. Here we give a proof for the last assertion only. For any t > s > 0 and u ∈ X we set

F (t, s; u) = R 1 e

log st A

t

log t u − R1

N (log t − τ, u) dτ.

t

log s

From the definition of etA and (48) we have N(t, f ) = Ret N et − 1, f .

elog s A = R t e( s −1)A , t

t

s

Hence F (t, s; u) is written as

F (t, s; u) = e

(t−s)A

t R1 u − s

1 t −r R1 N , u dr r r r

s

=e

(t−s)A

t R1 u −

N (t − r, R 1 u) dr.

s

r

s

In the last line we used Lemma 3.1. Note that u ∈ X is a stationary solution to (51) if and only if u satisfies

u=e

log st A

log t u−

N (log t − τ, u) dτ = Rt F (t, s; u),

log s

for any t > s > 0. From the above calculation this is equivalent with

R 1 u = F (t, s; u) = e t

(t−s)A

t R1 u −

N (t − r, R 1 u) dr,

s

r

s

i.e., R 1 u is a self-similar solution to (E). This completes the proof. t

2

4. Existence of self-similar solutions In this section we prove the existence of self-similar solutions to (E), which is equivalent with the existence of stationary solutions to (51) by Lemma 3.2. For this purpose we look for a stationary solution Uδ of the form Uδ = δw0 + v where v belongs to Q0,0 X. Then the equation for v is −Av + N (δw0 + v) = 0.

(52)

3054


Here we used the fact that Aδw0 = 0. Noting that N (·) ∈ Q0,0 X, (52) in Q0,0 X can be solved by the usual contraction mapping theorem for the map Φ(δ, f ) = −(−A)−1 N (δw0 + f ).

(53)

The main result of this section is as follows. Theorem 4.1. Assume that (E1), (A1), (N1), (N2), and (N3) hold. Then there is a positive number δ0 such that if |δ| δ0 then there is a unique solution vδ to (52) in Q0,0 X satisfying the estimate vδ Dom(A) C|δ|1+q .

(54)

Moreover, vδ is C 1+α in X with respect to δ, |δ| < δ0 . Thus, we have a self-similar solution to (E) which takes the form R 1 Uδ with Uδ = δw0 + vδ . t

Proof. Let δ0 be a sufficiently small positive number. We show that Φ(δ, ·) is a contraction mapping on the closed ball Bδ0 = {f ∈ Q0,0 X | f Dom(A) δ0 }, if |δ| δ0 1. Indeed, from (N1) we easily see that Φ(δ, f )

Dom(A)

1+q C N (δw0 + f )X Cδ0 ,

if f ∈ Bδ0 .

Moreover, since the Laplace formula yields the representation ∞ 1 Φ(δ, f ) − Φ(δ, g) =

N t, δw0 + θf + (1 − θ )g (f − g) dθ dt,

0 0

we have from (N2) that Φ(δ, f ) − Φ(δ, g)X Cδ0α f − gX . Combining these above, we observe that Φ(δ, ·) is a contraction mapping on Bδ0 if |δ| δ0 and δ0 is small enough. Thus there is a unique fixed point of Φ(δ, ·) in Bδ0 . Let vδ be the fixed point. Then 1+q vδ Dom(A) N (δw0 + vδ )X C|δ|1+q + C|δ|q vδ X . Hence if |δ| is sufficiently small, then vδ Dom(A) 2|δ|1+q . Finally, since Φ(δ, ·) is C 1+α from X into Q0,0 X, by the uniform contraction mapping principle, vδ is C 1+α with respect to δ in X. This completes the proof. 2 5. Global solvability of the evolution equations In this section we prove the global existence of mild solutions to (51) for sufficiently small initial data. Again by Lemma 3.2 this means the existence of time global solutions to (E) for small initial data. The next result shows that the self-similar solution Uδ plays important roles for the large time behavior of solutions to (51).


3055

Theorem 5.1. Assume that (E1), (A1), (A2), (N1), (N2), and (N3) hold. Set δ = u0 , w0∗ . If u0 X is sufficiently small, then there is a unique mild solution u(t) ∈ C([0, ∞); X) to (51) such that

sup e 2 t u(t) − Uδ X Cu0 − Uδ X .

(55)

t>0

Proof. We first note that the smallness of u0 X leads to the smallness of |δ|, which guarantees the existence of the stationary solution Uδ by Theorem 4.1. Let us consider the equation for ω(t) = u(t) − Uδ ∈ Q0,0 X: ∂t ω − Aω = −N (Uδ + ω) + N (Uδ ),

t > 0.

(56)

The associated integral equation in X is t ω(t) =: Υ (ω) = e ω0 − tA

N t − s, Uδ + ω(s) − N (t − s, Uδ ) ds,

(57)

0

where ω0 = u0 − Uδ . Clearly this is equivalent with (51). For θ ∈ [0, 1] and the Banach space B let us introduce the function space Cθ ([0, ∞); B) as follows:

Cθ [0, ∞ ; B) = f ∈ C [0, ∞); B f Cθ (B) := sup eθt f (t)B < ∞ .

(58)

t>0

Eq. (57) can be solved by the usual contraction mapping theorem on the ball BR = {f ∈ C ([0, ∞); Q0,0 X) | f C (X) R}, when R = 2Υ (0)C (Q0 X) . Note that Υ (0)(t) = etA ω0 . 2

2

Since

2

N t − s, Uδ + ω1 (s) − N t − s, Uδ + ω2 (s) 1 =

N t − s, Uδ + σ ω1 + (1 − σ )ω2 dσ ω1 (s) − ω2 (s) ,

0

from Lemma 2.2 we have the estimates for Υ such as Υ (ω1 ) − Υ (ω2 ) (t) X t C

e 0

− 3 4 (t−s)

1+t −s t −s

β

α Uδ X + ω1 (s)X + ω2 (s)X ω1 (s) − ω2 (s)X ds

C |δ|α + R α e− 2 t ω1 − ω2 C (X) . 2

Since ω0 ∈ Q0,0 X, the function Υ (0)(t) is estimated as Υ (0)C (X) Cu0 − Uδ X by Propo2

sition 2.3. Hence Υ is a contraction mapping on BR for sufficiently small |δ| and R. Note that |δ|

3056


and R can be taken small enough if u0 X is sufficiently small. Thus there is a unique solution to (57) in BR if the initial data is sufficiently small. The proof is complete. 2 As a corollary of Theorem 5.1 we have from Lemma 3.2: Corollary 5.1. Assume that (E1), (A1), (A2), (N1), (N2), and (N3) hold. Set δ = Ω0 , w0∗ . If Ω0 X is sufficiently small, then there is a unique mild solution Ω(t) ∈ C([0, ∞); X) to (E) such that

R1+t Ω(t) − Uδ C(1 + t)− 2 Ω0 − Uδ X , X

t > 0.

(59)

This corollary immediately leads to Theorem 2.2. 6. More precise asymptotic profile by shift In this section we study the asymptotic behavior of solutions obtained by Theorem 5.1 in more details by assuming that (E) possesses an invariance with respect to translations. The key step in our arguments is to introduce a linear transform with n + 2 parameters y = (y0 , y), y = (y1 , . . . , yn+1 ): H (y; Uδ ) = S(y; Uδ+y0 ),

(60)

where S(y; f ) = τy(1) · · · τy(n) R n ,1+yn+1 1 ,1+yn+1

1 1+yn+1

(61)

f.

Proposition 6.1. Let f ∈ Dom(A). Assume that there is an open ball O ⊂ Rn+1 centered at the origin such that S(·; f ) : O → X is C 1 . Then S(y; f ) ∈ Dom(A) ∩ Dom(B) ∩ (k) Dom(D1+yn+1 ) for all k = 1, . . . , n. Moreover, for j = 1, . . . , n and y = (y1 , . . . , yn+1 ) ∈ O (k)

(l)

we have S(y; f )|y1 =···=yj =0 ∈ Dom(A) ∩ Dom(B) ∩ Dom(D1+yn+1 ) ∩ Dom(Γyl ,1+yn+1 ) for all k = 1, . . . , n and l = 1, . . . , j . Proof. For j = 0, 1, . . . , n we set Sj (yj +1 , . . . , yn+1 ; f ) = S(y; f )|y1 =···=yj =0 if j 1 and S0 (y; f ) = S(y; f ). Since S(y; f ) is assumed to be C 1 in O, each Sj is differentiable if y ∈ O. (k) Especially, we have Sj ∈ Dom(D1+y ) for all k = 1, . . . , n by the independent property of n+1 (j )

(l)

(l)

(j )

τaj ,θ τal ,θ = τal ,θ τaj ,θ for 1 j, l n. Now we will show by the backward induction on j that each Sj belongs to Dom(A) ∩ (k) (l) Dom(B) ∩ Dom(D1+yn+1 ) ∩ Dom(Γyl ,1+yn+1 ) for all k = 1, . . . , n and l = 1, . . . , j when y ∈ O. Let us consider Sn (yn+1 ; f ) = R 1 f . From the C 1 regularity of Sn we have f ∈ Dom(B) 1+yn+1

and so is true for Sn (yn+1 ; f ). Then from the relation of e(e −1)A f = Re−t etA f and the assumption of f ∈ Dom(A), we have f ∈ Dom(A). So from the invariant property of Rλ eλt A = et A Rλ , we also have Sn (yn+1 ; f ) = R 1 f ∈ Dom(A). Since both S(y; f )|yi =0,i=l,n+1 = t

1+yn+1

(l)

τyl ,1+yn+1 R

1 1+yn+1

f and R

1 1+yn+1

f are C 1 with respect to yn+1 for each l = 1, . . . , n, we can


check that R

1 1+yn+1

3057

(l)

f ∈ Dom(Γyl ,1+yn+1 ). This gives the desired regularity for Sn . Suppose that

(k) ) ∩ Dom(Γy(l) ) for all k = 1, . . . , n Sj +1 belongs to Dom(A) ∩ Dom(B) ∩ Dom(D1+y l ,1+yn+1 n+1 and l = 1, . . . , j + 1. Then we have from (T1) that (j +1)

Sj (yj +1 , . . . , yn+1 ; f ) = τyj +1 ,1+yn+1 Sj +1 (yj +2 , . . . , yn+1 ; f ) ∈ Dom(B). Furthermore, from the invariant property of (17) we see (j +1)

(j +1)

et A τyj +1 ,1+yn+1 Sj +1 = τyj +1 ,t+1+yn+1 et A Sj +1 . (j +1)

(j )

Hence, τyj +1 ,1+yn+1 Sj +1 ∈ Dom(A) from the assumption of Sj +1 ∈ Dom(A)∩Dom(Γyj ,1+yn+1 ). (l)

Now it remains to show Sj ∈ Dom(Γyl ,1+yn+1 ) for all l = 1, . . . , j when j 1. But this can Sj (yj +1 , . . . , yn+1 ; f ) = S(y; f )|yi =0,i=l,j +1,...,n , be verified from the fact that both τy(l) l ,1+yn+1 1 l = 1, . . . , j , and Sj are C with respect to yn+1 . The regularity of S0 follows from the ones of S1 , similarly. This completes the proof. 2 Let H (·; Uδ ) : (−δ0 + δ, δ0 − δ) × O → X be C 1 , where O ⊂ Rn+1 is an open ball centered at the origin. Then we observe that ∂y0 H (0; Uδ ) = ∂δ Uδ , (l)

∂yl H (0; Uδ ) = D1 Uδ ,

(62) 1 l n,

∂yn+1 H (0; Uδ ) = BUδ .

(63) (64)

Our aim is to determine the parameters y(t) = (y0 (t), y(t)) so that u(t) − H (y(t); Uδ )X decays faster than u(t) − Uδ X . Let us formulate our problem precisely. Recall that A−1 is the generator of etA in the negative order space X−1 with the domain X. We consider the equation for v(t) = u(t) − H (y(t); Uδ ) in X−1 where y(t) ∈ Rn+2 is determined later. Then we obtain the equation in X−1 such as ∂t v − A−1 − N (Uδ ) v = Tδ v + Fδ (v) + J (Vδ ), where Vδ (t) = H y(t); Uδ , Tδ v(t) = N (Uδ ) − N Vδ (t) v(t), 1 Fδ (v)(t) = − 0

N Vδ (t) + τ v(t) − N Vδ (t) v(t) dτ,

J (Vδ )(t) = −∂t Vδ (t) + A−1 Vδ (t) − N Vδ (t) ,

(65)

3058


and ∞

N (f ) = (−A−1 + I )

e−t N (t, f ) dt.

0

Let us consider the linearized operator A−1 − N (Uδ ) in X−1 with the domain

f ∈ X−1 A−1 − N (Uδ ) f ∈ X−1 .

It is not difficult to see that Dom(A−1 − N (Uδ )) = X. Indeed, by Lemma 2.2 we can write ∞

N (Uδ ) = Cθ (−A−1 + I )

1−θ

t −θ e−t N (t, Uδ ) dt

(66)

0

with θ ∈ [0, 1 − β). Thus the interpolation inequality yields that N (Uδ ) is relatively A−1 bounded in X−1 with the bound 0, and hence, Dom(A−1 − N (Uδ )) = Dom(A−1 ) = X. Let Lδ be the part of A−1 − N (Uδ ) in X, that is,

Dom(Lδ ) = f ∈ X A−1 − N (Uδ ) f ∈ X , Lδ f := A−1 − N (Uδ ) f.

(67)

Then by using Lemma 2.2 we can apply the perturbation theory of Desch–Schappacher to Lδ ; see [6, III-3-3]. Lemma 6.1. The operator Lδ above generates the strongly continuous semigroup etLδ in X, and satisfies the equation t e

tLδ

=e

tA

−

e(t−s)A−1 N (Uδ )esLδ ds.

(68)

0

Moreover, the estimate etLδ f X C(t)f X follows with a constant C(t) 1 which is independent of δ such as |δ| δ0 . Proof. From Lemma 2.2 we have for any f (s) ∈ C([0, t0 ); X), t0 t0 1 + t0 − s β − (t0 −s) (t0 −s)A−1

α f (s) ds e N (U )f (s) ds C|δ| e 2 δ X t0 − s 0

X

0

C(t0 )|δ0 |α sup f (s)X , 0<s μj then −μj is a semisimple eigenvalue of Lδ with multiplicity {μk | μk = μj }. For this purpose we first observe that −μj is a semisimple eigenvalue of A, and hence, of A−1 , with multiplicity {μk | μk = μj } when ν0 > μj . Indeed, since ν0 > μj the space Ker(A + μj I ) is spanned by {e0,k | μk = μj }, otherwise there is an eigenfunction in Q0 X (see (37) for the definition of Q0 ) of the eigenvalue −μj to Q0 AQ0 , which contradicts with ν0 > μj by (38) and the spectral mapping theorem. Hence the geometric multiplicity of the eigenvalue −μj to A is {μk | μk = μj }. Assume that there is a nontrivial function f ∈ such that f ∈ / Ker(A + μj I ). Then since (A + μj I )f ∈ Ker(A + μj I ) we have Ker(A + μj I )2 (A + μj I )f = μk =μj ak e0,k for some ak ∈ C, which yields (Q0 AQ0 + μj I )Q0 f = 0. Since f∈ / Ker(A + μj I ) we have Q0 f = 0. Hence −μj must be an eigenvalue of Q0 AQ0 , which again contradicts with ν0 > μj by (38). Thus we have Ker(A + μj I )2 = Ker(A + μj I ). On the other hand, since the rank of the eigenprojection around the eigenvalue −μj of A is finite by (A2) and [6, Corollary IV-2-11], −μj must be a pole of the resolvent of A; see [6, Section IV-1-17]. Thus by [19, Remark A.2.4], −μj is a semisimple eigenvalue of A. Finally we prove that the eigenvalue −μj of Lδ is semisimple if ν0 > μj and |δ| is sufficiently small. From Corollary 6.1 −μj is an isolated eigenvalue with finite algebraic multiplicity. So it suffices to show Ker(Lδ + μj I )2 = Ker(Lδ + μj I ) as above. We note that, since Lδ is a part of A−1 − N (Uδ ) in X, the eigenvalues of Lδ are also eigenvalues of A−1 − N (Uδ ) to

3062


which the general perturbation theory for linear operators is easily applied. For the operator A−1 − N (Uδ ) the rank of the eigenprojection around −μj is invariant if |δ| is sufficiently small, so it is {μk | μk = μj } < ∞ and −μj is a pole of the resolvent of A−1 − N (Uδ ) by [6, Section IV-1-17]. On the other hand, the geometric multiplicity of the eigenvalue −μj to A−1 − N (Uδ ) is large than or equal to {μk | μk = μj }, for we have found corresponding eigenfunctions. Hence the rank of the eigenprojection around −μj coincides with the geometric multiplicity of −μj . This implies Ker(Lδ + μj I )2 = Ker(Lδ + μj I ) since Lδ is a part of A−1 − N (Uδ ) in X. This completes the proof. 2 Proposition 6.3. The map Nδ (t, ·) = etLδ N : Dom(A) → X can be extended as a C 1+α map on X and satisfies the estimate β

N (t, f )h − N (t, g)h C 1 + t e− 2 t f − gα hX , δ δ X X t

(75)

for any f, g, h ∈ X. Here Nδ (t, f ) is the Fréchet derivative of Nδ (t, ·) at f . Especially, N can be extended as a C 1+α map from X into the negative order space Dom(Lδ,−1 ) with · Dom(Lδ,−1 ) = (−Lδ + I )−1 · X , and the relation etLδ,−1 N (f ) = Nδ (t, f )

(76)

holds for any f ∈ X. Proof. The assertion that N is extended as a C 1+α map from X into Dom(Dδ,−1 ) is shown as in Lemma 2.2. Let f, g ∈ Dom(A). Then from Lemma 6.1 we have etLδ N (f )h − etLδ N (g)h

t

= e N (f )h − e N (g)h − tA

tA

e(t−s)A−1 N (Uδ ) esLδ N (f )h − esLδ N (g)h ds.

0

Then using (31), we get tL

e δ N (f )h − etLδ N (g)h

X

etA N (f )h − etA N (g)hX + C|δ|α

t 0

(1 + t − s)(1 + s) (t − s)s

β

t e 2 t etLδ N (f )h − etLδ N (g)hX , × sup 1 + t t>0 which implies from (31) that

β

3

e− 4 (t−s) e− 2 s ds


t sup 1 + t t>0

β

C sup t>0

3063

e 2 t etLδ N (f )h − etLδ N (g)hX

t 1+t

β

e 2 t etA N (f )h − etA N (g)hX

Cf − gαX hX . This proves (75). The relation (76) follows from the argument just as in the proof of (30) in Lemma 2.2. This completes the proof. 2 From Lemma 6.1 and Proposition 6.3 we convert (65) to the integral equation t v(t) = e

tLδ

e(t−s)Lδ Tδ v(s) + Fδ (v)(s) + J Vδ (s) ds.

v0 +

(77)

0

6.2. Determination of y(t) In this section we find suitable parameters y(t) = (y0 (t), . . . , yn+1 (t)) such that u(t) − H (y(t); Uδ )X decays faster than u(t) − Uδ X . For this purpose the next lemma for J (Vδ )(t) = −∂t Vδ (t) + A−1 Vδ (t) − N (Vδ (t)) with Vδ (t) = H (y(t); Uδ ) is important. Lemma 6.3. Assume that H (y; Uδ ) is C 1 near y = 0. Let y(t) ∈ C 1 ((0, ∞); Rn+2 ) with supt>0 |y(t)| 1 be given. Set μ0 = 0 and μn+1 = 1, and let μj be the number in Lemma 6.2 for j = 1, . . . , n. Then for any stationary solution Uδ of (51) with |δ| 1, we have J (Vδ )(t) = −

n+1

∂yj H y(t); Uδ · yj (t) + μj yj (t) .

(78)

j =0

Proof. From Proposition 6.1 we have H (y; Uδ ) ∈ Dom(A) ∩ Dom(B) and (j +1)

(n)

τyj +1 ,1+yn+1 · · · τyn ,1+yn+1 R

1 1+yn+1

(j ) Uδ+y0 ∈ Dom(A) ∩ Dom(B) ∩ Dom Γyj ,1+yn+1

for each j = 1, . . . , n − 1. We start from the equality ∂yn+1 H (y; Uδ ) = AH (y; Uδ ) − N H (y; Uδ ) ,

(79)

which follows by regarding yn+1 as a time variable. Indeed, since Uδ+y0 is a stationary solution to (51), R 1 Uδ+y0 is a self-similar solution to (E) from Lemma 3.2. Especially, R 1 Uδ+y0 is a mild 1 +t 2

t

solution to (E) with initial data R2 Uδ+y0 . Then from Corollary 3.1 we see a mild solution to (E) with initial data τ

(n) R U . yn , 12 2 δ+y0

(n) τ 1 R yn , 2 +t

1 1 +t 2

Uδ+y0 is

Thus again from Corollary 3.1 we have

3064

τ


(n−1) (n) τ R yn−1 , 12 +t yn , 12 +t

1 1 +t 2

Uδ+y0 is a mild solution to (E) with initial data τ

peating this, we observe that τ data τ

(1) y1 , 12

···τ

(n) R U . yn , 12 2 δ+y0

(1) y1 , 12 +t

···τ

(n) R yn , 12 +t

1 1 +t 2

(n−1) (n) τ R U . yn−1 , 12 yn , 12 2 δ+y0

Re-

Uδ+y0 is a mild solution to (E) with initial

Hence, by setting t = yn+1 +

1 2

we get (79). On the other hand, the

direct calculation yields that ∂yn+1 H (y; Uδ ) =

n j =1

−

(j −1)

(1)

(j +1)

(j )

(n)

τy1 ,1+yn+1 · · · τyj −1 ,1+yn+1 Γyj ,1+yn+1 τyj +1 ,1+yn+1 · · · τyn ,1+yn+1 R

1 1+yn+1

Uδ

1 τ (1) · · · τy(n) BR 1 Uδ . n ,1+yn+1 1+yn+1 1 + yn+1 y1 ,1+yn+1

Hence ∂yn+1 H (y; Uδ ) + =

n j =1

1 BH (y; Uδ ) 1 + yn+1 (j −1)

(1)

(j +1)

(j )

(n)

τy1 ,1+yn+1 · · · τyj −1 ,1+yn+1 Γyj ,1+yn+1 τyj +1 ,1+yn+1 · · · τyn ,1+yn+1 R

1 1+yn+1

Uδ

n

(j +1) 1 (j −1) (j ) (1) τy1 ,1+yn+1 · · · τyj −1 ,1+yn+1 B, τyj ,1+yn+1 τyj +1 ,1+yn+1 · · · 1 + yn+1

+

j =1

(n) × τyn ,1+yn+1 R 1 1+y

Uδ .

n+1

By (T1) we get (1 + yn+1 )∂yn+1 H (y; Uδ ) + BH (y; Uδ ) =−

n j =1

=−

n j =1

=−

n

(j −1)

(j )

(j )

τy(1) · · · τyj −1 ,1+yn+1 yj μj Dyj ,1+yn+1 τyj ,1+yn+1 · · · τy(n) R n ,1+yn+1 1 ,1+yn+1

1 1+yn+1

Uδ

(j )

μj yj Dyj ,1+yn+1 H (y; Uδ )

μj yj ∂yj H (y; Uδ ).

j =1 (j )

Here we used the independent property of {Tθ } assumed in (E2). Combining this with (79), we have AH (y; Uδ ) − N H (y; Uδ ) = AH (y; Uδ ) − N H (y; Uδ ) + BH (y; Uδ )


= ∂yn+1 H (y; Uδ ) − (1 + yn+1 )∂yn+1 H (y; Uδ ) −

n

3065

μj yj ∂yj H (y; Uδ )

j =1

= −yn+1 ∂yn+1 H (y; Uδ ) −

n

μj yj ∂yj H (y; Uδ ).

j =1

Then (78) easily follows. This completes the proof.

2

Now let us derive the ODEs which determines y(t). Below we assume that δ = 0. As in Lemma 6.3 we put μ0 = 0 and μn+1 = 1. Recalling Lemma 6.2, we set eδ,0 = cδ,0 ∂δ Uδ ,

(80)

(j ) eδ,j = cδ,j D1 Uδ ,

1 j n,

eδ,n+1 = cδ,n+1 BUδ .

(81) (82)

Here each cδ,j is taken so that eδ,j X = 1. Let us introduce the eigenprojections Pδ,j , j = 0, . . . , n + 1 by ∗ eδ,j . Pδ,j f = f, eδ,j

(83)

∗ }n+1 satisfy the relation e , e∗ = δ and e∗ is the eigenfunction Here the functions {eδ,j δ,l δ,j jl j =0 δ,0 ∗ is one of functions which form of the adjoint operator L∗δ for the simple eigenvalue 0, eδ,j the generalized eigenspace for the eigenvalue −μj of L∗δ when j 1. Note that if −μj is a ∗ is also an eigenfunction of L∗ for the eigenvalue −μ . semisimple eigenvalue of Lδ then eδ,j j δ Then we introduce the projection

Pδ =

n+1

Pδ,j ,

Qδ = I − Pδ .

(84)

j =0

Then from Lemma 6.2 and Proposition 6.3 we have Lemma 6.4. Let ν0 and η(δ) be the numbers in Lemma 6.2. Let Nδ (t, f ) = etLδ N (f ) be the nonlinear operator in Proposition 6.3. For any > 0 set ν,δ = ν0 − η(δ) − . Then there is a positive constant C such that Qδ etLδ f C e−ν,δ t Qδ f X X

(85)

β

Qδ N (t, f ) − N (t, g) h C 1 + t e−ν,δ t f − gα hX δ δ X X t

(86)

and

hold for any f, h and g ∈ X.

3066


Proof. We note that, although Qδ etLδ = Qδ etLδ Qδ holds, Qδ X is not invariant under the action of etLδ in general, for each −μj is not assumed to be a semisimple eigenvalue of Lδ in this lemma. From the semigroup property Qδ e(t+s)Lδ = Qδ etLδ Qδ esLδ the growth bound of Qδ etLδ is given by the spectral radious of Qδ etLδ , which is denoted by r(Qδ etLδ ). Hence it suffices to show that r(Qδ etLδ ) is less than or equal to e−tν0,δ . To prove this we recall that if ν0 > μj then −μj is a semisimple eigenvalue of Lδ with multiplicity {μk | μk = μj } by Lemma 6.2. Especially, for such μj the corresponding eigenspace is spanned by {eδ,k | μk = μj } and it must be included in Pδ X. By considering the spectral projection P δ on the eigenspace for the eigenvalues {−μj | ν0 > μj }, we may consider the problem in Q δ X where Q δ = I − P δ since Q δ X is invariant under the action of etLδ and Qδ X ⊂ Q δ X. Then from Lemma 6.2 and the spectral mapping theorem for eigenvalues, we have r Qδ etLδ r Q δ etLδ max ress Q δ etLδ , e− min{μj |ν0 μj }t , e−ν0,δ t = max ress etLδ , e− min{μj |ν0 μj }t , e−ν0,δ t .

(87)

Assume that there is no μj satisfying ν0 μj . By (74) we have ress (etLδ ) e−(ζ −η (δ))t e−(ν0 −η(δ))t . Hence r(Qδ etLδ ) e−ν0,δ t in this case by (87), which gives (85). Next we assume that there is a μj such that ν0 μj . Then ζ > μ∗ ν0 by (A2), and so we may assume that |η (δ)| + |η(δ)| < ζ − μ∗ . Hence we have

ress etLδ e−(ζ −η (δ))t < e−(ν0 −η(δ))t ,

(88)

uniformly in |δ| 1 in this case. This implies that η(δ) is determined by the behavior of σ (Lδ ) around {μ ∈ σ (A) | Re(μ) = −ν0 } which consists of finite number of eigenvalues with finite multiplicity. If there is no μj satisfying μj = ν0 then we have r(Qδ etLδ ) e−ν0,δ t from (87) and (88), and thus (85) follows. Assume that there is a μj such that μj = ν0 . In this case we need to consider how η(δ) is determined. Since the eigenvalues of Lδ are continuously depending on δ by (70), η(δ) is taken as the maximum of the following two quantities l1 , l2 when |δ| 1: ⎧ max{Re(μ(δ)) − ν0 | μ(δ) ∈ σ (Lδ )\{−ν0 } is a bifurcation from ⎪ ⎨ some eigenvalue μ(0) of A with Re(μ(0)) = −ν0 }, l1 = −∞ if the set {μ(δ) ∈ σ (Lδ )\{−ν0 } | μ(δ) is a bifurcation from ⎪ ⎩ some eigenvalue μ(0) of A with Re(μ(0)) = −ν0 } is empty, l2 =

if the rank of the total eigenprojection for the eigenvalue − ν0 of Lδ is strictly larger than {μk | μk = ν0 }, −∞ otherwise.

0

We note that the value of max{l1 , l2 } cannot be −∞; otherwise we must have {μ ∈ σ (A) | Re(μ) = −ν0 } = {−ν0 } from l1 = −∞ and its algebraic multiplicity is {μk | μk = ν0 } from l2 = −∞. Thus the eigenspace of the eigenvalues {μ ∈ σ (A) | Re(μ) = −ν0 } is just spanned by {e0,k | μk = ν0 }, which contradicts with the definition of ν0 in (38). From the definitions of l1 and l2 the inequality r(Qδ etLδ ) e−ν0,δ t holds. Indeed, if η(δ) is negative then l2 = −∞ and thus −ν0 is a semisimple eigenvalue of Lδ whose eigenspace is spanned by {eδ,k | μk = ν0 }. So from (88) we have instead of (87),


3067

r Qδ etLδ max ress etLδ , e− min{μj |ν0 1 we first note the relation Nδ (t + t0 , f ) = e(t+t0 )Lδ,−1 N (f ) = etLδ,−1 Nδ (t0 , f ) = etLδ Nδ (t0 , f ) where 0 < t0 < 12 . Here we used (76). Then we get (86) from (85) and (75). This completes the proof. 2 Let η > 0 be a sufficiently small number satisfying η > η (δ), where η (δ) is the number in Corollary 6.1. Let Eδ be the total eigenprojection for the eigenvalues {μ ∈ σ (Lδ ) | Re(μ) > −ζ + η }. Then from (77) we have t Eδ v(t) = e

(t−T )Lδ

Eδ v(T ) +

e(t−s)Lδ Eδ Tδ v(s) + Fδ (v)(s) + J Vδ (s) ds.

(89)

T

Since Eδ {Tδ v(t) + Fδ (v)(t) + J (Vδ (t))} ∈ C([0, ∞); X), Eδ v(t) is differentiable in X with respect to t and we obtain d Eδ v(t) − Lδ Eδ v(t) = Eδ Tδ v(t) + Fδ (v)(t) + J Vδ (t) . dt

(90)

Our aim is to construct (n + 2) parameters y(t) so that v(t) belongs to Qδ X for all t T with

large T . From now on we assume that {−μj }n+1 j =1 are semisimple eigenvalues of A − N (Uδ ). Then we have Pδ Lδ Qδ = 0. By the fact that Pδ Eδ = Eδ Pδ = Pδ , from (90) the requirement Pδ v(t) = 0 leads to the equation −Pδ J (Vδ ) = Pδ Tδ v + Pδ Fδ (v),

t > T,

(91)

and the initial condition at the initial time T 0, Pδ u(T ) = Pδ H y(T ); Uδ .

(92)

Eq. (91) and (92) are equivalent with the ODE system as follows. For 0 j, l n + 1, let kj,l (t), Tδ,j (t) and Fδ,j (t) be functions defined by ∗ , kj,l (t) = ∂yl H y(t); Uδ − ∂yl H (0; Uδ ), eδ,j ∗ Tδ,j (t) = Tδ v, eδ,j , ∗ . Fδ,j (t) = Fδ (v), eδ,j As in Lemma 6.3 we set μ0 = 0 and μn+1 = 1. Let Π˜ = (d˜ij )1i,j n be an n × n matrix whose component d˜ij is given by d˜ij = μj δij ,

1 i, j n.

(93)

3068


Set kj (t) = (kj,1 (t), . . . , kj,n (t)) . From (91), (62)–(64), and Lemma 6.3, we get the equation for y0 (t) such as

1 + k0,0 (t) y0 = −k0 (t) · y˜ + Π˜ μ y˜ − k0,n+1 (t) yn+1 + yn+1 + Tδ,0 (t) + Fδ,0 (t).

(94)

Similarly we have for y(t), ˜

˜ I + K(t) y˜ + Π˜ y˜ = −k˜ (1) (t)y0 − k˜ (n) (t) yn+1 + yn+1 + T˜δ (t) + F˜ δ (t),

(95)

˜ where T˜δ (t) = (Tδ,1 (t), . . . , Tδ,n (t)) , F˜ δ (t) = (Fδ,1 (t), . . . , Fδ,n (t)) , and K(t) = (k1 (t), . . . , kn (t)) is an n × n matrix with the vector kj (t). The vectors k˜ (1) (t) and k˜ (n) (t) are 1-st and n-th ˜ Finally we have for yn+1 (t), columns of the transposed matrix of K.

1 + kn+1,n+1 (t) yn+1 + yn+1 = −kn+1,0 (t)y0 − kn+1 (t) · y˜ + Π˜ μ y˜ + Tδ,n+1 (t) + Fδ,n+1 (t).

(96)

We consider the ODE system for y(t) with the initial time T 0 under the initial condition

∗ ∗ u(T ), eδ,j = H y(T ); Uδ , eδ,j ,

(97)

for each j = 0, . . . , n + 1. We set Tδ (t) = Tδ,0 (t), T˜ (t), Tδ,n+1 (t) , Fδ (t) = Fδ,0 (t), F˜ δ (t), Fδ,n+1 (t) . Then the system (94)–(96) can be written in the form I + K(t) y (t) + Πy(t) = Tδ (t) + Fδ (t), where K(t) is an (n + 2) × (n + 2) matrix whose components are given by linear combinations of kj,l (t), and Π = (dij )0i,j n+1 is an (n + 2) × (n + 2) matrix whose component dij is given by dij = μj δij ,

0 i, j n + 1.

(98)

By the definitions of {kj,l (t)}0j,ln+1 , it is not difficult to prove Proposition 6.4. Assume that H (y; Uδ ) is C 1+γ in X near y = 0. Let kj,l (t) be given in (94). Then n+1 n+1

kj,l (t) C y(t) γ , j =0 l=0

where C does not depend on |δ| 1 and t > 0. Especially, we have

(99)


K(t)

Rn+2 →Rn+2

γ

C0 y(t) ,

3069

(100)

for |y(t)| 1, where C0 is independent of t and δ. We expect that |y(t)| is sufficiently small. If this is true, the inverse of I + K(t) exists. Then we have the equation −1 y (t) + Πy(t) = I + K(t) Tδ (t) + Fδ (t) =: W t, y(t) .

(101)

Let us write W(t, y(t)) = (W0 (t, y(t)), W (t, y(t))) , where W t, y(t) = W1 t, y(t) , . . . , Wn+1 t, y(t) . The representation (101) is useful since the right-hand side of (101) does not depend on the time derivative of y(t). In order to solve (101) we derive some estimates of W(t, y(t)). Proposition 6.5. Let Tδ and Fδ be given by (65). Assume that H (y; Uδ ) is C 1 in X near y = 0. Then for any f ∈ X we have

(−Lδ + I )−1 Tδ f C y(t) α f X , X (−Lδ + I )−1 Fδ (f ) Cf 1+α . X X

(102) (103)

Here C is independent of |δ| 1 and t > 0. As a consequence, we have

Tδ (t) C y(t) α v(t)

X

1+α

and Fδ (t) C v(t)X .

Especially, if H (y; Uδ ) is C 1+γ in X near y = 0 then we have

W t, y(t) C y(t) α + v(t)α v(t) . X X Proof. We give the proof of (102) only. By the Laplace formula we have −1

(−Lδ + I )

∞ Tδ f =

e−s esLδ N (Uδ ) − N Vδ (t) f ds.

0

Hence from (75) and the C 1 regularity of H (y; f ) with respect to y, we get (−Lδ + I )−1 Tδ f C X

∞ 0

1+s s

β

α e−s ds Uδ − Vδ (t)X f X

α

C y(t) f X . This completes the proof.

2

(104)

3070


We look for a solution y(t) which decays at time infinity. However, the equation for y0 (t) does not lead to the time decay apparently. To overcome this difficulty we consider the following integral equation ∞ y0 (t) = −

W0 s, y(s) ds,

(105)

t

yi (t) = e

−μi (t−T )

t yi (T ) +

e−μi (t−s) Wi s, y(s) ds,

1 i n,

(106)

T

yn+1 (t) = e−t+T yn+1 (T ) +

t

e−(t−s) Wn+1 s, y(s) ds,

(107)

T

which is equivalent with (94)–(96) if limt→∞ |y0 (t)| = 0. The problem here is that solutions to (105)–(107) might not satisfy the initial condition (97) for j = 0. In fact, we can show the initial condition for j = 0 is automatically satisfied if y(t) is a solution to (105)–(107) decaying at time infinity. The main result in this section is as follows. Proposition 6.6. Let u(t) be the solution in Theorem 5.1 with the initial data u0 ∈ X and δ = 0. Then for sufficiently large T > 0 there is a solution y(t) ∈ C 1 ([T , ∞); Rn+2 ) to (105)–(107) satisfying the initial condition (97) and the estimate

y (t) + y(t) c e− 3 (1+α)(t−T ) ,

(108)

t > T,

where c > 0 is a sufficiently small constant depending only on u(T ) − Uδ X and T > 0.

Proof. From Theorem 5.1 we have u(t) − Uδ X e− 2 t u0 − Uδ X . Especially, by taking T large enough we may assume that u(T ) − Uδ X κ where κ is so small as we want if needed. −1

Let C0 be the constant in (100). Let r1 ∈ (0, (2C0 ) γ ) be a small number and consider the closed

convex set Br1 = {y(t) ∈ C([T , ∞); Rn+2 ) | supt>T e 3 (t−T ) |y(t)| r1 }. For y ∈ Br1 we consider the map Y(t; y) = (Y0 (t; y), . . . , Yn+1 (t; y)) defined by ∞ Y0 (t; y) = −

W0 s, y(s) ds,

t

Yj (t; y) = e

−μj (t−T )

t Yj (T ; y) +

e−μj (t−s) Wj s, y(s) ds,

j 1

T

where the initial data Y (T ; y) = (Y1 (T ; y), . . . , Yn+1 (T ; y)) is determined by the relation

∗ ∗ u(T ), eδ,j = H Y0 (T ; y), Y (T ; y); Uδ , eδ,j ,

for j = 1, . . . , n + 1. The existence of such Y (T ; y) will be proved later.

(109)


3071

Our aim is to find a fixed point of the map Y(t; y) on Br1 by the Schauder fixed point theorem. For this purpose, let us first estimate W(t, y(t)) for y ∈ Br1 . By the estimate of Uδ and the definition of H (y; ·), we see that

Uδ − H y(t); Uδ C y(t) . X

(110)

Since we already have u(t) − Uδ X κe− 2 (t−T ) for any t T with sufficiently small κ > 0, it follows that v(t) = u(t) − H y(t); Uδ X X u(t) − Uδ X + Uδ − H y(t); Uδ X

κe− 2 t + C y(t) . Hence we have

y(t) α + v(t)α v(t) C r 1+α + κ 1+α e− 3 (1+α)(t−T ) . 1 X X

(111)

Then from Proposition 6.5 we have |W(t, y(t))| C(r11+α + κ 1+α )e− 3 (1+α)(t−T ) . This yields that

Y0 (t; y) C r 1+α + κ 1+α e− 3 (1+α)(t−T ) , 1

Yj (t; y) e−μj (t−T ) Yj (T ; y) + C r 1+α + κ 1+α e− 3 (1+α)(t−T ) . 1

(112) (113)

Here we used the fact min{μ1 , . . . , μn+1 } and thus μj > 3 (1 + α) for each j = 1, . . . , n + 1. Next we prove the existence and the uniqueness of Y (T ; y) = Y1 (T ; y), . . . , Yn+1 (T ; y) by the implicit function theorem. Note that u(T ) is written in the form u(T ) = Uδ + ξ0 ω(T ¯ ) ¯ )X = 1, and sufficiently small ξ0 0. For φ = (φ1 , . . . , φn+1 ) ∈ where ω(T ¯ ) ∈ Q0,0 X with ω(T Rn+1 and ξ1 ∈ R we consider the vector function R(ξ0 , ξ1 , φ) = R1 (ξ0 , ξ1 , φ), . . . , Rn+1 (ξ0 , ξ1 , φ) where ∗ ∗ Rj (ξ0 , ξ1 , φ) = Uδ + ξ0 ω(T − H (ξ1 , φ; Uδ ), eδ,j . ¯ ), eδ,j Clearly R(ξ0 , ξ1 , φ) is C 1 near the origin. Since H (0; Uδ ) = Uδ , we see that Rj (0, 0, 0) = 0 ∗ = c−1 δ , we have | det(∇ R(0, 0, 0))| = for each j . Moreover, since ∂yl H (0; Uδ ), eδ,j φ δ,l j l

−1 1 |Πjn+1 =1 cδ,j | > 0. Hence by the implicit function theorem there is a C function φ(ξ0 , ξ1 ) such

3072


that R(ξ0 , ξ1 , φ(ξ0 , ξ1 )) = 0 if |ξ0 | and |ξ1 | are sufficiently small. Thus if u(T ) − Uδ X and |Y0 (T ; y)| are sufficiently small, then there is a unique Y (T , y) ∈ Rn+1 satisfying (109). Moreover, from the equality ∂ξi Rj +

n+1

∂φl Rj ∂ξi φl = 0,

i = 0, 1, j = 1, . . . , n,

l=1

1 and φl (ξ0 , ξ1 ) = 1i=0 0 (∂ξi φl )(θ ξ0 , θ ξ1 )ξi dθ , we have |φl (ξ0 , ξ1 )| C(|δ|)(|ξ0 | + |ξ1 |). Noting that |ξ0 | = u(T ) − Uδ X κ and |ξ1 | = |Y0 (T ; y)| C(r11+α + κ 1+α ) from (112), we get |Y (T ; y)| C(κ + r11+α ). Hence by combining this with (113) it is easy to see that Y(t; y) is a completely continuous mapping from Br1 into itself by the Ascoli–Arzelá theorem for sufficiently small κ and r1 . Then by the Schauder fixed point theorem we have y ∈ Br1 such that y(t) = Y(t; y) for any t T . It is clear that the fixed point y(t) satisfies the estimate (108). It remains to prove this fixed point y(t) satisfies the initial condition (97). From the definition of Y(t; y), (97) holds for j = 1, . . . , n + 1. So it suffices to check (97) for j = 0. Note that we already know Eδ v(t) = Eδ (u(t) − H (y(t); Uδ )) vanishes at time infinity and satisfies Eq. (90). By the construction of y(t) and (91), the right-hand side of (90) belongs to (I − Pδ,0 )X. Thus Eδ v(t) satisfies the integral equation t Eδ v(t) = e

(t−T )Lδ

Eδ v(T ) +

e(t−s)Lδ Eδ (I − Pδ,0 )S(s) ds

(114)

T

where S(t) = Tδ v(t) + Fδ (v(t)) + J (Vδ )(t). By Proposition 6.5, Lemma 6.3, and (101) it follows that

Eδ (I − Pδ,0 )S(t) C v(t) y(t) α + v(t)α + W t, y(t)

(115) X X X for a constant C independent of t > T . In particular, we have Eδ (I − Pδ,0 )S(t)X t

Ce− 3 (1+α)t . Hence 0 e(t−s)Lδ Eδ (I − Pδ,0 )S(s) dsX converges to zero at time infinity. Since Eδ v(t)X also converges to zero at time infinity, we have limt→∞ e(t−T )Lδ Eδ v(T )X = 0. ∗ = H (y(T ); U ), e∗ , otherwise e(t−T )Lδ E v(T ) cannot But this implies that u(T ), eδ,0 δ δ X δ,0 vanish at time infinity. The proof of Proposition 6.6 is now complete. 2 6.3. Estimate of u(t) − H (y(t); Uδ ) In this section we calculate v(t)X more precisely by using the integral equation (77) for the initial data v(T ) = u(T ) − H (y(T ); Uδ ) ∈ Qδ X, which leads to Theorem 2.3. The main result in this section is as follows. Theorem 6.1. Let u(t) be the mild solution to (E) obtained in Theorem 5.1 with the initial data u0 ∈ X and δ = 0. Let y(t) ∈ C 1 ([T , ∞); Rn+2 ) be the parameters in Proposition 6.6. Then for any > 0, v(t) = u(t) − H (y(t); Uδ ) satisfies v(t) Ce−(ν0 −η(δ)−)(t−T ) , X Here C depends only on , δ, and u(T ) − Uδ X .

∀t > T .

(116)


3073

Proof. From the construction of y(t) we have Qδ v(t) = v(t) for t T . Hence v(t) satisfies t v(t) = Qδ e

(t−T )Lδ

v(T ) +

Qδ e(t−s)Lδ Tδ v(s) + Fδ v(s) + J (Vδ )(s) ds.

(117)

T

From Lemma 6.4 we have for any > 0, β

Qδ e(t−s)Lδ Tδ v(s) C 1 + t − s e−ν,δ (t−s) y(s) α v(s) , X X t −s β Qδ e(t−s)Lδ Fδ v(s) C 1 + t − s e−ν,δ (t−s) v(s)1+α . X X t −s Moreover, from Lemma 6.3 and (101) we have Qδ J (Vδ )(t) = −Qδ ∇y H y(t); Uδ · W t, y(t) = −Qδ ∇y H y(t); Uδ − ∇y H (0; Uδ ) · W t, y(t) . This gives from (104) that

Qδ e(t−s)Lδ J (Vδ )(s) C e−ν,δ (t−s) y(s) γ y(s) α + v(s)α v(s) . X X X Then, combining these above with Proposition 6.6, we get Qδ e(t−s)Lδ Tδ v(s) + Fδ v(s) + J (Vδ )(s) C

1+t −s t −s

β

e−ν,δ (t−s) e

− 3 α(s−T )

X

v(s) . X

This yields for any t T T , v(t) C e−ν,δ (t−T ) v T X X + C e

− 6 α(T −T )

t T

1+t −s t −s

β

e−ν,δ (t−s) e− 6 α(s−T ) v(s)X ds.

By taking T sufficiently large, we can take C v(T )X and C e− 6 α(T −T ) sufficiently small. Then it is not difficult to get the estimate (116). We omit the details here. This completes the proof. 2 Using Theorem 6.1, we can improve the decay estimate of W(t, y(t)). Let us recall that μn+1 = 1 and μ∗ , μ∗ are defined by (22).

3074


Proposition 6.7. Let y(t) ∈ C 1 ([T , ∞); Rn+2 ) be the parameters constructed in Proposition 6.6. Then for sufficiently small > 0 we have

W t, y(t) Ce−(ν,δ +α min{μ∗ ,ν,δ })(t−T ) .

(118)

Proof. From Proposition 6.6 and Theorem 6.1 we have

y(t) α + v(t)α v(t) C e− 3 α(1+α)(t−T ) + e−αν,δ (t−T ) e−ν,δ (t−T ) . X X Hence we first get the estimate |W(t, y(t))| Ce−ν,δ (t−T ) from (104). Then from (105)–(107) we get the estimates for y(t) such as |y(t)| Ce− min{μ∗ ,ν,δ }(t−T ) if > 0 is sufficiently small. Then again by (104) we have the improved estimate (118). This completes the proof. 2 From Proposition 6.7 we obtain the precise decay estimates for y(t). Corollary 6.2. Let y(t) ∈ C 1 ([T , ∞); Rn+2 ) be the parameters constructed in Proposition 6.6. Assume that ν0 μj . Then the limit yj∗ = limt→∞ eμj t yj (t) exists and it follows that

μt

e j yj (t) − y ∗ Ce−(ν,δ +α min{μ∗ ,ν,δ }−μj )(t−T ) .

(119)

j

Proof. From (118) we observe that eμj t Wj (t, y(t)) is integrable over (T , ∞) since the value ν,δ + α min{μ∗ , ν,δ } − μj is strictly positive if ν0 μj and , |δ| are sufficiently small by the definition ν,δ = ν0 − η(δ) − . Hence we have t e

−μj (t−s)

Wj s, y(s) ds = e−μj t

T

∞ e

μj s

Wj s, y(s) ds −

∞ T

e

μj s

Wj s, y(s) ds .

t

T

Especially, for yj∗ = eμj T yj (T ) + pletes the proof. 2

∞

eμj s Wj (s, y(s)) ds we have the estimate (119). This com-

Proof of Theorem 2.3. Let Ω(t) ∈ C([0, ∞); X) be the solution to (E) with initial data Ω0 with Ω0 X 1. Then by Theorem 2.2 and Lemma 3.2, u(τ ) = Reτ Ω(eτ − 1) is the unique solution to (51) with initial data Ω0 . Let y(τ ) = (y0 (τ ), y(τ )) ∈ C 1 ([T , ∞); Rn+2 ) be the parameters constructed in Proposition 6.6. We note that S(y(τ ); Uδ ) = H (0, y(τ ); Uδ ). Then from Theorem 6.1 and Corollary 6.2 with j = 0 we have u(τ ) − S y(τ ); Uδ u(τ ) − H y(τ ); Uδ + H y(τ ); Uδ − H 0, y(τ ); Uδ X X X

−(ν0 −η(δ)−)(τ −T ) Ce + C y0 (τ )

Ce−(ν0 −η(δ)−)(τ −T ) ,

τ > T 1.

Thus from u(τ ) = Reτ Ω(eτ − 1) we have Reτ Ω eτ − 1 − S y(τ ); Uδ Ce−(ν0 −η(δ)−)(τ −T ) X

for τ > T 1,


3075

that is, R1+t Ω(t) − S ξ(t); Uδ C(1 + t)−ν0 +η(δ)+ , X

t 1,

(120)

where ξ(t) = y(log(1 + t)). Assume now that ν0 μ∗ . Then by Corollary 6.2 there are y ∗ = ∗ ) such that (119) holds. Then from (120) we conclude that (y1∗ , . . . , yn+1 R1+t Ω(t) − S

∗ yn+1 y1∗ yn∗ C(1 + t)−ν0 +η(δ)+ , , . . . , , ; U δ (1 + t)μ1 (1 + t)μn 1 + t X

(121)

for t 1. If ν0 > μ∗ then by Lemma 6.2 {−μj }n+1 j =1 must be semisimple eigenvalues of Lδ if |δ| is sufficiently small. Hence (121) holds in this case. This completes the proof of Theorem 2.3. 2 Remark 6.3. Let η (δ) be the number in (71). By Corollary 6.1 the set {μ ∈ σ (Lδ ) | Re(μ) > −ζ + η (δ)} consists of isolated eigenvalues with finite multiplicity. Hence, if ζ > ν0 and |δ| is sufficiently small then the set {μ ∈ σ (Lδ ) | Re(μ) −ν0 + η(δ) = −ν0,δ } consists of finite number of eigenvalues with finite multiplicity. In this case if all eigenvalues in {μ ∈ σ (Lδ ) | Re(μ) −ν0,δ } are semisimple then we can take = 0 in (116) and (118), and hence, in (121). Indeed, by considering the spectral decomposition for the semisimple eigenvalues {μ ∈ σ (Lδ ) | Re(μ) −ν0,δ } it is not difficult to verify Qδ etLδ f X Ce−ν0,δ t Qδ f X in Lemma 6.4. Then the calculations of v(t)X above imply v(t)X Ce−ν0,δ (t−T ) , and thus, we first obtain the estimate |W(t, y(t))| Ce−(α min{μ∗ ,ν,δ }+ν0,δ )(t−T ) by (118). This yields |y(t)| Ce− min{μ∗ ,ν0,δ }(t−T ) from (105)–(107). Then again by (118) we have the improved estimate |W(t, y(t))| Ce−(ν0,δ +α min{μ∗ ,ν0,δ })(t−T ) as desired. 6.4. Comments on Theorem 2.4 Theorem 2.4 is proved in the same way as in Theorem 2.3. Indeed, it suffices to determine y(t) = (y0 (t), y(t)) ˜ ∈ C 1 ([T , ∞); Rn+1 ) so that v(t) ˜ = u(t) − H (y0 (t), y(t), ˜ 0; Uδ ) belongs to ˜ Q0 X for all t T , which leads to the ODEs for (y0 (t), y(t)) ˜ as in the case of Theorem 2.3. We can solve this ODEs by using the equality in Lemma 6.3: J (V˜δ )(t) = −

n

∂yj H y0 (t), y(t), ˜ 0; Uδ · yj (t) + μj yj (t) ,

(122)

j =0

where V˜δ (t) = H y0 (t), y(t), ˜ 0; Uδ ,

J (V˜δ )(t) = −∂t J (V˜δ )(t) + A−1 V˜δ (t) − N V˜δ (t) . Under the assumptions of Theorem 2.4 we can show, instead of (69), that

ν0 + η(δ) σ (Lδ ) ⊂ {−μj }n+1 ˜ j =0 ∪ μ ∈ C Re(μ) −˜

(123)

3076


for some η(δ) ˜ ∈ R satisfying limδ→0 η(δ) ˜ = 0. Then if we set μ˜ ∗ = max{μ1 , . . . , μn }, μ˜ ∗ = min{μ1 , . . . , μn }, and ν˜ δ, = ν˜ 0 − η(δ) ˜ − for > 0, as in Theorem 6.1 and Corollary 6.2, we obtain the estimates

y(t) Ce− min{μ˜ ∗ ,˜νδ, }(t−T ) .

v(t) ˜ X Ce−˜νδ, (t−T ) ,

(124)

Then by the relation R1+t Ω(t) = u(log(1 + t)) we have R1+t Ω(t) − S˜ ξ˜ (t); Uδ C(1 + t)−νδ, , X

t 1,

(125)

where ξ˜ (t) = y(log(1 ˜ + t)). If ν˜ 0 μ˜ ∗ then the limit yj∗ = limt→∞ eμj t yj (t) exists and

μt

e j yj (t) − y ∗ Ce−(ν˜ ,δ +α min{μ˜ ∗ ,˜ν,δ }−μj )(t−T )

(126)

j

holds. Hence we have R1+t Ω(t) − S˜ (1 + t)−μ1 y ∗ , . . . , (1 + t)−μn y ∗ ; Uδ C(1 + t)−˜νδ, , n 1 X

t 1.

(127)

The details are omitted here. Especially, if ν˜ 0 > μ˜ ∗ and |δ| is sufficiently small, then {−μj }nj=1 must be semisimple eigenvalues of Lδ as in the proof of Theorem 2.3. Hence (127) holds in this case. This completes the proof of Theorem 2.4. 7. Applications In this section we consider several nonlinear equations to illustrate that our method systematically yields an asymptotic description of the large time behavior in terms of shifted self-similar solutions. Especially, except for the results on the two-dimensional vorticity equations in Section 7.2, the results below improve the known results, which are new contributions of this paper. 7.1. Convection–diffusion equations (n-B) We first consider the convection–diffusion equation (n-B) with p = 1 ∂t Ω − Ω + a · ∇ |Ω| n Ω = 0,

1 n

for n 2, i.e.,

t > 0, x ∈ Rn .

(128)

In contrast to the case of (1-B), the Hopf–Cole transformation does not work for (128) with n 2. By applying our method we establish the following result which improves the estimate (1) obtained in [8]. For m 0 and s ∈ N let L2m , Hms be Hilbert spaces defined by L2m

2 n

2 2 2 m

f (x) dx < ∞ , 1 + |x| = f ∈ L R f L2 = m

(129)

Rn

Hms = f ∈ L2m ∂xβ f ∈ L2m , |β| s .

(130)


3077

We will apply Theorem 2.4 to (128) and obtain the following Theorem 7.1. Let m2 > n4 +12 . Assume that Ω(t) ∈ C([0, ∞); L2m ) be the solution to (128) satisfying Ω(0)L2m 1 and Rn Ω(0, x) dx = δ = 0. Then there are η(δ) ˜ ∈ R and y˜ ∗ ∈ Rn such that limδ→0 η(δ) ˜ = 0 and n 1 m n · + y˜ ∗ ˜ C (1 + t)− 2 (1− p )−min{1, 2 − 4 }+η(δ)+ Ω(t) − (1 + t)− n2 Uδ √ 1 + t Lp

(131)

holds for all t 1, > 0, and 1 p 2. Remark 7.1. When n = 1 if the initial data Ω0 belongs to L2m with m > 52 then we can apply Theorem 2.3, which recovers the results of (2). The details are omitted here. In order to prove Theorem 7.1 we have to check the conditions stated in Theorem 2.4 for (128). First we observe that the operator A for (128) is Af = f,

Dom(A) = f ∈ L2m f ∈ L2m .

(132)

So (E0 ) is the linear heat equation (H) in this case. As is well known, the associated (C0 ) semigroup et is given by t e f (x) =

1 (4πt)

n 2

e−

|x−y|2 4t

f (y) dy.

(133)

Rn

From the Calderón–Zygmund inequality and the interpolation inequality it is not difficult to see Proposition 7.1. Let m 0. Then Dom() = Hm2 . In particular, ρ m ∂xj f ∈ W 1,2 (Rn ) for each j . The proof is omitted here. For each λ > 0, a ∈ R, and j = 1, . . . , n, we set n 1 (Rλ f )(x) = λ 2 f λ 2 x , (j ) τa f (x) = f (x1 , . . . , xj −1 , xj + a, xj +1 , . . . , xn ).

(134) (135)

(j )

Then by the density arguments R = {Rλ }λ∈R× and T (j ) = {τa }a∈R are shown to be a scaling (j ) and a translation in L2m , respectively. Moreover, by setting Tθ = T (j ) for each θ ∈ R, we have (j ) independent one parameter families of translations {Tθ }θ∈R , j = 1, . . . , n; see Definition 2.1. (j ) The generators of R and Tθ are respectively given by B= (j )

n x ·∇ + , 2 2

Dθ = D (j ) = ∂xj ,

Dom(B) = f ∈ L2m x · ∇f ∈ L2m ,

(j ) Dom Dθ = f ∈ L2m ∂xj f ∈ L2m .

(136) (137)

3078

Y. Kagei, Y. Maekawa / Journal of Functional Analysis 260 (2011) 3036–3096 τ

(j )

(j )

(f )−τ

(j )

(f )

Clearly Γa,θ (f ) = limh→0 a,θ+h h a,θ = limh→0 case. Let G be the n-dimensional Gaussian n

(j )

(j )

τa (f )−τa (f ) h

G(x) = (4π)− 2 e−

|x|2 4

= 0 for each a and θ in this

.

Then we have Proposition 7.2. Let m2 > n4 + 1 and X = L2m . Then under the setting of (132), (134), and (135), the conditions (E1), (E2), (T1), (T2), (A1), and (A2) are satisfied with μj = 12 , = 12 , and ζ = m2 − n4 . The eigenprojection P0,0 for the eigenvalue 0 of A (see (23)) is given by

P0,0 f =

(138)

f (y) dy G, Rn

and the number ν0 defined by (38) is 1 when n 2 and min{2, m2 − 14 } when n = 1. Moreover, if m n 1 2 2 > 4 + 2 and X = Lm then the conditions (E1), (E2), (T1), (T2), (A1), and (A2)’ are satisfied 1 1 with μj = 2 , = 2 , and ζ = m2 − n4 . The number ν˜ 0 defined by (45) is min{1, m2 − n4 } in this case. (j )

(j )

Proof. It is easy to check Rλ eλt = et Rλ and τa et = et τa , which gives (E1) and (E2). The condition (T1) follows with μj = 12 from (136) and (137). To check (T2) we first note that (j ) |x|f (x) ∈ L2 (Rn ) if f ∈ L2m with m2 > n4 + 1. Let f ∈ Dom(B) ∩ nj=1 Dom(D1 ) and suppose that a0 Bf +

n

(j ) aj D1 f

= a0

j =1

x n · ∇f + f 2 2

+

n

aj ∂xj f = 0,

(139)

j =1

where aj ∈ R for each j = 0, . . . , n. If a0 = 0 then multiplying both sides above by f and integrating over Rn , we get from the integration by parts, a04n f 2L2 = 0, i.e., f = 0. If a0 = 0 and there is an aj with j 1, then multiplying both sides of (139) by xj f and integrating over Rn , we have aj f 2L2 = 0. That is, f = 0. Hence (T2) holds. We note that the semigroup etA = e(1−e tA e f (x) =

nt

e2

(4πa(t))

n 2

−t )A

e−

Ret is explicitly given by

|x−y|2 4a(t)

f yet dy,

a(t) = 1 − e−t ,

(140)

Rn

and A=+

n x ·∇ + , 2 2

x

Dom(A) = f ∈ L2m + · ∇ f ∈ L2m . 2

In [9] Gallay and Wayne proved that the bound of the essential spectrum of etA and the spectrum of A in L2m are given by


3079

m n ress etA = e−( 2 − 4 )t ,

m n k

∪ − k = 0, 1, 2, . . . . σ (A) = μ ∈ C Re(μ) − + 2 4 2

(141) (142)

Moreover, if k ∈ N ∪ {0} satisfies − k2 > − m2 + n4 then − k2 is a semisimple eigenvalue with mulβ tiplicity (n+k−1)! k!(n−1)! and the associated eigenspace is spanned by the Hermite functions {∂x G}|β|=k . This gives (A1) and (A2) when m2 > n4 + 1 and (A2)’ when m2 > n4 + 12 . The eigenprojection P0,0 is easily calculated and given by (138). The proof is completed. 2 For later use we consider the relations between the domains Dom(A), Dom(A), and Dom(B) in the case of A = and B = x2 · ∇ + n2 in L2m . As stated in (20), the inclusion Dom(A) ∩ Dom(B) ⊂ Dom(A) holds in general. Proposition 7.3. Let m 0. Then Dom(A) = Dom(A) ∩ Dom(B) and Af L2m + Bf L2m C(Af L2m + f L2m ) hold for all f ∈ Dom(A). 1

Proof. Set ρ(x) = (1 + |x|2 ) 2 . For any function f in the Schwartz class the direct calculation yields the equality Rn

2

2

x x 4m dx ρ 2m

f + · ∇f

dx = ρ 2m |f |2 dx + ρ 2m

· ∇f

1 − 2 2 1 + |x|2 Rn

+

ρ

2m

|∇f |

2

Rn

Rn

n m dx. −1+m− 2 1 + |x|2

Combining this with the equality

ρ 2m |∇f |2 dx = −

Rn

ρ 2m ff dx + 2m

Rn

|f |2 ∇ · xρ 2m−2 dx,

Rn

we get Af L2m + Bf L2m C(Af L2m + f L2m ), which holds for all f ∈ Dom(A) since the Schwartz class is a core of A. This completes the proof. 2 ˜ yn+1 ) ∈ O with y˜ = Let O ⊂ Rn be a small open ball centered at the origin. For y = (y, (y1 , . . . , yn ), we set S(y; f ) = τy(1) · · · τy(n) R n 1

n

1 1+yn+1

f = (1 + yn+1 )− 2 f

Since G is rapidly decreasing and smooth we have Proposition 7.4. The map S(·; G) : O → L2m is C ∞ .

· + y˜ 1

(1 + yn+1 ) 2

.

(143)

3080


Especially, the facts ∂xj G and (x · ∇ + n)G are the eigenfunctions of the eigenvalues − 12 and −1 are interpreted as the symmetry of the heat equation from Lemma 2.3. 1 1 Let X = L2m . Then N (f ) = a · ∇(|f | n f ) = (1 + n1 )|f | n a · ∇f makes sense for f ∈ Dom(). Indeed, we have from the Hölder inequality, N (f )

L2m

1 1 1+ |a|f n 1+ 1 ρ m ∇f 1+ n1 , L n L n

(144)

which is bounded if f ∈ Dom() by Proposition 7.1 and the Gagliardo–Nirenberg inequality. Now we prove 1

Proposition 7.5. Let m2 > n4 + 12 and X = L2m . Then N (f ) = a · ∇(|f | n f ) satisfies (N1), (N2), and (N3) with q = α = n1 , β = 34 , and 0 = 0. Proof. It is easy to check (N3). We will show (N1) and (N2). From (144), Proposition 7.1, 1+ 1

n . Proposition 7.3, and the Gagliardo–Nirenberg inequality, we have N (f )L2m Cf Dom(A) Moreover, N maps Dom(A) into Q0,0 X by (138). Hence (N1) follows. To prove (N2) we recall the estimates for etA (see (140)) obtained in [9]:

tA ∇e f

L2m

m ρ f q , L

C

a(t)

n 1 1 1 2 ( q − 2 )+ 2

1 q 2,

(145)

t

where a(t) = 1 − e−t . Then by the relation ∂xj etA = e 2 etA ∂xj we have tA e ∂x f 2 j L m

t

Ce− 2 a(t)

n 1 1 1 2 ( q − 2 )+ 2

m ρ f q , L

1 q 2.

(146)

Thus N(t, f ) = etA N (f ) is estimated as N(t, f ) − N (t, g) 2 L

m+ m n

t

Ce− 2

Ce− 2

a(t)

t

a(t)

m 1 1 ρ m+ n |f | n f − |g| n g

3 4

3 4

2n

L n+1

1 m ρ n |f | + |g| n ρ m (f − g)

2n

L n+1

t

1 Ce− 2 f L2m + gL2m n f − gL2m . 3 a(t) 4

(147)

Note that we have the additional decay for N (t, f ) if m > 0. Similarly, by the relation 1 N (t, f )h = (1 + n1 )etA a · ∇(|f | n h) we have

N (t, f )h − N (t, g)h

t

L2m+ m

n

We omit the details here. This completes the proof.

Ce− 2 a(t) 2

3 4

1

f − gLn 2 hL2m . m

(148)


Let

m 2

>

n 4

3081

+ 12 . Then by Theorem 2.1 there is a δ0 > 0 such that for each δ with |δ| δ0 there 1

is a Uδ ∈ L2m which gives a self-similar solution R 1 Uδ to (128) and is C 1+ n in L2m with respect 1+t to δ. Note that Uδ is a solution to −U −

1 x n · ∇U − U = a · ∇ |U | n U , 2 2

x ∈ Rn ,

(149)

with Rn U (x) dx = δ by the definition of the projection P0,0 . On the other hand, in [1] it is proved that there is a unique solution U˜ δ to (149) with Rn U˜ δ (x) dx = δ for all δ ∈ R which belongs to HG2 ∩ W 2,p (Rn ) with 1 p < ∞, where HG2 is the Gaussian weighted L2 space defined by

2 dx

n 4

1

+ 12 . Then H (y0 , y; Uδ ) is C 1+ n as a mapping from (−δ0 + δ,

Proof. Since HG1 is included in Dom(B) in L2m , the fact Uδ ∈ HG2 implies that H (y0 , y; Uδ ) is 1

in fact C 2 in L2m with respect to y for each fixed y0 . Since we already know that Uδ is C 1+ n 1 with respect to δ in L2m , H (y0 , y; Uδ ) is also C 1+ n with respect to y0 in L2m . Hence it suffices to show, for example, Uδ is differentiable in HG1 with respect to δ. To prove this, let us recall that the operator A = + x2 · ∇ + n2 is realized as a self-adjoint operator in L2G , denoted by 1

A∞ in order to avoid confusions, and HG1 = Dom((−A∞ + I ) 2 ) with equivalent norms; see [7]. 1

Moreover, the duality arguments as in [11, Proposition 2.1] shows that (−A∞ )− 2 ∂xj is extended

3082


as a bounded operator in L2G . Now let us consider the operator (−A∞ )−1 ∂xj . By decomposing 1

1

(−A∞ )−1 ∂xj = (−A∞ )− 2 (−A∞ )− 2 ∂xj , we have the estimate (−A∞ )−1 ∂x f 1 Cf 2 , j L H

(154)

G

G

for all f ∈ L2G . Since Uδ = δG + vδ solves vδ = (−A∞ )−1 N (Uδ ) in L2G , we have for sufficiently small |h| |δ|, −1

1

vδ+h − vδ = (−A∞ )

N τ Uδ+h + (1 − τ )Uδ (vδ+h − vδ + hG) dτ,

0 1

where N (f )g = (1 + n1 )a · ∇(|f | n g). Then from (152) and (154) we have 1 vδ+h − vδ H 1 C G

1 τ Uδ+h + (1 − τ )Uδ n (vδ+h − vδ + hG)

L2G

dτ

0

1 C|δ| n vδ+h − vδ L2 + |h| , G

and hence, vδ+hh−vδ H 1 C|δ| n with C independent of h. Set ωh = 1

G

−1

1

ωh − ωh = (−A∞ )

vδ+h −vδ . h

Then we have

N τ Uδ+h + (1 − τ )Uδ (ωh + G)

0

− N τ Uδ+h + (1 − τ )Uδ (ωh + G) dτ.

Similar calculations as above yields 1 ωh − ωh H 1 C |Uδ+h − Uδ+h | n (ωh + G)L2 G

G

CUδ+h − Uδ+h 1 ωh H 1 + C 1 n

HG

G

1

C h − h n . Thus ωh converges to ∂δ vδ is HG1 as h → 0, which completes the proof.

2

Proof of Theorem 7.1. We first note that − 12 is a semisimple eigenvalue of Lδ . Indeed, for (128) we have ν˜ 0 = min{1, m2 − n4 } and μ˜ ∗ = μ˜ ∗ = μj = 12 with j = 1, . . . , n, by Proposition 7.2. Then by Lemma 6.2, − 12 is a semisimple eigenvalue of Lδ if |δ| is sufficiently small. Hence from Theorem 2.3 there is y˜ ∗ = (y1∗ , . . . , yn∗ ) ∈ Rn such that


R1+t Ω(t) − S˜

y1∗

(1 + t) 2 n

Thus from Rλ f Lp = λ 2 Ω(t) − R

1 1+t

S˜

y1∗ (1 + t)

for 1 p 2. Since R rem 7.1 is completed.

,..., 1

1 1+t

2

1 2

(1− p1 )

,...,

˜ S(

y1∗

yn∗

; U δ 1

(1 + t) 2

L2m

n

C(1 + t)− min{1, 2 − 4 }+η(δ)+ .

f Lp we have yn∗

; Uδ 1

(1 + t) 2

1 (1+t) 2

m

3083

,...,

yn∗

n − n2 (1− p1 )−min{1, m 2 − 4 }+η(δ)+

C(1 + t)

∗

x+y˜ ; Uδ ) = (1 + t)− 2 Uδ ( √ ), the proof of Theon

1 (1+t) 2

,

Lp

1+t

7.2. Two-dimensional vorticity equations In this section we consider the two-dimensional vorticity equations for viscous incompressible flows: ∂t Ω − Ω + ∇ · Ω∇ ⊥ (−)−1 Ω = 0,

t > 0, x ∈ R2 ,

(2-V)

where ∇ ⊥ = (∂x2 , −∂x1 ) and ∇ ⊥ (−)−1 f is explicitly given as ∇ ⊥ (−)−1 f =

1 2π

R2

(x − y)⊥ f (y) dy, |x − y|2

where x ⊥ = (−x2 , x1 ) . In this case A = and N (f ) = ∇ · (f ∇ ⊥ (−)−1 f ). Proposition 7.7. Let m > 3 and X = L2m . Then N (f ) = ∇ · (f ∇ ⊥ (−)−1 f ) satisfies (N1), (N2), and (N3) with q = α = 1, β = 34 , and 0 = 0. Proof. Since (N3) is easy to see, we will check only (N1) and (N2). We note that N (f ) = (∇ ⊥ (−)−1 f, ∇)f . From the Hardy–Littlewood–Sobolev inequality we have ⊥ ∇ (−)−1 f 4 Cf 4 Cf 2 if m > 1. Lm L L3

Hence by Proposition 7.1 and Proposition 7.3 we have N (f )

L2m

= ρ m N (f )L2 C ∇ ⊥ (−)−1 f L4 ρ m ∇f L4 Cf L2m f Dom(A) f 2Dom(A) .

Together with (138), this shows (N1). Next we consider (N2). Since N is a bilinear form, it suffices to give the estimate for etA M(f, g) := etA ∇ · (g∇ ⊥ (−)−1 f ). Then from (146) and the Hölder inequality we have

3084


tA e M(f, g)

t

L2m

Ce− 2 a(t)

3 4

ρ m g∇ ⊥ (−)−1 f

4

L3

t

This completes the proof.

Ce− 2 3

a(t) 4

gL2m f L2m .

2

For (2-V) the equation for Uδ is −U −

x n · ∇U − U + ∇ · U ∇ ⊥ (−)−1 U = 0, 2 2

x ∈ R2 ,

(155)

with R2 U (x) dx = δ. In [10] Gallay and Wayne proved that Uδ = δG is the unique solution to (155) in L1 (R2 ) by proving the global stability of δG; see also [13,12,20]. Especially, the function H (y0 , y; Uδ ) defined by (153) is C ∞ as a mapping from R × O ⊂ Rn+2 to L2m . When Ωδ (t) ∈ C([0, ∞); L1 (Rn )) ∩ C((0, ∞); L∞ (R2 )) is a solution to (2-V) with Ω(0, x) dx = δ, R2

the large time behavior of Ωδ (t) is described as Ωδ (t) − δt −1 G √· t

−1+ 1 p , =o t

t → ∞, 1 p ∞.

(156)

Lp

This was proved by [13,3,12] for sufficiently small |δ|, and the smallness of δ was removed by [10]. Applying Theorem 2.3, we have the following Theorem 7.2. Let m > 3. Then for any δ ∈ R with 0 < |δ| 1 there exists a negative number η(δ) such that limδ→0 η(δ) = 0 and the following statements hold. Assume that Ω(t) ∈ C([0, ∞); L2m ) be the solution to (2-V) satisfying R2 Ω(0, x) dx = δ. Then there is a y ∗ = (y˜ ∗ , y3∗ ) ∈ R2 × R such that Ω(t) satisfies ∗ Ω(t) − δ 1 + t + y ∗ −1 G · + y˜ 3 ∗ 1+t +y 3

−2+ p1 +η(δ)+

C (1 + t)

,

(157)

Lp

for all t 1, > 0, and 1 p 2. We note that, the second and third asymptotic expansions to (2-V) are established in [10] by obtaining the estimates of the type (157). So Theorem 7.2 here is a restatement of their results from the abstract framework. Proof of Theorem 7.2. Since ν0 = 1 > 12 = μj , j = 1, 2, − 12 is a semisimple eigenvalue of Lδ by Lemma 6.2. Furthermore, from [10, Remark 4.9] we observe that −1 is a simple eigenvalue of Lδ and η(δ) is strictly negative if δ is not zero. Hence from Theorem 2.3 there are y3∗ ∈ R3 such that


R1+t Ω(t) − S (1 + t)− 12 y ∗ , (1 + t)− 12 y ∗ , (1 + t)−1 y ∗ ; Uδ 1

2

3

where Uδ = δG, which implies (157). This completes the proof.

L2m

3085

C(1 + t)−1+η(δ)+ ,

2

7.3. Keller–Segel systems We consider the two-dimensional parabolic systems modeling chemotaxis:

∂t Ω (1) − Ω (1) + ∇ · Ω (1) ∇Ω (2) = 0, t > 0, x ∈ R2 , ∂t Ω (2) − Ω (2) − Ω (1) = 0,

t > 0, x ∈ R2 .

(KS)

For (KS) the existence of self-similar solutions is proved in [2] and the stability estimate (158) below is obtained by [23] when the initial data (Ω (1) (0), Ω (2) (0)) satisfies (1 + |x|2 )Ω (1) (0) ∈ L1 (R2 ), ∂xj Ω (2) (0) ∈ L1 (R2 ) for each j , and Ω (1) (0)L1 and ∇Ω (2) (0)L2 are sufficiently small: (1) Ω (t, ·) − t −1 U (1) √· δ t

Lp (R2 )

−1+ 1 −σ p , =O t

τ → ∞,

3 p 2. 4

(158)

(1) (2) (1) Here (t −1 Uδ ( √xt ), Uδ ( √xt )) is the self-similar solution to (KS) with R2 Uδ (x) dx = 1 (1) R2 Ω (x, 0) dx =: δ, and σ is a constant in (0, 2 ). The value of σ is not explicitly determined in [23]. As is shown in [15], one can apply our method to obtain the following result which improves the estimate (158). Theorem 7.3. Let m > 2. Assume that (1) Ω (0), Ω (2) (0) 2 L ×H 1 m

m−2

1 and

Ω (1) (0, x) dx = δ = 0. R2

1 ) to (KS) such Then there exists a unique solution (Ω (1) (t), Ω (2) (t)) ∈ C([0, ∞); L2m × Hm−2 that the following statements hold. There are η(δ) ˜ ∈ R and y˜ ∗ ∈ R2 such that limδ→0 η(δ) ˜ = 0 and

(1) 1 m−1 · + y˜ ∗ ˜ C (1 + t)−1+ p −min{1, 2 }+η(δ)+ Ω (t) − (1 + t)−1 U (1) √ δ 1 + t Lp

(159)

holds for all t 1, > 0, and 1 p 2. Furthermore, if m > 3 then there are η(δ) ∈ R and (y˜ ∗ , y3∗ ) ∈ R2 × R such that limδ→0 η(δ) = 0 and ∗ (1) 1 C (1 + t)−2+ p +η(δ) Ω (t) − 1 + t + y ∗ −1 U (1) · + y˜ 3 δ 1 + t + y3∗ Lp

(160)

holds for all t 1 and 1 p 2. Moreover, η(δ) is positive (negative) if δ is positive (negative).

3086


Remark 7.2. Let m > 3. Estimate (160) implies that η(δ) ˜ + = 0 if δ < 0 and η(δ) ˜ + = η(δ) if δ > 0 in (159). Therefore, (160) gives more precise asymptotic profile than (159) if m > 3 and δ < 0; see Remark 2.5. One can see from the estimate (160) that the convergence rate to a shifted self-similar solution depends on the sign of δ. The argument to check the conditions stated in Section 2.2 is not so straightforward, and, furthermore, it is needed a much more detailed argument to analyze the value η(δ). So we give a proof of Theorem 7.3 in the separate paper [15]. Remark 7.3. Instead of (KS) we can also apply our abstract results to the Keller–Segel system of a parabolic–elliptic type:

∂t Ω (1) − Ω (1) + ∇ · Ω (1) ∇Ω (2) = 0, t > 0, x ∈ R2 , −Ω (2) = Ω (1) ,

t > 0, x ∈ R2 .

(KS’)

We can show the estimate like (159) for solutions to (KS’) when Ω (1) (0)L2m is sufficiently small for some m > 2, but the details are omitted here. 7.4. One-dimensional Vlasov–Poisson–Fokker–Planck equations In this section we consider the one-dimensional Vlasov–Poisson–Fokker–Planck equations without friction: ∂t Ω + u∂x Ω + E± (Ω)∂u Ω − ∂u2 Ω = 0,

t > 0, (x, u) ∈ R × R.

(161)

Here Ω = Ω(t, x, u), and 1 E± (Ω) = ± sgn(x) ∗x 2

Ω(t, x, u) du.

(162)

R

For simplicity we consider the case E(Ω) = E+ (Ω) here. In [5] the existence of global and classical solutions to (161) is proved for some class of initial data. We here restrict ourselves to the case where the mass of initial data is zero, which seems to be less physical but still has an interesting aspect. In fact, from a scaling point of view Eq. (161) is strongly nonlinear, that is, the nonlinear term would dominate the large time behavior of solutions. This implies that one cannot expect the self-similar behavior of solutions in general. However, when the mass of solutions is zero the nonlinearity of (161) is shown to be critical, and we can prove that there exists a selfsimilar solution with “zero-mass” and the large time behavior of solutions in a class of zero-mass functions is described by the self-similar solution. Another interesting feature of (161) is that it is not invariant with respect to a usual translation of a spatial variable. But our abstract method is still applicable. By using the Fourier transform the semigroup associated with ∂u2 − u∂x is given by e

tA

√ 2 3 − 33 {x−y− 2t (u+v)}2 − (u−v) 4t t f= e f (y, v) dy dv. 2πt 2 R2

(163)


3087

It is not difficult to see that for any compactly supported function f , the function et A f is smooth as a function of (t, x, u) if t > 0, and satisfies ∂t Ω + u∂x Ω − ∂u2 Ω = 0,

t > 0, (x, u) ∈ R × R,

(164)

pointwisely. Indeed, this can be verified by using the results on fundamental solutions to the linear Vlasov–Poisson–Fokker–Planck equation obtained by Victory and O’Dwyer [25]. In this section the spaces L2m and Hms are defined by

2

m

L2m = f ∈ L2 R2 f 2L2 = 1 + x 2 + u2 f (x, u) dx du < ∞ , m

R2

Hms = f ∈ L2m ∂xβ1 ∂uβ2 f ∈ L2m , 0 β1 + β2 s . By setting e0A = I , we can check that {et A }t0 is a strongly continuous semigroup in L2m for each m 0. The associated generator is again denoted by A. The following estimates for et A are useful. Proposition 7.8. Assume that m 0. Let s be a nonnegative integer and j1 , j2 ∈ {0, 1}. Then for any > 0 we have tA e f j1 j2 t A ∂x ∂u e f tA e ∂u f

Hms

C et f Hms ,

Hms

Hms

C et 3j1 +j2

t 2 C et 1

t2

t > 0,

f Hms ,

f Hms ,

t > 0, t > 0.

(165) (166) (167)

If m = 0 and s = 0 then we can take = 0 in the above estimates. As a consequence, we have ∂u (I − A)−1 f 2 Cf 2 . Lm L m

(168)

Proof. We rewrite (163) as e

tA

√ 2 2 3 t − 3y3 − v4t t f= e f x − y − (2u − v), u − v dy dv. 2 2πt 2 R2

Set t f˜(x, u; y, v, t) = f x − y − (2u − v), u − v . 2 Then we can check the equality ∂xβ1 ∂uβ2 f˜(x, u; y, v, t) =

0lβ2

β +l β −l t 1 2 cl t ∂x ∂u f x − y − (2u − v), u − v , 2 l

(169)

3088


where each cl is a suitable constant and c0 = 1. Then by the inequality m ρ m = 1 + x 2 + u2 2

m

m t m m m m

C 1 + x − y − (2u − v) + |y| + 1 + t |u − v| + 1 + t |v| , 2 and the Minkovski inequality, we have m β β tA ρ ∂ 1 ∂ 2 e f x

u

L2

C 1 + t m f H β1 +β2 , m

t > 0,

(170)

for some m m, which gives (165). Estimate (166) is proved similarly from j j ∂x1 ∂u2 et A f

√ t 3 t j1 j1 − 3y32 − v4t2 j2 t ∂ ∂ f x − y − ∂ (2u − v), u − v dy dv. = − e y y v 2 2 2πt 2 R2

To prove (167) we note that β β +1 t ∂x 1 ∂u 2 f x − y − (2u − v), u − v 2 t β1 β2 +1 ˜ l β1 +l β2 +1−l cl t ∂x ∂u f x − y − (2u − v), u − v . = ∂x ∂u f (x, u; y, v, t) − 2 1lβ2 +1

Thus we have from (166) and ∂x et A f = et A ∂x f , β β tA ∂ 1 ∂ 2 e ∂u f x

u

L2m

∂xβ1 ∂uβ2 +1 et A f L2 + C m

1lβ2 +1

1

C 1 + t m t − 2 f H β1 +β2 + C m

t l ∂x et A ∂xβ1 +l−1 ∂uβ2 +1−l f L2

1lβ2 +1

m

3

t l 1 + t m t − 2 f H β1 +β2 m

1

C 1 + t m t − 2 f H β1 +β2 , m

for some m

m . Estimate (168) is obtained from (166) and the Laplace formula ∂u (I − ∞ A)−1 f = 0 e−t ∂u et A f dt. This completes the proof. 2 For each λ > 0 and a, θ ∈ R, set 5 3 1 (Rλ f )(x) = λ 2 f λ 2 x, λ 2 u ,

(171)

(τa,θ f )(x) = f (x + θ a, u + a).

(172)

Then R = {Rλ }λ∈R× and {Tθ }θ∈R with Tθ = {τa,θ }a∈R+ are a scaling and a strongly continuous one parameter family of translations in L2m , respectively. The generators of R and Tθ are given by


B=

3089

3x u

Dom(B) = f ∈ L2m

∂x + ∂u f ∈ L2m , 2 2

2

Dom(Dθ ) = f ∈ Lm (θ ∂x + ∂u )f ∈ L2m .

3x u 5 ∂x + ∂u + , 2 2 2 Dθ = θ ∂x + ∂u ,

Furthermore, Γa,θ is given by Γa,θ = a∂x τa,θ ,

Dom(Γa,θ ) = f ∈ L2m a∂x τa,θ f ∈ L2m .

For j, k ∈ N ∪ {0} we introduce a function Hj,k by j Hj,k = cj,k ∂x (∂x

k −3(x− u2 )2 − 14 u2

+ ∂u ) e

,

1 cj,k = − 3

|j | √ 3 . 2πj !k!

(173)

Let m > 1 and let L2m,0 be a subspace of L2m defined by L2m,0

2

= f ∈ Lm f (x, u) dx du = 0 .

(174)

R2

Proposition 7.9. Let q ∈ N with q 3. Let m q2 + 74 and X = L2m,0 . Then under the setting of (164), (171), and (172), the conditions (E1), (E2), (T1), (T2), (A1), and (A2) hold with n = 1, μ1 = 12 , = 12 , and ζ = q2 . The eigenprojection P0,0 for the eigenvalue 0 of A (see (23)) is given by P0,0 f =

uf (x, u) dx du H0,1 ,

(175)

R2

and the number ν0 defined by (38) is 1. Moreover, if q ∈ N with q 2 and m > q2 + 74 and X = L2m,0 then the conditions (E1), (E2), (T1), (T2), (A1), and (A2)’ are satisfied with μj = 12 ,

= 12 , and ζ = q2 . The number ν˜ 0 defined by (45) is 1 also in this case. Proof. It is easy to see from (163), (171), and (172) that Rλ eλt A = et A Rλ and τa,θ+t et A = et A τa,θ for each t > 0, which implies (E1) and (E2). Let f ∈ Dom(B) ∩ Dom(Dθ ) ∩ Dom(Γa,θ ). We will show τa,θ f ∈ Dom(B). Indeed, if a = 0 then τ0,θ = I and thus f ∈ Dom(B). If a = 0 then ∂x f ∈ L2m by (173). Thus ∂u f also belongs to L2m since Dθ f = (θ ∂x + ∂u )f ∈ L2m . The assertion τa,θ f ∈ Dom(B) follows from the equality

3x u 3(x + θ a) u+a 3θ a a ∂x + ∂u τa,θ f = ∂x + ∂u τa,θ f − ∂x + ∂u τa,θ f. 2 2 2 2 2 2

The condition (T1) is now verified from the above equality and (173). Suppose that f ∈ Dom(A) ∩ Dom(B) ∩ Dom(D1 ) satisfies a1 Bf + a2 D1 f = 0.

(176)

3090


Then by (168) we have ∂u f ∈ L2m , and hence, ∂x f ∈ L2m by the definition of D1 . If a1 = 0 then we multiply both sides of (176) by f and integrate over R2 , which yields by integration by parts that 3a21 f 2L2 = 0, i.e., f = 0. If a1 = 0 then multiplying both sides of (176) by xf and integrating over R2 , we have is expressed as

a2 2 2 f L2

= 0. This implies (T2). We note that etA = e(1−e

√ 5t 2 (u−v)2 3t t 3e 2 − 3 3 {x−y− a(t) 2 (u+v)} − 4a(t) a(t) e f ye 2 , ve 2 dy dv, e f= 2 2πa(t) tA

−t )A

Re t

(177)

R2 qt

where a(t) = 1 − e−t . Let m q2 + 74 . In [14] it is proved that ress (etA ) e− 2 in L2m and the spectrum of A in L2m is estimated as 1 l

q ∪ − l = 1, 2, . . . . σ (A) ⊂ Re(μ) − 2 2 2

(178)

Moreover, if q + 1 > l and l ∈ N ∪ {0} then 12 − 2l is a semisimple eigenvalue and its eigenspace is spanned by {Hj,k }3j +k=l ; the associated eigenprojection is given by

∗ f, Hj,k Hj,k ,

Pl f =

(179)

3j +k=l

where

f, g =

u 2 1 2 +4u

f (x, u)g(x, u)e3(x− 2 )

dx du,

R2 u 2 1 2 −4u

∗ = (∂ + 3∂ )j (∂ + 2∂ )k e−3(x− 2 ) and Hj,k x u x u

P0 f =

f (x, u) dx du H0,0 ,

R2

. Especially, the direct calculations show that

P1 f =

uf (x, u) dx du H0,1 .

R2

So the first and the second eigenvalues of A in L2m,0 are simple and given by 0 and − 12 , respectively, and the eigenprojection for the eigenvalue 0 is P0,0 f = P1 f =

uf (x, u) dx du H0,1 .

(180)

R2

Since the third eigenvalue of A in L2m,0 is −1, we observe that ν0 defined by (38) is equal to 1. This completes the proof. 2 Let O ⊂ R2 be a small open ball centered at the origin. Let f ∈ L2m,0 . For (161) the map S(·; f ) : O → L2m,0 is defined by


S(y; f )(x, u) = (τy1 ,1+y2 R − 52

= (1 + y2 )

1 1+y2

f

3091

f )(x, u) x + y1 (1 + y2 ) 3

(1 + y2 ) 2

,

u + y1 1

(1 + y2 ) 2

.

(181)

From the definition of H0,1 we have Proposition 7.10. Let m 0. Then the map S(·; H0,1 ) : O → L2m,0 is C ∞ . Let us give the estimates for derivatives of etA f , which are essentially obtained in [14]. Proposition 7.11. Let m 94 . Then for any nonnegative integer s and j1 , j2 ∈ {0, 1} we have for any f ∈ L2m,0 ∩ Hms , j1 j2 tA ∂x ∂u e f

Hms

3j1 +j2 C 1 + t − 2 f Hms ,

t > 0.

(182)

Moreover, we have for any f ∈ L2m ∩ Hms , tA e ∂u f Remark 7.4. If m

11 4

Hms

1 C 1 + t − 2 f Hms ,

t > 0.

(183)

and if f ∈ Q0,0 L2m,0 ∩ Hms , then (182) is replaced by

j1 j2 tA ∂x ∂u e f

Hms

3j1 +j2 t C 1 + t − 2 e− 2 f Hms ,

t > 0.

(184)

t

This is proved from the estimate etA f L2m Ce− 2 f L2m for f ∈ Q0,0 L2m,0 , (182), and the semigroup property of etA . We omit the details here. Proof of Proposition 7.11. Estimate (182) is already observed in [14], but we give the proof −t for convenience to the reader. Let 0 < t 2. Recalling the relation etA = e(1−e )A Ret , we have from (166) that j1 j2 tA ∂x ∂u e f

Hms

Ct −

3j1 +j2 2

Ret f Hms Ct −

3j1 +j2 2

f Hms ,

0 < t 2.

For the case t 2 we use the semigroup property and obtain j1 j2 tA ∂x ∂u e f

Hms

C e(t−1)A f L2 Cf L2m . m

Let us prove (183). It suffices to consider the case t 2 by the semigroup property. Then from (167) we get tA e ∂u f

Hms

t 1 t = Ret e(e −1)A ∂u f H s C e(e −1)A ∂u f H s Ct − 2 f Hms .

This completes the proof.

m

2

m

3092


Next we consider the nonlinear term N (f ) = E(f )∂u f . 2 Proposition 7.12. Let m 11 4 and X = Lm,0 . Then N (f ) = E(f )∂u f satisfies (N1), (N2), and 1 (N3) with q = α = 2, β = 2 , and 0 = 0.

Proof. From (162) it is easy to see that for f ∈ L2m with m > 1, N (f )

L2m

E(f )L∞ ∂u f L2m f L1 ∂u f L2m Cf L2m ∂u f L2m .

(185)

On the other hand, from (182) with l = 0 and k = 1 we see that ∂u (I − A)−1 is a bounded operator in L2m,0 as in the proof of (168). So (185) yields N (f )L2m Cf L2m f Dom(A) . Next we claim that N maps Dom(A) ∩ L2m,0 into

Q0,0 L2m,0 = f ∈ L2m,0 uf (x, u) dx du = 0 . R2

x Indeed, since R2 f (x, u) dx du = 0 we have E(f ) = −∞ R f (y, v) dv fy =: F (x). Then it follows that by integration by parts,

uE(f )∂u f dx du = −

R2

E(f )f dx du

R2

=− R

=−

F (x) f (x, u) du dx R

F (x)F (x) dx = 0.

R

This proves the claim, and (N1) holds. The condition (N3) is easily checked, so we omit the proof of it. Finally we consider (N2). Let N (t, f ) = etA N (f ) with f ∈ L2m,0 . If t 1 then we have from (183), N(t, f ) − N (t, g)

L2m

1 Ct − 2 E(f − g)f + E(g)(f − g)L2 Ct

− 12

f − gL2m f L2m + gL2m .

m

When t 1 we first note that ∂u (E(f − g)f + E(g)(f − g)) ∈ Q0,0 L2m,0 if f, g ∈ L2m,0 ∩ t

Dom(A). Thus in this case, by using the estimate etA f L2m Ce− 2 f L2m for f ∈ Q0,0 L2m,0 and (183), we have N(t, f ) − N (t, g)

L2m

t Ce− 2 e1A ∂u E(f − g)f + E(g)(f − g) L2 m − 2t Ce f − gL2m f L2m + gL2m .


3093

From the density arguments the above inequality is valid for any f, g ∈ L2m,0 . Since the Fréchet derivative of N(t, ·) is formally given by N (t, f )h = etA ∂u (E(f )h + E(h)f ), by the similar arguments as above, we have

N (t, f )h − N (t, g)h

L2m

C

1+t t

1 2

t

e− 2 f − gL2m hL2m .

Hence (N2) follows and the proof of Proposition 7.12 is completed.

2

Let m 11 4 . Then by Theorem 2.1 there is a δ0 > 0 such that for each δ with |δ| δ0 there is 2 a Uδ ∈ Lm which gives a self-similar solution R 1 Uδ to (161) and is C 2 in L2m with respect to 1+t

δ. Note that Uδ is of the form Uδ = δH0,1 + vδ from Theorem 4.1, and vδ ∈ Q0,0 L2m,0 ∩ Dom(A) solves −1

vδ = (−A)

∞ N (δH0,1 + vδ ) =

etA N (δH0,1 + vδ ) dt.

(186)

0

Since vδ ∈ Q0,0 L2m,0 ∩ Dom(A) implies ∂u vδ ∈ L2m , N (δH0,1 + vδ ) belongs to Q0,0 L2m,0 by 2 . For this purpose we (185). In order to apply Theorem 2.3 we need the regularity of vδ ∈ Hm+2 2 resolve (186) in Hm+2 below. Proposition 7.13. Let m 11 4 . If |δ| is sufficiently small, then there is a solution vδ to (186) in 2 ¯ 2 for some constant C¯ > 0. Moreover, this solution is unique in Hm+2 such that vδ H 2 C|δ| m+2 ¯ the set {f ∈ L2m | f L2 C|δ|}, and vδ is C 2 in H 2 with respect to δ. m+2

m

Proof. We first prove the uniqueness. Let v˜1 and v˜2 be two solutions to (186) with v˜i L2m ¯ C|δ|. Recalling the definition N (t, f ) = etA N (f ), we have ∞ v˜1 − v˜2 =

∞

N (t, δH0,1 + v˜1 )(v˜1 − v˜2 ) dt − 0

N (t, v˜1 − v˜2 ) dt. 0

Then from (N2) we observe that ¯ v˜1 − v˜2 L2m C C|δ|

∞

¯ t − 2 e− 2 v˜1 − v˜2 L2m dt C C|δ| v˜1 − v˜2 L2m . 1

t

0

Thus if |δ| is small enough, we have v˜1 = v˜2 . Next we find a solution to (186) in the ball Bδ = 2 ¯ 2 }. The proof is just as same as in Theorem 4.1. For v ∈ Bδ we set | f H 2 C|δ| {f ∈ Hm+2 m+2

∞ Ψ (v) =

∞ e N (δH0,1 + v) dt = tA

0

0

etA ∂u E(δH0,1 + v)(δH0,1 + v) dt.

(187)

3094


From (183) and (184) we can see that tA e ∂u E(f )f

2 Hm+2

1+t C t

1

m+2

holds. Combining with the estimate E(f )f H 2

m+2

CδH0,1 + v2

2 Hm+2

t e− 2 E(f )f H 2

2

Cf 2

2 Hm+2

, we get Ψ (v)H 2

m+2

. Similarly we also have

Ψ (v1 ) − Ψ (v2 )

2 Hm+2

C |δ| + v1 H 2

m+2

+ v2 H 2

m+2

v1 − v2 H 2 . m+2

These estimates are enough to conclude that Ψ is a contraction mapping in Bδ , and hence there is a unique fixed point vδ in Bδ . Since N is a bilinear form, it is also easy to see that vδ is C 2 in 2 with respect to δ. This completes the proof. 2 Hm+2 By Theorem 2.2 and the spectral property of A it is not difficult to show that the solution Ω(t) ∈ C([0, ∞); L2m,0 ) to (161) with Ω(0)L2m 1 satisfies Ω(t) − (1 + t)− 52 Uδ

x

u

1

−3+ p2 +

C (1 + t)

, (188) (1 + t) 2 Lp for all t 1, > 0, and 1 p 2, where δ = R2 uΩ(0, x, u) dx du. We will improve the asymptotic profile by considering a shift of the self-similar solution. Let |δ| < δ0 1. For (y0 , y) ∈ (−δ0 + δ, δ0 − δ) × O ⊂ R3 we set (1 + t)

3 2

,

− 52

H (y0 , y; Uδ ) = S(y; Uδ+y0 ) = (1 + y2 )

Uδ+y0

x + y1 (1 + y2 ) 3

(1 + y2 ) 2

,

u + y1

. 1

(1 + y2 ) 2

(189)

Then Proposition 7.13 immediately leads to Corollary 7.1. Let m into L2m .

11 4 .

Then H (y0 , y; Uδ ) is C 2 as a mapping from (−δ0 + δ, δ0 − δ) × O

2 2 Proof. From Proposition 7.13 we see Uδ = δH0,0 + vδ ∈ Hm+2 and is C 2 in Hm+2 with respect to δ. Then from the definition of (189) we have the claim. The proof is completed. 2

Now we can apply Theorem 2.4 to (161) and obtain 2 Theorem 7.4. Let m > 11 4 . Assume that Ω(t) ∈ C([0, ∞); Lm,0 ) is the solution to (161) satisfying Ω(0)L2m 1 and R2 uΩ(0, x, u) dx du = δ = 0. Then there are η(δ) ˜ ∈ R and y1∗ ∈ R ˜ = 0 and such that limδ→0 η(δ)

∗ ∗ 7 2 ˜ Ω(t) − (1 + t)− 52 Uδ x + (1 + t)y1 , u + y1 C (1 + t)− 2 + p +η(δ)+ 3 1 p (1 + t) 2 (1 + t) 2 L holds for all t 1, > 0, and 1 p 2.

(190)


Proof of Theorem 7.4. Since ν˜ 0 = 1 > that there are η(δ) ˜ and y1∗ ∈ R such that

1 2

3095

= μ1 by Proposition 7.9, we see from Theorem 2.4

˜ R1+t Ω(t) − S˜ (1 + t)− 12 y ∗ ; Uδ 2 C(1 + t)−1+η(δ)+ , 1 L m

if t 1. Hence from Rλ f Lp = λ Ω(t) − R

1 1+t

2(1− p1 )+ 12

f Lp we have

1 − 7 + 2 +η(δ)+ ˜ , S˜ (1 + t)− 2 y1∗ ; Uδ Lp C(1 + t) 2 p

for 1 p 2, which gives (190). The proof is completed.

2

References [1] J. Aguirre, M. Escobedo, E. Zuazua, Self-similar solutions of a convection–diffusion equation and related elliptic problems, Comm. Partial Differential Equations 15 (2) (1990) 139–157. [2] P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (2) (1998) 715–743. [3] A. Carpio, Asymptotic behavior for the vorticity equations in dimensions two and three, Comm. Partial Differential Equations 19 (1994) 827–872. [4] I.-L. Chern, T.-P. Liu, Convergence to diffusion waves of solutions for viscous conservation laws, Comm. Math. Phys. 110 (1987) 503–517. [5] P. Degond, Global existence of smooth solutions for the Vlasov–Fokker–Planck equation in 1 and 2 space dimensions, Ann. Sci. Ec. Norm. Super. (4) 19 (4) (1986) 519–542. [6] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., SpringerVerlag, New York, 2000. [7] M. Escobedo, O. Kavian, Variational problems related to self-similar solutions for the heat equation, Nonlinear Anal. 11 (1987) 1103–1133. [8] M. Escobedo, E. Zuazua, Large time behavior for convection–diffusion equations in Rn , J. Funct. Anal. 100 (1991) 119–161. [9] Th. Gallay, C.E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier–Stokes and vorticity equations on R2 , Arch. Ration. Mech. Anal. 163 (2002) 209–258. [10] Th. Gallay, C.E. Wayne, Global Stability of vortex solutions of the two dimensional Navier–Stokes equation, Comm. Math. Phys. 255 (1) (2005) 97–129. [11] Th. Gallay, C.E. Wayne, Existence and stability of asymmetric Burgers vortices, J. Math. Fluid Mech. 9 (2007) 243–261. [12] M.-H. Giga, Y. Giga, J. Saal, Nonlinear Partial Differential Equations: Asymptotic Behaviors of Solutions and Self-Similar Solutions, Birkhäuser, 2010. [13] Y. Giga, T. Kambe, Large time behavior of the vorticity of two dimensional viscous flow and its application to vortex formation, Comm. Math. Phys. 117 (1988) 549–568. [14] Y. Kagei, Invariant manifolds and long-time asymptotics for the Vlasov–Poisson–Fokker–Planck equation, SIAM J. Math. Anal. 33 (2) (2001) 489–507. [15] Y. Kagei, Y. Maekawa, On asymptotic behaviors of solutions to parabolic systems modelling chemotaxis, MI Preprint Series 2009-30, Kyushu University, 2009. [16] T. Kato, Perturbation Theory for Linear Operators, reprint of the 1980 edition Classics Math., Springer-Verlag, Berlin, 1995. [17] M. Kato, Sharp asymptotics for a parabolic system of chemotaxis in one space dimension, Osaka Univ. Research Report in Math. 07-03. [18] S. Kawashima, Self-similar solutions of a convection–diffusion equation, Lecture Notes in Num. Appl. Anal. 12 (1993) 123–136. [19] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progr. Nonlinear Differential Equations Appl., Birkhäuser, 1995. [20] Y. Maekawa, Spectral properties of the linearization at the Burgers vortex in the high rotation limit, J. Math. Fluid Mech., doi:10.1007/s00021-010-0048-4, in press.

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[21] J.C. Miller, A.J. Bernoff, Rates of convergence to self-similar solutions of Burgers’s equation, Stud. Appl. Math. 111 (1) (2003) 29–40. [22] T. Nagai, T. Yamada, Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space, J. Math. Anal. Appl. 336 (2007) 704–726. [23] Y. Naito, Asymptotically self-similar solutions for the parabolic system modelling chemotaxis, in: Self-Similar Solutions of Nonlinear PDE, in: Banach Center Publ., vol. 74, Polish Acad. Sci., Warsaw, 2006, pp. 149–160. [24] K. Nishihara, Asymptotic profile of solutions to a parabolic system of chemotaxis in one dimensional space, preprint. [25] H.D. Victory, B.P. O’Dwyer, On classical solutions of Vlasov-Poisson Fokker–Planck systems, Indiana Univ. Math. J. 39 (1990) 105–156. [26] T.P. Witelski, A.J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math. 100 (1998) 153–193. [27] T. Yanagisawa, Asymptotic behavior of solutions to the viscous Burgers equation, Osaka J. Math. 44 (1) (2007) 99–119.


Regularity theory for the fractional harmonic oscillator ✩ Pablo Raúl Stinga a , José Luis Torrea b,∗ a Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain b Departamento de Matemáticas and ICMAT-CSIC-UAM-UCM-UC3M, Facultad de Ciencias,

Universidad Autónoma de Madrid, 28049 Madrid, Spain Received 28 April 2010; accepted 1 February 2011 Available online 12 February 2011 Communicated by Gilles Godefroy

Abstract In this paper we develop the theory of Schauder estimates for the fractional harmonic oscillator H σ = k,α is defined, in which we (− + |x|2 )σ , 0 < σ < 1. More precisely, a new class of smooth functions CH study the action of H σ . In fact these spaces are those adapted to the operator H , hence the suited ones for this type of regularity estimates. In order to prove our results, an analysis of the interaction of the Hermite– k,α is needed, that we believe of independent interest. Riesz transforms with the Hölder spaces CH © 2011 Elsevier Inc. All rights reserved. Keywords: Fractional harmonic oscillator; Schauder estimate; Fractional integral; Riesz transform

1. Introduction For a given partial differential operator L, the analysis of its regularity properties with respect to Hölder classes is one of the tools employed in the theory to prove important facts about partial differential equations. Indeed, being a bit imprecise, it is well known that if f is a Hölder continuous function with exponent α, then the equation −u = f has a unique solution u, whose second order derivatives belong to C α , and uC 2,α is controlled by f C α . This result was first applied to obtain classical solutions of second order elliptic equations of the form Lu = f (see for instance [7, Chapter 6]). Recently, and motivated by the obstacle problem for the fractional ✩

Research supported by Ministerio de Ciencia e Innovación de España under project MTM2008-06621-C02-01.


E-mail addresses: [email protected] (P.R. Stinga), [email protected] (J.L. Torrea). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.02.003

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P.R. Stinga, J.L. Torrea / Journal of Functional Analysis 260 (2011) 3097–3131

Laplacian, L. Silvestre proved in [11] (see also his thesis [10]) the regularity properties for the operator (−)σ , 0 < σ < 1, when acting on Hölder spaces. For more applications see [5] and [6]. Let H be the most basic Schrödinger operator in Rn , n 1, the harmonic oscillator: H = − + |x|2 . The fractional powers H σ , 0 < σ < 1, were introduced in [13]. The aim of this paper is to prove regularity estimates in Hölder classes for the fractional k,α , different than harmonic oscillator H σ . For this purpose, we define new Hölder spaces CH k,α σ the classical Hölder spaces C , in which the smoothness properties of H are analyzed, see Definition 1.1 and Theorem A. k,α The classes CH are the natural spaces associated to H . This becomes evident, for instance, in the fact that the Hermite–Riesz transforms have the expected behavior: they preserve them, see Theorem 4.1. Also the fractional integrals produce a kind of “inverse fractional derivative” k,α , see Theorem B. process when acting in CH Our estimates, together with Harnack’s inequality for H σ proved in [13], are the basic regularity estimates one expects to get for the fractional powers of a second order operator. Moreover, we found the right spaces for which Schauder estimates are appropriated. We expect to obtain the correct regularity estimates for nonlinear problems related to the fractional harmonic oscillator in these spaces. Applications will appear elsewhere. Let us introduce the definition of H σ . For a function f in Schwartz’s class S and 0 < σ < 1, the fractional harmonic oscillator H σ is given by the classical formula 1 H f (x) = Γ (−σ ) σ

∞ −tH dt e f (x) − f (x) 1+σ , t

(1.1)

0

where v(x, t) = e−tH f (x) is the solution of the heat-diffusion equation ∂t v + H v = 0 in Rn × (0, ∞), with initial datum v(x, 0) = f (x) on Rn . In [13] it is shown that H σ f (x) =

f (x) − f (z) Fσ (x, z) dz + f (x)Bσ (x),

x ∈ Rn , f ∈ S,

(1.2)

Rn

where 1 Fσ (x, z) = −Γ (−σ ) 1 Bσ (x) = Γ (−σ )

∞ Gt (x, z)

dt t 1+σ

0

∞

Gt (x, z) dz − 1

0

Rn

,

dt t 1+σ

,

(1.3)

and Gt (x, z) is the kernel of the heat-diffusion semigroup generated by H , see (3.1). Next we define the Hölder spaces in which the regularity properties of the operators will be considered.


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Definition 1.1. Let 0 < α 1. A continuous function u : Rn → R belongs to the Hermite–Hölder 0,α associated to H , if there exists a constant C, depending only on u and α, such that space CH u(x1 ) − u(x2 ) C|x1 − x2 |α ,

and u(x)

C , (1 + |x|)α

for all x1 , x2 , x ∈ Rn . With the notation [u]C 0,α =

sup x1 ,x2 ∈Rn x1 =x2

|u(x1 ) − u(x2 )| , |x1 − x2 |α

α and [u]M α = sup 1 + |x| u(x), x∈Rn

0,α we define the norm in the spaces CH to be

uC 0,α = [u]C 0,α + [u]M α . H

When working with the harmonic oscillator H some special first order partial differential k,α operators are considered, see (3.3), which are the natural derivatives. Then the classes CH can be defined in a usual way, see Definition 3.1. We present now our first main result. Theorem A. Let α ∈ (0, 1] and σ ∈ (0, 1). 0,α 0,α−2σ and 2σ < α. Then H σ u ∈ CH and (A1) Let u ∈ CH

σ H u

0,α−2σ CH

CuC 0,α . H

1,α 1,α−2σ (A2) Let u ∈ CH and 2σ < α. Then H σ u ∈ CH and

σ H u

1,α−2σ CH

CuC 1,α . H

1,α 0,α−2σ +1 (A3) Let u ∈ CH and 2σ α, with α − 2σ + 1 = 0. Then H σ u ∈ CH and

σ H u

0,α−2σ +1 CH

CuC 1,α . H

l,β

k,α and assume that k + α − 2σ is not an integer. Then H σ u ∈ CH where l is (A4) Let u ∈ CH the integer part of k + α − 2σ and β = k + α − 2σ − l. k,α are the reasonable The last theorem can be interpreted as saying that the Hölder spaces CH σ classes in order to obtain Schauder type estimates for H . Indeed, if we define the negative powers of H , i.e. the fractional integral operators

H

−σ

1 f (x) = Γ (σ )

∞

e−tH f (x)

0

then we are able to prove our second main result:

dt t 1−σ

,

0 < σ 1,

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0,α Theorem B. Let u ∈ CH , for some 0 < α 1, and 0 < σ 1. 0,α+2σ and (B1) If α + 2σ 1, then H −σ u ∈ CH

−σ H u

0,α+2σ CH

CuC 0,α . H

1,α+2σ −1 (B2) If 1 < α + 2σ 2, then H −σ u ∈ CH and

−σ H u

1,α+2σ −1 CH

CuC 0,α . H

2,α+2σ −2 and (B3) If 2 < α + 2σ 3, then H −σ u ∈ CH

−σ H u

2,α+2σ −2 CH

CuC 0,α . H

It is worth noting that, if in Theorem B(B3) we take σ = 1, we get the Schauder estimate for 0,α . the solution to H u = f , in Rn , with f ∈ CH To prove the two main theorems of this paper we need a study of the first and second order 0,α Hermite–Riesz transforms, Ri and Rij , acting on the spaces CH , that we believe of independent interest. See the final part of Section 3 and Theorem 4.1. The first main task in our paper is to obtain explicit pointwise expressions for all the operators k,α , and the second one involved, when they are applied to functions belonging to the spaces CH is to actually prove the regularity estimates. Section 2 contains two abstract propositions dealing with these two aspects: Proposition 2.1 takes care of the pointwise formulas and Proposition 2.3 contains a regularity result. We will apply, in a systematic way, both propositions in order to reach our objectives: see Section 3 for all the pointwise formulas, and Section 4 for the proofs of Theorems A and B. In Section 5 we collect all the computational lemmas used in the previous sections. In some recent papers, B. Bongioanni, E. Harboure and O. Salinas studied the boundedness of fractional integrals (see [2]) and Riesz transforms (see [3]), associated to a certain class of β Schrödinger operators L = − + V , in spaces of BMOL type, 0 β < 1, using Harmonic β Analysis techniques. In [2, Proposition 4], they showed that the spaces BMOL coincide with a β 0,β Hölder type space ΛL , 0 < β < 1, with equivalent norms. In the case V = |x|2 , our space CH β coincides with their space ΛH , for 0 < β < 1. A natural question to think about is the possibility of getting (at least the local part of) our results by modifying, in an appropriate way, the kernel of the classical fractional Laplace operator. In our opinion our procedure is the natural one and we haven’t found a smooth bridge to pass from one case to the other. Even more, some recent (local) results by R.F. Bass in [1] about stable-like operators cannot be applied in our case because, clearly, his assumption on the kernel A(x, h) [1, Assumption 1.1] is not fulfilled by our kernel Fσ (x, z), see Lemma 5.4 below. Moreover, he does not allow for α + β to be an integer [3, Assumption 1.1], but we do. We want to complete the thought of this paragraph by establishing the parallelism with possible definitions of Sobolev spaces for the harmonic oscillator that were considered by R. Radha and S. Thangavelu in [9], by S. Thangavelu in [15] and also by B. Bongioanni and J.L. Torrea in [4].


3101

In that case natural definitions of Sobolev spaces were given in order to get the results for the harmonic oscillator. Throughout this paper, the letter C denotes a positive constant that may change in each occurrence and it will depend on the parameters involved (whenever it is necessary, we point out this dependence with subscripts), and Γ stands for the Gamma function. Recall that Γ (−σ ) = −Γ (1 − σ )/σ , 0 < σ < 1. Without mentioning it, we will repeatedly apply the inequality r η e−r Cη e−r/2 , η 0, r > 0. 2. Two abstract results Proposition 2.1. Let T be a bounded operator on S such that Tf, g = f, T g , for all f, g ∈ S. Assume that Tf (x) =

f (x) − f (z) K(x, z) dz + f (x)B(x),

x ∈ Rn ,

Rn

where the kernel K verifies K(x, z)

2 C − |x||x−z| − |x−z| C C , e e |x − z|n+γ

x, z ∈ Rn ,

(2.1)

for some −n γ < 1, and B is a continuous function with polynomial growth at infinity. Let 0,γ +ε u ∈ CH , with 0 < γ + ε 1, ε > 0. Then T u is well defined as a tempered distribution and it coincides with the continuous function T u(x) =

u(x) − u(z) K(x, z) dz + u(x)B(x),

x ∈ Rn .

(2.2)

Rn

Proof. By (2.1) and the smoothness of u, the integral in (2.2) is absolutely convergent. Since B has polynomial growth at infinity, the right-hand side of (2.2) defines a tempered distribution. n Let us take γ +ε 0. Let fj (x) := ζ (x/j )(u ∗ W1/j )(x), j ∈ N, where Wt (z) = (4πt)−n/2 e−|z| /(4t) is the Gauss–Weierstrass kernel and ζ is a nonnegative smooth cutoff function (that is, ζ ∈ Cc∞ (Rn ), 0 ζ 1, ζ ≡ 1 in B1 (0), ζ ≡ 0 in B2c (0), and |∇ζ | < C in Rn ). Note that each fj belongs to S. It is easy to check that the sequence {fj }j ∈N converges to u in Lp (Rn ) and uniformly in BR (x) for each x ∈ Rn , and [fj ]C 0,γ +ε CuC 0,γ +ε =: M. As j → ∞, Tfj → T u in S . Since H

B is a continuous function, fj B converges uniformly to uB in BR (x0 ), x0 ∈ Rn . There exists 0 < δ < R/2 such that M Bδ (0)

η |z|ε−n dz . 3

3102


For x ∈ BR/2 (x0 ), we write

fj (x) − fj (z) K(x, z) dz =

Rn

+

Bδ (x)

= I + II.

Bδc (x)

Then, by the choice of δ, 2 |I | + u(x) − u(z) K(x, z) dz η. 3 Bδ (x)

We also have

1/p II − C fj (x) − u(x) + C fj (z) − u(z)p dz u(x) − u(z) K(x, z) dz Bδc (x)

Bδc (x)

η , 3 for sufficiently large j , uniformly in x ∈ BR/2 (x0 ). Therefore,

fj (x) − fj (z) K(x, z) dz ⇒

Rn

u(x) − u(z) K(x, z) dz,

j → ∞,

Rn

in BR/2 (x0 ). Hence, by uniqueness of the limits, T u is a function that coincides with (2.2), and it is a continuous function because it is the uniform limit of continuous functions. 2 Remark 2.2. In the context of Proposition 2.1, assume that, instead of having estimate (2.1) on the kernel, we just know that |K(x, z)| Φ(x − z), where Φ ∈ Lp (Rn ), and p is the conjugate 0,α n < p < ∞. Then, it is enough to take u ∈ CH , for some exponent of some p such that γ +ε 0 < α 1, to get the same conclusion, since the approximation procedure given in the proof above can also be applied in this situation. Proposition 2.3. Let T be an operator satisfying the hypotheses of Proposition 2.1, with 0 γ < 1 and 0 < γ + ε 1, for some 0 < ε < 1. Assume that the kernel K and the function B also satisfy: |z||x2 −z|2

|x2 −z|2

− 1 −x2 | C (a) |K(x1 , z) − K(x2 , z)| C |x |x−z| e− C , when |x1 − z| > 2|x1 − x2 |. n+1+γ e 2 (b) There exists a constant C > 0 such that | |x−z|>r K(x, z) dz| Cr −γ , for all x ∈ Rn . (c) For all x ∈ Rn , |B(x)| C(1 + |x|)γ , and ∇B ∈ L∞ (Rn ). 0,γ +ε

Then T maps CH

0,ε into CH continuously.


3103

= B(x2 , 4|x1 − x2 |) and B = Proof. Given x1 , x2 ∈ Rn , let B = B(x1 , 2|x1 − x2 |), B B(x2 , |x1 − x2 |). We write T u(x) = u(x) − u(z) K(x, z) dz + u(x) − u(z) K(x, z) dz + u(x)B(x) Bc

B

= I (x) + II(x) + III(x). By (2.1) we have I (x1 ) − I (x2 ) u(x1 ) − u(z) K(x1 , z) dz + u(x2 ) − u(z) K(x2 , z) dz B

B

− z|γ +ε

|x1 dz + |x1 − z|n+γ

C[u]C 0,γ +ε B

B

|x2 − z|γ +ε dz = C[u]C 0,γ +ε |x1 − x2 |ε . |x2 − z|n+γ

For the difference II(x1 )−II(x2 ), we add the term ±u(x2 )K(x1 , z) and we use the smoothness and cancellation properties of the kernel K(x, z) (hypotheses (a) and (b)) to get: II(x1 ) − II(x2 ) u(x2 ) − u(z) K(x1 , z) − K(x2 , z) dz + u(x1 ) − u(x2 ) K(x1 , z) dz Bc

Bc

γ +ε

C[u]C 0,γ +ε

|x2 − z| Bc

C[u]C 0,γ +ε (B )c

If |x1 − x2 |
2. We write

dz. u(x) − u(z) K(x, z) dz = + Rn

On the one hand,

1 |x−z|< 1+|x|

1 |x−z| 1+|x|

u(x) − u(z)K(x, z) dz C[u]

|x − z|γ +ε dz |x − z|n+γ

C 0,γ +ε

1 |x−z|< 1+|x|

1 |x−z|< 1+|x|

= [u]C 0,γ +ε On the other hand, by (b), u(x) K(x, z) dz 1 |x−z| 1+|x|

Since |x − z|

1 1+|x|

implies that

1 1+|z|

C . (1 + |x|)ε

γ [u]M γ +ε C C 1 + |x| = [u]M γ +ε . γ +ε (1 + |x|) (1 + |x|)ε 2|x − z|, applying (2.1) we get

u(z)K(x, z) dz C[u]M γ +ε

1 |x−z| 1+|x|

|x||x−z|

1 |x−z| 1+|x|

γ +ε e

C[u]M γ +ε

|x − z|

− |x||x−z| − |x−z| C C

1 |x−z| 1+|x|

= C[u]M γ +ε

∞

j =0 2j |x−z|∼ 1+|x|

[u]M γ +ε

|x−z|2

1 e− C e− C (1 + |z|)γ +ε |x − z|n+γ

e−

|x||x−z| C

e |x − z|n+γ

dz

2

dz

|x−z|2

e− C |x − z|n−ε

dz

∞

2j C C 2j ε e− C = [u]M γ +ε , ε (1 + |x|) (1 + |x|)ε j =0

where in the last line the constant C appearing in the exponential is independent of x because |x||x − z| ∼ 2j . By pasting the estimates above the result is proved. 2


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3. The operators k,α , of all the operators involved. In this section we give the pointwise definitions, in the class CH

3.1. The heat-diffusion semigroup: e−tH In our paper, the kernel of the heat-diffusion semigroup generated by H will play an essential role. We shall need the pointwise formula for it. Recall (see [14]) that the eigenfunctions of H are the multi-dimensional Hermite functions 2 hν (x) = e−|x| /2 Ψν (x), ν = (ν1 , . . . , νn ) ∈ Nn0 , where Ψν are the multi-dimensional Hermite polynomials, with positive eigenvalues: H hν = (2|ν| + n)hν , for all ν ∈ Nn0 , |ν| = ν1 + · · · + νn . Moreover, span{hν : ν ∈ Nn0 } = L2 (Rn ). The heat-diffusion semigroup generated by H is given as an integral operator: for u ∈ 1p∞ Lp (Rn ), e−tH u(x) =

(3.1)

Gt (x, z)u(z) dz Rn

=

∞

Rn

= Rn

e

−t (2j +n)

hν (x)hν (z) u(z) dz

|ν|=j

j =0 1

e−[ 2 |x−z| coth 2t+x·z tanh t] u(z) dz. (2π sinh 2t)n/2 2

Note that, for all ν ∈ Nn0 , e−tH hν (x) = e−t (2|ν|+n) hν (x), t 0. With the following change of parameters (due to S. Meda) t=

1+s 1 log , 2 1−s

t ∈ (0, ∞), s ∈ (0, 1),

(3.2)

the heat-diffusion kernel can be expressed as Gt (s) (x, z) =

∞

1 − s j +n/2 j =0

=

1+s

1 − s2 4πs

n/2

hν (x)hν (z)

|ν|=j 1

2 + 1 |x−z|2 ] s

e− 4 [s|x+z|

,

s ∈ (0, 1).

3.2. The fractional operators: H σ and (H ± 2k)σ , k ∈ N Let f, hν = Rn f (z)hν (z) dz. Let us first analyze the fractional harmonic oscillatorH σ . If f ∈ S, the Hermite series expansion ν f, hν hν = ∞ |ν|=k f, hν hν converges to f k=0 n 2 n ∞ n uniformly in R (and also in L (R )), since hν L (R ) C, for all ν ∈ Nn0 , and, for each m ∈ N, −t (2|ν|+n) mf −m . As e−tH f (x) = f, hν hν , from we have |f, hν | H L2 (Rn ) (2|ν| + n) νe (1.1) we get H σ f = ν (2|ν| + n)σ f, hν hν , and the series converges uniformly in Rn . As a consequence of the last reasonings, H σ is a bounded operator in S. Note that, by using Hermite

3106


series expansions, we can check that H σ f, g = f, H σ g , for all f, g ∈ S, and H 1 f = Hf , H 0f = f . The proof of the identity (1.2) can be found in [13]. We sketch it here for completeness. Since e−tH 1(x) is not a constant function, we have 1 Γ (−σ )

∞

dt e−tH f (x) − f (x) 1+σ t

0

1 = Γ (−σ )

∞ 0

=

1 Γ (−σ )

∞ 0

1 = Γ (−σ ) =

Rn

Rn

∞ 0 Rn

dt Gt (x, z)f (z) dz − f (x) 1+σ t

dt Gt (x, z) f (z) − f (x) dz + f (x) Gt (x, z) dz − 1 t 1+σ Rn

dt 1 Gt (x, z) f (z) − f (x) dz 1+σ + f (x) Γ (−σ ) t

∞ −tH dt e 1(x) − 1 1+σ t 0

f (x) − f (z) Fσ (x, z) dz + f (x)Bσ (x).

Rn

The subtle point in the calculations above is to justify the last equality. If 0 < σ < 1/2, the last integral is absolutely convergent. In the case 1/2 σ < 1, a cancellation is involved (which is also exploited in the proof of Theorem 3.2 below), that allows to show that the integral converges as a principal value. As we said in the Introduction, some type of derivatives (first order partial differential operators) are usually considered when working with the operator H . Recall the factorization 1 (Ai A−i + A−i Ai ), 2 n

H=

i=1

where Ai = ∂xi + xi ,

A−i = A∗i = −∂xi + xi ,

i = 1, . . . , n.

(3.3)

In the Harmonic Analysis associated to H , the operators Ai , 1 |i| n, play the role of the classical partial derivatives ∂xi in the Euclidean Harmonic Analysis (see [14,4,8,12]). Now, it is 0,α natural to consider the classes of functions whose k-th derivatives are in CH . k,α Definition 3.1. For each k ∈ N, we define the Hermite–Hölder space CH , 0 < α 1, as the set k n of all functions u ∈ C (R ) such that the following norm is finite:

uC k,α = [u]M α + H

1|i1 |,...,|im |n 1mk

[Ai1 · · · Aim u]M α +

[Ai1 · · · Aik u]C 0,α .

1|i1 |,...,|ik |n


3107

We are ready to show that the pointwise formula for H σ u, when u belongs to the Hölder k,α , is the same as (1.2). class CH Theorem 3.2. Let 0 < α 1 and 0 < σ < 1. 0,α , then (1) If 0 < α − 2σ < 1 and u ∈ CH

H σ u(x) =

u(x) − u(z) Fσ (x, z) dz + u(x)Bσ (x),

x ∈ Rn ,

(3.4)

Rn

and the integral converges absolutely. 1,α (2) If −1 < α − 2σ 0 and u ∈ CH , then H σ u(x) is given by (3.4), where the integral converges as a principal value. 1,1 to have the conclusion of (2). (3) When −2 < α − 2σ −1, it is enough to take u ∈ CH In the three cases above, H σ u ∈ C(Rn ). Proof. If 0 < α − 2σ < 1, then σ < 1/2. The properties of Fσ and Bσ established in Lemmas 5.4 and 5.5 (see Section 5), allow us to apply Proposition 2.1, with K(x, z) = Fσ (x, z), B = Bσ and γ = 2σ < 1, to get (1). Under the hypotheses of (2), we will take advantage of a cancellation to show that the integral in (3.4) is well defined. Suppose that δ > 0. By Lemma 5.4,

u(x) − u(z)Fσ (x, z) dz Cδ uL∞ (Rn )

|x−z|δ

e−

|x−z|2 C

dz < ∞.

|x−z|δ

For ρ ∈ R, the change of parameters (3.2) produces dt ds = dμρ (s) := , 1 2 1+ρ t 1+ρ (1 − s )( 2 log 1+s 1−s )

t ∈ (0, ∞), s ∈ (0, 1),

so that, 1 Fσ (x, z) = −Γ (−σ )

1

1 − s2 4πs

n/2

1

2 + 1 |x−z|2 ] s

e− 4 [s|x+z|

dμσ (s),

(3.5)

0

which, up to the multiplicative constant 1/(−Γ (−σ )), gives

u(x) − u(x − z) Fσ (x, x − z) dz

I= |z| 2σ , an application of Proposition 2.1 with γ = −2σ and ε = α + 2σ gives the result. For the case n 2σ , we use Remark 2.2. 2 We shall also need to work with the derivatives of H −σ u. The following theorem gives the pointwise formula that will be used along the paper. 0,α then, for each Theorem 3.8. Take 0 < α 1 and 0 < σ 1, such that α + 2σ > 1. If u ∈ CH −σ n 1 |i| n, we have Ai H u ∈ S ∩ C(R ), and u(z) − u(x) Ai F−σ (x, z) dz + u(x)Ai H −σ 1(x), x ∈ Rn . Ai H −σ u(x) = Rn

Proof. Let us first prove the result when u = f ∈ S. It is enough to consider 1 i n. We have Ai H −σ f (x) = Ai f (z) − f (x) F−σ (x, z) dz + ∂xi f (x)H −σ 1(x) + f (x)Ai H −σ 1(x). Rn

We want to put the Ai inside the integral. In order to do that, we apply a classical approximation argument given in the proof of Lemma 4.1 of [7], that we sketch here. By estimate (5.10), Lemma 5.7 (see Section 5), and the fact that α + 2σ > 1, the function g(x) = ∂xi f (z) − f (x) F−σ (x, z) dz Rn

= Rn

f (z) − f (x) ∂xi F−σ (x, z) dz − ∂xi f (x)H −σ 1(x)

3114


is well defined. Fix a function φ ∈ C 1 (R) satisfying 0 φ 1, φ(t) = 0 for t 1, φ(t) = 1 for t 2, and 0 φ 2. Define, for 0 < ε < 1/2, hε (x) =

f (z) − f (x) F−σ (x, z)φ ε −1 |x − z| dz.

Rn

Then estimate (5.9) implies that, as ε → 0, hε (x) converges uniformly in Rn to

f (z) − f (x) F−σ (x, z) dz.

Rn

Moreover, hε ∈ C 1 (Rn ), and, again by (5.9) and (5.10), g(x) − ∂x hε (x) i

∂x f (z) − f (x) F−σ (x, z) 1 − φ ε −1 |x − z| dz i

Rn

Cf |x−z| 2σ − 1, then, by (5.10), Proposition 2.1 can be applied with γ = 1 − 2σ and ε = α + 2σ − 1, and, if n = 2σ − 1, we can use Remark 2.2. 2

3.4. The Hermite–Riesz transforms: Ri and Rij The first order Hermite–Riesz transforms are given by Ri = Ai H −1/2 ,

1 |i| n.

These operators were first introduced and studied by Thangavelu [14]. The second order Hermite–Riesz transforms are (see [8,12]) Rij = Ai Aj H −1 ,

1 |i|, |j | n.

Using Hermite series expansions it is easy to check that the first and second order Hermite–Riesz transforms are symmetric operators in S and that they map S into S continuously. Taking σ = 1/2 in Theorem 3.8 we get


3115

0,α Theorem 3.9. If u ∈ CH , 0 < α 1, then, for all 1 |i| n, we have Ri u ∈ S ∩ C(Rn ), and

Ri u(x) =

u(z) − u(x) Ai F−1/2 (x, z) dz + u(x)Ai H −1/2 1(x),

x ∈ Rn .

Rn

Using the properties of the kernel of the second order Riesz transform Rij (x, z) = Ai Aj F−1 (x, z) (see Lemma 5.8 in Section 5), it is easy to get a pointwise description of Rij f , f ∈ S. Hence, we can apply Proposition 2.1, with γ = 0 and ε = α, to have the following theorem. 0,α , 0 < α 1, then, for all 1 |i|, |j | n, we have Rij u ∈ S ∩ C(Rn ), Theorem 3.10. If u ∈ CH and u(z) − u(x) Rij (x, z) dz + u(x)Ai Aj H −1 1(x), x ∈ Rn . Rij u(x) = Rn

4. Proofs of the main results 4.1. Regularity properties of the Hermite–Riesz transforms As we already said in the Introduction, a study of the action of the Hermite–Riesz transforms k,α in the Hölder spaces CH is needed. Theorem 4.1. The Hermite–Riesz transforms Ri and Rij , 1 |i|, |j | n, are bounded opera0,α 0,α 0,α tors on the spaces CH : if u ∈ CH , for some 0 < α < 1, then Ri u, Rij u ∈ CH , and Ri uC 0,α + Rij uC 0,α CuC 0,α . H

H

H

Proof. By Lemmas 5.6, 5.7, and 5.10 of Section 5, and Theorem 3.9, the result for Ri can be deduced applying Proposition 2.3, with γ = 0 and ε = α. Let us consider the operator Rij , for some j ∈ {1, . . . , n}. Then, by Remark 3.4, Rij = Ai Aj H −1 = Ai Aj H −1/2 H −1/2 = Ai (H + 2)−1/2 Aj H −1/2 = Ai (H + 2)−1/2 ◦ Rj . 0,α . When Therefore, it is enough to prove that Ai (H + 2)−1/2 is a continuous operator on CH f ∈ S, we can write −1/2

Ai (H + 2)

f (x) =

f (z) − f (x) Ai F2,−1/2 (x, z) dz + f (x)Ai (H + 2)−1/2 1(x),

Rn

where 1 F2,−1/2 (x, z) = Γ (1/2)

1 0

1−s Gt (s) (x, z) dμ−1/2 (s), 1+s

3116


and (H + 2)−1/2 1(x) =

F2,−1/2 (x, z) dz. Rn

Following the proof of Lemmas 5.6 and 5.7 given in Section 5, it can be checked that the kernel Ai F2,−1/2 (x, z) and the function (H + 2)−1/2 1(x) share the same size and smoothness properties than the kernel Ai F−1/2 (x, z) and the function H −1/2 1(x) stated in the mentioned lemmas (the details are left to the reader). Thus, as a consequence of the results of Section 2, 0,α 0,α 0,α → CH continuously. Therefore Rij is a bounded operator on CH , when Ai (H + 2)−1/2 : CH j ∈ {1, . . . , n}. Note that Rij = ∂x2i ,xj H −1 + xj ∂xi H −1 + xi ∂xj H −1 + xi xj H −1 + δij H −1 , which, at the level of kernels, means that Rij (x, z) = ∂x2i ,xj F−1 (x, z) + xj ∂xi F−1 (x, z) + xi ∂xj F−1 (x, z) + xi xj F−1 (x, z) + δij F−1 (x, z). By the estimates given in Lemmas 5.6, 5.8 and 5.9 of Section 5, we can apply the statements 0,α of Section 2 to show that the operators xi ∂xj H −1 , xi xj H −1 and H −1 are bounded on CH . 0,α 0,α 2 −1 Hence, ∂xi ,xj H maps CH into CH continuously. Observe now that the operator Ri,−j , for j ∈ {1, . . . , n}, can be written as Ri,−j = −∂x2i ,xj H −1 + xj ∂xi H −1 − xi ∂xj H −1 + xi xj H −1 + δij H −1 . The observations above give the conclusion for Ri,−j , j ∈ {1, . . . , n}.

2

For technical reasons we have to consider the first order adjoint Hermite–Riesz transforms, that are defined by R∗i f (x) = H −1/2 Ai f (x) =

F−1/2 (x, z)(Ai f )(z) dz,

f ∈ S, x ∈ Rn , 1 |i| n.

Rn 0,α Theorem 4.2. The operators R∗i , 1 |i| n, are bounded operators on CH , 0 < α < 1. ∗ = H −1/2 A −1/2 . Proof. Observe that, if 1 i n, then, by Remark 3.4, R−i −i = A−i (H + 2) This operator already appeared in the proof of Theorem 4.1, and there we showed that it is a 0,α . bounded operator on CH ∗ On the other hand, R−i = −H −1/2 ∂xi + H −1/2 xi . But by Lemmas 5.6, 5.7, and 5.9 of Section 5, and Proposition 2.3, we can see that the operator f → H −1/2 xi f , initially defined on S, 0,α into itself continuously. Therefore, we obtain the same conclusion for the operator maps CH ∗ f − H −1/2 x f = −H −1/2 ∂ f . Consequently, R ∗ = H −1/2 A = H −1/2 ∂ + H −1/2 x f → R−i i xi i xi i i 0,α is a bounded operator on CH . 2


3117

4.2. Proof of Theorem A We start with (A1). By recalling the results in Lemmas 5.4 and 5.5, if we put γ = 2σ < 1 and ε = α − 2σ in Proposition 2.3, we get the conclusion. Consider now (A2). Using Remark 3.4 and Theorem 3.6, we have 1,α−2σ H σ u ∈ CH

⇐⇒

0,α−2σ Ai H σ u, A−i H σ u ∈ CH

⇐⇒

0,α−2σ (H + 2)σ Ai u, (H − 2)σ A−i u ∈ CH .

By Theorem 3.5, together with Lemmas 5.4 and 5.5, we can apply Proposition 2.3, with 0,α 0,α−2σ γ = 2σ < 1 and ε = α − 2σ , in order to get (H + 2)σ : CH → CH continuously, and then σ (H + 2) Ai uC 0,α−2σ CAi uC 0,α CuC 1,α . Applying Theorem 3.6, Lemmas 5.4 and 5.5, H H H and Proposition 2.3, we get (H − 2)σ A−i uC 0,α−2σ CuC 1,α . Thus, H σ uC 1,α−2σ H H H CuC 1,α . H Let us prove (A3). We can write H σ = H σ −1/2 ◦ H −1/2 ◦ H = H σ −1/2 ◦

1 ∗ R−i A−i + R∗i Ai , 2 n

i=1

0,α where R∗±i are the adjoint Hermite–Riesz transforms, that are bounded operators on CH (Theorem 4.2). Consequently, n 1 ∗ 0,α R−i A−i u + R∗i Ai u =: v ∈ CH . 2 i=1

Now we distinguish two cases. If σ − 1/2 > 0, then 0 < α − 2(σ − 1/2) < 1 by hypothesis, 0,α−2σ +1 , and H σ uC 0,α−2σ +1 CuC 1,α . so we can apply (A1) to obtain that H σ −1/2 v ∈ CH H

H

0,α−2σ +1 , If σ − 1/2 < 0, then 0 < α + 2(−σ + 1/2) < 1, and we will get H −(−σ +1/2) v ∈ CH and H σ uC 0,α−2σ +1 CuC 1,α , as soon as we have proved Theorem B(B1). If σ = 1/2, the H

H

0,α , Theresult just follows from the boundedness of the adjoint Hermite–Riesz transforms on CH orem 4.2. By iteration of (A1), (A2) and (A3), and using Remark 3.4 and Theorems 3.5 and 3.6, we can derive (A4). The rather cumbersome details are left to the interested reader.

4.3. Proof of Theorem B To prove (B1) note that, if α + 2σ 1 then 0 < σ < 1/2. Let us write u(z) − u(x1 ) F−σ (x1 , z) − F−σ (x2 , z) dz H −σ u(x1 ) − H −σ u(x2 ) = Rn

+ u(x1 ) H −σ 1(x1 ) − H −σ 1(x2 ) .

By Lemma 5.7, the second term above is bounded by C[u]M α |x1 − x2 |α+2σ . Split the remaining integral on B = B(x1 , 2|x1 − x2 |) and on B c . By Lemma 5.6,

3118


u(z) − u(x1 )F−σ (x1 , z) dz C[u]

C 0,α

B

B

|x1 − z|α dz = C[u]C 0,α |x1 − x2 |α+2σ . |x1 − z|n−2σ

Let B = B(x2 , 4|x1 − x2 |). Then, by the triangle inequality,

u(z) − u(x1 )F−σ (x2 , z) dz

B

C[u]C 0,α B

|x1 − z|α dz |x2 − z|n−2σ

α C[u]C 0,α |x1 − x2 | B

1 dz + |x2 − z|n−2σ

|x2 − z|

α−n+2σ

dz

B

= C[u]C 0,α |x1 − x2 |α+2σ . c , |z − x1 | < the ball with center x2 and radius |x1 − x2 |. Note that, for z ∈ B Denote by B 2|z − x2 |. Then, apply Lemma 5.6 to get

u(z) − u(x1 )F−σ (x1 , z) − F−σ (x2 , z) dz

Bc

C[u]C 0,α |x1 − x2 | c B

2 |z − x2 |α − |x−z| C e dz |z − x2 |n+1−2σ

C[u]C 0,α |x1 − x2 |α+2σ . Thus, [H −σ u]C 0,α+2σ CuC 0,α . For the decay, we put H

H −σ u(x) =

u(z) − u(x) F−σ (x, z) dz +

u(z) − u(x) F−σ (x, z) dz + u(x)H −σ 1(x),

Bc

B 1 ). We have where B = B(x, 1+|x|

u(z) − u(x)F−σ (x, z) dz C[u]

C 0,α

B

|z − x|α C dz [u]C 0,α , |x − z|n−2σ (1 + |x|)α+2σ

B

and −σ u(z) − u(x) F−σ (x, z) dz + u(x)H 1(x) u(z)F−σ (x, z) dz + 2u(x)H −σ 1(x). Bc

Bc

To estimate the very last integral, we can proceed as we did for the integral part of the operator T in the proof of Proposition 2.3, splitting the integral in annulus (the details are left to the


3119

reader). By Lemma 5.7, |u(x)H −σ 1(x)| C[u]M α (1 + |x|)−(α+2σ ) . This concludes the proof of Theorem B(B1). In order to prove (B2), we observe that, by using the boundedness of the first order Hermite– 0,α Riesz transforms on CH (Theorem 4.1), we get Ai H −σ u 0,α+2σ −1 = Ri H −σ +1/2 u 0,α+2σ −1 C H −σ +1/2 u 0,α+2σ −1 Cu 0,α , C C C C H

H

H

H

where in the last inequality we applied Theorem A(A1) if −σ + 1/2 > 0, and the case (B1) just proved above if −σ + 1/2 < 0. The case σ = 1/2 is contained in Theorem 4.1. 0,α+2σ −2 Under the hypotheses of (B3), we have to prove that Ai Aj H −σ u belongs to CH . But −σ 1−σ u. Therefore, Theorem A(A1) and Theorem 4.1 give the result. Ai Aj H u = Rij H 5. Computational lemmas Lemma 5.1. For each positive number a, let 2 + 1 |x−z|2 ] s

a ψs,z (x) = e−a[s|x+z|

,

x, z ∈ Rn , s ∈ (0, 1).

Then, a |x−z|2 s

a

a ψs,z (x) e− 4 |x||x−z| e− 4

(5.1)

.

Proof. We have a |x−z|2 s

a ψs,z (x) e− 2

a

2 + 1 |x−z|2 ] s

e− 2 [s|x+z|

a |x−z|2 s

e− 2

a

e− 2 |x−z||x+z| .

The first inequality above is obvious. For the second one, we argue as follows: if |x + z| |x − z| then it is clearly valid; when |x − z| < |x + z| we minimize the function θ (s) = a2 [s|x + z|2 + 1 a 2 s |x − z| ], s ∈ (0, 1), to get θ (s) 2 |x − z||x + z|. To obtain the desired estimate let us first a |x−z|2 s 2 − a2 |x−z| s

assume that x · z > 0. Then |x + z| |x|, and e− 2 x · z 0, then |x − z| |x|, and e e

− a4

|x−z|2 s

e

− a4 |x||x−z|

. Thus, (5.1) follows.

a |x−z|2 s 2 − a4 |x−z| s

e− 2 |x−z||x+z| e− 4 a

a

e− 2 |x−z||x+z| e

a

e− 4 |x||x−z| . If a |x||x−z| s

e− 4

2

n/2 e− Remark 5.2. Note that Lemma 5.1 gives the estimate Gt (s) (x, z) C( 1−s s ) n s ∈ (0, 1), x, z ∈ R , which appeared in Lemma 5.10 of [13].

|x||x−z| C

Lemma 5.3. Let η, ρ ∈ R. Then, for all x, z ∈ Rn , 1 0

1−s s

n/2

|x||x−z| |x−z|2 1 −C[s|x+z|2 + 1 |x−z|2 ] s e dμρ (s) Ce− C e− C · Iη,ρ (x, z), η s

e−

|x−z|2 Cs

,

3120


where ⎧ 1 , ⎪ ⎪ ⎨ |x−z|n+2η+2ρ Iη,ρ (x, z) = 1 + log( C 2 )χ{ C >1} (x − z), |x−z| ⎪ |x−z|2 ⎪ ⎩ 1,

if n/2 + η + ρ > 0, if n/2 + η + ρ = 0, if n/2 + η + ρ < 0.

Proof. By (5.1), we get 1

1−s s

n/2

1 −C[s|x+z|2 + 1 |x−z|2 ] s e dμρ (s) sη

0

Ce

− |x||x−z| C

1/2

1 s n/2+η+ρ

e

− |x−z| Cs

2

(1 − s)

n/2

0

= Ce

− |x||x−z| C

Ce

− |x||x−z| C

= Ce−

|x||x−z| C

1

|x−z|2 ds + e− C s

dμρ (s)

1/2

C |x − z|n+2η+2ρ |x−z|2

e− C |x − z|n+2η+2ρ

∞ r

|x−z|2 dr + Ce− C r

n/2+η+ρ −2r

e

|x−z|2 C

∞

r n/2+η+ρ e−r

|x−z|2 dr + e− C r

|x−z|2 C

· I.

If n/2 + η + ρ > 0, we have |x−z|2

e− C I |x − z|n+2η+2ρ

∞

r n/2+η+ρ e−r

0

|x−z|2 |x−z|2 dr C + e− C e− C . n+2η+2ρ r |x − z|

Suppose now that n/2 + η + ρ 0. Consider two cases: if

I e

− |x−z| C

= Ce

and, when

|x−z|2 C

2

2 − |x−z| C

< 1, we have

1 |x − z|n+2η+2ρ

∞ r 1

|x−z|2 C

n/2+η+ρ −r

e

C + 1 C, |x − z|n+2η+2ρ

1, then,

dr +1 r


I e

− |x−z| C

Ce

2

1

1

r

|x − z|n+2η+2ρ

− |x−z| C

2

·

n/2+η+ρ

3121

dr +C +1 r

|x−z|2 C

C 1 + log( |x−z| 2 ),

if n/2 + η + ρ = 0,

1,

if n/2 + η + ρ < 0.

2

Lemma 5.4. Denote by F any of the kernels Fσ (x, z) (defined in (1.3)), or F±2k,σ (x, z) (given in Theorems 3.5 and 3.6). Then, F (x, z)

2 C − |x||x−z| − |x−z| C C , e e |x − z|n+2σ

(5.2)

for all x, z ∈ Rn , and F (x1 , z) − F (x2 , z)

C|x1 − x2 | − |z||x2 −z| − |x2 −z|2 C C e e , |x2 − z|n+1+2σ

(5.3)

for all x1 , x2 ∈ Rn such that |x1 − z| > 2|x1 − x2 |. Proof. Let us first consider F = Fσ . The estimate in (5.2) is already stated in [13, Lemma 5.11]. Nevertheless, we can prove it here quickly by using, in (3.5), Lemma 5.3, with η = 0 and ρ = σ . To get (5.3), we observe that, by the Mean Value Theorem,

n/2 1 − 1 [s|ξ +z|2 + 1 |ξ −z|2 ] Gt (s) (x1 , z) − Gt (s) (x2 , z) C|x1 − x2 | 1 − s s e 8 , s s 1/2

(5.4)

for some ξ = (1 − λ)x1 + λx2 , λ ∈ [0, 1]. Then, by Lemma 5.3, with η = 1/2 and ρ = σ , Fσ (x1 , z) − Fσ (x2 , z) |x1 − x2 |

sup

{ξ =(1−λ)x1 +λx2 : λ∈[0,1]}

1 C|x1 − x2 | sup ξ

C|x1 − x2 | sup ξ

C

1−s s

n/2

∇x Fσ (ξ, z)

1 s 1/2

1

2 + 1 |ξ −z|2 ] s

e− 8 [s|ξ +z|

dμσ (s)

0 |z||ξ −z| |ξ −z|2 1 e− C e− C n+1+2σ |ξ − z|

|z||x −z| |x −z|2 |x1 − x2 | − C2 − 2C e e , |x2 − z|n+1+2σ

where in the last inequality we used that |ξ − z| 12 |x2 − z|, since |x1 − z| > 2|x1 − x2 |. In a similar way we can prove both estimates for F = F2k,σ , because 0 F2k,σ (x, z) Fσ (x, z), and the details are left to the reader. Note that, by (3.8)–(3.9), up to a multiplicative constant we have

3122


F−2k,σ (x, z) 1/2

1 1+s k 1 + s k = Gt (s) (x, z) − φ2k (x) dμσ (s) Gt (s) (x, z) dμσ (s) + 1−s 1−s 0

1/2

C Fσ (x, z) + e

− |x||x−z| C

e

1

2 − |x−z| C

1/2

1−s 1+s

n/2

dμσ (s) ,

and therefore, (5.2) is valid for F−2k,σ . By (5.4), 1/2 0

1+s 1−s

k

Gt (s) (x1 , z) − Gt (s) (x2 , z) dμσ (s) 1


1−s s

n/2

1 s 1/2

1

2 + 1 |ξ −z|2 ] s

e− 8 [s|ξ +z|

dμσ (s).

(5.5)

0

Recall the definition of Mr (x, z) given in (3.7). It can be checked that k d |x−z|2 − |x||x−z| C e− C , dr k ∇x Mr (x, z) Ce

r ∈ (0, 1/3).

Thus, by Taylor’s formula,

k+n/2 |x||x−z| |x−z|2 ∇x Gt (s) (x, z) − φ2k (x, z, s) C 1 − s e− C e− C , 1+s

s ∈ (1/2, 1),

(5.6)

and, consequently, when |x1 − z| > 2|x1 − x2 |, 1 1/2

1+s 1−s

k

Gt (s) (x1 , z) − φ2k (x1 , z, s) − Gt (s) (x2 , z) − φ2k (x2 , z, s) dμσ (s) 1


1/2

C|x1 − x2 | sup e−

1+s 1−s

|z||ξ −z| C

k

e−

∇x Gt (s) (ξ, z) − φ2k (ξ, z, s) dμσ (s)

|ξ −z|2 C

C|x1 − x2 |e−

|z||x2 −z| C

e−

ξ

Pasting estimates (5.5) and (5.7), (5.3) follows for F = F−2k,σ .

2

|x2 −z|2 C

.

(5.7)


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Lemma 5.5. Denote by B any of the functions Bσ or B±2k,σ defined in (1.3) and in Theorems 3.5 and 3.6. Then B ∈ C ∞ (Rn ) and, for all x ∈ Rn , |x|, if |x| 1, B(x) C 1 + |x|2σ , and ∇B(x) C (5.8) |x|2σ −1 , if |x| > 1. Proof. The first inequality in (5.8) for the case B = Bσ is contained in [13, Lemma 5.11]. The identity tanh 2t 1 2 −tH e 1(x) = Gt (x, z) dz = e− 2 |x| (2π cosh 2t)n/2 Rn

(stated in [8]) and Meda’s change of parameters (3.2) give 1 Bσ (x) = Γ (−σ )

1 0

1 − s2 2π(1 + s 2 )

n/2 e

−

s |x|2 1+s 2

− 1 dμσ (s).

We differentiate under the integral sign to see that Bσ ∈ C ∞ (Rn ), and 1 2 n/2 s 2 2x 1 − s s − |x| 2 ∇Bσ (x) = 1+s e dμ (s) σ Γ (−σ ) 1 + s 2 2π(1 + s 2 ) 0

1 C|x|

se− 2 |x| dμσ (s) =: I (x). s

2

0

By (3.6), 1/2

I (x) C|x|

s

e− 2 |x|

2

ds +e sσ

2 − |x| C

0

1

|x|2

dμσ (s) = C|x|2σ −1

1/2

4

e−r

0

If |x| 1, |x|2

4

|x|2

e−r

dr rσ

0

4

dr = C|x|2−2σ , rσ

0

and, if |x| > 1, |x|2

4 0

e−r

dr rσ

1/4 0

Hence, (5.8) with B = Bσ is proved.

|x|2

dr + rσ

4 1/4

e−r dr = C − e−

|x|2 C

C.

2 dr − |x| C . + C|x|e rσ

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We can write 1 B2k,σ (x) = Γ (−σ )

1 0

1 + Γ (−σ )

1

1−s 1+s

k

1 − s2 2π(1 + s 2 )

n/2

− s |x|2 − 1 e 1+s 2 dμσ (s)

− s 2 |x|2 e 1+s − 1 dμσ (s) =: I + II.

0

The bounds for I and II can be deduced as in the proof of Lemma 5.11 of [13]. We give the calculation for completeness. For both terms we use (3.6) and the Mean Value Theorem. That is,

n/2 1/2 1 1/2 1−s k ds 1 − s2 ds |I | C − 1 σ +1 + dμσ (s) C s 1+σ + C = C. 2 1+s 2π(1 + s ) s s 0

1/2

0

For II we have to consider two cases. Assume first that |x|2 2. Then, 1/2 1 1/2 ds − s |x|2 ds − 1 1+σ + dμσ (s) C |x|2 s 1+σ + C C. |II| C e 1+s 2 s s 0

1/2

0

In the case |x|2 > 2, 1

|x|2 |II| |x|

2

s

ds s 1+σ

1

1 +

0

|x|2 dμσ (s) |x|

2

s

−σ

1 ds +

0

1 |x|2

−σ 1 = C|x|2σ + C −log 1 − 2 C|x|2σ , |x|

1 |x|2

ds (1 − s)(−log(1 − s))1+σ

since −log(1 − s) ∼ s, as s ∼ 0. On the other hand, observe that |∇B2k,σ (x)| I (x). Thus, (5.8) follows with B = B2k,σ . When B = B−2k,σ , 1 B−2k,σ (x) = Γ (−σ )

1/2 0

1 + Γ (−σ )

1 1/2

= III + IV.

1+s 1−s

k

1+s 1−s

1 − s2 2π(1 + s 2 )

k Rn

n/2 e

−

s |x|2 1+s 2

− 1 dμσ (s)

Gt (s) (x, z) − φ2k (x, z, s) dz − 1 dμσ (s)


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If we write III as 1 Γ (−σ )

1/2 0

1+s 1−s

k

1 − s2 2π(1 + s 2 )

n/2

1 − 1 dμσ (s) + Γ (−σ )

1/2 − s 2 |x|2 − 1 dμσ (s), e 1+s 0

then we can handle these two terms as we did for I and II above to get |III| C(1 + |x|2σ ). By (3.8)–(3.9), |IV| C. For the gradient of B−2k,σ , similar estimates to those used for ∇Bσ can be applied for the term ∇x III. Finally, (5.6) implies that |∇x IV| C. The proof is complete. 2 The following lemma contains a small refinement of the estimate for the kernel F−σ (x, z) given in [4, Proposition 2]. Lemma 5.6. Take σ ∈ (0, 1]. Then, for all x, z ∈ Rn , ⎧ 2 ⎪ 1 − |x||x−z| − |x−z| ⎪ C C , e e ⎪ n−2σ ⎪ |x−z| ⎪ ⎨ |x||x−z| |x−z|2 0 F−σ (x, z) C e− C e− C [1 + log( C 2 )χ C { >1} (x − z)], |x−z| ⎪ ⎪ |x−z|2 ⎪ ⎪ ⎪ |x||x−z| |x−z|2 ⎩ e− C e− C ,

if n > 2σ, if n = 2σ,

(5.9)

if n < 2σ.

If F(x, z) denotes any of the kernels ∇x F−σ (x, z), xi F−σ (x, z) or zi F−σ (x, z), then F(x, z) C

⎧ |x||x−z| |x−z|2 1 ⎪ e− C e− C , ⎨ |x−z|n+1−2σ |x||x−z| |x−z|2 ⎪ ⎩ e− C e− C [1 + log(

if n > 2σ − 1, (5.10)

C )χ C (x |x−z|2 { |x−z|2 >1}

− z)],

if n = 2σ − 1.

Moreover, when |x1 − z| 2|x1 − x2 |, F−σ (x1 , z) − F−σ (x2 , z) ⎧ |z||x2 −z| |x2 −z|2 ⎪ ⎪ ⎨ |x −z|1n+1−2σ e− C e− C , 2 C|x1 − x2 | |z||x2 −z| |x2 −z|2 ⎪ C ⎪ ⎩ e− C e− C [1 + log( |x−z| 2 )χ{

if σ = 1, (5.11) C >1} |x−z|2

(x − z)],

if σ = 1,

and, F(x1 , z) − F(x2 , z) C

|z||x −z| |x −z|2 |x1 − x2 | − C2 − 2C e e . |x2 − z|n+2−2σ

(5.12)

Proof. By (3.2), 1 F−σ (x, z) = Γ (σ )

1

1 Gt (s) (x, z) dμ−σ (s) = C 0

1 − s2 s

n/2

1

2 + 1 |x−z|2 ] s

e− 4 [s|x+z|

dμ−σ (s).

0

(5.13)

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Then apply Lemma 5.3, with η = 0 and ρ = −σ , to get (5.9). Differentiation with respect to x inside the integral in (5.13) gives ∇x F−σ (x, z) C

1

1−s s

n/2

1 s 1/2

1

2 + 1 |x−z|2 ] s

e− 8 [s|x+z|

dμ−σ (s),

(5.14)

0

and then Lemma 5.3, with η = 1/2 and ρ = −σ , implies (5.10) with F(x, z) = ∇x F−σ (x, z). Take x, z ∈ Rn . If x · z 0, then |x| |x + z| and, in this situation, |x|F−σ (x, z) is bounded by the RHS of (5.14). If x · z < 0, we have |x| |x − z|, and in this case

|x|F−σ (x, z) C|x − z|e

− 18 |x−z|2

1

1−s s

n/2

1

2 + 1 |x−z|2 ] s

e− 8 [s|x+z|

dμ−σ (s).

0

Therefore, by Lemma 5.3, we obtain (5.10) for F(x, z) = xi F−σ (x, z). The same reasoning applies to F(x, z) = zi F−σ (x, z), since |z| |z − x| + |x|. To derive (5.11), we follow the proof of (5.3) in Lemma 5.4, with −σ in the place of σ , and we use Lemma 5.3. Estimate (5.12) for F(x, z) = ∇x F−σ (x, z) can be deduced by using the Mean Value Theorem and Lemma 5.3, since

∂x2i ,xj F−σ (x, z) =

1 Γ (σ )

1

1 − s2 4πs

n/2

1

2 + 1 |x−z|2 ] s

e− 4 [s|x+z|

0

s 1 s 1 × − (xi + zi ) − (xi − zi ) − (xj + zj ) − (xj − zj ) 2 2s 2 2s

s 1 dμ−σ (s) + δij − − 2 2s gives that 2 D F−σ (x, z) C

1

x

1−s s

n/2

1 −C[s|x+z|2 + 1 |x−z|2 ] s e dμ−σ (s). s

(5.15)

0

Similar ideas can also be used to prove (5.12) when F(x, z) is either xi F−σ (x, z) or zi F−σ (x, z). We skip the details. 2 Lemma 5.7. The function H −σ 1 belongs to the space C ∞ (Rn ), and −σ H 1(x)

C , (1 + |x|)2σ

and ∇H −σ 1(x)

C . (1 + |x|)1+2σ


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Proof. Observe that (3.2) applied to (3.11) gives

H

−σ

1 1(x) = Γ (σ )

1 e

−t (s)H

1 1(x) dμ−σ (s) = C

0

0

1 − s2 1 + s2

n/2 e

−

s |x|2 1+s 2

dμ−σ (s).

(5.16)

Since ∇x e−t (s)H 1(x) = 2|x|

n/2 s s 1 − s2 2 − s |x|2 e 1+s 2 Cs 1/2 e− C |x| , 2 2 1 + s 2π(1 + s )

(5.17)

differentiation inside the integral sign in (5.16) is justified. By repeating this argument we obtain H −σ 1 ∈ C ∞ (Rn ). To study the size of H −σ 1, note that we can restrict to the case |x| > 1 because H −σ 1 is a continuous function. By (3.6), we have 1/2 0

1 − s 2 n/2

|x|2

1 + s2

1/2 2C s dr 2 ds − s 2 |x|2 − |x| −2σ e 1+s dμ−σ (s) C e C = C|x| e−r 1−σ C|x|−2σ , 1−σ s r

1

n/2

0

0

and

1/2

1 − s2 1 + s2

e

−

s |x|2 1+s 2

dμ−σ (s) Ce

−C|x|2

1 1/2

(1 − s)n/2−1 2 ds = Ce−C|x| . 1−σ (−log(1 − s))

Plugging these two estimates into (5.16) we get the bound for H −σ 1. For the growth of the gradient, we can use (5.17) and similar estimates as above to obtain the result. 2 Lemma 5.8. For 1 |i|, |j | n, denote by R(x, z) any of the kernels ∂x2i ,xj F−1 (x, z), xi ∂xj F−1 (x, z) or xi xj F−1 (x, z). Then R(x, z)

2 C − |x||x−z| − |x−z| C C , e e |x − z|n

(5.18)

and, when |x1 − z| 2|x1 − x2 |, |z||x −z| |x −z|2 R(x1 , z) − R(x2 , z) C |x1 − x2 | e− C2 e− 2 C . n+1 |x2 − z|

(5.19)

As a consequence, the kernel of the second order Hermite–Riesz transforms Rij (x, z) = Ai Aj F−1 (x, z) also satisfies these size and smoothness estimates. Proof. We put σ = 1 in (5.15) and we use Lemma 5.3, with η = 1 and ρ = −1, to obtain the desired estimate for Dx2 F−1 . From (5.14),

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xi ∂x F−1 (x, z) C|x| j

1

1−s s

n/2

1 s 1/2

1

2 + 1 |x−z|2 ] s

e− 8 [s|x+z|

dμ−1 (s).

(5.20)

0

If |x| 2, then Lemma 5.3, with η = 1/2 and ρ = −1, applied to (5.20) gives xi ∂x F−1 (x, z) j

2 C − |x||x−z| − |x−z| C C . e e |x − z|n−1

Assume that |x| > 2 in (5.20). Consider first the case |x| < 2|x − z|, then by Lemma 5.3, xi ∂x F−1 (x, z) C j

1

1−s s

n/2

2 + 1 |x−z|2 ] s

e−C[s|x+z|

dμ−1 (s)

0

2 C − |x||x−z| − |x−z| C C . e e |x − z|n−2

In the other case, namely |x| 2|x − z|, we use the fact that |x| > 2 to see that |x + z|2 = 2|x|2 − |x − z|2 + 2|z|2 > |x|2 . Hence, xj ∂x F−1 (x, z) C j

1

1−s s

n/2

1 s 1/2

2 + 1 |x−z|2 ] s

e−C[s|x+z|

dμ−1 (s)

0

|x||x−z| |x−z|2 C e− C e− C . n |x − z|

Collecting terms we have (5.18) for R(x, z) = xj ∂xj F−1 (x, z). Finally, to obtain (5.18) with R(x, z) = xi xj F−1 (x, z), we note that by (5.13), xi xj F−1 (x, z) C|x|2

1

1−s s

n/2

1

2 + 1 |x−z|2 ] s

e− 8 [s|x+z|

dμ−1 (s),

0

and we consider the cases |x| 2 and |x| > 2 as before. In the second situation, we assume first that |x| 2|x − z| and, then, that |x| 2|x − z| (which implies |x| |x + z|), and we use the method of the proof given for xj ∂xi F−1 above. To prove (5.19) we can use the Mean Value Theorem and Lemma 5.1 (see the proof of (5.3) and (5.12)). We omit the details. 2 Lemma 5.9. Denote by K(x, z) any of the functions |x|2σ F−σ (x, z), |z|2σ F−σ (x, z), 0 < σ 1, or the kernel xi ∂xj F−1 (x, z). Then sup x

Rn

K(x, z) dz C.


3129

Proof. Consider the function |x|2σF−σ (x,z). If |x| 2 then, by (5.9), |x|2σ Rn F−σ (x,z) dz C. If |x| > 2, we consider two regions of integration: |x| < |x − z| and |x| |x − z|. In the first region, by Lemma 5.3, 1 |x| F−σ (x, z) C 2σ

1−s s

n/2

1

2 + 1 |x−z|2 ] s

e−C[s|x+z|

s −σ

dμ−σ (s) CΦ(x − z),

0

with Φ ∈ L1 (Rn ). To study the second region of integration, namely |x| |x − z|, we use the fact that |x + z| > |x| and we split the integral defining F−σ into two intervals: (0, 1/2) and (1/2, 1). To estimate the part of the integral over the interval (0, 1/2) we note that, by using (3.6) and three different changes of variables, we have

1/2 Gt (s) (x, z) dμ−σ (s) dz C

|x||x−z| 0

1/2

|x||x−z| 0

1 − 1 [s|x|2 + 1 |x−z|2 ] ds s e 4 dz s n/2 s 1−σ |x|2

2

1 − 1 [r+ 1 |x|2 |x−z|2 ] dr e 4 r dz r n/2 r 1−σ

= C|x|n−2σ |x||x−z| 0 |x|2

= C|x|−2σ

2

1

1

r n/2

1

e− 4 [r+ r |w|

2]

dr r 1−σ

dw

|x|2 |w| 0 2 2 |x|

= C|x|−2σ

|x| 2 0

−2σ

1

1

r n/2 0

∞

C|x|

∞

r

e− 4

0

∞

= C|x|

e

dr r r

− 4r σ

ρ2 r ]

2

e

r n/2−σ 0

−2σ

e− 4 [r+

− ρ4r

dρ ρ ρ

∞

0

dr n−1 ρ dρ r 1−σ

n

−t n/2

e t

dr r dt t

= C|x|−2σ .

0

The integral over the interval (1/2, 1) is bounded by 1 |x|

2σ

(1 − s)n/2 e−C|x| e− 2

|x−z|2 C

dμ−σ (s) Ce−

|x−z|2 C

∈ L1 R n .

1/2

Hence we get the conclusion for K(x, z) = |x|2σ F−σ (x, z). To prove the result for the function |z|2σ F−σ (x, z), observe that |z|2σ C(|z − x|2σ + |x|2σ ), so we can apply the estimates

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above. When F(x, z) = xi ∂xj F−1 (x, z) we can argue as we did for |x|2σ F−σ (x, z) above, because of (5.20). 2 Lemma 5.10. For all 1 |i| n, and 0 < r1 < r2 ∞, sup

Ai F−1/2 (x, z) dz C,

x

r1 0 is independent of r1 and r2 . Proof. By estimate (5.10) given in Lemma 5.6, it is enough to consider r2 < 1. From Lemma 5.9, with σ = 1/2, we have that Rn xi F−1/2 (x, z) dz C. We can write

∂xi F−1/2 (x, z) dz = r1 0) if a − 0 for some real > 0.

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J. Cimpriˇc / Journal of Functional Analysis 260 (2011) 3132–3145

Theorem E. Let A be a C ∗ -algebra and let p ∈ R[x] ⊗ Ah and pk ∈ R[x] ⊗ Mνk (C)h , k = 1, . . . , m, νk ∈ N, be homogeneous polynomials of even degree. Assume that K0 := {t ∈ S d−1 | p1 (t) 0, . . . , pm (t) 0} is nonempty and p(t) > 0 for all t ∈ K0 . Then there are homogeneous polynomials qj ∈ R[x] ⊗ A, qj k ∈ R[x] ⊗ Mνk ×1 (A), j ∈ J , J finite, and an integer θ such that m 2 2 θ ∗ ∗ q j k pk q j k . x1 + · · · + xd p = qj qj + j ∈J

k=1

Our interest in this subject stems from the following generalization of Theorem A which is a reformulation of a result of Scherer and Hol. See [17, Corollary 1] for the original result and [10, Theorem 13] for the reformulation and extension to infinite S. Theorem F. For a finite subset S = {p1 , . . . , pm } of Mν (R[x])h , ν ∈ N, write KS := {t ∈ Rd | ∗ ∗ p1 (t) 0, . . . , pm (t) 0} and MS := { j ∈J (qj qj + m k=1 qj k pk qj k ) | qj , qj k ∈ Mν (R[x]), j ∈ J, J finite}. If there is g ∈ MS ∩ R[x] such that the set {x ∈ Rd | g(x) 0} is compact (i.e. the quadratic module MS ∩ R[x] in R[x] is archimedean) then for every p ∈ Mν (R[x])h such that p(t) > 0 on KS we have that p ∈ MS . Finally, we mention an interesting generalization of Theorem C which was proved by Dritschel and Rovnyak in [6, Theorem 5.1]. Theorem G. Let A be the ∗-algebra of all bounded operators on a Hilbert space. If an element p ∈ R[cos φ1 , sin φ1 , . . . , cos φn , sin φn ] ⊗ Ah is strictly positive for every φ1 , . . . , φn then p = sin φ1 , . . . , cos φn , sin φn ] ⊗ A.

j ∈J

qj∗ qj for some finite J and qj ∈ R[cos φ1 ,

The aim of this paper is to prove the following very general operator-theoretic positivstellensatz and show that it implies generalizations of Theorems E, F and G. (They will be extended from finite to arbitrary S, from C ∗ -algebras to algebraically bounded ∗-algebras A and from (trigonometric) polynomials to affine commutative real algebras. Theorem F will also be extended from matrices to more general operators.) Theorem H. Let R be a commutative real algebra, A a real or complex ∗-algebra and M a quadratic module (cf. Section 2) in R ⊗ A. If M is archimedean then for every p ∈ R ⊗ Ah the following are equivalent: (1) p ∈ + M for some real > 0. (2) For every multiplicative state φ on R, there exists real φ > 0 such that (φ ⊗ idA )(p) ∈ φ + (φ ⊗ idA )(M). If A is algebraically bounded (cf. Section 4) and the quadratic module M ∩ R in R is archimedean (cf. Theorem D) then M is archimedean.


3135

One of the main differences between the operator case and the scalar case is that in the operator case an element of Ah that is not 0 is not necessarily > 0. We would like to give an algebraic characterization of operator-valued polynomials that are not 0 in every point from a given set. Every theorem of this type is called a nichtnegativsemidefinitheitsstellensatz. We will prove variants of Theorems F and G that fit into this context. Finally, we use our results and the main theorem from [9] to get a generalization of the existence result for operator-valued moment problems from [1] to algebraically bounded ∗-algebras. 2. Factorizable states Associative unital algebras with involution will be called ∗-algebras for short. Let B be a ∗algebra over F ∈ {R, C} where F always comes with complex conjugation as involution. Write Z(B) for the center of B and write Bh = {b ∈ B | b∗ = b} for its set of hermitian elements. Note that the set Bh is a real vector space; we assume that it is equipped with the finest locally convex topology, i.e. the coarsest topology such that every convex absorbing set in Bh is a neighborhood of zero. Clearly, every linear functional on Bh is continuous with respect to the finest locally convex topology. In other words, the algebraic and the topological dual of Bh are the same; we will write (Bh ) for both. We assume that (Bh ) is equipped with the weak*-topology, i.e. topology of pointwise convergence. We say that ω ∈ (Bh ) is factorizable if ω(xy) = ω(x)ω(y) for every x ∈ Bh and y ∈ Z(B)h . Clearly, the set of all factorizable linear functionals on Bh is closed in the weak*-topology. We say that a subset M of Bh is a quadratic module if 1 ∈ M, M + M ⊆ M and b∗ Mb ⊆ M for every b ∈ B. The smallest quadratic module in B is the set Σ 2 (B) which consists of all finite sums of elements b∗ b with b ∈ B. The largest quadratic module in B is the set Bh . A quadratic module M in B is proper if M = Bh (or equivalently, if −1 ∈ / M). Proper quadratic modules in B exist iff −1 ∈ / Σ 2 (B). We say that an element b ∈ Bh is bounded w.r.t. a quadratic module M if there exists a number l ∈ N such that l ± b ∈ M. A quadratic module M is archimedean if every element b ∈ Bh is bounded w.r.t. M (or equivalently, if 1 is an interior point of M). For every subset M of Bh write M ∨ for the set of all f ∈ (Bh ) which satisfy f (1) = 1 and f (M) 0. The set of all extreme points of M ∨ will be denoted by ex M ∨ . Elements of M ∨ will be called M-positive states and elements of ex M ∨ extreme M-positive states. A Σ 2 (B)-positive state will simply be called a state. Suppose now that M is an archimedean quadratic module. Applying the Banach–Alaoglu Theorem to V = (M − 1) ∩ (1 − M) which is a neighborhood of zero, we see that M ∨ is compact. The Krein–Milman theorem then implies, that M ∨ is equal to the closure of the convex hull of the set ex M ∨ . We will show later (see Corollary 4) that M ∨ is nonempty iff M is proper. Recall that a (bounded) ∗-representation of B is a homomorphism of unital ∗-algebras from B to the algebra of all bounded operators on some Hilbert space Hπ . We say that a ∗-representation π of B is M-positive for a given subset M of Bh if π(m) is positive semidefinite for every m ∈ M. For every such π and every v ∈ Hπ of norm 1, ωπ,v (x) := π(x)v, v belongs to M ∨ . Conversely, if M is an archimedean quadratic module, then every ω ∈ M ∨ is of this form by the GNS construction. The equivalence of (1)–(4) in the following result is sometimes referred to as archimedean positivstellensatz for ∗-algebras. It originates from the Vidav–Handelmann theory, cf. [7, Section 1] and [20]. Our aim is to add assertions (5) and (6) to this equivalence.

3136


Proposition 1. For every archimedean quadratic module M in B and every element b ∈ Bh the following are equivalent: (1) (2) (3) (4) (5) (6)

b ∈ M ◦ (the interior w.r.t. the finest locally convex topology), b ∈ + M for some real > 0, π(b) is strictly positive definite for every M-positive ∗-representation π of B, f (b) > 0 for every f ∈ M ∨ , f (b) > 0 for every f ∈ ex M ∨ , f (b) > 0 for every factorizable f ∈ M ∨ .

Proof. (1) implies (2) because the set M − b is absorbing, hence −1 ∈ t (M − b) for some t > 0. Clearly (2) implies (3). (3) implies (4) because the cyclic ∗-representation that belongs to f by the GNS construction clearly has the property that π(m) is positive semidefinite for every m ∈ M. (4) implies (1) by the separation theorem for convex sets. The details can be found in [4, Theorem 12] or [18, Proposition 15] or [5, Proposition 1.4]. If (5) is true then, by the compactness of ex M ∨ , there exists > 0 such that f (b) for every f ∈ ex M ∨ , hence (4) is true by the Krein–Milman theorem. Clearly, (4) implies (6). By Proposition 3 below and the fact that the set of all factorizable M-positive states is closed, (6) implies (5). 2 Similarly, we have the following: Proposition 2. For every archimedean quadratic module M in B and every element b ∈ Bh the following are equivalent: (1) (2) (3) (4) (5) (6)

b ∈ M (the closure w.r.t. the finest locally convex topology), b + ∈ M for every > 0, π(b) is positive semidefinite for every M-positive ∗-representation π of B, f (b) 0 for every f ∈ M ∨ , f (b) 0 for every f ∈ ex M ∨ , f (b) 0 for every factorizable f ∈ M ∨ .

The following proposition which extends [19, Ch. IV, Lemma 4.11] was used in the proof of equivalences (4)–(6) in Propositions 1 and 2. Its proof depends on the equivalence of (2) and (3) in Proposition 2. Proposition 3. If M is an archimedean quadratic module in B then all extreme M-positive states are factorizable. Proof. Pick any ω ∈ ex M ∨ and y ∈ Z(B)h . We claim that ω(xy) = ω(x)ω(y) for every x ∈ Bh . Since y = 14 ((1 + y)2 − (1 − y)2 ) and (1 ± y)2 ∈ M, we may assume that y ∈ M. Since M is archimedean, we may also assume that 1 − y ∈ M. Claim. If ω(y) = 0, then ω(y 2 ) = 0. (Equivalently, if ω(1 − y) = 0, then ω((1 − y)2 ) = 0.) Since y, 2 − y ∈ M, it follows that 1 − (1 − y)2 = 12 (y(2 − y)2 + (2 − y)y 2 ) ∈ M. Since ω is an M-positive state, it follows that ω((1 − y)2 ) 1. On the other hand, ω((1 − y)2 )ω(12 )


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|ω((1 − y) · 1)|2 by the Cauchy–Schwartz inequality. Now, ω(y) = 0 implies that ω((1 − y)2 ) = 1, hence ω(y 2 ) = 0. Case 1: If ω(y) = 0, then ω(xy) = 0 for every x ∈ Bh . (Equivalently, if ω(1 − y) = 0, then ω(x(1 − y)) = 0 for every x ∈ Bh .) Namely, by the Cauchy–Schwartz inequality and the Claim, |ω(xy)|2 ω(x 2 )ω(y 2 ) = 0. It follows that ω(xy) = ω(x)ω(y) if ω(y) = 0 or ω(y) = 1. Case 2: If 0 < ω(y) < 1, then ω1 and ω2 defined by ω1 (x) :=

1 ω(xy) ω(y)

and ω2 (x) :=

1 ω x(1 − y) ω(1 − y)

(x ∈ Bh ) are M-positive states on Bh . Namely, for every M-positive ∗-representation π of B and every x ∈ M, we have that π(xy) = π(x)π(y) is a product of two commuting positive semidefinite bounded operators, hence a positive semidefinite bounded operator. By the equivalence of assertions (2) and (3) in Proposition 2, xy + ∈ M for every > 0. Since ω is M-positive, it follows that ω(xy) 0 as claimed. Similarly, we prove that ω2 is M-positive. Clearly, ω = ω(y)ω1 + ω(1 − y)ω2 . Since ω is an extreme point of the set of all M-positive states on Bh , it follows that ω = ω1 = ω2 . In particular, ω(xy) = ω(x)ω(y). 2 If we apply Proposition 1 or 2 to b = −1, we get the following corollary, parts of which were already mentioned above. Corollary 4. For every archimedean quadratic module M in B the following are equivalent: (1) (2) (3) (4) (5)

−1 ∈ / M, there exists an M-positive ∗-representation of B, there exists an M-positive state on B, there exists an extreme M-positive state on B, there exists a factorizable M-positive state on B.

The following variant of Proposition 1 which follows easily from Corollary 4 was proved in [3, Theorem 5]. We could call it archimedean nichtnegativsemidefinitheitsstellensatz for ∗-algebras. Proposition 5. For every archimedean proper quadratic module M on a real or complex ∗algebra B and for every x ∈ Bh , the following are equivalent: (1) For every M-positive ∗-representation ψ of B, ψ(x) is not negative semidefinite (i.e. ψ(x)v, v > 0 for some v ∈ Hψ ). (2) There exist k ∈ N and c1 , . . . , ck ∈ B such that ki=1 ci xci∗ ∈ 1 + M. 3. Theorems H and F The aim of this section is to prove Theorem H (see Theorem 6) and show that it implies a generalization of Theorem F to compact operators. We also prove a concrete version of Proposition 5.

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Theorem 6. Let R be a commutative real algebra with trivial involution, A a ∗-algebra over F ∈ {R, C} and M an archimedean quadratic module in B := R ⊗ A. For every element p of Bh = R ⊗ Ah , the following are equivalent: (1) p ∈ + M for some real > 0. (2) For every multiplicative state φ on R, there exists real φ > 0 such that (φ ⊗ idA )(p) ∈ φ + (φ ⊗ idA )(M). The following are also equivalent: (1 ) p + ∈ M for every real > 0. (2 ) For every multiplicative state φ on R and every real > 0 we have that (φ ⊗ idA )(p) + ∈ (φ ⊗ idA )(M). Moreover, the following are equivalent: (1 ) There exist finitely many ci ∈ B such that i ci∗ pci ∈ 1 + M. (2 ) For every multiplicative state φ on R there exist finitely many di ∈ A such that i di∗ (φ ⊗ idA )(p)di ∈ 1 + (φ ⊗ idA )(M). Proof. Clearly (1) implies (2). We will prove the converse in several steps. Note that for every multiplicative state φ on R, the mapping φ ⊗ idA : B → A is a surjective homomorphism of ∗algebras, hence (φ ⊗ idA )(M) is an archimedean quadratic module in A. Replacing B, M, f and p in Proposition 1 with A, (φ ⊗ idA )(M), σ and (φ ⊗ idA )(p), we see that (2) is equivalent to: (A) For every multiplicative state φ on R and every state σ on Ah such that σ ((φ ⊗idA )(M)) 0 we have that σ ((φ ⊗ idA )(p)) > 0. Note that (φ ⊗ σ )(r ⊗ a) = φ(r)σ (a) = σ (φ(r)a) = σ ((φ ⊗ idA )(r ⊗ a)) for every r ∈ R and a ∈ Ah . It follows that φ ⊗ σ = σ ◦ (φ ⊗ idA ). Thus, (A) is equivalent to: (B) For every M-positive state on R ⊗ Ah of the form ω = φ ⊗ σ where φ is multiplicative, we have that ω(p) > 0. Since R ⊗ 1 ⊆ Z(B), every factorizable state ω satisfies ω(r ⊗ a) = ω(r ⊗ 1)ω(1 ⊗ a) and ω(rs ⊗ 1) = ω(r ⊗ 1)ω(s ⊗ 1) for any r, s ∈ R and a ∈ Ah . Hence ω = φ ⊗ σ where φ is a multiplicative state on R and σ is a state on Ah . Therefore, (B) implies that: (C) ω(p) > 0 for every factorizable ω ∈ M ∨ . By Proposition 1, (C) is equivalent to (1). The equivalence of (1 ) and (2 ) can be proved in a similar way using Proposition 2. It can also be easily deduced from the equivalence of (1) and (2). / N where N := {m − (1 ) implies (2 ). Conversely, if (1 ) is false, then −1 ∈ Clearly ∗ ci pci | m ∈ M, ci ∈ B} is the smallest quadratic module in B which contains M and −p. By Corollary 4, there exists a factorizable state ω ∈ N ∨ . From the proof of (1) ⇔ (2), we know that ω = φ ⊗ σ = σ ◦ (φ ⊗ idA ) for a multiplicative state φ on R and a state σ on A.


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Since σ ((φ ⊗ idA )(N )) = ω(N ) 0, it follows that −1 ∈ / (φ ⊗ idA )(N ). Since (φ ⊗ idA )(N ) = {(φ ⊗ idA )(m) − j dj∗ (φ ⊗ idA )(p)dj | m ∈ M, dj ∈ A}, we get that (2 ) is false. 2 For every Hilbert space H we write B(H ) for the set of all bounded operators on H , P (H ) = Σ 2 (B(H )) for the set of all positive semidefinite operators on H and K(H ) for the set of all compact operators on H . Lemma 7. Let H be a separable Hilbert space and M a quadratic module in B(H ) which is not contained in P (H ). Then M contains all hermitian compact operators, i.e. K(H )h ⊆ M. Proof. Let M be a quadratic module in B(H ) which is not contained in P (H ). Pick an arbitrary operator L in M \ P (H ) and a vector v ∈ H such that v, Lv < 0. Write P for the orthogonal projection of H on the span of v. Clearly, P LP = λP where λ < 0, hence −P ∈ M. If Q is an orthogonal projection of rank 1, then Q = U ∗ P U for some unitary U , hence −Q ∈ M. Since also Q = Q∗ Q ∈ M, M contains all hermitian operators of rank√1. Therefore, M contains all finite rank operators. √ Pick any K ∈ K(H )h ∩ P (H ) and note that K ∈ K(H )h ∩ P (H ) as well. Clearly, −K + K ∈ M for every > 0 since it is a sum of an element from K(H )h ∩P (H ) and a finite rank operator (check the eigenvalues). It follows that −K ∈ M. It is also clear that every element of K(H )h is a difference of two elements from K(H )h ∩P (H ), hence K(H )h ⊆ M. 2 As the first application of Theorem 6 and Lemma 7, we prove the following generalization of Theorem F. By Lemma 11 below, Theorem F corresponds to the case R = R[x] and H finitedimensional, i.e. in the finite-dimensional case we can omit the assumption on p. Theorem 8. Let R be a commutative real algebra with trivial involution, H a separable Hilbert space, M an archimedean quadratic module in R ⊗ B(H ) and p an element of R ⊗ B(H ). If for every multiplicative state φ on R there exists a real ηφ > 0 such that (φ ⊗ idB(H ) )(p) ∈ ηφ + P (H ) + K(H )h (e.g. if p ∈ η + R ⊗ K(H )h for some η > 0) then the following are equivalent: (1) p ∈ M ◦ . (2) For every multiplicative state φ on R such that (φ ⊗ idB(H ) )(M) ⊆ P (H ), there exists φ > 0 such that (φ ⊗ idB(H ) )(p) ∈ φ + P (H ). If (φ ⊗ idB(H ) )(p) ∈ P (H ) + K(H )h for every multiplicative state φ on R (e.g. if p ∈ R ⊗ K(H )h ) then the following are equivalent: (1 ) p ∈ M. (2 ) For every multiplicative state φ on R such that (φ ⊗ idB(H ) )(M) ⊆ P (H ), we have that (φ ⊗ idB(H ) )(p) ∈ P (H ). Proof. Suppose that (1) is true, i.e. p ∈ + M for some > 0. For every multiplicative state φ on R such that (φ ⊗ idB(H ) )(M) ⊆ P (H ) we have that (φ ⊗ idB(H ) )(p) ∈ (φ ⊗ idB(H ) )( + M) ⊆ + P (H ), hence (2) is true. Conversely, suppose that (2) is true. We claim that for every multiplicative state φ on R there exists φ > 0 such that (φ ⊗idB(H ) )(p) ∈ φ +(φ ⊗idB(H ) )(M). Then it follows by Theorem 6 that (1) is true. If (φ ⊗ idB(H ) )(M) ⊆ P (H ), then (φ ⊗ idB(H ) )(p) ∈

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φ + P (H ) ⊆ φ + (φ ⊗ idB(H ) )(M) by (2) and the fact that (φ ⊗ idB(H ) )(M) is a quadratic module in B(H ). On the other hand, if (φ ⊗ idB(H ) )(M) P (H ), then K(H )h ⊆ (φ ⊗ idB(H ) )(M) by Lemma 7. The assumption (φ ⊗ idB(H ) )(p) ∈ ηφ + P (H ) + K(H )h for some ηφ > 0 then η implies that (φ ⊗ idB(H ) )(p) ∈ 2φ + (φ ⊗ idB(H ) )(M) as claimed. The proof of the equivalence (1 ) ⇔ (2 ) is similar. 2 In the infinite-dimensional case, the assumption on p cannot be omitted as the following example shows: Example. Let H be an infinite-dimensional separable Hilbert space, 0 = E ∈ B(H )h an or/ P (H ) + K(H )h . thogonal projection of finite rank and T an element of B(H )h such that T ∈ Since the quadratic module Σ 2 (B(H )/K(H )) is closed, also P (H ) + K(H )h is closed, hence there exists a real > 0 such that T + ∈ / P (H ) + K(H )h . Write p1 = −x 2 E, p2 = 1 − x 2 2 and p = + x T . Let M be the quadratic module in R[x] ⊗ B(H ) generated by p1 and p2 . Since p2 ∈ M, it follows from Lemma 11 below that M is archimedean. For every point a ∈ R such that p1 (a) 0 and p2 (a) 0 we have that a = 0, hence p(a) = . Therefore, assertion (2) of Theorem 8 is true for our M and p. Assertion (1), however, fails for our M and p. If it was true then therewould exist finitely manyqi , uj , vk ∈ R[x] ⊗ B(H ) and a real η > 0 such that p = η + i qi∗ qi + j u∗j p1 uj + k vk∗ p2 vk . For x = 1, we get + T = η + i qi (1)∗ qi (1) − j uj (1)∗ Euj (1). The first two terms belong to P (H ) and the last term belongs to K(H )h , a contradiction with the choice of T . We finish this section with a concrete version of Proposition 5 in the spirit of Theorem F. For R = R[x], we get [10, Corollary 22]. Theorem 9. Let R be a commutative real algebra with trivial involution, ν ∈ N, and M an archimedean quadratic module in Mν (R). For every element p ∈ Mν (R)h , the following are equivalent: (1) There are finitely many ci ∈ Mν (R) such that i ci∗ pci ∈ 1 + M. (2) For every multiplicative state φ on R such that (φ ⊗ id)(m) is positive semidefinite for all m ∈ M, we have that the operator (φ ⊗ id)(p) is not negative semidefinite. Proof. Write A = Mν (R). Clearly, a matrix C ∈ Ah is not negative semidefinite (i.e. it has at least one strictly positive eigenvalue) iff there exist matrices Di ∈ A such that i Di∗ CDi − I is positive semidefinite. It follows that a quadratic module M in A which is different from Σ 2 (A) contains −I , hence it is equal to Ah . (This also follows from Lemma 7.) Now we use equivalence (1 ) ⇔ (2 ) of Theorem 6. 2 4. Theorem E Recall that a ∗-algebra A is algebraically bounded if the quadratic module Σ 2 (A) is archimedean. For an element a ∈ Ah we say that a 0 iff a + ∈ Σ 2 (A) for all real > 0 (i.e. iff a ∈ Σ 2 (A)) and that a > 0 iff a ∈ + Σ 2 (A) for some real > 0 (i.e. iff a ∈ Σ 2 (A)◦ ). It is well known that every Banach ∗-algebra is algebraically bounded. The aim of this section is to deduce the following theorem from Theorem 6 and to show that it implies Theorem E. Other applications of Theorem 6 will be discussed in Section 5.


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Theorem 10. Let R be a commutative real algebra with trivial involution and A an algebraically bounded ∗-algebra over F ∈ {R, C}. Let U be an inner product space over F , L(U ) the ∗algebra of all adjointable linear operators on U , L(U )+ its subset of positive semidefinite operators, and M an archimedean quadratic module in R ⊗ L(U ). Write B := R ⊗ A and consider the vector space B ⊗ U as left R ⊗ L(U ) right B bimodule which is equipped with the biadditive form ·,· defined by b1 ⊗ u1 , b2 ⊗ u2 := b1∗ b2 u1 , u2 U . Write M for the subset of Bh which consists of all finite sums of elements of the form q, mq where m ∈ M and q ∈ B ⊗ U . We claim that the set M is an archimedean quadratic module and that for every element p ∈ R ⊗ Ah the following are equivalent: (1) p ∈ + M for some real > 0. (2) For every multiplicative state φ on R such that (φ ⊗ idL(U ) )(M) ⊆ L(U )+ , we have that (φ ⊗ idA )(p) > 0. Moreover, the following are equivalent: (1 ) p ∈ M . (2 ) For every multiplicative state φ on R such that (φ ⊗ idL(U ) )(M) ⊆ L(U )+ , we have that (φ ⊗ idA )(p) 0. Finally, the following are equivalent: (1 ) There exist finitely many ci ∈ B such that i ci∗ pci ∈ 1 + M . such that (φ ⊗ idL(U ) )(M) ⊆ L(U )+ , there exist (2 ) For every multiplicative state φ on R finitely many elements di ∈ A such that i di∗ (φ ⊗ idA )(p)di − 1 0. We will need the following observation which follows from the fact that the set of bounded elements w.r.t. a given quadratic module is closed for addition and multiplication of commuting elements. Lemma 11. Let R be a commutative algebra with trivial involution and A an algebraically bounded ∗-algebra. A quadratic module N in R ⊗ A is archimedean if and only if N ∩ R is archimedean in R. If x1 , . . . , xd are generators of R, then N is archimedean if and only if it contains K 2 − x12 − · · · − xd2 for some real K. Proof of Theorem 10. To prove that (1) implies (2), it suffices to prove: Claim 1. For every multiplicative state φ on R such that (φ ⊗ idL(U ) )(M) ⊆ L(U )+ , we have that (φ ⊗ idA )(M ) ⊆ Σ 2 (A). For every q ∈ B ⊗ U and m ∈ M we have that (φ ⊗ idA )(q, mq) = s, (φ ⊗ idL(U ) )(m)s where s = (φ ⊗ idA ⊗ idU )(q) ∈ A ⊗ U . If s = ki=1 ai ⊗ ui , then ⎤ a1 ⎢ ⎥ ak∗ T ⎣ ... ⎦ ⎡

s, (φ ⊗ idL(U ) )(m)s = a1∗

...

ak

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where T = [ui , (φ ⊗ idL(U ) )(m)uj ]ki,j =1 ∈ Mk (F ). Since (φ ⊗ idL(U ) )(m) is positive semidefinite for every m ∈ M, T is also positive semidefinite. To prove that (2) implies (1), consider the following statement: (3) For every multiplicative state φ on R there exists a real φ > 0 such that (φ ⊗ idA )(p) ∈ φ + (φ ⊗ idA )(M ). We claim that (2) implies (3) and (3) implies (1). Clearly, b∗ q, mqb = qb, mqb ∈ M for every m ∈ M, q ∈ B ⊗ U and b ∈ B, hence the set M is a quadratic module in B. Clearly, M ∩ R is archimedean since it contains M ∩ R. By Lemma 11, M is also archimedean. Hence (3) implies (1) by Theorem 6. Suppose that (2) is true and pick a multiplicative state φ on R. Clearly, (φ ⊗ idL(U ) )(M) is a quadratic module in L(U ) and (φ ⊗ idA )(M ) is a quadratic module in A. If (φ ⊗ idL(U ) )(M) ⊆ L(U )+ , then (2) implies that (φ ⊗ idA )(p) ∈ φ + Σ 2 (A) ⊆ φ + (φ ⊗ idA )(M ) for some real φ > 0, hence (3) is true. If (φ ⊗ idL(U ) )(M) L(U )+ then (3) follows from: Claim 2. For every multiplicative state φ on R such that (φ ⊗ idL(U ) )(M) L(U )+ we have that (φ ⊗ idA )(M ) = Ah . We could use Lemma 7 but we prefer to prove this claim from scratch. Pick any C ∈ (φ ⊗ idL(U ) )(M) \ L(U )+ . There exists u ∈ U of length 1 such that u, Cu < 0. Write P for the orthogonal projection of U on the span of {u}. Clearly, P ∗ CP = −λP for some λ > 0, hence −P ∈ (φ ⊗ idL(U ) )(M). Also, P = P ∗ P ∈ (φ ⊗ idL(U ) )(M). Let m± ∈ M be such that (φ ⊗ idL(U ) )(m± ) = ±P . Pick any a ∈ Ah and write q± = 1R ⊗ 1±a 2 ⊗ u where 1 = 1A . The element m = q+ , m+ q+ + q− , m− q− belongs to M and, by the proof of Claim 1, (φ ⊗ idA )(m ) = s+ , (φ ⊗ idL(U ) )(m+ )s+ + s− , (φ ⊗ idL(U ) )(m− )s− where s± = (φ ⊗ 1+a 2 1−a 2 idA ⊗ idU )(q± ) = 1±a 2 ⊗ u. Therefore, (φ ⊗ idA )(m ) = ( 2 ) u, P u + ( 2 ) u, −P u = 1−a 2 2 ( 1+a 2 ) − ( 2 ) = a. Claim 1 also gives implications (1 ) ⇒ (2 ) and (1 ) ⇒ (2 ) and Claim 2 gives their converses. Note that assertion (3) must be replaced with suitable assertions (3 ) and (3 ) to which Theorem 6 can be applied. 2 For U = F ν , we have that B ⊗ U ∼ = Bν ∼ = Mν (R) ⊆ Mν (B) and = Mν×1 (B), R ⊗ L(U ) ∼ q, mq = q ∗ mq in Theorem 10. However, if also A = Mν (F ) (i.e. B = Mν (R)), we do not get Theorem F. Combining both theorems, we get that archimedean quadratic modules M and M in Mν (R) have the same interior and the same closure. Finally, we would like to show that Theorem 10 implies Theorem E. The proof also works for algebraically bounded ∗-algebras. Proof of Theorem E. Write ν = 2 + ν1 + · · · + νm , x =

x12 + · · · + xd2 and p0 = [1 −

x2 ] ⊕ [x2 − 1] ⊕ p1 ⊕ · · · ⊕ pm ∈ Mν (R[x]). Clearly, K0 = {t ∈ Rd | p0 (t) 0}. Let M0 be the quadratic module in Mν (R[x]) generated by p0 . Since M0 contains (1 − x2 )Iν , it is archimedean by Lemma 11. By Theorem 10 applied to U = F ν , every element p ∈ R[x] ⊗ A which positive definite on K0 belongs to M0 . From the definition of M0 , we get that is strictly ∗ p = j ∈J (sj sj + qj∗ p0 qj ) for a finite J , sj ∈ R[x] ⊗ A and qj ∈ Mν×1 (R[x] ⊗ A), hence


p=

sj∗ sj

+ wj∗

m 1 − x2 wj + zj∗ x2 − 1 zj + sj∗k pk sj k

j ∈J

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k=1

for a finite J , sj , wj , zj ∈ R[x] ⊗ A and sj k ∈ Mνk ×1 (R[x] ⊗ A). Replacing x by tiplying with a large power of x2 we get that

x x

and mul-

∗ x p(x) = uj (x) + xvj (x) uj (x) + xvj (x) 2θ

j ∈J m ∗ uj k (x) + xvj k (x) pk (x) uj k (x) + xvj k (x) +

k=1

where uj , vj ∈ R[x] ⊗ A and uj k , vj k ∈ Mνk ×1 (R[x] ⊗ A) for every j ∈ J . Finally, we can get rid of the terms containing x by replacing x with −x and adding the old and the new equation. 2 5. Theorem G and moment problems Our next result, Theorem 12, is a special case of Theorem 10 for U = F . The proof can be shortened in this case because both claims become trivial. For R = R[cos φ1 , sin φ1 , . . . , cos φn , sin φn ], M = Σ 2 (R) and A = B(H ) it implies Theorem G. For R = R[x] and A a finitedimensional C ∗ -algebra it implies [17, Theorem 2], a step in the original proof of Theorem F. Both special cases can also be obtained from the original proof of Theorem E, namely, Theorem 3 and Lemma 5 from [1] imply Theorem 12 for R = R[x] and A a C ∗ -algebra. Theorem 12. Let R be a commutative real algebra with trivial involution, A an algebraically bounded ∗-algebra over F ∈ {R, C} and M an archimedean quadratic module in R. Write M = M · Σ 2 (R ⊗ A) for the quadratic module in R ⊗ A which consists of all finite sums of elements of the form mq ∗ q with m ∈ M and q ∈ R ⊗ A. For every element p ∈ R ⊗ Ah , the following are equivalent: (1) p ∈ + M for some real > 0. (2) For every multiplicative state φ on R such that φ(M) 0, we have that (φ ⊗ idA )(p) > 0. If we combine Theorem 12 with a suitable version of the Riesz representation theorem (namely, Theorem 1 in [9]) we get the following existence result for operator-valued moment problems which extends Theorem 3 and Lemma 5 from [1]. Theorem 13. Let A be an algebraically bounded ∗-algebra, R a commutative real algebra and M an archimedean quadratic module on R. For every linear functional L : R ⊗ A → R such that L(mq ∗ q) 0 for every m ∈ M and q ∈ R ⊗ A, there exists an A -valued nonnegative measure ∨ m on M such that L(p) = M ∨ (dm, p) for every p ∈ R ⊗ A. (Note that p defines a function φ → (φ ⊗ idA )(p) from M ∨ to A.) Proof. Recall that the set M ∨ is compact in the weak*-topology. We assume that A is equipped with its natural C ∗ -seminorm induced by the archimedean quadratic module Σ 2 (A), see [4,

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Section 3], hence it is a locally convex ∗-algebra. We will write C + (M ∨ , A) := C(M ∨ , Σ 2 (A)) for the positive cone of C(M ∨ , A). Let i be the mapping from R ⊗ A to C(M ∨ , A) defined by i(p)(φ) = (φ ⊗ idA )(p) for every p ∈ R ⊗ A and φ ∈ M ∨ . By Theorem 12, we have that C + (M ∨ , A) ∩ i(R ⊗ A) = i(M ) where M = M · Σ 2 (R ⊗ A). Note that L is an M -positive functional on R ⊗ A and that is defines in the natural way an i(M )-positive functional L on i(R ⊗ A). By the Riesz extension theorem for positive functionals, L extends to a C + (M ∨ , A)positive functional on C(M ∨ , A) which has the required integral representation by Theorem 1 in [9]. Hence L also has the required integral representation. 2 Finally, we would like to prove a nichtnegativsemidefinitheitsstellensatz that corresponds to Theorem 12. Theorem 14. Let H be a separable infinite-dimensional complex Hilbert space and R a commutative real algebra with trivial involution. Let M be an archimedean quadratic module in R and M = M · Σ 2 (R ⊗ B(H )). For every p ∈ R ⊗ B(H )h , the following are equivalent: (1) There are finitely many ci ∈ R ⊗ B(H ) such that i ci∗ pci ∈ 1 + M . (2) For every multiplicative state φ on R such that φ(M) 0, the operator (φ ⊗ idB(H ) )(p) is not the sum of a negative semidefinite and a compact operator. Note that for finite-dimensional H , (1) is equivalent to the following: For every multiplicative state φ on R such that φ(M) 0, the operator (φ ⊗ idB(H ) )(p) is not negative semidefinite; cf. Theorem 9. Proof. The equivalence (1 ) ⇔ (2 ) of Theorem 10 (with U = C and A = B(H )) says that our assertion (1) is equivalent to the following: (3) For every multiplicative state φ on R such that φ(M) 0, there exist finitely many operators Di ∈ B(H ) such that i Di∗ (φ ⊗ idA )(p)Di ∈ 1 + P (H ). Therefore it suffices to prove the following claim: Claim. For every operator C ∈ B(H )h , the following are equivalent: (i) (ii) (iii) (iv)

C is not the sum of a negative semidefinite and a compact operator, the positive part of C is not compact, there exists an operator D such that D ∗ CD = 1, there exist finitely many Di ∈ B(H ) such that i Di∗ CDi ∈ 1 + P (H ).

The implications (i) ⇒ (ii), (iii) ⇒ (iv) and (iv) ⇒ (i) are clear. To prove that (ii) implies (iii) we first note that C+ := E0 C = E0∗ CE0 where E0 is the spectral projection belonging to the interval [0, ∞). Since C+ is not compact, there exists by the spectral theorem a real number γ > 0 such that the spectral projection Eγ belonging to the interval [γ , ∞) has infinitedimensional range. The operator Cγ := Eγ C+ = Eγ∗ CEγ has decomposition Cγ = C˜ γ ⊕ 0 with respect to H = Eγ H ⊕ (1 − Eγ )H where C˜ γ γ . Write F = (C˜ γ )−1/2 ⊕ 0 and note that (Eγ F )∗ C(Eγ F ) = 1 ⊕ 0. Since Eγ H is infinite-dimensional, it is isometric to H . If G is an isometry from H onto Eγ H then D := Eγ F G satisfies (iii). 2


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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

C.-G. Ambrozie, F.-H. Vasilescu, Operator-theoretic Positivstellensätze, Z. Anal. Anwend. 22 (2) (2003) 299–314. R. Berr, T. Wörmann, Positive polynomials and tame preorderings, Math. Z. 236 (4) (2001) 813–840. J. Cimpriˇc, Maximal quadratic modules on ∗-rings, Algebr. Represent. Theory 11 (1) (2008) 83–91. J. Cimpriˇc, A representation theorem for archimedean quadratic modules on ∗-rings, Canad. Math. Bull. 52 (2009) 39–52. J. Cimpriˇc, M. Marshall, T. Netzer, Closures of quadratic modules, Trans. Amer. Math. Soc., in press. M.A. Dritschel, J. Rovnyak, The operator Fejer–Riesz theorem, arXiv:0903.3639v1. D. Handelman, Rings with involution as partially ordered abelian groups, Rocky Mountain J. Math. 11 (3) (1981) 337–381. T. Jacobi, A representation theorem for certain partially ordered commutative rings, Math. Z. 237 (2) (2001) 259– 273. G.W. Johnson, The dual of C(S, F ), Math. Ann. 187 (1970) 1–8. I. Klep, M. Schweighofer, Pure states, positive matrix polynomials and sums of hermitian squares, arXiv:0907.2260. M. Marshall, Positive polynomials and sums of squares, in: Math. Surveys Monogr., vol. 146, American Mathematical Society, Providence, RI, 2008, xii+187 pp., ISBN 978-0-8218-4402-1; 0-8218-4402-4. M. Marshall, A general representation theorem for partially ordered commutative rings, Math. Z. 242 (2) (2002) 217–225. A. Naftalevich, B. Schreiber, Trigonometric polynomials and sums of squares, in: Number Theory, New York, 1983–1984, in: Lecture Notes in Math., vol. 1135, Springer, Berlin, 1985, pp. 225–238. M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J. 42 (3) (1993) 969–984. M. Putinar, F.-H. Vasilescu, Solving moment problems by dimensional extension, Ann. of Math. (2) 149 (3) (1999) 1087–1107. B. Reznick, Uniform denominators in Hilbert’s seventeenth problem, Math. Z. 220 (1) (1995) 75–97. C.W. Scherer, C.W.J. Hol, Matrix sum-of-squares relaxations for robust semi-definite programs, Math. Program. 107 (1–2 (B)) (2006) 189–211. K. Schmüdgen, Noncommutative real algebraic geometry—some basic concepts and first ideas, in: Emerging Applications of Algebraic Geometry, in: IMA Vol. Math. Appl., vol. 149, Springer, New York, 2009, pp. 325–350. M. Takesaki, Theory of Operator Algebras, I, Springer, New York, Heidelberg, ISBN 0-387-90391-7, 1979, vii+415 pp. I. Vidav, On some ∗-regular rings, Acad. Serbe Sci. Pubi. Inst. Math. 13 (1959) 73–80.

Further reading [21] K. Schmüdgen, The K-moment problem for compact semi-algebraic sets, Math. Ann. 289 (2) (1991) 203–206.


Lp self-improvement of generalized Poincaré inequalities in spaces of homogeneous type ✩ Nadine Badr a , Ana Jiménez-del-Toro b , José María Martell c,∗ a Université Claude Bernard, Lyon I et CNRS UMR 5208, Bâtiment Doyen Jean Braconnier, 43 boulevard du 11

novembre 1918, 69622 Villeurbanne Cedex, France b Departamento de Matemáticas, Universidad Autónoma de Madrid, E-28049 Madrid, Spain c Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas,

C/Nicolás Cabrera 13-15, E-28049 Madrid, Spain Received 28 July 2010; accepted 21 January 2011 Available online 18 February 2011 Communicated by Gilles Godefroy

Abstract In this paper we study self-improving properties in the scale of Lebesgue spaces of generalized Poincaré inequalities in spaces of homogeneous type. In contrast with the classical situation, the oscillations involve approximation of the identities or semigroups whose kernels decay fast enough and the resulting estimates take into account their lack of localization. The techniques used do not involve any classical Poincaré or Sobolev–Poincaré inequalities and therefore they can be used in general settings where these estimates do not hold or are unknown. We apply our results to the case of Riemannian manifolds with doubling volume form and assuming Gaussian upper bounds for the heat kernel of the semigroup e−t with being the Laplace–Beltrami operator. We obtain generalized Poincaré inequalities with oscillations that involve the semigroup e−t and with right hand sides containing either ∇ or 1/2 . © 2011 Elsevier Inc. All rights reserved. Keywords: Semigroups; Heat kernels; Self-improving properties; Generalized Poincaré–Sobolev and Hardy inequalities; Pseudo-Poincaré inequalities; Dyadic cubes; Weights; Good-λ inequalities; Riemannian manifolds

✩

The second and third authors are supported by MEC Grant MTM2010-16518. The third author was also supported by CSIC PIE 200850I015. We warmly thank P. Auscher for his interest and helpful discussions. We also want to thank the anonymous referee for the suggestions that enhanced the presentation of this article. * Corresponding author. E-mail addresses: [email protected] (N. Badr), [email protected] (A. Jiménez-del-Toro), [email protected] (J.M. Martell). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.01.014

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N. Badr et al. / Journal of Functional Analysis 260 (2011) 3147–3188

1. Introduction In analysis and PDEs we can find various estimates that encode self-improving properties of the integrability of the functions involved. For instance, the John–Nirenberg inequality establishes that a function in BMO, which is a priori in L1loc (Rn ), is indeed exponentially integrable p which in turn implies that it is in Lloc (Rn ) for any 1 p < ∞. Another situation where functions self-improve their integrability comes from the classical (p, p)-Poincaré inequality in Rn , n 2, 1 p < n, p − |f − fQ | dx C(Q) − |∇f |p dx. Q

Q p

p

It is well known that this estimate yields that for any function f ∈ Lloc (Rn ) with ∇f ∈ Lloc (Rn ), 1/p∗ 1/p ∗ − |f − fQ |p dx C(Q) − |∇f |p dx Q

Q p∗

pn where p ∗ = n−p . Thus, f ∈ Lloc (Rn ) and f has self-improved its integrability. Both situations have something in common: they involve the oscillation of the functions on some cube Q via f − fQ . In [16], general versions of these estimates are considered. They start with inequalities of the form

− |f − fQ | dx a(Q, f ),

(1.1)

Q

where a is a functional depending on the cube Q, and sometimes on the function f . There, the authors present a general method based on the Calderón–Zygmund theory and the good-λ inequalities introduced by Burkholder and Gundy [7] that allows them to establish that under mild geometric conditions on the functional a, inequality (1.1) encodes an intrinsic self-improvement on Lp for p > 1. On the other hand, in [27] a new sharp maximal operator associated with an approximation of the identity {St }t>0 is introduced: MS# f (x) =

sup − |f − StQ f | dy,

Qx

Q

where tQ is a parameter depending on the side-length of the cube Q. This operator allows one to define the space BMOS , for which the John–Nirenberg inequality also holds (see [15]). In this way, starting with an estimate as (1.1) where the oscillation f − fQ is replaced by f − StQ f , and a(Q, f ) = C, a self-improving property is obtained. This new way of measuring the oscillation allows one to define new function spaces as the just mentioned BMOS of [15] and the Morrey– Campanato associated with an approximation of the identity of [14,34].


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In [23] and [24] self-improving properties related to this new way of measuring oscillation are under study. The starting estimate is as follows − |f − StQ f | dx a(Q, f ), (1.2) Q

with St being a family of operators (e.g., semigroup) with fast decay kernel. By analogy to (1.1), we will refer to these estimates as generalized Poincaré inequalities. The case a increasing, considered in [23] both in the Euclidean setting and also in spaces of homogeneous type, yields local exponential integrability of the new oscillation f − St f . In [24] functionals satisfying a weaker r -summability condition (see Dr below) are studied in the Euclidean setting and Lr,∞ local integrability of the oscillation is obtained. Taking [24] as a model and motivation, in this paper we consider (1.2) in the setting of the spaces of homogeneous type for functionals satisfying some summability conditions. The proofs of this paper and [24] are built upon the same ideas. However, the easier-to-handle Euclidean setting in [24] gives cleaner arguments that help to understand the present paper, and also that could be of interest to those readers that do not want to get into the technicalities that involve this less friendly setting of the spaces of homogeneous type. We present extensions of the Poincaré–Sobolev inequalities for the oscillations f − StQ f in Q that are valid in settings where the classical Poincaré–Sobolev inequalities (for the oscillations f − fQ ) do not hold or are unknown — this should be compared with the Euclidean setting where classical Poincaré– Sobolev inequalities are always at our disposal. That is the case of some Riemannian manifolds assuming only doubling volume form and Gaussian upper bounds for the heat kernel associated to the semigroup generated by the Laplace–Beltrami operator. As a consequence of the (local) Poincaré–Sobolev inequalities just mentioned we also obtain global pseudo-Poincaré (see Section 4 below), e.g., f − St f Lp (X) t 1/m hLp (X) where m is some scaling parameter and h plays the role of the gradient of f . In order to present the applications on Riemannian manifolds, which are the main motivation of the general results presented here, we need to introduce some notation, see Section 4.4 for more details. Let M be a complete non-compact connected Riemannian manifold with d its geodesic distance. Assume that volume form μ is doubling and let n be its doubling order (see (2.1) below). Then M equipped with the geodesic distance and the volume form μ is a space of homogeneous type. Let be the positive Laplace–Beltrami operator on M given by f, g = ∇f · ∇g dμ M

where ∇ is the Riemannian gradient on M and · is an inner product on T M. We assume that the heat kernel pt (x, y) of the semigroup e−t has Gaussian upper bounds if for some constants c, C > 0 and all t > 0, x, y ∈ M, pt (x, y)

d 2 (x,y) C e−c t . √ μ(B(x, t ))

We define q˜+ as the supremum of those p ∈ (1, ∞) such that for all t > 0, −t ∇e f Lp Ct −1/2 f Lp .

(UE)

(Gp )

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If the Riesz transform |∇−1/2 | is bounded in Lp , by analyticity of the heat semigroup, then (Gp ) holds. Therefore, q˜+ is greater than the supremum on the exponents p for which the Riesz transform is bounded on Lp . In particular q˜+ 2 by [11]. As a consequence of our main results and in the absence of Poincaré inequalities we obtain the following (see Corollary 4.6 below for the precise statement): Theorem 1.1. Let M be complete non-compact connected Riemannian manifold satisfying the doubling volume property and (UE). Given 1 p < ∞ we set p ∗ = np/(n − p) if 1 p < n and p ∗ = ∞ otherwise. (a) Given N 1 (N is taken large enough when 1 < p < n), let StN = I − (I − e−t )N and 1 < q < p ∗ . Then, for any smooth function with compact support f we have 1/q 1/p q p − f − StNB f dμ C φ(k)r σ k B − 1/2 f dμ , k1

B

σ kB

where φ(k) = σ −kθ and θ depends on m, n and p. (b) For any p ∈ ((q˜+ ) , ∞) ∪ [2, ∞), any 1 < q < p ∗ and any smooth function with compact support f we have 1/q 1/p k −tB q −cσ k p − f −e f dμ C e r σ B − |∇f | dμ . k1

B

σ kB

In this result σ is a large constant depending on the doubling condition (see Section 2 below). The plan of the paper is as follows. In Section 2 we give some preliminaries and definitions. The main result and its different extensions are in Section 3. Applications are considered in Section 4. In particular, we devote Sections 4.1 and 4.3 to study various Poincaré type inequalities in general spaces of homogeneous type. In the former we start from an estimate whose right hand side is localized to a given ball B, in the latter we take into account the lack of localization of the approximation of the identity or the semigroup and the right hand side contains a series of terms as in the applications to manifolds stated above. As a consequence, in Section 4.2 we obtain some global pseudo-Poincaré inequalities. In Section 4.4 we consider the application above and obtain generalized Poincaré inequalities in Riemannian manifolds. The subsequent sections contain the proofs of our results. 2. Preliminaries 2.1. Spaces of homogeneous type For full details and references we refer the reader to [10] and [9]. Let (X, d, μ) be a space of homogeneous type: X is a set equipped with a quasi-metric d and a non-negative Borel measure μ satisfying the doubling condition μ B(x, 2r) cμ μ B(x, r) < ∞,


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for some cμ 1, uniformly for all x ∈ X and r > 0, and where B(x, r) = {y ∈ X: d(y, x) < r}. We note that, in general, different centers and radii can define the same ball. Therefore, given a ball B we implicitly assume that a center and a radius are specified: B = B(xB , r(B)) where xB is the center and r(B) is the radius. The doubling property implies μ B(x, λr) cμ λn μ B(x, r)

and

r(B2 ) n μ(B2 ) cμ , μ(B1 ) r(B1 )

(2.1)

for some cμ , n > 0 and for all x, y ∈ X, r > 0 and λ 1, and for all balls B1 and B2 with B1 B2 . Let us recall that d being a quasi-metric on X means that d is a function from X × X to [0, +∞) satisfying the same conditions as a metric, except for the triangle inequality that is weakened to d(x, y) D0 d(x, z) + d(z, y) ,

(2.2)

for all x, y, z ∈ X and where 1 D0 < ∞ is a constant independent of x, y, z. Unfortunately, when D0 > 1 it does not follow, in general, that the balls are open. However, Macías and Segovia [26] proved that given any quasi-metric d, there exists another quasi-metric d equivalent to d such that the metric balls defined with respect to d are open. Thus, without loss of generality, from now on we assume that the metric balls are open sets. Also, in order to simply the computations, we assume that X is unbounded and therefore μ(X) = ∞, see for instance [28]. We make some conventions: A B means that the ratio A/B is bounded by a constant that does not depend on the relevant variables in A and B. Throughout this paper, the letter C denotes a constant that is independent of the essential variables and that may vary from line to line. Given a ball B = B(xB , r(B)) and λ > 0, we write λB = B(xB , λr(B)). For any set E we write diam(E) = supx,y∈E d(x, y). The average of f ∈ L1loc in B is denoted by fB = − f (x) dμ(x) = B

1 μ(B)

f (x) dμ(x) B

and the localized and normalized norm of a Banach or a quasi-Banach function space A by f A,B = f A(B,μ/μ(B)) . Examples of spaces A are Lp,∞ , Lp or more general Marcinkiewicz and Orlicz spaces. 2.2. Dyadic sets We take the dyadic structure given in [9] (here we use the notation in [23]). Theorem 2.1. (See [9].) There exist σ > 4D03 > 1 large enough, 0 < c1 , C1 , C2 < ∞ and D =

k∈Z Dk a countable collection of open sets Q with the following properties:

(i) Dk is a countable collection of disjoint sets such that X = Q∈Dk Q μ-a.e. (ii) If Q ∈ Dk , then diam(Q) C1 σ k . (iii) If Q ∈ Dk , then there exist xQ ∈ Q and balls BQ = B(xQ , c1 σ k ) and Bˆ Q = B(xQ , C1 σ k ) such that BQ ⊂ Q ⊂ Bˆ Q .

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(iv) If Q1 ∈ Dk1 and Q2 ∈ Dk2 with k1 k2 , then either Q1 ∩ Q2 = ∅ or Q1 ⊂ Q2 . We will refer to Q as dyadic cubes and to Dk as the k-th generation of D. In what follows, we fix σ > 4D03 large enough and consider the dyadic structure given by Theorem 2.1.

We will use the following decomposition of X in dyadic annuli: given Q ∈ D, we write X = k1 Ck (Q) with C1 (Q) = σ Bˆ Q and Ck (Q) = σ k Bˆ Q \ σ k−1 Bˆ Q , k 2. Also, given

a ball B, we write X = k1 Ck (B) with C1 (B) = σ B and Ck (B) = σ k B \ σ k−1 B, k 2. 2.3. Muckenhoupt weights A weight w is a non-negative locally integrable function. For any measurable set E, we write w(E) = E w(x) dμ(x). Also, we set − f dw = − f (x) dw(x) = B

B

1 w(B)

f (x)w(x) dμ(x). B

As before, we write f A(w),B = f A(B,w/w(B)) to denote the localized and normalized weighted norm of a Banach or a quasi-Banach function space A. We say that a weight w ∈ Ap (μ), 1 < p < ∞, if there exists a positive constant C such that for every ball B p−1 1−p − w dμ −w dμ C. B

B

For p = 1, we say that w ∈ A1 (μ) if there is a positive constant C such that for every ball B, − w dμ Cw(y), for μ-a.e. y ∈ B. B

We write A∞ (μ) =

p1 Ap (μ).

See [33] for more details and properties.

2.4. Functionals Let a : B × F −→ [0, +∞), where B is the family of all balls in X and F is some family of functions. When the dependence on the functions is not of our interest, we simply write a(B). We say that a is doubling if there exists some constant Ca > 0 such that for every ball B, a(σ B) Ca a(B). We recall the definition of the classes Dr introduced in [16]: given a Borel measure ν and 1 r < ∞, a satisfies the Dr (ν) condition (we simply write a ∈ Dr (ν)), if there exists 1 Ca < ∞ such that for each ball B and any family of pairwise disjoint balls {Bi }i ⊂ B, the following holds i

a(Bi )r ν(Bi ) Car a(B)r ν(B).


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We write aDr (ν) for the infimum of the constants Ca . By simplicity, we write Dr or Dr (w), when ν = μ or w is a weight. Note that, by Hölder’s inequality, the Dr (ν) conditions are decreasing: Dr (ν) ⊂ Ds (ν) and aDs (ν) aDr (ν) , for 1 s < r < ∞. On the other hand, if a is quasi-increasing (that is, a(B1 ) Ca a(B2 ), for all B1 ⊂ B2 ) then, a ∈ Dr (ν) for any Borel measure ν and 1 r < ∞. 2.5. Approximations of the identity and semigroups We work with families of linear operators {St }t>0 that play the role of generalized approximations of the identity. The reader may find convenient to think of {St }t>0 as being a semigroup since this is our main motivation. We assume from now on that these operators commute (that is, St ◦ Ss = Ss ◦ St for every s, t > 0). Families of operators that form a semigroup (that is, Ss St = Ss+t for all s, t > 0) satisfy this property. We assume that these operators admit an integral representation: St f (x) = st (x, y)f (y) dμ(y), X

where st (x, y) is a measurable function such that st (x, y)

1 d(x, y)m , g t μ(B(x, t 1/m ))

(2.3)

for some positive constant m and a positive, bounded and non-increasing function g. Observe that (2.3) leads to a rescaling between the parameter t and the space variables. Thus, given a ball B, we write tB = r(B)m in such a way that the parameter t and St are “adapted” or “scaled” to B. We also assume that for all N 0, lim r N g(r) = 0.

r→∞

We can relax the decay on g by fixing N > 0 large enough (in such a way that the estimates obtained below are not trivial). Further details are left to the reader. Let us note that the decay of g yields that the integral representation of St makes sense for all functions f ∈ Lp (X) and that ∞. As in [15], we consider a the operators St are uniformly bounded on Lp (X) for all 1

p

wider class of functions for which St is well defined: M = x∈X β>0 M(x,β) , where M(x,β) is the set of measurable functions f such that |f (y)| dμ(y) < ∞. f M(x,β) = 2n+β (1 + d(x, y)) μ(B(x, 1 + d(x, y))) X

It is shown in [15] that (M(x,β) , · M(x,β) ) is a Banach space, and if f ∈ M then, St f and Ss (St f ) are well defined and finite almost everywhere for all t, s > 0. As examples of semigroups we can consider second order elliptic form operators in Rn , Lf = − div(A∇f ), with A being an elliptic n × n matrix with complex L∞ -valued coefficients. The operator −L generates a C 0 -semigroup {e−tL }t>0 of contractions on L2 (Rn ). Under further

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assumptions (for instance, real A in any dimension; complex A in dimensions n = 1 or n = 2, etc.) the heat kernel has Gaussian bounds, that is, the above estimates hold with m = 2 and g(t) = 2 ce−ct . In this way we can take St = e−tL or St = I − (I − e−tL )N for some fixed N 1. Note that for the latter we lose the semigroup property, however, we still have the commutation rule and the Gaussian decay. Thus we can apply our results to that families. In some applications it is interesting to have N large enough so that one obtains extra decay in the resulting estimates (see [22,1,4] and the references therein). Similar examples could be considered in smooth domains of Rn since these are spaces of homogeneous type. Another examples of interest are the Riemannian manifolds X with the doubling property. In such a situation we can consider the Laplace–Beltrami operator . We assume that the heat kernel pt (x, y) of the semigroup e−t has Gaussian upper bounds (UE). As before, this allows us to use our results both for St = e−t or St = I − (I − e−t )N for some fixed N 1. Note 2 that the Gaussian upper bounds imply (2.3) with m = 2 and g(t) = ce−ct . See Section 4.4 for applications of our main results to this setting. 3. Main results Theorem 3.1. Let {St }t>0 be as above, 1 < r < ∞ and a ∈ Dr (μ). Let f ∈ M be such that − |f − StB f | dμ a(B),

(3.1)

B

for all balls B and where tB = r(B)m . Then for any ball B, we have f − StB f Lr,∞ ,B C

σ 2nk g cσ mk a σ k B

(3.2)

k0

with C 1 and 0 < c < 1. Furthermore, if a is doubling, then f − StB f Lr,∞ ,B a(B). The previous theorem can be extended to spaces with A∞ (μ) weights as follows: Theorem 3.2. Let {St }t>0 be as above, w ∈ A∞ (μ), 1 r < ∞ and a ∈ Dr (w) ∩ D1 (μ). If f ∈ M satisfies (3.1) then, f − StB f Lr,∞ (w),B C

σ 2nk g cσ mk a σ k B

k0

for all balls B with C 1 and 0 < c < 1. Further, if a is doubling, we can write Ca(B) in the right hand side. Remark 3.3. We would like to call attention to the fact that (3.1) is an unweighted estimate and that from it we obtain a weighted estimate for the oscillation f − StB f .


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Remark 3.4. We notice that we have imposed the mild condition D1 (μ), since in the proof we are going to use Lemma 5.1 and Proposition 5.3 below. Observe that if we assume w ∈ Ar (μ), then a ∈ Dr (w) implies a ∈ D1 (μ), see [24]. We would like to point out that one could have removed the condition a ∈ D1 (μ) in the particular case where St is a semigroup. The argument of the proof is somehow different and more technical as one needs an alternative proof for Lemma 5.1 and Proposition 5.3. We leave the details to the reader. As in [16,24], we extend Theorems 3.1 and 3.2. We change the hypothesis on the functional a so that the Dr (μ) condition allows a different functional in the right hand side. Theorem 3.5. Let {St }t>0 be as above and f ∈ M be such that (3.1) holds. Given 1 < r < ∞, and functionals a and a¯ we assume the following Dr (μ) type condition:

r a(Bi )r μ(Bi ) a(B) ¯ μ(B),

(3.3)

i

for each ball B and any family of pairwise disjoint balls {Bi }i ⊂ B. Then, we have f − StB f Lr,∞ ,B C

σ 2nk g cσ mk a¯ σ k B

(3.4)

k0

for all balls B with C 1 and 0 < c < 1. Furthermore, if a¯ is doubling, we can write C a(B) ¯ in the right hand side. Remark 3.6. Given two functionals a and a, ¯ abusing the notation, we say that (a, a) ¯ ∈ Dr (μ) if (3.3) holds. As in Theorem 3.2 we can consider a weighted extension of the previous result: we assume that (a, a) ¯ ∈ Dr (w) ∩ D1 (μ) and obtain the corresponding Lr,∞ (w) estimate. Details are left to the reader. 4. Applications We present some applications to the main results in the previous section. Some of the applications considered are analogous to those from [24] in the Euclidean setting. We would like to point out that although the underlying measure of the given space of homogeneous might be non-isotropic (i.e., we lose the property |Q| = (Q)n ), we will have at our disposal estimates (4.3) and (4.4). Examples 1, 2, 3, 4, 6 are essentially contained in [24] and therefore we sill skip some details. Examples 5, 7, 8 are new. We recall that Kolmogorov’s inequality implies that for any 0 < q < r < ∞ f Lq ,B

r r −q

1/q f Lr,∞ ,B .

(4.1)

This means that whenever we apply the previous results, we can replace Lr,∞ by Lq for every 0 < q < r. Note that the same occurs in the weighted situations.

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Example 1 (BMO and Morrey–Campanato spaces). We set a(B) = Cμ(B)α , α 0, and note that a is clearly increasing and doubling (because so is μ). Thus, a ∈ Dr (μ) for every 1 r < ∞. Consequently if f ∈ M is such that 1 − |f − StB f | dμ C, μ(B)α

(4.2)

B

we can conclude by Theorem 3.1 and Kolmogorov’s inequality (4.1), f − StB f Lr ,B μ(B)α , for every 1 < r < ∞ and for all balls B. Also all these estimates hold in Lr (w) with w ∈ A∞ (μ). Under the additional assumption that {St }t > 0 is a semigroup, (4.2) defines the spaces BMOS for α = 0 (see [15]) and the Morrey–Campanato LS (α) for α > 0 (see [34]). The reader is referred to those references for the corresponding self-improvement results (see also [14,23,24]). A unified approach to these examples is given in [23] where exponential self-improvement is obtained for general quasi-increasing functionals (and this is stronger than what ones obtained here). Analogously, one can consider the spaces BMOϕ,S (μ) that generalize those defined by S. Spanne [32] in Rn (see [24] and [23] for further details). For the following examples we assume that all annuli are non-empty, i.e., B(x, R) \ B(x, r) = ∅ for all 0 < r < R < ∞. This implies that r(B) ≈ diam(B) and also that B1 ⊂ B2 clearly yields r(B1 ) 2D0 r(B2 ) — we notice that these two properties fail to hold in general. In particular, r(B2 ) n μ(B2 ) cμ , μ(B1 ) r(B1 )

(4.3)

for every B1 ⊂ B2 . Also, in the examples below, r(B) can be replaced by diam(B) which is univocally determined (we however keep r(B) to emphasize the analogy with the Euclidean case). The non-empty annuli property implies that μ satisfies the reverse doubling condition (see [35]): there exist n¯ > 0 and c¯μ > 0 such that r(B1 ) n¯ μ(B1 ) c¯μ , μ(B2 ) r(B2 )

(4.4)

for all balls B1 and B2 with B1 ⊂ B2 . Example 2 (Fractional averages). Given λ 1, 0 < α < n, 1 p < n/α and a weight u, we set a(B) = r(B)

α

u(λB) μ(B)

1/p .

This functional is connected to the concept of higher gradient in [20,21]. Note that if p n/α, by (4.3) a is increasing; therefore, a ∈ Dr (μ) ∩ Dr (w), for every r 1 and w ∈ A∞ (μ). Thus,


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Theorem 3.1 together with (4.1) give self-improvement in all the range 1 r < ∞ for Lr (μ) and Lr (w) with w ∈ A∞ (μ). By [16] (see also [24]), we have that a ∈ Dr (μ) for 1 < r < pn/(n − αp). Thus, if f ∈ M satisfies u(λB) 1/p − |f − StB f | dμ r(B)α μ(B) B

for all balls B, then 1/r α u(σ k λB) 1/p σ 2nk g cσ mk r σ k B , − |f − StB f |r dx μ(σ k B) k0

B

for every 1 < r < pn/(n − αp). If in addition we assume that u ∈ A∞ (μ), [16] shows that a ∈ D pn + (μ) ( > 0 depends on u ∈ A∞ (μ)). Also we trivially have a doubling since so n−αp is u (and then we can take λ = 1). Therefore, in the previous estimate we reach the end-point r = pn/(n − αp) and furthermore on the right hand side we can write a(Q). See [24] for more details. 4.1. Reduced Poincaré type inequalities As in the previous examples and motivated by the classical (1, 1)-Poincaré inequality, one could consider estimates as follows: let f ∈ M be such that − |f − StB f | dμ r(B) − h dμ, B

(4.5)

B

for all balls B and where h is some non-negative measurable function: Typically h depends on f . For instance, in Rn one can take h = C|∇f |. However, in the computations below we can work with any given function h. We call this estimate a reduced Poincaré type inequality, in contrast with the expanded estimates (4.18) that we consider in Section 4.3 below. In this context it is more natural to relax (4.5) and take as an initial estimate 1/p p − |f − StB f | dμ r(B) − h dμ , B

(4.6)

B

with 1 p < ∞. We would like to apply our results to obtain self-improvement from (4.6). Example 3 (Poincaré–Sobolev inequality). If 1 p < n we show that (4.6) yields f − StB f Lp∗ ,∞ ,B

k0

1/p p φ(k)r σ B − h dμ ,

k

σ kB

(4.7)

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np for all balls B, for some sequence {φ(k)}k0 and where p ∗ = n−p . By Kolmogorov’s inequalr ∗ ity (4.1), we get strong-type estimates on L for every 1 < r < p . 1/p We set a(B) = r(B)( −B hp dμ) . Note that when p n, by (4.3) it follows that a is quasiincreasing. Thus we have strong-type estimates for all 1 < r < ∞. This case is studied in [23] and a stronger exponential integrability is proved. In our case, 1 < p < n, it suffices to see that a ∈ Dp∗ (μ) and to apply Theorem 3.1. Let B be a ball and {Bi }i ⊂ B a family of pairwise disjoint balls. Then, we use (4.3) (let us notice that in the Euclidean setting it suffices to use that |Bi | = cn r(Bi )n ) and the fact that p ∗ > p:

p∗

a(Bi ) μ(Bi ) =

i

r(Bi )n p∗ /n i

h dμ

μ(Bi ) Bi

∗ r(B)n p /n μ(B)

p∗ /p p

r(B)n μ(B)

p∗ /p p

h dμ

i B i

p∗ /p

p∗ /n hp dμ B

p∗

= a(B) μ(B).

(4.8)

Example 4 (Poincaré–Sobolev inequality for A1 (μ) weights). Given w ∈ A1 (μ) and 1 p < n, (4.6) implies f − StB f Lp∗ ,∞ (w),B

1/p φ(k)r σ k B − hp dw .

k0

(4.9)

σ kB

As a consequence of the previous inequality and the weighted version of Kolmogorov’s inequality, we get the strong norm Lr (w, B) for every 1 < r < p ∗ . In order to show (4.9) we use Theorem 3.2. First, using that w ∈ A1 (μ), we have that (4.6) gives 1/p p − |f − StB f | dμ r(B) − h dw = a(B). B

B

1 Let us recall the notation introduced above −B · · · dw = w(B) B · · · w dμ. To show that a ∈ Dp∗ (w) we proceed as in (4.8) replacing everywhere μ by w and using that w(B) μ(B) w(Bi ) μ(Bi )

r(B) r(Bi )

n ,

where the first estimate follows from the left hand side of inequality (5.14) and the fact that w ∈ A1 (μ), and the second inequality is (4.3). On the other hand, notice that w ∈ A1 (μ) ⊂ Ap∗ (μ) and therefore a ∈ D1 (μ) (see Remark 3.4). Thus, applying Theorem 3.2, we obtain (4.9).


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As before, when p n, we can obtain exponential type self-improvement since the functional is increasing (see [23]). Example 5 (Poincaré–Sobolev inequality for Ar (μ) weights, r > 1). We show that (4.6) with nrp 1 p < n implies that for every r > 1 and w ∈ Ar (μ), there exists q > n−p (depending on p, n, w) such that the following holds f − StB f Lq (w),B

1/(rp) rp φ(k)r σ B − h dw .

k0

σ kB

k

(4.10)

To check (4.10), we first see that (4.6) and w ∈ Ar (μ) give 1/(rp) rp = a(B). − |f − StB f | dμ r(B) − h dw B

B

The openness property of the Ar (μ) class gives that w ∈ Aτ r (μ) for some 0 < τ < 1. Without loss of generality, τ can be chosen so that pn < τ < 1. Hence, for any Bi ⊂ B we have, by (5.14) below and (4.3), w(B) w(Bi )

μ(B) μ(Bi )

τ r

r(B) r(Bi )

nτ r .

nrp We pick q0 = (nτ rp)/(nτ − p) and observe that q0 > n−p . Using this and proceeding as in the two previous examples we can easily see that a ∈ Dq0 (w) which by using Theorem 3.2 nrp < q < q0 , and Remark 3.4 (since q0 > r) leads to an estimate in Lq0 ,∞ (w). Next taking n−p Kolmogorov’s inequality gives (4.10).

Example 6 (Two-weight Poincaré inequality). Given 1 p q r < ∞, let (w, v) be a pair of weights with w ∈ Ar (μ), v ∈ Aq/p (μ) such that the following balance condition holds r(B1 ) w(B1 ) 1/r v(B1 ) 1/q , r(B2 ) w(B2 ) v(B2 )

for all B1 , B2 with B1 ⊂ B2 .

(4.11)

Then, (4.6) allows us to obtain f − StB f Lr,∞ (w),B

1/q φ(k)r σ k B − hq dv .

k0

(4.12)

σ kB

Consequently by Kolmogorov’s inequality, we obtain strong type estimates in the range 1 < s < r. In order to obtain (4.12), note that by (4.6) and using that v ∈ Aq/p (μ), we get 1/q = a(B). − |f − StB f | dμ r(B) − hq dv B

B

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Using the balance condition together with r/q 1, it is not difficult to see that a ∈ Dr (w). Hence, applying Remark 3.4 and Theorem 3.2, we obtain the desired inequality. Example 7 (Generalized Hardy inequality). We take 1 < p < n¯ (where n¯ is the exponent given in (4.4)) and fix x0 ∈ X. Let us consider wx0 (x) = d(x, x0 )−p . Then from (4.6) we obtain f − StB f Lp,∞ (wx

0

,B)

φ(k)

k0

1 wx0 (σ k B)

1/p

hp dμ

.

(4.13)

σ kB

As a consequence of (4.1), we automatically obtain strong type estimates in the range 1 < r < p. Note that the claimed estimate implies 1/p

1/p p ˜ φ(k) h dμ sup λwx0 x ∈ B: f (x) − StB f (x) > λ

λ>0

k0

σ kB

and this should be compared with the classical Hardy inequality f (x) − fB 2 dx ∇f (x)2 dx. |x|2 B

B

To obtain (4.13) we first observe that it is easy to see that for every ball B = B(xB , r(B)) / 2D0 B, α ∈ R, (4.14) − d(x, x0 )α dμ(x) ≈ d(x0 , xB )α , x0 ∈ B

and − d(x, x0 )α dμ(x) ≈ r(B)α ,

x0 ∈ 2D0 B, α > −n. ¯

(4.15)

B

Using these estimates it follows that wx0 ∈ A1 (μ) and r(B)(wx0 (B)/μ(B))1/p 1. Then we readily obtain that (4.6) yields − |f − StB f | dμ B

1 wx0 (B)

1/p

= a(B).

hp dμ

(4.16)

B

It is trivial to show that a ∈ Dp (wx0 ) and also that a ∈ D1 (μ) by Remark 3.4 and the fact that wx0 ∈ Ap (μ). Thus Theorem 3.2 gives as desired (4.13). Example 8 (Generalized two weights Hardy inequality). We take 1 < p < n¯ and 0 q p. Fixed x0 ∈ X we set wx0 (x) = d(x, x0 )−p and w¯ x0 (x) = d(x, x0 )−q . Then from (4.6) we obtain f − StB f Lp,∞ (w¯ x

0 ),B

k0

φ(k)

1 wx0 (σ k B)

1/p

p

h dμ σ kB

.

(4.17)


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As a consequence of the weighted version of (4.1), we automatically prove estimates in Lr (w¯ x0 ) for every 1 r < p. Taking the functional from the previous example, we have already shown (4.16) and a ∈ D1 (μ). Using (4.14) and (4.15) we obtain the following balance condition w¯ x0 (B1 ) wx0 (B2 ) 1, w¯ x0 (B2 ) wx0 (B1 )

B1 ⊂ B2 .

This easily gives a ∈ Dp (w¯ x0 ). Note also that w¯ x0 ∈ A1 (μ). Thus, Theorem 3.2 yields (4.17). 4.2. Global pseudo-Poincaré inequalities As a consequence of our results and using some ideas from [24], we are going to obtain the following generalized global pseudo-Poincaré inequalities, see [31]. These are of interest to obtain interpolation and Gagliardo–Nirenberg inequalities, see [31,6,25,29]. Assume that f ∈ M satisfies (4.6) with 1 p < n. Then for all t > 0: • Global pseudo-Poincaré inequalities: f − St f Lp (X) t 1/m hLp (X) . • Global weighted pseudo-Poincaré inequalities: for every w ∈ Ar (μ), 1 r < ∞, f − St f Lpr (w) t 1/m hLpr (w) . • Global pseudo-Hardy inequalities: let 1 0 and take k0 ∈ Z such that C1 σ t k +1 0 C1 σ . Then, we write X = Q∈Dk Q a.e. Note that for each Q ∈ Dk0 , there exists τ with 0 1 τ < σ m such that t = τ tBˆ Q . As in Lemma 5.5, we fix Q0 ∈ Dk0 and consider the family Jk = {Q ∈ Dk0 : σ k+1 Bˆ Q ∩ σ k+1 Bˆ Q0 = ∅}. It is easy to see that each Q ∈ Jk satisfies Q ⊂ σ k+2 Bˆ Q0 ⊂ σ k+3 Bˆ Q . This and the fact that μ is doubling imply #Jk cμ (C1 /c1 )n σ n(k+3) . On the other hand, Example 3 easily gives Lp strong-type estimates. Then, Minkowski’s inequality and Lemma 5.4 imply

f − St f Lp (X) =

1/p |f − St f | dμ p

Q∈Dk0 Q

Q∈Dk0

1/p

|f − Sτ tBˆ f |p dμ Q

τ 1/m Bˆ Q

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1/m k 1/m ˆ ˆ μ τ φ(k)r σ τ BQ BQ

Q∈Dk0

t

1/m

k0

φ(k)σ

k(1−n/p) ¯

t 1/m

Q∈Dk0 ¯ φ(k)σ k(1+n/p−n/p)

1/p p 1/p p

h dμ

σ k τ 1/m Bˆ Q

k0

−

p

h dμ σ k+1 Bˆ Q

k0

1/p

1/p t 1/m hLp (X) ,

hp dμ X

where we have used that {φ(k)}k0 (given in Theorem 3.1) is a fast decay sequence by the decay of g. In the weighted case with w ∈ Ar (μ), we use Example 4 for r = 1 and Example 5 for r > 1. nrp > rp. Thus, in both cases we obtain For r = 1 we have p ∗ > p, and if r > 1 we observe that n−p 1/(rp) 1/(rp) k rp rp − |f − StB f | dw φ(k)r σ B − h dw . k0

B

σ kB

Proceeding as before and using that the w dμ is doubling we obtain the desired inequality. For the pseudo-Hardy inequalities one uses the same ideas with the weak-type norm in the left hand side. 4.3. Expanded Poincaré type inequalities We introduce some notation: given 1 p, q < ∞ we say that f ∈ M satisfies an expanded Lq − Lp Poincaré inequality if for all balls B ⊂ X 1/q 1/p k q p − |f − StB f | dμ α(k)r σ B − h dμ , k0

B

σ kB

where {α(k)}k0 is a sequence of non-negative numbers and h is some non-negative measurable function. In this section we start with an expanded L1 − Lp Poincaré inequality and show that it selfimproves to an expanded Lq − Lp Poincaré inequality for q in the range (1, p ∗ ). More precisely, our starting estimate is the following: let p 1 and f ∈ M be such that 1/p k p − |f − StB f | dμ α(k)r σ B − h dμ , B

k0

(4.18)

σ kB

for all balls B ⊂ X and where {α(k)}k0 is a sequence of non-negative numbers and h is some non-negative measurable function. In the classical situation, replacing StB f by fB and taking h = C|∇f | and α(k) = 0 for k 1, this inequality is nothing but the L1 − Lp Poincaré–Sobolev inequality. Let us also observe that if α(k) = 0 for k 1, we get back to (4.5) in the previous section. On the other hand, if hp is


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doubling and {α(k)}k0 decays fast enough, then (4.18) leads us again to (4.6). As mentioned in [24] and [23], we believe that the estimates (4.18) are more natural than (4.5) or (4.6) in the sense that they take into account the tail effects of the semigroup in place of looking only at a somehow local term. As done in [24], (4.18) with h = |Df |, where D is some (differential) operator, can be obtained if we further assume that St 1 ≡ 1 a.e. in X and for all t > 0, and the following L1 − Lp Poincaré–Sobolev inequality 1/p − |f − fB | dμ Cr(B) − |Df |p dμ . B

B

As we show below, under some conditions on a Riemannian manifold we can obtain (4.18) without any kind of Poincaré–Sobolev inequality, thus our results are applicable in situations where such estimates do not hold or are unknown. Starting with (4.18) we are going to apply our main results to obtain a self-improvement on the integrability of the left hand side. For the sake of simplicity, we are going to treat only the unweighted Poincaré–Sobolev inequality analogous to those in Example 3. We notice that the same ideas can be used to consider Example 4 and obtain (4.9) with Lr (w), 1 < r n−p (here one can show that a ∈ Dq0 − (w) and this allows us to pick such value of q); and Example 6 for which we can show (4.12) with Ls (w), 1 < s < r, in place of Lr,∞ (w) if we further assume that 1 p q < r (here one can show that a ∈ Dr− (w)). Further details are left to the interested reader. We borrow some ideas from [24, Section 4.2]. We fix 1 p < n and define a(B) =

α(k)a0 σ k B

1/p with a0 (B) = r(B) − hp dμ .

k0

B

We are going to find another functional a¯ with a similar expression so that (a, a) ¯ satisfies a Dq condition as in Theorem 3.5. Proposition 4.1. Given a as above, let 1 p < n and 1 < q < p ∗ . There exists a sequence of non-negative numbers {α(k)} ¯ k0 , so that if we set a(B) ¯ =

k α(k)a ¯ 0 σ B ,

k0

we have that (a, a) ¯ ∈ Dq . The proof of this result is postponed until Section 5.4. From the proof we obtain that α(0) ¯ = k( pn − qn¯˜ ) l n/ ¯ q ˜ Cα(0) and α(l) ¯ = Cσ α(k) for l 1 with q˜ = max{q, p}. kmax{l−2,0} σ This result, Theorem 3.5, and Kolmogorov’s inequality (4.1) readily lead to the following corollary:

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Corollary 4.2. Given 1 p < n, let f ∈ M satisfy (4.18). Then, for all 1 < q < p ∗ there exists another sequence of non-negative numbers {α(k)} ˜ k0 so that 1/q 1/p k − |f − StQ f |q dμ α(k) ˜ σ Q − hp dμ . Q

k0

It is straightforward to show that α(k) ˜ =C

σ kQ

k

j =0 σ

2nj g(cσ mj )α(k ¯

− j ).

Remark 4.3. We would like to call the reader’s attention to the fact that in the case p n, the functional a defined above is increasing since so it is a0 . Therefore the previous estimate holds for all 1 < q < ∞ with a sequence α˜ defined as before and where α¯ = α. As in [24, Section 4.2] one can consider generalized Poincaré inequalities at the scale p ∗ . More precisely, one can push the exponent q to p ∗ and obtain an estimate in the Marcinkiewicz ∗ ∗ space associated with ϕ(t) ≈ t 1/p (1 + log+ 1/t)−(1+ )/p , > 0. Notice that ϕ is the fun∗ damental function of the Orlicz space Lp (log L)−(1+ ) , and the Marcinkiewicz space is the corresponding weak-type space (as Lq,∞ is for Lq ). Further details are left to the reader, see [24]. Given 1 p < ∞, by Corollary 4.2 and Remark 4.3 both particularized to q = p, we immediately get that f ∈ M satisfies an expanded L1 − Lp Poincaré inequality (4.18) (with a fast decay sequence) if and only if it satisfies an expanded Lp − Lp Poincaré inequality. Notice also that an expanded L1 − Lp Poincaré inequality implies trivially an expanded L1 − Lq (equivalently Lq − Lq ) Poincaré inequality for every q p. As a consequence of this and repeating the argument in the previous section we obtain the following global pseudo-Poincaré inequalities: Corollary 4.4. Assume that (4.18) holds with a fast decay sequence {α(k)}k0 . Then, for all q p and all t > 0 f − St f Lq (X) t 1/m hLq (X) . 4.4. Expanded Poincaré type inequalities on manifolds In this section we show that on Riemannian manifolds we can obtain expanded Poincaré type inequalities as (4.6) with different functions h on the right hand side. As observed before (see [24]), assuming that St 1 = 1 μ-a.e., classical Poincaré–Sobolev inequalities imply (4.18). There are situations where such Poincaré inequalities do not hold or are unknown. However the arguments below lead us to obtain generalized expanded Poincaré type inequalities to whom the self-improving results are applicable. We refer the reader to [3] and the references therein for a complete account of this topic. Let M be a complete non-compact connected Riemannian manifold with d its geodesic distance. Assume that the volume form μ is doubling. Then M equipped with the geodesic distance and the volume form μ is a space of homogeneous type. Non-compactness of M implies infinite diameter, which together with the doubling volume property yields μ(M) = ∞ (see for instance [28]). Notice that connectedness implies that M has the non-empty annuli property, therefore we are in a setting where we can apply all the previous applications.


3165

Let be the positive Laplace–Beltrami operator on M given by ∇f · ∇g dμ

f, g = M

where ∇ is the Riemannian gradient on M and · is an inner product on TM. The Riesz transform is the tangent space valued operator ∇−1/2 and it is bounded from L2 (M, μ) into L2 (M; TM, μ) by construction. One says that the heat kernel pt (x, y) of the semigroup e−t has Gaussian upper bounds if for some constants c, C > 0 and all t > 0, x, y ∈ M, pt (x, y)

d 2 (x,y) C √ e−c t . μ(B(x, t ))

(UE)

It is known that under doubling it is a consequence of the same inequality only at y = x [18, Theorem 1.1]. Notice that (UE) implies that pt (x, y) satisfies (2.3) with m = 2 (therefore 2 tB = r(B)2 ) and g(t) = ce−ct . Thus our results are applicable to the semigroup St = e−t and to the family of commuting operators St = I − (I − e−t )N with N 1 — expanding the latter one trivially sees that its kernel satisfies (UE). Under doubling and (UE), [11] shows that ∇−1/2 f

Lp

Cp f Lp

(Rp )

holds for 1 < p < 2 and all f bounded with compact support. Here, | · | is the norm on TM associated with the inner product. We define

q+ = sup p ∈ (1, ∞): (Rp ) holds which satisfies q+ 2 under doubling and (UE). It can be equal to 2 [11]. It is bigger than 2 assuming further the stronger L2 -Poincaré inequalities [2] and in some situations q+ = ∞. We also define q˜+ as the supremum of those p ∈ (1, ∞) such that for all t > 0, −t ∇e f

Lp

Ct −1/2 f Lp .

(Gp )

By analyticity of the heat semigroup, one always has q˜+ q+ ; indeed (Rp ) implies (Gp ): −t ∇e f

Lp

Cp 1/2 e−t f Lp Cp t −1/2 f Lp .

As we always have (R2 ) then this estimate implies (G2 ). Under the doubling volume property and L2 -Poincaré inequalities, q+ = q˜+ , see [3, Theorem 1.3]. It is not known if the equality holds or not under doubling and Gaussian upper bounds. Proposition 4.5. Let M be complete non-compact connected Riemannian manifold satisfying the doubling volume property and (UE).

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(a) Given N 1, let StN = I − (I − e−t )N . For any smooth function with compact support f we have k 1/2 N −k(2N −n) σ r σ B − f dμ. − f − StB f dμ C k1

B

σ kB

(b) For any p ∈ ((q˜+ ) , ∞) ∪ [2, ∞) and any smooth function with compact support f we have 1 1/p p k −tB p −cσ 2k p − f −e f dμ C e r σ B − |∇f | dμ . k1

B

σ kB

As a consequence of this result (whose proof is given below) and by Corollary 4.2 and Remark 4.3 we obtain Theorem 1.1 whose precise statement is given next: Corollary 4.6. Let M be complete non-compact connected Riemannian manifold satisfying the doubling volume property and (UE). Given 1 p < ∞ we set p ∗ = np/(n − p) if 1 p < n and p ∗ = ∞ otherwise. (a) Given N 1, let StN = I − (I − e−t )N and 1 < q (n + n/p − n/ ¯ max{q, p})/2 if 1 < p < n. Then, for any smooth function with compact support f we have 1/q 1/p 1/2 p k N q C φ(k)r σ B − f dμ , − f − StB f dμ k1

B

σ kB

where φ(k) = σ −k(2N −D−n/p) if 1 < p < n and φ(k) = σ −k(2N −D) if p n. (b) For any p ∈ ((q˜+ ) , ∞) ∪ [2, ∞), any 1 < q < p ∗ and any smooth function with compact support f we have 1/q 1/p k −tB q −cσ k p − f −e f dμ C e r σ B − |∇f | dμ . k1

B

σ kB

Remark 4.7. As mentioned before we can also get similar estimates assuming further local Poincaré–Sobolev inequalities. Notice that our assumptions guarantee that e−t 1 ≡ 1. Let us suppose that M satisfies the L1 − Lp Poincaré inequality, 1 p < ∞, that is, for every ball B p and every f ∈ L1loc (M), |∇f | ∈ Lloc (M) 1/p p − |f − fB | dμ r(B) − |∇f | dμ . B

B

Then, 1/p k −cσ k p e r σ B − |∇f | dμ − |f − St f | dμ C B

k1

σ kB


3167

with either St = e−t or St = I − (I − e−t )m . Notice that Proposition 4.5 establishes this estimate for some values p, and for the first choice of St , without assuming any kind of Poincaré inequalities. We would like to call the reader’s attention to the fact that, as mentioned before, one could prove similar estimates in the spirit of Examples 4, 5 and 6. Besides, global pseudo-Poincaré inequalities can be derived in the same manner. We finish this section exhibiting some examples of manifolds where the previous results can be applied. The most interesting example, where our results seem to be new is the following: Consider two copies of Rn minus the unit ball glued smoothly along their unit circles with n 2. It is shown in [11] that this manifold has doubling volume form and Gaussian upper bounds. L2 − L2 Poincaré does not hold: in fact, it satisfies Lp − Lp Poincaré if and only if p > n (see [19] in the case of a double-sided cone in Rn , which is the same). If n = 2, (Rp ) holds if and only if p 2 [11]. If n > 2, (Rp ) holds if and only if p < n [8]. In any case, we have q+ = n, hence q˜+ n. We can apply Corollary 4.6 and obtain (a) and (b). Notice that although classical Lp − Lp Poincaré holds if and only if p > n, (b) yields in particular expanded Lp − Lp Poincaré estimates for all n < p < ∞. There are many examples of manifolds or submanifolds satisfying the doubling property and the classical L1 − L1 Poincaré. Since doubling and L1 − L1 Poincaré imply (UE), we can apply Proposition 4.5 and Corollary 4.6 on such manifolds. Note that in this case, (b) of Proposition 4.5 and (b) of Corollary 4.6 are not new since, as mentioned before, Poincaré inequalities are stronger than expanded Poincaré inequalities. However, (a) yields new expanded Poincaré inequalities involving the square root of the Laplace–Beltrami operator on the right hand side. From these manifolds, we would like to mention the following: • Complete Riemannian manifolds M that are quasi-isometric to a Riemannian manifold with non-negative Ricci curvature (in particular every Riemannian manifold with non-negative Ricci curvature) have doubling volume form and admit classical L1 − L1 Poincaré. • Singular conical manifolds with closed basis admit classical L2 − L2 Poincaré inequalities for C ∞ functions (see [12]). Using the methods of [17] one can also see that classical L1 −L1 Poincaré holds. Such manifolds do not necessarily satisfy the doubling property, but they do, if for instance, one assumes that the basis is compact. • Co-compact covering manifolds with polynomial growth deck transformation group satisfy the doubling property and the classical L1 − L1 Poincaré (see [30]). • Nilpotent Lie groups have polynomial growth, then they satisfy the doubling property and the classical L1 − L1 Poincaré inequality. Among the important nilpotent Lie groups we mention the Carnot groups. 5. Proofs of the main results In this section we give the proof of the main results. For ease of reference we recall the meaning of some geometric constants that will appear several times in the proofs: cμ and n refer to the doubling constants for μ in (2.1); D0 is the constant in the quasi-distance condition (2.2); and σ , C1 , c1 are taken from Chirst’s dyadic construction in Theorem 2.1.

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5.1. Proof of Theorem 3.1 We split the proof in two parts. 5.1.1. Step I : Dyadic case We use some ideas from [24]. First, we fix σ > 4D03 large enough and take the dyadic structure given by Theorem 2.1. In this part of the proof, we show that for every 1 τ < σ m and for every Q ∈ D, f − Sτ tBˆ f Q

Lr,∞ ,Q

σ 2nk g σ m(k−8) a σ k Bˆ Q .

k0

In order to get it, we define a functional a˜ : B × F −→ [0, +∞) given by a(B) ˜ =

σ 2nk g σ m(k−8) a σ k B .

k0

Fix Q ∈ D and assume that a( ˜ Bˆ Q ) < ∞, otherwise, there is nothing to prove. Let G(x) = |f (x) − Sτ tBˆ f (x)|χσ 2 Bˆ Q (x). The Lebesgue differentiation theorem implies that it sufficies to Q

estimate MGLr,∞ ,Q . Thus, we study the level sets Ωt = {x ∈ X: MG(x) > t}, t > 0. We split the proof in two cases. When t is large, we use the Whitney covering lemma (Theorem 5.2 below). When t is small, the estimate is straightforward. The following auxiliary result will be very useful. Its proof is postponed until Section 5.1.3. Lemma 5.1. Assume that a ∈ D1 and (3.1). For every 1 τ < σ m , k 0 and R ∈ D, we have 2 5n − |f − Sτ tBˆ f | dμ aD1 (μ) cμ σ (C1 /c1 )n a σ k+2 Bˆ R . R

σ k Bˆ R 3 (C /c )2n g(1)−1 , where c is the constant of the weak-type (1, 1) Take c0 = cM aD1 (μ) cμ 1 1 M of M. Then, using the previous lemma, (2.1) and Theorem 2.1 we have

GL1 (X) =

2 5n |f − Sτ tBˆ f | dμ aD1 (μ) cμ σ Q

σ 2 Bˆ Q

C1 c1

n

a σ 4 Bˆ Q μ σ 2 Bˆ Q

c0 7n c0 σ g(1)a σ 4 Bˆ Q μ(Q) a( ˜ Bˆ Q )μ(Q). cM cM

(5.1)

˜ Bˆ Q ) < ∞. Also, M is of weak type (1, 1) with constant cM , and then Then G ∈ L1 (X) since a( we obtain μ(Ωt )

cM c0 ˆ GL1 (X) a( ˜ BQ )μ(Q). t t

(5.2)


3169

Next, let q > 1 be large enough, to be chosen. Our goal is to show the following good-λ inequality: given 0 < λ < 1, for all t > 0

a( ˜ Bˆ Q ) μ(Ωqt ∩ Q) λμ(Ωt ∩ Q) + λt

r (5.3)

μ(Q).

If 0 < t c0 cμ (C1 /c1 )n σ 2n a( ˜ Bˆ Q ) and 0 < λ < 1 then (5.3) is trivial: μ(Ωqt ∩ Q) μ(Q)

a( ˜ Bˆ Q ) λt

r

μ(Q) λμ(Ωt ∩ Q) +

a( ˜ Bˆ Q ) λt

r μ(Q).

In order to consider the other case, we need to state the following version of the Whitney covering lemma whose proof is given in Section 5.1.3 below. Theorem 5.2. Let t > 0 and G ∈ L1 (X). Let Ωt = {x ∈ X: MG(x) > t} be a proper subset of X. Then, there is a family of Whitney cubes {Qti }i such that

(a) Ωt = i Qti μ-almost everywhere. (b) {Qti }i ⊂ D, these cubes are maximal with respect to the inclusion and therefore they are pairwise disjoint. (c) 0 < (C1 /c1 )σ 6 r(Bˆ Qt ) < d(Qti , Ωtc ) (1/2)(C1 /c1 )σ 8 r(Bˆ Qt ) and as a consequence i

i

σ 9 (C1 /c1 )2 BQt ∩ Ωtc = ∅. i (d) −σ k Bˆ t G dμ t, for all k 1. Qi

(e) M(Gχ(σ Bˆ

Qti )

c

)(x) t, for all x ∈ Qti .

Suppose that t > c0 cμ (C1 /c1 )n σ 2n a( ˜ Bˆ Q ). Note that Ωt is a level set of the lower semicontinuous function MG. Moreover, as we have already seen, G ∈ L1 (X) and μ(Ωt ) < ∞. Thus, Ωt is an open proper subset of X. Therefore, the set Ωt can be covered by the family of Whitney cubes {Qti }i , by applying Theorem 5.2. From now on we restrict our attention to those cubes Qti with Qti ∩ Q = ∅. Notice that as a consequence of (5.2) and t > c0 cμ (C1 /c1 )n σ 2n a( ˜ Bˆ Q ), we have μ(Ωt ) < μ(Q) and therefore Qti Q for every Qti ∩ Q = ∅. Also for such cubes, by (5.2), (2.1) and Theorem 2.1 we obtain c0 ˆ ˜ BQ )μ(Q) μ Qti μ(Ωt ) a( t ˆ n ˆ n r(BQ ) r(BQ ) c0 ˆ ˜ BQ )cμ a( μ Qti σ −2n μ Qti t r(BQt ) r(Bˆ Qt ) i

i

and therefore r(Bˆ Qt ) σ −2 r(Bˆ Q ) i

and σ 2 Bˆ Qt ⊂ σ Bˆ Q . i

We have the following estimate, its proof is given in Section 5.1.3.

(5.4)

3170


Proposition 5.3. For every x ∈ Qti , MG(x) M |f − Sτ tBˆ f |χσ Bˆ t (x) + c1 t + c2 a( ˜ Bˆ Q ). Qti

Qi

Using this and the fact that a(Q) ˜ t we conclude that MG(x) M |f − Sτ tBˆ f |χσ Bˆ t (x) + C0 t. Qti

Qi

We choose q large enough so that q > C0 and take 0 < λ < 1. Using that the level sets are nested, we write μ(Ωqt ∩ Q) =

i: Qti ⊂Q

μ x ∈ Qti : MG(x) > qt

μ x ∈ Qti : M |f − Sτ tBˆ f |χσ Bˆ t (x) > (q − C0 )t Qti

i: Qti ⊂Q

=

··· +

Γ1

Qi

· · · = I + II,

(5.5)

Γ2

where Γ1 = Qti ⊂ Q: − |f − Sτ tBˆ f | dμ λt , Qti

σ Bˆ Qt

i

Γ2 = Qti ⊂ Q: − |f − Sτ tBˆ f | dμ > λt . Qti

σ Bˆ Qt

i

Applying that M is of weak type (1, 1), μ doubling, and Theorems 2.1 and 5.2, we estimate I : 1 I t Γ1

|f − Sτ tBˆ f | dμ λ Qti

σ Bˆ Qt

μ Qti λμ(Ωt ∩ Q).

i: Qti ⊂Q

i

In order to estimate II, we first observe that if Qti ∈ Γ2 (by Lemma 5.1), we have λt < − |f − Sτ tBˆ f | dμ a σ 3 Bˆ Qt . σ Bˆ Qt

Qti

i

i

Thus, II

1 r r μ Qti a σ 3 Bˆ Qt μ Qti . i λt t Γ2

i: Qi ⊂Q


3171

In principle, it is not possible to apply the condition Dr (μ) since the balls of the family {σ 3 Bˆ Qt } i i may not be pairwise disjoint. Note that by (5.4) we have {σ 3 Bˆ Qt } ⊂ σ 2 Bˆ Q . Next, we claim that i

i

3n 13n {σ 3 Bˆ Qt } splits in N families {Ej }N . j =1 of pairwise disjoint balls with N cμ (C1 /c1 ) σ i i Assuming this, we use that a ∈ Dr (μ) over each Ej , the fact that μ is doubling and Theorem 2.1 to obtain

II

1 λt 1 λt

r N r

r a σ 3 Bˆ Qt μ σ 3 Bˆ Qt i

j =1 i: Qti ∈Ej

i

1 λt

r

r a σ 2 Bˆ Q μ σ 2 Bˆ Q

a( ˜ Bˆ Q )r μ(Q).

Plugging the estimates for I and II into (5.5), we conclude

1 μ(Ωqt ∩ Q) λμ(Ωt ∩ Q) + λt

r

a( ˜ Bˆ Q )r μ(Q),

˜ Bˆ Q ) provided we check the previous claim. Note that by for all t > c0 cμ (C1 /c1 )n σ 2n a( Lemma 5.4 below it suffices to fix Qtj and show that

#Ej := # Qti : σ 3 Bˆ Qt ∩ σ 3 Bˆ Qt = ∅ cμ (C1 /c1 )3n σ 13n . i

j

As a consequence of Theorems 2.1 and 5.2, for any Qti ∈ Ej we have 0 < σ 5 r(Bˆ Qt ) < d σ 3 Bˆ Qt , Ωtc σ 8 (C1 /c1 )r(Bˆ Qt ). i

i

i

Then it is easy to see that σ −4 (C1 /c1 )r(Bˆ Qt ) r(Bˆ Qt ) σ 4 (C1 /c1 )r(Bˆ Qt ) i

j

i

and Qti ⊂ σ 8 (C1 /c1 )Bˆ Qt ⊂ σ 13 (C1 /c1 )2 Bˆ Qt . j

i

Using these estimates, (2.1) and Theorem 2.1 we obtain μ σ 13 (C1 /c1 )2 Bˆ Qt cμ σ 13n (C1 /c1 )3n μ Qti μ σ 8 (C1 /c1 )Bˆ Qt #Ej j

Qti ∈Ej

cμ σ 13n (C1 /c1 )3n μ

i

Qti ∈Ej

Qti

Qti ∈Ej

cμ σ 13n (C1 /c1 )3n μ σ 8 (C1 /c1 )Bˆ Qt j

and this readily leads to the desired bound for #Ej .

3172


Next, we fix N > 0. Note that the good-λ inequality (5.3) implies ˆ r μ(Ωqt ∩ Q) a( ˜ BQ ) r μ(Ωt ∩ Q) sup t cλ sup t +c μ(Q) μ(Q) λ 0 0 and y ∈ M, with γ > 0 depending on p . Using this estimate and Minkowski’s inequality we can control Ik : Ik =

p 1/p ∇x ps (x, y)ϕ(y) dμ(y) dμ(x) Ck (B) B

e

−c σ

2k r(B)2 s

B

s

s

s

−1/2

−1/2

−1/2

e

Ck (B)

2k 2 −c σ r(B) s

1/p d 2 (x,y) ∇x ps (x, y)p eγ s dμ(x) ϕ(y) dμ(y)

1 ϕ(y) dμ(y) √ μ(B(y, s ))1/p

B

r(B) √ s r(B) √ s

n/p

e n/p

−c σ

2k r(B)2 s

1 μ(B)1/p

ϕ(y) dμ(y) B

e−c

σ 2k r(B)2 s

μ(B)1/p ,

√ √ where we have used that μ(B) ≈ μ(B(y, rB )) cμ (rB / s )n μ(B(y, s )) since 0 < s < tB = r(B)2 and y ∈ B. Then, I s −1/2

1/p 1/p ∞ k σ r(B) n/p −c σ 2k r(B)2 s − |∇f |p dμ − |∇f |p dμ + s −1/2 e . √ s k=2

σB

σ kB

Taking the supremum over all such functions ϕ we obtain 1/p 1/p tB −tB p − f −e f dμ − |∇f | dμ s −1/2 ds B

σB

0

k 1/p tB ∞ σ r(B) n/p −c σ 2k r(B)2 p s − |∇f | dμ + s −1/2 e ds √ s k=2

0

σ kB

∞

1/p k −cσ 2k p e r σ B . − |∇f | dμ

k=1

σ kB

It remains to prove (a). We write h = 1/2 f and h = ∞ √ −t dt 1/2 =c 0 te t we obtain N − f − StNB f dμ = − I − e−tB f dμ B

B

∞

k=1 hk

with hk = hχCk (B) . Since


3185

tB d −s −tB N −1 = − I − e f (x) ds dμ − e ds B

0

tB N −1 −s 1/2 − I − e−tB e h dμ ds 0

B

tB ∞ √ dt N −1 −(s+t) − I − e−tB e h dμ t ds t 0 0

B

∞ tB ∞ k=1 0 0

√ dt N −1 −(s+t) ds. − I − e−tB e hk dμ t t B

One has that t∂t pt (x, y) satisfies also (UE) (see [13, Theorem 4] or [18, Corollary 3.3]) and this easily implies that {e−t (t)}t>0 satisfies L1 − L1 full off-diagonal estimates (see [5] for a discussion of off-diagonal estimates associated to semigroups): given E, F closed sets and t > 0 −t e (t)(f χE )

L1 (F )

Ce−c

d(E,F )2 t

f L1 (E) .

(5.22)

This and (UE) imply that e−t (t) and (I − e−t )N −1 are uniformly bounded on L1 . These facts allow us to estimate the term k = 1: tB ∞ √ dt N −1 −(s+t) ds − I − e−tB e h1 dμ t t 0 0

B

tB ∞ √ t dt ds r(σ B) − |h| dμ. − |h| dμ t +s t σB

σB

0 0

For k 2 we split the t-variable integral in two pieces: 0 < t < N tB and t N tB . We first fix 0 < t < N tB and 0 < s < tB . Observe that N −1 N −1 −(s+t) e = Cj,N e−(j tB +t+s) I − e−tB j =0

and that t + s j tB + t + s 2N tB . Then (5.22) implies N −1 −(s+t) − I − e−tB e hk dμ

N −1 σ 2k r(B)2 1 −1 − j tB +t+s (j tB + t + s) e |h| dμ μ(B) j =0

B

e−cσ (t + s)−1 2k

− |h| dμ.

σ kB

σ kB

3186


Hence, we conclude that tB N tB √ dt N −1 −(s+t) ds − I − e−tB e hk dμ t t 0

0

B

e−cσ

2k

tB N tB √ t dt − |h| dμ ds t +s t

σ kB

0

0

2k e−cσ r σ k B − |h| dμ. σ kB

Next for the case t N tB we make the changes of variables t = t/(tB N ) and s = s/tB : tB ∞ √ dt N −1 −(s+t) ds − I − e−tB e hk dμ t I= t 0 N tB B

1 ∞ √ dt N −1 −tt (N −1) −(s+t)t B (t )h dμ ds r(B) − I − e−tB e B e t B k t 0 1

B

1 ∞ r(B) 0 1

N −1 −(s+t)t dt B (s + t)tB hk dμ 3 ds. − e−ttB − e−(ttB +tB ) e t2 B

We need the following lemma whose proof is below. Lemma 5.9. Given given E, F closed sets and 0 < v u, we have u −u d(E,F )2 −(u+v) e (f χ − e ) Ce−c u f L1 (E) . E v L1 (F ) Using this result, (5.22) and [22, Lemma 2.3] we have for every 0 < s < 1 < t < ∞ N −1 −(s+t)t dt B − e−ttB − e−(ttB +tB ) (s + t)tB hk dμ 3 ds e t2 B

t

−(N −1)

σ 2k r(B)2 1 −c max{tt B ,(s+t)tB } e μ(B)

|h| dμ t σ kB

Thus,

2k

−(N −1) kn −c σ t

σ e

− |h| dμ. σ kB

(5.23)


3187

1 ∞ σ 2k dt I r(B)σ − |h| dμ t −(N −1) e−c t 3 ds t2 0 1 σ kB σ −k(2N −n) r σ k B − |h| dμ. kn

σ kB

Gathering the obtained estimates the proof is complete. Proof of Lemma 5.9. We proceed as in [22, p. 504]: u v d u −u −(u+v) −(u+s) e − e (f χ − e ) = (f χ ) ds E E 1 v v ds L (F ) 0

u v

v

L1 (F )

−(u+s) e (u + s) (f χE )

0

u Cf L1 (E) v

v

e−

cd(E,F )2 u+s

0

Ce

)2 − cd(E,F u

L1 (F )

ds u+s

ds u+s

f L1 (E) ,

where we have used (5.22) and that u u + s u + v 2v.

2

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The inclusion relation between Sobolev and modulation spaces Masaharu Kobayashi a,∗ , Mitsuru Sugimoto b a Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjuku-ku, Tokyo 162-8601, Japan b Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya 464-8602, Japan

Received 5 September 2010; accepted 17 February 2011 Available online 25 February 2011 Communicated by I. Rodnianski

Abstract The inclusion relations between the Lp -Sobolev spaces and the modulation spaces is determined explicα itly. As an application, mapping properties of unimodular Fourier multiplier ei|D| between Lp -Sobolev spaces and modulation spaces are discussed. © 2011 Elsevier Inc. All rights reserved. Keywords: Modulation space; Lp -Sobolev space; Inclusion; Unimodular Fourier multiplier

1. Introduction p,q

The modulation spaces Ms are one of the function spaces introduced by Feichtinger [6] in 1980’s to measure the decaying and regularity property of a function or distribution in a way p p,q different from Lp -Sobolev spaces Ls or Besov spaces Bs . The precise definitions of these function spaces will be given in Section 2, but the main idea of modulation spaces is to consider the space variable and the variable of its Fourier transform simultaneously, while they are treated independently in Lp -Sobolev spaces and Besov spaces. Because of this special nature, modulation spaces are now considered to be suitable spaces in the analysis of pseudo-differential operators after a series of important works [4,9–11,24, 25] and so on. “Modulation spaces and pseudo-differential operators” is still an active fields * Corresponding author.

E-mail addresses: [email protected] (M. Kobayashi), [email protected] (M. Sugimoto). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.02.015

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of research (see, for example, [5,12,13,17,23,28]). On the other hand, modulation spaces have also remarkable applications in the analysis of partial differential equations. For example, the p,q Schrödinger and wave propagators, which are not bounded on neither Lp nor Bs , are bounded p,q on Ms [2]. Modulation spaces are also used as a regularity class of initial data of the Cauchy problem for nonlinear evolution equations, and in this way the existence of the solution is shown under very low regularity assumption for initial data (see [30–32]). In the last several years, many basic properties of modulation spaces are established. In particular, the inclusion relation between Besov spaces and modulation spaces has been completely determined. Let us define the indices ν1 (p, q) and ν2 (p, q) for 1 p, q ∞ in the following way: ⎧ ⎨0 ν1 (p, q) = 1/p + 1/q − 1 ⎩ −1/p + 1/q ⎧ ⎨0 ν2 (p, q) = 1/p + 1/q − 1 ⎩ −1/p + 1/q

if (1/p, 1/q) ∈ I1∗ : min(1/p, 1/p ) 1/q, if (1/p, 1/q) ∈ I2∗ : min(1/q, 1/2) 1/p , if (1/p, 1/q) ∈ I3∗ : min(1/q, 1/2) 1/p, if (1/p, 1/q) ∈ I1 : max(1/p, 1/p ) 1/q, if (1/p, 1/q) ∈ I2 : max(1/q, 1/2) 1/p , if (1/p, 1/q) ∈ I3 : max(1/q, 1/2) 1/p,

where 1/p + 1/p = 1 = 1/q + 1/q . We remark ν2 (p, q) = −ν1 (p , q ). 1/q

1/q

1

1 I1

I3∗

1/2

1/2 I1∗

I3

I2

0

I2∗

1/2

1

1/p

0

1/2

1

1/p

Then the following result is known: Theorem 1.1. (See Sugimoto and Tomita [22], Toft [25].) Let 1 p, q ∞ and s ∈ R. Then we have p,q

(1) Bs (Rn ) → M p,q (Rn ) if and only if s nν1 (p, q); p,q (2) M p,q (Rn ) → Bs (Rn ) if and only if s nν2 (p, q). As for the inclusion relation between Lp -Sobolev spaces and modulation spaces, the following result (see also [27]) is immediately obtained from Theorem 1.1 if we notice the inclusion p p,q p property Ls+ε → Bs → Ls−ε for ε > 0 (see, [29, p. 97]):


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Corollary 1.2. Let 1 p, q ∞ and s ∈ R. Then we have p

p

(1) Ls (Rn ) → M p,q (Rn ) if s > nν1 (p, q). Conversely, if Ls (Rn ) → M p,q (Rn ), then s nν1 (p, q); p p (2) M p,q (Rn ) → Ls (Rn ) if s < nν2 (p, q). Conversely, if M p,q (Rn ) → Ls (Rn ), then s nν2 (p, q). But in Corollary 1.2, there still remains a question whether the critical case s = nν1 (p, q) or s = nν2 (p, q) is sufficient or not for the inclusion. The objective of this paper is to answer this basic question and complete the picture of inclusion relations between the Lp -Sobolev spaces and the modulation spaces. The following theorems are our main results: p

Theorem 1.3. Let 1 p, q ∞ and s ∈ R. Then Ls (Rn ) → M p,q (Rn ) if and only if one of the following conditions is satisfied: (1) (2) (3) (4)

q p > 1 and s nν1 (p, q); p > q and s > nν1 (p, q); p = 1, q = ∞, and s nν1 (1, ∞); p = 1, q = ∞ and s > nν1 (1, q). p

Theorem 1.4. Let 1 p, q ∞ and s ∈ R. Then M p,q (Rn ) → Ls (Rn ) if and only if one of the following conditions is satisfied: (1) (2) (3) (4)

q p < ∞ and s nν2 (p, q); p < q and s < nν2 (p, q); p = ∞, q = 1, and s nν2 (∞, 1); p = ∞, q = 1, and s < nν2 (∞, q).

It should be mentioned that Kobayashi, Miyachi and Tomita [14] determines the inclusion p,q relation between modulation spaces Ms and local Hardy spaces hp for 0 1 since we have hp = Lp then. As a matter of fact, the proof of Theorems 1.3 and 1.4 heavily depends on the results and arguments established in [14]. As an application of our main theorems, we also consider mapping properties of unimoduα lar Fourier multiplier ei|D| , α 0, which is a generalization of wave (α = 1) and Schrödinger (α = 2) propagators. See Corollaries 5.2 and 5.4 in Section 5. As Theorem A and Theorem B α there say, the operator ei|D| (0 α 2) is bounded on modulation spaces while not on Lp Sobolev spaces. Theorems 1.3 and 1.4 help us to understand what happen if we consider the operator between Lp -Sobolev spaces and modulation spaces. We explain the organization of this paper. After the next preliminary section devoted to the definitions and basic properties of function spaces treated in this paper, we give a proof of Theorem 1.4 in Sections 3 and 4. We remark that Theorem 1.3 is just the dual statement of Theorem 1.4. In Section 5, we consider mapping properties of unimodular Fourier multipliers between Lp -Sobolev spaces and modulation spaces, as well as those of invertible pseudo-differential operators.

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2. Preliminaries 2.1. Basic notation The following notation will be used throughout this article. We write S(Rn ) to denote the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on on Rn , i.e., the topological dual of Rn and S (Rn ) to denote the space of tempered distributions S(Rn ). The Fourier transform is defined by fˆ(ξ ) = Rn f (x)e−ix·ξ dx and the inverse Fourier transform by f ∨ (x) = (2π)−n fˆ(−x). We define f Lp =

f (x)p dx

1/p

Rn

for 1 p < ∞ and f L∞ = ess. supx∈Rn |f (x)|. We also define the Lp -Sobolev norm · Lps by

1/2 ∨

f Lps = ·s fˆ(·) Lp with · = 1 + | · |2 following the notation of Sogge [20] and Stein [21]. Let (X, · X ) and (Y, · Y ) be two Banach spaces, which include S(Rn ), respectively. We say that an operator T from X to Y is bounded if there exists a constant C > 0 such that Tf Y C f X for all f ∈ S(Rn ), and we set

T X→Y = sup Tf Y f ∈ S Rn , f X = 1 . We use the notation I J if I is bounded by a constant times J , and we denote I ≈ J if I J and J I . 2.2. Modulation spaces We recall the modulation spaces. Let 1 p, q ∞, s ∈ R and ϕ ∈ S(Rn ) be such that supp ϕ ⊂ [−1, 1]n and ϕ(ξ − k) = 1 for all ξ ∈ Rn .

(1)

k∈Zn

Then the modulation space Ms (Rn ) consists of all tempered distributions f ∈ S (Rn ) such that the norm q/p 1/q p sq ϕ(D − k)f (x) dx f Msp,q =

k p,q

k∈Zn

Rn

is finite, with obvious modifications if p or q = ∞. Here we denote ϕ(D − k)f (x) = (ϕ(· − k)fˆ(·))∨ (x). p,q p,q We simply write M p,q (Rn ) instead of M0 (Rn ). The space Ms (Rn ) is a Banach space n which is independent of the choice of ϕ ∈ S(R ) satisfying (1) [6, Theorem 6.1]. If 1 p, p,q q < ∞, then S(Rn ) is dense in Ms (Rn ) [6, Theorem 6.1]. If 1 p1 p2 ∞, 1 p1 ,q1 p ,q q1 q2 ∞ and s1 s2 then Ms1 (Rn ) → Ms22 2 (Rn ) [6, Proposition 6.5]. Let us dep,q n n fine by Ms (R ) the completion of S(R ) under the norm · Msp,q . If 1 p, q < ∞, then

M. Kobayashi, M. Sugimoto / Journal of Functional Analysis 260 (2011) 3189–3208 p,q

p,q

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p,q

Ms (Rn ) = Ms (Rn ) [1, Lemma 2.2] and the dual of Ms (Rn ) can be identified with p ,q M−s (Rn ), where 1/p + 1/p = 1 = 1/q + 1/q . Moreover, the complex interpolation theory for these spaces reads as follows: Let 0 < θ < 1 and 1 p1 , p2 , q1 , q2 ∞, s1 , s2 ∈ R. Set 1/p = (1 − θ )/p1 + θ/p2 , 1/q = (1 − θ )/q1 + θ/q2 and s = (1 − θ )s1 + θ s2 , then p ,q p ,q p,q (Ms11 1 , Ms22 2 )[θ] = Ms ([6, Theorem 6.1], [30, Theorem 2.3]). We recall the following lemmas. Lemma 2.1. (See [6, Proposition 6.7].) Let 1 p ∞, 1/p + 1/p = 1 and s ∈ R. Then p,min(p,p ) n

R

Ms

p p,max(p,p ) n → Ls Rn → Ms R .

Let Uλ : f (x) → f (λx) be the dilation operator. Then the following dilation property of M p,q is known. Lemma 2.2. (See [22, Theorem 3.1].) Let 1 p, q ∞. We have, for C1 , C2 > 0, Uλ f M p,q C1 λnμ1 (p,q) f M p,q , Uλ f M p,q C2 λnμ2 (p,q) f M p,q ,

∀λ 1, ∀f ∈ M p,q Rn , ∀λ 1, ∀f ∈ M p,q Rn ,

where ⎧ ⎨ −1/p μ1 (p, q) = 1/q − 1 ⎩ −2/p + 1/q ⎧ ⎨ −1/p μ2 (p, q) = 1/q − 1 ⎩ −2/p + 1/q

if (1/p, 1/q) ∈ I1∗ : min(1/p, 1/p ) 1/q, if (1/p, 1/q) ∈ I2∗ : min(1/q, 1/2) 1/p , if (1/p, 1/q) ∈ I3∗ : min(1/q, 1/2) 1/p, if (1/p, 1/q) ∈ I1 : max(1/p, 1/p ) 1/q, if (1/p, 1/q) ∈ I2 : max(1/q, 1/2) 1/p , if (1/p, 1/q) ∈ I3 : max(1/q, 1/2) 1/p.

p,q Let Is0 : f → ( ·s0 fˆ(·))∨ , s0 ∈ R. Then following lifting property of Ms is known. p,q

Lemma 2.3. (See [25].) Let 1 p, q ∞, s ∈ R. Then Is0 maps Ms p,q Ms−s0 (Rn ).

(Rn ) isomorphically onto

2.3. Besov spaces We recall the Besov spaces. Let 1 p, q ∞ and s ∈ R. Suppose that ψ0 , ψ ∈ S(Rn ) satisfy j supp ψ0 ⊂ {ξ | |ξ | 2}, supp ψ ⊂ {ξ | 1/2 |ξ | 2} and ψ0 (ξ ) + ∞ j =1 ψ(ξ/2 ) = 1 for all p,q n j n ξ ∈ R . Set ψj (·) = ψ(·/2 ) if j 1. Then the Besov space Bs (R ) consists of all f ∈ S (Rn ) such that f Bsp,q =

∞

2

j =0

with usual modification again if q = ∞.

∨ q fˆ(·)ψj (·) Lp

j sq

1/q < ∞,

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M. Kobayashi, M. Sugimoto / Journal of Functional Analysis 260 (2011) 3189–3208 p ,q

p,q

If 1 p, q < ∞, then the dual of Bs (Rn ) can be identified with B−s (Rn ), where 1/p + 1/p = 1 = 1/q + 1/q . 2.4. Local Hardy spaces n We recall the local Hardy spaces. Let 0 p< pn < ∞, and let Ψ ∈ S(Rn ) be such that Rn Ψ (x) dx = 0. Then the local Hardy space h (R ) consists of all f ∈ S (R ) such that

f hp = sup |Ψt ∗ f |

0 n, that is, s > n(1/p − 1/q). 2 p,q

Lemma 4.3. Let 1 q < p < ∞ and s ∈ R. If Lp (Rn ) → Ms 1/q − 1).

(Rn ), then s < −n(1/p +

p,q

Lemma 4.4. Let 1 p, q < ∞ and s ∈ R. If Lp (Rn ) → Ms (Rn ), then q/p 1/q 1/p |k|(n(1/p−1)+s)q |c |p |ck |p k=0

|k|/2| |2|k|

k=0

for all finitely supported sequences {ck }k∈Zn \{0} . Proof of Lemma 4.4.. Let 0 < δ < 1 and a ∈ S(Rn ) be such that ˆ ) C > 0 supp a ⊂ [−δ/8, δ/8]n , a L∞ 1, and a(ξ

on |ξ | 2

(see, for example, [14, Lemma 4.3]). For a finitely supported sequence {c } ∈Zn \{0} , we define f ∈ S(Rn ) by f (x) = c | |n/p a | |(x − ) .

=0

We first estimate f Lp . Since n supp a | |(· − ) ⊂ + −δ 8| | , δ 8| | , we have p f Lp

p n/p = c | | a | |(x − ) dx Rn

=

=0

p p |c |p | |n a | |(x − ) dx = a Lp |c |p .

Rn =0

Next, we estimate f Msp,q . We note the following facts:

=0

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Fact 1. Let Ψ ∈ S(Rn ) be such that Ψ = 1 on [−δ/4, δ/4]n , supp Ψ ⊂ [−3δ/8, 3δ/8]n , and | C > 0 on [−2, 2]n . Then we have |Ψ f Msp,q ≈

q 1/q sq

1 + |k| f ∗ (Mk Ψ ) Lp , k∈Zn

where Mk Ψ (x) = eik·x Ψ (x). Fact 2. For all = 0, we have n supp a | |(· − ) ⊂ + −δ/8| |, δ/8| | ⊂ + [−δ/8, δ/8]n . Fact 3. For all x ∈ m + [−δ/8, δ/8]n , m ∈ Zn , we have supp Ψ (x − ·) ⊂ x + [−3δ/8, 3δ/8]n ⊂ m + [−δ/2, δ/2]n . From these facts, we have

(Mk Ψ ) ∗ f p p L (Mk Ψ ) ∗ f (x)p dx m∈Zn (m,δ)

=

n p eik·(x−y) Ψ (x − y) p c | | a | |(y − ) dy dx

m∈Zn (m,δ) Rn

=

=0

n p e−ik·y Ψ (x − y)cm |m| p a |m|(y − m) dy dx,

m=0 (m,δ) Rn

where we set (m, δ) = m + [−δ/8, δ/8]n . If x ∈ m + [−δ/8, δ/8]n and y ∈ supp a(|m|(· − m)), then x − y ∈ m + [−δ/8, δ/8]n − m + [−δ/8, δ/8]n = [−δ/4, δ/4]n , and so Ψ (x − y) = 1. Hence,

(Mk Ψ ) ∗ f p p L n p e−ik·y Ψ (x − y)cm |m| p a |m|(y − m) dy dx m=0 (m,δ) Rn

=

n p e−ik·y cm |m| p a |m|(y − m) dy dx

m=0 (m,δ) Rn

n k p δ p n−pn = |cm | |m| aˆ |m| . 4 m=0


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Moreover, using |a(ξ ˆ )| C > 0 for all 1/2 |ξ | 2, we obtain

p

(Mk Ψ ) ∗ f p p (δ/4)n |cm |p |m|n−pn aˆ k/|m| L m=0

(δ/4)n

p |cm |p |m|n−pn aˆ k/|m|

|k|/2|m|2|k|

|cm |p |m|n−pn |k|n−pn

|k|/2|m|2|k|

|cm |p

|k|/2|m|2|k|

for all k = 0. Then f Msp,q ≈

q sq

1 + |k| (Mk Ψ ) ∗ f Lp

1/q

k∈Zn

sq |k|n−pn 1 + |k| k=0

|k|

p,q

|cm |

.

(Rn ), we have

k=0

q/p 1/q

|k|(n(1/p−1)+s)q

|cm |p

|k|/2|m|2|k|

f

q/p 1/q p

|k|/2|m|2|k|

Therefore, by our assumption Lp (Rn ) → Ms

|cm |p

|k|/2|m|2|k|

(n(1/p−1)+s)q

k=0

q/p 1/q

p,q Ms

f

Lp

|c |

p

1/p .

2

=0

Proof of Lemma 4.3. Suppose that s −n(1/p + 1/q − 1) contrary to our claim. Noting that q/p < 1 from the assumption q < p, take ε > 0 such that (1 + ε)q/p < 1 and define {ck }k∈Zn \{0} by ck =

|k|−n/p (log |k|)−(1+ε)/p 0

if |k| N, if |k| < N,

where N is sufficiently large. Note also that {|k|−n/r (log |k|)−α/r }|k|N ∈ r if α > 1, and / r if α 1, where r < ∞ (see, for example, [23, Remark 4.3]). {|k|−n/r (log |k|)−α/r }|k|N ∈ Thus k=0

1/p |ck |p

=

−n/p −(1+ε)/p p 1/p |k| log |k| < ∞. |k|N

On the other hand, since n(1/p − 1) + s −n/q and (1 + ε)q/p < 1, we see that

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|k|

k=0

|c |

p

|k|/2| |2|k|

|k|(n(1/p−1)+s)q

|k|2N

q/p 1/q

(n(1/p−1)+s)q

−n/p −(1+ε)/p p | | log | |

|k|/2| |2|k|

|k|

(n(1/p−1)+s)q

−(1+ε)q/p log |k|

q/p 1/q

1/q

|k|2N

−n/q −{(1+ε)q/p}/q q 1/q |k| log |k| = ∞. |k|2N

This contradicts Lemma 4.4.

2 p

Proof of Theorem 1.4 (“only if” part). Suppose M p,q (Rn ) → Ls (Rn ). Then we have s nν2 (p, q) by Corollary 1.2. Particularly in the case p < q, we have s < −n(1/p − 1/q) = nν2 (p, q) for p 2 by Lemma 4.1, and s < −n(1/p + 1/q − 1) = n(1/p + 1/q − 1) = nν2 (p, q) for 2 p by the dual statement of Lemma 4.3. In the case p = ∞, we must have L1−s (Rn ) → M 1,q (Rn ), since otherwise the fact S(Rn ) is dense in both L1−s (Rn ) and M1,q (Rn ) 1,q

n 1 n n implies that M ∞,q (Rn ) ⊂ L∞ s (R ), contrary to the assumptions. Hence L (R ) → Ms (R ). Then we have s < −n/q = n(1/q − 1) = nν2 (∞, q) for q = 1 by Lemma 2.4. All of these results yields the necessity of conditions (1)–(4). 2

5. Applications α

We consider the unimodular Fourier multiplier ei|D| , α 0, defined by α i|D|α f (x) = ei|ξ | fˆ(ξ )eix·ξ dξ, f ∈ S Rn . e Rn α

The operator ei|D| has an intimate connection with the solution u(t, x) of initial value problem for the dispersive equation

i∂t u + ||α/2 u = 0, u(0, x) = f (x), α

(t, x) ∈ R × Rn . The boundedness of ei|D| on several function spaces has been studied extenp p,q sively by many authors. Concerning the Lp -Sobolev spaces Ls and the modulation spaces Ms , the following theorems are known. α

Theorem A. (See Miyachi [15].) Let 1 1. Then ei|D| is bounded from p Ls (Rn ) to Lp (Rn ) if and only if s αn|1/p − 1/2|. α

Theorem B. (See Bényi, et al. [2].) Let 1 p, q ∞ and 0 α 2. Then ei|D| is bounded from M p,q (Rn ) to M p,q (Rn ).

M. Kobayashi, M. Sugimoto / Journal of Functional Analysis 260 (2011) 3189–3208 α

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p,q

The boundedness of ei|D| with 0 α 2 on weighted modulation spaces Ms (Rn ) follows from Theorem B and the lifting property of modulation spaces (Lemma 2.3). Indeed, since the α operator T = ei|D| is translation invariant, it commutes with Is and we have Tf Msp,q ≈ Is Tf M p,q ≈ T Is f M p,q Is f M p,q ≈ f Msp,q , p,q

which means the boundedness of T on Ms . We remark that Theorem B with α = 2 is established in a more general form by [26]. We also remark that Bényi and Okoudjou [3] extends Theorem B to the case 1 p ∞, 0 < q ∞, 0 α 2 and the case n/(n + 1) p < 1, 0 < q ∞, α = 1, 2. For α > 2, we have a different type of boundedness: α

Theorem C. (See Miyachi, et al. [16].) Let 1 p, q ∞, s ∈ R and α > 2. Then ei|D| is p,q bounded from Ms (Rn ) to M p,q (Rn ) if and only if s (α − 2)n|1/p − 1/2|. α

Theorem A says that the operator ei|D| is not bounded on Lp (Rn ), and we have generally a loss of regularity of the order up to αn|1/p − 1/2|. Theorems B and C describe an advantage of modulation spaces because we have no loss in the case 0 α 2 or smaller loss in the case α α > 2 if we consider the operator ei|D| on these spaces. α Then what is the exact order of the loss when we consider the operator ei|D| between Lp spaces and modulation spaces. We can answer this question by using our main theorem. The case 0 α 2 is rather simple, and we have the following results: α

p,q

Theorem 5.1. Let 1 p, q ∞, s ∈ R and 0 α 2. Then ei|D| is bounded from Ms p,q to Lp (Rn ) if and only if Ms (Rn ) → Lp (Rn ). α

p,q

Proof. Assume that ei|D| is bounded from Ms

(Rn )

(Rn ) to Lp (Rn ):

i|D|α

e f Lp f Msp,q . Note that e−i|D| is also bounded from Ms have α

p,q

(Rn ) to Lp (Rn ). Then by taking f = e−i|D| g we α

α

g Lp e−i|D| g M p,q g Msp,q , s

α p,q α which means that Ms (Rn ) → Lp (Rn ) by the equation ei|D| f = e−i|D| f¯. Conversely assume α p,q p,q that Ms (Rn ) → Lp (Rn ). Since ei|D| is bounded on Ms (Rn ) by Theorem B, we have

i|D|α

e f

α

Lp

α

ei|D| f M p,q f Msp,q , s

p,q

which means that ei|D| is bounded from Ms

(Rn ) to Lp (Rn ).

2

The following corollary is straightforwardly obtained from Theorem 5.1 and Theorem 1.4. The second part is just the dual statement of the first part:

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M. Kobayashi, M. Sugimoto / Journal of Functional Analysis 260 (2011) 3189–3208 α

p,q

Corollary 5.2. Let 1 p, q ∞, s ∈ R and 0 α 2. Then ei|D| is bounded from Ms to Lp (Rn ) if and only if one of the following conditions is satisfied: (1) (2) (3) (4)

(Rn )

q p < ∞ and s −nν2 (p, q); p < q and s > −nν2 (p, q); p = ∞, q = 1, and s −nν2 (∞, 1); p = ∞, q = 1, and s > −nν2 (∞, q), p

and from Ls (Rn ) to M p,q (Rn ) if and only if one of the following conditions is satisfied: (5) (6) (7) (8)

q p > 1 and s nν1 (p, q); p > q and s > nν1 (p, q); p = 1, q = ∞, and s nν1 (1, ∞); p = 1, q = ∞ and s > nν1 (1, q). For α > 2, we have the following results: α

Theorem 5.3. Let 1 p, q ∞, s ∈ R and α > 2. Then ei|D| p,q p,q Ms+(α−2)n|1/p−1/2| (Rn ) to Lp (Rn ) if Ms (Rn ) → Lp (Rn ). α

p,q

Proof. Assume that Ms (Rn ) → Lp (Rn ). Since p,q p,q Ms+(α−2)n|1/p−1/2| (Rn ) to Ms (Rn ) by Theorem C, we have

i|D|α

e f

α

Lp

ei|D|

α

ei|D| f M p,q f M p,q s

s+(α−2)n|1/p−1/2|

is

is bounded from

bounded

from

,

p,q

which means that ei|D| is bounded from Ms+(α−2)n|1/p−1/2| (Rn ) to Lp (Rn ).

2

The following corollary is obtained from Theorem 5.3, Theorem 1.4 and the duality argument again: α

p,q

Corollary 5.4. Let 1 p, q ∞, s ∈ R and α > 2. Then ei|D| is bounded from Ms Lp (Rn ) if one of the following conditions is satisfied: (1) (2) (3) (4)

q p < ∞ and s −nν2 (p, q) + (α − 2)n|1/p − 1/2|; p < q and s > −nν2 (p, q) + (α − 2)n|1/p − 1/2|; p = ∞, q = 1, and s −nν2 (∞, 1) + (α − 2)n|1/p − 1/2|; p = ∞, q = 1, and s > −nν2 (∞, q) + (α − 2)n|1/p − 1/2|, p

and from Ls (Rn ) to M p,q (Rn ) if one of the following conditions is satisfied: (5) (6) (7) (8)

q p > 1 and s nν1 (p, q) + (α − 2)n|1/p − 1/2|; p > q and s > nν1 (p, q) + (α − 2)n|1/p − 1/2|; p = 1, q = ∞, and s nν1 (1, ∞) + (α − 2)n|1/p − 1/2|; p = 1, q = ∞ and s > nν1 (1, q) + (α − 2)n|1/p − 1/2|.

(Rn ) to


3203

Remark 5.5. For pseudo-differential operators 1 x x +y , ξ ei(x−y)·ξ f (y) dy dξ σ (X, D)f (x) = σ (2π)n 2n 2 W

R

with the symbol σ ∈ M ∞,1 (R2n ), it is well known that it is bounded on each M p,q (Rn ) [8, Theorem 14.5.2] (see also [18]). Furthermore, this class of pseudo-differential operators is a Wiener algebra, meaning that if in addition σ W (X, D) is invertible on L2 (Rn ) = M 2,2 (Rn ), then its inverse is of the form τ W (X, D) with τ ∈ M ∞,1 (R2n ) (see [9,19]). From these observation it follows if s ∈ R and σ ∈ M ∞,1 (R2n ), then the following is true: p

p

(1) If M p,q (Rn ) → Ls (Rn ), then σ W (X, D) is bounded from M p,q (Rn ) to Ls (Rn ). (2) Suppose that σ W (X, D) is invertible on L2 (Rn ). If σ W (X, D) is bounded from M p,q (Rn ) p p to Ls (Rn ), then Mp,q (Rn ) → Ls (Rn ). We may now combine (1) and (2) with Theorem 1.4 to obtain similar results to Corollaries 5.2 and 5.4. We omit the details. Although the converse of Theorem 5.3 is not true, we believe that the converse of Corollary 5.4 is still true. In fact we have at least the following result: Theorem 5.6. Let 1 p, q ∞, s ∈ R and α > 2. α

p,q

(1) Suppose that ei|D| is bounded from Ms (Rn ) to Lp (Rn ). Then we have s −nν2 (p, q) + (α − 2)n|1/p − 1/2|. α p (2) Suppose that ei|D| is bounded from Ls (Rn ) to M p,q (Rn ) instead. Then we have s nν1 (p, q) + (α − 2)n|1/p − 1/2|. To prove Theorem 5.6, we use the following lemma. Lemma 5.7. Let 1 p, q ∞, s ∈ R and α 0. Then

α

α

sup k−s ϕ(D − k)ei|D| Lp →Lp ei|D| M p,q →Lp , s

k∈Zn

where ϕ is a function satisfying (1). Proof. Let N be a positive integer such that ϕ(· − k) =

ϕ(· − k)ϕ · − (k + )

| |N

for all k ∈ Zn . Then we have

ϕ(D − k)ei|D|α f p L

i|D|α

ϕ(D − k)f p,q p,q e M →Lp M s

s

(6)

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α

= ei|D|

p,q

→Lp

Ms

m

p,q

→Lp

Ms

1/q

L

m∈Zn

α

= ei|D|

q ϕ(D − m)ϕ(D − k)f p

sq

q

k + ϕ D − (k + ) ϕ(D − k)f Lp sq

| |N

1/q

α ks ei|D| M p,q →Lp f Lp s

for f ∈ S(Rn ) and q = ∞. We have easily the same conclusion for q = ∞. Hence, we obtain the desired result. 2 Remark 5.8. We remark that since

i|D|α

e

p,q˜

Ms

→M p,q˜

α

≈ sup k−s ϕ(D − k)ei|D| Lp →Lp k∈Zn

for 1 p, q˜ ∞, s ∈ R (see [16, Lemma 2.2]), we have

i|D|α

e

p,q˜ Ms →M p,q˜

α

ei|D| M p,q →Lp s

for all 1 p, q, q˜ ∞. Now, we prove Theorem 5.6. Proof of Theorem 5.6. Since the latter is just the dual statement of the former, we prove only α p,q the former. Suppose that ei|D| is bounded from Ms (Rn ) to Lp (Rn ):

i|D|α

e f

Lp

f Msp,q ,

f ∈ S Rn .

(7)

(i) Let q min(p, p ). By the necessary condition of Theorem C and Remark 5.8, we have s (α − 2)n|1/p − 1/2|. Since ν2 (p, q) = 0, we obtain the desired result. (ii) Let 1 p 2 and p q p . Note that inequality (7) can be written as

i|D|α

e

D−s f

Lp

f M p,q ,

f ∈ S Rn

(8)

by the lifting property. Here, we denote D−s f = ( ·−s fˆ(·))∨ for s ∈ R. Let g ∈ S(Rn ) be such that

supp gˆ ⊂ ξ 2−1 < |ξ | < 2 and g(ξ ˆ ) = 1 on ξ 2−1/2 < |ξ | < 21/2 , and test (8) with a specific f = Uλ g, λ 1. Since α α ei|D| D−s Uλ g = Uλ ei|λD| λD−s g , it follows from Theorem 2.2 that

α λ−n/p ei|λD| λD−s g Lp λnμ1 (p,q) g M p,q .

(9)


3205

On the other hand, by the change of variable x → λα x and the method of stationary phase, we obtain

i|λD|α

e

λD−s g

Lp

ix·ξ +i|λξ |α −s

= e

λξ g(ξ ˆ ) dξ

Lp

Rn

iλα (x·ξ +|ξ |α ) −s

= λαn/p

e

λξ g(ξ ˆ ) dξ

Lp

Rn

λαn/p−αn/2−s . Combining these two estimates, we get λnμ1 (p,q)+n/p−αn/p+αn/2+s 1 for all λ 1. Letting λ → ∞ yields the necessary condition s −n(1/q − 1) − n/p + αn/p − αn/2 = (α − 2)n(1/p − 1/2) + n/p − n/q = (α − 2)n|1/p − 1/2| − nν2 (p, q), since μ1 (p, q) = 1/q − 1 and ν2 (p, q) = −1/p + 1/q. (iii) Let 2 p ∞ and p q p. By duality, we have

−i|D|α

e f

Lp

f

p ,q

M−s

,

f ∈ S Rn .

So, we have only to prove the following lemma. p

Lemma 5.9. Let 1 p 2, p q p and s ∈ R. If ei|D| is bounded from Ls (Rn ) to M p ,q (Rn ), then s (α − 2)n|1/p − 1/2| − nν2 (p, q). α

Proof of Lemma 5.9. Set f = Uλ g, λ 1, where g is a function satisfying (9). Then, by Lemma 2.2, we have

i|D|α

e f

Mp

,q

α = ei|D| Uλ g M p ,q

α

= Uλ ei|λD| g M p ,q

α

λnμ2 (p ,q ) ei|λD| g

Mp

,q

.

In the same way as (ii), we obtain, by the change of variable x → λα x and the method of stationary phase,

3206


i|λD|α

e g

Mp

,q

1/q

i|λD|α q

ϕ(D − k)e = g Lp

k∈Zn

ix·ξ +i|λξ |α

ϕ(ξ )e g(ξ ˆ ) dξ

Lp

Rn

=λ

iλα (x·ξ +|ξ |α ) ϕ(ξ )g(ξ ˆ ) dξ

e

αn/p

Lp

Rn

λαn/p −αn/2 . Hence, we have

i|D|α

e f

Mp

,q

λnμ2 (p ,q ) λαn(1/p −1/2) .

On the other hand, we have f

p

Ls

= Uλ g

p

Ls

≈ λ−n/p λs .

Combining these two estimates, we obtain

λs−n/p −nμ2 (p ,q )−αn(1/p −1/2) 1 for all λ 1. Letting λ → ∞ yields the necessary condition s αn 1/p − 1/2 + n/p + n −2/p + 1/q = (α − 2)n 1/p − 1/2 + 2n 1/p − 1/2 + n/p + n −2/p + 1/q = (α − 2)n 1/p − 1/2 + n 1/p + 1/q − 1 = (α − 2)n(1/2 − 1/p) + n(1 − 1/p − 1/q) = (α − 2)n|1/p − 1/2| − nν2 (p, q), since μ2 (p , q ) = −2/p + 1/q and ν2 (p, q) = 1/p + 1/q − 1.

2

(iv) Let 2 p ∞ and p < q. Contrary to our claim, suppose that there exists ε > 0 such that s = (α − 2)n|1/p − 1/2| − nν2 (p, q) − ε implies (7). Then, by interpolation with the estimate for a point Q(1/p1 , 1/q1 ) with 2 < p1 < ∞, p1 < q1 < p1 and s = (α − 2)n|1/p1 − 1/2| − nν2 (p1 , q1 ) (which holds by Corollary 5.4), one would obtain an improved estimates of the


3207

segment joining P (1/p, 1/q) and Q(1/p1 , 1/q1 ), which is not possible. In the same way as above, we can treat the case 1 p 2 and p < q, and we have the conclusion. 1/q 1

1/2

Q P

0

1/2

1

1/p

2

Acknowledgments The authors are grateful to the referees for several valuable comments and suggestions. In particular, the original proof of Theorem 1.4 (“only if” part) for p = ∞, which contained an error, was corrected due to one of them. References [1] Á. Bényi, K. Gröchenig, C. Heil, K. Okoudjou, Modulation spaces and a class of bounded multilinear pseudodifferential operators, J. Operator Theory 54 (2005) 387–399. [2] Á. Bényi, K. Gröchenig, K. Okoudjou, L.G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal. 246 (2007) 366–384. [3] Á. Bényi, K. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc. 41 (2009) 549–558. [4] E. Cordero, K. Gröchenig, Time-frequency analysis of localization operators, J. Funct. Anal. 205 (2003) 107–131. [5] E. Cordero, F. Nicola, Strichartz estimates in Wiener amalgam spaces for the Schrödinger equation, Math. Nachr. 281 (2008) 25–41. [6] H.G. Feichtinger, Modulation spaces on locally compact abelian groups, in: M. Krishna, R. Radha, S. Thangavelu (Eds.), Wavelets and Their Applications, Chennai, India, Allied Publishers, New Delhi, 2003, pp. 99–140, updated version of a technical report, University of Vienna, 1983. [7] D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979) 27–42. [8] K. Gröchenig, Foundations of Time–Frequency Analysis, Birkhäuser, 2001. [9] K. Gröchenig, Time–frequency analysis of Sjöstrand’s class, Rev. Mat. Iberoam. 22 (2006) 703–724. [10] K. Gröchenig, C. Heil, Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory 34 (1999) 439–457. [11] C. Heil, J. Ramanathan, P. Topiwala, Singular values of compact pseudodifferential operators, J. Funct. Anal. 150 (1997) 426–452. [12] A. Holst, J. Toft, P. Wahlberg, Weyl product algebras and modulation spaces, J. Funct. Anal. 251 (2007) 463–491. [13] M. Izuki, Y. Sawano, Greedy bases in weighted modulation spaces, Nonlinear Anal. 71 (2009) e2045–e2053. [14] M. Kobayashi, A. Miyachi, N. Tomita, Embedding relations between local Hardy and modulation spaces, Studia Math. 192 (2009) 79–96. [15] A. Miyachi, On some Fourier multipliers for H p (R n ), J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980) 157–179. [16] A. Miyachi, F. Nicola, S. Rivetti, A. Tabacco, N. Tomita, Estimates for unimodular Fourier multipliers on modulation spaces, Proc. Amer. Math. Soc. 137 (2009) 3869–3883. s with 0 < p, q ∞, s ∈ R, Proc. A. Razmadze [17] Y. Sawano, Atomic decomposition for the modulation space Mp,q Math. Inst. 145 (2007) 63–68. [18] J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett. 1 (1994) 185–192.

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[19] J. Sjöstrand, Wiener Type Algebras of Pseudodifferential Operators, Sémin. Équ. Dériv. Partielles, vol. IV, École Polytech., Palaiseau, 1994/1995, 21 pp. [20] C.D. Sogge, Fourier Integrals in Classical Analysis, Cambridge University Press, 1993. [21] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. [22] M. Sugimoto, N. Tomita, The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal. 248 (2007) 79–106. [23] M. Sugimoto, N. Tomita, A remark on fractional integrals on modulation spaces, Math. Nachr. 281 (2008) 1372– 1379. [24] K. Tachizawa, The boundedness of pseudodifferential operators on modulation spaces, Math. Nachr. 168 (1994) 263–277. [25] J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus, I, J. Funct. Anal. 207 (2004) 399–429. [26] J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II, Ann. Global Anal. Geom. 26 (2004) 73–106. [27] J. Toft, Convolutions and embeddings for weighted modulation spaces, in: Advances in Pseudo-Differential Operators, in: Oper. Theory Adv. Appl., vol. 155, Birkhäuser, 2004, pp. 165–186. [28] J. Toft, Multiplication properties in pseudo-differential calculus with small regularity on the symbols, J. PseudoDiffer. Oper. Appl. 1 (2010) 101–138. [29] H. Triebel, Theory of Function Spaces. II, Birkhäuser, 1992. [30] B. Wang, C. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations 239 (2007) 213–250. [31] B. Wang, H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations 232 (2007) 36–73. λ and applications to nonlinear [32] B. Wang, L. Zhao, B. Guo, Isometric decomposition operators, function spaces Ep,q evolution equations, J. Funct. Anal. 233 (2006) 1–39.


Strong solidity of group factors from lattices in SO(n, 1) and SU(n, 1) Thomas Sinclair Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, United States Received 17 September 2010; accepted 17 December 2010 Available online 28 December 2010 Communicated by S. Vaes

Abstract We show that the group factors LΓ , where Γ is an ICC lattice in either SO(n, 1) or SU(n, 1), n 2, are strongly solid in the sense of Ozawa and Popa (2010) [13]. This strengthens a result of Ozawa and Popa (2010) [14] showing that these factors do not have Cartan subalgebras. © 2010 Elsevier Inc. All rights reserved. Keywords: Strong solidity; Lattices

0. Introduction In their breakthrough paper [13], Ozawa and Popa brought new techniques to bear on the study of free group factors which allowed them to show that these factors possess a powerful structural property, what they called “strong solidity.” Definition 0.1. (See Ozawa and Popa [13].) A II1 factor M is strongly solid if for any diffuse amenable subalgebra P ⊂ M we have that NM (P ) is amenable. As usual, NM (P ) = {u ∈ U(M): uP u∗ = P } denotes the normalizer of P in M. It can be seen that every nonamenable II1 subfactor of a strongly solid II1 factor is non-Gamma, prime and has no Cartan subalgebras. Thus, Ozawa and Popa’s result broadened and offered a unified approach to the two main results on the structure of free group factors hitherto known: Voiculescu’s [29] E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.017

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T. Sinclair / Journal of Functional Analysis 260 (2011) 3209–3221

pioneering result, which showed that the free group factors LFn , 2 n ∞, have no Cartan subalgebras, and Ozawa’s [12] seminal work on “solid” von Neumann algebras, which showed that every nonamenable II1 subfactor of a free group factor is non-Gamma and prime. Moreover, they exhibited the first, and so far only, examples of II1 factors with a unique Cartan up to unitary conjugacy; namely, the group-measure space constructions of free ergodic profinite actions of groups with property (HH)+ [14]; e.g., nonamenable free groups. This improved on the groundbreaking work of Popa [20], which gave examples of II1 factors with a unique “HT-Cartan” subalgebra up to unitary equivalence; e.g., L(Z2 SL2 (Z)). By incorporating ideas and techniques of Peterson [17], Ozawa and Popa [14] were later able to extend the class of strongly solid factors to, in particular, all group factors of ICC lattices in PSL(2, R) or PSL(2, C). Other examples of strongly solid factors were subsequently constructed by Houdayer [9] and by Houdayer and Shlyakhtenko [10]. By a lattice we mean a discrete subgroup Γ < G of some Lie group with finitely many connected components such that G/Γ admits a regular Borel probability measure invariant under left translation by G. The main goal of this paper will be to demonstrate the following result: Theorem 0.2. If Γ is an ICC lattice in SO(n, 1) or SU(n, 1), then LΓ is strongly solid. These factors are already known by the work of Ozawa and Popa [14] to have no Cartan subalgebras. Since SO(n, 1) and SU(n, 1) are simple Lie groups with finite center, Borel’s density theorem via Theorem 6.5 in [5] shows that every γ ∈ Γ which is not in the center of G has infinite Γ -conjugacy class, so examples of ICC lattices abound. In the SO(n, 1) case, the restriction of the lattice subgroup Γ to the connected component of the identity SO(n, 1)0 is always ICC, SO(n, 1)0 having trivial center, and all results in our paper will hold for these groups as well. In particular, we have that PSL(2, R) ∼ = SO(2, 1)0 ∼ = SU(1, 1) and PSL(2, C) ∼ = SO(3, 1)0 , so Theorem 0.2 recovers the main result in Ozawa–Popa [14]. Finally, notice that if G is a Lie group with finite center and finitely many connected components which is locally isomorphic to SO(n, 1), then it is a finite-to-one covering of—hence, a finite extension of—SO(n, 1)0 . Cohomological induction, combined with the techniques below, will then be sufficient to show that the group von Neumann algebra of any ICC lattice in such a Lie group is also strongly solid. The proof follows the same strategy as Ozawa and Popa’s in [13,14]. Though, instead of working with closable derivations, we use a natural one-parameter family of deformations first constructed by Parthasarathy and Schmidt [15]. The derivations Ozawa and Popa consider appear as the infinitesimal generators of these deformations (so, the approaches are largely equivalent), but by using the deformations we avoid some of the technical issues which arise when working with derivations. The main difficulty in obtaining Theorem 0.2 for lattice factors in SO(n, 1) or SU(m, 1) when n 4 or m 2 is that the bimodules which admit good deformations/derivations are themselves too weak to allow one to deduce the amenability of the normalizer algebra e.g., strong solidity. However, sufficiently large tensor powers of these bimodules can be used to deduce strong solidity. Unfortunately, derivation techniques perturb the original bimodules slightly, and the behavior of tensor powers of the perturbed bimodules becomes unclear. To circumvent this problem, we first notice that Ozawa and Popa’s techniques actually allow one to deduce a kind of relative amenability of the normalizer subalgebra with respect to the bimodule, given in terms of an “invariant mean”. We then use a result of Sauvageot [25] to obtain from the invariant mean an almost invariant sequence of vectors in the bimodule. Since the property of having an almost invariant sequence of vectors is stable under taking tensor powers, we are able to transfer rela-


3211

tive amenability to a large tensor power of the bimodule in order to deduce amenability of the normalizer algebra. We remark that as a corollary to the techniques used in the proof of Theorem 0.2, we are able to strengthen a result of Houdayer [9] on free product group factors admitting no Cartan subalgebras. Theorem 0.3. Let Γ be a nonamenable, countable, discrete group which has the complete metric approximation property (Definition 1.6). If Γ ∼ = Γ1 ∗Γ2 decomposes as a non-trivial free product, then LΓ has no Cartan subalgebras. Moreover, if N ⊂ LΓ is a nonamenable subfactor which has a Cartan subalgebra, then there exists projections p1 , p2 in the center of N ∩ LΓ such that p1 + p2 = 1 and unitaries u1 , u2 ∈ U(M) such that ui Npi u∗i ⊂ LΓi ⊂ LΓ , i ∈ {1, 2}. 1. Preliminaries We collect in this section the necessary definitions, concepts and results needed for the proofs of Theorems 0.2 and 0.3. 1.1. Representations, correspondences and weak containment Let Γ be a countable discrete group and π, ρ be unitary representations of Γ into separable Hilbert spaces Hπ and Hρ , respectively. Definition 1.1. We say that ρ is weakly contained in π if for any ε > 0, ξ ∈ Hρ and any finite subset F ⊂ Γ , there exist vectors ξ1 , . . . , ξn ∈ Hπ such that | ρ(γ )ξ, ξ − ni=1 π(γ )ξi , ξi | < ε for all γ ∈ F . A representation π is said to be tempered if it is weakly contained in the left-regular representation, and strongly p [27] if for any ε > 0, there exists a dense subspace H0 ⊂ H such that for all ξ, η ∈ H0 the matrix coefficient π(γ )ξ, η belongs to p+ε (Γ ). By a theorem of Cowling, Haagerup and Howe [6], a representation which is strongly 2 is tempered. As was pointed out in [27], applying standard Hölder estimates to the matrix coefficients, we obtain that if π is strongly p for some p 2, then for all n > p/2, π ⊗n is strongly 2 , hence tempered. In the theory of von Neumann algebras, correspondences (also called Hilbert bimodules) play an analogous role to unitary representations in the theory of countable discrete groups. For von Neumann algebras N and M, recall that an N –M correspondence is a ∗-representation π of the algebraic tensor N M o into the bounded operators on a Hilbert space H which is normal when restricted to both N and M o . We will denote the restrictions of π to N and M o by πN and πM o , respectively. When the N –M correspondence π is implicit for the Hilbert space H, we will use the notation xξy to denote π(x ⊗ y o )ξ , for x ∈ N , y ∈ M and ξ ∈ H. Definition 1.2. Let π : N M o → B(Hπ ), ρ : N M o → B(Hρ ) be correspondences. We say any finite subsets F1 ⊂ N , F2 ⊂ M, that ρ is weakly contained in π if for any ε > 0, ξ ∈ Hρ and there exist vectors ξ1 , . . . , ξn ∈ Hπ such that | xξy, ξ − ni=1 xξi y, ξi | < ε for all x ∈ F1 , y ∈ F2 . There is a well-known functor from the category whose objects are (separable) unitary representations of Γ and morphisms weak containment to the one of LΓ –LΓ correspondences and

3212


weak containment, cf. [22], which translates the representation theory of Γ into the theory of LΓ –LΓ correspondences. The construction is as follows. Given π : Γ → U(Hπ ) a unitary representation, let Hπ be the Hilbert space Hπ ⊗ 2 Γ . Then, the maps uγ (ξ ⊗ η) = π(γ )ξ ⊗ uγ η, (ξ ⊗ η)uγ = ξ ⊗ (ηuγ ) extend to commuting normal representations of LΓ and (LΓ )o on Hπ : the former by Fell’s absorption principle, the latter trivially. This functor is well behaved with respect to tensor products; i.e., Hπ⊗ρ ∼ = Hπ ⊗LΓ Hρ as LΓ –LΓ correspondences for any unitary Γ -representations π and ρ. We refer the reader to [1,19] for the theory of tensor products of correspondences and the basic theory of correspondences in general. For a II1 factor M there are two canonical correspondences: the trivial correspondence, L2 (M) with M acting by left and right multiplication, and the coarse correspondence, L2 (M) ⊗ ¯ with M acting by left multiplication of the left copy of L2 (M) and right multiplicaL2 (M) tion on the right copy. When M = LΓ for some countable discrete group, the trivial and coarse correspondences are the correspondences induced respectively by the trivial and left regular representations of Γ . 1.2. Cocycles and the Gaussian construction In this section, H will denote a real Hilbert space which we will fix along with an orthogonal representation π : Γ → O(H) of some countable discrete group Γ . Definition 1.3. A cocycle is a map b : Γ → H satisfying the cocycle relation b γ γ = b(γ ) + π(γ )b γ ,

for all γ , γ ∈ Γ.

Given π and H, there is a canonical standard probability space (X, μ) and a canonical measure-preserving action Γ σ (X, μ) such that there is a Hilbert space embedding of H into L2R (X, μ) intertwining π and the natural representation induced on L2R (X, μ) by σ . This is know as the Gaussian construction, cf. [18] or [26]. It is well known that the natural Γ representation σ0 on L20 (X, μ) = L2 (X, μ) C1X inherits all “stable” properties from π , cf. [18]. In particular, σ0⊗n is tempered if and only if π ⊗n is tempered for any n 1. It was discovered by Parthasarathy and Schmidt [15] that cocycles also fit well into the framework of the Gaussian construction, inducing one-parameter families of deformations (i.e., cocycles) of the action σ . To be precise: Theorem 1.4. (See Parthasarathy and Schmidt [15].) Let b : Γ → H be a cocycle, then there exists a one-parameter family ωt : Γ → U(L∞ (X, μ)), t ∈ R such that: ωt γ γ = ωt (γ )σγ ωt γ ,

for all γ , γ ∈ Γ,

(1.1)

and

2 ωt (γ ) dμ = exp − t b(γ ) ,

for all γ ∈ Γ.

(1.2)


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1.3. Weak compactness and the CMAP Definition 1.5. (See Ozawa and Popa [13].) Let (P , τ ) be a finite von Neumann algebra equipped with a trace τ , and G σ P be an action of a group G on A by τ -preserving ∗automorphisms. We say that the action σ is weakly compact if there exists a net of unit vectors ¯ P¯ , τ ⊗ τ¯ )+ such that: (ηk ) ∈ L2 (P ⊗ (1) ηk − (v ⊗ v)η ¯ k → 0, for all v ∈ U(P ); (2) ηk − (σg ⊗ σ¯ g )ηk → 0, for all g ∈ G; and (3) (x ⊗ 1)ηk , ηk = τ (x) = (1 ⊗ x)η ¯ k , ηk , for all x ∈ P . Definition 1.6. A II1 factor M is said to have the complete metric approximation property (CMAP) if there exists a net (ϕi ) of finite-rank, normal, completely bounded maps ϕi : M → M such that lim sup ϕi cb 1 and such that ϕi (x) − x2 → 0, for all x ∈ M. If Γ is an ICC countable discrete group, then LΓ has the CMAP if and only if the Cowling– Haagerup constant of Γ , Λcb (Γ ), equals 1, and if Γ is a lattice in G, then Λcb (Γ ) = Λcb (G), cf. Section 12.3 of [3] and [8]. Theorem 1.7. (See Ozawa and Popa, Theorem 3.5 of [13].) Let M be a II1 factor which has the CMAP. Then for any diffuse amenable ∗-subalgebra A ⊂ M, NM (A) acts weakly compactly on A by conjugation. 2. Amenable correspondences Definition 2.1. (See Anantharaman-Delaroche [1].) An N –M correspondence H is called (left) amenable if H ⊗M H¯ weakly contains the trivial N –N correspondence. The concept of amenability for correspondences is the von Neumann algebraic analog of the concept of amenability of a unitary representation of a locally compact group due to Bekka [2]. As was observed by Bekka, amenability of the representation π is equivalent to the existence of a state Φ on B(H) satisfying Φ(π(g)T ) = Φ(T π(g)) for all g ∈ G, T ∈ B(H). One can ask if a similar criterion holds for amenable correspondences. When M is a II1 factor, we will show that this indeed is the case if we replace B(H) with the von Neumann algebra N = B(H)∩πM o (M o ) . That is, we obtain the following characterization of amenable correspondences: Theorem 2.2. (Compare with Theorem 2.1 in [13].) Let H be an N –M correspondence with N finite with normal faithful trace τ and M a II1 factor. Let P ⊂ N be a von Neumann subalgebra. Then the following are equivalent: (1) there exists a net (ξn ) in H⊗M H¯ such that xξn , ξn → τ (x) for all x ∈ N and [u, ξn ] → 0 for all u ∈ U(P ); (2) there exists a P -central state Φ on N such that Φ is normal when restricted to N and faithful when restricted to Z(P ∩ N ); (3) there exists a P -central state Φ on N which restricts to τ on N .

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We do so by constructing a normal, faithful, semi-finite (tracial) weight τ¯ on N which canonically realizes H ⊗M H¯ as L2 (N , τ¯ ). An identical construction to the one we propose has already appeared in the work of Sauvageot [25] for an arbitrary factor M. However, we present an elementary approach in the II1 case. Before presenting the details, we pause here to illustrate how Theorem 2.2 generalizes (relative) amenability for II1 factors. Let M be a II1 factor and H = L2 M ⊗ L2 M¯ be the coarse ¯ M–M correspondence. We then have explicitly that M = B(H) ∩ (M o ) = B(L2 M)⊗M; hence, M has an M-central state which restricts to the trace on M if and only if the trace on M extends to a hypertrace on B(L2 M), i.e., M is amenable. Similarly, if P , Q ⊂ M are von Neumann subalgebras and H = L2 M, eQ , where M, eQ denotes the basic construction of Jones, then M, eQ ⊂ B(H) ∩ (M o ) . So, if P ⊂ M satisfies one of the conditions in the above theorem, then P M Q in the sense of Theorem 2.1 in [13]. Conversely, if P M Q, then using condition (4) in Theorem 2.1 of [13], one can construct such a functional Φ on B(H) ∩ (M o ) as in condition (3) in the above theorem for P ⊂ M. Recall that a vector ξ ∈ H is right bounded if there exists C > 0 such that for all x ∈ M, ξ x Cx2 . The right-bounded vectors form a dense subspace of H which we will denote by Hb . Regarding H as a right Hilbert M-module, we can define a natural M-valued inner product on Hb , which we will denote (ξ |η) ∈ M for ξ, η ∈ Hb , by setting (ξ |η) to be the Radon– Nikodym derivative of the normal functional x → ξ x, η . Then it is easy to see that (·|·) satisfies the following properties for all ξ, η ∈ Hb , x, y ∈ M: (1) (ξ |ξ ) 0, (2) (η|ξ ) = (ξ |η)∗ , (3) (ξ x|ηy) = y ∗ (ξ |η)x (i.e., (Hb , (·|·)) is an M-rigged space in the sense of Rieffel [24]). Trivially, we have that ξ, η = τ ((ξ |η)). Also, by the non-degeneracy of πM o , (ξ |ξ ) = 0 only if ξ = 0. Moreover, (·|·) satisfies a noncommutative Cauchy–Schwartz inequality: (4) (η|ξ )(ξ |η) (ξ |ξ )∞ (η|η) (cf. [23], Proposition 2.9). Let N = B(H) ∩ πM o (M o ) . (For instance, if M˜ ⊃ M is a tracial inclusion of II1 factors and ˜ considered as a Hilbert M–M bimodule in the natural way, then we have that (x| H = L2 M, ˆ y) ˆ = ∗ ˜ eM .) For ξ, η ∈ Hb , let Tξ,η : Hb → Hb EM (y x) for all x, y ∈ M˜ and B(H) ∩ (M o ) = M, be the “rank-one operator” given by Tξ,η (·) = ξ(·|η). Then Tξ,η extends to a bounded operator with Tξ,η 2∞ (ξ |ξ )∞ (η|η)∞ [23]. Notice that Tξ,ξ 0 and that Tξ,ξ is a projection if (ξ |ξ ) ∈ P(M). Since Tξ,η πM o (x) = πM o (x)Tξ,η for all x ∈ M o , we have that Tξ,η ∈ N . It is easy to see that the span of all such operators Tξ,η is a ∗-subalgebra of B(H) which we will denote by Nf . Noticing that for any S ∈ N , S(Hb ) ⊂ Hb , we have that STξ,η = TSξ,η and Tξ,η S = Tξ,S ∗ η . It follows that Nf is an ideal of N which can be considered as the analog of the finite-rank operators in B(H). The following lemma further cements this analogy. Lemma 2.3. If M is a II1 factor, we have that Nf ∩ B(H) = πM o (M o ); hence, Nf + C1B(H) is weakly dense in N .


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Proof. The inclusion πM o (M o ) ⊂ Nf ∩ B(H) is trivial. Conversely, let T ∈ Nf ∩ B(H) and choose a non-zero ζ ∈ Hb such that (ζ |ζ ) = p ∈ P(M). (One can always find such a ζ as H has an orthonormal basis of right bounded vectors as a right Hilbert M-module.) Choose a sequence ηi ∈ Hb such that ηi − T ζ → 0 and let yi = (ηi |ζ ). Then for every ξ ∈ Hb we have 1/2 that T (ξp) − ξyi Tξ,ζ ∞ ηi − T ζ (ξ |ξ )∞ ηi − T ζ so, the sequence (πM o (yio )) o converges in the strong topology to T ◦ πM o (p ). Hence, T ◦ πM o (p o ) ∈ πM o (M o ) = πM o (M o ). Since M is a II1 factor, by repeating the argument with ζ = ζ u for u ∈ U(M) and using standard averaging techniques, we conclude that there exists yT ∈ M such that T ξ = ξyT for all ξ ∈ H. Thus T = πM o (yTo ). 2 Now, consider an element ϕ ∈ M∗ , and define a functional ϕ¯ ∈ (Nf )∗ by ϕ(T ¯ ξ,η ) = ϕ((ξ |η)). It is easy to see that ϕ¯ is normal on Nf and so, by the preceding lemma, may be extended to a normal semi-finite weight on N . Hence, we may construct for each such ϕ a noncommutative Lp -space over N , Lp (N , ϕ) ¯ = {T ∈ N : T p = ϕ(|T ¯ |p )1/p < ∞}. If M is a II1 factor with trace τ , then τ¯ is a normal, faithful, semi-finite trace on N and we denote Lp (N , τ¯ ) simply by Lp (N ). In the case of L2 (N ), we compute that Tξ,η 22 = τ ((ξ |ξ )(η|η)) = ξ(η|η), ξ . This shows that the map which sends Tξ,η to the elementary M-tensor ξ ⊗M η¯ ∈ H ⊗M H¯ extends to ¯ We are now ready to prove an N –N bimodular Hilbert space isometry from L2 (N ) to H ⊗M H. the motivating result in this section. Proof of Theorem 2.2. The proof of (1) ⇔ (3) follows the usual strategy. For (1) ⇒ (3), we have that there exists a net (ξn ) of vectors in H ⊗M H¯ such that xξn , ξn → τ (x) for all x ∈ N and [u, ξn ] → 0 for all u ∈ U(P ). Viewing ξn as an element of L2 (N ), let Φn ∈ N∗ be given by Φn (T ) = τ¯ (ξn ξn∗ T ) for any T ∈ N . Then, by the generalized Powers–Størmer inequality (Theorem IX.1.2 in [28]), we have that |Φn (x) − τ (x)| → 0 for all x ∈ N and Ad(u)Φn − Φn 1 → 0 for all u ∈ U(P ). Taking a weak cluster point of (Φn ) in N ∗ gives the required N tracial P -central state on N . Conversely, given such a state Φ, we can find a net (ηn ) in L1 (N )+ such that Φn (T ) = τ¯ (ηn T ) weakly converges to Φ. In fact, by passing to convex combinations we may assume [u, ηn ]1 → 0 for all u ∈ U(P ). By another application of the generalized 1/2 Powers–Størmer inequality, it is easy to check that ξn = ηn ∈ L2 (N ) ∼ = H ⊗M H¯ satisfies the requirements of (1). We now need only show (2) ⇒ (3) as (3) ⇒ (2) is trivial. But this is exactly the averaging trick found in the proof of (2) ⇒ (1) in Theorem 2.1 of [13]. We repeat the argument here for the sake of completeness. Since Φ is normal on N , we have that for some η ∈ L1 (N )+ , Φ(x) = τ (ηx) for all x ∈ N . In fact, η ∈ L1 (P ∩ N )+ since Φ is P -central. Denoting by F the net of finite subsets of U(P ∩ N ) under inclusion, for any ε > 0, we set ηF = |F1 | u∈F uηu∗ −1/2 and ξF,ε = (χ[ε,∞) (ηF ))ηF . We now let ΨF,ε (T ) = |F1 | u∈F Φ(uξF,ε T ξF,ε u∗ ). Note that ΨF,ε is still P -central. Now it is easy to see that limF ,ε χ[ε,∞) (ηF ) = z, where z is the central support of η. But by the faithfulness of Φ on Z(P ∩ N ), we see that z = 1. Hence, any weak cluster point of (ΨF,ε )F ,ε in N ∗ is a P -central state which when restricted to N is τ . 2 Corollary 2.4 (Generalized Haagerup’s criterion for amenability). Let N , M be II1 factors and H an N –M correspondence. If P ⊂ N is a von Neumann subalgebra, then H is left amenable over P (in the sense of Theorem 2.2) if and only if for every non-zero projection p ∈ Z(P ∩ N ) and finite subset F ⊂ U(P ), we have

3216


up ⊗ up u∈F

H⊗M H¯ ,∞

= |F |,

¯ where · H⊗M H¯ ,∞ denotes the operator norm on B(H ⊗M H). Proof. Since we have obtained a “hypertrace” characterization of amenability for correspondences in Theorem 2.2, the result follows by the same arguments as in Lemma 2.2 in [7]. 2 Definition 2.5. (Cf. Definition 1.3 in [18].) Let M be a II1 factor, H an M–M correspondence and P c the orthogonal projection onto the subspace Hc = {ξ ∈ H: xξ = ξ x, ∀x ∈ M}. The correspondence H has spectral gap if for every ε > 0 there exist δ > 0 and x1 , . . . , xn ∈ M such that if xi ξ − ξ xi < δ, i = 1, . . . , n, then ξ − P c ξ < ε. The correspondence H has stable spectral gap if H ⊗M H¯ has spectral gap. ¯ c = {0}. Note that if H has stable spectral gap, then H is amenable if and only if (H ⊗M H) Hence, we say an M–M correspondence H is nonamenable if it has stable spectral gap and ¯ c = {0}. The following theorem is the analog of Lemma 3.2 in [21] for the category (H ⊗M H) of correspondences. N.B. Stable spectral gap as defined in [21] corresponds to our definition of nonamenability. Theorem 2.6. Let M be a II1 factor and H an M–M correspondence. Then H is nonamenable if and only if H ⊗M K¯ has spectral gap and for any M–M correspondence K. Proof. Let Hb , Kb denote subspaces of right-bounded vectors in H and K, respectively. Given ξ ∈ Hb and η ∈ Kb , by the same arguments as above we can define a bounded operator ¯ = Tξ,η : K → H by Tξ,η (·) = ξ(·|η). As above, one may check that (T ∗ T )1/2 2 = ξ ⊗M η (T T ∗ )1/2 2 so that H ⊗M K is isometric to a Hilbert-normed subspace of the bounded right M-linear operators from H to K, which we denote L2 (H, K). Moreover, this identification is natural with respect to the M–M bimodular structure on L2 (H, K) given by xTξ,η y = Txξ,y ∗ η . We need now only prove the forward implication, as the converse is trivial. Let us fix some ¯ c= arbitrary M–M correspondence K. From Proposition 1.4 in [18], we have that if (H ⊗M H) c ¯ {0}, then (H ⊗M K) = {0}. So, by way of contradiction, we may assume that for every ε > 0 and x1 , . . . , xn ∈ M, there exists a unit vector ξ ∈ H ⊗M K¯ such that xi ξ − ξ xi 2 ε, i = 1, . . . , n. Without loss of generality, we may assume x1 , . . . , xn are unitaries. Viewing ξ as an element of L2 (H, K), let η = (ξ ∗ ξ )1/2 ∈ L2 (H). By the generalized Powers–Størmer inequality, we have xi ηxi∗ − η22 2xi ξ xi∗ − ξ 2 2ε, i = 1, . . . , n. Hence, η ∈ L2 (H) ∼ = H ⊗M H¯ is a unit √ vector such that xi η − ηxi 2 2ε, i = 1, . . . , n. Thus, H ⊗M H¯ does not have spectral gap, a contradiction. 2 3. Proofs of main theorems In this section we prove our main result, from which will follow Theorems 0.2 and 0.3. To begin, let Γ be an ICC countable discrete group which admits an unbounded cocycle b : Γ → K for some orthogonal representation π : Γ → O(K). Let Γ σ (X, μ) be the Gaussian construction associated to π as described in Section 1.2 and {ωt : t ∈ R} be the one-parameter family of cocycles associated to b as given by Theorem 1.4. Let αt be the ∗-automorphism of M˜ = L∞ (X, μ) Γ defined by αt (auγ ) = aωt (γ )uγ for all a ∈ L∞ (X, μ), γ ∈ Γ . Finally,


3217

we set M = LΓ , and we denote by H the M–M bimodule L20 (X, μ) ⊗ 2 Γ with the usual bimodule structure; i.e., the one defined by uγ (ξ ⊗ η) = σ (γ )ξ ⊗ uγ η, (ξ ⊗ η)uγ = ξ ⊗ (ηuγ ) for all ξ ∈ L20 (X, μ), η ∈ 2 Γ and γ ∈ Γ . Theorem 3.1. With the assumptions and notations as above, suppose P ⊂ M is a diffuse von Neumann subalgebra such that NM (P ) acts weakly compactly on P via conjugation. Let Q = NM (P ) . If either: (1) b is a proper cocycle; or (2) π is a mixing representation and αt does not converge · 2 -uniformly to the identity on (Qp)1 for any projection p ∈ Z(Q ∩ M) as t → 0, then the M–M correspondence H is left amenable over Q in the sense of satisfying Theorem 2.2 for Q ⊂ M. Proof. In the case of (1), since b is proper, it is easy to see by formula 1.2 that ELΓ ◦ αt restricted to LΓ is compact for all t > 0. Hence, by the proof of Theorem 4.9 in [13], for any K 8, any non-zero projection p ∈ Z(Q ∩ M), and any finite subset F ⊂ NM (P ), we can find a vector ¯ such that xξp,F x2 for all x ∈ M, pξp,F p2 /K and [u ⊗ ξp,F ∈ H ⊗ L2 (M) u, ¯ ξp,F ] < 1/|F | for all u ∈ F . In the case of (2), we need only demonstrate that our assumptions imply the existence of such a net (ξp,F ) as in case (1) for some K 8 then argue commonly for both sets of assumptions. By contradiction if such a net (ξp,F ) did not exist for any K 8, the proof of Theorem 4.9 in [13] shows that for every 0 < δ = K −1 1/8 and for any t > 0 sufficiently small we have EM ◦ αt (up) (1 − 6δ)p2 2

(3.1)

for all u ∈ U(P ). Now, the operators (EM ◦ αt )t0 can be seen to form a one-parameter semigroup of unital, tracial completely-positive maps (cf. Example 2.2 in [17]) such that 0 EM ◦ αt id for all 0 t < ∞. This implies that 2 2 x22 − EM ◦ αt (x)2 x − EM ◦ αt (x)2 .

(3.2)

Hence, if αt does not converge uniformly on (Pp)1 , we have that αt cannot converge uniformly on U(P )p, so there exists c > 0 such that for every t > 0 sufficiently small, there exists ut ∈ U(P ) such that EM ◦ αt (ut p) 1 − c2 p2 . 2 However, this contradicts the inequality 3.1 for δ sufficiently small. To conclude the discussion of case (2), we have that π is mixing and, by the previous paragraph, αt converges · 2 -uniformly on (Pp)1 . We will show that αt converges uniformly on (Qp)1 , which contradicts our assumptions on αt . Since there is a natural trace˜ which pointwise fixes M and such that β ◦ αt = α−t ◦ β preserving automorphism β ∈ Aut(M) for all t ∈ R (cf. [18]), by Popa’s transversality lemma [21], it is enough to show that αt (x) − EM ◦ αt (x)2 → 0 uniformly on (Qp)1 . Notice that δt (x) = αt (x) − EM ◦ αt (x) = (1 − EM )(αt (x) − x) is a (bounded) derivation δt : M → H. Since π is mixing, by [16] we have that H is a compact correspondence; hence, by Theorem 4.5 in [17] δt → 0 uniformly in · 2 -norm on (Qp)1 as t → 0.

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Now, fixing a suitable K 8 and proceeding commonly for both cases, let F be the net of finite subsets of NM (P ) under inclusion. We define the state Φp on N = B(H) ∩ πM o (M o ) by Φp (T ) = LimF

1 pξp,F 22

(T ⊗ 1)pξp,F , pξp,F ,

where LimF is an arbitrary Banach limit. It is easy to see by the properties of ξp,F that Φp is normal on M. Proceeding as in Lemma 5.3 in [14], we then have that for all u ∈ NM (P ), Φp u∗ T u = LimF = LimF = LimF

1

(T ⊗ 1)upξp,F , upξp,F

pξp,F 22

1

(T ⊗ 1)p(u ⊗ u)ξ ¯ p,F , p(u ⊗ u)ξ ¯ p,F

pξp,F 22

1 pξp,F 22

(T ⊗ 1)p(u ⊗ u)ξ ¯ p,F (u ⊗ u) ¯ ∗ , p(u ⊗ u)ξ ¯ p,F (u ⊗ u) ¯ ∗

= Φp (T ).

(3.3)

Hence, we have that Φp ([x, T ]) = 0 for all x in the span of NM (P ) and T ∈ N . But we have that 1 Φp (T x) T ∞ LimF xpξp,F , pξp,F 2 pξp,F 2 T ∞ LimF

1 pξp,F 2

K T ∞ x2 p2

xpξp,F 2 (3.4)

and similarly for |Φp (xT )|. Thus, by Kaplansky’s density theorem we have that Φp is a Qcentral state. To summarize, for every non-zero projection p ∈ Z(Q ∩ M), we have obtained a state Φp on N such that Φp (p) = 1, Φp is normal on M and Φp is Q-central. A simple maximality argument then shows that there exists a state Φ on N which is normal on M, Q-central, and faithful on Z(Q ∩ M). Thus, Φ satisfies condition (2) of Theorem 2.2, and we are done. 2 Keeping with the same notations, assume now that the orthogonal representation b : Γ → O(K) is such that there exists a K > 0 such that π ⊗K is weakly contained in the left regular representation. As was pointed out in Section 1.2, the representation induced on L20 (X, μ) by Γ σ (X, μ) also has this property. Let Hσ = L20 (X, μ) so that H = Hσ ⊗ 2 Γ is the M– M correspondence induced by the representation σ . Denote by H˜ n the M–M correspondence ((Hσ ⊗ H¯ σ )⊗n ) ⊗ 2 Γ with the natural bimodule structure. It is straightforward to check that ¯ ⊗M · · · ⊗M (H ⊗M H) ¯ for n + 1 copies is H ⊗M H¯ ∼ = Hσ ⊗ 2 Γ ⊗ H¯ σ and that (H ⊗M H) n ¯ ˜ isomorphic to H ⊗M (H ) ⊗M H as M–M bimodules. Hence, the M-tensor product of K copies of H ⊗M H¯ is weakly contained in the coarse M–M correspondence.


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Theorem 3.2. With the assumptions and notations as above, including those assumed for Theorem 3.1, suppose P ⊂ M is a diffuse von Neumann subalgebra such that NM (P ) acts weakly compactly on P via conjugation. Then Q = NM (P ) is amenable. Proof. Let p be a non-zero projection in Z(Q ∩ M). By the proofs of Theorems 3.1 and 2.2, it follows that we can find a net (ξn ) in H ⊗M H¯ such that xξn , ξn → τ (pxp)/τ (p) for all x ∈ M and [u, ξn ] → 0 for all u ∈ U(Q). In fact, without loss of generality we may assume that xξn , ξn = τ (pxp)/τ (p) = ξn x, ξn for all x ∈ M (see the proof of Proposition 3.2 in [13]). In particular, (ξn ) is uniformly left and right bounded. Let ξñ be the M-tensor product of K copies of ξn . Then the net (ξñ ) may be seen to satisfy the same properties. Since (ξñ ) are vectors in a correspondence weakly contained in the coarse M–M correspondence, we have that for any finite subset F ⊂ U(Q) that up ⊗ up u∈F

¯ M¯ M⊗

∗ ˜ lim uξn u = |F |. n u∈F

Hence, by Haagerup’s criterion [7], Q is amenable.

2

Remark 3.3. Let M be a II1 factor and δ a closable real derivation from M into an M–M correspondence H (cf. [17]). Suppose P ⊂ M is a von Neumann subalgebra and NM (P ) acts weakly compactly on P by conjugation. Let Q = NM (P ) . One can show that if δ ∗ δ¯ has compact resolvents, then H is left amenable over Q. The proof is identical to the proof of Theorem B in [14], using the generalized Haagerup’s criterion (Corollary 2.4). In particular, this sharpens Theorem A in [14]: any II1 factor M with the CMAP admitting such a derivation into a nonamenable correspondence has no Cartan subalgebras. We are now ready to prove Theorems 0.2 and 0.3. Proof of Theorem 0.2. We need only check the cases SO(n, 1), n 4, and SU(m, 1), m 2, as SO(1, 1) is amenable and the remaining cases were dealt with by Ozawa and Popa [14]. If Γ is an ICC lattice in SO(n, 1) for n 3 or SU(m, 1) for m 2, then Theorems 1.9 in [27] shows that Γ possesses an unbounded cocycle into some strongly p representation for p 2. By Theorem 3.4 in the same, any unbounded cocycle for such a lattice is proper. Since LΓ has the CMAP by [4] and [8], by Theorems 1.7 and 3.2 the result obtains. 2 Proof of Theorem 0.3. For first assertion, since Γ ∼ = Γ1 ∗ Γ2 , Γ admits a canonical un the ∞ 2 bounded cocycle b : Γ → Γ into a direct sum of left regular representations. The left regular representation is mixing and LΓ –LΓ correspondence associated to the left regular representation (the coarse correspondence) is amenable if and only if Γ is amenable. So, if LΓ did admit a Cartan subalgebra, then by Theorems 1.7 and 3.1 the deformation αt of LΓ obtained from the cocycle b would have to converge uniformly on (LΓ )1 as t → 0. But this contradicts that b is unbounded. For the second assertion, if a nonamenable II1 subfactor N ⊂ LΓ admits a Cartan subalgebra, then we have that αt converges · 2 -uniformly on the unit ball of N , since N has the CMAP and the coarse LΓ –LΓ correspondence viewed an N –N correspondence embeds into a direct sum of coarse N –N correspondences. Let Γ˜ = Γ ∗ F2 , where F2 is the free group on

3220


two generators. Let u1 , u2 ∈ LF2 be the canonical generating unitaries and h1 , h2 ∈ LF2 selfadjoint elements such that uj = exp(πihj ), j = 1, 2. Define utj = exp(πithj ), j = 1, 2, and let θt be the ∗-automorphism of LΓ˜ given by θt = Ad(ut1 ) ∗ Ad(ut2 ). It follows from Lemma 5.1 in [17] and Corollary 4.2 in [18] that θt converges uniformly in · 2 -norm on the unit ball of N ⊂ LΓ ⊂ LΓ˜ as t → 0. An examination of the proof of Theorem 4.3 in [11] shows that this is the only condition necessary for the theorem to obtain. Our result then follows directly from Theorem 5.1 in [11]. 2 Acknowledgments The author is grateful to Jesse Peterson for suggesting the problem and making several helpful comments and suggestions. The author also thanks Jesse Peterson and Ionut Chifan for many stimulating discussions on the work of Ozawa and Popa and the referee for catching several errors in the original manuscript as well as for suggesting several improvements in the exposition. References [1] C. Anantharaman-Delaroche, Amenable correspondences and approximation properties for von Neumann algebras, Pacific J. Math. 171 (2) (1995) 309–341. [2] B. Bekka, Amenable representations of locally compact groups, Invent. Math. 100 (2) (1990) 383–401. [3] N. Brown, N. Ozawa, C ∗ -Algebras and Finite-Dimensional Approximations, Grad. Stud. Math., vol. 88, Amer. Math. Soc., 2008. [4] J. de Cannière, U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985) 455–500. [5] M. Cowling, U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (3) (1989) 507–549. [6] M. Cowling, U. Haagerup, R. Howe, Almost L2 matrix coefficients, J. Reine Angew. Math. 387 (1988) 97–110. [7] U. Haagerup, Injectivity and decomposition of completely bounded maps, in: Operator Algebras and Their Connections with Topology and Ergodic Theory, Bu¸steni, 1983, in: Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 170–222. [8] U. Haagerup, Group C∗ -algebras without the completely bounded approximation property, preprint, 1986. [9] C. Houdayer, Strongly solid group factors which are not interpolated free group factors, Math. Ann. 346 (2010) 969–989. [10] C. Houdayer, D. Shlyakhtenko, Strongly solid II1 factors with an exotic masa, preprint, arXiv:0904.1225, 2009. [11] A. Ioana, J. Peterson, S. Popa, Amalgamated free products of weakly rigid factors and calculation of their symmetry groups, Acta Math. 200 (2008) 85–153. [12] N. Ozawa, Solid von Neumann algebras, Acta Math. 192 (2004) 111–117. [13] N. Ozawa, S. Popa, On a class of II1 factors with at most one Cartan subalgebra, Ann. of Math. (2) 172 (1) (2010) 713–749. [14] N. Ozawa, S. Popa, On a class of II1 factors with at most one Cartan subalgebra, II, Amer. J. Math. 132 (2010) 841–866. [15] K.R. Parthasarathy, K. Schmidt, Infinitely divisible projective representations, cocycles and the Levi–Khinchine formula on locally compact groups, preprint, 1970. [16] J. Peterson, Examples of group actions which are virtually W∗ -superrigid, preprint, arXiv:1002.1745, 2009. [17] J. Peterson, L2 -rigidity in von Neumann algebras, Invent. Math. 175 (2009) 417–433. [18] J. Peterson, T. Sinclair, On cocycle superrigidity for Gaussian actions, preprint, arXiv:0910.3958, 2009. [19] S. Popa, Correspondences, INCREST, preprint, 1986. [20] S. Popa, On a class of type II1 factors with Betti number invariants, Ann. of Math. (2) 163 (3) (2006) 809–899. [21] S. Popa, On the superrigidity of malleable actions with spectral gap, J. Amer. Math. Soc. 21 (2008) 981–1000. [22] S. Popa, S. Vaes, Group measure space decomposition of II1 factors and W ∗ -superrigidity, preprint, arXiv: 0906.2765 [math.OA], 2009. [23] M.A. Rieffel, Induced representations of C ∗ -algebras, Adv. Math. 13 (1974) 176–257. [24] M.A. Rieffel, Morita equivalence for C ∗ -algebras and W ∗ -algebras, J. Pure Appl. Algebra 5 (1974) 51–96.


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[25] J.-L. Sauvageot, Sur le produit tensoriel relatif d’espaces de Hilbert, J. Operator Theory 9 (2) (1983) 237–252. [26] K. Schmidt, From infinitely divisible representations to cohomological rigidity, in: Analysis, Geometry and Probability, in: Texts Read. Math., vol. 10, Hindustan Book Agency, Dehli, 1996, pp. 173–197. [27] Y. Shalom, Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group, Ann. of Math. (2) 152 (1) (2000) 113–182. [28] M. Takesaki, Theory of Operator Algebras II, Encyclopaedia Math. Sci., vol. 127, Springer, 2003. [29] D. Voiculescu, The analogues of entropy and of Fischer’s information measure in free probability theory, III. The absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1) (1996) 172–199.


Paley–Wiener spaces with vanishing conditions and Painlevé VI transcendents Jean-François Burnol Université Lille 1, UFR de Mathématiques, Cité scientifique M2, F-59655 Villeneuve d’Ascq, France Received 17 September 2010; accepted 1 December 2010 Available online 9 December 2010 Communicated by Alain Connes

Abstract We modify the classical Paley–Wiener spaces PW x of entire functions of finite exponential type at most x > 0, which are square integrable on the real line, via the additional condition of vanishing at finitely many complex points z1 , . . . , zn . We compute the reproducing kernels and relate their variations with respect to x to a Krein differential system, whose coefficient (which we call the μ-function) and solutions have determinantal expressions. Arguments specific to the case where the “trivial zeros” z1 , . . . , zn are in arithmetic progression on the imaginary axis allow us to establish for expressions arising in the theory a system of two non-linear first order differential equations. A computation, having this non-linear system at his start, obtains quasi-algebraic and among them rational Painlevé transcendents of the sixth kind as certain quotients of such μ-functions. © 2010 Elsevier Inc. All rights reserved. Keywords: Paley–Wiener; Painlevé; Krein

1. Introduction and summary of results ∞ Let φ ∈ L2 (R, dt) and F (φ)(z) = −∞ φ(t)eizt dt its Fourier transform. When φ is supported in (−x, x), f (z) = F (φ)(z) is an entire function of exponential type at most x. Conversely the Paley–Wiener theorem identifies the vector space PW x of entire functions of exponential type at most x, square-integrable on the real line, as the Hilbert space of such Fourier transforms. Our convention for our scalar products is for them to be conjugate linear in the first factor and complex E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.002

J.-F. Burnol / Journal of Functional Analysis 260 (2011) 3222–3251

3223

linear in the second factor. Specifically (φ, ψ) = R φ(t)ψ(t) dt, hence for the transforms f 1 and g: (f, g) = 2π R f (z)g(z) dz = (φ, ψ). The evaluator Zz is the element of PW x such that x ∀g ∈ PW x ,

g = F (ψ),

(Zz , g) = g(z) =

eizt ψ(t) dt = F e−izt (−x,x) , g .

(1)

−x

Hence: Zz (w) = 2

sin((z − w)x) eizx e−iwx − e−izx eiwx = . z−w i(z − w)

(2)

Let E(w) = e−ixw and E ∗ (w) = E(w). The evaluators in PW x are given by Zz (w) = (Zw , Zz ) =

E(z)E(w) − E ∗ (z)E ∗ (w) . i(z − w)

(3)

Let us also define: 1 E(w) + E ∗ (w) , 2 i B(w) = E(w) − E ∗ (w) . 2

A(w) =

(4a) (4b)

Then E = A − iB, A = A∗ , B = B ∗ and: (Zw , Zz ) = Zz (w) = 2

B(z)A(w) − A(z)B(w) . z−w

(5)

For the Paley–Wiener spaces, A(w) = cos(xw) is even and B(w) = sin(xw) is odd. Let us consider generally a Hilbert space H , whose vectors are entire functions, and such that the evaluations at complex numbers are continuous linear forms, hence correspond to specific vectors Zz . Let σ = (z1 , . . . , zn ) be a finite sequence of distinct complex numbers. We let H σ be the closed subspace of H of functions vanishing at the zi ’s. Let γ (z) =

1 (z − z1 ) . . . (z − zn )

(6)

and define H (σ ) = γ (z)H σ : H (σ ) = F (z) = γ (z)f (z) f ∈ H, f (z1 ) = · · · = f (zn ) = 0 .

(7)

We introduced this notion in [5]. We say that F (z) = γ (z)f (z) is the “complete” form of f , and refer to z1 , . . . , zn as the “trivial zeros” of f . We give H (σ ) the Hilbert space structure which makes f → F an isometry with H σ . Let us note that evaluations F → F (z) are again continuous

3224


linear forms on this new Hilbert space of entire functions: this is immediate if z ∈ / σ and follows from the Banach–Steinhaus theorem if z ∈ σ . We thus define Kz in H (σ ) to be the evaluator at z: Kz ∈ H (σ ),

∀F ∈ H (σ ),

F (z) = (Kz , F )H (σ ) .

(8)

Here is a summary of the results presented here. We start by showing how to find entire functions Aσ and Bσ , real on the real line, such that: (Kw , Kz ) = Kz (w) = 2

Bσ (z)Aσ (w) − Aσ (z)Bσ (w) . z−w

(9)

This will be done under the following hypotheses: (1) the initial Hilbert space of entire functions H satisfies the axioms of [2], hence its evaluators Zz are given by a formula (5) for some entire functions A and B which are real on the real line, (2) A can be chosen even and B can be chosen odd, and (3) the added “trivial zeros” are purely imaginary. The produced functions Aσ and Bσ giving the reproducing kernel (9) of the modified space H (σ ) will be respectively even and odd. The restrictive hypotheses (2) and (3) can be disposed of, as is explained in companion paper [5]. We follow here another method, which proves formulas of a different type than those available from the general treatment [5]. The interested reader will find in [5] the easy arguments establishing that H (σ ) verifies the axioms of [2] if the initial space H does: this explains a priori why indeed a formula of the type (9) has to exist if (5) holds for H . Then, we examine the case of a dependency of the initial space on a parameter x. Assuming that the initial Ax and Bx obey a first order differential system of the Krein type [10,11] as functions of x (involving as coefficient what we call a μ-function) we prove that the new Aσ,x and Bσ,x do as well (in other words we compute the μσ -function in terms of the initial μ-function). The result is already notable when we start from the classical Paley–Wiener spaces for which the initial μ(x)-function (x > 0) vanishes identically. It will be achieved through establishing a “pre-Crum formula” for the effect of Darboux transformations on Schrödinger equations linked into Krein systems. The final part of the paper establishes the main result. We consider the classical Paley–Wiener spaces PW x modified by imaginary trivial zeros in an arithmetic progression σ . We prove that certain quotients of the μ-functions associated to the spaces PW x (σ ) obey the Painlevé VI differential equation. 2. A determinantal identity The following identity is quasi-identical with a formula of Okada [13, Theorem 4.2] and immediately equivalent to it (see also Lascoux [12] for more general determinants). We give a different proof. Theorem 1. Let there be given indeterminates ui , vi , ki , xi , yi , li , for 1 i n. We define the following n × n matrices ⎛ u 1 ⎜ k1 v 1 2 Un = ⎜ ⎝ k1 u1 .. .

u2 k2 v 2 k22 u2

... ... ...

...

...

un ⎞ kn v n ⎟ kn2 un ⎟ ⎠, .. .

⎛ v 1 ⎜ k1 u1 2 Vn = ⎜ ⎝ k1 v 1 .. .

v2 k2 u2 k22 v2

... ... ...

...

...

vn ⎞ kn un ⎟ kn2 vn ⎟ ⎠ .. .

(10)


3225

where the rows contain alternatively u’s and v’s. Similarly: ⎛ x 1 ⎜ l1 y 1 2 Xn = ⎜ ⎝ l1 x 1 .. .

x2 l2 y 2 l22 x2

... ... ...

...

...

⎛ y 1 ⎜ l1 x 1 2 Yn = ⎜ ⎝ l1 y 1 .. .

xn ⎞ ln y n ⎟ ln2 xn ⎟ ⎠, .. .

y2 l2 x 2 l22 y2

... ... ...

...

...

yn ⎞ ln x n ⎟ ln2 yn ⎟ ⎠. .. .

(11)

There holds: det

1i,j n

ui yj − vi xj lj − k i

Un 1 = Vn i,j (lj − ki )

Xn . Yn 2n×2n

Proof. Let A, B, C, D be n × n matrices, with A and C invertible. Using A 0 I A−1 B we obtain −1 0 C I C

(12) A B CD

=

D

A C

B = |A||C|C −1 D − A−1 B D

(13)

where vertical bars denote determinants. Let d(u) = diag(u1 , . . . , un ) and pu = 1in ui . We define similarly d(v), d(x), d(y) and pv , px , py . From the previous identity we get Ad(u) Cd(v)

Bd(x) = |A||C|pu pv d(v)−1 C −1 Dd(y) − d(u)−1 A−1 Bd(x) Dd(y) = |A||C|d(u)C −1 Dd(y) − d(v)A−1 Bd(x).

(14)

The special case A = C, B = D, gives Ad(u) Bd(x) = det(A)2 det (ui yj − vi xj ) A−1 B ij . Ad(v) Bd(y) 1i,j n 2n×2n

(15)

Let W (k) be the Vandermonde matrix with rows (1 . . . 1), (k1 . . . kn ), (k12 . . . kn2 ), . . . , and (k) = det W (k) its determinant. Let K(t) =

(t − km )

(16)

1mn

and let C be the n × n matrix (cim )1i,mn , where the cim ’s are defined by the partial fraction expansions: 1 i n,

t i−1 cim = . K(t) t − km 1mn

(17)

3226


We have the two matrix equations:

C·

1 lj − km

C = W (k) diag K (k1 )−1 , . . . , K (kn )−1 ,

(18a)

= W (l) diag K(l1 )−1 , . . . , K(ln )−1 .

(18b)

1m,j n

This gives the (well-known) identity:

1 lj − km 1m,j n = diag K (k1 ), . . . , K (kn ) W (k)−1 W (l) diag K(l1 )−1 , . . . , K(ln )−1 .

(19)

We can thus rewrite the determinant we want to compute as: ui yj − vi xj l −k j

i

1i,j n

=

m

K (km )

K(lj )−1 (ui yj − vi xj ) W (k)−1 W (l) ij n×n .

(20)

j

We shall now make use of (15) with A = W (k) and B = W (l): ui yj − vi xj −2 −1 W (k)d(u) = (k) K (k ) K(l ) m j W (k)d(v) l −k j i 1i,j n m j

n(n−1) W (k)d(u) (−1) 2 = W (k)d(v) (l − k ) i i,j j

W (l)d(x) W (l)d(y)

W (l)d(x) . W (l)d(y) 2n×2n

(21)

n

The sign (−1)n(n−1)/2 = (−1)[ 2 ] is the signature of the permutation which exchanges rows Uni Xand n n + i for i = 2, 4, . . . , 2[ 2 ] and transforms the determinant on the right-hand side into Vn Ynn . This concludes the proof. 2 3. A and B for spaces with imaginary trivial zeros Just using the existence of continuous evaluators but not yet (5), we have by a simple argument of orthogonal projection (see [5]): Proposition 2. Let H be a Hilbert space of entire functions with continuous evaluators Zz : ∀f ∈ H , f (z) = (Zz , f ). Let σ = (z1 , . . . , zn ) be a finite sequence of distinct complex numbers with associated evaluators Z1 , . . . , Zn , assumed to be linearly independent. Let H (σ ) be the Hilbert space of entire functions which are complete forms of the elements of H vanishing on σ . The evaluators of H (σ ) are given by


(Z1 , Z1 ) γ (w)γ (z) (Z2 , Z1 ) Kz (w) = .. Gn . (Zw , Z1 )

... ... ... ...

(Z1 , Zn ) (Z2 , Zn ) .. . (Zw , Zn )

(Z1 , Zz ) (Z2 , Zz ) , .. . (Zw , Zz )

3227

(22)

where Gn > 0 is the principal n × n minor of the matrix. Recalling the form (5) of the reproducing kernel: (Zw1 , Zw2 ) = 2

B(w2 )A(w1 ) − A(w2 )B(w1 ) w 2 − w1

(23)

we see that the choices: 1 i n:

ui = A(zi ),

vi = B(zi ),

un+1 = A(w), 1 j n:

xj = A(zj ), xn+1 = A(z),

ki = zi ,

vn+1 = B(w), yj = B(zj ), yn+1 = B(z),

kn+1 = w, lj = zj , ln+1 = z

(24a) (24b) (24c) (24d)

allow to make use of Theorem 1. This gives: 2n+1 γ (w)γ (z)(−1)n γ ∗ (w)γ (z) Un,w Kz (w) = Gn · 1i,j n (zj − zi ) · (z − w) Vn,w

Xn,z Yn,z (2n+2)×(2n+2)

(25)

with ⎛ A(z ) 1 ⎜ z1 B(z1 ) 2 Un,w = ⎜ ⎝ z1 A(z1 ) .. . ⎛ B(z ) 1

⎜ z1 A(z1 ) 2 Vn,w = ⎜ ⎝ z1 B(z1 ) .. .

A(z2 ) z2 B(z2 ) z22 A(z2 )

... ... ...

...

...

B(z2 ) z2 A(z2 ) z22 B(z2 ) ...

A(zn ) zn B(zn ) zn2 A(zn ) .. .

A(w) ⎞ wB(w) ⎟ w 2 A(w) ⎟ ⎠, .. . B(w) ⎞

. . . B(zn ) . . . zn A(zn ) wA(w) ⎟ . . . zn2 B(zn ) w 2 B(w) ⎟ ⎠, .. .. ... . .

(26)

(27)

Xn,z = Un,z ,

(28)

Yn,z = Vn,z .

(29)

We shall now make the following hypotheses: (1) the zi ’s are purely imaginary, (2) A is even and B is odd. Then A(zi ) = A(zi ) = A(−zi ) = A(zi ), and B(zi ) = B(zi ) = B(−zi ) = −B(zi ). The first n columns of the matrix Un,w are thus real and identical with the first n columns of Xn,z . The first n columns of the matrix Vn,w are purely imaginary and thus the opposite of the first n columns of Yn,z .

3228


Un,w Xn,z In order to compute the determinant Vn,w Yn,z (2n+2)×(2n+2) , we substract, for 1 j n, column j to column j + n + 1. This sets to zero all columns of Xn,z except its last and multiplies by 2 the n first columns of Yn,z . We then apply a Laplace expansion using the (n + 1) × (n + 1) minors built with the first and last (n + 1) rows. If the top minor has both the w and z columns, its complementary bottom minor will have two proportional columns hence vanish. There are thus only two contributions, and taking (various) signs into account we obtain: Un,w Vn,w

Xn,z = 2n det(Un,w ) det(Yn,z ) − det(Xn,z )(−2)n det(Vn,w ). Yn,z (2n+2)×(2n+2)

(30)

So: Kz (w) =

22n+1 γ (w)γ (z) γ (z)(−1)n γ ∗ (w) Gn i,j n (zj − zi ) ·

det(Un,w )det(Vn,z ) − det(Un,z )(−1)n det(Vn,w ) . z−w

(31)

Let us also compute the Gram determinant Gn . The determinantal identity gives: Un 2n Gn = Vn i,j (zj − zi )

22n (−1)n Un det(Un ) det(Vn ). = −Vn 2n×2n 1i,j n (zj − zi )

(32)

Finally: Kz (w) = 2γ (w)γ ∗ (w)γ (z)γ ∗ (z)

det(Un,w )det(Vn,z ) − det(Un,z )(−1)n det(Vn,w ) . det(Un ) det(Vn )(z − w)

(33)

Taking into account that i n det(Vn ) is real we get: det(Un,w )i n det(Vn,z ) − det(Un,z )i n det(Vn,w ) (34a) det(Un )(−i)n det(Vn )(z − w) γ (w)γ ∗ (w)γ (z)γ ∗ (z) det(Vn,z ) det(Un,w ) det(Un,z ) (−1)n det(Vn,w ) . =2 (−1)n − z−w det Vn det Un det Un det Vn (34b)

Kz (w) = 2γ (w)γ ∗ (w)γ (z)γ ∗ (z)

The following has been obtained: Theorem 3. Let H be a Hilbert space of entire functions with reproducing kernel Zz (w) = , where the entire functions A and B are real on the real line and respectively 2 B(z)A(w)−A(z)B(w) z−w even and odd. Let σ = (zi )1in be a finite sequence of distinct purely imaginary numbers. We assume that the associated evaluators are linearly independent, and also that zi + zj = 0 1 for all i, j . Let H (σ ) be the Hilbert space of the functions γ (z)f (z), where γ (z) = i z−z and i f is in H with f (z) = 0 for z ∈ σ . Let


Aσ (w) =

(−1)

n(n−1) 2

γ (w)γ ∗ (w) det Un

A(z ) 1 z B(z ) 1 1 × z2 A(z1 ) 1 .. . Bσ (w) =

(−1)

3229

A(z2 ) z2 B(z2 ) z22 A(z2 )

... ... ...

A(zn ) zn B(zn ) zn2 A(zn )

...

...

...

B(z2 ) z2 A(z2 ) z22 B(z2 )

... ... ...

B(zn ) zn A(zn ) zn2 B(zn )

...

...

...

A(w) wB(w) , w 2 A(w) .. . (n+1)×(n+1)

(35a)

B(w) wA(w) w 2 B(w) .. . (n+1)×(n+1)

(35b)

n(n+1) 2

γ (w)γ ∗ (w) det Vn

B(z ) 1 z A(z ) 1 1 × z2 B(z1 ) 1 .. .

where the denominators det Un and det Vn are the principal n × n minors of the numerators. The space H (σ ) has evaluators Kz satisfying the formula: Kz (w) = (Kw , Kz ) = 2

Bσ (z)Aσ (w) − Aσ (z)Bσ (w) . z−w

(36)

The functions Aσ and Bσ are entire, real on the real line, Aσ is even and Bσ is odd. n(n−1)

Remark 1. The additional (−1) 2 is to make Aσ (it) > 0 and −iBσ (it) > 0 for t > 0, at least in the case of the Paley–Wiener spaces (this sign is easily determined from the asymptotics as t → +∞; let us also mention that −iBσ (it)Aσ (it) > 0 for t > 0, from (36) and if H (σ ) = {0}). We observe that if the initial A and B verify the normalization −iB(it) A(it) →t→+∞ 1 then the new Aσ and Bσ also. This normalization (when possible) has proven to be more natural in this and other investigations, than other normalizations such as, for example, A(0) = 1. Remark 2. A formula of the type (36) for evaluators in a Hilbert space of entire functions is guaranteed by the axiomatic framework of [2]. The passage from an H to an H (σ ) is compatible to these axioms (cf. [5]), hence existence of an Eσ = Aσ − iBσ function was known in advance. Determination of a suitable Eσ , without any of the restrictive hypotheses made here, is achieved in [5] with another method. Let us record a special case of the computation (32) of the Gram determinant Gn , using the notation W (f1 , . . . , fn ) for Wronskian determinants det(fj(i) )1i,j n (derivatives with respect to x): Proposition 4. The following identity holds: sh((κi + κj )x) W (ch(κ1 x), . . . , ch(κn x)) · W (sh(κ1 x), . . . , sh(κn x)) = . 2 κi + κj 1i,j n 1in κi 1i<j n (κi + κj )

(37)

3230


Proof. Let ui = ch(κi x), vi = sh(κi x), ki = κi , xj = ch(κj x) = uj , yj = − sh(κj x) = −vj , lj = −κj = −kj . With these choices: sh((κi + κj )x) ui yj − vi xj = . κi + κj lj − k i

(38)

By Theorem 1: det

1i,j n

ui yj − vi xj lj − k i

Un 1 Vn i,j (lj − ki )

=

Xn Yn 2n×2n

(−1)n (−2)n det(Un ) det(Vn ) = i,j (κi + κj )

(39)

where we used Un = Xn , Vn = −Yn . As det Un = W (ch(κ1 x), . . . , ch(κn x)) and det Vn = W (sh(κ1 x), . . . , sh(κn x)), this completes the proof. 2 4. Crum formulas for Darboux transformations of Krein systems All derivatives in this chapter will be with respect to a variable x. We are interested in differential systems of the Krein type: (S)

a − μa = −kb, b + μb = +ka.

(40)

Here a, b, and μ are functions of a variable x and k is a scalar. Krein uses systems of this type in particular in his approach [10] to Inverse Scattering Theory and in his continuous analogues to topics of Orthogonal Polynomial Theory [11]. The system couples two Schrödinger equations: −a + V + a = k 2 a

with V + = μ2 + μ ,

(41a)

−b + V − b = k 2 b

with V − = μ2 − μ .

(41b)

It proves quite convenient to introduce the notion of a tau-function, which is a function such that: μ2 = −(log τ ) .

(42)

We shall also use the notation λ = (log τ ) , so that μ2 = −λ . The well-known Darboux transformation [7, §6] transforms the solutions of a Schrödinger equation −f + Vf = Ef into solutions of another one, and the formulas of Crum [6] give Wronskian expressions for both solutions and potentials after successive such Darboux transformations. In this chapter we introduce a notion of “simultaneous” or “linked” such transformations which act at the level of the Krein system (40). This provides a kind of refinement to the formula of Crum, the change of the two potentials being lifted to the change of the “tau” function. We did not find in the literature the results we prove here, but it is so extensive that we may have missed some important contributions.


3231

We make use also of couples (α, β) of the type (a, −ib). Hence we also consider the differential systems: (T )

α − μα = +κβ, β + μβ = +κα.

(43)

It corresponds to (S) (40) via k = iκ, a = α, b = iβ. The Schrödinger equations become: α = κ 2 + V + α, β = κ 2 + V − β.

(44a) (44b)

Theorem 5. Let κ ∈ C and let (α, β) be a solution of the differential system (T ) (43), with neither α nor β identically zero. The simultaneous Darboux transformations: α a, α β b → b1 = b − b β

a → a1 = a −

(45a) (45b)

transform any solution (a, b, k) of the differential system (S) (40) into a solution (a1 , b1 , k) of a transformed system: (S1 )

a1 − μ1 a1 = −kb1 , b1 + μ1 b1 = +ka1

(46)

where the new coefficient μ1 is μ1 = μ −

α d log . dx β

(47)

If μ2 = −(log τ ) then μ21 = −(log τ1 ) with τ1 = τ αβ.

(48)

α a α a α a Proof. From αa1 = α a , we get (αa1 ) = α a = (V + +κ 2 )α (V + −k 2 )a = −(k 2 + κ 2 )aα, which we rewrite as a1 +

α a1 = − k 2 + κ 2 a = −k b + μb − κ 2 a. α

(49)

Further αa1 = α(−kb + μa) − (κβ + μα)a = −kαb − κβa.

(50)

Eliminating a gives: a1 +

α α α a1 − κ a1 = −k b + μb + kκ b. α β β

(51)

3232


Using κ βα =

β β

+ μ: a1

α β β − − μ a1 = −k b − b . + α β β

With the definitions μ1 = μ −

α α

+

β β

and b1 = b −

β βb

(52)

this gives indeed:

a1 − μ1 a1 = −kb1 .

(53)

β b β b β b From βb1 = β b , we get (βb1 ) = β b = (V − +κ 2 )β (V − −k 2 )b = −(k 2 + κ 2 )bβ, which gives: β b1 = − k 2 + κ 2 b = k a − μa − κ 2 b. β

(54)

βb1 = β(ka − μb) − (κα − μβ)b = kβa − καb.

(55)

b1 + On the other hand

Eliminating b gives: b1 + Using κ βα =

α α

b1

(56)

− μ finally leads to: β α α − + μ b1 = k a − a + β α α

Let λ1 = λ + λ1

β β β b1 − κ b1 = k a − μa − kκ a. β α α

α α

+

β β

⇒

b1 + μ1 b1 = +ka1 .

(57)

= (log τ αβ) . We must also verify μ21 = −λ1 .

2 2 2 2 + − α β α β 2 2 2 2 =λ + V +κ − + V +κ − = μ + 2κ − − , (58) α β α β

β α (α − μα)(β + μβ) α β = − μ + μ − μ2 , αβ α β β α β α α 2 α β β 2 − 2μ − − λ1 = −μ2 + 2 − = −μ21 . α β β α α β κ2 =

(59) 2

(60)

Remark 3. A solution (a, b, k) of system (S) (40) corresponds to a solution (α, β, κ) = (a, −ib, −ik) of system (T ) (43), and from a solution (α, β, κ) of (T ) we can switch to the solution (α, iβ, iκ) of (S), having the same logarithmic derivatives with respect to x. Hence it is just a matter of arbitrary choice to consider the Darboux transformations to be associated to a specific solution of (T ) rather than to a specific solution of (S). Moreover, the same Darboux


3233

transformations (45a), (45b) which are associated to a given (α, β, κ) but now applied to a triple (γ , δ, ξ ), solution of (T ), produce a solution of the transformed system (T1 )

γ1 − μ1 γ1 = ξ δ1 , δ1 + μ1 δ1 = ξ γ1

(61a)

of type (T ) associated to the new coefficient μ1 . Theorem 6. Let there be given n triples (αj , βj , κj ), solutions of the differential system (T ) (43). We assume that α1 , . . . , αn are linearly independent, and β1 , . . . , βn as well. To each solution (a, b, k) of the system (S)

a − μa = −kb, b + μb = +ka

(62)

we associate W (α1 , . . . , αn , a) , W (α1 , . . . , αn ) W (β1 , . . . , βn , b) . bn = W (β1 , . . . , βn ) an =

(63a) (63b)

Going from (a, b) to (an , bn ) is the result of the n successive simultaneous Darboux transformations (45a) and (45b) associated to (α1 , β1 ), . . . , (αn , βn ) (themselves transformed along the way). There holds: (Sn )

an − μn an = −kbn , bn + μn bn = +kan

(64)

W (α1 , . . . , αn ) d log . dx W (β1 , . . . , βn )

(65)

where the coefficient μn is given by μn = μ −

If furthermore one chooses a tau-function such that μ2 = −(log τ ) then μ2n = −

d2 log τn dx 2

(66)

where τn = τ · W (α1 , . . . , αn )W (β1 , . . . , βn ).

(67)

Proof. Let us consider first the simultaneous Darboux transformations of system (S) (40) and of (1) (1) its partner (T ) (43), defined by (α1 , β1 ). Let us write in particular (α2 , β2 ) for the transform of the couple (α2 , β2 ). We then apply the associated Darboux transformations to (S1 ) giving (2) (2) rise to (S2 ). The couple (α3 , β3 ) is transformed into a solution (α3 , β3 ) of partner (T2 ). Etc.

3234


Although we speak of transformed systems to keep track of the coupling, each of the associ ated Schrödinger equation −f + V ± f = k 2 f is transformed by f → f − gg f where g is a solution of −g + V ± g = −κ 2 g, hence independently of what happens to the other equation −φ + V ∓ φ = k 2 φ. One part of the theorem of Crum [6] (which we do not reprove here) tells us that if we apply n successive Darboux transformations f → f − gg f , first by g1 , then by the transformed g2 , then by the transformed g3 , etc., the final action can be written directly as: f →

W (g1 , . . . , gn , f ) . W (g1 , . . . , gn )

(68)

Hence definitions (63a) and (63b) of an and bn can be viewed as the final result of the n successive simultaneous Darboux transformations. Theorem 5 tells us how μ changes when system (S) is transformed once, hence iterative use of the theorem gives a formula for μn involving in fact telescopic products of quotients of Wronskians, hence Eq. (65). Moreover if a tau-function is initially chosen with −(log τ ) = μ2 , Theorem 5 can again be applied iteratively, leading to a function τn given by (67), and verifying −(log τn ) = μ2n . 2

μ

Let us take note that μ2n + μn = −(log τ ) − (log W (α1 , . . . , αn )) − (log W (β1 , . . . , βn )) + W (α1 ,...,αn ) d2 d2 2 2 2 − dx 2 log W (β1 ,...,βn ) = μ + μ − 2 dx 2 log W (α1 , . . . , αn ). And similarly μn − μn = μ − 2

d μ − 2 dx 2 log W (β1 , . . . , βn ). Thus:

Corollary 7. Using the notations of Theorem 6, there holds −an + Vn+ an = k 2 an ,

(69a)

−bn

2

(69b)

Vn+ = V + − 2

d2 log W (α1 , . . . , αn ), dx 2

(70a)

Vn− = V − − 2

d2 log W (β1 , . . . , βn ). dx 2

(70b)

+ Vn− bn

= k bn

with

These formulas are the part of Crum’s theorem [6] regarding the effect of successive Darboux transformations on the potentials of Schrödinger equations. 5. Modification of μ-functions by trivial zeros We are interested in Hilbert spaces Hx of entire functions in the sense of [2], whose reproducing kernels are given by formula (5), where the functions A (= Ax ) and B (= Bx ) are real-valued on the real line, respectively even and odd, and obey a first order differential system with respect to x of the Krein type [10], with a real-valued coefficient function μ(x):


d Ax (w) − μ(x)Ax (w) = −wBx (w), dx d Bx (w) + μ(x)Bx (w) = wAx (w). dx

3235

(71a) (71b)

Remark 4. More general integral equations play the important general structural role in [2]. We have found that the above restricted type arises naturally in the study of some specific instances of Hilbert spaces of entire functions [3]. It turns out to be well adapted to the present study of the classical Paley–Wiener spaces modified by adding trivial zeros on the imaginary axis. If we remove the restriction for the zeros to lie on the imaginary axis, the functions Ax and Bx real on the real line will (generally speaking) cease to be respectively even and odd and they obey the more general type of Krein system from [11] which has both the real and imaginary parts of a complex-valued μ-function as coefficients. We want to combine Theorems 3 and 6. We will suppose that the functions Ax are even, the functions Bx odd, and the trivial zeros zi , 1 i n, are purely imaginary and verify zi = ±zj for all i, j . From (71a) and (71b):

d −μ dx

d +μ dx d +μ dx

d −μ dx d −μ dx

2p Ax = (−1)p w 2p Ax ,

(72a)

Ax = (−1)p+1 w 2p+1 Bx .

(72b)

2p

d (2p) d (2p+1) By recurrence the left side of (72a) (resp. (72b)) is ( dx ) Ax (resp. ( dx ) Ax ) up to a finite linear combination of lower derivatives of Ax with coefficients being function of x (independent of w). Hence, for n = 2m:

W Ax (z1 ), . . . , Ax (zn ), Ax (w) A (z ) . . . Ax (zn ) x 1 z B (z ) . . . z B (z ) 1 x 1 n x n z2 A (z ) . . . z2 A (z ) x 1 n x n = 1 .. . ... ... 2m z1 Ax (z1 ) . . . ...

Ax (w) wBx (w) w 2 Ax (w) .. . w 2m Ax (w) (2m+1)×(2m+1)

(73)

and for n = 2m + 1: W Ax (z1 ), . . . , Ax (zn ), Ax (w) Ax (z1 ) ... z1 Bx (z1 ) ... 2 z1 Ax (z1 ) ... = (−1)m+1 .. ... 2m . z Ax (z1 ) . . . 1 z2m+1 B (z ) . . . x 1 1

Ax (zn ) zn Bx (zn ) zn2 Ax (zn ) .. . ... ...

Ax (w) wBx (w) w 2 Ax (w) . .. . w 2m Ax (w) w 2m+1 Bx (w) (2m+2)×(2m+2)

(74)

3236


If we divide the Wronskians (constructed with derivations with respect to x) and the (n + 1) × (n + 1) determinants at the right by their respective n × n principal minors, the resulting fractions will thus coincide up to (−1)m for n = 2m and (−1)m+1 for n = 2m + 1, hence in both cases up 1 to (−1) 2 n(n+1) . We can thus rewrite the function Aσ of Theorem 3 as: Aσ (w) = (−1)n γ (w)γ ∗ (w)

W (Ax (z1 ), Ax (z2 ), . . . , Ax (zn ), Ax (w)) . W (Ax (z1 ), Ax (z2 ), . . . , Ax (zn ))

(75)

In the same manner

d +μ dx

d −μ dx d −μ dx

d +μ dx d +μ dx

2p Bx = (−1)p w 2p Bx ,

(76a)

Bx = (−1)p w 2p+1 Ax .

(76b)

2p

d (2p) d (2p+1) By recurrence the left side of (76a) (resp. (76b)) is ( dx ) Bx (resp. ( dx ) Bx ) up to a finite linear combination of lower derivatives of Bx with coefficients being function of x (independent of w). Hence, for n = 2m:

W Bx (z1 ), . . . , Bx (zn ), Bx (w) B (z ) . . . Bx (zn ) x 1 z A (z ) . . . z A (z ) 1 x 1 n x n 2 m z2 Bx (z1 ) . . . z n Bx (zn ) = (−1) 1 .. . ... ... 2m z1 Bx (z1 ) . . . ...

Bx (w) wAx (w) w 2 Bx (w) .. . w 2m Bx (w) (2m+1)×(2m+1)

(77)

Bx (w) wAx (w) w 2 Bx (w) . .. . w 2m Bx (w) w 2m+1 Ax (w) (2m+2)×(2m+2)

(78)

and for n = 2m + 1: W Bx (z1 ), . . . , Bx (zn ), Bx (w) Bx (z1 ) . . . Bx (zn ) z1 Ax (z1 ) . . . zn Ax (zn ) 2 z1 Bx (z1 ) . . . zn2 Bx (zn ) = .. .. ... . 2m . z Bx (z1 ) . . . . .. 1 z2m+1 A (z ) . . . . . . x 1 1

If we divide the Wronskians and the determinants at the right by their respective n × n principal minors, the results will coincide up to (−1)m for n = 2m and (−1)m for n = 2m + 1, hence in 1 both cases up to (−1) 2 n(n−1) . We can rewrite the function Bσ of Theorem 3 as: Bσ (w) = (−1)n γ (w)γ ∗ (w)

W (Bx (z1 ), Bx (z2 ), . . . , Bx (zn ), Bx (w)) W (Bx (z1 ), Bx (z2 ), . . . , Bx (zn ))

where the Wronskians are constructed with derivations with respect to x.

(79)


3237

Taking into account that (−1)n γ ∗ (w) = γ (−w) we can thus sum up these computations in the following: Theorem 8. Let there be given Hilbert spaces Hx of entire functions, with functions Ax (even, real on the real line) and Bx (odd, real on the real line) computing the evaluators in Hx by formula (5), and whose variations with respect to the parameter x are given by d Ax (w) − μ(x)Ax (w) = −wBx (w), dx d Bx (w) + μ(x)Bx (w) = wAx (w). dx

(80a) (80b)

Let σ = (z1 , . . . , zn ) be a finite sequence of purely imaginary numbers (the associated evaluators in Hx being supposed linearly independent) with zi = ±zj for 1 i, j n and let Hx (σ ) be the Hilbert space Hx modified by σ . Its evaluators Kz are given by Kz (w) = (Kw , Kz ) = 2

Bx,σ (z)Ax,σ (w) − Ax,σ (z)Bx,σ (w) z−w

(81)

with W (Ax (z1 ), Ax (z2 ), . . . , Ax (zn ), Ax (w)) , W (Ax (z1 ), Ax (z2 ), . . . , Ax (zn )) W (Bx (z1 ), Bx (z2 ), . . . , Bx (zn ), Bx (w)) Bx,σ (w) = γ (w)γ (−w) W (Bx (z1 ), Bx (z2 ), . . . , Bx (zn ))

Ax,σ (w) = γ (w)γ (−w)

(82a) (82b)

where the Wronskians involve derivatives with respect to the variable x. The entire functions Ax,σ and Bx,σ are real on the real line, and respectively even and odd. Taking into account Theorem 6 we thus learn that: Theorem 9. Let there be given Hilbert spaces Hx of entire functions, functions Ax and Bx , imaginary numbers z1 , z2 , . . . verifying the hypotheses of Theorem 8. Let Hx (n) = Hx (z1 , . . . , zn ) and let the functions Ax,n and Bx,n computing the reproducing kernel in Hx (n) be provided by Theorem 3. They are obtained by successive transformations (essentially) of Darboux type: d d 1 dx Ax,n (zn+1 ) Ax,n (w) − Ax,n (w) , Ax,n+1 (w) = 2 Ax,n (zn+1 ) zn+1 − w 2 dx d Bx,n (zn+1 ) d 1 Bx,n (w) − dx Bx,n (w) Bx,n+1 (w) = 2 Bx,n (zn+1 ) zn+1 − w 2 dx

(83a)

(83b)

and verify the equations d Ax,n (w) − μn (x)Ax,n (w) = −wBx,n (w), dx d Bx,n (w) + μn (x)Bx,n (w) = +wAx,n (w) dx

(84a) (84b)

3238


with μn = μ −

W (Ax (z1 ), . . . , Ax (zn )) d log . dx W (Bx (z1 ), . . . , Bx (zn )) 2

(85)

2

d d 2 If a function τ is chosen with μ2 = − dx 2 log τ then μn = − dx 2 log τn with

τn = τ · W Ax (z1 ), . . . , Ax (zn ) W −iBx (z1 ), . . . , −iBx (zn ) .

(86)

In the following, the index x shall be dropped from the notations. Combining (83a) with (84a) we obtain: 2 Bn (zn+1 ) 2 An (w) zn+1 − w An+1 (w) = μn An (w) − wBn (w) − μn − zn+1 An (zn+1 ) zn+1 = −(zn+1 + w)Bn (w) + Bn (zn+1 )An (w) + An (zn+1 )Bn (w) An (zn+1 ) zn+1 = −(zn+1 + w)Bn (w) + (zn+1 + w)K n (zn+1 , w). (87) 2An (zn+1 ) We have written K n (z, w) = Kzn (w) for the evaluator in H (n) = H (z1 , . . . , zn ). Combining (83b) with (84b) gives:

2 zn+1

An (zn+1 ) Bn (w) − w Bn+1 (w) = −μn Bn (w) + wAn (w) − −μn + zn+1 Bn (zn+1 ) zn+1 = (zn+1 + w)An (w) − An (zn+1 )Bn (w) + Bn (zn+1 )An (w) Bn (zn+1 ) zn+1 = (zn+1 + w)An (w) − (zn+1 + w)K n (zn+1 , w). (88) 2Bn (zn+1 ) 2

We thus have the identities: Theorem 10. Let H = Hx , An , Bn , for n 1 be as in Theorem 9. There holds: zn+1 K n (zn+1 , w), 2An (zn+1 ) zn+1 (w − zn+1 )Bn+1 (w) = −An (w) + K n (zn+1 , w) 2Bn (zn+1 )

(w − zn+1 )An+1 (w) = Bn (w) −

(89a) (89b)

where K n (z, w) = Kzn (w) is the reproducing kernel in H (n) = H (z1 , . . . , zn ). From formula (22) in Proposition 2 we know that 1in (w − zi ) · K n (zn+1 , w) is a linear combination of the initial evaluators Zi (w) (= Zzi (w)), 1 i n + 1. Hence by induction we obtain the following:


3239

Theorem 11. Let H = Hx , An , Bn , for n 1 be as in Theorem 9. Let En = An − iBn and Fn = En∗ = An + iBn . The function (−i)n 1in (w − zi ) · En (w) differs from the initial E = A − iB function by a finite linear combination of the initial evaluators Zi (w), 1 i n. Also the function i n 1in (w − zi ) · Fn (w) (Fn = En∗ ) differs from the initial F = A + iB function by a finite linear combination of the initial evaluators Zi (w), 1 i n. Remark 5. Let us note that this characterizes uniquely the En (and Fn ) provided by Theorem 8, as the unknown linear combinations of (linearly independent) evaluators will be constrained by their values at the zi ’s. This theorem for the transition from H to H (σ ) holds with much greater generality than achieved here (see the companion article [5]): it suffices for H to verify the axioms of [2]. Thus, reverting the steps we could have started from the results proven in [5] and, under the additional hypotheses made here (existence of a parameter x and of a differential system of Krein type, imaginary trivial zeros, etc.), obtain the Darboux transformations ((83a) and (83b) in Theorem 9) and later the Wronskian formulas (Theorem 8) as corollaries. 6. Non-linear equations for Paley–Wiener spaces with trivial zeros On the basis of Theorem 11 it is convenient to work with the “incomplete” forms of the various objects encountered. As the main results of this chapter are for the classical Paley–Wiener spaces PW x , we will from the start assume H = PW x . We consider its modification H (σ ) by finitely many “trivial” distinct zeros σ = (z1 , . . . , zn ) (the associated evaluators in H are always linearly 1 independent). Let γ (w) = 1j n w−z be the corresponding gamma factor. We define the j incomplete version K σ (z, w) of the reproducing kernel K(z, w) in H (σ ) via the relation K(z, w) = Kz (w) = (Kw , Kz ) = γ (w)γ (z)K σ (z, w).

(90)

Proposition 2 is the statement that K σ (z, w) is the unique entire function of w which vanishes at z1 , . . . , zn and differs additively from the initial evaluator Z(z, w) by a finite linear combination of the initial evaluators Z(z1 , w), . . . , Z(zn , w). Let us now consider the functions Eσ and Fσ characterized as in Theorem 11. We consider their incomplete versions, up to a factor i n : Eσ (w) = i n γ (w)Eσ (w),

Fσ (w) = i n γ (w)Fσ (w).

(91)

Of course, there does not hold (for n 1) Fσ = Eσ∗ (this last function has its trivial zeros not at the zi ’s but at the zi ’s). The formula for the incomplete reproducing kernel is K σ (z, w) =

Eσ (z)Eσ (w) − Fσ (z)Fσ (w) . i(z − w)

(92)

The rationale for the i n in (91) is twofold: first Theorem 11, second the fact that if the zi ’s are imaginary the function Aσ and iBσ obtained in Theorem 3 are real on iR, hence Eσ and Fσ are real on iR, hence Eσ (it) and Fσ (it) as defined by (91) are real for t real. The differential system with respect to x for Eσ and Fσ (as for their complete versions Eσ , Fσ ) is:

3240


d Eσ (it) = tEσ (it) + μσ (x)Fσ (it), dx d Fσ (it) = −tFσ (it) + μσ (x)Eσ (it). dx

(93a) (93b)

We introduce the coefficients c1 , . . . , cn , d1 , . . . , dn which are the functions of x and of the imaginary points z1 = −iκ1 , . . . , zn = −iκn such that, according to Theorem 11, the following holds: Eσ (it) = ext +

cj

1j n

Fσ (it) = (−1)n e−xt +

2 sh((t − κj )x) , t − κj

1j n

dj

2 sh((t − κj )x) . t − κj

(94a)

(94b)

The identity following from Fσ = Eσ∗ is (we use that Eσ and Fσ are real-valued on iR): (t − κj )Fσ (it) = (−1)n (t + κj )Eσ (−it). j

(95)

j

If one is interested in explicit formulas for the cj ’s and dj ’s, the initial recipe is to put t = −κ1 , . . . , t = −κn in (94a) (resp. (94b)) and to use the trivial zeros Eσ (−iκj ) = 0 (resp. Fσ (−iκj ) = 0). Cramer’s formulas thus lead to determinantal representations for the cj ’s and dj ’s (which are seen to be real-valued). Remark 6. We pause here to explain how to remove the restrictions κi + κj = 0. These constraints go back to Theorem 3. They were necessary to avoid vanishing of the denominators Un and Vn , in the formulas for An , Bn . But (94a) and (94b) define Eσ and Fσ , and the validity of K σ (z, w) =

Eσ (z)Eσ (w) − Fσ (z)Fσ (w) i(z − w)

(96)

follows by continuity (for real κi ’s), as there is no singularity arising in the formulas for the coefficients c1 , . . . , cn , d1 , . . . , dn . The same remark applies to the μ-function μσ which will be expressed below in terms of these coefficients. Hence by continuity we again have a μ-function and a differential system (93a), (93b) even when κi + κj = 0 for some (i, j ). The conditions κi + κj = 0 were inforced only in order to facilitate the writing of explicit formulas of Wronskian type for the A’s and B’s. There is a plethora of various algebraic and differential identities involving the cj ’s and dj ’s. We propose a basic selection, sufficient for our goal in this chapter. From (94a), the value of d ( dx − t)Eσ (it) is 2 sh((t − κj )x) cj − κj cj + cj 2 ch (t − κj )x − 2 sh (t − κj )x . t − κj

1j n

1j n

(97)


3241

Comparison with (93a) gives:

μσ (x) = (−1)n

2cj eκj x

and

(98)

1j n

1j n

d cj − κj cj = μσ dj . dx

⇒

(99)

d + t)Fσ (it) is Similarly, from (94b), the value of ( dx

2 sh((t − κj )x) dj + κj dj + dj 2 ch (t − κj )x + 2 sh (t − κj )x . t − κj

1j n

(100)

1j n

Thus:

μσ (x) =

2dj e−κj x

and

(101)

1j n

1j n

d dj + κj dj = μσ cj . dx

⇒

(102)

We take note of the asymptotics: ασ (x) + O t −2 , Eσ (it) =t→+∞ ext 1 − 2t

ασ (x) = −2

cj e−κj x ,

1j n

−2 δσ (x) n −xt 1+ +O t , Fσ (it) =t→−∞ (−1) e 2t

(103a) δσ (x) = −(−1)n 2

d j e κj x .

1j n

(103b) Using (95) we obtain δσ (x) = ασ (x) + 4

1j n κj .

Further,

d ασ (x) = −2 cj − κj cj e−κj x = −2μσ dj e−κj x = −μ2σ . dx 1j n

(104)

1j n

Using either the differential equations or the identities already known provides the two further asymptotics:

−2 μσ (x) Fσ (it) =t→+∞ e +O t , 2t −μσ (x) Eσ (it) =t→−∞ (−1)n e−xt + O t −2 . 2t xt

(105a) (105b)

We definitely switch to viewing functions depending on x as functions of the variable a = e−x . d d = − dx . We also fix once For example we write μσ (a), rather than μσ (− log(a)). We have a da

3242


and for all an integer n 1, and will study the spaces PW x (ν, n) associated to a sequence of trivial zeros z1 , . . . , zn , in arithmetic progression: κ1 =

ν+1 , 2

κ2 =

ν +1 + 1, 2

...,

κn =

ν +1 + n − 1, 2

zj = −iκj .

(106)

The transition from n to n + 1 is described by Theorem 10. Here n is fixed, and we shall study the relation between ν and ν + 1. We will use the notations Eν , Eν+1 , Fν , Fν+1 , μν , μν+1 , and K ν , K ν+1 for the incomplete reproducing kernel. Neither the dependency on a nor on n is explicitly recalled in the notation. Also we shall write c1ν , . . . , cnν and d1ν , . . . , dnν , respectively c1ν+1 , . . . , cnν+1 , and d1ν+1 , . . . , dnν+1 , for the coefficients appearing in Eqs. (94a) and (94b) for ν and ν + 1. These coefficients are functions of a (depending on n). We rewrite (94a) and (94b) as x Eν (it) = e + xt

e−ty e

ν+1 2 y

Fν (it) = (−1) e

cjν e(j −1)y dy,

(107a)

1j n

−x n −xt

x +

e−ty e

ν+1 2 y

djν e(j −1)y dy.

(107b)

1j n

−x

According to (107a): x ν+2 1 xt a Eν i t + = e + e−ty e 2 y 2 1 2

−x

1

0j n−1

a 2 cjν +1 e(j −1)y dy.

(108)

1

So the function w → a 2 Eν (w + i 12 ) − Eν+1 (w) is a finite linear combination of the n + 1 initial ν+2 Paley–Wiener evaluators Z(−i ν2 , w), Z(−i ν+2 2 , w), . . . , Z(−i 2 − i(n − 1), w). Moreover it 1

has trivial zeros at the trivial zeros of Eν+1 . By Proposition 2 this identifies a 2 Eν (w + i 12 ) − 1

Eν+1 (w) with a constant multiple of K ν+1 (−i ν2 , w), the factor being precisely a 2 c1ν . We prove in this manner the first of the following identities: Proposition 12. There holds:

1 w+i 2

ν −i , w , = Eν+1 (w) + a a Eν 2 1 1 1 ν a − 2 Fν w + i = Fν+1 (w) + a − 2 d1ν K ν+1 −i , w , 2 2 1 ν +1 − 12 − 12 ν+1 ν a Eν+1 w − i = Eν (w) + a cn K −i − in, w , 2 2 1 1 1 ν +1 ν+1 ν 2 2 = Fν (w) + a dn K −i − in, w . a Fν+1 w − i 2 2 1 2

1 2

c1ν K ν+1

(109a) (109b) (109c) (109d)


3243

Proof. The additional three identities are proved by the same method as the first. We will not make direct use of the ensuing relation between the two kinds of evaluators. 2 Let us recall that: Eν+1 (−i ν2 )Eν+1 (it) − Fν+1 (−i ν2 )Fν+1 (it) ν , K ν+1 −i , it = 2 t − ν2 ν+1 Eν (−i( ν+1 ν +1 ν 2 + n))Eν (it) − Fν (−i( 2 + n))Fν (it) K −i . + n , it = 2 t − ν+1 2 −n

(110a) (110b)

In order to shorten the formulas we adopt the notations: ν+1 − in , eν = Eν −i 2 ν+1 fν = Fν −i − in , 2

ν gν+1 = Eν+1 −i , 2 ν hν+1 = Fν+1 −i . 2

Combining (109a) and (110a) gives (with the notation shx (y) = ν t− 2

0j n−1

(111a) (111b)

sh(xy) y )

1 ν ν a 2 cjν +1 2 shx t − − j − cjν+1 2 shx t − − j 2 2

= a c1ν gν+1 Eν+1 (it) − hν+1 Fν+1 (it) .

1j n

1 2

(112)

Using (t − ν2 ) shx (t − ν2 − j ) = sh(x(t − ν2 − j )) + j shx (t − ν2 − j ) we can rearrange and obtain identities by termwise identifications. We record only the identity corresponding to the term with the highest value of j (j = n): 1 n cnν+1 = a 2 c1ν hν+1 dnν+1 − gν+1 cnν+1 .

(113)

Combining similarly (109b) with (110a) we obtain (among other identities!) 1 ndnν+1 = a − 2 d1ν hν+1 dnν+1 − gν+1 cnν+1 .

(114)

Dealing in the same manner with (109c), (109d) and (110b) gives two further identities of a symmetric type. Summing up, the four obtained relations are: Proposition 13. There holds: 1 ncnν+1 = a 2 c1ν hν+1 dnν+1 − gν+1 cnν+1 , 1 ndnν+1 = a − 2 d1ν hν+1 dnν+1 − gν+1 cnν+1 , 1 −nc1ν = a − 2 cnν+1 fν d1ν − eν c1ν , 1 −nd1ν = a 2 dnν+1 fν d1ν − eν c1ν .

(115a) (115b) (115c) (115d)

3244


We will need the following: Proposition 14. None of the eight quantities eν , fν , gν+1 , hν+1 , c1ν , d1ν , cnν+1 , and dnν+1 can vanish. Proof. The proof introduces tacitly a link with techniques of orthogonal polynomial theory, which leads to some results we will expose elsewhere. Let us write explicitly the system of linear j equations for the cν ’s:

∀i = 1 . . . n,

x cjν

1j n

where we recall that κj =

ν+1 2

e(κi +κj )y dy = −e−xκi

(116)

−x

+ j − 1. Letting G be the matrix of this system, we have:

−xκ1 −e −e−xκ2 det(G)c1ν = ... −xκn −e

x (κ +κ )y e 1 2 dy −x x (κ2 +κ2 )y dy −x e . . . x (κ +κ )y n 2 dy −x e

x (κ +κ )y e 1 n dy −x x (κ2 +κn )y dy −x e . x (κ +κ )y n n dy −x e

... ... ... ...

(117)

We now exploit the relation κj +1 = κj + 1 to transform by row manipulations the determinant on the right into: −xκ1 −e 0 0 0

x

(κ1 +κ2 )y dy −x e x (κ +κ 2 2 )y (1 − e −x e −y ) dy −x e

... ... ... ...

x (κ +κ )y . . . −x −y n 2 (1 − e e ) dy −x e x (κ +κ )y y 2 2 (e − e−x )e−y dy −x e −xκ1 = −e ... x (κn +κ2 )y (ey − e−x )e−y dy e −x

x

x

(κ1 +κn )y dy −x e (κ +κ )y −x −y n 2 e (1 − e e ) dy

. . . x (κ +κ )y n n (1 − e −x e −y ) dy −x e x . . . −x e(κ2 +κn )y (ey − e−x )e−y dy . ... ... x (κn +κn )y y −x −y . . . −x e (e − e )e dy −x

(118)

This new determinant is a Gramian for a positive measure, it is strictly positive. So c1ν < 0. It can be proven in the exact same manner d1ν > 0, (−1)n cnν+1 > 0, (−1)n dnν+1 < 0, (−1)n eν > 0, (−1)n fν > 0, gν+1 > 0, and hν+1 > 0. 2 Using the asymptotics for t → −∞ (103b) and (105b) in (109a) (and (110a)) and also in (109c) (and (110b)), and the asymptotics (103a), (105a) (t → +∞) in (109b) and (109d) gives the following proposition: Proposition 15. There holds: 1

1

aμν − μν+1 = 2a 2 c1ν hν+1 = −2a 2 cnν+1 fν , 1 2

1 2

aμν+1 − μν = −2a d1ν gν+1 = 2a dnν+1 eν .

(119a) (119b)


3245

Proposition 16. There holds: c1ν hν+1 = −cnν+1 fν ,

(120a)

d1ν gν+1 ac1ν gν+1

= −dnν+1 eν ,

(120b)

= −cnν+1 eν ,

(120c)

d1ν hν+1

= −adnν+1 fν .

(120d)

Proof. The first two follow from the previous proposition. The last two follow from the first acν

cν+1

and the relation d ν1 = nν+1 (from (115a), (115b)). Alternatively they can be deduced from a dn 1 look at the asymptotics in (109a), (109c) for w = it, t → +∞ and in (109b) and (109d) for t → −∞. 2 Definition 1. We define the quantities X = X(ν, n, a) and Y (ν, n, a) by the following expressions: X :=

gν+1 1 eν = , hν+1 a fν

Y := − From Proposition 15 we have μν = tions in Proposition 13 gives μν =

(121a)

c1ν 1 cnν+1 = − . a dnν+1 d1ν

1

2a 2 (−ahν+1 c1ν 1−a 2

+ gν+1 d1ν ) and using the first two rela-

2na −hν+1 cnν+1 + gν+1 dnν+1 2na X + aY . = 1 − a 2 hν+1 dnν+1 − gν+1 cnν+1 1 − a 2 1 + aXY

Similarly from (119a) and (119b) μν+1 = sition 13 and Definition 1 μν+1 =

(121b)

1

2a 2 (−hν+1 c1ν 1−a 2

(122)

+ agν+1 d1ν ) which gives using Propo-

2na aX + Y . 1 − a 2 1 + aXY

(123)

Theorem 17. The quantities X and Y from Definition 1 are related to the μ-functions μν and μν+1 by the equations: μν =

2na X + aY , 1 − a 2 1 + aXY

μν+1 =

2na aX + Y . 1 − a 2 1 + aXY

(124)

They obey the following non-linear differential system: 2na aX + Y d X = νX − 1 − X 2 , da 1 − a 2 1 + aXY 2na X + aY d . a Y = −(ν + 1)Y + 1 − Y 2 da 1 − a 2 1 + aXY

a

(125a) (125b)

3246


Proof. From X =

gν+1 hν+1

and (93a), (93b): d ν gν+1 = − gν+1 + μν+1 hν+1 , da 2 ν d −a hν+1 = hν+1 + μν+1 gν+1 . da 2 −a

(126a) (126b)

cν

d Hence: a da X = νX − μν+1 (1 − X 2 ). And from Y = − d1ν and (99), (102) we have: 1

d ν ν+1 ν c = c + μν d1ν , da 1 2 1 ν +1 ν d d + μν c1ν −a d1ν = − da 2 1 −a

d Y = −(ν + 1)Y + μν (1 − Y 2 ). and this gives a da

(127a) (127b)

2

Theorem 18. Let (X, Y ) be two functions of a variable a. If they obey the differential system (VI ν,n ): 2na aX + Y d X = νX − 1 − X 2 , da 1 − a 2 1 + aXY 2na X + aY d a Y = −(ν + 1)Y + 1 − Y 2 da 1 − a 2 1 + aXY

a

(128a) (128b)

2 then the quantity q = a aX+Y X+aY satisfies as function of b = a the PVI differential equation:

2 1 1 dq 1 1 dq d 2q 1 1 1 + + + + = − 2 2 q q −1 q −b db b b − 1 q − b db db βb γ (b − 1) δb(b − 1) q(q − 1)(q − b) α+ 2 + + + b2 (b − 1)2 q (q − 1)2 (q − b)2

(129)

−(ν+n+1) n 1−n with parameters (α, β, γ , δ) = ( (ν+n) , 2 , 2 ). 2 , 2 2

2

2

2

Proof. The computation being lengthy we only give some brief indications. It is useful to introduce the variable T :=

1 . 1 + aXY

(130)

It verifies the differential equation: a

2a 2 n 2 2 d T= T Y − X2 . 2 da 1−a

(131)


3247

Let b = a 2 . One has n(q 2 − b) d T = T (T − 1) . db b(q − b)(q − 1)

(132)

On the other hand one establishes: 2n + ν + (ν + 1)b (ν + n)q 2 + (ν + 1 + n)b 2nqT dq = q− + . db b(b − 1) b(b − 1) b

(133)

This equation gives an expression of T in terms of dq db , q, and b. Substituting this value of T in (132) gives a second order differential equation for q. With some tenacity one finally realizes 2 −(ν+n+1)2 2 2 that it is nothing else than PVI with parameters (α, β, γ , δ) = ( (ν+n) , n2 , 1−n 2 , 2 2 ). 2 Remark 7. One also establishes that as functions of b = a 2 , the expressions PVI -transcendents.

−aY X

and a 2 X 2 are

We mention finally the following: Theorem 19. Let (X, Y ) be a solution of the non-linear system (VI ν,n ). Let Z be defined by the following relation (n = 1):

aY + Z = 1 + aY Z

1 ν+1 Y aY + X n − −a . 2 a n−11−Y (n − 1) 1 + aXY

(134)

Then (Y, Z) is a solution of system (VI ν+1,n−1 ). Let W be such that W + aX = 1 + aW X

1 ν X aX + Y n −a . − a n + 1 1 − X 2 (n + 1) 1 + aXY

(135)

Then (W, X) is a solution of system (VI ν−1,n+1 ). Proof. One needs to eliminate X (resp. Y ) from the result of computing a long computation. It can also be confirmed by formal algebra software.

dZ da

(resp. 2

dW da ).

This is

7. Conclusion We assemble some of our main results in the following summary: Theorem 20. Let x > 0, a = e−x , and let PW x be the Paley–Wiener space of entire functions which are square integrable on the real line and of finite exponential type at most x > 0. Let σ = (z1 = −iκ1 , . . . , zn = −iκn ) be a finite sequence of n distinct purely imaginary numbers and

(136)

3248


PW x (σ ) =

f (z) f ∈ PW x , f (z1 ) = · · · = f (zn ) = 0 . 1j n (z − zj )

(137)

The modified space PW x (σ ) is a Hilbert space through its identification with the closed subspace of functions in PW x vanishing on σ . There is a unique entire function Eσ verifying the conditions: 1. Eσ (it) is real for t real, 2. Eσ (0) > 0, 3. limt→+∞ EEσσ(−it) (it) = 0, and in terms of which the scalar products of evaluators in PW x (σ ) are: Kσ (z, w) =

Eσ (z)Eσ (w) − Eσ (z)Eσ (w) . i(z − w)

(138)

Let Fσ (w) = Eσ (w) (= Eσ (−w)). There holds: −a

d Eσ (w) = −iwEσ (w) + μσ (a)Fσ (w). da

(139)

The function μσ (a) admits (among others) the two representations: μσ (a) = a

W (ch(κ1 x), . . . , ch(κn x)) d log da W (sh(κ1 x), . . . , sh(κn x))

(140)

where W (f1 , . . . , fn ) is the Wronskian determinant (with respect to x = log a1 ) and x (κ1 +κ1 )y dy −x e x (κ +κ )y 2 1 dy 2(−1)n −x e μσ (a) = . . . gσ (a) x (κn +κ1 )y dy −x e exκ1

x (κ +κ )y e 1 2 dy −x x (κ2 +κ2 )y dy −x e . . . x (κ +κ )y n 2 dy −x e ...

... ... ... ... ...

x (κ +κ )y e 1 n dy −x x (κ2 +κn )y dy −x e . . . x (κ +κ )y n n dy −x e xκ n e

e−xκ1 e−xκ2 . . . (141) e−xκn 0

x where gσ (a) = det( −x e(κi +κj )y dy)1i,j n is the principal n × n minor of the (n + 1) × (n + 1) determinant. There also holds μσ (a)2 = −a

d d a log gσ (a). da da

(142)

In the specific case where σ is an arithmetic progression: κ1 =

ν +1 , 2

κ2 =

ν+1 + 1, 2

...,

κn =

ν +1 + n − 1, 2

the function μν,n (a) can be expressed as a quotient of two multiple integrals:

(143)


μν,n (a) = 2n

...

− ti )2 i tiν (ti − a)( a1 − ti ) dt1 . . . dtn−1 . . . [a, 1 ]n i<j (tj − ti )2 i tiν dt1 . . . dtn

3249

[a, a1 ]n−1

i<j (tj

(144)

a

and the expression qν,n =

aμν+1,n , μν,n

(145)

as a function of a 2 , verifies the Painlevé VI equation with parameters (α, β, γ , δ) =

(ν + n)2 −(ν + n + 1)2 n2 1 − n2 , , , . 2 2 2 2

(146)

1 Proof. We have established (138) with Eσ (it) = 1in t+κ Eσ (it) and Eσ (it) given by equai tion (94a). In particular Eσ hence Eσ is real on the imaginary axis. One has Eσ (it) ∼t→+∞ ext , hence Eσ (it), which by (138) cannot vanish for t > 0, is positive for t > 0. It is easy to prove that given arbitrary points w1 , . . . , wm and non-negative integers n1 , . . . , nm there is in PW x a function vanishing exactly to the order nj at wj for all j . So in PW x (σ ) the evaluator at z = 0 is non-zero, which proves Eσ (0) = 0, hence Eσ (0) > 0. As Eσ is real on the imaginary line one has σ (it) Fσ (w) = Eσ (−w). And from the representations (94a) and (94b) we know F Eσ (it) →t→+∞ 0. Let E be another function computing the reproducing kernel and with the properties (1), (2) and (3). Let A = 12 (E + E ∗ ), B = 2i (E − E ∗ ), which are respectively even and odd, real on the real line. The evaluator at the origin (as said above, necessarily non-zero) is Kσ (0, w) = 2A(0) w1 B(w), hence the function B is known up to a positive real multiple, then the function A is known up to (it) (the inverse of) this multiple. Condition (3) can be written as −iAB(it) →t→+∞ 1, and this finally identifies A = Aσ and B = Bσ . Formula (140) (in which κi + κj = 0 is assumed) follows from Theorem 9, and formula (142) from Theorem 9 and Proposition 4. According to (98) one has μσ (a) = 2(−1)n 1j n cjσ eκj x , the coefficients cjσ solving:

∀i = 1 . . . n,

1j n

x cjσ

e(κi +κj )y dy = −e−xκi .

(147)

−x

This gives the representation (141). In the case of the arithmetic progression of reason one, row manipulations replace the (n + 1) × (n + 1) determinant by x (κ1 +κ1 )y (1 − a1 ey ) dy −x e x (κ2 +κ1 )y (1 − 1 ey ) dy −x e a −xκn −e ... x e(κn−1 +κ1 )y (1 − 1 ey ) dy −x a exκ1

... ... ... ... ...

x

e(κ1 +κn )y (1 − a1 ey ) dy (κ2 +κn )y (1 − 1 ey ) dy −x e a . . . . x (κ +κ )y 1 y n n−1 e (1 − e ) dy −x a xκ n e −x x

(148)

3250


Column manipulations lead to: x (κ1 +κ1 )y (1 − aey )(1 − a1 ey ) dy −x e x e(κ2 +κ1 )y (1 − aey )(1 − a1 ey ) dy − −x ... x (κn−1 +κ1 )y e (1 − aey )(1 − a1 ey ) dy −x a1 = (−1)

n

det

t

1i,j n−1

ν+i+j −2

...

x

e(κ1 +κn−1 )y (1 − aey )(1 − −x x (κ2 +κn−1 )y (1 − aey )(1 − −x e

1 y a e ) dy 1 y a e ) dy

... ... ... x . . . −x e(κn−1 +κn−1 )y (1 − aey )(1 − a1 ey ) dy

1 − t (t − a) dt . a

(149)

a

So we have the representation of μν,n as a quotient of two Gramians (for n = 1 the determinant at the numerator is taken to be 1): 1 det1i,j n−1 ( aa t ν+i+j −2 ( a1 − t)(t − a) dt) . μν,n (a) = 2 1 det1i,j n ( aa t ν+i+j −2 dt) This gives formula (144). Finally the Painlevé VI assertion follows from Theorems 17 and 18.

(150)

2

Remark 8. The change of variables tj = a + ( a1 − a)u transforms the multiple integrals at the numerator and denominator of expression (144) into, respectively:

1 −a a

n2 −1

a (n−1)ν

...

[0,1]n−1 i<j

(uj − ui )2

(1 + αui )ν ui (1 − ui ) du1 . . . dun−1 i

(151) and

1 −a a

n2

a nν

...

[0,1]n i<j

(uj − ui )2

(1 + αui )ν du1 . . . dun i

(152) with α = a12 − 1. Similar multiple integrals arise in the work of Forrester and Witte [8] relating the “Jacobi unitary ensemble” of random matrices to the Okamoto hamiltonian formulation of the Painlevé VI equation [14]. It seems however (cf. the introduction of [9]) that in this context of multiple integrals one had so far not yet encountered directly Painlevé VI transcendents per se, but rather solutions to Okamoto’s “σ -equation”. Remark 9. We see from (150) or (141) that μν,n (a) is the product of a ν+1 with a rational function μ is a rational function (with rational coefficients) of b of a 2 and a 2ν . The quantity q = a μν+1,n ν,n and bν (b = a 2 ). In particular, for ν ∈ Z, q is a rational function of b.

Remark 10. We have left aside most of the developments which have their origins in [3,4] and which put the objects studied here in another context (which leads in particular to various further representations for the μ-functions), indeed a context which presided over their introduction. We have also left aside a number of other developments related to techniques of orthogonal


3251

polynomial theory, multiple integrals and non-linear relations. We hope to address these topics in further publications. Acknowledgments All results included in this manuscript were obtained between January and late April 2008, while I was spending a seven months visit at the I.H.E.S. Numerous conversations on various (mostly other) topics with permanent members, visitors and staff made this a particularly stimulating and memorable residence. I also benefited during this period from electronic exchanges with Philippe Biane. I had found non-linear equations in the framework I had developed to establish the Fourier transform as a scattering [3,4] and Philippe Biane’s question about what this would give when the Gamma function was replaced by a finite rational expression gave a welcome impetus to my efforts of that time. I discovered in the process of answering this question the non-linear equations (125a), (125b), from which the previous ones derived by a process of confluence, and I related them to Painlevé VI. I thus thank him for many interesting interactions (for example, in relation with the Wronskians built with the trigonometric functions, which he had also encountered in his own enterprises), and for having then kept me informed of some of his works [1]. I also thank the organizers Joaquim Burna, Håkan Hedenmalm, Kristian Seip, and Mikhail Sodin of the mini-symposium on Hilbert spaces of entire functions (EMS congress, Amsterdam 2008) for having given me the opportunity to present there aspects of this material. Finally, I would like to express special thanks to Michel Balazard for having made possible my participation to the Zeta Functions congresses I, II, and III in Moscow, in 2006, 2008, and 2010, which gave me also opportunities to present related material. References [1] Ph. Biane, Orthogonal polynomials on the unit circle, q-Gamma weights, and discrete Painlevé equations, arXiv: 0901.0947, July 2010 (v1 January 2009), 26 pp. [2] L. de Branges, Hilbert Spaces of Entire Functions, Prentice Hall Inc., Englewood Cliffs, 1968. [3] J.-F. Burnol, Des équations de Dirac et de Schrödinger pour la transformation de Fourier, C. R. Acad. Sci. Paris Ser. I 336 (2003) 919–924. [4] J.-F. Burnol, Scattering, determinants, hyperfunctions in relation to (1−s) (s) , arXiv:math.NT/0602425, February 2006, 63 pp. [5] J.-F. Burnol, Hilbert spaces of entire functions with trivial zeros, arXiv:1008.0518 [math.FA], July 2010, 10 pp. [6] M.M. Crum, Associated Sturm–Liouville systems, Quart. J. Math. Oxford (2) 6 (1955) 121–127. [7] G. Darboux, Sur une proposition relative aux équations linéaires, C. R. Acad. Sci. 94 (1882) 1456–1459. [8] P.J. Forrester, N.S. Witte, Application of the τ -function theory of Painlevé equations to random matrices: PVI , the JUE, CyUE, cJUE and scaled limits, Nagoya Math. J. 174 (2004) 29–114. [9] P.J. Forrester, N.S. Witte, Random matrix theory and the sixth Painlevé equation, J. Phys. A 39 (2006) 12211–12233. [10] M.G. Krein, On the determination of the potential of a particle from its S-function, Dokl. Akad. Nauk SSSR 105 (3) (1955) 433–436. [11] M.G. Krein, Continual analogues of propositions on polynomials orthogonal on the unit circle, Dokl. Akad. Nauk SSSR 105 (4) (1955) 637–640. [12] A. Lascoux, Pfaffians and representations of the symmetric group, Acta Math. Sinica 25 (2009) 1929–1950. [13] S. Okada, Applications of minor summation formulas to rectangular-shaped representations of classical groups, J. Algebra 205 (2) (1998) 337–367. [14] K. Okamoto, Studies on the Painlevé equations. I. Sixth Painlevé equation PVI , Ann. Mat. Pura Appl. (4) 146 (1987) 337–381.


Two-parameter families of quantum symmetry groups Teodor Banica a , Adam Skalski b,c,∗ a Department of Mathematics, Cergy-Pontoise University, 95000 Cergy-Pontoise, France b Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, United Kingdom c Mathematical Institute of the Polish Academy of Sciences, ul. Sniadeckich ´ 8, 00-956 Warszawa, Poland

Received 24 September 2010; accepted 29 November 2010 Available online 16 December 2010 Communicated by D. Voiculescu

Abstract We introduce and study natural two-parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups. Specifically, for each pair (p, q) of non-negative integers we define and investigate quantum groups O + (p, q), B + (p, q), S + (p, q) and H + (p, q) corresponding to, respectively, orthogonal groups, bistochastic groups, symmetric groups and hyperoctahedral groups. In the first three cases the new quantum groups turn out to be related to the (dual free products of ) free quantum groups studied earlier. For H + (p, q) the situation is different and we show that H + (p, 0) ≈ QISO(F p ), where the latter can be viewed as a liberation of the classical isometry group of the p-dimensional torus. © 2010 Elsevier Inc. All rights reserved. Keywords: Quantum symmetry groups; Quantum isometry groups; Liberation; Representation theory of quantum groups; Tannakian categories

0. Introduction Compact quantum groups have entered mathematics in late 1980s (see [27,17] and references therein). Recent years have brought an increased interest in investigating quantum groups as quantum symmetry or isometry groups of classical or quantum spaces [2,5,16,13,8]. One particular approach to constructing quantum symmetry groups is the so-called ‘liberation’ of classical * Corresponding author at: Mathematical Institute of the Polish Academy of Sciences, ul. Sniadeckich ´ 8, 00-956 Warszawa, Poland. E-mail addresses: [email protected] (T. Banica), [email protected] (A. Skalski).


T. Banica, A. Skalski / Journal of Functional Analysis 260 (2011) 3252–3282

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Table 1 Quantum group G Algebra C(G) Cat. of reps. Classical version

O + (p, q) Ao (2p + q) NC2 O2p+q

B + (p, q) Ab (p, q) NC12 O2p+q−i(p,q)

S + (p, q) Ah (p) As (q) NC• NC Hp × Sq

H + (p, q) Ah (p, q) NCeven (Tp Hp ) × Hq

Table 2 Quantum group G

O + (p, 0)

B + (p, 0)

S + (p, 0)

H + (p, 0)

Quantum symmetry group of Classical version

S2p

2p S−

[0, 1]p

O2p

O2p−1

Hp

F p Tp Hp

Classical symmetry group of

S 2p

S−

[0, 1]p

Tp

2p

compact groups. This technique, developed by the first named author and his collaborators, is based on choosing a suitable collection of relations satisfied by the functions on the group in question and then relaxing the commutativity assumptions [9]. On the other hand in recent work of Bhowmick and the second named author (motivated by [16]) a quantum isometry group has been associated with the (dual of ) each finitely generated discrete group; in particular the quantum isometry groups of the duals of the free groups were computed. The last result and the form of the obtained quantum isometry groups suggested considering a general framework in which a variation of universal quantum orthogonal groups of [21] is realised by replacing the usual selfadjointness of entries by imposing specific relations between entries and their adjoints. Although it turns out that this new choice actually does not introduce nontrivial modifications on the level of quantum orthogonal groups, the situation changes if we consider quantum symmetric groups [24] or quantum hyperoctahedral groups [6]. In this paper we present a full study of these deformations and cast them in the language of ‘liberations’ studied in [9]. As we are going to consider at the same time deformed adjoint relations and the usual selfadjointness conditions on some other entries, we will throughout the paper work with two parameters, p (representing the deformed relations) and q (the standard ones). These parameters are assumed to be non-negative integers, not simultaneously equal to 0. As the deformed relations involve pairs of coordinates, the quantum groups that we study will have ‘rank’ (the dimension of the fundamental unitary representation) equal to 2p + q. We will consider deformed quantum versions of orthogonal, bistochastic, symmetric and hyperoctahedral groups. Main body of the results obtained in this paper can be summarised in Table 1 (the categories of representations are described in terms of non-crossing, possibly ‘bulleted’, partitions, with the details given in Section 4; i(p, q) is defined to be equal to 1 if pq = 0, and to 2 if pq > 0). As explained earlier, if p = 0 we obtain the ‘liberated objects’ studied in [9]. On the other hand when q = 0 we obtain Table 2 (in particular H + (p, 0) is the quantum isometry group of the C ∗ -algebra of the free group Fp discovered in [14] and in a way providing a starting point for the considerations in this work). Above S 2p and S2p denote respectively the usual sphere in R2p and the free sphere studied 2p 2p in [8], and S− and S− denote the respective spheres with one coordinate fixed. With the isomorphisms established in this paper the first two columns can be deduced from results in [8] (and the third is a consequence of [6]). The detailed plan of the paper is as follows: in Section 1 we quote basic definitions, establish some terminology related to compact quantum groups and recall the quantum (free) symme-

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try groups corresponding to orthogonal, symmetric, bistochastic and hyperoctahedral groups. In Section 2 the study of their two-parameter counterparts begins with the analysis of the orthogonal, bistochastic and symmetric quantum groups denoted respectively O + (p, q), B + (p, q) and S + (p, q). It turns out that all of them can be described in terms of (the free products of ) the one-parameter versions. Section 3 contains a detailed analysis of the two-parameter quantum hyperoctahedral group H + (p, q); in particular we show that H + (p, 0) coincides with the quantum isometry group of the dual of the free group discovered in [14]. Section 4 is devoted to establishing the description of the categories of representations of our quantum groups in terms of non-crossing (marked) partitions. Further in Section 5 theorems proved in Sections 2–4 are used to analyse the relations between the quantum groups studied in the paper: we investigate the generation results, intersection results and inclusions of the form G(p, 0) ˆ G(0, q) ⊂ G(p, q). We also show there a fact conjectured in [14]: the two-parameter quantum hyperoctahedral group + . In H + (p, q) may be viewed as a quantum extension of the quantum symmetric group S2p+q Section 6 we describe the classical versions of the quantum groups we study to show that each of these quantum groups has a natural description as a liberation of a classical symmetry group. Finally in the last section we discuss in what sense the family of quantum groups described in this paper exhausts the natural two-parameter construction presented in Sections 2–3 and in the process discover another two-parameter quantum group, Hs+ (p, q) which turns out to be isomorphic to the dual free product of Hp+4 and Hq+ . 1. Compact quantum groups – definitions and notation In this section we recall the definition of a compact quantum group due to Woronowicz and introduce quantum orthogonal, symmetric, hyperoctahedral and bistochastic groups. The minimal tensor product of C ∗ -algebras will be denoted by ⊗, algebraic tensor products by . For n ∈ N we denote the algebra of n by n complex matrices by Mn . Definition 1.1. Let A be a unital C ∗ -algebra and : A → A ⊗ A be a unital ∗ -homomorphism satisfying the coassociativity condition: ( ⊗ idA ) = (idA ⊗ ). If additionally (A)(1 ⊗ A) = (A)(A ⊗ 1) = A ⊗ A we say that A is the algebra of continuous functions on a compact quantum group G and usually write A = C(G). A unique dense Hopf ∗ -subalgebra of C(G), the algebra of coefficients of finite-dimensional unitary representations of G, will be denoted by R(G). If G is a compact quantum group and n ∈ N then a unitary matrix U = (Uij )ni,j =1 ∈ Mn (C(G)) is called a fundamental representation of G (or a fundamental corepresentation of C(G)) if for each i, j = 1, . . . , n (Uij ) =

n

Uik ⊗ Ukj

k=1

and the entries of U generate C(G) as a C ∗ -algebra. If G admits a fundamental representation, it is called a compact matrix quantum group. This will be the case for all quantum groups considered in this paper; in fact they will be defined via their respective fundamental representations.


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In general there is an ambiguity in passing from R(G) to C(G) related to the fact that not all compact quantum groups are coamenable. As all quantum groups studied in this paper will be defined by universal properties, we will assume that C(G) is the universal completion of R(G) [11]. We will later need a free product construction introduced in [23]. If G1 , G2 are compact quantum groups, then the C ∗ -algebraic free product C(G1 ) C(G2 ) has a natural structure of the algebra of continuous functions on a compact quantum group, to be denoted G1 ˆ G2 . In particular if U1 ∈ Mn (C(G1 )) and U2 ∈ Mn (C(G2 )) are respective fundamental corepresenta tions, then U01 U02 ∈ Mn (C(G1 ) C(G2 )) is the fundamental corepresentation of C(G1 ˆ G2 ). The construction is dual to the usual free product of discrete groups: when the quantum groups in question are duals of classical discrete groups, G1 = Γ1 , G2 = Γ2 , then G1 ˆ G2 ≈ Γ 1 Γ2 . Let n ∈ N; it will denote the dimension of the fundamental representation of the compact quantum groups defined below. The following definition comes from [21]; here we recast it in the language described above. Definition 1.2. Let F ∈ Mn be an invertible matrix such that F F¯ = cIn for some c ∈ C. Let Ao (F ) denote the universal C ∗ -algebra generated by the entries of a unitary U ∈ Mn ⊗ Ao (F ) such that U = (F ⊗ 1)U¯ F −1 ⊗ 1

(1.1)

(here and in what follows a bar over the matrix denotes a matrix obtained by an entrywise conjugation of entries). When U ∈ Mn ⊗ Ao (F ) is interpreted as the fundamental unitary corepresentation, we can view Ao (F ) as the algebra of continuous functions on the compact quantum group denoted by O + (F ). In particular if F = In , we write simply Ao (n) and On+ instead of Ao (In ) and O + (In ). Recall that if A is a C ∗ -algebra then a unitary matrix U ∈ Mn (A) is called a magic unitary if each entry of U is a self-adjoint projection. A unitary U ∈ Mn (A) is called cubic if its entries are selfadjoint and the products of each pair different entries lying in the same row or column are 0. The following definitions come respectively from [24] and [6]. Definition 1.3. Denote by As (n) the universal C ∗ -algebra generated by the entries of an n by n magic unitary U . When U ∈ Mn ⊗ As (n) is interpreted as the fundamental unitary corepresentation, we view As (n) as the algebra of continuous functions on the quantum permutation group on n elements, Sn+ . Definition 1.4. Denote by Ah (n) the universal C ∗ -algebra generated by the entries of an n by n cubic unitary U . When U ∈ Mn ⊗ Ah (n) is interpreted as the fundamental unitary corepresentation, we view Ah (n) as the algebra of continuous functions on the quantum hyperoctahedral group on n coordinates, Hn+ . The quantum groups On+ , Sn+ and Hn+ are also called the free orthogonal quantum group, the free symmetric quantum group and the free hyperoctahedral quantum group and can be respectively viewed as liberations of the compact groups On , Sn and Hn [9]. More information about their properties, including their interpretations as quantum symmetry groups can be found in that paper and also respectively in [8,5,6].

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The following definition was introduced in [9]. Definition 1.5. Let n ∈ N. Denote by Ab (n) the universal C ∗ -algebra generated by the entries of an n by n unitary U , which has selfadjoint entries which sum to 1 in each row/column. When U ∈ Mn ⊗ Ab (n) is interpreted as the fundamental unitary corepresentation, we view Ab (n) as the algebra of continuous functions on the quantum bistochastic group in n dimensions, Bn+ . The condition on the sum of entries in each row/column being equal to 1 is equivalent to stating that the vector [1, . . . , 1]t is an eigenvector for both U and U t . This observation leads to + [19]. a natural isomorphism Bn+ ≈ On−1 ∗ Relations between the C -algebras and quantum groups introduced above can be expressed via the following diagrams (arrows on the level of algebras denote surjective unital ∗ -homomorphisms intertwining the respective coproducts): Ao (n)

↓

∪

As (n)

Hn+

↓ Ah (n)

On+

→ Ab (n)

→

⊃ Bn+ ∪ ⊃

(1.2)

Sn+

The diagram on the right suggests a number of questions, for instance whether On+ = or whether Sn+ = Bn+ ∩ Hn+ (at the level of classical versions, the answers are yes and yes). Once these questions are properly formulated, for instance in terms of tensor categories, the answers turn out to be positive, and can be deduced from [9]. We will describe the details later, when we discuss similar problems in a more general 2-parameter framework. Bn+ , Hn+ ,

2. Quantum groups O + (p, q), B + (p, q) and S + (p, q) In this section we describe the two-parameter families of quantum groups generalising the quantum orthogonal, bistochastic and symmetric groups described in Section 1. We begin by introducing necessary notations. ρ ρ7 2πi Let p, q ∈ N0 := N ∪ {0}, p 2 + q 2 > 0 and let ρ = e 8 . Put F = 01 10 , C = √1 3 5 , let 2 ρ ρ Fp,0 ∈ M2p be the matrix given by ⎛

F ⎜0 . Fp,0 = ⎜ ⎝ ..

0 F .. .

0

0

··· ··· .. . ···

Fp,0 0

0 Iq

⎞ 0 0⎟ .. ⎟ . ⎠ F

and define Fp,q ∈ M2p+q by Fp,q =

.

(2.1)

The matrix Fp,q is a selfadjoint unitary, and a permutation matrix; moreover Fp,q = (Fp,q )t = (Fp,q )∗ . The matrix Cp,q ∈ M2p+q is defined in an analogous way, replacing F by C in appropriate matrix blocks. Note that C, so also Cp,q , is unitary.


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Whenever we consider matrices of the size 2p + q we will denote the indices corresponding to the ‘p-part’ by pairs iα, where i ∈ {0, 1} and α ∈ {1, . . . , p} and to the ‘q-part’ by capital Latin letters running from 1 to q. Moreover we will use the notation i¯ (where 0¯ = 1, 1¯ = 0, so that i¯¯ = i). This facilitates the description of the fact that coordinates in the ‘p-part’ come naturally in pairs. To simplify the notation we will also write Jp = {iα: i ∈ {0, 1}, α ∈ {1, . . . , p}}, ¯ if Jp,q = Jp ∪ {1, . . . , q}, and extend the ‘bar’ notation to indices in Jp,q , writing z¯ = iα z = iα ∈ Jp and z¯ = M if z = M ∈ {1, . . . , q}. The canonical basis in C2p+q will be often denoted by (ez )z∈Jp,q . + 2.1. Quantum group O + (p, q) ≈ O2p+q

As the matrix Fp,q defined in (2.1) satisfies the conditions listed in Definition 1.2 we can consider Ao (p, q) := Ao (Fp,q ) and O + (p, q) := O + (Fp,q ). Clearly O + (0, q) ≈ Oq+ ; in fact + the discussion in Section 5 of [15] implies that O + (p, q) ≈ O2p+q . From our point of view it is important to consider the following rephrasing of the above definitions: Theorem 2.1. The algebra Ao (p, q) is the universal C ∗ -algebra generated by elements {Uz,y : z, y ∈ Jp,q } such that the resulting 2p + q by 2p + q matrix U is unitary and for each iα, jβ ∈ Jp , M, N ∈ {1, . . . , q} we have ∗ = Uiα, Uiα,jβ ¯ j¯β ,

(2.2)

∗ = Uiα,N , Uiα,N ¯

(2.3)

∗ = UM,j¯β , UM,jβ

(2.4)

∗ = UM,N . UM,N

(2.5)

Moreover Ao (p, q) with U viewed as a fundamental corepresentation is the algebra of continu+ ous functions on a compact quantum group O + (p, q) ≈ O2p+q . Proof. The first part is a direct consequence of the fact that Fp,q has a block-matrix form and formulas:

0 1 1 0

A∗ C∗

∗ C B∗ = D∗ A∗

D∗ , B∗

A∗ C∗

B∗ D∗

∗ 0 1 B = 1 0 D∗

A∗ C∗

and

0 1 1 0

A∗ C∗

B∗ D∗

∗ 0 1 D = 1 0 B∗

C∗ , A∗

which imply that if U = (Uz,y )z,y∈Jp,q then U = (Fp,q ⊗ 1)U¯ (Fp,q ⊗ 1) if and only if the entries of U satisfy relations (2.2)–(2.5). The second part follows from the discussion before the theorem, but can be also seen directly: indeed, exploiting the equalities of the type

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ρ ρ7 ρ(A + ρ 6 C) A , = C ρ 3 (A + ρ 2 C) ρ3 ρ5 7 ρ ρ5 = ρ(ρ 6 A + B) ρ 3 (ρ 2 A + B) [A B ] 3 ρ ρ

and

ρ ρ3

ρ7 ρ5

A C

B D

ρ7 ρ

ρ5 A − iC + iB + D = iA − C − B − iD ρ3

−iA − C − B + iD A + iC − iB + D

one can check that if A is a C ∗ -algebra and U ∈ M2p+q (A) is a unitary matrix, then the entries of U satisfy conditions displayed in the theorem if and only if the entries of the unitary matrix (Cp,q ⊗ 1A )U (Cp,q ∗ ⊗ 1A ) are selfadjoint. 2 Note that in terms of the notation introduced earlier the relations (2.2)–(2.5) can be summarised by saying Uz,y = Uz¯∗,y¯ ,

z, y ∈ Jp,q .

(2.6)

It is well known that algebraic relations between entries of the fundamental representation U of a given quantum group can be interpreted as declaring certain scalar matrices to be elements of Hom(U ⊗k ; U ⊗l ) for some k, l ∈ N (see for example [1]). Proposition 2.2. The algebra Ao (p, q) is the universal C ∗ -algebra generated by the entries of a q unitary 2p + q by 2p + q matrix U such that the vector ξ := iα∈Jp eiα ⊗ eiα ¯ + M=1 eM ⊗ eM

is a fixed vector for U ⊗2 (in other words the map 1 → ξ is an element of Hom(1; U ⊗2 )). Proof. Let (Uz,y )z,y∈Jp,q be unitary. Then U ⊗2 ξ = ξ if and only if

z,y∈Jp,q

Uz,iα Uy,iα ¯ +

iα∈Jp

q

Uz,M Uy,M ez ⊗ ey = ξ.

M=1

This means that for example for each jβ ∈ Jq and y ∈ Jp,q we must have

Ujβ,iα Uy,iα ¯ +

iα∈Jp

q

j¯β

Ujβ,M Uy,M = δy .

M=1

∗ on the right and sum over y ∈ J Fix z ∈ Jp,q , multiply the last formula by Uy,z p,q . As U is unitary, this yields

iα∈Jp

z Ujβ,iα δiα ¯ +

q M=1

z Ujβ,M δM = Uj∗¯β,z .


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Considering respectively z ∈ {1, . . . , q} and z ∈ Jp yields the first two displayed formulas in Theorem 2.1. It is easy to see that the third and fourth formulas can be obtained in an analogous way. 2 The last proposition allows one to describe the tensor category corresponding to the dual of O + (p, q) in terms of Temperley–Lieb diagrams [1] or non-crossing pair partitions [9]. We refer to these papers for details; in Section 4 below we state the corresponding results in terms of partitions. + 2.2. Quantum group B + (p, q) ≈ Op+q−i(p,q)

Definition 2.3. The algebra Ab (p, q) is the universal C ∗ -algebra generated by elements {Uz,y , z, y ∈ Jp,q } which satisfy all the relations required of generators of Ao (p, q) and additionally are such that entries in each row/column of the resulting unitary U sum to 1: for all y ∈ Jp,q z∈Jp,q

Let η = tion.

z∈Jp,q ez

Uz,y =

Uy,z = 1.

z∈Jp,q

∈ C2p+q and recall the matrix Cp,q defined in the beginning of this sec-

Theorem 2.4. The algebra Ab (p, q) is the algebra of continuous functions on a compact quantum group, denoted further B + (p, q). The unitary U = (Uz,y )z,y∈Jp,q ∈ M2p+q ⊗ Ab (p, q) is the fundamental representation of B + (p, q). The algebra Ab (p, q) is isomorphic to the universal C ∗ -algebra generated by entries of a 2p + q by 2p + q unitary V which is orthogonal and satisfies the condition V (Cp,q η ⊗ 1) = Cp,q η ⊗ 1. Proof. As explained after Definition 1.5 the condition that the entries in each row and column of a matrix U sum to 1 are equivalent to the fact that the vector η is fixed both by U and U t . This observation (or a direct computation) implies that Ab (p, q) is the algebra of continuous functions on a compact quantum group. Further note that as Fp,q η = η and Fp,q is a selfadjoint unitary we have the following string of equivalences for a unitary U ∈ Mn (Ao (p, q)) satisfying the condition (1.1) with F = Fp,q : U (η ⊗ 1) = η ⊗ 1

⇔

U ∗ (η ⊗ 1) = η ⊗ 1

⇔

U t (η ⊗ 1) = η ⊗ 1,

⇔

(Fp,q ⊗ 1)U t (Fp,q ⊗ 1)(η ⊗ 1) = η ⊗ 1

so that the condition on the sum of entries in each column of a unitary U as above being equal to 1 follows from the analogous condition for rows. In the last part of the proof of Theorem 2.1 we noticed that the transformation between the + fundamental unitary in O + (p, q) and that of O2p+q is implemented by conjugating with the unitary matrix Cp,q . Hence to prove the last statement it suffices to note that a unitary U fixes ∗ ⊗ 1) fixes C the vector η if and only if (Cp,q ⊗ 1)U (Cp,q p,q η. 2

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When q = 0 or p = 0 the above theorem implies that the ‘deformed’ quantum bistochastic group coincides with the one studied in [9]. Theorem 2.5. The following isomorphisms hold: + + ≈ O2p−1 , B + (p, 0) ≈ B2p

+ B + (0, q) ≈ Bq+ ≈ Oq−1 ,

+ B + (p, q) ≈ O2p+q−2

for pq > 0,

B + (p, 1) ≈ B + (p, 0). + (the Proof. The only isomorphism in the first line that needs to be proved is B + (p, 0) ≈ B2p ones involving the quantum orthogonal groups are the consequences of [19], as explained after Definition 1.5). Due to Theorem 2.4 the fundamental representation of B + (p, 0) can be defined as a 2p by 2p unitary matrix U with selfadjoint entries which satisfies the condition U (Cp,0 η ⊗ 1) = Cp,0 η ⊗ 1. It is clear that in the above condition Cp,0 η can be replaced by a non-zero scalar multiple; in particular by the vector [1, −1, . . . , 1, −1]t . But that vector can be mapped by a real orthogonal matrix onto η, so also onto [1, 0, 0, . . . , 0]t , and the argument of [19] implies that the desired isomorphism holds. Assume now that pq > 0. Using once again Theorem 2.4 a fundamental representation of B + (p, q) can be defined as a 2p + q by 2p + q unitary matrix U with selfadjoint entries which satisfies the condition U (Cp,q η ⊗ 1) = Cp,q η ⊗ 1. One can check that the vector Cp,q η is proportional to the vector ηp,q ∈ C2p+q given by

ηp,q = [ 1 − i, i − 1, . . . , 1 − i, i − 1, 1 + i, . . . , 1 + i ]T 2p times

q times

= [ 1, −1, . . . , 1, −1, 1, . . . , 1 ] + i[ −1, 1, . . . , −1, 1, 1, . . . , 1 ]T . T

2p times

q times

2p times

q times

As U has selfadjoint entries, it preserves ηp,q if and only if it preserves its real and imaginary parts; equivalently, it preserves vectors [ 1, −1, . . . , 1, −1, 0, . . . , 0 ]T 2p times

q times

and [ 0, 0, . . . , 0, 0, 1, . . . , 1 ]T . 2p times

q times

Repeating an earlier argument we find a matrix in O2p × Oq ⊂ O2p+q mapping these vectors respectively to [1, 0, . . . , 0]T and [0, . . . , 0, 1]T ; this provides the isomorphism B + (p, q) ≈ + + O2p+q−2 , which implies in particular that B + (p, 1) ≈ Bp,0 . 2


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Note that the argument in the second part of the above proof can be framed in general terms – for n 3 the group of n by n real orthogonal matrices preserving a fixed non-zero vector v ∈ Cn is either isomorphic to On−1 (when the real and imaginary parts of v are proportional to each other) or to On−2 (when they are not); the same applies to On+ . The following result is a consequence of Proposition 2.2 and the arguments in the proof of Theorem 2.4. Corollary 2.6. The algebra Ab (p, q) is the universal C ∗ -algebra generated by the entries of a unitary 2p + q by 2p + q matrix Usuch that the vector ξ defined in Proposition 2.2 is a fixed vector for U ⊗2 and the vector η = z∈Jp,q ez is a fixed vector for U . The above corollary implies that the diagrammatic description of the category of representations of B + (p, q) coincides with that of B + (2p + q) [9], see Section 4. 2.3. Quantum group S + (p, q) ≈ Hp+ ˆ Sq+ The free quantum group of permutations of n elements may be viewed as the universal C ∗ -algebra generated by entries of an orthogonal matrix which are additionally required to be orthogonal projections. This motivates the following definition. Definition 2.7. The algebra As (p, q) is the universal C ∗ -algebra generated by self-adjoint projections {Uz,y , z, y ∈ Jp,q } which satisfy all the relations required of generators of Ao (p, q). As for every magic unitary entries lying in the same row or column are pairwise orthogonal, the generators of As (p, q) satisfy the following relations: ∗ Uz,y Uz,x = Uz,y Uz,x = 0,

∗ Uy,z Ux,z = Uy,z Ux,z = 0,

z, y, x ∈ Jp,q , x = y.

Proposition 2.8. The algebra As (p, q) is the universal C ∗ -algebra generated by two families of self-adjoint projections {Uiα,jβ : iα, jβ ∈ Jp } and {UM,N ; M, N ∈ {1, . . . , q}}, such that both q matrices (Uiα,jβ )iα,jβ∈Jp and (UM,N )M,N =1 are magic unitaries and moreover Uiα,jβ = Uiα, ¯ j¯β ,

iα, jβ ∈ Jp .

(2.7)

Proof. It suffices to show that whenever iα ∈ Jp and M ∈ {1, . . . , q} then Uiα,M = 0 = UM,iα . But the matrix {Uz,y , z, y ∈ Jp,q } ∈ M2p+q (As (p, q)) is a magic unitary, so each of its columns consists of mutually orthogonal projections, and as we have Uiα,M = Uiα,M , it follows that ¯ Uiα,M = 0. The second equality follows from the orthogonality of projections in each row of a magic unitary. 2 Theorem 2.9. The algebra As (p, q) is the algebra of continuous functions on a compact quantum group, denoted further S + (p, q). The unitary U = (Uz,y )z,y∈Jp,q ∈ M2p+q ⊗ As (p, q) is the fundamental representation of S + (p, q). The algebra As (p, q) is isomorphic to the free product Ah (p) As (q); on the level of quantum groups we have S + (p, q) ≈ Hp+ ˆ Sq+ .

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Proof. Due to Proposition 2.8 it suffices to consider separately the cases p = 0 and q = 0; as we have As (0, q) ≈ As (q) (with the isomorphism preserving natural fundamental corepresentations), we can assume that q = 0. Further it suffices to show that if we define for each iα, jβ ∈ Jp iα,jβ := U

Uiα,kγ ⊗ Ukγ ,jβ

kγ ∈Jp

iα,jβ is a self-adjoint projection (the fact that the conditions in (2.7) will then be then each U satisfied follows from Theorem 2.1; and analogous statements for the potential antipode and counit follow from the fact that the adjoint of a projection is a projection and 0, 1 ∈ C are projections). The last statement is however a direct consequence of orthogonality of the rows/columns iα,jβ is a of the magic unitary; it can also be deduced from the fact that the map Uiα,jβ → U ∗ -homomorphism. Proposition 2.8 implies also that As (p, q) ≈ As (p, 0) As (0, q). Therefore it suffices to show that As (p, 0) ≈ Ah (p). This is however an immediate consequence of the fact that Ah (p) can be defined via the requirement that its fundamental corepresentation is a 2p by 2p sudoku unitary, a b i.e. a magic unitary which has a block-matrix form b a (Definition 5.2 and Theorem 6.2 of [6]). Indeed, formulas (2.2)–(2.5) imply that the fundamental unitary representation of S + (p, 0) is a magic unitary of the form ⎤ ⎡ A B C D E F ··· ⎢ B A D C F E ···⎥ ⎥ ⎢ ⎢ G H I J K L ···⎥ ⎥ ⎢ ⎢H G J I L K ···⎥, ⎥ ⎢ ⎢M N O P Q R ···⎥ ⎥ ⎢ ⎣ N M P O R Q ···⎦ .. .. .. .. .. .. . . . . . . . . . which can be transformed into a sudoku unitary by permuting rows and columns (so that odd rows/columns remain in the same order but are shifted to the left/up so that they become first p rows/columns). 2 The next proposition facilitates the description of the category of the representations of S + (p, q) in terms of the non-crossing partitions, see Section 4. Proposition 2.10. The algebra As (p, q) is the universal C ∗ -algebra generated by the entries of a unitary 2p + q by 2p + q matrix U such that the vector ξ defined in Proposition 2.2 is a fixed vector for U ⊗2 and the map eiα → eiα ⊗ eiα ¯ , eM → eM ⊗ eM defines a morphism in Hom(U ; U ⊗2 ). Proof. If U is a unitary 2p + q by 2p + q matrix such that ξ is a fixed vector for U ⊗2 , then by Proposition 2.2 the entries of U satisfy the relations (2.2)–(2.5). Further the condition that the map described in this proposition is an intertwiner between U and U ⊗2 is satisfied if and only if for each iα ∈ Jp jβ∈Jp

ejβ ⊗ ej¯β ⊗ Ujβ,iα +

q N =1

eN ⊗ eN ⊗ UN,iα =

z,y∈Jp+q

ez ⊗ ey ⊗ Uz,iα Uy,iα ¯


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and for each M ∈ {1, . . . , q}

ejβ ⊗ ej¯β ⊗ Ujβ,M +

jβ∈Jp

q

eN ⊗ eN ⊗ UN,M =

ez ⊗ ey ⊗ Uz,M Uy,M .

z,y∈Jp+q

N =1

The above conditions hold if and only if for each iα, jβ ∈ Jp , M, N ∈ {1, . . . , q}, y ∈ Jp+q j¯β

Ujβ,iα Uy,iα ¯ = δy Ujβ,iα , N UN,iα Uy,iα ¯ = δy UN,iα , j¯β

Ujβ,M Uy,M = δy Ujβ,M , UN,M Uy,M = δyN UN,M . Bearing in mind the relations (2.2)–(2.5) we see that all entries of U are orthogonal projections; hence U is a magic unitary. Conversely it is easy to check that if U is a magic unitary whose entries satisfy (2.2)–(2.5) then the last four displayed formulas automatically hold (see also the comment after Definition 2.7). 2 3. Quantum group H + (p, q) The quantum groups we defined in the last section have all been shown to be closely related to well-studied objects. The generalised quantum hyperoctahedral groups to be introduced here are genuinely new quantum groups, connected to quantum hyperoctahedral groups [6] and quantum isometry groups of C ∗ -algebras of free groups [14]. The fact that an element x of a C ∗ -algebra is an orthogonal projection can be written as x = x ∗ x. It is natural to consider the condition x = xx ∗ x, which of course means that x is a partial isometry. This leads to the following definition. Definition 3.1. The algebra Ah (p, q) is the universal C ∗ -algebra generated by partial isometries {Uz,y : z, y ∈ Jp+q } which satisfy all the relations required of generators of Ao (p, q). Here we also have some automatic ‘orthogonality’, which will be described by the next proposition and its corollary. Proposition 3.2. Let A be a C ∗ -algebra, n ∈ N and let U ∈ Mn (A) be a unitary matrix whose entries are partial isometries. Then ∗ ∗ Uy,z Ux,z = Uz,y Uz,x = 0,

z, y, x ∈ {1, . . . , n}, x = y.

Proof. For each y, z ∈ {1, . . . , n} denote the initial and range projections of Uy,z respectively by Py,z and Qy,z , so that ∗ Py,z = Uy,z Uy,z ,

∗ Qy,z = Uy,z Uy,z .

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Fix z ∈ {1, . . . , n}. The unitarity of U implies that ny=1 Qz,y = ny=1 Py,z = 1A , so that for y, x ∈ {1, . . . , n}, x = y there is Qz,y Qz,x = Py,z Px,z = 0. The initial/range projection interpretation ends the proof. 2 Application of the above proposition and the equality (2.6) immediately gives the following corollary. Corollary 3.3. Let U ∈ M2p+q (Ah (p, q)) be the unitary matrix ‘generating’ Ah (p, q). Then ∗ ∗ ∗ ∗ Uy,z Ux,z = Uz,y Uz,x = Uy,z Ux,z = Uz,y Uz,x = 0,

z, y, x ∈ Jp,q , x = y.

The last observations lead to the following theorem. Theorem 3.4. The algebra Ah (p, q) is the algebra of continuous functions on a compact quantum group, denoted further H + (p, q). The unitary U = (Uz,y )z,y∈Jp,q ∈ M2p+q ⊗ Ah (p, q) is the fundamental representation of H + (p, q). The quantum group H + (0, q) is the quantum hyperoctahedral group Hq+ studied in [6]. Proof. The first statement can be deduced as in the proof of Theorem 2.9, using the fact that a image (and the adjoint) of a partial isometry is a partial isometry. It remains to observe that the algebra Ah (0, q) is isomorphic to Ah (q). But this is a natural consequence of Corollary 3.3 and the fact that if a unitary is cubic then its entries satisfy the condition uij = u3ij = u∗ij (easy to show and noted implicitly in [6]). 2

∗ -homomorphic

As stated in the introduction, the quantum group H + (p, 0) is in fact the quantum isometry group of the dual of the free group Fp [14]. Let us quickly recall the general notion of a quantum isometry group of the dual of a finitely generated discrete group. Let Γ be a finitely generated discrete group with a fixed finite symmetric set of generators S ⊂ Γ . The choice of a generating set S determines a word-length function l on Γ . Denote the universal group C ∗ -algebra of Γ by C(Γ) and let the group ring C[Γ ] ⊂ C(Γ) be denoted by R(Γ). Then the multiplication by the length function defines an operator Dˆ : R(Γ) → R(Γ), ˆ γ ) = l(γ )λγ , D(λ

γ ∈ Γ.

We say that a quantum group G acts on the dual of Γ by orientation preserving isometries if there exists a unital ∗ -homomorphism α : C(Γ) → C(Γ) ⊗ C(G) such that (α ⊗ idC(G) )α = (idC(Γ) ⊗ )α and moreover α restricts to a unital ∗ -homomorphism α0 : R(Γ) → R(Γ) R(G) satisfying the commutation relation α0 Dˆ = (Dˆ ⊗ idR(G) )α0

(3.1)

and preserving the canonical trace on R(Γ). For the motivation behind this definition and connections with spectral triples and noncommutative manifolds we refer to [14] and references therein.


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Theorem 3.5. (See [14].) Let Γ be a discrete group with a fixed finite symmetric set of generators S. The category of all compact quantum groups acting on Γ by orientation preserving isometries admits a universal (initial) object, denoted further by QISO+ (Γ, S) and called the quantum group of orientation preserving isometries of Γ. When the choice of the generating set is clear, we write simply QISO+ (Γ). In particular if Γ = Fp , the free group on p generators x1 , . . . , xp , we use the generating set S = {x1 , x1−1 , . . . , xp , xp−1 }. The following result is essentially a rephrasing of Theorem 5.1 of [14]; for the convenience of the reader we sketch the proof. Theorem 3.6. The quantum group H + (p, 0) is isomorphic to the quantum group of orientation preserving isometries of F p. Proof. Discussion after Theorem 2.6 of [14] implies that G := QISO+ (Fp ) is a compact matrix quantum group with a fundamental representation determined by a unitary U = (Ut,s )t,s∈S ∈ M2p (C(G)) such that the map α0 (λt ) =

λs ⊗ Us,t ,

t ∈ S,

s∈S

extends to a ∗ -homomorphism α0 : R(F p ) → R(F p ) R(G) satisfying (3.1). Relabel generators −1 in S so that S = {xiα , iα ∈ Jp } and xiα = xiα for each iα ∈ Jp . Then the fact that α0 is a ¯ ∗ -map implies that the entries of U satisfy the relations (2.2) (where we write U iα,jβ := Uxiα ,xjβ ). Further unitality of α0 (specifically the conditions α0 (xt )α0 (xt −1 ) = 1C(Fp ) ⊗ 1G for each t ∈ S) together with unitarity of U imply that each Uiα is a partial isometry. Finally a combinatorial argument shows that no additional relations are implied by the fact that the ∗ -homomorphism α0 satisfies (3.1); hence the universal properties defining the standard fundamental representations of G and H + (p, 0) coincide. 2 Finally we describe Ah (p, q) in terms of the intertwiners between the tensor powers of the fundamental corepresentation. Proposition 3.7. The algebra Ah (p, q) is the universal C ∗ -algebra generated by the entries of a unitary 2p + q by 2p + q matrix U such that the vector ξ defined in Proposition 2.2 is a fixed vector for U ⊗2 and the map eiα → eiα ⊗ eiα ¯ ⊗ eiα , eM → eM ⊗ eM ⊗ eM defines a morphism in Hom(U ; U ⊗3 ). Proof. The proof is very similar to that of Proposition 2.10. The difference lies in the fact that we obtain conditions of the type (iα, jβ ∈ Jp , M, N ∈ {1, . . . , q}, y, x ∈ Jp+q ) j¯β jβ

Ujβ,iα Uy,iα ¯ Ux,iα = δy δx Ujβ,iα , N N UN,iα Uy,iα ¯ Ux,iα = δy δx UN,iα , j¯β jβ

Ujβ,M Uy,M Ux,M = δy δx Ujβ,M , UN,M Uy,M Ux,M = δyN δxN UN,M .

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These together with relations (2.2)–(2.5) imply that all Uz,y (z, y ∈ Jp,q ) are partial isometries. 2 4. Categories of representations via partitions In this section we describe the categories of representations of the quantum groups we are considering in this paper in terms of (marked) partitions. Let P (k, l) (k, l ∈ N0 ) be the set of partitions between k upper points and l lower points. Given π ∈ P (k, l) and two multi-indices k and j = (j , . . . , j ) ∈ J l , we define a number δ (i ) ∈ {0, 1} in the i = (i1 , . . . , ik ) ∈ Jp,q 1 l π j p,q following way: first place the indices (i1 , . . . , ik ) and (j1 , . . . , jl ) respectively on the upper and lower points and then put δπ (ij ) = 1 if in any block of π the upper and lower sequences of indices r and (y , . . . , y ) ∈ J s , satisfy x = x¯ = x = x¯ = · · · , y = in Jp,q , say (x1 , . . . , xr ) ∈ Jp,q 1 s 1 2 3 4 1 p,q y¯2 = y3 = y¯4 = · · · , x1 = y1 (the last condition disappearing if either the top or the bottom part of the block is empty), and δπ (ij ) = 0 otherwise. Thus δπ (ij ) = 1 if and only if indices in each block of the partition π have the following pattern: x¯

x

x

x

x¯

x¯

x

x

Further consider the following operator in B((C2p+q )⊗k ; (C2p+q )⊗l ): Tπ (ei1 ⊗ · · · ⊗ eik ) =

δπ

i j e j1 ⊗ · · · ⊗ e jl ,

k (i1 , . . . , ik ) ∈ Jp,q .

(4.1)

j1 ,...,jl ∈Jp,q

We denote by P2 , P12 , Peven ⊂ P respectively the pairings, the singletons and pairings, and the partitions having even blocks. Let also NCx = NC ∩ Px , for any x ∈ {., 2, 12, even}, where NC ⊂ P denotes the set of all non-crossing partitions. It turns out that such collections of partitions, known to correspond to representations of the orthogonal, symmetric, bistochastic and hyperoctahedral (quantum) groups [9], can be also used to describe the categories in our twoparameter context. Indeed, one can check, similarly as it was done in [9], that for P12 and Peven the usual operations of juxtaposition, concatenation (with the appropriate multiplication factor added for deleted closed blocks), and turning the partition upside-down correspond to tensoring, composing and passing to the adjoint on the level of the associated operators Tπ defined in (4.1). Note that such a statement fails when we consider the whole category P – this explains why the case of S + (p, q), to be discussed later on, cannot be included in the following theorem. Theorem 4.1. For G = O + (p, q) (respectively, G = B + (p, q), H + (p, q)) let U denote the fundamental representation introduced earlier. Then for all k, l ∈ N0 ! Hom U ⊗k ; U ⊗l = span Tπ ! π ∈ D(k, l) , with D = NC2 (respectively, D = NC12 , NCeven ). Proof. In the one-parameter case (p = 0) the result was proved in [9]; it turns out that the methods used there can be easily adopted to our framework when we work with the quantum groups


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listed in the theorem. Details will be provided only for the case of O + (p, q), as the other two follow similarly. In fact for O + (p, q) the result is a consequence of the general facts about Ao (F ). Indeed, let F ∈ Mn (R) satisfy F = F t and F 2 = 1, and consider the algebra Ao (F ). Our claim is that we have Hom(U ⊗k ; U ⊗l ) = span(Tπ | π ∈ NC2 (k, l)), where Tπ (ei1 ⊗ · · · ⊗ eik ) = i ˜ i j1 ,...,jl δπ (j )ej1 ⊗ · · · ⊗ ejl , with δπ (j ) ∈ R being given by δ˜π

i1 j1

· · · ik · · · jl

=

" (ir is )∈π

Fi r i s

" (jr js )∈π

F jr js

"

δi r j s .

(ir js )∈π

In other words, δ˜π is obtained by taking the product of Fab ’s, over all “horizontal” strings of p, and of δab ’s, over all “vertical” strings of π (note that as we only consider here pair partitions, in fact only one delta factor will appear for any given block of the partition). Observe further that in ˜ it therefore suffices the case of the matrix F = Fp,q used for defining O + (p, q), we have δ = δ; to establish the claim. The proof of the claim has two steps. First, we prove that π → Tπ transforms the categorical operations for partitions into the categorical operations for the linear maps. This is indeed clear for the tensor product and for the duality. Regarding now the composition axiom, the only problem might come from the closed circles that appear in the middle: but here we can use the formula T∪ T∩ = (2p + q) · id, with 2p + q = i,j ∈Jp,q Fij2 . Summarising, the spaces span(Tπ | π ∈ NC2 (k, l)) form a tensor category in the sense of Woronowicz. Now since this tensor category is generated by T∩ (that is because NC2 is generated by ∩) the corresponding Hopf algebra is the one obtained by using the relation T∩ ∈ Fix(U ⊗2 ). And since T∩ = Fij ei ⊗ ej , this algebra is Ao (F ), and the proof is finished. Note in passing, that when F is not necessarily real, the result remains the same, but with the δ numbers obtained by making the product of Fab ’s over “oriented” horizontal strings, and then the product of δab ’s over vertical strings. The cases of B + (p, q) and H + (p, q) follow in a similar way, first adding the singletons and then considering all partitions with blocks of even size. 2 Remark 4.2. The corresponding result holds also for the classical versions of O + (p, q), S + (p, q) and H + (p, q) (see Section 6), with the non-crossing partitions replaced by all partitions; the proof follows in a similar way. For S + (p, q) the above proof does not work: as already mentioned above, the problem is that, since some of the blocks have an odd number of entries, the composition axiom does not hold for the implementation provided by the operators Tπ described in (4.1). We therefore need to provide a new combinatorial description of the category of representations. From Theorem 2.9 we know that S + (p, q) ≈ Hp+ ˆ Sq+ ; hence we begin by considering separately the cases p = 0 and q = 0. When p = 0, the quantum group S + (0, q) is a quantum symmetric group with the usual fundamental representation, so the corresponding category is NC, with the standard implementation providing the isomorphism (there are no ‘conjugate’ coordinates to deal with), as shown in [1] or in [9]. The situation for q = 0 is more complicated, as the defining fundamental representation of S + (p, 0) corresponds rather to the ‘sudoku’ representation of Hp+ than to the representation studied from the categorical point of view in [6] and [9]. We begin by defining the relevant category of ‘bulleted’ partitions.

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Definition 4.3. Let (for k, l ∈ N0 ) P• (k, l) denote the collection of all partitions between k upper and l lower points, such that each point is marked either with a black or a white circle and we identify the partitions which differ by ‘mirrored markings’ in some of the blocks, so that for • • ◦ • # example • ◦ • equals ◦ • • in P• (2, 3). The category P• = k,l∈N0 P• (k, l) is defined similarly to P , with the result of the concatenation operation being non-zero only if the corresponding markings match, remembering that we are allowed to replace colours of markings in any block of any given partition. Denote by NC• the subcategory of all bulleted non-crossing partitions. Given π ∈ P• (k, l) and multi-indices i = (i1 , . . . , ik ) ∈ Jpk , j = (j1 , . . . , jl ) ∈ Jpl define a number δπ (ij ) ∈ {0, 1} in the following way: δπ (ij ) = 1 if for any pair of vertices in a fixed block of the partition π the corresponding indices are equal if the bullets on the vertices have the same colour and are ‘conjugate’ if the colours of the respective bullets are different. Define further Tπ : (C2p )⊗k → (C2p )⊗l by the usual formula (i1 , . . . , ik ∈ Jp ):

Tπ (ei1 ⊗ · · · ⊗ eik ) =

δπ

i j e j1 ⊗ · · · ⊗ e jl .

(4.2)

j1 ,...,jl ∈Jp

We are ready to formulate the result describing the representation theory of S + (p, 0). Theorem 4.4. Let U denote the fundamental representation of S + (p, 0) introduced in Definition 2.7. We have for all k, l ∈ N0 ! Hom U ⊗k ; U ⊗l = span Tπ ! π ∈ NC• (k, l) . Proof. It is standard to check that NC• is a tensor category satisfying the properties needed to apply the Tannaka–Krein duality of [26] and that the map π → Tπ transforms the natural operations (concatenation with deletion of the closed blocks, tensoring, turning upside-down) to the corresponding operations on the level of linear maps. Proposition 2.10 implies that the category of representations of S + (p, 0) is generated by the morphisms Tπ1 and Tπ2 associated with the partitions π1 =

, •

◦

•

π2 = •

. ◦

The partitions π1 and π2 generate the whole NC• . Indeed, it is well known that the analogous ‘non-bulleted’ partitions generate NC, and it suffices to notice that π1 and π2 can be first combined to obtain singletons and then the ‘exchange’ partition π3 =◦• (the fact that Tπ3 intertwines U is in a sense a consequence of the fact that entries of u are selfadjoint). Once we know that the category generated by π1 and π2 contains partitions of all shapes and we can exchange colours of markings at every vertex using π3 the generation statement is clear. The argument above shows that the category of representations of S + (p, 0) is at least as big as NC• ; so there is an inclusion G ⊃ S + (p, 0), where G is the compact quantum group arising as the Tannaka–Krein dual of NC• via the described implementation. Note that the inclusion G ⊃ S + (p, 0) can be described as the surjection on the level of function algebras: C(G)


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C(S + (p, 0)). To conclude the proof, by using a standard Peter–Weyl argument, it suffices to show that for each k ∈ N the number dk := card(Hom(1; U ⊗k )) is equal to card(NC• (0, k)). The numbers dk are expressed by the formula dk = hS + (p,0) Tr U ⊗k , where hS + (p,0) denotes the Haar state. It follows from the proof of Theorem 2.9 that when we compute the trace of U we obtain the two copies of the elements which turn up as the trace of the top-left block of U in the ‘sudoku’ picture. By Theorem 7.2 and Corollary 7.4 of [7] we know that the corresponding single random variable has (with respect to the Haar state of Hp+ ) the free Poisson distribution with parameter 12 . As the Haar state is obviously preserved by the isomorphism between S + (p, 0) and Hp+ we have dk = 2

k

π∈NC(0,k)

ν(π) 1 = 2

2k−ν(π)

(4.3)

π∈NC(0,k)

(where ν(π) denoted the number of blocks in the partition π and we used the formula for the moments of the free Poisson distribution with parameter t, see e.g. [18]). It remains to check that the number in (4.3) equals card(NC• (0, k)). But this follows from the fact that for every non-bulleted block of a partition in NC(0, k) which has l legs we have 2l−1 choices of the bullet pattern. 2 Remark 4.5. Again we have a corresponding statement for the classical version of S + (p, 0), with the category NC• replaced by P• . Note that the categories of representations listed in Theorem 4.1 can also be described in terms of the bulleted partitions with the fixed colouring pattern consistent with the implementation in (4.1) (so for example if a block in a pair partition is ‘vertical’, both ends are given the same colour, and if it is ‘horizontal’, then we have a colour exchange). 4.1. ‘Free product’ of categories described by partitions Recall that for p, q > 0 we have S + (p, q) = S + (p, 0) ˆ S + (0, q), and Theorem 4.4 together with the discussion before it describe the categories of representations of both S + (p, 0) and S + (0, q). Before we use this decomposition to provide a description of the category for S + (p, q), let us discuss a general free product framework. If G1 , G2 are compact quantum groups and G = G1 ˆ G2 , although the knowledge of irreducible representations of G1 and G2 suffices to describe the irreducible representations of G, the description on the level of categories of representations seems to be quite difficult. If however respective categories for G1 and G2 , denoted say by C1 and C2 , are given by (non-crossing) partitions, there is a natural candidate for the category corresponding to G – it should be given by all non-crossing partitions which decompose into blocks of two colours, where the sub-partition obtained by looking only at the blocks of the first colour belongs to C1 and that for the second colour belongs to C2 . The tensoring and reflection operations are defined as for usual partitions, and so is the concatenation, with the caveat that to obtain a non-zero outcome the colours must fit together. The resulting category will be denoted by C1 C2 (note that the construction works well

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also for bulleted partitions). As in the proof of Theorem 4.4, using the Tannakian duality we can establish the existence of a compact quantum group G , whose category of representations equals C1 C2 , and deduce the existence of the inclusion G ⊃ G. To show that this inclusion is in fact an equality, we need to compare the dimensions of the fixed point spaces (or, equivalently, the moments of the laws describing the characters of the defining fundamental representations). As the Haar state of G was shown in [23] to be the free product of the Haar states of G1 and G2 , the law related to G arises as the free convolution of the corresponding distributions for G1 and G2 . Thus if this free convolution can be easily computed, and the categories C1 and C2 are identical, the corresponding count can be performed without much difficulty. In particular using the natural ‘free product’ implementations π → Tˆπ defined in the spirit of (4.1) we obtain the following result. Theorem 4.6. Let n, m ∈ N. The category corresponding to the standard fundamental represen+ (respectively, H + + + ˆ B + and O + + tations of Sn+ ˆ Sm n ˆ Hm , Bn m n ˆ Om ) is equal to the span of {Tˆπ : π ∈ C C}, where C = NC (respectively, NC12 , NCeven and NC2 ). Proof. The proof follows as in Theorem 4.4, so we only provide the argument for the equality of the cardinality of partitions in (C C)(0, k) and the dimension of the fixed point spaces for tensor powers of the defining fundamental representations. Fix x $∈ {., 2, 12, even} and for each t > 0 consider the probability measure μt having as moments λk dμt (λ) = π∈NCx t ν(π) , where ν(π) denotes the number of blocks in π . The results in [9] imply that μ1 is the measure describing the distribution of the character of the defining representation of the quantum group corresponding to x, and also that the measures {μt : t > 0} form a semigroup with respect to the free convolution (see [22]). A standard argument using the definition of the free product ˆ G+ construction implies that the character of the representation for G+ n m follows the $ defining k ν(π) . But this is exactly the law μ1 μ1 = μ2 , with moments given by λ dμ2 (λ) = π∈NCx 2 number of partitions featuring in (C C)(0, k) – using the surjective map C C → C we see that each element in the preimage of a given π ∈ C(0, k) is the partition π with each block given one of the two colours; thus the preimage has 2ν(π) elements. 2 Before we prove an analogous, more technical, result for the quantum group S + (p, q) isomorphic to S + (p, 0) ˆ S + (0, q) we need a simple observation related to the free cumulants of probability measures. All relevant definitions can be found in Lectures 11 and 12 in [18]. Lemma 4.7. Assume that ν is a compactly supported probability measure on R. Let X be a realvalued random variable with distribution ν, let μ = ν ν and let ν be the law of 2X. Denote ∞ the respective free cumulants of μ and of ν by (κk (μ))∞ k=1 and (κk (ν ))k=1 . Then for each k ∈ N κk ν = 2k−1 κk (μ). Proof. It follows from Proposition 12.3 in [18] that κn (ν) = 12 κn (μ). Further the fact that the moments and cumulants are related by the Speicher’s moment-cumulant formula [20]: dk =

"

π∈NC(0,k) b: block in π

implies that κk (ν ) = 2k κk (ν).

2

κcard(b) ,

(4.4)


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The following theorem completes the description of the categories for the main family of quantum groups studied in this paper. Theorem 4.8. Let u denote the fundamental representation of S + (p, q) introduced in Definition 2.7. Then for all k, l ∈ N0 ! Hom U ⊗k ; U ⊗l = span Tπ ! π ∈ (NC• NC)(k, l) . Proof. As in the proof of Theorem 4.6 we only provide the argument for the equality of the number of partitions in (NC• NC)(0, k) and the k-th moment of the character of the defining representation of S + (p, q). The latter can be expressed via the moment-cumulant formula (4.4), with the free cumulants (κk )∞ k=1 given by the sum of the free cumulants of the corresponding laws of the characters for S + (p, 0) and S + (0, q) (this is a consequence of the discussion before Theorem 4.6 and Proposition 12.3 in [18]). The free cumulants for the second law (which is the free Poisson distribution) are equal to 1. As the free Poisson laws form a free convolution semigroup, the proof of Theorem 4.4 and Lemma 4.7 imply that the free cumulants for the first law are equal to 2k−1 . Hence the moment-cumulant formula for the k-th moment of the law we are interested in yields: "

dk =

card(b)−1 2 +1 .

π∈NC(0,k) b: block in π

It remains to note that the above number indeed corresponds to the number of partitions in (NC• NC)(0, k): for each block b in a given non-crossing partition π of k-points we can choose whether it ‘comes’ from NC• or NC and in the first case we have additionally 2card(b)−1 choices of inequivalent bulleting patterns. 2 5. Relations between the two-parameter families Recall the diagram (1.2) describing the relations between On+ , Bn+ , Hn+ and Sn+ . Universal properties imply that the corresponding diagram can be drawn in the two-parameter case; for each p, q ∈ N0 we have Ao (p, q)

→

↓ Ah (p, q) →

Ab (p, q)

O + (p, q)

↓

∪

As (p, q)

⊃

H + (p, q) ⊃

B + (p, q) ∪

(5.1)

S + (p, q)

Below we describe further connections between the quantum groups studied above, first discussing the general framework. Definition 5.1. Assume that we have surjective morphisms of Hopf C ∗ -algebras, corresponding to inclusions of compact quantum groups, as follows:

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A

→

↓ C

B ↓

→

D

:

G

⊃ H

∪

∪

K

⊃

(5.2)

L

(1) We write G = H, K if the ∗-algebra ideal ker(A → B) ∩ ker(A → C) contains no non-zero Hopf ideal. (2) We write L = H ∩ K when ker(A → D) is the Hopf ideal generated by ker(A → B) and ker(A → C). The definitions above can be easily seen to coincide with the standard ones in the classical case. More generally, given morphisms α : A → B and β : A → C as above, one can define a quantum group H, K ⊂ G by the formula C( H, K ) = A/I , where I is the biggest Hopf ideal contained in ker(α) ∩ ker(β). Once again, this definition agrees with the usual one in the classical case. These notions are best understood in terms of the Hopf image formalism, introduced in [4]. Consider indeed the morphism (α, β) : A → B × C. Then C( H, K ) is the Hopf image of (α, β), and we have H, K = G if and only if (α, β) is inner faithful. Similarly given a 4-term diagram as in the above definition, we can form the Hopf ideal J = ker(A → B), ker(A → C) , and define a quantum group H ∩ K by C(H ∩ K) = A/J . We have L = H ∩ K if and only if the above condition (2) is satisfied. In order to deal effectively with part (1) of Definition 5.1, we use the following Tannakian reformulation. Lemma 5.2. Consider the diagram (5.2) and denote the Hopf morphism between A and B (respectively, between A and C) by α (respectively, β). We have H, K = G if and only if Fix(R) = Fix (id ⊗ α)R ∩ Fix (id ⊗ β)R for any representation R arising as a tensor product between U and U¯ ’s, with U being the fundamental representation of G. Proof. The result is a consequence of general considerations in [4]; we sketch the proof below. Let I be the biggest Hopf ideal contained in ker(α) ∩ ker(β), whose existence follows from [4], and let G1 = H, K be the compact quantum group given by C(G1 ) = C(G)/I . By Frobenius duality, the collection of equalities in the statement is equivalent to the following collection of equalities: Hom(R; S) = Hom (id ⊗ α)R; (id ⊗ α)S ∩ Hom (id ⊗ β)R; (id ⊗ β)S , where now both R and S are representations arising as tensor products of several copies of U and U¯ . According to the general results of Woronowicz in [26], the Hom spaces on the right form a Tannakian category, which should therefore correspond to a certain compact quantum group K2 . Our claim is that we have G1 = G2 . Indeed, this follows from Theorem 8.4 of [4] applied to the


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morphism (α, β) : A → B × C (and can be viewed as a consequence of the general Peter–Weyl type results in [25] and the Tannakian duality results in [26]). In particular G = G1 is equivalent to G = G2 , and we are done. 2 Before we formulate the specific intersection/generation results in the cases in which we are interested, we need another lemma: Lemma 5.3. Let A be a C ∗ -algebra, n ∈ N and let U ∈ Mn (A) be a unitary matrix whose entries are partial isometries. If for each z ∈ {1, . . . , n} there is ny=1 Uy,z = 1, then each Uy,z is an orthogonal projection. Proof. Fix y, z ∈ {1, . . . , n}. We have Uy,z = 1 − x=y Ux,z . Proposition 3.2 implies that mul∗ on the right yields U U ∗ = U ∗ . Hence U tiplying the last equality by Uy,z y,z y,z y,z is a selfadjoint y,z projection. 2 Theorem 5.4. The following relations hold: (1) O + (p, q) = B + (p, q), H + (p, q) . (2) S + (p, q) = B + (p, q) ∩ H + (p, q). Proof. (1) follows from Lemma 5.2, Theorem 4.1, and the equality NC2 = NC12 ∩ NCeven . (2) is a direct consequence of Lemma 5.3. 2 5.1. Free products We recall from Section 2 that we have a canonical isomorphism Ao (p, q) Ao (2p + q). It is well known that for any n, m ∈ N there exists a natural surjective map Ao (m + n) → Ao (m) Ao (n). Similar maps turn out to exist in all the cases considered in this paper. Proposition 5.5. Let g ∈ {o, b, h, s}. There exists a natural surjective map Ag (p, q) → Ag (p, 0) Ag (0, q) such that the following diagram (in which the vertical maps are the ones introduced earlier in this section) is commutative: Ao (p, q)

→

↓ Ag (p, q) →

Ao (p, 0) Ao (0, q) ↓ Ag (p, 0) Ag (0, q)

The horizontal maps in the diagram above intertwine respective comultiplications. Proof. Let U1 ∈ M2p (Ag (p, 0)) and U2 ∈ Mq (Ag (0, q)) be fundamental unitary representa tions of the respective compact quantum groups. Consider the unitary matrix U01 U02 viewed as an element of M2p+q (Ag (p, 0) Ag (0, q)). It is easy to see that it satisfies all the conditions required of the generating matrix in M2p+q (Ag (p, q)), so the universality implies the existence of a ∗ -homomorphism requested by the proposition. It is easily seen to be surjective, as entries

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of U1 (respectively, U2 ) generate Ag (p, 0) (respectively, Ag (0, q)). An explicit description of the comultiplications involved implies the last statement of the proposition. Recall that the existence of surjective (compact quantum group) morphisms from Ao (p, q) to Ag (p, q) followed in an analogous way; the fact that the diagram above is commutative is thus a direct consequence of the definitions of the maps considered. 2 Using directly the language of compact quantum groups we obtain the following corollary: Corollary 5.6. Let G ∈ {O + , H + , B + , S + }. There exist natural inclusions O + (p, q) ⊃ O + (p, 0) ˆ O + (0, q) ∪ G(p, q)

∪ ⊃

G(p, 0) ˆ G(0, q)

In the case of Ao (p, q) the map described in Proposition 5.5 and the isomorphisms/surjections recalled before it can be combined into the following commutative diagram: Ao (p, q)

→

↓ Ao (2p + q) →

Ao (p, 0) Ao (0, q) ↓ Ao (2p) Ao (q)

(the commutativity is the consequence of the fact how the isomorphism was described in the proof of Theorem 2.1 and the block-diagonal form of the matrices Cp,q implementing it). Theorem 2.9 implies that the inclusion S + (p, q) ⊃ S + (p, 0) ˆ S + (0, q) is in fact an isomorphism. In all the other cases when both p and q are non-zero (for B + (p, q) we actually need to assume q > 1 to avoid trivialities) G(p, 0) ˆ G(0, q) is a proper quantum subgroup of G(p, q). This is the content of the next proposition. Proposition 5.7. If p, q = 0 then the inclusions O + (p, 0) ˆ O + (0, q) ⊂ O + (p, q) and H + (p, 0) ˆ H + (0, q) ⊂ H + (p, q) are proper. If p > 0 and q > 1 then the inclusion B + (p, 0) ˆ B + (0, q) ⊂ B + (p, q) is proper. Proof. It is enough to show that the corresponding homomorphisms Ao (p, q) → Ao (p, 0) Ao (0, q), Ah (p, q) → Ah (p, 0) Ah (0, q) and Ab (p, q) → Ab (p, 0) Ab (0, q) are not injective; in other words it suffices to find unitary matrices satisfying the defining conditions for Ao (p, q), Ah (p, q) and Ab (p, q) which are not of the form U01 U02 with U1 a 2p by 2p matrix. In the case of the orthogonal and the bistochastic group the existence of such matrices is visible already at the commutative level. Consider then the case of H + . It suffices to find suitable matrices for p = q = 1. Let then QA , QB , PA , PB be non-zero orthogonal projections on some Hilbert space H such that QA + QB + PA + PB = 1 and let A, B, C, D be partial isometries in B(H) such that AA∗ = QA , A∗ A = PA , BB ∗ = QB , B ∗ B = PB , CC ∗ = 1 − C ∗ C = PB + QA , DD ∗ = 1 − D ∗ D = PA + QB . The existence of such partial isometries can be assured by choosing all the projections to have infinitedimensional ranges. Then the matrix


%

A B∗ D

B A∗ D∗

C C∗ 0

3275

&

can be checked to be a unitary satisfying assumptions listed in Definition 3.1.

2

Note that the above construction excludes the possibility of the partial isometries A, B, C, D generating a commutative algebra. This is a general fact – in Theorem 6.1 below we will see that if we additionally request commutativity then the ‘off-diagonal’ terms of a unitary satisfying the defining conditions of Ah (p, q) must vanish. 5.2. Quantum permutation groups as quotient quantum groups of H + (p, q) Definition 5.8. If G1 and G2 are compact quantum groups and π : C(G1 ) → C(G2 ) is a unital injective ∗ -homomorphism intertwining the respective coproducts and injective on the Hopf∗ algebra R(G1 ), then G1 is said to be a quotient quantum group of G2 (alternatively, G2 is said to be a quantum group extension of G1 ). Consider once again the quantum group H + (p, q). As the generators of Ah (p, q), denoted ∗ U by Uz,y (z, y ∈ Jp,q ) are partial isometries, the operators Pz,y := Uz,y z,y are orthogonal projections. Theorem 5.9. The Hopf ∗ -subalgebra of Ah (p, q) generated by Pz,y (z, y ∈ Jp,q ) is isomorphic + ) ⊂ As (2p + q). to R(S2p+q Proof. Let A denote the C ∗ -subalgebra of Ah (p, q) generated by Pz,y (z, y ∈ Jp,q ). Note first that the generators {Pz,y : z, y ∈ Jp,q } satisfy the same relations as generators of As (2p + q); indeed for z, y ∈ Jp,q

Pz,x Py,x =

x∈Jp,q

∗ ∗ Uz,x Uz,x Uy,x Uy,x = δz

y

x∈Jp,q y

= δz

∗ Uz,x Uz,x = δz

y

x∈Jp,q

∗ ∗ Uz,x Uz,x Uz,x Uz,x

x∈Jp,q

Uz¯ ,x¯ Uz¯∗,x¯ = δz 1. y

x∈ ¯ Jp,q

Similarly it follows from Corollary 3.3 that x∈Jp,q

Px,z Px,y =

∗ ∗ Ux,z Ux,z Ux,y Ux,y = δz

y

x∈Jp,q y

= δz

x∈Jp,q

∗ Ux,z Ux,z = δz 1. y

x∈Jp,q

If denotes the coproduct of Ah (p, q) then for z, y ∈ Jp,q (Pz,y ) =

x∈Jp,q

Pz,x ⊗ Px,y .

∗ ∗ Ux,z Ux,z Ux,z Ux,z

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+ Hence A is isomorphic to C(G), where G is a quantum subgroup of S2p+q (precisely speaking + ∗ A is isomorphic to some C -completion of R(G)). It remains to check that G is actually S2p+q . For that it suffices to analyse the corresponding categories of representations, and even more specifically to check that cardinalities of fixed point sets of tensor powers of the fundamental + (see Section 4, where this method representations of G coincide with those appearing for S2p+q was applied several times). In the partition picture described in Section 4 the passage from H + (p, q) to G corresponds to considering only partitions with even number of upper and lower points and composing them (both from above and from below) with a suitable number of copies of the partition . In terms of the fundamental representations it corresponds to observing that the fundamental representation of G arises as a specific subrepresentation of U ⊗2 . This means that we are left only with these partitions in NC(2k, 2l) in which the pairs of points beginning at an odd place are necessarily joined (it is easy to see that all such partitions arise in this procedure); we denote them by NCjoin (2k, 2l). Now collapsing the pairs listed above we obtain a bijection between NCjoin (2k, 2l) and NC(k, l). This suffices to perform the count needed to finish the proof. 2

Using the dual language and noticing that the map morphism constructed above is injective and preserves the respective coproducts, we obtain the following corollary. + . Corollary 5.10. The quantum group H + (p, q) is an extension of S2p+q

6. Classical versions and interpretations in terms of classical/quantum symmetries Each of the quantum groups studied above has a classical version. These are understood as follows: if G is a compact quantum group then the quotient of the algebra C(G) by its commutator ideal is isomorphic to the algebra of continuous functions on a certain uniquely determined compact group G. Then we call G the classical version of G. Note that if we use the notation G(p, q) for the classical version of the quantum group G+ (p, q), then Corollary 5.6 and a straightforward analysis of classical versions of free products described in Section 1 implies that we have the following inclusions: O(p, q)

⊃ O(p, 0) × O(0, q)

∪ G(p, q)

∪ ⊃

G(p, 0) × G(0, q)

The combination of results from the previous section and [9] allow us to identify the classical version of O + (p, q) with O2p+q , the classical version of B + (p, q) with O2p+q−i(p,q) (where i(p, q) = 1 if pq = 0 and i(p, q) = 2 if p, q > 0) and the classical version of S + (p, q) with Hp × Sq . Theorem 6.1. The classical version of H + (p, q) is (Tp Hp ) × Hq (recall that Tp Hp is the usual isometry group of Tp ). Proof. Observe first that if {Uz,y : z, y ∈ Jp,q } are commuting partial isometries which satisfy the condition (2.6) and form a unitary matrix, then Uiα,M = UM,iα = 0 for all iα ∈ Jp and


3277

∗ ∗ ∗ M ∈ {1, . . . , q}. Indeed, the projections Uiα,M Uiα,M and Uiα,M Uiα,M = Uiα,M Uiα,M are then ¯ ¯ equal; as due to Proposition 3.2 they are pairwise orthogonal to each other, they must vanish (similar argument applies to UM,iα ). Hence to prove the theorem it suffices to consider separately cases p = 0 and q = 0. The first one was treated in [6]. Assume then that q = 0. The result can be deduced from calculations in [12], but we can also offer an explicit isomorphism. Denote the images of elements of {Uiα,jβ : iα, jβ ∈ Jp } in the quotient space with respect to the commutator ideal of Ah (p, 0) by {Uˆ iα,jβ : iα, jβ ∈ Jp } and p ∗ let the quotient C ∗ -algebra be denoted by Acom h (p, 0). Identify C(T Hp ) as a C -algebra in p a natural way with C(T ) ⊗ C(Sp ) ⊗ C( Z2 × · · · × Z2 ). Let χα,β ∈ C(Sp ) be the characteristic p times

function of the set of these permutations which map α into β and let κ0,α , κ1,α ∈ C(Z2 ×· · ·×Z2 ) be characteristic functions of sets which have respectively 0 or 1 in the α-coordinate. One can then check that the map Uˆ 0α,0β → zα ⊗ χα,β ⊗ κ0,α , and Uˆ 0α,1β → zα ⊗ χα,β ⊗ κ1,α p extends to an isomorphism of Acom h (p, 0) with C(T Hp ); this isomorphism preserves also the respective coproducts. 2

Note that from the point of view of the philosophy presented in [9] H + (p, 0) can therefore be viewed as the liberation of a group of isometries of p-copies of the circle. This is consistent with the results of Section 5 of [14], where H + (p, 0) was first discovered as the quantum isometry ∗ group of C(F p ) – in other words, of the ‘free torus’ (note that C(F p ) is the universal C -algebra p ∗ generated by p unitaries, whereas C(T ) is the universal C -algebra generated by p commuting unitaries). Similarly the quantum group Op+ can be viewed as the quantum symmetry group of S2p , the free sphere studied in [8]. To strengthen this analogy we observe the following fact. Proposition 6.2. There is a natural surjection from C(S2p ) to C(F p ). Proof. Recall that C(S2p ) is theuniversal C ∗ -algebra generated by selfadjoint operators p 2 2 ∗ {xi , yi : i = 1, . . . , p} such that i=1 xi + yi = 1 and C (Fp ) ≈ C(Fp ) is the universal ∗ C -algebra generated by unitaries {ui : i = 1, . . . , p}. It suffices to observe that the quotient of C(S2p ) by the closed two-sided ideal generated by expressions xi yi = yi xi , xi2 + yi2 = p1 (i = 1, . . . , p) is naturally isomorphic to C(F p ) – the isomorphism maps images of xi and yi in 1 ∗ the quotient respectively to 2√p (ui + ui ) and 2√i p (ui − u∗i ). 2 7. Quantum group Hs+ (p, q) It is natural to ask whether the two-parameter families studied in this paper in a sense exhaust all possible natural choices in the ‘Fp,q ’ framework. When we studied Ah (p, q) and As (p, q) we required the entries of the fundamental corepresentation U to satisfy respectively the conditions of the type x = xx ∗ x and x = x ∗ x, which have natural descriptions in terms of the intertwiners

3278


between U and its tensor powers (see Propositions 2.10 and 3.7). We could equally well describe in terms of the intertwiners conditions on the entries of the type say x = xx ∗ xx ∗ . These introduce only one new possibility, as explained by the next proposition. Proposition 7.1. Let x be a bounded operator on a Hilbert space, k ∈ N. If x = (x ∗ x)k or x = (xx ∗ )k then x is an orthogonal projection. If x = x(x ∗ x)k , then x is a partial isometry. If x = x ∗ (xx ∗ )k , then x is a partial symmetry (a selfadjoint partial isometry). Proof. We only prove the last statement, leaving the first two to the reader. Suppose x = x ∗ (xx ∗ )k . Then x ∗ = x(x ∗ x)k , so x 2 = (xx ∗ )k+1 and (x ∗ )2 = (x ∗ x)k+1 . Taking the adjoint of the last relation we obtain that (x ∗ x)k+1 = (xx ∗ )k+1 . As both xx ∗ and x ∗ x are positive, the last equality holds if and only if x is normal. Now it remains to observe that for z ∈ C the equality z = z¯ |z|k holds if and only if z ∈ {−1, 0, 1} and apply the spectral theorem to end the proof. 2 The above proposition motivates the next definition. Definition 7.2. The algebra Ahs (p, q) is the universal C ∗ -algebra generated by partial symmetries {Uz,y : z, y ∈ Jp+q } which satisfy all the relations required of generators of Ao (p, q). Proposition 7.3. The algebra Ahs (p, q) is the universal C ∗ -algebra generated by two families of partial symmetries {Uiα,jβ : iα, jβ ∈ Jp } and {UM,N ; M, N ∈ {1, . . . , q}}, such that both q matrices (Uiα,jβ )iα,jβ∈Jp and (UM,N )M,N =1 are cubic unitaries and moreover Uiα,jβ = Uiα, ¯ j¯β ,

iα, jβ ∈ Jp .

(7.1)

Proof. It suffices to show that whenever iα ∈ Jp and M ∈ {1, . . . , q} then Uiα,M = 0 = UM,iα . Due to Proposition 3.2 we have (Uiα,M )2 = 0 = (UM,iα )2 = 0, and the proof is finished. 2 Before we formulate the main result we need to recall the definition of a ‘higher-order’ quantum hyperoctahedral group first defined in [6] and later studied for example in [7] and [3]. Definition 7.4. Denote by Ah (n) the universal C ∗ -algebra generated by the entries of an n by n unitary U , such that U¯ is also unitary, for each i, j = 1, . . . , n the entry Uij is normal, and Uij4 = Uij Uij∗ is an orthogonal projection. When U ∈ Mn ⊗ As (n) is interpreted as the fundamental (4)

(4)

unitary corepresentation, we view Ah (n) as the algebra of continuous functions on the quantum hyperoctahedral group of order 4 on n coordinates, Hn4+ . Note that if in the above definition 4 is replaced by 2 we obtain the quantum hyperoctahedral group Hn+ introduced in Definition 1.4. The factors 2 and 4 can be replaced by arbitrary s ∈ N, as shown in the papers cited above. Theorem 7.5. The algebra Ahs (p, q) is the algebra of continuous functions on a compact quantum group, denoted further Hs+ (p, q). The unitary U = (Uz,y )z,y∈Jp,q ∈ M2p+q ⊗ Ahs (p, q) is the fundamental representation of Hs+ (p, q). The algebra Ahs (p, q) is isomorphic to the free (4) product Ah (p) Ah (q); on the level of quantum groups we have Hs+ (p, q) ≈ Hp4+ ˆ Hq+ .


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Proof. By Proposition 7.3 it suffices to consider separately the cases p = 0 and q = 0. The fact that Ahs (0, q) ≈ Ah (q) (with the isomorphism preserving natural fundamental corepresentations) is immediate. Assume then that q = 0. Note that the fundamental corepresentation has then the form ⎡

pA − q A ⎣ pB − qB .. .

pB − q B pA − q A .. .

pC − q C pD − q D .. .

pD − q D pC − q C .. .

⎤ ··· ···⎦, .. .

(7.2)

where for each X ∈ {A, B, C, D, . . .} the pair (pX , qX ) is a pair of mutually orthogonal projections. Following the permutation procedure described in the proof of Theorem 2.9 we can describe the fundamental representation as a ‘sudoku’ unitary, whose entries are partial symmetries; and further considering separately positive and negative part of each partial symmetry express the relations between them by placing them in a ‘double sudoku’, i.e. a magic unitary of dimension 4p by 4p, which has the following block-matrix structure: ⎡

P ⎢Q ⎣ R S

Q R P S S P R Q

⎤ S R⎥ ⎦, Q P

(7.3)

with P , Q, R, S being p by p matrices whose entries are orthogonal projections. Note that this + fits with the fact that Hs+ (p, 0) is a quantum subgroup of H2p . We can now apply Theorem 2.3 (2) of [10] to deduce that the algebra Ahs (p, 0) is isomorphic to C(Hp4+ ); one can check that this isomorphism is also a Hopf ∗ -algebra morphism. 2 Note that the fundamental corepresentation of C(Hp4+ ) in (7.2) is in a sense intermediate between that of [3] (which has dimension p and is given by the matrix with entries of the type pA − qA + i(pB − qB )) and that given in (7.3) considered in [10] (which has dimension 4p). Once again we describe the quantum group studied in terms of the intertwiners between the tensor powers of the fundamental corepresentation. Proposition 7.6. The algebra Ahs (p, q) is the universal C ∗ -algebra generated by the entries of a unitary 2p + q by 2p + q matrix U such that the vector ξ defined in Proposition 2.2 is a fixed vector for U ⊗2 and the map eiα → eiα ¯ ⊗ eiα ⊗ eiα ¯ , eM → eM ⊗ eM ⊗ eM defines a morphism in Hom(U ; U ⊗3 ). Proof. Identical to that of Proposition 3.7.

2

It remains to describe the category describing the representations of Hs+ (p, q). Once again we will first separately consider the case p = 0 (when everything reduces to the usual computations with Hq+ and the corresponding category is NCeven ) and the case q = 0. For the second case recall Definition 4.3 and denote by NC•,even the subcategory given by the bulleted partitions of even size.

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Theorem 7.7. Let U denote the fundamental representation of Hs+ (p, 0) introduced in Definition 7.2. We have ! Hom U ⊗k ; U ⊗l = span Tπ ! π ∈ NC•,even (k, l) , where the implementing maps are defined as in (4.2). Proof. The proof is very similar to that of Theorem 4.4. It is standard to check that NC•,even is a tensor category satisfying the properties needed to apply the Tannaka–Krein duality of [26] and that the map π → Tπ transforms the natural operations (concatenation, tensoring, adjoint) to the corresponding operations on the level of linear maps. Proposition 7.6 implies that the category of representations of S + (p, 0) is generated by the morphisms Tπ1 and Tπ2 associated with the partitions π1 =

, •

π2 =

◦

• ◦ • ◦

.

The partitions π1 and π2 generate the whole NC•,even . Indeed, the analogous ‘non-bulleted’ partitions generate NCeven , and it suffices to notice that π1 and π2 can be composed to obtain the ‘exchange’ partition π3 =•. The generation statement is now clear. ◦ The argument above shows that the category of the representations of Hs+ (p, 0) is at least as big as NC•,even ; hence there is an inclusion G ⊃ S + (p, 0), where G is the compact quantum group arising as the Tannaka–Krein dual of NC•,even via the described implementation. To conclude the proof it suffices to show that for each k ∈ N the number dk := card(Hom(1; U ⊗2k )) is equal to card(NC•,even (0, 2k)). The numbers dk are expressed by the formula dk = hHs+ (p,0) Tr U ⊗2k , where hHs+ (p,0) denotes the Haar state. Using the argument similar to that of Corollary 7.4 of [7] and the identification in the proof of Theorem 7.5 one can show that the variables p = pA + pB + pC + pD + · · · and q = qA + qB + qC + qD + · · · (see the matrix (7.2)) are free Poisson variables of parameter 14 – note that this provides a concrete realisation of the variables found in [3]. Thus we are left with the computation of the moments of 2 copies of a difference of such two free Poisson variables. This can be deduced from the results in [3]: first note that according to Theorem 7.1 of [3] the 2k-th moment of p − q is the k-th moment of the free Bessel law with parameters (2, 12 ), and then combine the observation in the proof of Theorem 5.2 of [3] with Theorem 4.3 of the same paper that the latter is equal to π∈NCeven (0,2k) ( 12 )ν(π) , so that dk = 2k

π∈NCeven (0,2k)

ν(π) 1 = 2

2k−ν(π)

(7.4)

π∈NCeven (0,2k)

(where ν(π) denoted the number of blocks in the partition π ). As in the proof of Theorem 4.4 we observe that the number in (7.4) indeed equals card(NC• (0, 2k)) due to the multiplication factor expressing the choice of colourings of bullets in each block of π . 2


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We are now ready to describe the category of partitions corresponding to the representation theory of Hs+ (p, q). Theorem 7.8. Let U denote the fundamental representation of Hs+ (p, q) introduced in Definition 7.2. Then for all k, l ∈ N0 ! Hom U ⊗k ; U ⊗l = span Tˆπ ! π ∈ (NC•,even NCeven )(k, l) . Proof. We only provide the argument for the equality of the number of partitions in (NC•,even NCeven )(0, k) and the k-th moment of the character of the defining representation of Hs+ (p, q). The latter can be expressed via the moment-cumulant formula (4.4), with the free cumulants (κk )∞ k=1 given by the sum of the free cumulants of the corresponding laws of the characters for Hs+ (p, 0) and Hs+ (0, q). As the second law is the free Bessel distribution, another use of the moment-cumulant formula (4.4) and Theorem 4.3 of [3] implies that the even free cumulants for it are equal to 1, and the odd ones vanish. As the free Bessel laws form a free convolution semigroup, Lemma 4.7 and the proof of Theorem 7.7 imply then that the odd free cumulants related to Hs+ (0, q) vanish and even ones are equal to 2k−1 . Hence the moment-cumulant formula for the k-th moment of the law we are interested in yields: dk =

"

card(b)−1 2 +1 .

π∈NCeven (0,k) b: block in π

The rest of the proof follows as in Theorem 4.8.

2

Acknowledgments Part of the work on this paper was began during the visit of the second named author to the Université Cergy-Pontoise in April 2010. The work of the first named author was supported by the ANR grant ‘Galoisint’. We would like to thank the referee for many useful comments and suggestions. References [1] T. Banica, Quantum groups and Fuss–Catalan algebras, Comm. Math. Phys. 226 (1) (2002) 221–232. [2] T. Banica, Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2) (2005) 243–280. [3] T. Banica, S. Belinschi, M. Capitaine, B. Collins, Free Bessel laws, Canad. J. Math., in press, available at arXiv: 0710.5931. [4] T. Banica, J. Bichon, Hopf images and inner faithful representations, Glasg. Math. J. 52 (2010) 677–703. [5] T. Banica, J. Bichon, B. Collins, Quantum permutation groups: a survey, Banach Center Publ. 78 (2007) 13–34. [6] T. Banica, J. Bichon, B. Collins, The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007) 345–384. [7] T. Banica, S. Curran, R. Speicher, Stochastic aspects of easy quantum groups, Probab. Theory Related Fields, in press, available at arXiv:0909.0188. [8] T. Banica, D. Goswami, Quantum isometries and noncommutative spheres, Comm. Math. Phys. 298 (2010) 343– 356. [9] T. Banica, R. Speicher, Liberation of orthogonal Lie groups, Adv. Math. 222 (4) (2009) 1461–1501. [10] T. Banica, R. Vergnioux, Fusion rules for quantum reflection groups, J. Noncommut. Geom. 3 (2009) 327–359. [11] E. Bédos, G.J. Murphy, L. Tuset, Co-amenability of compact quantum groups, J. Geom. Phys. 40 (2) (2001) 130– 153. [12] J. Bhowmick, Quantum isometry group of the n-tori, Proc. Amer. Math. Soc. 137 (9) (2009) 3155–3161.

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[13] J. Bhowmick, D. Goswami, A. Skalski, Quantum isometry groups of 0-dimensional manifolds, Trans. Amer. Math. Soc. 363 (2011) 901–921. [14] J. Bhowmick, A. Skalski, Quantum isometry groups of noncommutative manifolds associated to group C ∗ -algebras, J. Geom. Phys. 60 (10) (2010) 1474–1489. [15] J. Bichon, A. De Rijdt, S. Vaes, Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys. 262 (3) (2006) 703–728. [16] D. Goswami, Quantum group of isometries in classical and noncommutative geometry, Comm. Math. Phys. 285 (1) (2009) 141–160. [17] A. Maes, A. Van Daele, Notes on compact quantum groups, Nieuw Arch. Wisk. (4) 16 (1–2) (1998) 73–112. [18] A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability, London Math. Soc. Lecture Note Ser., vol. 335, Cambridge University Press, 2006. [19] S. Raum, Isomorphisms and fusion rules of orthogonal free quantum groups and their complexifications, preprint available at arXiv:1006.2979. [20] R. Speicher, Multiplicative functions on the lattice of noncrossing partitions and free convolution, Math. Ann. 298 (1994) 611–628. [21] A. Van Daele, S. Wang, Universal quantum groups, Internat. J. Math. 7 (2) (1996) 255–264. [22] D.V. Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66 (1986) 323–346. [23] S. Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (3) (1995) 671–692. [24] S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1) (1998) 195–211. [25] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987) 613–665. [26] S.L. Woronowicz, Tannaka–Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988) 35–76. [27] S.L. Woronowicz, Compact quantum groups, in: A. Connes, K. Gawedzki, J. Zinn-Justin (Eds.), Symétries Quantiques, Les Houches, Session LXIV, 1995, pp. 845–884.


On the essential spectrum of complete non-compact manifolds ✩ Zhiqin Lu a,∗ , Detang Zhou b a Department of Mathematics, University of California, Irvine, CA 92697, USA b Instituto de Matematica, Universidade Federal Fluminense, Niterói, RJ 24020, Brazil

Received 28 September 2010; accepted 15 October 2010 Available online 15 December 2010 Communicated by Daniel W. Stroock

Abstract In this paper, we prove that the Lp essential spectra of the Laplacian on functions are [0, +∞) on a noncompact complete Riemannian manifold with non-negative Ricci curvature at infinity. The similar method applies to gradient shrinking Ricci soliton, which is similar to non-compact manifold with non-negative Ricci curvature in many ways. © 2010 Elsevier Inc. All rights reserved. Keywords: Essential spectrum; Shrinking soliton

1. Introduction The spectra of Laplacians on a complete non-compact manifold provide important geometric and topological information of the manifold. In the past two decades, the essential spectra of Laplacians on functions were computed for a large class of manifolds. When the manifold has a soul and the exponential map is a diffeomorphism, Escobar [11], Escobar and Freire [12] proved that the L2 spectrum of the Laplacian is [0, +∞), provided that the sectional curvature is nonnegative and the manifold satisfies some additional conditions. In [18], the second author proved that those “additional conditions” are superfluous. When the manifold has a pole, J. Li [14] ✩

The first author is partially supported by the NSF award DMS-0904653. The second author is partially supported by CNPq and Faperj of Brazil. * Corresponding author. E-mail addresses: [email protected] (Z. Lu), [email protected] (D. Zhou). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.10.010

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Z. Lu, D. Zhou / Journal of Functional Analysis 260 (2011) 3283–3298

proved that the L2 essential spectrum is [0, +∞), if the Ricci curvature of the manifold is nonnegative. Z. Chen and the first author [7] proved the same result when the radical sectional curvature is non-negative. Among the other results in his paper [10], Donnelly proved that the essential spectrum is [0, +∞) for manifold with non-negative Ricci curvature and Euclidean volume growth. In 1997, J.-P. Wang [17] proved that, if the Ricci curvature of a manifold M satisfies Ric(M) −δ/r 2 , where r is the distance to a fixed point, and δ is a positive number depending only on the dimension, then the Lp essential spectrum of M is [0, +∞) for any p ∈ [1, +∞]. In particular, for a complete non-compact manifold with non-negative Ricci curvature, all Lp essential spectra are [0, +∞). Complete gradient shrinking Ricci soliton, which was introduced as singularity model of type I singularities of the Ricci flow, has many similar properties to complete non-compact manifold with non-negative Ricci curvature. From this point of view, we expect the conclusion of Wang’s result is true for a larger class of manifolds, including gradient shrinking Ricci solitons. The first result of this paper is a generalization of Wang’s theorem [17]. Theorem 1. Let M be a complete non-compact Riemannian manifold. Assume that lim RicM (x) = 0.

(1)

x→∞

Then the Lp essential spectrum of M is [0, +∞) for any p ∈ [1, +∞]. It should be pointed out that, contrary to the L2 spectrum, the Lp spectrum of Laplacian may contain non-real numbers. Our proof made essential use of the following result due to Sturm [16]: Theorem 2 (Sturm). Let M be a complete non-compact manifold whose Ricci curvature has a lower bound. If the volume of M grows uniformly sub-exponentially, then the Lp spectra are the same for all p ∈ [1, ∞]. We say that the volume of M grows uniformly sub-exponentially, if for any ε > 0, there exists a constant C = C(ε) such that, for all r > 0 and all p ∈ M, vol Bp (r) C(ε)eεr vol Bp (1) ,

(2)

where we denote Bp (r) the ball of radius r centered at p. Remark 1. Note that by the above definition, a manifold with finite volume may not automatically be a manifold of volume growing uniformly sub-exponentially. For example, consider a manifold whose only end is a cusp and the metric dr 2 + e−r dθ 2 on the end S 1 × [1, +∞). The volume of such a manifold is finite. However, since the volume of the unit ball centered at any point p decays exponentially, it doesn’t satisfy (2). Remark 2. The assumption that the Ricci curvature has a lower bound is not explicitly stated in Sturm’s paper, but is needed in the proof of Theorem 2. Remark 3. Under the assumptions of Theorem 2, all Lp spectra are contained in [0, ∞). Thus for a fixed p, the Lp spectrum being equal to [0, ∞) is equivalent to the Lp essential spectrum


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being equal to [0, ∞). For the sake of simplicity, in this paper, we don’t distinguish the two concepts: the spectrum and the essential spectrum. In [16, Proposition 1], it is proved that if (1) is true, then the volume of the manifold grows uniformly sub-exponentially. Thus in order to prove Theorem 1, we only need to compute the L1 spectrum of the manifold. Using the recent volume estimates obtained by H. Cao and the second author [3], we proved that the essential L1 spectrum of any complete gradient shrinking soliton contains the half line [0, +∞) (see Theorem 6). Combining with Sturm’s Theorem we have Theorem 3. Let M be a complete non-compact gradient shrinking Ricci soliton. If the conclusion of Theorem 2 holds for M, then the Lp essential spectrum of M is [0, +∞) for any p ∈ [1, +∞]. Finally, under additional curvature conditions, we proved Theorem 4. Let (M, gij , f ) be a complete shrinking Ricci soliton. If lim

R

x→+∞ r 2 (x)

= 0,

then the L2 essential spectrum is [0, +∞), where R is the scalar curvature and r(x) is the distance function. We believe that the scalar curvature assumption in the above theorem is technical and could be removed. From [3] the average of scalar curvature is bounded and we know no examples of shrinking solitons with unbounded scalar curvature. 2. Preliminaries Let p0 be a fixed point of M. Let ρ be the distance function to p0 . Let δ(r) be a continuous function on R+ such that (a) limr→∞ δ(r) = 0; (b) δ(r) > 0; (c) Ric(x) −(n − 1)δ(r), if ρ(x) r. Note that δ(r) is a decreasing continuous function. The following lemma is standard: Lemma 1. With the assumption (1), we have lim ρ 0

x→∞

in the sense of distribution. Proof. Let g be a smooth function on R+ such that ⎧ ⎨ g (r) − δ(r)g(r) = 0, g(0) = 0, ⎩ g (0) = 1.

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Then by the Laplacian comparison theorem, we have ρ(x) (n − 1)g ρ(x) /g ρ(x) in the sense of distribution. The proof of the lemma will be completed if we can show that g (r) = 0. r→∞ g(r) lim

By the definition of g(r), we have g(r) 0 and g(r) is convex. Thus g(r) → +∞, as r → +∞. By the L’Hospital Principal, we have (g (r))2 2g (r)g (r) = lim δ(r) = 0, = lim r→+∞ (g(r))2 r→+∞ 2g(r)g (r) r→+∞ lim

2

and this completes the proof of the lemma.

Without loss of generality, for the rest of this paper, we assume that g (r) δ(r) g(r) for all r > 0. The following result is well-known: Proposition 1. There exists a C ∞ function ρ˜ on M such that (a) |ρ˜ − ρ| + |∇ ρ˜ − ∇ρ| δ(ρ(x)), and (b) ρ˜ 2δ(ρ(x) − 1) for any x ∈ M with ρ(x) > 2. Proof. Let {Ui } be a locally finite cover of M and let {ψi } be the partition of unity subordinating to the cover. Let xi = (xi1 , . . . , xin ) be the local coordinates of Ui . Define ρi = ρ|Ui . Let ξ(x) be a non-negative smooth function whose support is within the unit ball of Rn . Assume that ξ(x) dx. Rn

Without loss of generality, we assume that all Ui are open subsets of the unit ball of Rn with coordinates xi . Then for any ε > 0 small enough, ρi,ε =

1 εn

ξ Rn

xi − yi ρi (yi ) dyi ε


3287

is a smooth function whose support is within Ui . Let K(x) =

|ψi | + 2|∇ψi | + 1. i

Then K(x) is a smooth positive function on M. On each Ui , we choose εi small enough such that supp{ρi,εi } ⊂ Ui , |ρi,εi − ρi | δ ρ(x) /K(x), |∇ρi,εi − ∇ρi | δ ρ(x) /K(x), ρi,εi δ ρ(x) − 1 , for ρ(x) > 1.

(3)

Here Lemma 1 is used in the last inequality above. We define ρ˜ =

ψi ρi,εi .

i

The proof follows from the standard method: let’s only prove (b) of the proposition. Since ρ˜ =

ψi ρi,εi + 2∇ψi ∇ρi,εi + ψi ρi,εi ,

i

we have ρ˜ =

ψi (ρi,εi − ρi ) + 2∇ψi (∇ρi,εi − ∇ρi ) + ψi ρi,εi .

i

By (3), we have ρ˜ δ ρ(x) + δ ρ(x) − 1 , and the proposition is proved.

2

Let p0 ∈ M, and let V (r) = vol Bp0 (r) for any r > 0. The main result of this section is (cf. [5,8]). Lemma 2. Assume that (1) is valid. Then for any ε > 0, there is an R1 > 0 such that for r > R1 , we have

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(a) if vol(M) = +∞, then

|ρ| ˜ 2εV (r) + 2vol ∂Bp0 (R1 ) ;

Bp0 (r)\Bp0 (R1 )

(b) if vol(M) < +∞, then

|ρ| ˜ 2ε vol(M) − V (r) + 2vol ∂Bp0 (r) .

M\Bp0 (r)

Proof. By Proposition 1, for any ε > 0 small enough, we can find R1 large enough such that ρ˜ < ε for x ∈ M\Bp0 (R1 ). Thus |ρ| ˜ 2ε − ρ, ˜ and we have Bp0 (R2 )\Bp0 (r)

for any R2 > r > R1 by the Stokes’ Theorem, where direction of the boundary ∂Bp0 (r). By (3), we get

|ρ| ˜ 2ε V (R2 ) − V (r) −

∂ ρ˜ + ∂n

∂Bp0 (R2 ) ∂ ∂n

∂ ρ˜ ∂n

∂Bp0 (r)

is the derivative of the outward normal

1 |ρ| ˜ 2ε V (R2 ) − V (r) − vol ∂Bp0 (R2 ) + 2vol ∂Bp0 (r) . 2

(4)

Bp0 (R2 )\Bp0 (r)

If vol(M) = +∞, then we take R2 = r, r = R1 in the above inequality and we get (a). If vol(M) < +∞, taking R2 → +∞ in (4), we get (b). 2 3. Proof of Theorem 1 In this section we prove the following result which implies Theorem 1. Theorem 5. Let M be a complete non-compact manifold satisfying (1) the volume of M grows uniformly sub-exponentially; (2) the Ricci curvature of M has a lower bound; (3) M satisfies the assertions in Lemma 2. Then the L1 essential spectrum of the Laplacian is [0, ∞). Proof. We essentially follow Wang’s proof [17]. First, using the characterization of the essential spectrum (cf. Donnelly [9, Proposition 2.2]), we only need to prove the following: for any λ ∈ R positive and any positive real numbers ε, μ, there exists a smooth function ξ = 0 such that


3289

(1) supp(ξ ) ⊂ M\Bp0 (μ) and is compact; (2) ξ + λξ L1 < ε ξ L1 . Let R, x, y be big positive real numbers. Assume that y > x + 2R and x > 2R > 2μ + 4. Define a cut-off function ψ : R → R such that (1) supp ψ ⊂ [x/R − 1, y/R + 1]; (2) ψ ≡ 1 on [x/R, y/R], 0 ψ 1; (3) |ψ | + |ψ | < 10. For any given ε, μ and λ, let φ=ψ

√ ρ˜ i λρ˜ e . R

A straightforward computation shows that √ √ 2 √ 1 ψ 2 2 ρ˜ ei λρ˜ φ + λφ = ψ |∇ ρ| ˜ + i λ ψ |∇ ρ| ˜ + i λψ + R R R2 + λφ −|∇ ρ| ˜ 2+1 .

By Proposition 1, |φ + λφ|

C + C|ρ| ˜ + Cδ ρ(x) , R

where C is a constant depending only on λ. Thus we have C + Cδ(x − R) V (y + R) − V (x − R) R +C |ρ|. ˜

φ + λφ L1

(5)

Bp0 (y+R)−Bp0 (x−R)

Case 1: vol(M) = +∞. By Lemma 2, if we choose ε/C small enough and R, x big enough and then assume y is large if necessary, we get

φ + λφ L1 4εV (y + R).

(6)

Note that φ L1 V (y) − V (x). If we choose y big enough, then we have 1

φ L1 V (y). 2

(7)

We claim that there exists a sequence yk → ∞ such that V (yk + R) 2V (yk ). If not, then for a fixed number y, we have V (y + kR) > 2k V (y)

3290


for any k ∈ Z positive. On the other hand, by the uniform sub-exponentially growth of the volume, we have 2k V (y) V (y + kR) C(ε)V (1)eε(y+kR) for any k large and for any ε > 0. This is a contradiction if εR < log 2. Thus there is a y such that V (y + R) 2V (y), and thus by (6), (7), we have

φ + λφ L1 16ε φ L1 . The case when M is of infinite volume is proved. Case 2: vol(M) < +∞. By Lemma 2, 1 + 2ε + δ(x − R) vol(M) − V (x − R) C R + 2Cvol ∂Bp0 (x − R) .

ψ + λψ L1

Let f (r) = vol(M) − V (r). Like above, we choose ε small and R, x big. Then

φ + λφ L1 4εf (x − R) − 2Cf (x − R) for any x, y large enough. On the other hand, we always have

φ L1 f (x) − f (y). Since the volume is finite, we choose y large enough such that 1

φ L1 f (x). 2 Similar to the case of vol(M) = +∞, the theorem is proved if the following statement is true: there is a sequence xk → +∞ such that 2εf (xk − R) − Cf (xk − R) 4εf (xk ) for all k. If there doesn’t exist such a sequence, then for x large enough, we have 2εf (x − R) − Cf (x − R) 4εf (x). Replacing ε by ε/C, we have 2εf (x − R) − f (x − R) 4εf (x), which is equivalent to − e−2εx f (x − R) 4εe−2εx f (x).


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Integrating the expression from x to x + R, using the monotonicity of f (x), we get −e−2ε(x+R) f (x) + e−2εx f (x − R) 2e−2εx 1 − e−2εR f (x + R), which implies f (x − R) 2 1 − e−2εR f (x + R). Let R be big so that 5 2 1 − e−2εR > . 4 Then we have 5 f (x − R) f (x + R) 4 for x large enough. Iterating the inequality, we get k 5 f (x − R) f x + (2k − 1)R 4

(8)

for all positive integer k. On the other hand, we pick points pk so that dist (pk , p0 ) = x + (2k − 1)R + 1. Then by the uniform sub-exponential growth of the volume, for any ε > 0, since Bpk (1) ⊂ M\Bp0 (x + (2k − 1)R), we have f x + (2k − 1)R vol Bpk (1)

1 −ε(x+(2k−1)R+2) e vol Bpk x + (2k − 1)R + 2 . C(ε)

But Bpk (x + (2k − 1)R + 2) ⊃ Bp0 (1) so that there is a constant C, depending on ε and x only such that f x + (2k − 1)R CV (1)e−2εkR . Choosing ε small enough such that 2εR < log 54 , we get a contradiction to (8) when k → ∞.

2

4. Gradient shrinking Ricci soliton A complete Riemannian metric gij on a smooth manifold M is called a gradient shrinking Ricci soliton, if there exists a smooth function f on M n such that the Ricci tensor Rij of the metric gij is given by Rij + ∇i ∇j f = ρgij

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for some positive constant ρ > 0. The function f is called a potential function. Note that by scaling gij we can rewrite the soliton equation as 1 Rij + ∇i ∇j f = gij 2

(9)

without loss of generality. The following basic result on Ricci soliton is due to Hamilton (cf. [13, Theorem 20.1]). Lemma 3. Let (M, gij , f ) be a complete gradient shrinking Ricci soliton satisfying (9). Let R be the scalar curvature of gij . Then we have ∇i R = 2Rij ∇j f, and R + |∇f |2 − f = C0 for some constant C0 . By adding the constant C0 to f , we can assume R + |∇f |2 − f = 0.

(2.1)

We fix this normalization of f throughout this paper. Definition 1. We define the following notations: (i) since R 0 by Lemma 4 below, f (x) 0. Let ρ(x) = 2 f (x); (ii) for any r > 0, let

D(r) = x ∈ M: ρ(x) < r

and V (r) =

dV ;

D(r)

(iii) for any r > 0, let χ(r) =

R dV .

D(r)

The function ρ(x) is similar to the distance function in many ways. For example, by [3, Theorem 20.1], we have r(x) − c ρ(x) r(x) + c, where c is a constant and r(x) is the distance function to a fixed reference point.


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We summarize some useful results of gradient shrinking Ricci soliton in the following lemma without proof: Lemma 4. Let (M, gij , f ) be a complete non-compact gradient shrinking Ricci soliton of dimension n. Then (1) The scalar curvature R 0 (B.-L. Chen [6], see also Proposition 5.5 in [2]). (2) The volume is of Euclidean growth. That is, there is a constant C such that V (r) Cr n (Theorem 2 of [3]). (3) We have 4 nV (r) − 2χ(r) = rV (r) − χ (r) 0. r In particular, the average scalar curvature over D(r) is bounded by (Lemma 3.1 in [3]).

n 2,

i.e. χ(r) n2 V (r)

(4) We have ∇f ∇ρ = √ f

and |∇ρ|2 =

R |∇f |2 = 1 − 1. f f

Using the above lemma, we prove the following result which is similar to Lemma 2. Lemma 5. Let (M, gij , f ) be a complete non-compact gradient shrinking Ricci soliton of dimension n. Then for any two positive numbers x, r with x > r, we have |ρ|

2n V (x) − V (r) + V (r), r

D(x)\D(r)

|ρ| 2

n2 R + 2n max 2 V (x). ρ∈[r,x] ρ r2

D(x)\D(r)

Proof. Since R + f =

n 2

and R 0, we have 1 |∇f |2 n f f ρ = √ − √ 3 √ . 2( f) ρ f f

By the Co–Area formula (cf. [15]), we have,

r V (r) =

ds 0

∂D(s)

1 dA. |∇ρ|

(10)

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Therefore,

1 r dA = |∇ρ| 2

V (r) = ∂D(r)

∂D(r)

1 dA. |∇f |

Thus we have

D(x)\D(r)

n − ρ

|ρ| 2 D(x)\D(r)

D(x)\D(r)

n − ρ

=2 D(x)\D(r)

∂ρ + ∂ν

∂D(x)

n + ρ

2

ρ

D(x)\D(r)

∂D(r)

∂ρ ∂ν

∂D(r)

1 |∇ρ|

2n V (x) − V (r) + V (r), r

(11)

∇ρ where ν = |∇ρ| is the outward normal vector to ∂D. This completes the proof of the first part of the lemma. Now we prove the second part of the lemma. From (10), we have

|∇ρ|2 2f − ρ ρ 1 R 2 n −R − 1− = ρ 2 ρ f

ρ =

=

n − 1 2R 4R − + 2 ρ ρ ρ

−

2R . ρ

(12)

Then

|ρ|2

D(x)\D(r)

D(x)\D(r)

n2 + ρ2

4R 2 ρ2

D(x)\D(r)

4R n2 2 V (x) − V (r) + max 2 χ(x) ρ∈[r,x] ρ r 2 n R + 2n max 2 V (x), ρ∈[r,x] ρ r2

where in the last inequality above we used (3) of Lemma 4.

2

(13)


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Now we are ready to prove Theorem 6. Let (M, gij , f ) be a complete gradient shrinking Ricci soliton. Then the L1 essential spectrum contains [0, +∞). Proof. Similar to that of Theorem 1, we only need to prove the following: for any λ ∈ R positive and any positive real numbers ε, μ, there exists a smooth function ξ = 0 such that (1) supp(ξ ) ⊂ M\Bp0 (μ) and is compact; (2) ξ + λξ L1 < ε ξ L1 . Let a 2 be a positive number. Define a cut-off function ψ : R → R such that (1) supp ψ ⊂ [0, a + 2]; (2) ψ ≡ 1 on [1, a + 1], 0 ψ 1; (3) |ψ | + |ψ | < 10. For any given b 2 + μ, l 2 and λ > 0, let φ=ψ

ρ − b i √λρ e . l

(14)

A straightforward computation shows that φ + λφ =

√ √ √ 2ψ √ ψ ψ 2 2 i λρ i λρ e |∇ρ| ρe |∇ρ| + i λ + i λψ + l l l2 2 + λφ −|∇ρ| + 1 .

By Lemma 4, we have |φ + λφ|

C R + C|ρ| + λ , l f

(15)

where C is a constant depending only on λ. By Lemma 5, we have

φ + λφ L1

C V b + (a + 2)l − V (b) + C l +λ

|ρ|

D(b+(a+2)l)\D(b)

4R ρ2

D(b+(a+2)l)\D(b)

C 2nC V b + (a + 2)l − V (b) + CV (b) + l b 4λ + 2 R b

D(b+(a+2)l)\D(b)

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C 2nC + V b + (a + 2)l − V (b) + CV (b) l b 4λ + 2 χ b + (a + 2)l . b

(16)

From Lemma 4, we can choose l and b large enough so that

φ + λφ L1 εV b + (a + 2)l + CV (b). By a result of Cao–Zhu (cf. [1, Theorem 3.1]), the volume of M is infinite. Therefore we can fix b and let l be large enough so that

φ + λφ L1 2εV b + (a + 2)l .

(17)

On the other hand, note that φ L1 V (b + (a + 1)l) − V (b + l). If we choose a large enough, then we have 1

φ L1 V b + (a + 1)l . 2

(18)

We claim that there exists a sequence ak → ∞ such that V (b + (ak+1 + 2)l) 2V (b + (ak + 1)l). Otherwise for some fixed number a, we have V b + (a + k)l > 2k−1 V b + (a + 1)l for any k 2, which contradicts to the fact that the volume is of Euclidean growth (Lemma 4). Let a be a constant large enough such that V (b + (a + 2)l) 2V (b + (a + 1)l). By (17), (18), we have

φ + λφ L1 8ε φ L1 , and the proof is complete.

2

Proof of Theorem 4. The proof is similar to that of Theorem 6: it suffices to prove the following: for any λ ∈ R positive and any positive real numbers ε, μ, there exists a smooth function ξ = 0 such that (1) supp(ξ ) ⊂ M\Bp0 (μ) and is compact; (2) ξ + λξ L2 < ε ξ L2 . Let a 2 be a positive number. For any given b 2 + μ, l 2 and λ > 0, let φ be defined as in (14). By (15), we have |φ + λφ|2

C R2 2 + C|ρ| + C , l2 f2

where C is a constant depending only on λ. Thus we have


φ + λφ 2L2

C V b + (a + 2)l − V (b) 2 l |ρ|2 + C +C D(b+(a+2)l)\D(b)

3297

16R 2 ρ4

D(b+(a+2)l)\D(b)

R 1 n2 V b + (a + 2)l + + 2n max ρ∈[b,b+(a+2)l] ρ 2 l 2 b2 4C + 2 R b

C

D(b+(a+2)l)\D(b)

C +

R n2 1 V b + (a + 2)l + + 2n max 2 2 2 ρ∈[b,b+(a+2)l] ρ l b

4C χ b + (a + 2)l , 2 b

(19)

where we used Lemma 5 and the fact R f = 14 ρ 2 . From Lemma 4, we can choose l and b large enough so that

φ + λφ 2L2 εV b + (a + 2)l . Note that φ 2L2 V (b + (a + 1)l) − V (b + l). If we choose a big enough, then we have 1

φ 2L2 V b + (a + 1)l . 2

(20)

Since the volume of M is of Euclidean growth, there is a positive number a > 0 such that 1 V b + (a + 1)l V b + (a + 2)l , 2 and therefore we have

φ + λφ 2L2 4ε φ 2L2 . The theorem is proved.

2

5. Further discussions As can be seen clearly in the above context, the key of the proof is the L1 boundedness of ρ. The Laplacian comparison theorem implies the volume comparison theorem. The converse is, in general, not true. On the other hand, the formula1 1 In the sense of distribution.

3298


ρ = vol ∂B(R) − vol ∂B(r)

B(R)\B(r)

clearly shows that volume growth restriction gives the bound of the integral of ρ. Based on this observation, we make the following conjecture Conjecture 1. Let M be a complete non-compact Riemannian manifold whose Ricci curvature has a lower bound. Assume that the volume of M grows uniformly sub-exponentially. Then the Lp essential spectrum of M is [0, +∞) for any p ∈ [1, +∞]. Such a conjecture, if true, would give a complete answer to the computation of the essential spectrum of non-compact manifold with uniform sub-exponential volume growth. The parallel Sturm’s theorem on p-forms was proved by Charalambous [4]. Using that, a similar result of Theorem 1 also holds for p-forms under certain conditions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

[16] [17] [18]

H.D. Cao, Geometry of complete gradient shrinking solitons, arXiv:0903.3927v2. H.D. Cao, Recent Progress on Ricci Solitons, Adv. Lectures Math., vol. 11, 2009, pp. 1–38. H.D. Cao, D. Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010) 175–185. Nelia Charalambous, On the Lp independence of the spectrum of the Hodge Laplacian on non-compact manifolds, J. Funct. Anal. 224 (1) (2005) 22–48, MR2139103 (2006e:58044). Jeff Cheeger, Tobias H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1) (1996) 189–237, doi:10.2307/2118589, MR1405949 (97h:53038). Bing-Long Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2) (2009) 363–382, MR2520796. Zhi Hua Chen, Zhi Qin Lu, Essential spectrum of complete Riemannian manifolds, Sci. China Ser. A 35 (3) (1992) 276–282, MR1183713 (93k:58221). Tobias H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 (3) (1997) 477–501, doi:10.2307/2951841, MR1454700 (98d:53050). Harold Donnelly, On the essential spectrum of a complete Riemannian manifold, Topology 20 (1) (1981) 1–14, doi:10.1016/0040-9383(81)90012-4, MR592568 (81j:58081). Harold Donnelly, Exhaustion functions and the spectrum of Riemannian manifolds, Indiana Univ. Math. J. 46 (2) (1997) 505–527, doi:10.1512/iumj.1997.46.1338, MR1481601 (99b:58230). José F. Escobar, On the spectrum of the Laplacian on complete Riemannian manifolds, Comm. Partial Differential Equations 11 (1) (1986) 63–85, doi:10.1080/03605308608820418, MR814547 (87a:58155). José F. Escobar, Alexandre Freire, The spectrum of the Laplacian of manifolds of positive curvature, Duke Math. J. 65 (1) (1992) 1–21, doi:10.1215/S0012-7094-92-06501-X, MR1148983 (93d:58174). Richard S. Hamilton, The formation of singularities in the Ricci flow, in: Surveys in Differential Geometry, vol. II, Cambridge, MA, 1993, Int. Press, Cambridge, MA, 1995, pp. 7–136, MR1375255 (97e:53075). Jia Yu Li, Spectrum of the Laplacian on a complete Riemannian manifold with nonnegative Ricci curvature which possess a pole, J. Math. Soc. Japan 46 (2) (1994) 213–216, doi:10.2969/jmsj/04620213, MR1264938 (95g:58248). R. Schoen, S.-T. Yau, Lectures on differential geometry, in: Conference Proceedings and Lecture Notes in Geometry and Topology, I, Int. Press, Cambridge, MA, 1994, Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; translated from the Chinese by Ding and S.Y. Cheng; Preface translated from the Chinese by Kaising Tso, MR1333601 (97d:53001). Karl-Theodor Sturm, On the Lp -spectrum of uniformly elliptic operators on Riemannian manifolds, J. Funct. Anal. 118 (2) (1993) 442–453, doi:10.1006/jfan.1993.1150, MR1250269 (94m:58227). Jiaping Wang, The spectrum of the Laplacian on a manifold of nonnegative Ricci curvature, Math. Res. Lett. 4 (4) (1997) 473–479, MR1470419 (98h:58194). De Tang Zhou, Essential spectrum of the Laplacian on manifolds of nonnegative curvature, Int. Math. Res. Not. 5 (1994), doi:10.1155/S1073792894000231, 209 ff., approx. 6 pp. (electronic) MR1270134 (95g:58250).


Generalized coorbit space theory and inhomogeneous function spaces of Besov–Lizorkin–Triebel type Holger Rauhut, Tino Ullrich ∗ Hausdorff Center for Mathematics & Institute for Numerical Simulation, Endenicher Allee 60, 53115 Bonn, Germany Received 29 September 2010; accepted 7 December 2010 Available online 21 December 2010 Communicated by J. Coron Dedicated to Hans Georg Feichtinger on the occasion of his 60th birthday

Abstract Coorbit space theory is an abstract approach to function spaces and their atomic decompositions. The original theory developed by Feichtinger and Gröchenig in the late 1980ies heavily uses integrable representations of locally compact groups. Their theory covers, in particular, homogeneous Besov–Lizorkin–Triebel spaces, modulation spaces, Bergman spaces and the recent shearlet spaces. However, inhomogeneous Besov–Lizorkin–Triebel spaces cannot be covered by their group theoretical approach. Later it was recognized by Fornasier and Rauhut (2005) [24] that one may replace coherent states related to the group representation by more general abstract continuous frames. In the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces. A unified approach towards atomic decompositions and Banach frames with new results for general coorbit spaces is presented. In the second part we apply the abstract setting to a specific framework and study coorbits of what we call Peetre spaces. They allow to recover inhomogeneous Besov–Lizorkin–Triebel spaces of various types of interest as coorbits. We obtain several old and new wavelet characterizations based on explicit smoothness, decay, and vanishing moment assumptions of the respective wavelet. As main examples we obtain results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal spaces, Besov–Lizorkin–Triebel–Morrey spaces, spaces of dominating mixed smoothness and even mixtures of the mentioned ones. Due to the generality of our approach, there are many more examples of interest where the abstract coorbit space theory is applicable. © 2010 Elsevier Inc. All rights reserved.


E-mail addresses: [email protected] (H. Rauhut), [email protected] (T. Ullrich). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.006

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H. Rauhut, T. Ullrich / Journal of Functional Analysis 260 (2011) 3299–3362

Keywords: Coorbit space theory; Besov–Lizorkin–Triebel type spaces; Peetre maximal function; 2-microlocal spaces; Muckenhoupt weights; Doubling weights; Morrey spaces; Dominating mixed smoothness; Atomic decomposition; Wavelet bases

1. Introduction Coorbit space theory was originally developed by Feichtinger and Gröchenig [20,27,28] with the aim to provide a unified approach for describing function spaces and their atomic decompositions, that is, characterizations via (discrete) sequence spaces. Their theory uses locally compact groups together with an integrable group representation as a key ingredient. The idea is to measure smoothness via properties of an abstract wavelet transform (the voice transform) associated to the integrable group representation. More precisely, one asks whether the transform is contained in certain function spaces (usually Lp -spaces) on the index set of the transform, which is the underlying group. As main examples classical homogeneous Besov–Lizorkin– Triebel spaces [53–55] can be identified as coorbit spaces [61], and the abstract theory provides characterizations via wavelet frames. Also modulation spaces and characterizations via Gabor frames [29,18], Bergman spaces [20], and the more recent shearlet spaces [13] can be treated via classical coorbit space theory. In [42] this theory was extended in order to treat also quasi-Banach function spaces. Later it was recognized that certain transforms and associated function spaces of interest do not fall into the classical group theoretical setting, and the theory was further generalized from groups to the setting of homogeneous spaces, that is, quotients of groups via subgroups [14, 15,12]. Examples of spaces that fall into this setup are modulation spaces on the sphere [14], as well as α-modulation spaces [12]. The latter were originally introduced by Feichtinger and Gröbner as “intermediate” spaces (but not interpolation spaces) between modulation spaces and Besov spaces [19,26]. In another direction, the first named author developed a coorbit theory in the setup of spaces of functions obeying symmetries such as radiality [39,40]. Here, one takes the set of residue classes of the locally compact group modulo a symmetry group leading to a hypergroup structure. In concrete setups, the theory provides then frames of radial wavelets (that is, each frame element is a radial function) for radial homogeneous Besov–Lizorkin–Triebel spaces, as well as, radial Gabor frames for radial modulation spaces. Coorbit space theory can then be used to show compactness of certain embeddings when restricting modulation spaces to radial functions [41]. As the next step, the first named author together with Fornasier realized that group theory is not needed at all in order to develop a coorbit space theory [24]. The starting point is now an abstract continuous frame [1], which induces an associated transform. Then one measures “smoothness” via the norm of the transform in suitable function spaces on the index set of the continuous frame. Under certain integrability and continuity properties of the continuous frame, again discrete Banach frames for the associated coorbit spaces can be derived via sampling of the continuous frame. All the setups of coorbit space theory mentioned above fall into this generalization (except that the theory for quasi-Banach spaces still needs to be extended). The advantage of the group theoretical setup is only that some of the required conditions are automatically satisfied, while in this general context they enter as additional assumptions, which means that they have to be checked in a concrete situation. While the theory in [24] essentially applies only to coorbit spaces with respect to weighted Lebesgue spaces, we extend this abstract theory in the present paper in order to treat a wider va-


3301

riety of coorbit spaces. Our main motivation is to cover inhomogeneous Besov–Lizorkin–Triebel spaces and generalizations thereof. Those spaces indeed do not fit into any of the group theoretical approaches which were available before. In order to handle them in full generality, one needs to take coorbits with respect to more complicated spaces rather than only weighted Lebesgue spaces. Indeed, we will need (weighted) mixed Lp,q -spaces. We derive characterizations of such general coorbits via discrete Banach frames and atomic decomposition, i.e., characterizations using discrete sequence spaces. Such are very useful in order to study embeddings, s-numbers, interpolation properties, etc., because the structure of sequence spaces is usually much easier to investigate. We further treat the identification as coorbits of inhomogeneous Besov–Lizorkin–Triebel type spaces in detail (Section 4). The application of our general abstract coorbit space theory from Section 3 leads to concrete atomic decompositions and wavelet characterizations of the mentioned spaces. Such discretizations have a certain history. A remarkable breakthrough in the theory was achieved by Frazier, Jawerth [25] with the invention of the ϕ-transform. They fixed the notion of smooth atoms and molecules as building blocks for classical function spaces. Afterwards many authors have dealt with wavelet characterizations of certain generalizations of Besov–Lizorkin–Triebel spaces in the past. To mention all the relevant contributions to the subject would go beyond the scope of this paper. We rather refer to the monograph [56, Chapt. 2, 3], the references given there and to our overview Section 2. Our results on wavelet basis characterizations in this paper rely on the abstract discretization result in Theorem 3.14 below, which allows to use orthogonal and even biorthogonal wavelets as well as tight (discrete) wavelet frames. We are able to come up with a suitable definition of weighted Besov–Lizorkin–Triebel spaces and their wavelet characterizations when the weight is only assumed to be doubling. Muckenhoupt Ap -weights fall into this class of weights, but there exist doubling weights for which a proper notion of Besov–Lizorkin–Triebel spaces was more or less unavailable before, although there exist certain attempts, see for instance [4]. In addition, we treat generalized 2-microlocal spaces, Morrey–Besov–Lizorkin–Triebel spaces, and Besov–Lizorkin–Triebel spaces of dominating mixed smoothness and their characterizations via wavelet bases. The treatment of spaces with variable integrability, or more general, with parameters p, q, s depending on x, will be considered in a subsequent contribution. As another main feature we also provide a better way to identify Lizorkin–Triebel type spaces as coorbits. So far, the (homogeneous) Lizorkin–Triebel spaces have been identified as coorbits of so-called tent spaces [27,28] on the ax + b group. However, tent spaces [11] are rather complicated objects. In this paper, we proceed by introducing a Peetre type maximal function, related to the one introduced in [38], as well as corresponding function spaces on the index set of the (continuous wavelet) transform. Then Lizorkin–Triebel spaces can also be identified as coorbits with respect to these new spaces, which we call Peetre spaces. This was recently accomplished for the homogeneous spaces [61]. It turns out that Peetre spaces are much easier to handle than tent spaces. In the present paper we restrict our considerations to coorbit space theory for Banach spaces. While an extension to the setting of quasi-Banach spaces is available for classical coorbit space theory [42], such extension is more technical for general coorbit spaces, and currently under development. We expect similar results also in this situation. In order to be well prepared for applying this generalized coorbit space theory for quasi-Banach spaces once it is developed in detail, we state certain characterizations of generalized Besov–Lizorkin–Triebel type spaces also for the quasi-Banach space cases p, q < 1 – although we do not need such cases in the present contribution.

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We hope to convince the reader with this paper that the abstract coorbit space theory is a very powerful tool, and allows a unified treatment of function spaces. In contrast, the theory leading to atomic decompositions, wavelet characterizations of several newly introduced function spaces is often developed from scratch. We believe, that most of these spaces can be interpreted as coorbit spaces in our setting. Once this is established, then one has to follow an easy recipe checking only basic properties, in order to come up with corresponding discrete characterizations. These can be widely applied for approximation issues, to prove certain embeddings, interpolation formulas, etc. While our main focus in this paper is on inhomogeneous Besov–Lizorkin–Triebel spaces, we expect that the principles of the abstract coorbit space theory apply also to other setups. To be more precise, we expect that our theory can be used to introduce also inhomogeneous shearlet spaces, and their atomic decompositions, and α-modulation spaces with different p, q-indices (the paper [12] only treats the case p = q). The paper is structured as follows. After setting some basic notation we give an overview over the main results and achievements of the paper in Section 2. Section 3 is devoted to the extension of the abstract generalized coorbit space theory from [24]. In Section 4 we apply this abstract theory to the specific situation of coorbits with respect to Peetre type spaces. We study several examples in Section 5 and give concrete discretizations for generalized inhomogeneous Besov–Lizorkin–Triebel spaces of various type in terms of wavelet bases with corresponding sufficient conditions for admissible wavelets. Appendix A contains some basic facts concerning orthonormal wavelet bases on R and Rd , in particular, orthonormal spline wavelets. 1.1. Notation To begin with we introduce some basic notation. The symbols R, R+ , C, N, N0 and Z denote the real numbers, positive real numbers, complex numbers, natural numbers, natural numbers including 0 and the integers. Let us emphasize that Rd has the usual meaning and d is reserved for its dimension. The elements are denoted by x, y, z, . . . and |x| is used for the Euclidean norm. We use |k|1 for the d1 -norm of a vector k. Sometimes the notation a¯ is used to indicate that we deal with vectors a¯ = (a1 , . . . , ad ) taken from Rd . The notation a¯ > b, where b ∈ R, means ai > b for every i = 1, . . . , d. If X is a (quasi-)Banach space and f ∈ X we use f |X or simply f for its (quasi-)norm. The class of linear continuous mappings from X to Y is denoted by L(X, Y ) or simply L(X) if X = Y . Operator (quasi-)norms of A ∈ L(X, Y ) are denoted by A : X → Y , or simply by A. As usual, the letter c denotes a constant, which may vary from line to line but is always independent of f , unless the opposite is explicitly stated. We also use the notation a b if there exists a constant c > 0 (independent of the context dependent relevant parameters) such that a cb. If a b and b a we write a b. For a real number t, we denote t+ = max{t, 0} and t− = min{t, 0}. The ball in Rd with center x ∈ Rd and radius r > 0 is denoted by B(x, r) = {y ∈ Rd , |x − y| r}, while |B(x, r)| is its volume. 1.2. Lebesgue spaces and tempered distributions For a measure space (X, μ) and a positive measurable weight function w : X → R, we define the space Lw p (X, μ), 1 p < ∞, as usual by F |Lw (X, μ) :=

p

X

1/p w(x)F (x)p dμ(x) < ∞.


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A function F belongs to Lw ∞ (X, μ) if and only if F w is essentially bounded with respect to the measure μ. If w ≡ 1 we simply write Lp (X, μ) instead of Lw p (X, μ). Moreover, the space loc L1 (X, μ) contains all functions F for which the integral over all subsets of finite measure K ⊂ X is finite. If X = Rd and the measure μ is the Lebesgue measure dx then we write Lp (Rd ). For a measurable weight function v > 0, the space Lp (Rd , v), 0 < p ∞, is the collection of all functions F such that F |Lp Rd , v :=

F (x)p v(x) dx

1/p < ∞,

(1.1)

Rd

i.e., it coincides with Lp (X, μ) where X = Rd and dμ(x) = v(x) dx. As usual S(Rd ) is used for the locally convex space of rapidly decreasing infinitely differentiable functions on Rd where its topology is generated by the family of semi-norms ϕk, =

sup

α¯ D ϕ(x) 1 + |x| k ,

x∈Rd , |α| ¯ 1

ϕ ∈ S Rd ,

where k, ∈ N0 . The space S (Rd ), the topological dual of S(Rd ), is also referred to as the set of tempered distributions on Rd . Indeed, a linear mapping f : S(Rd ) → C belongs to S (Rd ) if and only if there exist numbers k, ∈ N0 and a constant c = cf such that f (ϕ) cf

sup x∈Rd , |α| ¯ 1

α¯ D ϕ(x) 1 + |x| k

for all ϕ ∈ S(Rd ). The space S (Rd ) is equipped with the weak∗ -topology. The convolution ϕ ∗ ψ of two integrable (square integrable) functions ϕ, ψ is defined via the integral (ϕ ∗ ψ)(x) =

ϕ(x − y)ψ(y) dy.

(1.2)

Rd

If ϕ, ψ ∈ S(Rd ) then (1.2) still belongs to S(Rd ). The convolution can be extended to S(Rd ) × S (Rd ) via (ϕ ∗ f )(x) = f (ϕ(x − ·)). It is a pointwise defined C ∞ -function in Rd of at most polynomial growth. The Fourier transform defined on both S(Rd ) and S (Rd ) is given by f (ϕ) := f ( ϕ ), where f ∈ S (Rd ), ϕ ∈ S(Rd ), and ϕ (ξ ) := (2π)−d/2

e−ix·ξ ϕ(x) dx.

Rd

ϕ (−·). The Fourier transform is a bijection (in both cases) and its inverse is given by ϕ ∨ =

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1.3. The continuous wavelet transform Many considerations in this paper are based on decay results for the continuous wavelet transform Wg f (x, t). A general reference for this notion is provided by the monograph [16, 2.4]. In [61, App. A] the second named author provided decay results based on the following setting. For x ∈ Rd and t > 0 we define the unitary dilation and translation operators DtL2 and Tx by DtL2 g := t −d/2 g

· and Tx g := g(· − x), t

g ∈ L2 R d .

The wavelet g is said to be the analyzing vector for a function f ∈ L2 (Rd ). The continuous wavelet transform Wg f is then defined by

Wg f (x, t) = Tx DtL2 g, f ,

x ∈ Rd , t > 0,

where the bracket ·,· denotes the inner product in L2 (Rd ). We call g an admissible wavelet if cg := Rd

| g (ξ )|2 dξ < ∞. |ξ |d

If this is the case, then the family {Tx DtL2 g}t>0, x∈Rd represents a tight continuous frame in L2 (R) where C1 = C2 = cg (see Section 2.1). The decay of the function |Wg f (x, t)| mainly depends on the number of vanishing moments of the wavelet g as well as on the smoothness of g and the function f to be analyzed, as is made precise in the following definition. Definition 1.1. Let L + 1 ∈ N0 , K > 0. We define the properties (D), (ML ) and (SK ) for a function f ∈ L2 (Rd ) as follows. (D) For every N ∈ N there exists a constant cN such that f (x)

cN . (1 + |x|)N

(ML ) All moments up to order L vanish, i.e., x α f (x) dx = 0 Rd

for all α ∈ Nd0 such that |α|1 L. (SK ) The function K 1 + |ξ | D α f (ξ ) belongs to L1 (Rd ) for every multi-index α ∈ Nd0 .


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Property (SK ) is rather technical. Suppose we have a function f ∈ C K+d+1 (Rd ) for some K ∈ N such that f itself and all its derivatives satisfy (D). The latter holds, for instance, if f is compactly supported. Then this function satisfies (SK ) by elementary properties of the Fourier transform. Conversely, if a function g ∈ L2 (Rd ) satisfies (SK ) for some K > 0 then we have g ∈ C K (Rd ). However, in case of certain wavelet functions ψ where the Fourier transform F ψ is given explicitly (see Appendix A.1) we can verify (SK ) directly. Depending on these conditions we state certain decay results for the function |Wg f (x, t)| in Lemma 4.17 below. 2. Overview on main results As suggested in [24] coorbit space theory can be generalized to settings without group structure, and thereby allows the treatment of even more function spaces via coorbit space theory. We follow this path and develop the theory even further. Our main application are inhomogeneous Besov–Lizorkin–Triebel spaces with several generalizations and their wavelet characterizations. 2.1. Abstract coorbit space theory In this section we give a brief overview before going into details later in Section 3. Assume H to be a separable Hilbert space and X be a locally compact Hausdorff space endowed with a positive Radon measure μ with supp μ = X. A family F = {ψx }x∈X of vectors in H is called a continuous frame if there exist constants 0 < C1 , C2 < ∞ such that C1 f |H2

f, ψx 2 dμ(x) C2 f |H2

for all f ∈ H.

(2.1)

X

For the sake of simplicity, we assume throughout this paper that ψx |H C, x ∈ X, and that the continuous frame is tight, i.e., C1 = C2 . After a possible re-normalization we may assume that C1 = C2 = 1. We note, however, that non-tight frames appear also in several relevant examples and the associated coorbit theory is worked out in [24] – at least to a significant extent. (The generalizations in this paper can also be developed in the setting of non-tight frames.) Associated to a continuous frame we define the transform V = VF : H → L2 (X, μ) by VF f (x) = f, ψx ,

f ∈ H, x ∈ X,

and its adjoint VF∗ : L2 (X, μ) → H, ∗

VF F =

F (y)ψy dμ(y). X

Since we assume the frame F to be tight, i.e., C1 = C2 = 1 in (2.1), the operator VF∗ VF is the identity. Hence, f=

(VF f )(y)ψy dμ(y) and VF f (x) =

X

VF f (y) ψy , ψx dμ(y). X

(2.2)

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It follows from the tightness of the frame that VF∗ VF is a multiple of the identity, so that the transform VF is invertible (more precisely, has a left-inverse). The second identity in (2.2) is the crucial reproducing formula F = R(F ) on the image of H under VF , where R(x, y) = RF (x, y) = ψy , ψx ,

x, y ∈ X,

is an integral kernel (operator). The idea of coorbit space theory is to measure “smoothness” of f via properties, i.e., suitable norms of the transform VF f . Under certain integrability properties of the kernel R(x, y), see (3.1) in Section 3.1, one can introduce a suitable space Hv1 of test functions and its dual (Hv1 )∼ (which plays the role of the tempered distributions in this abstract context, see (3.6)), and extend the definition of the transform VF to (Hv1 )∼ in (3.7). Then associated to a solid Banach space Y of locally integrable functions on X (see Definition 3.5), one defines the coorbit space

Co Y = f ∈ Hv1

∼

: VF f ∈ Y ,

f |Co Y := VF f |Y ,

provided that, additionally, the kernel RF acts continuously from Y into Y as an integral operator. The latter is expressed as RF being contained in an algebra BY,m of kernels, see (3.4). Then Co Y is a Banach space, and one can show that “similar” frames (in the sense that their cross Gramian kernel satisfies suitable integrability properties) define the same coorbit spaces, see Lemma 3.6. A key feature of coorbit space theory is the discretization machinery, which provides discrete frames, and characterizations of coorbit spaces Co Y via suitable sequence spaces Y and Y . This is, of course, very useful because many properties, such as embeddings, s-numbers, etc., are much easier to analyze for sequence spaces. Here, the starting point is a suitable covering U = {Ui }i∈I of the space X, of compact subsets Ui ⊂ X. One defines the U -oscillation kernel oscU (x, y) := sup ϕx , ϕy − ϕz z∈Qy

where Qy = y∈Ui Ui . This kernel can be viewed as a sort of modulus of continuity associated to the frame F and the covering U . If osc together with its adjoint osc∗ is also contained in the algebra BY,m , see (3.4), then one obtains a discrete Banach frame and atomic decompositions by subsampling the continuous frame at points xi ∈ Ui , that is, Fd = {ϕxi }i∈I , see Theorem 3.11 for details. In particular, the coorbit space Co Y is discretized by the sequence space Y with norm {λi }i∈I |Y = |λi |χUi Y . i∈I

Another new important key result of the abstract theory is that orthogonal and biorthogonal basis expansions, as well as tight frame expansions, where the basis/frame elements are sampled from a continuous frame, extend automatically from the Hilbert space H to coorbit spaces under certain natural conditions, see Theorem 3.14. In addition, these basis/frame expansions characterize the respective coorbit space. For the concrete setup of characterizing generalized Besov–Lizorkin–Triebel spaces these “natural conditions” reduce to certain moment, decay and smoothness conditions (Definition 1.1) on the used wavelet and dual wavelet. In Section 5 we give sufficient conditions for the orthonormal wavelet characterization of several common as well as new generalizations of the inhomogeneous Besov–Lizorkin–Triebel spaces.


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The described setup indeed generalizes the original group theoretical one due to Feichtinger and Gröchenig, see [24] for details. For convenience of the reader we briefly summarize the main innovations and advances with respect to previous results. 2.1.1. Main contribution and novelty • Feichtinger and Gröchenig used group representations as an essential ingredient in their initial work on coorbit space theory [20–22,28,23]. The formulation of the theory in the present paper gets completely rid of group theory and uses general continuous frames instead. This general approach was initiated in [24]. The present paper even removes certain strong restrictions on the spaces Y to treat a wider variety of coorbit spaces. • We provide characterizations of general coorbit spaces Co Y by (discrete) Banach frames or atomic decompositions under suitable conditions (Theorem 3.11). There is an easy and explicit connection of the corresponding sequence space Y to the function space Y . This discretization machinery may be useful in situations, where it is even hard to construct a related basis or frame for the underlying Hilbert space H. • In several cases, an orthonormal basis, a Riesz basis or a tight frame for the Hilbert space H that arises from samples of a continuous frame can be constructed directly via methods outside coorbit space theory. Then under natural conditions on the continuous frame, Theorem 3.14 below shows that the corresponding expansions and characterizations automatically extend to the coorbit spaces. This represents one of the core results in the present paper. It generalizes a result from classical coorbit theory (associated to group representations) due to Gröchenig [27]. In contrast to his approach, our proof is independent from the discretization machinery in Theorem 3.11. • Our extended coorbit theory allows to identify a large class of function spaces as coorbits. Therefore, the abstract discretization machinery is available to such function spaces. We emphasize that due to this unified approach, the theory leading to atomic decompositions for several classes of spaces does not have to be developed from scratch over and over again for each new class of function spaces. From this point of view there are numerous previous results on atomic decompositions, which are partly recovered as well as extended by our theory. With a similar intention Hedberg and Netrusov gave an axiomatic approach to function spaces of Besov–Lizorkin–Triebel type in their substantial paper [32]. Their approach is different from ours but also leads to atomic decompositions in a unified way. In a certain sense our approach is more flexible since the abstract theory in Section 3 is also applicable to, e.g., the recent shearlet spaces [13] as well as modulation spaces. 2.2. Inhomogeneous Besov–Lizorkin–Triebel type spaces In order to treat inhomogeneous spaces of Besov–Lizorkin–Triebel type, see [53,55,56] and the references given there, we introduce the index set X = Rd × [(0, 1) ∪ {∞}], where “∞” denotes an isolated point, and define the Radon measure μ by 1

F (x) dμ(x) = X

F (x, s) Rd 0

ds s d+1

dx + Rd

F (x, ∞) dx.

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The main ingredient is a solid Banach space Y of functions on X. We use two general scales w PB,q,a (X) and Lw B,q,a (X) of spaces on X. Here, we have 1 q ∞ and a > 0. The parameter d B = B(R ) is a solid space of measurable functions on Rd in the sense of Definition 4.4 below, for instance, a weighted Lebesgue space. For a function F : X → C, the Peetre type maximal function Pa F defined on X is given by Pa F (x, t) := sup z∈Rd

Pa F (x, ∞) := sup z∈Rd

|F (x + z, t)| , (1 + |z|/t)a

x ∈ Rd , 0 < t < 1,

|F (x + z, ∞)| , (1 + |z|)a

x ∈ Rd .

(2.3)

The function w : X → R+ is a weight function satisfying the technical growth conditions (W1) and (W2) in Definition 4.1. Then the Peetre spaces and Lebesgue spaces are defined as w PB,q,a (X) := F : X → C: Lw B,q,a (X) := F : X → C:

F |P w < ∞ , B,q,a

F |Lw < ∞ , B,q,a

with respective norms 1 1/q d q dt d + w(·, t)Pa F (·, t) d+1 B R , B,q,a := w(·, ∞)Pa F (·, ∞)|B R t

F |P w

0

F |Lw

B,q,a

:= w(·, ∞)Pa F (·, ∞)|B Rd +

1

w(·, t)Pa F (·, t)|B Rd q dt t d+1

(2.4) 1/q .

0

(2.5) We give the definition of an admissible continuous frame F on X. Definition 2.1. A continuous (wavelet) frame F = {ϕx }x∈X , X = Rd × [(0, 1) ∪ {∞}], is admissible if it is of the form ϕ(x,∞) = Tx Φ0

and ϕ(x,t) = Tx DtL2 Φ,

> 0 on {x: 1/2 < |x| < 2} and where Φ denotes a radial function from S(Rd ) satisfying Φ Rd

)|2 |Φ(ξ dξ = 1. |ξ |d

(2.6)


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We further assume that Φ has infinitely many vanishing moments (see Definition 1.1). This vanishes on {x: |x| < 1/2}. The function Φ0 ∈ S(Rd ) is condition is satisfied, for instance, if Φ chosen such that Φ 0 (ξ )2 +

1

dt Φ(tξ )2 = 1. t

0

The functions Φ and Φ0 from Definition 2.1 satisfy (D) and (SK ) for every K > 0. Additionally, Φ satisfies (ML ) for any L ∈ N. Moreover, the continuous frame (2.6) represents a tight continuous frame in the sense of (2.1). Indeed, we apply Fubini’s and Plancherel’s theorem to get f |L2 Rd 2 =

f, ϕ(x,∞) 2 +

Rd

1

f, ϕ(x,t) 2 dt t d+1

dx =

f, ϕx 2 dμ(x).

X

0

The transform VF on H = L2 (Rd ) is then given by VF f (x) = f, ϕx , x ∈ X. With these ingredients at hand, the associated coorbit spaces are given as

w w w Co PB,q,a := Co PB,q,a , F = f ∈ S : VF f ∈ PB,q,a (X) , w

w Co Lw B,q,a := Co LB,q,a , F = f ∈ S : VF f ∈ LB,q,a (X) . w The spaces Co Lw B,q,a can be interpreted as generalized Besov spaces, while the spaces Co PB,q,a serve as generalized Lizorkin–Triebel spaces. Below we use the abstract machinery of coorbit space theory to show that these are Banach spaces, and we provide characterizations by wavelet bases, in particular, by orthonormal spline wavelets, see Appendix A.1. We will recover known and new spaces, as well as known and new wavelet characterizations. We shortly give some examples.

• Classical inhomogeneous Besov and Lizorkin–Triebel spaces. Here we take 1 p, q ∞, α ∈ R, wα (x, t) = t −α−d/2+d/q , B(Rd ) = Lp (Rd ) and a > d/ min{p, q}. Then

w

α α Rd , Co PB,q,a = Fp,q

w

α α Rd , Co LB,q,a = Bp,q

(2.7)

α (Rd ) is the classical Lizorkin–Triebel space and B α (Rd ) the classical Besov where Fp,q p,q space, see [53]. • Weighted inhomogeneous Besov and Lizorkin–Triebel spaces. For a doubling weight v and B(Rd ) = Lp (Rd , v) we obtain w

α α Rd , v , Co PB,q,a = Fp,q

w

α α Rd , v . Co LB,q,a = Bp,q

Note, that there are doubling weights which do not belong to the Muckenhoupt class A∞ . We provide a reasonable definition of the respective spaces (Definition 5.16) and atomic decompositions also in this situation, see Section 5.2.

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• Generalized 2-microlocal spaces. The identities in (2.7) remain valid if we replace the α (Rd ) and B α (Rd ) by the generalized 2-microlocal spaces F w (Rd ) and spaces Fp,q p,q p,q α w d Bp,q (R ) where w ∈ Wα13,α2 is an admissible 2-microlocal weight, see Definition 4.1. • Besov–Lizorkin–Triebel–Morrey type spaces. By putting B(Rd ) = Mu,p (Rd ), where the latter represents a Morrey space, we obtain a counterpart of (2.7) also for Besov–Lizorkin– Triebel–Morrey spaces. Furthermore, with a slightly different setup we also treat: • Besov–Lizorkin–Triebel spaces of dominating mixed smoothness. If 1 p, q ∞, r¯ ∈ Rd , and a¯ > 1/ min{p, q} we will show r¯ +1/2−1/q

Co Pp,q,a¯

r¯ = Sp,q F Rd ,

r¯ +1/2−1/q

Co Lp,q,a¯

r¯ = Sp,q B Rd ,

r¯ F (Rd ), S r¯ B(Rd ) are Lizorkin–Triebel and Besov spaces of mixed dominating where Sp,q p,q smoothness, see e.g. [48,62].

Let us summarize the innovations and main advances of our considerations with respect to the theory of Besov–Lizorkin–Triebel spaces. 2.2.1. Main contribution and novelty • We work out in detail the application of the general abstract coorbit theory to inhomogeneous function spaces of Besov–Lizorkin–Triebel type. We further create an easy recipe for finding concrete atomic decompositions which is applicable to numerous examples of well-known spaces on the one hand and new generalizations on the other hand. In Section 4 we give a very general definition of the family of Besov–Lizorkin–Triebel type spaces as coorbits of Peetre type spaces, see (2.4) and (2.5). These depend on a weight function w on X and a Banach space B on Rd . Indeed, our conditions on w and B (Definitions 4.1 and 4.4) are rather general but, however, allow for introducing and analyzing corresponding coorbit spaces. To the best knowledge of the authors Besov–Lizorkin–Triebel spaces have not yet been introduced in this generality. • In the classical literature on coorbit spaces, tent spaces [11] are used to identify homogeneous Lizorkin–Triebel spaces as coorbits. While tent spaces are rather complicated objects, our newly introduced Peetre type spaces are much easier to handle. Their structure (2.4), (2.5) allows for the definition of inhomogeneous spaces. Indeed, combined with Proposition 4.8 this represents one of the core ideas in the present paper. • The conditions on w and the space B(Rd ) in Definitions 4.1 and 4.4 involve parameters α1 , α2 , α3 , δ1 , δ2 , γ1 , γ2 . We identify explicit conditions on the smoothness K and number L of vanishing moments (see Definition 1.1) of wavelets in terms of these parameters, which allow to provide characterization of the generalized Besov–Lizorkin–Triebel spaces via wavelet bases (Theorem 4.25). While we state the result only for orthonormal wavelet bases it easily extends to biorthogonal wavelets. The corresponding sequence spaces are studied in detail. • In Section 5 we identify several known generalizations of inhomogeneous Besov–Lizorkin– Triebel type spaces as coorbits and generalize even further, see Theorems 5.7, 5.17, 5.21, and 5.30. This requires some effort since the spaces are usually not given in terms of continu-


3311

ous characterizations, see [61]. We provide these characterizations along the way, see Proposition 5.6 and the paragraphs before Theorems 5.17, 5.21. In particular, our analysis includes classical Besov–Lizorkin–Triebel spaces, 2-microlocal Besov–Lizorkin–Triebel spaces with Muckenhoupt weights and 2-microlocal Besov–Lizorkin–Triebel–Morrey spaces. Moreover, we introduce Besov–Lizorkin–Triebel spaces with doubling weights, which are not necessarily Muckenhoupt. In the latter case, a “classical” definition is not available (but see [4]), and we emphasize that coorbit space theory provides a natural approach for such spaces as well. • Special cases of our result concerning wavelet bases characterizations of Muckenhoupt weighted 2-microlocal Besov–Lizorkin–Triebel spaces in Theorems 5.8, 5.10, 5.12 already appeared in the literature. Indeed, see Theorem 3.10 in [31], Theorem 4 in [34], Theorem 1.20 in [57], Theorem 3.5 in [56], or Propositions 5.1, 5.2 in [36]. Our result concerning decompositions of Morrey type spaces, Theorem 5.22, has a special case in [44] and in the recent monograph [64, Thm. 4.1]. In the mentioned references the conditions on smoothness and cancellation (moment conditions) are often slightly less restrictive than ours for this particular case. But this fact might be compensated by the unifying nature of our approach. However, compared to the conditions in [56, Thm. 3.5] our restrictions in Theorem 5.12 are similar. Concerning characterizations of classical Besov spaces with orthonormal spline wavelets, see Appendix A.1, we refer to [3] for the optimal conditions with respect to the order m. • For technical reasons several authors restrict to compactly supported atoms [56, Sect. 3.1.3], [57, Sect. 1.2.2], [62, Sect. 2.2, 2.4], [64, Thm. 4.1], especially to wavelet decompositions using the well-known compactly supported but rather complicated Daubechies wavelet system [16]. In the literature more general atoms are called molecules. This term goes back to Frazier, Jawerth [25, Thm. 3.5]. Several authors [36,34,44,64,4] used their techniques in order to generalize results in certain directions. In this sense our approach is already sufficiently general because we allow arbitrary orthonormal (biorthogonal) wavelets having sufficiently large smoothness, vanishing moments, and decay. • By a slight variation of the setup of Section 4 we also identify inhomogeneous Besov– Lizorkin–Triebel spaces of dominating mixed smoothness as coorbit spaces and derive corresponding wavelet characterizations with explicit smoothness and moment conditions on the wavelets (Theorem 5.31). In contrast to most previous results [62,58], we are not restricted to compact support. In particular, we obtained characterizations via orthonormal spline wavelets in Corollary 5.32, which are comparable (with respect to the order of the splines) to the very recent results in the monograph [58, Sect. 2.5]. Furthermore, since our arguments are based on the abstract Theorem 3.14, our results extend in a straightforward way to discretization results using numerically convenient biorthogonal wavelets [10]. 2.3. Further extensions and applications We conclude this section with a list of further possible extensions and applications of our work. • The discrete wavelet characterizations derived in this paper allow to reduce many questions on function spaces to related questions on the associated sequence spaces. For instance, the study of embeddings or the computation of certain widths such as entropy, (non-)linear ap-

3312


proximation, Kolmogorov, Gelfand, . . . , are much more straightforward by using the stated sequence space isomorphisms. This can be seen as the major application of our theory. • Our abstract approach would clearly allow to incorporate further extensions of Besov– Lizorkin–Triebel type spaces. For instance, one might think of coorbits with respect to (weighted) Lorentz spaces or (weighted) Orlicz spaces, or one may introduce weights also in spaces of dominating mixed smoothness. Another recent development considers variable exponents where p, q are not constant but actually functions of the space variable. The general theory would then provide also wavelet characterizations of such spaces. • The abstract approach allows to handle also function spaces of different type than Besov– Lizorkin–Triebel spaces, such as (inhomogeneous) shearlet spaces or modulation spaces. For instance, it would be interesting to work out details for modulation spaces with Muckenhoupt weights [45]. • The abstract coorbit space theory in the present stage applies only to Banach spaces. An extension to quasi-Banach spaces, similar to the classical case in [42], is presently under investigation. 3. General coorbit space theory The classical coorbit space theory due to Feichtinger and Gröchenig [20,27,21,22,28] can be generalized in various ways. One possibility is to replace the locally compact group G by a locally compact Hausdorff space X without group structure equipped with a positive Radon measure μ that replaces the Haar measure on the group [24]. This section is intended to recall all the relevant background from [24] and to extend the available abstract theory. 3.1. Function spaces on X In order to define the coorbit space with respect to a Banach space Y of functions on X we need to require certain conditions on Y . (Y ) The space (Y, · |Y ) is a non-trivial Banach space of functions on X that is contained in Lloc 1 (X, μ) and satisfies the solidity condition, i.e., if F is measurable and G ∈ Y such that |F (x)| |G(x)| a.e., then F ∈ Y and F |Y G|Y . This property holds, for instance, for weighted Lw p (X, μ)-spaces. The classical theory by Feichtinger and Gröchenig [20,27,28] heavily uses the group convolution. Since the index space X does not possess a group structure in general we have to find a proper replacement for the convolution of functions on a group. Following [24] we use integral operators with kernels belonging to certain kernel algebras. Let

A1 := K : X × X → C: K is measurable and K|A1 < ∞ ,

(3.1)

K|A1 := max ess sup K(x, y) dμ(y), ess sup K(x, y) dμ(x) .

(3.2)

where

x∈X

X

y∈X

X


3313

The sub-index 1 indicates the unweighted case. We further consider weight functions v(x) 1 on X. The associated weight mv on X × X is given by mv (x, y) := max

v(x) v(y) , , v(y) v(x)

x, y ∈ X.

(3.3)

For a weight m on X × X the corresponding sub-algebra Am ⊂ A1 is defined as Am := {K : X × X → C: Km ∈ A1 } endowed with the norm K|Am := Km|A1 . Later we will need that the kernel R(x, y) from Section 2.1 and further related kernels (see Section 3.4) belong to Am for a proper weight function m. In order to define the coorbit of a given function space Y we will further need that these particular kernels act boundedly from Y to Y , i.e., the mapping K(F ) =

K(·, y)F (y) dμ(y) X

is supposed to be bounded. It is easy to check that the condition K ∈ Am is sufficient for K to map Y = Lvp (X) into Lvp (X) boundedly. This, however, is not the case in general and has to be checked for particular spaces Y . At this point we modify the setting in [24] according to Remark 2 given there. Associated to a space Y satisfying (Y ) and a weight m we introduce the sub-algebra BY,m := {K : X × X → C: K ∈ Am and K is bounded from Y into Y }, where

K|BY,m := max K|Am , K|Y → Y

(3.4)

defines its norm. 3.2. Associated sequence spaces Let us start with the definition of an admissible covering of the index space X. Definition 3.1. A family U = {Ui }i∈I of subsets of X is called admissible covering of X, if the following conditions are satisfied. (i) Each set Ui , i∈ I , is relatively compact and has non-void interior. (ii) It holds X = i∈I Ui .

3314


(iii) There exists some constant N > 0 such that sup {i ∈ I, Ui ∩ Uj = ∅} N < ∞.

(3.5)

j ∈I

Furthermore, we say that an admissible covering U = {Ui }i∈I is moderate with respect to μ, if it fulfills the following additional assumptions. (iv) There exists some constant D > 0 such that μ(Ui ) D for all i ∈ I . (v) There exists a constant C˜ such that ˜ μ(Ui ) Cμ(U j ),

for all i, j such that Ui ∩ Uj = ∅.

Based on this framework, we are now able to define sequence spaces associated to function spaces Y on the set X with respect to the covering U . Definition 3.2. Let U = {Ui }i be an admissible covering of X and let Y be a Banach function space satisfying (Y ), which contains all the characteristic functions χUi . We define the sequence spaces Y and Y associated to Y as < ∞ , Y |λ |χ Y = Y (U) := {λi }i∈I : {λi }i∈I |Y := i Ui i∈I

−1 Y = Y (U) := {λi }i∈I : {λi }i∈I |Y := |λi |μ(Ui ) χUi Y < ∞ . i∈I

Remark 3.3. Under certain conditions on the families U = {Ui }i∈I and V = {Vi }i∈I over the same index set I , the sequence spaces Y (U) and Y (V) coincide (similar for Y (U) and Y (V)), see Definition 7 and Lemma 6 in [24]. The following lemma states useful properties of these sequence spaces. Lemma 3.4. Let U = {Ui }i be an admissible covering of X and let Y be a Banach function space satisfying (Y ) which contains all the characteristic functions χUi . (i) If there exist constants C, c > 0 such that c μ(Ui ) C for all i ∈ I then the spaces Y and Y coincide in the sense of equivalent norms. (ii) If for all i ∈ I the relation χUi |Y vi holds, where vi = supx∈Ui v(x), then we have the continuous embeddings v

1i → Y → Y . Proof. The statement in (i) is immediate. The first embedding in (ii) is a consequence of the triangle inequality in Y , indeed |λi |χUi Y |λi |χUi |Y |λi |vi . i∈I

i∈I

i∈I

The second embedding is a consequence of the fact that U is an admissible covering.

2


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3.3. Coorbit spaces We now introduce properly our coorbit spaces and show some of their basic properties. To this end we fix a space Y satisfying (Y ), a weight v 1, and a tight continuous frame F = {ψx }x∈X ⊂ H which satisfies the following property (Fv,Y ). 1/v

(Fv,Y ) The image space RF (Y ) is continuously embedded into L∞ (X, μ) and RF belongs to the algebra BY,m , where m is the weight on X × X associated to v via (3.3). 1/v

The embedding RF (Y ) → L∞ (X, μ) might seem a bit strange at first glance. However, we will return to that point later on and reduce this question to conditions on the frame F and the sequence space associated to Y . The property (Fv,Y ) sets us in the position to define the coorbit space Co Y = Co(F , Y ). We first define the reservoir

Hv1 = f ∈ H: VF f ∈ Lv1 (X, μ)

(3.6)

endowed with the norm f |H1 = VF f |Lv . v

1

The space Hv1 is a Banach space, see [24]. By RF ∈ BY,m ⊂ Am we see immediately that ψx ∈ Hv1 for all x ∈ X. We denote by (Hv1 )∼ the canonical anti-dual of Hv1 . We may extend the transform V to (Hv1 )∼ by (VF f )(x) = f (ψx ),

∼ x ∈ X, f ∈ Hv1 .

(3.7)

The reproducing formula still holds true. If F = VF f for f ∈ (Hv1 )∼ then RF (F ) = F . Con1/v versely, if F ∈ L∞ satisfies the reproducing formula F = RF (F ) then there exists an f ∈ (Hv1 )∼ such that F = VF f . For more details see [24, Sect. 3]. Now we are able to give the crucial definition of the coorbit space Co Y . Definition 3.5. Let Y be a Banach function space on X satisfying (Y ). Let further F = {ψx }x∈X be a tight continuous frame on X with property (Fv,Y ). The coorbit Co(F , Y ) of Y with respect to F is given by

Co Y = Co(F , Y ) := f ∈ Hv1

∼

: VF f ∈ Y

with f |Co Y = Vf |Y .

For proofs of the following properties we refer to [24]. As a consequence of property (Fv,Y ) the space (Co Y, · |Co Y ) is a Banach space which is continuously embedded in (Hv1 )∼ and 1/v depends on the frame F . Moreover, we have the identities Co Lv1 = Hv1 , Co L∞ = (Hv1 )∼ , and Co L2 = H. Suppose that w is another weight function such that (Fw,Y ) is satisfied. Let mw (x, y) be the associated weight on X × X. If mw (x, y) Cmv (x, y) then the spaces Co Y (v) and Co Y (w) coincide and their norms are equivalent. Finally, we shall focus on the essential question of the coincidence of the two spaces Co(F , Y ) and Co(G, Y ), where F and G are two different continuous frames. One way to answer the above

3316


question is the following proposition which is essentially taken from [24]. Since we start with tight continuous frames the situation simplifies slightly here. Lemma 3.6. Let Y be a Banach function space on X satisfying property (Y ) and let v be a weight function. The tight continuous frames G = {gx }x∈X and F = {fx }x∈X on H are supposed to satisfy (Fv,Y ). Moreover, we assume that the Gramian kernel G(F , G)(x, y) := fy , gx ,

x, y ∈ X,

(3.8)

belongs to the algebra BY,m . Then it holds Co(F , Y ) = Co(G, Y )

in the sense of equivalent norms. We close this paragraph with a result concerning the independence of the coorbit space

Co(F , Y ) on the used reservoir (Hv1 )∼ . We state a version of Theorem 4.5.13 in [40].

Lemma 3.7. Let Y be a Banach function space on X satisfying (Y ) and let v 1 be a weight function. The definition of Co(F , Y ) is independent of the reservoir (Hv1 )∼ in the following sense: Assume that S ⊂ Hv1 is a non-trivial locally convex vector space and F ⊂ S be a tight continuous frame satisfying (Fv,Y ). Assume further that the reproducing formula VF f = RF (VF f ) extends to all f ∈ S ∼ (the topological anti-dual of S) then

Co(F , Y ) = f ∈ S ∼ : VF f ∈ Y .

Proof. Let f ∈ S ∼ such that VF f ∈ Y . Since the reproducing formula extends to S ∼ we have 1/v VF f = RF (VF f ) and hence VF f ∈ RF (Y ) ⊂ L∞ (X, μ) which gives f ∈ (Hv1 )∼ by definition of the latter space. 2 3.4. Discretizations Next we come to a main feature of coorbit space theory, the discretization machinery. It is based on the following definition, which is a slight modification of Definition 6 in [24] according to Remark 5 there. Definition 3.8. A tight continuous frame F = {ϕx }x∈X is said to possess property D[δ, m, Y ] for a fixed δ > 0 and a weight m : X × X → R if there exists a moderate admissible covering U = U δ = {Ui }i∈I of X such that sup sup m(x, y) Cm,U , i∈I x,y∈Ui

if the kernel RF belongs to BY,m , and if oscU (x, y) and osc∗U (x, y) satisfy oscU |BY,m < δ

and osc∗U |BY,m < δ.


3317

Here we put oscU (x, y) := sup ϕx , ϕy − ϕz = sup RF (x, y) − RF (x, z), z∈Qy

z∈Qy

osc∗U (x, y) = oscU (y, x) and Qy =

y∈Ui

Ui .

The following lemma states conditions on the frame F and the space Y which ensure that at least the test functions in Hv1 are contained in Co Y . Lemma 3.9. Let Y be a Banach function space satisfying (Y ). Let further v 1 be a weight function with the associated weight m = mv satisfying supi∈I supx,y∈Ui m(x, y) C and put vi = supx∈Ui v(x). The frame F is supposed to satisfy (Fv,Y ) as well as D[1, 1, Y ] with corresponding covering U = {Ui }i∈I . If χUi |Y vi then it holds ϕx |Co Y v(x) and ∼ Hv1 → Co Y → Hv1 .

(3.9)

Proof. For all i ∈ I and x ∈ Ui we have −1 μ(U χ ϕ ) sup , ϕ

(x) dμ(x) ϕx |Co Y = ϕx , ϕy |Y Y i z y Ui X

z∈Qx

osc + RF |Y → Y · μ(Ui )−1 χUi |Y vi v(x). 1/v

The second embedding in (3.9) follows from RF (Y ) ⊂ L∞ . By Theorem 1 in [24] an element f ∈ Hv1 can be written as a sum f = i∈I |ci |ϕxi , where I is a countable subset and f |H1 inf |ci |v(xi ), v

i∈I

where the infimum is taken over all representations of f in the above form. So let us take one of these representations and estimate by using the triangle inequality f |Co Y

|ci | · ϕxi |Co Y

i∈I

This concludes the proof.

|ci |v(xi ).

i∈I

2 1/v

We return to the question of ensuring RF (Y ) → L∞ (X). The following lemma states a sufficient condition. Lemma 3.10. Let Y be a Banach function space satisfying (Y ) and v 1 be a weight function with associated weight m satisfying supi∈I supx,y∈Ui m(x, y) C and put vi = supx∈Ui v(x). If U = {Ui }i∈I is a moderate admissible covering of X and χUi |Y 1/vi then we have the 1/v continuous embedding Y → (L∞ ) . If the frame F satisfies in addition D[1, 1, Y ] with respect 1/v to this covering then we even have RF (Y ) ⊂ L∞ (X, μ).

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Proof. Using the assumption χUi |Y 1/vi we get by the solidity of Y 1 −1 −1 −1 |λi |μ(Ui ) χUi Y μ(Ui ) |λi | · χUi |Y μ(Ui ) |λi | v i i∈I

for all i ∈ I . This yields 1/v 1 −1 −1 |λ |μ(U ) χ {λi }i∈I | L∞ , i i Ui Y sup μ(Ui ) |λi | vi i∈I i∈I

1/v

1/v

where we applied (3.5) in the last step. This proves Y → (L∞ ) . To show RF (Y ) ⊂ L∞ (X, μ) we start with F ∈ Y and estimate as follows, = sup RF (F )(x) 1 sup sup RF (x, y)F (y) dμ(y) 1 RF (F )|L1/v ∞ v(x) i∈I x∈Ui vi x∈X X

RF (z, y)F (y) dμ(y) χU (x)Y sup sup i i∈I z∈Ui

X

RF (z, y)F (y) dμ(y) χU (x)Y sup sup i i∈I

X

z∈Qx

∗ osc (x, y) + RF (x, y) · F (y) dμ(y)Y . X

Property D[1, 1, Y ] gives in particular the boundedness of the considered integral operator and we obtain RF (F )|L1/v cF |Y ∞ which concludes the proof.

2

The following abstract discretization results for coorbit spaces is a slight generalization of Theorem 5 in [24], see also Remark 5 there. We omit the proof since the necessary modifications are straightforward. Theorem 3.11. Let Y be a Banach space of functions on X satisfying (Y ) and let v 1 be a weight function with associated weight m. Assume that F = {ϕx }x∈X is a tight continuous frame satisfying (Fv,Y ) and D[δ, m, Y ] for some δ > 0 with corresponding moderate admissible covering U δ chosen in a way such that

δ R|BY,m + max Cm,U δ R|BY,m , R|BY,m + δ 1, where Cm,U δ is the constant from Definition 3.8. Choose points xi ∈ Ui . Then the discrete system Fd := {ϕxi }i∈I is both an atomic decomposition of Co Y with corresponding sequence space Y as well as a Banach frame with corresponding sequence space Y . This means that there exists a dual frame {ei }i∈I such that for all f ∈ Co Y :


3319

f |Co Y f, ϕxi i∈I |Y and f |Co Y f, ei i∈I |Y .

(a)

(b) If f ∈ Co Y then the series f=

i∈I

f, ei ϕxi =

f, ϕxi ei i∈I

converge unconditionally in the norm of Co Y if the finite sequences are dense in Y and with weak∗ -convergence induced by (Hv1 )∼ , in general. In the sequel we are interested in (wavelet) bases for the spaces Co Y . In many situations, such as in wavelet analysis, one often has an orthonormal basis, biorthogonal basis or discrete tight frame for the Hilbert space at disposal, which arises from sampling a continuous frame. (Of course, such an orthonormal basis has to be derived from different principles than available in the abstract situation of coorbit space theory.) Then the next main discretization result, Theorem 3.14 below, provides simple conditions, which ensure that the basis expansion extends to coorbit spaces, and characterizes them by means of associated sequence spaces. Our result generalizes one of Gröchenig in classical coorbit space theory, see [27] and also Theorem 5.7 in the preprint version of [42]. From an abstract viewpoint, extensions of basis expansions seem very natural. However, in classical function space theory usually much efforts are carried out in order to provide such wavelet basis characterization. In contrast, our discretization result provides a general approach, which requires to check only a single condition in a concrete setup. Before giving the precise statement of our result, we have to introduce some notation and state some auxiliary lemmas. Given a continuous frame F defining the coorbit space Co(F , Y ) we would like to discretize by a different frame G = {ψx }x∈X . Essentially this reduces to conditions on the Gramian kernel G(F , G)(x, y) introduced above. If U = {Ui }i∈I denotes a moderate admissible covering of X and xi ∈ Ui , i ∈ I , then we define the kernel (3.10) K(x, y) = sup G(F , G)(z, y) = sup ϕy , ψz , z∈Qx

where Qx =

i: x∈Ui

z∈Qx

Ui . Observe that K(x, y) depends on F , G and the covering U .

Lemma 3.12. Let Y , v, U = {Ui }i∈I be as above and xi ∈ Ui , i ∈ I . Let further F = {ϕx }x∈X be a tight continuous frame satisfying (Fv,Y ), and Co Y = Co(F , Y ). Assume that G = {ψx }x∈X ⊂ Hv1 is a further continuous frame such that the kernel K in (3.10) belongs to BY,m . Then there exists a constant C > 0 independent of f such that

f, ψx |Y C f |Co Y (F , Y ), f ∈ Co Y. i i∈I Proof. Since F is a tight continuous frame with frame constants one, we have VF∗ VF = Id, see Section 2.1. We conclude that VF f (y)ϕy dμ(y) (xi ) (VG f )(xi ) = VG VF∗ VF f (xi ) = VG =

X

VF f (y) ϕy , ψxi dμ(y). X

3320


This implies the relation |(VG f )(xi )| K(|VF f |)(xi ). We define the function H (x) =

(VG f )(xi )χUi (x),

i∈I (x)

where I (x) = {j ∈ I : x ∈ Uj }, and observe that by (3.5) H (x) χUi (x) (VF f )(y) · K(xi , y) dμ(y) i∈I (x)

N

X

(VF f )(y) · K(x, y) dμ(y).

X

Hence, |H | K(|(VF f )|) and together with (Y ) and our assumption on K we get finally

VG f (xi )

= H |Y N K|Y → Y · VF f |Y Cf |Co Y . |Y i∈I

2

We need a further technical lemma. 1/v

Lemma 3.13. Let Y , U , v, {xi }i∈I and m as above, such that Y → (L∞ ) . Let F = {ϕx }x∈X be a tight frame satisfying (Fv,Y ), put Co Y = Co(F , Y ), and assume G = {ψx }x∈X ⊂ Hv1 to be a continuous frame such that also K ∗ , see (3.10), belongs to BY,m . If {λi }i∈I ∈ Y then the sum f=

λi ψxi

i∈I

converges unconditionally in the weak∗ -topology of (Hv1 )∼ to an element f ∈ Co Y and there exists a constant c > 0 such that f |Co Y c{λi }i |Y .

(3.11)

If the finite sequences are dense in Y we even have unconditional convergence in the norm of Co Y . Proof. Step 1. We prove that i∈I |λi | · | ψxi , ϕx | converges pointwise for every x ∈ X and 1/v of partial sums that its pointwise limit function belongs to L∞ . This implies that the1 sequence of every rearrangement of i∈I λi ψxi is uniformly bounded in (Hv )∼ . Since by Theorem 1 in [24] span{ϕx : x∈ X} is dense in Hv1 we conclude with an analogous argument as used in [40, Lem. 4.5.8] that i∈I λi ψxi , ϕ converges unconditionally for every ϕ ∈ Hv1 . This defines the ∗ weak -limit of the expansion of i∈I λi ψxi . To show the necessary pointwise convergence we estimate as follows, |λi | vi 1 ψx , ϕx λi ψxi , ϕx · i v(x) vi v(x) i

i∈I


3321

|λi | sup μ(Ui )−1 m(x, y) sup ψz , ϕx dμ(y) vi i z∈Qy Ui

{λi }i |Y · K ∗ |Am . 1/v

In the last step we used the assumption Y ⊂ (L∞ ) . Step 2. We already know that i∈I λi ψxi =: f ∈ (Hv1 )∼ . We claim that f ∈ Co Y . Indeed, f |Co(F , Y ) = VF f |Y = λi ψxi , ϕx Y |λi | sup ψz , ϕx Y i∈I

z∈Ui

i∈I

−1 = |λ |μ(U ) sup , ϕ

χ (y) dμ(y) ψ Y i i z x Ui i∈I

X

z∈Qy

−1 K(y, x) |λi |μ(Ui ) χUi (y) dμ(y)Y . X

i∈I

By our assumption on K ∗ we obtain consequently −1 f |Co Y K ∗ |Y → Y · |λ |μ(U ) χ i i Ui Y , i∈I

which reduces to (3.11) using the definition of Y . This type of argument also implies the convergence in Co Y if the finite sequences are dense in Y . 2 Let now Gr = {ψxr }x∈X and G˜r = {ψ˜ xr }x∈X , r = 1, . . . , n, be continuous frames with associated Gramian kernels Kr (x, y) and K˜ r (x, y) defined by (3.10) for a moderate admissible covering U = {Ui }i∈I . Now we are prepared to state our next discretization result. In contrast to the proof of its predecessor in classical coorbit theory [27], we note, however, that it does not rely on our first discretization result Theorem 3.11. 1/v

Theorem 3.14. Let Y be as above and v and m such that Y ⊂ (L∞ ) . Let F = {ϕx }x∈X be a tight frame satisfying (Fv,Y ) and put Co Y = Co(F , Y ). The continuous frames Gr = {ψxr }x∈X , G˜r = {ψ˜ x }x∈X ⊂ Hv1 are such that the corresponding kernels Kr and K˜ r∗ belong to BY,m . Moreover, assume that f=

n

f, ψxri ψ˜ xri

(3.12)

r=1 i∈I

holds for all f ∈ H where xi ∈ Ui (the same covering which is used for the Gramian kernels Kr and K˜ r ). Then the expansion (3.12) extends to all f ∈ Co Y . Furthermore, f ∈ (Hv1 )∼ belongs to Co Y if and only if { f, ψxri }i∈I belongs to Y for each r = 1, . . . , n. Then we have f |Co Y

n

f, ψ r xi

r=1

i∈I

|Y .

(3.13)

3322


The convergence in (3.12) is in the norm of Co Y if the finite sequences are dense in Y . In general, we have weak∗ -convergence induced by (Hv1 )∼ . Proof. By Lemmas 3.13 and 3.12 the expansion n

f, ψxri ψ˜ xri

(3.14)

r=1 i∈I

converges in the weak∗ -topology of (Hv1 )∼ to an element f˜ ∈ Co Y ⊂ (Hv1 )∼ provided we assume that either f ∈ Co Y or { f, ψxri }i∈I belongs to Y for each r = 1, . . . , n. If the finite sequences are dense in Y we even have convergence in Co Y . It remains to show the identity f = f˜. Step 1. Let us start with a ϕ ∈ Hv1 . We apply Lemma 3.12 to the case Y = Lv1 and G = G˜r , r = 1, . . . , n. The assumption K˜ r ∈ Am implies then that K˜ r maps Lv1 boundedly into Lv1 . Therev fore, Lemma 3.12 yields that { ϕ, ψ˜ xri }i∈I belongs to 1i for all r = 1, . . . , n. Lemma 3.13 gives then that the expansion n

ϕ, ψ˜ xri ψxri

(3.15)

r=1 i∈I v

converges in the norm of Hv1 to an element g ∈ Hv1 since the finites sequences are dense in 1i . Observe that our global assumption v > 1 together with 2 h|H2 h, ϕx |L2 (X) h, ϕx |L∞ (X) · h, ϕx |L1 (X) h, ϕx |L∞ (X) · h|Hv1 and h, ϕx |L∞ (X) h|H · ϕx |H h|H, using ϕx |H C, imply the continuous embedding H1v → H. Hence, (3.15) converges also in H to g. On the other hand the identity in H η=

n

η, ψxri ψ˜ xri

r=1 i∈I

for arbitrary η ∈ H gives η, ϕ =

n

η, ψxri

r=1 i∈I

n

r r r ˜ ˜ ψxi , ϕ = η, ϕ, ψxi ψxi = η, g .

r=1 i∈I

Choosing η = ϕ − g gives ϕ = g. Step 2. Using that (3.15) converges to ϕ in Hv1 and that f˜ is the weak∗ -limit of (3.14), we finally obtain


f (ϕ) = f

n

ϕ, ψ˜ xri ψxri

=

r=1 i∈I

=

n

3323

n

ψ˜ xri , ϕ f ψxri

r=1 i∈I

f, ψxri ψ˜ xri , ϕ = f˜(ϕ).

r=1 i∈I

This implies f = f˜ since ϕ was chosen arbitrarily. The norm equivalence in (3.13) is a direct consequence of Lemmas 3.12, 3.13. 2 4. Peetre type spaces and their coorbits The generalized Besov–Lizorkin–Triebel spaces to be studied later in Section 5 are defined as coorbits of so-called Peetre spaces on the index set X = Rd × [(0, 1) ∪ {∞}] equipped with the Radon measure μ given by

1 F (x) dμ(x) =

X

F (y, s) Rd

ds s d+1

dy +

F (y, ∞) dy.

Rd

0

w We intend to define two general scales of Banach function spaces PB,q,a (X) and Lw B,q,a (X) d d on X. The parameter B(R ) is a Banach space of measurable functions on R , the parameter w : X → (0, ∞) represents a weight function on X, and 1 q ∞, a > 0. The letter P refers w to Peetre’s maximal function (2.3) which is always involved in the definition of PB,q,a (X), see Definition 4.6 below. Let us start with reasonable restrictions on the parameters w and B(Rd ). α We use the class Wα13,α2 of admissible weights introduced by Kempka [33]. α

Definition 4.1. A weight function w : X → R+ belongs to the class Wα13,α2 if and only if there exist non-negative numbers α1 , α2 , α3 0 such that, for x = (x, t) ∈ X,

(W1)

(W2)

( st )α1 w(x, s) w(x, t) ( st )−α2 w(x, s): 1 s t > 0, t α1 w(x, ∞) w(x, t) t −α2 w(x, ∞): s = ∞, 0 < t 1, (1 + |x − y|/t)α3 : t ∈ (0, 1), w(x, t) w(y, t) for all y ∈ Rd . t =∞ (1 + |x − y|)α3 :

Example 4.2. The main examples are weights of the form ws,s (x, t) =

|x−x0 | s

) : t

s − x0 |) :

t −s (1 +

t ∈ (0, 1),

(1 + |x

t = ∞,

where s, s ∈ R. The choice s = 0 is most common. Remark 4.3. The above considered weights are continuous versions of weights appearing in the definition of certain 2-microlocal function spaces of Besov–Lizorkin–Triebel type, see for instance [33–35].

3324


The second ingredient is a Banach space B(Rd ) of functions defined on Rd . Definition 4.4. A solid Banach space B = B(Rd ) of functions on Rd with norm · |B(Rd ) is called admissible if (B1) the elements of B(Rd ) are locally integrable functions with respect to the Lebesgue measure; (B2) there exist real numbers γ1 γ2 and δ2 δ1 with δ1 0 such that for every α > 0 there are constants Cα , cα with cα t

γ2

|x| 1+ t

δ2

d |x| δ1 γ 1 Cα t 1+ χQα(x,t) |B R , t

x ∈ Rd , t ∈ (0, 1],

where Qα(x,t) = x + t[−α, α]d denotes a d-dimensional cube with center x ∈ Rd . Example 4.5. If B(Rd ) = Lp (Rd ) is the classical Lebesgue space then χQα |B Rd = (2α)d t d/p . (x,t) Hence, the parameters in condition (B2) are given by Cα = cα = (2α)d , γ1 = γ2 = d/p, and δ1 = δ2 = 0. 4.1. Peetre type spaces on X Our key ingredient in recovering generalized Besov–Lizorkin–Triebel spaces are the following function spaces on X defined via the Peetre maximal function in (2.3). α

Definition 4.6. Let 1 q ∞, a > 0, and w ∈ Wα13,α2 be a weight function. Assume that B(Rd ) is a solid Banach space of functions on Rd satisfying (B1) and (B2). Then we define by w PB,q,a (X) := F : X → C: Lw B,q,a (X) := F : X → C:

F |P w < ∞ , B,q,a

F |Lw < ∞ B,q,a

two scales of function spaces on X, where the norms are given by (2.4) and (2.5). Remark 4.7. Assume that in addition the space B(Rd ) is uniformly translation invariant, i.e., the translation operators defined by Tx g = g(· − x) are uniformly bounded from B(Rd ) to B(Rd ), sup Tx : B Rd → B Rd < ∞. x∈Rd

Moreover, we assume that w(x, t) = w(t), ˜

(x, t) ∈ X.

(4.1)


3325

Under this stronger condition we define the scale of spaces Lw B,q (X), 1 q ∞, by · F (·, ∞)|B Rd + F |Lw := w(∞) ˜ B,q

1

q q dt w(t) ˜ F (·, t)|B Rd d+1 t

1/q .

0

These spaces can then also be taken in replacement of Lw B,q,a (X). An important class of examples of uniformly translation invariant spaces are the unweighted classical Lebesgue space Lp (Rd ). In the following we prove assertions on the boundedness of certain integral operators between these spaces. Recall that for a function G : X → C the action of a kernel K on G is defined by

K (x, t), (y, ∞) G(y, ∞) dy +

K(G)(x, t) = Rd

1

ds K (x, t), (y, s) G(y, s) d+1 dy. s

Rd 0

Condition (4.2) below will be satisfied for kernels associated to continuous wavelet transforms to be studied later. Proposition 4.8. Assume that K((x, t), (y, s)) denotes a kernel function on X × X such that ⎧ s G1 ( y−x ⎪ ⎪ t , t ): ⎪ ⎨ 1 G2 ( y−x t , t ): K (x, t), (y, s) ⎪ ⎪ G3 (y − x, s): ⎪ ⎩ G4 (y − x):

t, s ∈ (0, 1), t ∈ (0, 1), s = ∞, t = ∞, s ∈ (0, 1), t =s=∞

(4.2)

α

for some functions G1 , G2 , G3 , G4 . Let 1 q ∞, a > 0, and w ∈ Wα13,α2 . Assume B(Rd ) is a solid Banach function space satisfying (B1) and (B2) and suppose that the following quantities are finite, ∞ M1 :=

G1 (y, r) 1 + |y| a r d/q max 1, r −a max r −α1 , r α2 dy dr , r d+1

0 Rd

∞ M2 :=

a t α2 +d/q 1 + |y|

1 Rd

1 M3 := 0 Rd

M4 :=

r −(α1 +2a+d/q −d)

dt sup G2 y, t dy d+1 , t t/2t t a dr sup G3 y, r 1 + |y| dy d+1 , r r/2r r

G4 (y) 1 + |y| a dy,

Rd

where q is such that 1/q + 1/q = 1. Then

(4.3)

3326


K : P w (X) → P w (X) M1 + M2 + M3 + M4 , B,q,a B,q,a K : Lw (X) → Lw (X) M1 + M2 + M3 + M4 . B,q,a B,q,a

and

w . For Lw Proof. We prove the assertion only for the space PB,q,a B,q,a the calculation is simpler w and the modifications are straightforward. We first observe that, for a function F ∈ PB,q,a (X),

K(F )|P w

B,q,a

1 q 1/q 1 dr w(·, t) dt d K (· + z, t), (y, r) F (y, r) dy sup B R a d+1 d+1 (1 + |z|/t) r t d z∈R Rd 0

0

1 1/q !q w(·, t) dt d + sup K (· + z, t), (y, ∞) F (y, ∞) dy B R a t d+1 z∈Rd (1 + |z|/t) Rd

0

1 dr w(·, ∞) K (· + z, ∞), (y, r) F (y, r) dy + sup B R d z∈Rd (1 + |z|)a r d+1 0 Rd

w(·, ∞) K (· + z, ∞), (y, ∞) F (y, ∞) dy B Rd . sup + d (1 + |z|)a z∈R Rd

We denote the summands appearing on the right-hand side by S1 , S2 , S3 , S4 . Let us first treat S4 . We have d w(·, ∞) G4 y − (· + z) F (y, ∞) dy B R S4 sup a z∈Rd (1 + |z|) Rd

|F (· + y + z, ∞)| d G4 (y) w(·, ∞) sup dy B R (1 + |z|)a z∈Rd Rd

Rd

G4 (y) 1 + |y| a dy w(·, ∞) sup |F (· + z, ∞)| B Rd = M4 F |P w . B,a,q (1 + |z|)a z∈Zd

Similarly, we obtain 1 1/q !q y − (· + z) 1 w(·, t) dt d G2 S2 sup , · F (y, ∞) dy B R a d+1 (1 + |z|/t) t t t d z∈R 0

Rd

1 |F (· + z, ∞)| d −α2 w(·, ∞) sup R t B (1 + |z|)a z∈Rd 0

Rd

1/q !q dt G2 y , 1 1 + |y| a dy t t t d+1


∞

t

α2 −d+2d/q

G2 (y, t) 1 + |y| a dy

!q

dt t d+1

1/q

F |P w

B,q,a

3327

M2 F |P w

B,q,a

.

Rd

1

The next step is to estimate 1 q 1/q 1/t |F (· + z + ty, rt)| dt dr G1 (y, r) sup S1 dy d+1 w(·, t) B Rd a d+1 (1 + |z|/t) r t z∈Rd 0 Rd

0

1 q 1/q 1/t a |F (· + z, rt)| dt dr d G1 (y, r) 1 + |y| sup dy d+1 w(·, t) B R . a d+1 (1 + |z|/t) r t d z∈R 0 Rd

0

Minkowski’s inequality and a change of variable in the integral over t gives ∞ S1

G1 (y, r) 1 + |y| a max 1, r −a max r −α1 , r α2 r d/q

0 Rd

1 1/q ! dr |F (· + z, t)| q dt × w(·, t) sup B Rd dy d+1 a d+1 t r z∈Rd (1 + |z|/t) 0

M1 F |P w

B,q,a

.

It remains to estimate S3 . Using (W1) we get 1 dr w(·, ∞) G3 y − (· + z), r F (y, r) dy S3 sup B R d z∈Rd (1 + |z|)a r d+1 0 Rd

1 |F (· + y + z, r)| dr −α1 d G3 (y, r) sup w(·, r)r dy d+1 B R . a (1 + |z|) r d z∈R 0 Rd

For r ∈ (0, 1) we can estimate the supremum above by sup z∈Rd

|F (x + y + z, r)| |F (x + w, r)| (1 + |w|/r)a = sup · a a (1 + |w|/r)a (1 + |z|) w∈Rd (1 + |w − y|) sup w∈Rd

sup w∈Rd

a |F (x + w, r)| −a 1 + |y|/r · r (1 + |w|/r)a a |F (x + w, r)| −2a 1 + |y| . r (1 + |w|/r)a

(4.4)

3328


Plugging this into (4.4) and using Hölder’s inequality with 1/q + 1/q = 1 we finally get 1 S3

r

−(α1 +2a)

a G3 (y, r) 1 + |y| dy

!q

dr

1/q

r d+1

F |P w

B,a,q

M3 F |P w

B,a,q

.

Rd

0

2

This concludes the proof.

Remark 4.9. According to Remark 4.7 the conditions in (4.3) are simpler in the translation invariant case. The parameter a is then not required. We need a similar statement in order to guarantee that K belongs to Amv , where mv is the associated weight to v : X → R given by v(x, t) v(y, s) , mv (x, t), (y, s) := max v(y, s) v(x, t)

(4.5)

for the special choice v(x, t) :=

t −γ (1 + |x|/t)η : (1 + |x|)η :

t ∈ (0, 1], t = ∞,

(4.6)

where η, γ 0. Recall that we define K ∗ (x, y) = K(y, x). Proposition 4.10. Let K be a kernel function on X × X such that K and K ∗ satisfy (4.2) with functions Gi and G∗i , i = 1, . . . , 4, respectively. Let further v and mv be given by (4.6) and (4.5). If the quantities ∞ S1 :=

G1 (y, t) max t, t −1 |η|+|γ | 1 + |y| |η| dy dt , t d+1

0 Rd

S2 := ess sup t |η|+|γ |−d t>1

1 S3 :=

G2 (y, t) 1 + |y| |η| dy,

Rd

G3 (y, t)t −(|η|+|γ |) 1 + |y| |η| dy dt , t d+1

0 Rd

S4 :=

G4 (y) 1 + |y| |η| dy,

(4.7)

Rd

and the corresponding ones for K ∗ in terms of the function G∗i are finite then we have K, K ∗ ∈ Amv .


3329

Proof. A straightforward computation shows |x − y| |η| s t |γ |+|η| 1+ mv (x, t), (y, s) max , t s t

(4.8)

(obvious modification in case s = ∞ or t = ∞). According to (3.2) we have to show that K|Amv := max ess sup mv (x, y) K(x, y) dμ(y), ess sup mv (x, y) K(x, y) dμ(x) x∈X

y∈X

Y

X

(similar for K ∗ ) is finite. Combining (4.2), (4.7), and (4.8) finishes the proof.

2

4.2. Associated sequence spaces w As the next step we study the structure of the sequence spaces PB,q,a (X) and Lw B,q,a (X) associated to Peetre type spaces. We will use the following covering of the space X. For α > 0 and β > 1 we consider the family U α,β = {Uj,k }j ∈N0 , k∈Zd of subsets

U0,k = Q0,k × {∞}, k ∈ Zd , Uj,k = Qj,k × β −j , β −j +1 , j ∈ N, k ∈ Zd , where Qj,k = αk + αβ −j [0, 1]d . We will use the notation χj,k (x) =

"

1: x ∈ Uj,k , 0: otherwise.

Clearly, we have X ⊂ j ∈N0 , k∈Zd Uj,k and U α,β is a moderate admissible covering of X. We w ) and (Co Lw now investigate properties of the sequence spaces (Co PB,q,a B,q,a ) , recall Definition 3.2. α

Lemma 4.11. Let 1 q ∞, a > 0, and w ∈ Wα13,α2 . Let B(Rd ) be a solid Banach space satisfying (B1) and (B2). Then we have −1 w , vj,k χUj,k |PB,q,a

j ∈ N0 , k ∈ Zd ,

(4.9)

where vj,k = sup(x,t)∈Uj,k vw,B,q (x, t) with vw,B,q (x, t) =

t −|α1 +γ2 −d/q| (1 + |x|/t)|α3 −δ2 | : (1 + |x|)|α3 −δ2 | :

0 < t 1, t =∞

w for (x, t) ∈ X. The same holds for Lw B,q,a (X) in replacement of PB,q,a (X).

(4.10)

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Proof. Assume j 1, k ∈ Zd and choose (x, t) ∈ Uj,k arbitrarily. By (W2) we have χU |P w t −d/q w(x, t) sup χj,k (· + z) B Rd j,k B,q,a d (1 + |z|/t)a z∈R −α3 χj,k |B Rd t −d/q+α1 1 + |x|/t δ −α t −d/q+α1 +γ2 1 + |x|/t 2 3 . Hence, choosing vw,B,q (x, t) as above gives (4.9). Note that vw,B,q 1. In case j = 0 the modifications are straightforward. 2 Our next result provides equivalent norms of the sequence spaces associated to the Peetre type function spaces on X. Theorem 4.12. Let 1 q ∞, B(Rd ), w as in Lemma 4.11, and a > 0. Then {λj,k }j,k | P w B,q,a 1 sup w(αk, ∞)|λ |χ (· + z) B Rd 0,k 0,k d (1 + |z|)a z∈R d +

j ∈N0

k∈Z

!q 1/q β dj/q d −j −j sup |λj,k |χj,k (· + z) w αkβ , β B R j a z∈Rd (1 + β |z|) d k∈Z

(4.11) and {λj,k }j,k | Lw B,q,a 1 sup w(αk, ∞)|λ0,k |χ0,k (· + z)B Rd a z∈Rd (1 + |z|) d k∈Z

q 1/q β dj/q −j −j d sup + B R |λ w αkβ , β |χ (· + z) . j,k j,k d (1 + β j |z|)a z∈R d j ∈N0

k∈Z

(4.12) w w w Additionally, we have (Lw B,q,a ) = (LB,q,a ) and (PB,q,a ) = (PB,q,a ) , respectively.

Proof. According to Definition 3.2 the statement is a result of a straightforward computation taking (W2) into account. 2 If we have additional knowledge on the space B(Rd ), then the structure of the sequence w ) and (Lw spaces (PB,q,a B,q,a ) simplifies significantly. Indeed, under some additional conditions w and w of which the norms are given by (see below) they coincide with the spaces pB,q B,q


3331

{λj,k }j,k |p w = w(αk, ∞)|λ0,k |χ0,k B Rd B,q k∈Zd

!q 1/q dj d −j −j + |λj,k |χj,k β w αkβ , β B R , j ∈N0

k∈Zd

and {λj,k }j,k |w w(αk, ∞)|λ0,k |χ0,k B Rd B,q k∈Zd

+

j ∈N0

q 1/q −j −j d B R |λ β dj w αkβ , β |χ , j,k j,k k∈Zd

respectively, and get therefore independence of a. Before giving a precise statement we first d introduce the Hardy–Littlewood maximal function Mr f , r > 0. It is defined for f ∈ Lloc 1 (R ) via

1 (Mr f )(x) = sup x∈Q |Q|

f (y)r dy

1/r ,

x ∈ Rd ,

Q

where the sup runs over all rectangles Q containing x with sides parallel to the coordinate axes. The following majorant property of the Hardy–Littlewood maximal function is taken from [51, II.3]. d d Lemma 4.13. Let f ∈ Lloc 1 (R ) and ϕ ∈ L1 (R ) where ϕ(x) = ψ(|x|) with a non-negative decreasing function ψ : [0, ∞) → R. Then we have

(f ∗ ϕ)(x) (M1 f )(x)ϕ|L1 Rd for all x ∈ Rd . Proof. A proof can be found in [51, II.3], p. 59.

2

Let us further define the space B(q , Rd ) as the space of all sequences of measurable functions {fk }k∈I on Rd such that 1/q d q {fk }k∈I |B q , Rd := |fk | B R < ∞. k∈I

Corollary 4.14. Let 1 q ∞, a > 0, and B(Rd ), w as above. (i) If for some r > 0 with ar > d the Hardy–Littlewood maximal operator Mr is bounded on w w w B(Rd ) and on B(q , Rd ) then (Lw B,q,a ) = B,q and (PB,q,a ) = pB,q , respectively.

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w w (ii) If B(Rd ) is uniformly translation invariant, see (4.1), then (Lw B,q,a ) = (LB,q ) = B,q provided a > d.

Proof. For (x, t) ∈ Uj,k , we have |χj,k (w + z)| 1 t −d (w). χj,k ∗ sup j ar (1 + |w − x|/t)ar (1 + | · |/t)ar z (1 + β |z|)

(4.13)

Indeed, the first estimate is obvious. Let us establish the second one χj,k ∗

1 1 (w) = dy (1 + | · |/t)ar (1 + |w − y|/t)ar

|y|ct

Qj,k

= |y|ct

1 td 0

1 dy (1 + |w − x − y|/t)ar

1 dy (1 + |w − x|/t + |y|/t)ar s d−1 td ds . (1 + |w − x|/t + s)ar (1 + |w − x|/t)ar

Because of ar > d the functions gj = β j d (1 + β j | · |)−ar belong to L1 (Rd ) and the L1 (Rd )norms are uniformly bounded in j . (i) We use Lemma 4.13 in order to estimate the convolution on the right-hand side of (4.13) by the Hardy–Littlewood maximal function and obtain sup z

|χj,k (x + z)| M1 (χj,k )(x), (1 + β j |z|)ar

x ∈ Rd .

Hence, we can rewrite (4.11) as ! d M r B R (·) w(αk, ∞)|λ |χ 0,k 0,k B,q,a

{λj,k }j,k | P w

k∈Zd

!q 1/q d dj/q −j −j + Mr |λj,k |χj,k β w αkβ , β B R . j ∈N0

k∈Zd

Since by assumption Mr is bounded on B(q , Rd ) we obtain the desired upper estimate. The w corresponding estimate from below is trivial. The proof of the coincidence (Lw B,q,a ) = B,q is similar. For the proof of (ii) we do not need the Hardy–Littlewood maximal function. We use (4.13) with r = 1 and simply Minkowski’s inequality. This yields q 1/q β dj/q −j −j d sup |λj,k |χj,k (· + z)B R w αkβ , β d (1 + β j |z|)a z∈R d j ∈N0

k∈Z


3333

q 1/q j d/q −j −j d |λj,k |χj,k (·) ∗ gj (·)B R β w αkβ , β j ∈N0

j ∈N0

j ∈N0

k∈Zd

!q 1/q j d/q −j −j d dy B R |λ gj (y) β w αkβ , β |χ (· − y) j,k j,k k∈Zd

Rd

q 1/q −j −j d B R |λ βjd w αkβ , β |χ (· − y) . j,k j,k k∈Zd

The same argument works for the first summand in (4.12). The estimate from below is trivial.

2

Remark 4.15. The main examples for spaces B(Rd ) satisfying the assumptions in Corollary 4.14 are ordinary Lebesgue spaces Lp (Rd ), 1 p ∞, Muckenhoupt weighted Lebesgue space Lp (Rd , v), 1 p ∞, and Morrey spaces Mu,p (Rd ), 1 p u ∞, defined in Section 5.3. Remark 4.16. Corollary 4.14(ii) remains valid if we weaken condition (4.1) in the following sense, Tx : B Rd → B Rd < 1 + |x| η for some η > 0. One has to adjust the parameter a in this case. This setting applies to certain weighted Lp -spaces B(Rd ) = Lp (Rd , ω) with polynomial weight ω(y) = (1 + |y|)κ . w 4.3. The coorbits of PB,q,a (X) and Lw B,q,a (X)

Now we apply the abstract coorbit space theory from Section 3 to our concrete setup. We put H = L2 (Rd ) and fix an admissible continuous wavelet frame F in the sense of Definition 2.1. According to the abstract theory in Section 3 the operator RF is then given by RF (x, t), (y, s) = ϕ(x,t) , ϕ(y,s) ,

(x, t), (y, s) ∈ X.

The relevant properties of this kernel and the kernels below depend on smoothness and decay conditions of the wavelets, see Definition 1.1. The next result plays a crucial role and is proved in [61, Lem. A.3]. Similar results which are stated in a different language can be found for instance in [25, Lem. B1, B2], [43, Lem. 1], and [32, Lem. 1.2.8, 1.2.9]. Lemma 4.17. Let L ∈ N0 , K > 0, and g, f, f0 ∈ L2 (Rd ). (i) Let g satisfy (D), (ML−1 ) and let f0 satisfy (D), (SK ). Then for every N ∈ N there exists a constant CN such that the estimate min{L,K}+d/2 (Wg f0 )(x, t) CN t (1 + |x|)N

holds true for x ∈ Rd and 0 < t < 1.

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(ii) Let g, f satisfy (D), (ML−1 ) and (SK ). For every N ∈ N there exists a constant CN such that the estimate (Wg f )(x, t) CN

|x| −N t min{L,K}+d/2 1+ 1+t (1 + t)2 min{L,K}+d

holds true for x ∈ Rd and 0 < t < ∞. w (X). Based on this lemma, we can show that the kernel RF acts continuously on PB,q,a

Lemma 4.18. Let F be an admissible continuous wavelet frame. Then the operator RF belongs w to BY,m for Y = PB,q,a (X) or Y = Lw B,q,a (X) and every v, m given by (4.6) and (4.5). Moreover, 1/vw,B,q

it holds RF (Y ) ⊂ L∞

(X, μ) where vw,B,q is defined in (4.10).

Proof. We use that ⎧ s ⎪ W Φ( y−x ⎪ t , t ): ⎪ Φ ⎨ WΦ Φ0 (y − x, s): RF (x, t), (y, s) = y−x 1 ⎪ W Φ0 Φ( t , t ): ⎪ ⎪ ⎩ WΦ0 Φ0 (y − x, 1):

t, s ∈ (0, 1], t = ∞, s ∈ (0, 1], t ∈ (0, 1], s = ∞, t = s = ∞,

where the operator W denotes the continuous wavelet transform, see Section 1.3. Together with Propositions 4.8, 4.10 in combination with Lemma 4.17, this yields that RF belongs to BY,m . 1/v The embedding R(Y ) ⊂ L∞ w,B,q (X, μ) follows from the abstract result in Lemma 3.10 and the choice of the weight vw,B,q in Lemma 4.11. To prove that F satisfies the property D[1, 1, Y ] we refer to Section 4.4 below and Proposition 4.22. 2 w and Co Lw Now we are ready to define the coorbits Co PB,q,a B,q,a .

Definition 4.19. Let 1 q ∞, B(Rd ) and w as above, F be an admissible continuous frame in the sense of Definition 2.1, and a > 0. We define

∼

w : VF f ∈ PB,q,a (X) , w ∼ 1

Co Lw : VF f ∈ Lw B,q,a = Co LB,q,a , F := f ∈ Hvw,B,q B,q,a (X) . w w Co PB,q,a = Co PB,q,a , F := f ∈ Hv1w,B,q

Based on the abstract theory we immediately obtain the following basic properties of the introduced coorbit spaces. α

Theorem 4.20. Let 1 q ∞, a > 0, w ∈ Wα13,α2 , F be an admissible frame, and let B(Rd ) satisfy (B1) and (B2). Then we have the following properties. w (a) If a > 0 then the spaces Co Lw B,q,a and Co PB,q,a are Banach spaces. w w w (or Lw (b) A function F ∈ PB,q,a B,q,a ) is of the form VF f for some f ∈ Co PB,q,a (or Co LB,q,a ) if and only if F = RF (F ).


3335

w (c) The spaces Co PB,q,a and Co Lw B,q,a do not depend on the frame F in the sense that a different admissible frame in the sense of Definition 2.1 leads to the same space. Furthermore, if we use a weight of the form (4.6) satisfying v(x) vw,B,q (x) then the corresponding spaces coincide as well. We also have

w w , Co PB,q,a = f ∈ S Rd : VF f ∈ PB,q,a

and similarly for Co Lw B,q,a . Proof. Assertions (a), (b) follow from Proposition 2(a), (b) in [24] and Lemma 4.18. The assertion in (c) is a consequence of the abstract independence results in Lemmas 3.6, 3.7 together with Proposition 4.8. 2 4.4. Discretizations In the following we use a covering U = U α,β = {Uj,k }j,k as introduced in Section 4.2. Definition 4.21. The oscillation kernels oscα,β and osc∗α,β are given as follows oscα,β (x, t), (y, s) =

sup (z,r)∈Q(y,s)

where Q(y,s) =

(j,k): (y,s)∈Uj,k

RF (x, t), (y, s) − RF (x, t), (z, r) ,

Uj,k , and osc∗α,β is its adjoint.

Next, we show that the norms of these kernels can be made arbitrarily small by choosing a sufficiently fine covering. Proposition 4.22. Let F = {ϕx }x∈X be an admissible continuous frame in the sense of Definition 2.1. (i) Let α0 > 0 and β0 > 1 be arbitrary. The kernels oscα,β and osc∗α,β with 0 < α α0 and w 1 < β β0 are uniformly bounded operators on PB,q,a (X) and belong to Amv for every weight v of the form (4.6). (ii) If α ↓ 0 and β ↓ 1 then oscα,β : P w → P w → 0, B,q,a B,q,a

∗ osc

α,β

w w → 0. : PB,q,a → PB,q,a

Proof. Because of the particular structure of F , see Definition 2.1, we guarantee that Φ satisfies (D), (ML ) and (SL ) and that Φ0 satisfies (D) and (SL ) for all L > 0. Putting G1 (y, s) = WΦ Φ(y, s), G2 (y, s) = (WΦ0 Φ)(y, s), G3 (y, s) = (WΦ Φ0 )(y, s) and G4 (y) = (WΦ0 Φ0 )(y, 1) then Lemma 4.17 yields the following estimates for every L > 0 and every N ∈ N Gi (y, s) CN

s αi |y| −N , 1 + (1 + s)βi 1+s

y ∈ Rd , s ∈ (0, 1], i = 1, 2, 3,

(4.14)

3336


where α1 = L + d/2, β1 = 2L + d, α2 = 0, β2 = L + d/2 and α3 = L + d/2, β3 = 0. Moreover, we have G4 (y) CN 1 + |y| −N . For K = osc we choose the set U = [−α, α]d × [β −1 , β] and U0 = [−α, α]d and use the functions from (4.2),

G1 (y, s) =

sup (z,r)∈(y,s)·U

G2 (y, s) =

G2 (y, s) − G2 (z, s),

sup

(z,s)∈(y,s)·[U0 ×{1}]

G3 (y, s) =

G1 (y, s) − G1 (z, r),

sup

G3 (y, s) − G3 (z, r),

(z,r)∈(y,s)·U

G4 (y) = sup G4 (y) − G4 (z), z∈y+U0

where (y, s) · U = {(y + sx, st): (x, t) ∈ U }. Clearly, the functions Gi depend on α, β and obey a similar behavior as the functions Gi in (4.14) and moreover, the functions appearing in (4.3) possess this behavior for the same reason. The integrals in (4.14) are uniformly bounded in α α0 and β β0 . Using Propositions 4.8, 4.10 we obtain (i) for K = osc. For the kernel osc∗ we have to replace Gi (y, s) by Gi (y, s) defined via

G1 (y, s) =

G2 (y, s) =

sup (z,r)∈U −1 (y,s)


G3 (y, s) = G4 (y) =

G1 (y, s) − G1 (z, r), G2 (y, s) − G2 (z, r),

sup

(z,s)∈[U0 ×{1}](y,s)

G3 (y, s) − G3 (z, s),

sup G4 (y) − G4 (z),

z∈y+U0

where U −1 = {(−x/t, 1/t): (x, t) ∈ U } and U · (y, s) = {(x + ty, st): (x, t) ∈ U }. Analogous arguments give (i) for osc∗ . For the proof of (ii) we use the continuity of the functions Gi and argue analogously as in [28, Lem. 4.6(ii)]. 2 Let us state the first discretization result. α

Theorem 4.23. Let 1 q ∞, a > 0, w ∈ Wα13,α2 , B(Rd ) satisfying (B1) and (B2), and F = {ϕx }x∈X be an admissible continuous wavelet frame. There exist α0 > 0 and β0 > 1, such that for all 0 < α α0 and 1 < β β0 there is a discrete wavelet frame Fd = {ϕxj,k }j ∈N0 , k∈Zd with xj,k = (αkβ −j , β −j ) and a corresponding dual frame Ed = {ej,k }j ∈N0 , k∈Zd such that: (a)

f | Co P w

B,q,a

f, ϕx j,k j ∈N

d 0 , k∈Z

w

f, ej,k | PB,q,a j ∈N

d 0 , k∈Z

w . | PB,q,a


3337

w (b) If f ∈ Co PB,q,a then the series

f=

f, ej,k ϕxj,k =

j ∈N0 k∈Zd

f, ϕxj,k ej,k

j ∈N0 k∈Zd

w converge unconditionally in the norm of Co PB,q,a if the finite sequences are dense in w ∗ 1 (PB,q,a ) and with weak -convergence induced by (Hv )∼ otherwise.

Proof. The assertion is a consequence of our abstract Theorem 3.11. Due to the choice of w , v = vw,B,q , see (4.10), we know by Lemma 4.18 that RF belongs to BY,m for Y = PB,q,a 1/v

w w and that RF (PB,q,a ) → L∞ (X). Hence, we have that F satisfies (Fv,Y ) for Y = PB,q,a (X). ∗ As a consequence of Proposition 4.22, the kernels oscα,β and oscα,β are bounded operators from Y to Y , and the norms of oscα,β and osc∗α,β tend to zero when α → 0 and β → 1. Choosing v vw,B,q and the weight m accordingly, we obtain by analogous arguments that the norms of oscα,β and osc∗α,β in Am tend to zero. Therefore, we have F ∈ D[δ, m, Y ] for every δ > 0. In particular, F satisfies D[1, 1, Y ]. 2

4.5. Wavelet bases In the sequel we are interested in the discretization of coorbits with respect to Peetre type spaces via d-variate wavelet bases of the following type. According to Lemma A.2 we start E = {0, 1}d . For with a scaling function ψ 0 and wavelet ψ 1 belonging to L2 (R). Let further #d c d c c ∈ E we define the function ψ : R → R by the tensor product ψ = i=1 ψ ci , i.e., ψ c (x) = $d ci c c c i=1 ψ (xi ). The frame Ψ on X is given by Ψ = {ψz }z∈X , where for c = 0

c ψ(x,t)

L2 c = Tx Dtc ψ : 0 < t < 1, t = ∞, Tx ψ :

and 0 ψ(x,t)

=

0: Tx ψ 0 :

0 < t < 1, t = ∞.

This construction leads to a family of continuous systems Ψ c , c ∈ E. Our aim is to apply Theorem 3.14 in order to achieve wavelet basis characterizations of the Besov–Lizorkin–Triebel type w spaces Co PB,q,a and Co Lw B,q,a . In order to apply the abstract result in Theorem 3.14 we have to consider the Gramian cross kernels Kc and Kc∗ related to the covering U α,β defined by Kc (x, y) = sup G(F , Gc )(z, y),

x, y ∈ X, c ∈ E,

z∈Qx

and Kc∗ (x, y) = Kc (y, x), see (3.8) and (3.10). α

Lemma 4.24. Let 1 q ∞, a > 0, w ∈ Wα13,α2 , B(Rd ) satisfying (B1) and (B2). Let further F be an admissible continuous frame, Gc be the frames from above, and Kc , Kc∗ , c ∈ E, the corresponding Gramian cross kernels. The weight vw,B,q is given by (4.10) and mw,B,q denotes

3338


its associated weight. Assume further that the functions ψ 0 , ψ 1 satisfy (D) and (SK ), and that ψ 1 also satisfies (ML−1 ). (i) Under the assumption d d d d + |α3 − δ2 | + − α1 − γ2 , − + 2|α3 − δ2 | + − α1 − γ2 K, L > max 2 q 2 q (4.15) we have Kc , Kc∗ ∈ Amw,B,q for all c ∈ E. (ii) If d d d d − + α2 + a, α1 + 2a + − K, L > max q 2 2 q

(4.16)

w w then the kernels Kc , Kc∗ define bounded operators from PB,q,a (X) to PB,q,a (X).

Proof. We start with c ∈ E, c = 0. The following is analogous to the treatment of osc∗ in Proposition 4.22. As before we use the sets U = [−α, α]d × [β −1 , β] and U0 = [−α, α]d . A straightforward computation (analogously to the proof of Proposition 4.22) gives the bounds (4.2) for the kernel Kc with Gc1 (y, s) = Gc2 (y, s) = Gc3 (y, s) =



WΦ ψ c (z, r),

0 < s < ∞, y ∈ Rd ,

WΦ ψ c (z, r), 0

1 < s < ∞, y ∈ Rd ,

sup

(z,s)∈[U0 ×{1}](y,s)

WΦ ψ c (z, s),

Gc4 (y) = sup WΦ0 ψ c (z, 1), z∈U0 +y

0 < s < 1, y ∈ Rd ,

y ∈ Rd .

See the proof of Proposition 4.22 for the used notation. Since ψ c satisfies (D), (SK ), (ML−1 ), and Φ satisfies (MJ ) for all J ∈ N, we obtain with the help of Lemma 4.17 the following estimates, valid for all N ∈ N, |y| −N s K+d/2 1+ , 1 s +1 (1 + s)2K+d −N c G (y, s) CN s −(L+d/2) 1 + |y| , 2 s

c G (y, s) CN

K+d/2 c G (y, s) CN s , 3 (1 + |y|)N c CN G (y) . 4 (1 + |x|)N

Now we consider the kernels Kc∗ , c ∈ E. In this case we obtain (4.2) with

(4.17)


G∗,c 1 (y, s) = G∗,c 2 (y, s) = G∗,c 3 (y, s) = G∗,c 4 (y) =

sup (z,r)∈(y,s)·U

sup

(Wψ c Φ)(z, r),

0 < s < ∞, y ∈ Rd ,

(Wψ c Φ)(z, s),

1 < s < ∞, y ∈ Rd ,

(z,s)∈(y,s)·[U0 ×{1}]

sup

(Wψ c Φ0 )(z, s),

3339

0 < s < 1, y ∈ Rd ,

(z,r)∈(y,s)·U

sup (Wψ c Φ0 )(z, 1),

y ∈ Rd .

z∈y+U0

See again the proof of Proposition 4.22 for the used notation. The corresponding estimates are similar to (4.17), we just have to swap the role of K and L. Hence, Proposition 4.10 implies that Kc , Kc∗ belong to Amw,B,q if (4.15) is satisfied. Similar, Proposition 4.8 implies that the w w operators Kc , Kc∗ map PB,q,a boundedly into PB,q,a if (4.16) is satisfied. In case c = 0 we have 0 0 G1 = G2 = 0, and G3 (y, s), G4 (y). The same conditions on K and L lead to the boundedness of the operators K0 , K0∗ . 2 w Now we are ready for the discretization of Co PB,q,a and Co Lw B,q,a in terms of orthonormal w wavelet bases. We only state the results for Co PB,q,a . For Co Lw B,q,a it is literally the same. α

Theorem 4.25. Let 1 q ∞, a > 0, w ∈ Wα13,α2 , B(Rd ) satisfying (B1) and (B2), and F be an admissible continuous wavelet frame. Assume that ψ 0 , ψ 1 ∈ L2 (R) generate an orthonormal wavelet basis of L2 (Rd ) in the sense of Lemma A.2 where ψ 0 satisfies (D), (SK ), and ψ 1 satisfies (D), (SK ), (ML−1 ) such that d d d d K, L > max + |α3 − δ2 | + − α1 − γ2 , − + 2|α3 − δ2 | + − α1 − γ2 , 2 q 2 q d d d d − + α2 + a, α1 + 2a + − . (4.18) q 2 2 q w Then every f ∈ Co PB,q,a has the decomposition

f=

c∈E k∈Zd

λc0,k ψ c (· − k) +

jd λcj,k 2 2 ψ c 2j · −k ,

(4.19)

c∈E\{0} j ∈N k∈Zd

where the sequences λc = {λcj,k }j ∈N0 , k∈Zd defined by

jd λcj,k = f, 2 2 ψ c 2j · −k ,

j ∈ N0 , k ∈ Zd ,

w w ) = (PB,q,a ) (U) for every c ∈ E, where U = U 1,2 is the belong to the sequence space (PB,q,a covering introduced in Section 4.2 with α = 1, β = 2. w if all sequences λc (f ) belong to Conversely, an element f ∈ (Hv1w,B,q )∼ belongs to Co PB,q,a w w 1,2 if the finite sequences are (PB,q,a ) (U ). The convergence in (4.19) is in the norm of Co PB,q,a w ∗ dense in (PB,q,a ) . In general, we have weak -convergence.

3340


Proof. We apply the abstract Theorem 3.14. First of all, Lemma 4.18 shows that F satisfies 1/v w w and v = vw,B,q given by (4.10). The embedding (PB,q,a ) → (L∞ ) is (Fv,Y ) for Y = PB,q,a ensured by the abstract Lemma 3.10 in combination with Lemma 4.11. The required boundedness of the Gramian kernels is showed in Lemma 4.24. 2 Remark 4.26. The space Co Lw B,q can be discretized in the same way. According to Remark 4.7 the corresponding conditions in Proposition 4.8 are much weaker. The parameter a is not needed here. We leave the details to the reader. 5. Examples – generalized Besov–Lizorkin–Triebel spaces The main class of examples is represented by the scales of Besov–Lizorkin–Triebel–Morrey spaces and weighted Besov–Lizorkin–Triebel spaces, where we consider B(Rd ) to be a Morrey space Mu,p (Rd ), see Definition 5.19 below, or a weighted Lebesgue space Lp (Rd , v), see (1.1). In the sequel we consider only weight functions v such that (B2) is satisfied for Lp (Rd , v). 5.1. Generalized 2-microlocal Besov–Lizorkin–Triebel spaces with Muckenhoupt weights A large class of examples is given by the scales of inhomogeneous 2-microlocal Besov and w (Rd , v) and F w (Rd , v) on Rd with Muckenhoupt weights. These Lizorkin–Triebel spaces Bp,q p,q spaces represent a symbiosis of the spaces studied by Kempka [33–35], Bui [5,6], Bui et al. [7,8], w (Rd , v) and F w (Rd , v) contain the classical and Haroske and Piotrowska [31]. The scales Bp,q p,q inhomogeneous Besov–Lizorkin–Triebel spaces. For their definition, basic properties, and results on atomic decompositions we mainly refer to Triebel’s monographs [53,55–57]. Let us briefly recall the definition and some basic facts on Muckenhoupt weights. A locally integrable function v : Rd → R+ belongs to Ap , 1 0,

B

holds, where 1/p + 1/p = 1 and A is some constant independent of y and r. The Ap -condition implies the condition

−dp t + |x − y| v(y) dy ct −dp

Rd

v(y) dy,

x ∈ Rd , t > 0,

(5.1)

B(x,t)

where c is independent of x and t. See for instance [5] and the references given there. We further put A∞ :=

%

Ap .

p>1

Lemma 5.1. Let v ∈ Ap for some 1 < p < ∞. Then the space Lq (Rd , v), 1 q ∞, satisfies property (B2) with γ1 = 0, γ2 = dp/q and δ1 = dp/q and δ2 = −dp/q.


3341

Proof. Let x ∈ Rd and 0 < t < 1 arbitrary. Let further Q = [−1, 1]d . We estimate using (5.1) t dp/q t −dp/q

1/q t dp/q

v(y) dy

−dp 1 + |x − y| v(y) dy

Rd

x+tQ

t dp/q (1 + |x|)dp/q

1/q

−dp 1 + |y| v(y) dy

1/q

Rd

−dp/q t dp/q 1 + |x| , since v is supposed to be locally integrable. The estimate from above proceeds as follows,

1/q v(y) dy

1/q

v(y) dy √ B(0,|x|+ dt)

x+tQ

−dp dp 1 + |y| 1 + |y| v(y) dy

1/q

√ B(0,|x|+ dt)

dp/q 1 + |x|

1/q v(y) dy

dp/q 1 + |x| ,

B(0,1)

where we used (5.1) in the last step.

2

The crucial tool in the theory of Muckenhoupt weights is the vector-valued Fefferman–Stein maximal inequality, see for instance [5, Lem. 1.1] or [31, Thm. 2.11] and the references given there. Lemma 5.2. Let 1 < p < ∞, 1 < q ∞, v ∈ Ap , and {fj }j be a sequence in Lp (Rd , v). Then 1/q 1/q d d q q |Mfj | |fj | Lp R , v Lp R , v . j

j

w (Rd , v) and F w (Rd , v) relies on a dyadic decomposition of The definition of the spaces Bp,q p,q unity, see also [53, 2.3.1].

Definition 5.3. Let Φ(Rd ) be the collection of all systems {ϕj (x)}j ∈N0 ⊂ S(Rd ) with the following properties: (i) ϕj (x) = ϕ(2−j x), j ∈ N, (ii) supp ϕ0 ⊂ {x ∈ Rd : |x| 2}, supp ϕ ⊂ {x ∈ Rd : 1/2 |x| 2}, ∞ (iii) j =0 ϕj (x) = 1 for every x ∈ R.

3342

H. Rauhut, T. Ullrich / Journal of Functional Analysis 260 (2011) 3299–3362 α

w (Rd , v) and B w (Rd , v) we define for a weight w ∈ W 3 In order to define the spaces Fp,q α1 ,α2 p,q a semi-discrete counterpart {wj }j ∈N0 , corresponding to an admissible weight sequence in the sense of [33–35]. We put

wj (x) =

w(x, 2−j ): w(x, ∞):

j ∈ N, j = 0.

(5.2)

3 d Definition 5.4. Let v ∈ A∞ and {ϕj (x)}∞ j =0 ∈ Φ(R ). Let further w ∈ Wα1 ,α2 with associated j = ϕj . weight sequence {wj }j ∈N0 defined in (5.2) and 0 < q ∞. Put Φ

α

(i) For 0 < p ∞ we define (modification if q = ∞) w Bp,q

d R , v = f ∈ S Rd : f |B w Rd , v = p,q

∞ wj (x)(Φj ∗ f )(x)|Lp Rd , v q

1/q

& 0 and the sums by integrals over t. Characterizations of this type have some history and are usually referred to as characterizations via local means. For further references and some historical facts we mainly refer to [54,55,7,8,43] and in particular to the recent contribution [61], which provides a complete and self-contained reference. Essential for what follows are functions Φ0 , Φ ∈ S(Rd ) satisfying the so-called Tauberian conditions


3343

Φ 0 (x) > 0 on |x| < 2ε ,

Φ(x) > 0 on ε/2 < |x| < 2ε ,

(5.3)

= 0 for all |α| D α¯ Φ(0) ¯ 1R

(5.4)

for some ε > 0, and

for some R + 1 ∈ N0 . If R + 1 = 0 then the condition (5.4) is void. We call the functions Φ0 and Φ kernels for local means and use the notations Φk = 2kd Φ(2k ·), k ∈ N, as well as Ψt = Dt Ψ = t −d Φ(·/t), and the well-known classical Peetre maximal function ∗ |(Ψt ∗ f )(x + y)| Ψt f a (x) = sup , (1 + |y|/t)a y∈Rd

x ∈ Rd , t > 0,

originally introduced by Peetre in [38]. The second ingredient is a Muckenhoupt weight v ∈ A∞ . The critical index p0 is defined by p0 := inf{p: v ∈ Ap }.

(5.5)

α

Proposition 5.6. Let w ∈ Wα13,α2 and v belong to the class A∞ , where p0 is given by (5.5). Choose functions Φ0 , Φ ∈ S(Rd ) satisfying (5.3) and (5.4) with R + 1 > α2 . (i) If 0 < q ∞, 0 

dp0 p

+ α3 then, for both i = 1, 2,

d d

w w Bp,q R , v = f ∈ S Rd : f |Bp,q R , v i < ∞ ,

i = 1, 2,

where f |B w Rd , v = w(x, ∞)(Φ0 ∗ f )(x)|Lp Rd , v p,q 1 1 +

w(x, t)(Φt ∗ f )(x)|Lp Rd , v q dt t

1/q ,

0

f |B w Rd , v = w(x, ∞) Φ ∗ f (x)|Lp Rd , v p,q 0 2 a 1 +

w(x, t) Φ ∗ f (x)|Lp Rd , v q dt t a t

1/q .

0 w (Rd , v) , i = 1, 2, are equivalent quasi-norms in B w (Rd , v). Moreover, · |Bp,q i p,q (ii) If 0 < q ∞, 0 d max{p0 /p, 1/q} + α3 then

d d w w Fp,q R , v = f ∈ S Rd : f |Fp,q R , v i < ∞ , where

i = 1, 2,

(5.6)

3344


f |F w Rd , v = w(x, ∞)(Φ0 ∗ f )(x)|Lp Rd , v p,q 1 1 1/q q dt d w(x, t)(Φt ∗ f )(x) + Lp R , v , t

(5.7)

0

f |F w Rd , v = w(x, ∞) Φ ∗ f (x)|Lp Rd , v p,q 0 2 a 1 1/q q dt ∗ d w(x, t) Φ f (x) + Lp R , v . t a t

(5.8)

0 w (Rd , v) , i = 1, 2, are equivalent quasi-norms in F w (Rd , v). Moreover, · |Fp,q i p,q

Proof. We only prove (ii) since the proof of (i) is analogous. The arguments are more or less the same as in the proof of [61, Thm. 2.6]. Let us provide the necessary modifications. Step 1. At the beginning of Substep 1.3 in the proof of [61, Thm. 2.6] we proved a crucial inequality stating that for r > 0 and a < N ∗ |((Φk+ )t ∗ f )(y)|r 2−kN r 2(k+)d dy, (5.9) Φ2− t f a (x)r c (1 + 2 |x − y|)ar k∈N0

Rd

where c is independent of f , x, t and but may depend on N and a. In case = 0 we have to replace (Φ2∗− t f )a (x) by (Φ0∗ f )a (x) on the left-hand side and (Φk+ )t by Φk+ = Φ0 for k = 0 on the right-hand side. We modify (5.9) by multiplying with |w(x, 2− t)|r on both sides (|w(x, ∞)|r in case = 0). By using w(x, 2− t) 2kα1 (1 + 2 |x − y|)α3 w(y, 2−(k+) t), which follows from (W 1) and (W 2), this gives the following modified relation ∗ Φ − f (x)w x, 2− t r a 2 t c

k∈N0

2−k(N −α1 )r 2(k+)d

Rd

|((Φk+ )t ∗ f )(y)w(y, 2−(k+) t)|r dy. (1 + 2 |x − y|)(a−α3 )r

(5.10)

Now we choose r > 0 in a way such that (a) r(a − α3 ) > d, (b) p/r, q/r > 1, and (c) p0 < p/r. Let us shortly comment on these conditions. Condition (a) is needed in order to replace the convolution integral on the right-hand side of (5.10) by the Hardy–Littlewood maximal function M[|w(·, 2−(k+) t)((Φk+ )t ∗ f )|r ](x) via Lemma 4.13. Conditions (b) and (c) are necessary in order to guarantee the Fefferman–Stein maximal inequality, see Lemma 5.2, in the space Lp/r (q/r , Rd , v), where we use that v ∈ Ap/r as a consequence of (c) and (5.5). Since p0 1 the conditions (a), (b), (c) are satisfied if p d < r < min ,q a − α3 p0 which is possible if we assume a > α3 + d max{p0 /p, 1/q}. Now we can proceed analogously as done in Substep 1.3 of the proof of Theorem 2.6 in [61] and obtain the equivalence of w (Rd , v) and f |F w (Rd , v) on S (Rd ). With the same type of argument but some f |Fp,q 1 2 p,q


3345

minor modifications we show that their discrete counterparts (in the sense of Definition 5.4) and (5.7), (5.8) are equivalent as well. Step 2. It remains to show that we can change from the system from (Φ0 , Φ) to a system (Ψ0 , Ψ ) satisfying (5.3), (5.4). We argue as in Step 2 of the proof of [61, Thm. 2.6]. There we obtain the crucial inequality

Ψ∗ f

(x) a

∞ ∗ 2(−k)(L+1−a) : Φk f a (x) (k−)(R+1) : 2 k=0

k > , k,

where L can be chosen arbitrarily large. Multiplying both sides with w (x) and using 2(k−)α1 : k > , w (x) wk (x) (−k)α 2: k, 2 we obtain ∞ 2(−k)(L+1−a−α1 ) : w (x) Ψ∗ f a (x) wk (x) Φk∗ f a (x) (k−)(R+1−α ) 2 : 2 k=0

k > , k.

With our assumption R + 1 > α2 we obtain finally ∞ 2−|k−|δ wk (x) Φk∗ f a (x), w (x) Ψ∗ f a (x) k=0

where δ = min{1, R + 1 − α2 }. Now we use a straightforward generalization of the convolution Lemma 2 in [43] and obtain immediately the desired result ∗ w Ψ f |Lp q , Rd , v wk Φ ∗ f |Lp q , Rd , v . k a a This together with Step 1 and Definition 5.4 concludes the proof.

2

α

Theorem 5.7. Let w ∈ Wα13,α2 , v ∈ A∞ , and p0 given by (5.5). We choose an admissible continuous wavelet frame F according to Definition 2.1. Let further 1 p ∞ (p < ∞ in F -case) and 1 q ∞. Putting d/q−d/2 w(x, t): 0 < t 1, (5.11) w(x, ˜ t) := t w(x, ∞): t = ∞, and B(Rd ) = Lp (Rd , v) we have the following identities in the sense of equivalent norms d ˜ w Bp,q R , v = Co Lw B,q,a , F if a >

dp0 p

+ α3 and d w˜ w Fp,q R , v = Co PB,q,a ,F

if a > d max{p0 /p, 1/q} + α3 .

3346


Proof. For t ∈ (0, 1) and x ∈ Rd it holds (VF f )(x, t) = DtL2 Φ(−·) ∗ f¯ (x) = t d/2 Dt Φ(−·) ∗ f¯ (x) with an obvious modification in case t = ∞. Hence the identities are consequences of Definition 4.6, Theorem 4.20(c), and Proposition 5.6. 2 w (Rd , v). The Now we are prepared for the discretization result, which we state only for Fp,q w d 1,2 conditions for Bp,q (R , v) are the same. We use the covering U = U in Section 4.2 for α = 1, w (v) := p w˜ and bw (v) = w˜ where B(Rd ) = β = 2, and the associated sequence spaces fp,q p,q B,q B,q d Lp (R , v). We get

{λj,k }j,k |f w (v) p,q d = R L w(k, ∞)|λ |χ , v 0,k 0,k p k∈Zd

!q 1/q j d/2 −j −j d 2 w k2 , 2 |λj,k |χj,k + Lp R , v , j ∈N0

(5.12)

k∈Zd

as well as {λj,k }j,k |bw (v) p,q d = R w(k, ∞)|λ |χ , v 0,k 0,k Lp k∈Zd

j d/2 −j −j q 1/q d + w k2 , 2 |λj,k |χj,k Lp R , v . 2 j ∈N0

(5.13)

k∈Zd

Note, that Corollary 4.14 is applicable with p d < r < min ,q a − α3 p0 as a consequence of a > d max{p0 /p, 1/q} + α3 . α

Theorem 5.8. Let w ∈ Wα13,α2 , v ∈ A∞ , and p0 defined by (5.5). Let further 1 p ∞ (p < ∞ in F -case) and 1 q ∞. Assume that ψ 0 , ψ 1 ∈ L2 (R) generate a wavelet basis in the sense of Lemma A.2, where ψ 0 satisfies (D), (SK ), and ψ 1 satisfies (D), (SK ), (ML−1 ) such that 2p0 d d + d max 1, + α3 , K, L > max max α1 , − 2 q p max α1 , d − d + d max p0 , 3p0 − 1 + 2α3 , 2 q p p


p0 1 d d + d max , + α3 , max α2 , − q 2 p q p0 1 d d , max α1 , − + 2d max + 2α3 . 2 q p q

3347

(5.14)

w (Rd , v) has the decomposition Then every f ∈ Fp,q

f=

λc0,k ψ c (· − k) +

jd λcj,k 2 2 ψ c 2j · −k ,

(5.15)

c∈E\{0} j ∈N k∈Zd

c∈E k∈Zd

where the sequences λc = {λcj,k }j ∈N0 , k∈Zd defined by

jd λcj,k = f, 2 2 ψ c 2j · −k ,

j ∈ N0 , k ∈ Zd ,

w (v) for every c ∈ E. belong to the sequence space fp,q w (Rd , v) if all sequences λc (f ) belong Conversely, an element f ∈ (Hv1w,B,q )∼ belongs to Fp,q ˜

w (v). The convergence in (5.15) is in the norm of F w (Rd , v) if the finite sequences are to fp,q p,q w (v). In general, we have weak∗ -convergence induced by (H1 dense in fp,q )∼ . vw,B,q ˜

Proof. The statement is a consequence of Theorem 4.25, Lemma 5.1, Theorem 5.7, and Corollary 4.14. Indeed, the parameters α˜ 1 , α˜ 2 , and α˜ 3 according to w˜ are given by α˜ 1 = (α1 + d/q − d/2)+ , α˜ 2 = (α2 + d/2 − d/q)+ and α˜ 3 = α3 . 2 Remark 5.9. The conditions in the B-case are slightly weaker. Since we have then a > dp0 /p + α3 , see Theorem 5.7, we can replace the term d max{p0 /p, 1/q} by dp0 /p in (5.14). Without the weight v, i.e., v ≡ 1, we obtain wavelet characterizations for the generalized 2-microlocal spaces studied by Kempka in [33–35]. α

Theorem 5.10. Let w ∈ Wα13,α2 and 1 p ∞ (p < ∞ in F -case), 1 q ∞. Let further ψ 0 , ψ 1 ∈ L2 (R) generate a wavelet basis in the sense of Lemma A.2 where ψ 0 satisfies (D), (SK ), and ψ 1 satisfies (D), (SK ), (ML−1 ) such that 1 1 d d + d max , + α3 , K, L > max max α2 , − q 2 p q max α1 , d − d + d max 1 , 1 − 1 + α3 , 2 q p p 1 1 d d + 2d max , + 2α3 , max α1 , − 2 q p q max α1 , d − d + 2α3 . 2 q

(5.16)

w (Rd ) can be discretized in the sense of TheoThen the generalized 2-microlocal spaces Fp,q rem 5.8.

3348


Proof. We apply Theorem 4.25 with B = Lp (Rd ) in connection with Theorem 5.7 where in this case a > d max{1/p, 1/q} + α3 . We can use Example 4.5 instead of Lemma 5.1, where γ2 = d/p and δ2 = 0. 2 Remark 5.11. The conditions in the B-case are slightly weaker. Since we have then a > d/p + α3 , see Theorem 5.7 with p0 = 1, we can replace the term d max{1/p, 1/q} by d/p in (5.16). Finally, we obtain characterizations for the classical Besov–Lizorkin–Triebel spaces by putting w(x, t) = t −s . Theorem 5.12. Let 1 p, q ∞ (p < ∞ in the F -case) and s ∈ R. Assume that ψ 0 , ψ 1 ∈ L2 (R) generate a wavelet basis in the sense of Lemma A.2 and let ψ 0 satisfy (D), (SK ), and ψ 1 satisfy (D), (SK ), (ML−1 ). (i) Assuming that d d 1 1 1 1 d d + d max + d max , , min s, − ,1 − , K, L > max max s, − q 2 p q q 2 p p 1 1 d d + 2d max , , − min s, − q 2 p q s (Rd ) can be discretized in then the classical inhomogeneous Lizorkin–Triebel spaces Fp,q the sense of Theorem 5.8. (ii) In case

d d 1 d 1 d d + d max + , min s, − ,1 − , K, L > max max s, − q 2 p q 2 p p 2d d d + − min s, − q 2 p s (Rd ) can be discretized in the sense of Theorem 5.8. the classical Besov spaces Bp,q

Proof. We apply Theorem 4.25 with B = Lp (Rd ) and w(x, t) = t −s in connection with Theorem 5.7. This gives α˜ 2 = (s + d/2 − d/q)+ , α˜ 1 = −(s + d/2 − d/q)− , and α˜ 3 = 0. In the B-case we have therefore a > d/p, while a > d/ min{p, q} in the F -case. 2 Remark 5.13. Theorems 5.8, 5.10, and 5.12 provide in particular characterizations in terms of orthonormal spline wavelets, see Appendix A.1. Indeed, we have that ψ 1 = ψm satisfies (ML−1 ) for L = m and ψ 0 = ϕm , ψ 1 = ψm satisfy (D) and (SK ) for K < m − 1. Remark 5.14. Since all our results rely on the abstract Theorem 3.14 we are able to use even biorthogonal wavelets [10]. The conditions on the smoothness and the vanishing moments of the wavelet and dual wavelet are similar. See also [36] for earlier results in this direction.


3349

5.2. Besov–Lizorkin–Triebel spaces with doubling weights We intend to extend the definition of weighted Besov–Lizorkin–Triebel spaces also to general doubling weights and give corresponding atomic and wavelet decompositions. It is well known that, for general doubling weights, Lemma 5.2 does not apply. Therefore, the first challenge is to define certain spaces in a reasonable way, i.e., to get at least the independence of the definition of the used dyadic decomposition of unity. Bownik [4] approaches such definition by adapting the classical ϕ-transform due to Frazier and Jawerth [25] to the weighted anisotropic situation. A replacement of Lemma 5.2 is used to this end, where the classical Hardy–Littlewood maximal function is defined with respect to the doubling measure. Our approach is entirely different and relies on the fact that the spaces defined below can be interpreted as certain coorbits which allows to exploit our Theorems 4.23 and 4.25. A weight v : Rd → R+ is called doubling if, v(y) dy C v(y) dy, x ∈ Rd , r > 0, B(x,2r)

B(x,r)

for some positive constant C > 1 independent of r and x. Note that Muckenhoupt weights in A∞ are doubling, but there exist doubling weights which are not contained in A∞ . For a construction of such a weight we refer to [17]. However, doubling weights are suitable in our context. We start by proving that the weighted Lebesgue space Lp (Rd , v) satisfies property (B2) (note that (B1) is immediate). Lemma 5.15. Let v : Rd → (0, ∞) be a doubling weight with doubling constant C 1. Then Lp (Rd , v) satisfies property (B2) with γ1 = γ2 = 0,

δ2 =

− log2 C , p

δ1 =

log2 C . p

(5.17)

Proof. The idea is that, as a consequence of the doubling condition, v cannot decay and grow too fast. On the one hand, we have γ v(y) dy v(y) dy 1 + |x| , x ∈ Rd , (5.18) x+tQ

√ B(0,|x|+ d )

where γ = log2 C. On the other hand 1 v(y) dy n C x+tQ

√ B(x,|x|+ d )

1 v(y) dy n C

v(y) dy, B(0,1)

where n = log2 (c(1 + |x|)). Hence, we get −γ v(y) dy 1 + |x| .

(5.19)

x+Q

Finally, (5.18) and (5.19) imply that (B2) is satisfied with the parameters in (5.17).

2

3350


In order to avoid Fefferman–Stein maximal inequalities (which we indeed do not have here) we modify Definition 5.4 for a general doubling measure v as follows. d Definition 5.16. Let v be a doubling weight and {ϕj (x)}∞ j =0 ∈ Φ(R ), 0 < q ∞, and a > 0. α j = ϕj . Let w ∈ Wα13,α2 and define the weight sequence {wj }j ∈N0 as in (5.2). Put Φ

(i) For 0 < p ∞ we define (modification if q = ∞) d w R ,v Bp,q,a = f ∈ S Rd : f |B w

d = p,q,a R , v

& 1/q ∞ q |(Φ ∗ f )(x + z)| j d wj (x) sup 0) for the F - and B-spaces, respectively. To show this, we switch in a first step from one system Φ to another system Ψ in the discrete characterization given in Definition 5.16. Indeed, we argue analogously as in Step 2 of the proof of Proposition 5.6, see also [61, Thm. 2.6]. Note, that we did not use a Fefferman–Stein maximal inequality there. With a similar argument we switch in a second step from the discrete characterization to the continuous characterization (Proposition 5.6) using the same system Φ. Consequently, we identify Besov–Lizorkin–Triebel spaces with doubling weights as coorbits. α

Theorem 5.17. Let w ∈ Wα13,α2 and v be a general doubling weight with doubling constant C > 1. Choose F to be an admissible continuous wavelet frame according to Definition 2.1. Let further 1 p, q ∞ and a > 0. Putting w(x, ˜ t) as in (5.11) and B(Rd ) = Lp (Rd , v) we have the following identities in the sense of equivalent norms d ˜ w R , v = Co Lw Bp,q,a B,q,a , F , d w˜ w R , v = Co PB,q,a ,F . Fp,q,a Based on Theorem 4.25 we immediately arrive at one of our main discretization results. We state it only for the F -spaces. The conditions for the B-spaces are the same.


3351

α

Theorem 5.18. Let w ∈ Wα13,α2 and let v be a doubling weight with doubling constant C > 1. Assume 1 p, q ∞, a > 0, and let ψ 0 , ψ 1 ∈ L2 (R) generate an orthonormal wavelet basis in the sense of Lemma A.2 where ψ 0 satisfies (D), (SK ), and ψ 1 satisfies (D), (SK ), (ML−1 ) such that log2 C d d +d + + α3 , K, L > max max α1 , − 2 q p max α1 , d − d + 2 log2 C + α3 , 2 q p d d d d + a, max α1 , − + 2a . max α2 , − q 2 2 q w Then every f ∈ Fp,q,a (Rd , v) has the decomposition (5.15), where the sequences λc = {λcj,k }j ∈N0 , k∈Zd , c ∈ E, are contained in (PLw˜ (Rd ,v),q,a ) (U 1,2 ). The latter is equivalent to (4.11) p

with B(Rd ) replaced by Lp (Rd , v), β dj/q by β dj/2 , and with β = 2 and α = 1. w Conversely, an element f ∈ (Hv1w,B,q )∼ belongs to Fp,q,a (Rd , v) if all sequences λc (f ), ˜

w c ∈ E, belong to (PLw˜ (Rd ,v),q,a ) (U 1,2 ). The convergence in (5.15) is in the norm of Fp,q,a (Rd , v) p

if the finite sequences are dense in (PLw˜ (Rd ,v),q,a ) (U 1,2 ). In general, we have weak∗ p convergence. 5.3. Generalized 2-microlocal Besov–Lizorkin–Triebel–Morrey spaces Several applications in PDEs require the investigation of smoothness spaces constructed on Morrey spaces [37]. The spaces of Besov–Lizorkin–Triebel–Morrey type are currently a very active research area. We refer to Sawano [44], Sawano et al. [46,47], Tang and Xu [52] as well as to the recent monograph by Yuan et al. [64] and the references given there. Our intention in the current paragraph is to provide wavelet decomposition theorems as consequences of the fact that the mentioned spaces can be interpreted as coorbits. Note that [44,46,64] have already dealt with atomic and wavelet decompositions of these spaces. Our results have to be compared with the ones in there, see the list at the end of Section 2.2 above. We start with the definition of the Morrey space Mq,p (Rd ) on Rd . Definition 5.19. Let 0 0, x∈Rd

R

d(1/q−1/p)

f (y)p dy

1/p (5.20)

BR (x)

if p < ∞, where BR (x) denotes the Euclidean ball with radius R > 0 and center x ∈ Rd . In the case p = ∞ we put M∞,∞ (Rd ) := L∞ (Rd ). These spaces – studied first by Morrey [37] – generalize the ordinary Lebesgue spaces. Indeed, we have Mp,p (Rd ) = Lp (Rd ), 0 < p ∞. In the case q < p the quantity (5.20) is infinite as

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soon as f = 0, so that Mq,p (Rd ) = {0}. If p 1 then Mq,p (Rd ) is a Banach space, otherwise a quasi-Banach space. The following lemma ensures that Mq,p (Rd ) satisfies (B2). Lemma 5.20. Let 0 < p q ∞. Then the space Mq,p (Rd ) satisfies (B2) with γ1 = γ2 = d/q and δ1 = δ2 = 0. α of Proof. Fix x ∈ Rd and 0 < t < 1. We consider the norm of the characteristic function χ(x,t) α d d the cube Q(x,t) = x + t[−α, α] in Mq,p (R ). By (5.20) we obtain immediately

α χ

(x,t) |Mq,p

d R t d(1/q−1/p) t d/p = t d/q .

For the reverse estimate we use the well-known fact that Lq (Rd ) → Mq,p (Rd ), see [37]. Therefore, we have α χ

(x,t) |Mq,p

which concludes the proof.

d α R χ(x,t) |Lq Rd t d/q

2

w,u (Rd ) and Lizorkin–Triebel–Morrey We define the 2-microlocal Besov–Morrey spaces Bp,q w,u d d d spaces Fp,q (R ) by replacing Lp (R , v) by Mu,p (R ), u > p, in Definition 5.4. Here w ∈ α Wα13,α2 is a weight function and 0 < p, q ∞, 0 < p < u ∞, where p < ∞ in the F -case. This is a straightforward generalization of the spaces appearing in [52,44,46,47,64]. With exactly the same proof techniques we obtain a counterpart of Proposition 5.6 under the conditions a > d/p + α3 in the B-case and a > d max{1/p, 1/q} + α3 in the F -case. One uses a vector-valued Fefferman–Stein type maximal inequality for the space Mu,p (q , Rd ), where 1 < p u < ∞ and 1 < q ∞, see [52] and [9] for the case q = ∞. As a consequence, the Besov–Lizorkin–Triebel–Morrey spaces can be identified as coorbits, i.e., the following counterpart to Theorem 5.7 holds.

Theorem 5.21. Under the assumptions of Theorem 5.7 we have in the sense of equivalent norms d ˜ w,u R = Co Lw Bp,q Mu,p ,q,a , F if 1 p u ∞, 1 q ∞, and a >

d p

+ α3 as well as

d w˜ w,u R = Co PM ,F Fp,q u,p ,q,a for 1 p u < ∞, 1 q ∞, and a > d max{1/p, 1/q} + α3 . Since Corollary 4.14 is applies for the space Mu,p (q , Rd ) with d < r < min{p, q} a − α3 w,u w,u we may (and do) define the sequence spaces fp,q and bp,q just by replacing Lp (Rd , v) by d Mu,p (R ) in (5.12) and (5.13).


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The main result of this subsection is the following α

Theorem 5.22. Let w ∈ Wα13,α2 and 1 p u < ∞, 1 q ∞. Assume that ψ 0 , ψ 1 ∈ L2 (R) generate an orthonormal wavelet basis in the sense of Lemma A.2 and let ψ 0 satisfy (D), (SK ), and ψ 1 satisfy (D), (SK ), (ML−1 ) such that 1 1 d d + d max , + α3 , K, L > max max α2 , − q 2 p q max α1 , d − d + d max 1 , 1 − 1 + α3 , 2 q u u 1 1 d d + 2d max + 2α3 , max α1 , − , 2 q p q max α1 , d − d + 2α3 . 2 q w,u Then every f ∈Fp,q (Rd ) has the decomposition (5.15) where the sequences λc = {λcj,k }j ∈N0 , k∈Zd w,u belong to the sequence space fp,q for every c ∈ E. w,u Conversely, an element f ∈ (Hv1w,B,q )∼ belongs to Fp,q (Rd ) if all sequences λc (f ) belong ˜ w,u w,u . The convergence in (5.15) is considered in Fp,q (Rd ) if the finite sequences are dense to fp,q w,u d ∗ in fp,q (R ). In general, we have weak -convergence.

Remark 5.23. The modifications for the B-spaces are according to Remark 5.11. 5.4. Spaces of dominating mixed smoothness Recently, there has been an increasing interest in function spaces of dominating mixed smoothness, see [2,48–50,59,60,62]. Their structure is suitable for treating high-dimensional approximation and integration problems efficiently and overcome the so-called curse of dimensionality to some extent. These spaces can as well be treated in terms of our generalized coorbit space theory. We briefly describe this setting. In a certain sense these spaces behave like the isotropic ones for d = 1, and consequently, the proofs operate by iterating the techniques from Sections 4.1, 4.2, and 4.3. We start with a definition of mixed Peetre spaces on X¯ = X × · · · × X, where X = R × [(0, 1] ∪ {∞}], as a tensorized version of Definition 4.6. Our definition is motivated by equivalent characterizations of dominating mixed spaces, which are obtained by a combination of the techniques in [61] with [60,62]. Definition 5.24. Let 1 p, q ∞ and a¯ > 1. Let further r¯ ∈ Rd . We define by r¯ ¯ ¯ Pp,q, a¯ (X) = F : X → C: ¯ ¯ ¯ Lrp,q, a¯ (X) = F : X → C:

F |P w < ∞ , p,q,a¯

F |Lr¯ p,q,a¯ < ∞

¯ where we put two scales of Banach function spaces on X,

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:= sup |F ((x1 + z1 , ∞), . . . , (xd + zd , ∞))| Lp Rd p,q,a¯ d a a d 1 (1 + |z |) · · · (1 + |z |)

F |P r¯

1

z∈R

+

A⊂{1,...,d} A=∅

d

1 1 ! |F ((x1 + z1 , t1 ), . . . , (xd + zd , td ))| q $ sup $ ··· ai ai i∈A (1 + |zi |/ti ) i ∈A / (1 + |zi |) z∈Rd 0

'

dti × ti−ri q 2 ti i∈A

0

1/q

d Lp R

and := sup |F ((x1 + z1 , ∞), . . . , (xd + zd , ∞))| Lp Rd p,q,a¯ d a a (1 + |z |) 1 · · · (1 + |z |) d

F |Lr¯

1

z∈R

+

1

A⊂{1,...,d} A=∅

'

−r q dti × ti i 2 ti i∈A

0

d

q 1 |F ((x1 + z1 , t1 ), . . . , (xd + zd , td ))| d $ $ sup L R ··· p d (1 + |z |/t )ai (1 + |z |)ai 0

z∈R

i

i∈A

i

i ∈A /

i

1/q .

For fixed A ⊂ {1, . . . , d} we put ti = ∞ if i ∈ / A. In case q = ∞ the integrals over ti , i ∈ A, have to be replaced by a supremum over ti . 5.4.1. Associated sequence spaces We cover the space X¯ by the Cartesian product of the family from Section 4.2. For fixed α > 0 and β > 1 we consider the family U¯ α,β = {U¯ j¯,k¯ }j ∈Nd , k∈Zd of subsets 0

U¯ j¯,k¯ = Uj1 ,k1 × · · · × Ujd ,kd . Clearly, we have X¯ ⊂

¯ d j¯∈Nd0 , k∈Z

U¯ j¯,k¯ . We use the notation

χj¯,k¯ (x) = (χj1 ,k1 ⊗ · · · ⊗ χjd ,kd )(x) =

d '

χji ,ki (xi ),

x ∈ Rd .

i=1

Iterating dimensionwise the arguments leading to (4.13) gives the following description for the r¯ r¯ sequence spaces (Pp,q, a¯ ) and (Lp,q,a¯ ) . Theorem 5.25. If 1 p, q ∞, a¯ > 1/ min{p, q}, and r¯ ∈ Rd then !q 1/q |j | /q j¯r¯ d 1 {λ ¯ ¯ } ¯ ¯ | P r¯ β β |λj¯,k¯ |χj¯,k¯ (x) Lp R p,q,a¯ j ,k j ,k j¯∈Nd0

¯ d k∈Z

(5.21)


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and {λ ¯ ¯ } ¯ ¯ | Lr¯ p,q,a¯ j ,k j ,k

j¯∈Nd0

d '

β ji (ri +1/q−1/p)q

q/p 1/q |λj¯,k¯ |p

.

(5.22)

¯ d k∈Z

i=1

¯ r¯ r¯ r¯ We have (Lrp,q, a¯ ) = (Lp,q,a¯ ) and (Pp,q,a¯ ) = (Pp,q,a¯ ) , respectively.

Proof. Since we deal here with usual Lp (Rd ) and Lp (q )-spaces we can use the methods from Corollary 4.14 to obtain (5.21) and (5.22). The Hardy–Littlewood maximal operator then acts componentwise. For the corresponding maximal inequality see [62, Thm. 1.11]. 2 ¯ r¯ ¯ ¯ 5.4.2. The coorbits of Lrp,q, a¯ (X) and Pp,q,a¯ (X) We apply the abstract theory in a situation where the index set is given by

X¯ = X · · × X+ ( × ·)* d -times with X = R × [(0, 1) ∪ {∞}]. This space is equipped with the product measure μX¯ = μX ⊗ · · · ⊗ μX , i.e., F (z1 , . . . , zd ) μX¯ (dz) = . . . F (z1 , . . . , zd ) μX (dz1 ) . . . μX (dzd ). X¯

X

X

We put H = L2 (Rd ). We choose an admissible continuous frame F1 = {ϕx }x∈X according to Definition 2.1. For z = (z1 , . . . , zd ) ∈ X¯ we define ϕ¯z := ϕz1 ⊗ · · · ⊗ ϕzd . It is easy to see that the system F¯ = {ϕ¯ z }z∈X¯ represents a tight continuous frame indexed by X¯ in H. The corresponding frame transform is given by VF¯ f (z) = f, ϕ¯z ,

¯ z ∈ X.

For 1 p, q ∞ and r¯ ∈ Rd put for i = 1, . . . , d vp,q,ri (x, t) =

1: max{t −(1/q−1/p) t −ri , t −(1/p−1/q) t ri }:

t = ∞, t ∈ (0, 1)

and vp,q,¯r = vp,q,ri ⊗ · · · ⊗ vp,q,ri . Let us define the corresponding coorbit spaces. Definition 5.26. Let 1 p, q ∞, r¯ ∈ Rd , and a¯ > 0. We define

∼

r¯ ¯ : VF¯ f ∈ Pp,q, a¯ (X) , r¯ 1 ∼

¯ ¯ ¯ ¯ Co Lrp,q, : VF¯ f ∈ Lrp,q, a¯ = Co Lp,q,a¯ , F := f ∈ Hvp,q,¯r a¯ (X) . r¯ r¯ 1 ¯ Co Pp,q, a¯ = Co Pp,q,a¯ , F := f ∈ Hvp,q,¯r

An iteration of the techniques from Section 4 shows that all the conditions needed for the above definition are valid.

3356


Remark 5.27. It is also possible to define spaces with dominating mixed smoothness in a more general sense as done in the isotropic case, see Definition 4.6 and the corresponding coorbit spaces. Indeed, it is possible to treat even weighted spaces or 2-microlocal spaces of dominating mixed smoothness as in the previous subsections. r¯ r¯ Theorem 5.28. The spaces Co Pp,q, a¯ and Co Lp,q,a¯ are Banach spaces and do not depend on the frame F¯ . Furthermore, we have the identity d

r¯

r¯ Co Pp,q, a¯ = f ∈ S R : VF¯ f ∈ Co Pp,q,a¯ , ¯ respective for Co Lrp,q, a¯ .

5.4.3. Relation to classical spaces r¯ F (Rd ) and S r¯ B(Rd ). It is well known that these We give the definition of the spaces Sp,q p,q spaces can be characterized in a discrete way via so-called local means and Peetre maximal functions [62,60,30,61]. Recall the notion of decomposition of unity in Definition 5.3. We start with d systems ϕ i ∈ Φ(R) for i = 1, . . . , d and put 1 ϕ ⊗ · · · ⊗ ϕ d ¯(ξ1 , . . . , ξd ) := ϕ11 (ξ1 ) · · · ϕdd (ξd ), ξ ∈ Rd , ¯ ∈ Nd0 . Definition 5.29. Let r¯ = (r1 , . . . , rd ) ∈ Rd and 0 < q ∞. r¯ B(Rd ) is the collection of all f ∈ S (Rd ) such that (i) Let 0 < p ∞. Then Sp,q

f |S r¯ B Rd ϕ¯ = p,q

¯ d ∈N 0

2

¯ r¯ ·q 1

q ∨ ϕ ⊗ · · · ⊗ ϕ ¯(ξ )f (x)|Lp Rd d

1/q

is finite (modification if q = ∞). r¯ F (Rd ) is the collection of all f ∈ S (Rd ) such that (ii) Let 0 < p < ∞. Then Sp,q r¯ ·q ∨ q 1/q d ¯ 1 d f |S r¯ F Rd ϕ¯ = ϕ ⊗ · · · ⊗ ϕ ¯(ξ )f (x) 2 Lp R p,q ¯ d ∈N 0

is finite (modification if q = ∞). The following theorem states the relation between previously defined coorbit spaces and the classical spaces with dominating mixed smoothness. Theorem 5.30. Let 1 p, q ∞ (p < ∞ in the F -case), r¯ ∈ Rd , and a¯ > 1/ min{p, q}. Then we have in the sense of equivalent norms r¯ +1/2−1/q r¯ F Rd = Co Pp,q,a¯ , F¯ Sp,q and if a¯ > 1/p r¯ +1/2−1/q r¯ B Rd = Co Lp,q,a¯ , F¯ . Sp,q


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Proof. We apply the continuous characterization in terms of Peetre maximal functions of local means on the left-hand side, see [61] in connection with [60]. Then we apply Theorem 5.28 and get the equivalence. 2 It is also possible to obtain a “semi-discrete” characterization (in the sense of Definition 5.29) for the spaces on the right-hand side by using the abstract coorbit space theory from Section 3. 5.4.4. Wavelet bases Below we state a multivariate version of Theorem 5.12 on wavelet basis characterizations using tensor product wavelet frames. Let us start with a scaling function ψ 0 and a corresponding wavelet ψ 1 ∈ L2 (R) satisfying (D), (ML−1 ), and (SK ) for some K and L. In the sequel we use the tensor product wavelet system {ψj¯,k¯ }j¯,k¯ defined in Appendix A.2. r¯ F (Rd ) or We are interested in sufficient conditions on K, L such that every f ∈ Sp,q r ¯ d Sp,q B(R ), respectively, has the decomposition λj¯,k¯ ψj¯,k¯ (5.23) f= ¯ d j¯∈Nd0 k∈Z d d and the sequence λ = λ(f ) = {λj¯,k¯ }j¯∈Nd , k∈Z ¯ d defined by λj¯,k¯ = f, ψj¯,k¯ , j¯ ∈ N0 , k¯ ∈ Z , 0 belongs to the sequence spaces !q 1/q |j | /2 j¯r¯ d 1 {λ ¯ ¯ } ¯ ¯ |s r¯ f = , L 2 R 2 |λ |χ (x) p j ,k j ,k p,q j¯,k¯ j¯,k¯

{λ ¯ ¯ } ¯ ¯ |s r¯ b = j ,k j ,k p,q

¯ d k∈Z

j¯∈Nd0

j¯∈Nd0

d ' i=1

2

ji (ri +1/2−1/p)q

q/p 1/q |λj¯,k¯ |

p

¯ d k∈Z

r¯ +1/2−1/q r¯ +1/2−1/q corresponding to (Pp,q,a¯ ) , see (5.21), and (Lp,q,a¯ ) , see (5.22), where we used U¯ 1,2 . r¯ F (Rd ) or Furthermore, we aim at the converse that an element f ∈ (Hv1 )∼ belongs to Sp,q r ¯ d r ¯ r ¯ Sp,q B(R ) if the sequence λ(f ) belongs to sp,q f or sp,q b, respectively. The convergence r¯ F (Rd ) or S r¯ B(Rd ) if the finite sequences are in (5.23) is required to be in the norm of Sp,q p,q r ¯ r ¯ dense in sp,q f or sp,q b, respectively. In general we require weak∗ -convergence. The following theorem provides the corresponding wavelet basis characterization of spaces of mixed dominating smoothness.

Theorem 5.31. Let 1 p, q ∞ (p < ∞ in the F -case) and r¯ ∈ Rd . Let further ψ 0 , ψ 1 ∈ L2 (R) be a scaling function and associated wavelet where ψ 0 satisfies (D), (SK ), and ψ 1 satisfies (D), (SK ), (ML−1 ). (i) If, for i = 1, . . . , d, 1 1 1 1 1 1 1 1 + max + max , , min ri , − ,1 − , K, L > max max ri , − q 2 p q q 2 p p 1 1 1 1 − min ri , − , + 2 max (5.24) q 2 p q

3358


then the inhomogeneous Lizorkin–Triebel space with dominating mixed smoothness r¯ F (Rd ) can be discretized in the sense of (5.23) using the sequence space s r¯ f . Sp,q p,q (ii) If, for i = 1, . . . , d, 1 1 1 1 1 1 1 + max + , min ri , − ,1 − , K, L > max max ri , − q 2 p q 2 p p 2 1 1 − min ri , − + (5.25) q 2 p r¯ B(Rd ) can then the inhomogeneous Besov spaces with dominating mixed smoothness Sp,q r ¯ be discretized in the sense of (5.23) using the sequence space sp,q b.

Corollary 5.32. The wavelet basis characterization of the previous theorem holds for the choice r¯ B(Rd ) of an orthogonal spline wavelets system (ϕm , ψm ) of order m, see Appendix A.1. For Sp,q we need for i = 1, . . . , d m − 1 > rhs(5.25), r¯ F (Rd ) we need for i = 1, . . . , d whereas in case Sp,q

m − 1 > rhs(5.24). Proof. We apply Theorem 5.31 and the fact that ψ 1 = ψm satisfies (ML−1 ) for L = m and ψ 0 = ϕm , ψ 1 = ψm satisfy (D) and (SK ) for K < m − 1. 2 Remark 5.33. (i) Atomic decompositions of spaces with dominating mixed smoothness were already given by Vybíral [62]. He provides compactly supported atomic decompositions and in particular wavelet isomorphisms in terms of compactly supported Daubechies wavelet. Bazarkhanov [2] provided the ϕ-transform for dominating mixed spaces and obtained atomic decompositions in the sense of Frazier, Jawerth. (ii) Wavelet bases in terms of orthonormal spline wavelets with optimal conditions on the order m were given in [49] in case p = q. However, this restriction is due to the tensor product approach in [49], and is not needed in our result. Acknowledgments The authors acknowledge support by the Hausdorff Center for Mathematics, University of Bonn. In addition, they would like to thank Stephan Dahlke, Hans Feichtinger, Yoshihiro Sawano, Martin Schäfer, and Hans Triebel for valuable discussions, critical reading of preliminary versions of this manuscript and for several hints how to improve it. Appendix A. Wavelets For the notion of multi-resolution analysis, scaling function and associated wavelet we refer to Wojtaszczyk [63, 2.2] and Daubechies [16, Chapt. 5].


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A.1. Spline wavelets As a main example we use the spline wavelet system (ϕm , ψm ). Let us recall here the basic construction and refer to [63, Chapt. 3.3] for the properties listed below. The normalized cardinal B-spline of order m + 1 is given by Nm+1 (x) := Nm ∗ X (x),

x ∈ R, m ∈ N,

beginning with N1 = X , the characteristic function of the interval (0, 1). By !∨ m (ξ ) N 1 ϕm (x) := √ (x), ∞ m (ξ + 2πk)|2 )1/2 2π ( k=−∞ |N

x ∈ R,

we obtain an orthonormal scaling function which is again a spline of order m. Finally, by ψm (x) :=

∞

ϕm (t/2), ϕm (t − k) (−1)k ϕm (2x + k + 1)

k=−∞

the generator of an orthonormal wavelet system is defined. For m = 1 it is easily checked that −ψ1 (· − 1) is the Haar wavelet. In general, these functions ψm have the following properties: • • • •

Restricted to intervals [ k2 , k+1 2 ], k ∈ Z, ψm is a polynomial of degree at most m − 1; ψm ∈ C m−2 (R) if m 2; (m−2) The derivative ψm is uniformly Lipschitz continuous on R if m 2; The function ψm satisfies moment conditions of order up to m − 1, i.e., ∞ x ψm (x) dx = 0,

= 0, 1, . . . , m − 1.

−∞

In particular, ψm satisfies (ML−1 ) for L = m and ϕm , ψm satisfy (D) and (SK ) for K < m − 1. A.2. Tensor product wavelet bases on Rd There is a straightforward method to construct a wavelet basis on Rd from a wavelet basis on R. Putting ψj,k =

ψ 0 (· − k): 2j/2 ψ 1 (2j · −k):

j = 0, j 1,

j ∈ N0 , k ∈ Z

and ψj¯,k¯ = ψj1 ,k1 ⊗ · · · ⊗ ψjd ,kd , we obtain the following

j¯ = (j1 , . . . , jd ) ∈ Nd0 , k¯ = (k1 , . . . , kd ) ∈ Zd ,

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Lemma A.1. Let ψ 0 ∈ L2 (R) be an orthonormal scaling function with associated orthonormal wavelet ψ 1 ∈ L2 (R). Then the system

ψj¯,k¯ : j¯ ∈ Nd0 , k¯ ∈ Zd is an orthonormal basis in L2 (Rd ). The next construction is slightly more involved. The following lemma is taken from [57, 1.2.1]. Lemma A.2. Suppose, that we have a multi-resolution analysis in L2 (R) with scaling # function ψ 0 and associated wavelet ψ 1 . Let E = {0, 1}d , c = (c1 , . . . , cd ) ∈ E, and ψ c = dj =1 ψ cj . Then the system 0

jd

ψ (x − k): k ∈ Zd ∪ 2 2 ψ c 2j x − k : c ∈ E \ {0}, j ∈ N0 , k ∈ Zd is an orthonormal basis in L2 (Rd ). References [1] S.T. Ali, J.-P. Antoine, J.-P. Gazeau, Continuous frames in Hilbert space, Ann. Physics 222 (1) (1993) 1–37. [2] D.B. Bazarkhanov, Characterizations of the Nikol’skij–Besov and Lizorkin–Triebel function spaces of mixed smoothness, Proc. Steklov Inst. Math. 243 (2003) 53–65. [3] G. Bourdaud, Ondelettes et espaces de Besov, Rev. Mat. Iberoam. 11 (1995) 477–512. [4] M. Bownik, Anisotropic Triebel–Lizorkin spaces with doubling measures, J. Geom. Anal. 17 (3) (2007) 387–424. [5] H.-Q. Bui, Weighted Besov and Triebel spaces: interpolation by the real method, Hiroshima Math. J. 12 (3) (1982) 581–605. [6] H.-Q. Bui, Characterizations of weighted Besov and Triebel–Lizorkin spaces via temperatures, J. Funct. Anal. 55 (1) (1984) 39–62. [7] H.-Q. Bui, M. Paluszyński, M.H. Taibleson, A maximal function characterization of weighted Besov–Lipschitz and Triebel–Lizorkin spaces, Studia Math. 119 (3) (1996) 219–246. [8] H.-Q. Bui, M. Paluszyński, M.H. Taibleson, Characterization of the Besov–Lipschitz and Triebel–Lizorkin spaces, the case q < 1, J. Fourier Anal. Appl. (Special Issue) 3 (1997) 837–846. [9] F. Chiarenza, M. Frasca, Morrey spaces and Hardy–Littlewood maximal functions, Rend. Mat. 7 (1987) 273–279. [10] A. Cohen, I. Daubechies, J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (5) (1992) 485–560. [11] R.R. Coifman, Y. Meyer, E.M. Stein, Some new function spaces and their application to harmonic analysis, J. Funct. Anal. 62 (1985) 304–335. [12] S. Dahlke, M. Fornasier, H. Rauhut, G. Steidl, G. Teschke, Generalized coorbit theory, Banach frames, and the relation to alpha-modulation spaces, Proc. Lond. Math. Soc. (3) 96 (2008) 464–506. [13] S. Dahlke, G. Kutyniok, G. Steidl, G. Teschke, Shearlet coorbit spaces and associated Banach frames, Appl. Comput. Harmon. Anal. 27 (2) (2009) 195–214. [14] S. Dahlke, G. Steidl, G. Teschke, Coorbit spaces and Banach frames on homogeneous spaces with applications to the sphere, Adv. Comput. Math. 21 (1–2) (2004) 147–180. [15] S. Dahlke, G. Steidl, G. Teschke, Weighted coorbit spaces and Banach frames on homogeneous spaces, J. Fourier Anal. Appl. 10 (5) (2004) 507–539. [16] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. [17] C. Fefferman, B. Muckenhoupt, Two nonequivalent conditions for weight functions, Proc. Amer. Math. Soc. 45 (1974) 99–104. [18] H.G. Feichtinger, Modulation spaces on locally compact Abelian groups, technical report, January 1983. [19] H.G. Feichtinger, P. Gröbner, Banach spaces of distributions defined by decomposition methods. I, Math. Nachr. 123 (1985) 97–120.


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Compactness of the ∂-Neumann operator on singular complex spaces J. Ruppenthal Department of Mathematics, University of Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany Received 1 October 2010; accepted 21 December 2010 Available online 5 January 2011 Communicated by J. Coron

Abstract Let X be a Hermitian complex space of pure dimension n. We show that the ∂-Neumann operator on (p, q)-forms is compact at isolated singularities of X if p + q = n − 1, n and q 1. The main step is the construction of compact solution operators for the ∂-equation on such spaces which is based on a general characterization of compactness in function spaces on singular spaces, and that leads also to a criterion for compactness of more general Green operators on singular spaces. © 2011 Elsevier Inc. All rights reserved. Keywords: Green/Neumann operator; L2 -theory; Singular complex spaces

1. Introduction The Cauchy–Riemann operator ∂ and the related ∂-Neumann operator play a central role in complex analysis. Especially the L2 -theory for these operators is of particular importance and has become indispensable for the subject after the fundamental work of Hörmander on L2 -estimates and existence theorems for the ∂-operator (see [19] and [20]) and the related work of Andreotti and Vesentini (see [2]). By no means less important is Kohn’s solution of the ∂-Neumann problem (see [22–24]), which implies existence and regularity results for the ∂-complex, as well (see Chapter III.1 in [7]). Important applications of the L2 -theory are for instance the Ohsawa– Takegoshi extension theorem [28], Siu’s analyticity of the level sets of Lelong numbers [40] or the invariance of plurigenera [41]. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.022

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J. Ruppenthal / Journal of Functional Analysis 260 (2011) 3363–3403

Whereas the theory of the ∂-operator and the ∂-Neumann operator is very well developed on complex manifolds, not much is known about the situation on singular complex spaces which appear naturally as the zero sets of holomorphic functions. The further development of this theory is an important task since analytic methods have led to fundamental advances in geometry on complex manifolds (see Siu’s results mentioned above), but these analytic tools are still missing on singular spaces. After the first period of intensive research on the L2 -theory for the ∂-operator on singular spaces (see [27,31,14,26,32,11]), there has been good progress in this subject recently due to Pardon and Stern (see [33]), Diederich, Fornæss, Øvrelid and Vassiliadou (see [8,6,10,29,30]), Ruppenthal and Zeron (see [36–39]). On the other hand, the ∂-Neumann operator has not been studied on singular complex spaces yet. The purpose of the present paper is to initiate this branch of research in complex analysis on singular complex spaces. Let X be a Hermitian complex space1 of pure dimension n with isolated singularities only. Our intention is to study the behavior of the ∂-Neumann operator in the presence of these singularities. Let Ω ⊂⊂ X be a relatively compact domain, and assume that either X is compact and Ω = X, or that X is Stein and Ω has smooth strongly pseudoconvex boundary that does not contain singularities. Let Ω ∗ = Ω − Sing X and ∂ w the ∂-operator in the sense of distributions.2 Then there are only finitely many obstructions to solvability of the ∂ w -equation in the L2 -sense on Ω ∗ for forms of degree (p, q) with p +q = n, q 1.3 This can be deduced from L2 -regularity results for the ∂ w -equation at isolated singularities due to Fornæss, Øvrelid and Vassiliadou [10] by the use of Hironaka’s resolution of singularities (Theorem 4.1). Hence the ∂-operator in the sense of distributions ∂ w : L2p,q−1 Ω ∗ → L2p,q Ω ∗ has closed range R(∂ w ) in L2p,q (Ω ∗ ) for p + q = n. So, the densely defined closed self-adjoint ∂ w -Laplacian 2 = ∂ w ∂ ∗w + ∂ ∗w ∂ w has closed range in L2p,q (Ω ∗ ) for p + q = n − 1, n, and we obtain the orthogonal decomposition L2p,q Ω ∗ = ker 2p,q ⊕ R(2p,q ). Then the ∂ w -Neumann operator ∗ ∗ 2 2 Np,q = 2−1 p,q : Lp,q Ω → Dom(2p,q ) ⊂ Lp,q Ω 1 A reduced complex space with a Hermitian metric on the regular part which is induced by local embeddings into

complex number space, hence extends continuously into the singular set. 2 The ∂-operator in the sense of distributions is the maximal closed L2 -extension of the ∂-operator. The notation ∂ w refers to this as a weak extension. We will also use the notation ∂ s for the minimal (strong) closed L2 -extension of the ∂ cpt -operator on smooth forms with compact support (see Section 4.3). 3 If Ω = X is compact, we also have to assume that q = 1 in case p + q = n + 1. We keep this assumption throughout the text without mentioning it explicitly. However, this restriction can be omitted by the use of some additional arguments (see Acknowledgments).


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is well defined by the following assignment: let Np,q u = 0 if u ∈ ker 2p,q , and Np,q u be the uniquely defined preimage of u orthogonal to ker 2p,q if u ∈ R(2p,q ). The main result of the present paper is compactness of this operator Np,q : Theorem 1.1. Let X be a Hermitian complex space of pure dimension n with only isolated singularities, and Ω ⊂⊂ X a relatively compact domain such that either Ω = X is compact, or X is Stein and Ω has smooth strongly pseudoconvex boundary that does not contain singularities. Let p + q = n − 1, n and q 1. If Ω = X is compact and p + q = n + 1, let q = 1. Then the ∂-operator in the sense of distributions ∂ w has closed range in L2p,q (Ω ∗ ) and L2p,q+1 (Ω ∗ ) so that the ∂ w -Neumann operator −1 ∗ ∗ ∗ ∗ 2 2 Np,q = 2−1 p,q = ∂ w ∂ w + ∂ w ∂ w p,q : Lp,q Ω → Dom 2p,q ⊂ Lp,q Ω is well defined as above. Np,q is compact. s We remark that this also implies compactness of the ∂ s -Neumann operator Nn−p,n−q in degree (n − p, n − q) under the same assumptions (see Section 4.3). Note that compactness of the ∂ w -Neumann operator Np,q yields the important consequence that L2p,q (Ω ∗ ) has an orthonormal basis consisting of eigenforms of the ∂ w -Laplacian 2p,q , the eigenvalues are non-negative, have no finite limit point and appear with finite multiplicity. It might be an interesting question to study whether there is a nice connection between the eigenvalues and the structure of the singularities of X. Compactness of the ∂-Neumann operator is a classical topic of complex analysis on manifolds. A classical approach (e.g. on compact manifolds or on strongly pseudoconvex domains in complex manifolds) is to deduce compactness by the Rellich embedding theorem from subelliptic estimates for the complex Laplacian (see [42] for a recent comprehensive discussion of the ∂-Neumann problem). We choose a different approach to prove Theorem 1.1. It follows from the work of Fornæss, Øvrelid and Vassiliadou [10] that there are solution operators for the ∂ w -equation that have some gain of regularity at the isolated singularities (see Theorem 4.1). By the use of a Riesz characterization theorem for precompactness on arbitrary Hermitian manifolds (Theorem 2.5), we deduce that these operators are actually compact solution operators:

Theorem 1.2. Let X be a Hermitian complex space of pure dimension n with only isolated singularities, and Ω ⊂⊂ X a relatively compact domain such that either Ω = X is compact, or X is Stein and Ω has smooth strongly pseudoconvex boundary that does not contain singularities. Let p + q = n and q 1. If Ω = X is compact and p + q = n + 1, let q = 1. Then the range R(∂ w )p,q of the ∂-operator in the sense of distributions ∂ w : L2p,q−1 Ω ∗ → (ker ∂ w )p,q ⊂ L2p,q Ω ∗ has finite codimension in (ker ∂ w )p,q , and there exists a compact operator S : R(∂ w )p,q → L2p,q−1 Ω ∗ such that ∂ w Sf = f .

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That requires also Kohn’s subelliptic estimates and Hironaka’s resolution of singularities which we use to distinguish between the treatment of the isolated singularities on the one hand and the strongly pseudoconvex boundary of the domain Ω on the other hand. Compactness of the ∂ w -Neumann operator (i.e. Theorem 1.1) follows then by an argument of Hefer and Lieb since Np,q can be expressed in terms of the compact solution operators (see [17]). As a byproduct, we also obtain the following characterization of compactness of the ∂Neumann operator on singular spaces with arbitrary singularities (in the spirit of some recent work of Gansberger [13] and Haslinger [15] about compactness of the ∂-Neumann operator on domains in Cn ): Theorem 1.3. Let Z be a Hermitian complex space of pure dimension n, X ⊂ Z an open Hermitian submanifold and ∂ a closed L2 -extension of the ∂ cpt -operator on smooth forms with compact support in X, for example ∂ = ∂ w the ∂-operator in the sense of distributions. Let 0 p, q n. Assume that ∂ has closed range in L2p,q (X) and in L2p,q+1 (X). Then 2p,q = ∂∂ ∗ + ∂ ∗ ∂ p,q has closed range and the following conditions are equivalent: (i) The ∂-Neumann operator Np,q = 2−1 p,q is compact. (ii) For all > 0, there exists Ω ⊂⊂ X such that uL2p,q (X−Ω) < for all 2 u ∈ u ∈ Dom(∂) ∩ Dom ∂ ∗ ∩ R(2p,q ): ∂u2L2 + ∂ ∗ uL2 < 1 . (iii) There exists a smooth function ψ ∈ C ∞ (X, R), ψ > 0, such that ψ(z) → ∞ as z → bX, and (2p,q u, u)L2 ψ|u|2 dVX for all u ∈ Dom(2p,q ) ∩ R(2p,q ). X

The present paper is organized as follows. In Section 2, we give a criterion for L2 -precompactness of bounded sets of differential forms on arbitrary Hermitian manifolds in the spirit of the classical Riesz characterization (Theorem 2.5). This criterion is used to study compactness of general Green operators on singular spaces with arbitrary singularities (Theorem 3.6) in Section 3. Theorem 1.3 is an easy corollary from Theorem 3.6 in the special case of the unweighted ∂ w -Neumann operator. In Section 4.1, we use the results of Fornæss, Øvrelid and Vassiliadou to construct compact solution operators for the ∂ w -equation which are then used to show compactness of the ∂ w -Neumann operator by the method of Hefer and Lieb in Section 4.2. Note that the proof of Theorem 1.2 is contained in the proof of Theorem 4.4. Finally, we study the ∂ s -Neumann operator in Section 4.3. 2. Precompactness on Hermitian manifolds Let X be a Hermitian manifold. If f is a differential form on X, we denote by |f | its pointwise norm. For a weight function ϕ ∈ C 0 (X), we denote by L2p,q (X, ϕ) the Hilbert space of (p, q)-


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forms such that f 2L2

p,q

|f |2 e−ϕ dVX < ∞.

:= (X,ϕ) X

Note that we may take different weight functions for forms of different degree. We assume that X is connected. For two points p, q ∈ X, let distX (p, q) be the infimum of the length of curves connecting p and q in X. Let Φ : X → X be a diffeomorphism. Then we call md(Φ) := sup distX p, Φ(p) p∈X

the mapping distance of Φ. If Y is another Hermitian manifold and Φ : X → Y differentiable, the pointwise norm of the tangential map Φ∗ is defined by |Φ∗ |(p) := sup Φ∗ (v)T

Φ(p) Y

v∈Tp X |v|=1

.

This leads to the sup-norm of Φ∗ : Φ∗ ∞ := sup |Φ∗ |(p). p∈X

We also need to measure how far Φ∗ : T X → T X is from the identity mapping on tangential vectors (if Φ : X → X). As the total space T X inherits the structure of a Hermitian manifold, distT X is also well defined, and we set Φ∗ − id∞ = sup sup distT X (Φ∗ v, v). p∈X v∈Tp X |v|=1

Definition 2.1. Let Ω ⊂ X open. We call a diffeomorphism Φ : (Ω, X) → (Ω, X) a δ-variation of Ω in X if Φ|X−Ω is the identity map, mapping distance md(Φ) < δ and Φ∗ − id∞ , (Φ −1 )∗ − id∞ < 3δ. The set of all δ-variations of Ω in X will be denoted by Varδ (Ω, X). A δ-variation Φ ∈ Varδ (Ω, X) will be called δ-deformation, if it can be connected by a smooth path to the identity map in Varδ (Ω, X), i.e. if there exists a smooth map Φ· (·) : [0, 1] × X → X,

(t, x) → Φt (x) ∈ X,

such that Φt (·) ∈ Varδ (Ω, X) for all t ∈ [0, 1], Φ0 = id, Φ1 = Φ and ∂ Φt (x) 3δ ∂t

for all t ∈ [0, 1], x ∈ X.

The set of all δ-deformations of Ω in X will be denoted by Defδ (Ω, X).

(1)

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Note that Φ∗ − id∞ , (Φ −1 )∗ − id∞ < 3δ implies that Φ∗ ∞ , Φ −1 ∗ ∞ < 1 + 3δ.

(2)

A remark on condition (1) is in order: if Φ is a δ-deformation and x ∈ X, then Φ· (x) : [0, 1] → X is a path connecting x and Φ(x). Since distX (x, Φ(x)) < δ, condition (1) means that the path Φ· (x) is not too far away from a geodesic (of uniform velocity) connecting the two points if distX (x, Φ(x)) comes close to δ. Another useful observation is the following: Lemma 2.2. Let M1 and M2 be Hermitian manifolds, U1 ⊂ M1 and U2 ⊂ M2 open sets, and Γ : U1 → U2 a diffeomorphism. Let Ω1 ⊂⊂ U1 be an open subset of U1 and Ω2 = Γ (Ω1 ) ⊂⊂ U2 . For Φ ∈ Defδ (Ω2 , M2 ), we define the pullback as Γ # Φ = Γ −1 ◦ Φ ◦ Γ : (Ω1 , M1 ) → (Ω1 , M1 ). Let CΓ := max{1, Γ∗ 2∞,Ω , Γ∗−1 2∞,Ω }. Then 1

2

Γ # Φ ∈ DefCΓ δ (Ω1 , M1 ) for all δ > 0 and all Φ ∈ Defδ (Ω2 , M2 ). Proof. First off all, Γ −1 ◦ Φ ◦ Γ is only defined on U1 , but as it is the identity mapping on U1 − Ω1 , Γ # Φ is well defined as a map M1 → M1 if we extend it as the identity mapping to M 1 − U1 . Let Φ ∈ Varδ (Ω2 , M2 ). So md(Γ # Φ) < CΓ δ and # Γ Φ − id = Γ −1 ◦ (Φ∗ − id) ◦ Γ∗ CΓ 3δ. ∗ ∗ ∞ ∞ Moreover, if Φ ∈ Defδ (Ω2 , M2 ), then clearly (Γ # Φ)t = Γ # Φt connects Γ # Φ to the identity mapping and | ∂t∂ (Γ # Φ)t | CΓ 3δ. 2 Note that with the same constant CΓ > 0 also −1 # Γ : Defδ (Ω1 , M1 ) → DefCΓ δ (Ω2 , M2 ). Using δ-deformations on X, we can characterize precompact sets in the spaces of squareintegrable differential forms in the spirit of the classical Riesz characterization (see e.g. [1, 2.15]). Before, we need some preliminary considerations: ∞ Lemma 2.3. The space of smooth forms with compact support C(p,q),cpt (X) is dense in 2 Lp,q (X, ϕ).

Proof. This follows by the usual mollifier method with a suitable partition of unity. Let f ∈ L2p,q (X, ϕ) and > 0. Then there exists a compact subset K ⊂⊂ X such that fˆ − f L2p,q (X,ϕ) < if we denote by fˆ the trivial extension of f |K to X (see e.g. [1, A.1.16.2]).


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Now then, cover K by finitely many coordinate charts U1 , . . . , UN , and let {ψj } be a partition of unity for {Uj }. Then each ψj fˆ can be approximated by convolution with a Dirac sequence ∞ (X) such that ψj fˆ − hj L2p,q (X,ϕ) < /N . (mollifier method): there exist hj ∈ C(p,q),cpt N ∞ Letting h := j =1 hj ∈ C(p,q),cpt (X), it follows that f − hL2p,q (X,ϕ) < 2. 2 We observe that small deformations cannot disturb the L2 -norm arbitrarily: Lemma 2.4. Let f ∈ L2p,q (X, ϕ). Then: for all > 0 and all Ω ⊂⊂ X, there exists δf > 0 such that ∗ Φ f − f 2 L

p,q (X,ϕ)

0 and Ω ⊂⊂ X. For Φ ∈ Defδ (Ω, X), let Ψ = Φ −1 which is again in Defδ (Ω, X) by Definition 2.1, and consider ∗ Φ (f − fj )2 2 L

p,q

= (X,ϕ)

∗ Φ (f − fj )2 e−ϕ dVX

X

(1 + 3δ)

2(p+q)

Φ ∗ |f − fj |2 e−ϕ dVX ,

X

since f − fj is a (p, q)-form and Φ∗ ∞ < 1 + 3δ by (2). With Ψ∗ ∞ < 1 + 3δ and n = dimC X, it follows that ∗ Φ (f − fj )2 2 L

p,q

(1 + 3δ)2p+2q (X,ϕ) X

|f − fj |2 Ψ ∗ e−ϕ dVX

(1 + 3δ)2(p+q+n)

|f − fj |2 Ψ ∗ e−ϕ dVX

X

(1 + 3δ)

2(p+q+n)

sup eϕ(z)−ϕ(Ψ (z)) z∈Ω

|f − fj |2 e−ϕ dVX

X

= (1 + 3δ)2(p+q+n) sup eϕ(z)−ϕ(Ψ (z)) · f − fj 2L2 z∈Ω

p,q (X,ϕ)

.

Since ϕ ∈ C 0 (X), it is uniformly continuous on Ω. So, distX (z, Ψ (z)) < δ implies that there exists a δ0 > 0 such that

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∗ Φ (f − fj ) 2 L

p,q (X,ϕ)

2f − fj L2p,q (X,ϕ)

(3)

∞ (X), mapping distance md(Φ) < δ and Φ∗ − id∞ < 3δ, we also if δ < δ0 . Since fj ∈ C(p,q),cpt get (for fixed j ) that

2 sup e−ϕ(z) fj − Φ ∗ fj (z) → 0 as δ → 0.

(4)

z∈Ω

So then, choose fj such that f − fj < /4. It follows by (3) and (4) that ∗ Φ f − f f − fj + Φ ∗ f − Φ ∗ fj + fj − Φ ∗ fj 3f − fj + Vol(Ω)1/2 sup e−ϕ(z)/2 fj − Φ ∗ fj (z) z∈Ω

3/4 + /4 = if δ < δ0 is small enough, say δ < δf .

2

Now, precompact sets in L2p,q (X, ϕ) can be characterized by: Theorem 2.5. Let X be a Hermitian manifold and A a bounded subset of L2p,q (X, ϕ). Then A is precompact if and only if the following two conditions are fulfilled: (i) for all > 0 and all Ω ⊂⊂ X, there exists δ > 0 such that ∗ Φ f − f 2 L

p,q (X,ϕ)

0, there exists Ω ⊂⊂ X such that f L2p,q (X−Ω ,ϕ) < for all f ∈ A. Proof. First, assume that A is precompact. Let > 0. By definition, there exists an integer N and forms f1 , . . . , fN such that A⊂

N

B/4 (fj )

in L2p,q (X, ϕ).

j =1

To show (ii), choose Ω ⊂⊂ X so big that fj L2p,q (X−Ω ,ϕ) < /2 for j = 1, . . . , N ,


3371

which is possible because we have to consider only finitely many forms simultaneously (see e.g. [1, A.1.16.2]). For an arbitrary f ∈ A, there exists an index j0 such that f L2p,q (X−Ω ,ϕ) f − fj0 L2p,q (X−Ω ,ϕ) + fj0 L2p,q (X−Ω ,ϕ) < /4 + /2. So, property (ii) is valid. To show (i), we proceed analogously. Let Ω ⊂⊂ X. For each of the (finitely many) fj , there exists by Lemma 2.4 a δj > 0 depending on fj and such that ∗ Φ fj − fj 2 L

< /4

p,q (X,ϕ)

for all Φ ∈ Defδj (Ω, X). Set δ := min1j N {δj }. Now then, there exists for any f ∈ A an index j0 such that ∗ Φ f − f 2 L

p,q (X,ϕ)

Φ ∗ fj0 − fj0 L2 (X,ϕ) + fj0 − f L2p,q (X,ϕ) p,q ∗ + Φ (f − fj0 )L2 (X,ϕ) p,q ∗ < /4 + /4 + Φ (f − fj0 )L2 (X,ϕ) p,q

for all Φ ∈ Defδ (Ω, X). But ∗ Φ (f − fj ) 2 0 L

p,q (X,ϕ)

2f − fj0 L2p,q (X,ϕ) < 2/4

for all Φ ∈ Defδ (Ω, X) (independent of f , fj0 ) if we choose δ < δ small enough as in the proof of Lemma 2.4 (see estimate (3)). That proves property (i). Conversely, assume that conditions (i) and (ii) are satisfied. Our first objective is to show that under these circumstances the approximation by smooth compactly supported forms as in Lemma 2.3 can be made uniformly for all f ∈ A, i.e. we will construct a sequence of operators ∞ Tk : A → C(p,q),cpt (X),

k ∈ N,

such that Tk f − f L2p,q (X,ϕ) < 1/k

for all f ∈ A.

(5)

We start by using property (ii) to choose an exhaustion Ω1 ⊂⊂ Ω2 ⊂⊂ Ω3 ⊂⊂ · · · ⊂⊂ X of X by open relatively compact subsets Ωk such that f L2p,q (X−Ωk ,ϕ) < 1/ 3(k + 1)

for all f ∈ A.

(6)

3372


We will now define Tk . Let χ := χΩk be the characteristic function of Ωk . Cover Ωk by finitely many open sets U1 , . . . , UN ⊂⊂ Ωk+1 which are contained in coordinate charts, and ∞ (U ), 0 ψ 1, such that choose cut-off functions ψj ∈ Ccpt j j ψ :=

N

∞ ψj ∈ Ccpt (Ωk+1 )

(7)

j =1

satisfies 0ψ 1

and ψ|Ωk ≡ 1,

(8)

i.e. {ψj }j is a partition of unity on Ωk (subordinate to {Uj }). For f ∈ A, let fj := ψj f. Note that the fj have compact support in Ωk+1 and that χf = χ

N

fj

j =1

because ψj = ψ ≡ 1 on Ωk and χ is the characteristic function of Ωk . ∞ (X), we also observe that the forms f = ψ f still satisfy condition (i) as f Since ψj ∈ Ccpt j j runs through A: for all > 0 and all Ω ⊂⊂ X, there exists δj > 0 such that ∗ Φ f j − f j 2 L

p,q (X,ϕ)

= Φ ∗ (ψj f ) − ψj f L2

p,q (X,ϕ)

0.

Now then, define j Tk f (z) := (χfj ) ∗ hj (z) =

hj (z − ζ )(χψj f )(ζ ) dVCn Uj

where j > 0 has to be chosen later on (small enough), but at least j < distCn (supp ψj , bUj ),


3373 j

which also implies that j < distCn (supp fj , bUj ) distCn (supp χfj , bUj ). Hence Tk f ∈ ∞ (Uj ), and we can define C(p,q),cpt Tk f :=

N

j

Tk f =

j =1

N ∞ (χψj f ) ∗ hj ∈ C(p,q),cpt (X). j =1

Recall that χ = χΩk , the covering U1 , . . . , UN , the choice of local coordinates and the j clearly depend on k. We will now show that Tk f − f L2p,q (X,ϕ) < 1/k

for all f ∈ A

if we choose the j small enough. By (7), (8) and (6), we get Tk f − f L2p,q (X,ϕ) Tk f − ψf L2p,q (X,ϕ) + ψf − f L2p,q (X,ϕ) = Tk f − ψf L2p,q (X,ϕ) + (1 − ψ)f L2 (X−Ω p,q

k−1 ,ϕ)

Tk f − ψf L2p,q (X,ϕ) + f L2p,q (X−Ωk−1 ,ϕ) Tk f − ψf L2p,q (X,ϕ) + 1/(3k)

N j T f − ψj f 2 k L

p,q (X,ϕ)

j =1

+ 1/(3k).

j It remains to show that Tk f − ψj f L2p,q (X,ϕ) 2/(3k) for all f ∈ A if we choose the j > 0 small enough. So, consider

j

Tk f (z) − (ψj f )(z) =

hj (z − ζ )(χψj f )(ζ ) dVCn (ζ ) Uj

− (ψj f )(z) =

hj (z − ζ ) χ(ζ ) + (1 − χ)(ζ ) dVCn (ζ )

Uj

hj (z − ζ )χ(ζ ) (ψj f )(ζ ) − (ψj f )(z) dVCn (ζ )

Uj

− (ψj f )(z)

hj (z − ζ )(1 − χ)(ζ ) dVCn (ζ )

Uj j

j

=: 1 (z) − 2 (z). j

Since (1 − χ)(ζ ) = 0 for ζ ∈ Ωk , we can choose j (at least) so small that the integral in 2 (z) vanishes if z ∈ Ωk−1 . It follows that

3374


j (z) (ψj f )(z) 2

j

and 2 (z) = 0 if z ∈ Ωk−1 .

This yields by the use of (6) that N j T f − ψj f 2 k L

p,q

j =1

(X,ϕ)

N j

N j

j =1

j =1

0 j

small enough. Note that we already arranged j so small that 1 has support in Uj . By standard estimates for convolution integrals (see [1, 2.12.1], as it is applied in the proof of [1, 2.15]) and |χ| 1, j

1 L2p,q (Uj ,ϕ)

hj L1 (Cn ) sup fj (· − v) − fj L2

p,q (Uj ,ϕ)

v∈Cn |v|j

.

But translations by v in the coordinate chart Uj with |v| j can be extended to some δj deformation of Ωk+1 in X if j is small enough, because the connecting curves Φt (z) = z − tv, 0 t 1, behave more and more like geodesics as v → 0 and Ωk+1 is compact. That shows that (1) is fulfilled if v is small enough, the other conditions from Definition 2.1 are easy to check. With Lemma 2.2, we can assume that δj → 0 as j → 0. Hence j 2 1 L

p,q (Uj ,ϕ)

sup

∗ Φ fj − fj 2 L

Φ∈Defδj (Ωk+1 ,X)

p,q (X,ϕ)

< 1/(3kN )

for all f ∈ A by (9) if we choose first δj and then j small enough. This completes the proof of (5), i.e. the operators Tk give a uniform approximation of all f ∈ A by smooth forms with compact support in Ωk+1 . Since A is a bounded subset of L2p,q (X, ϕ), there exists a constant Ck > 0 such that j T f (z) h 2 n fj 2 Lp,q (Cn ) Ck , j L (C ) k and j P T f (z) ∇h 2 n fj 2 Lp,q (Cn ) Ck j L (C ) k for any first order differential operator with constant coefficients P . It follows that


3375

Ak := {Tk f : f ∈ A} 1 (Ω is a bounded subset of Cp,q k+1 ). 0 (Ω Let γ > 0. By the Arzela–Ascoli Theorem, there exist finitely many forms gl ∈ Cp,q k+1 ), l = 1, . . . , N(k, γ ), such that

Ak ⊂

N

(k,γ )

Bγ (gl )

l=1 0 (Ω with respect to the C 0 -norm in Cp,q k+1 ). But there exists a constant Dk > 0 such that

hL2

p,q (Ωk+1 ,ϕ)

Dk hC 0

p,q (Ωk+1 )

for all (p, q)-forms h. Taking into account that the forms Tk f have compact support in Ωk+1 , it follows that Ak ⊂

N

(k,γ )

Bγ Dk (gl )

l=1

in L2p,q (X, ϕ) if we extend the forms gl trivially to X. But Tk f − f L2p,q (X,ϕ) < 1/k for all f ∈ A. Hence A⊂

N

(k,γ )

Bγ Dk +1/k (gl )

l=1

in L2p,q (X, ϕ). This means that A is precompact because γ Dk + 1/k can be made arbitrarily small (by choosing first k big and then γ small enough). 2 We remark that the criterion carries over to Lp -forms, 0 p < ∞, without further difficulties. 3. Compactness of Green operators on Hermitian spaces Let X be a Hermitian manifold, ϕ ∈ C 0 (X) a weight function, and T : Dom T ⊂ L2∗ (X, ϕ) → L2∗ (X, ϕ) a densely defined closed linear partial differential operator such that T 2 = 0. We will always ∞ (X) are contained in the domain of assume that the smooth compactly supported forms C∗,cpt ∗ such an operator. The adjoint operator T is also closed and densely defined, T ∗∗ = T , (T ∗ )2 = 0 and (ker T )⊥ = R T ∗ ,

⊥ ker T ∗ = R(T ),

3376


where we denote by R(T ), R(T ∗ ) the range of T and T ∗ , respectively. Then we define P : L2∗ (X, ϕ) → L2∗ (X, ϕ) by Dom P = u ∈ Dom T ∩ Dom T ∗ : T u ∈ Dom T ∗ , T ∗ u ∈ Dom T , P = T ∗T + T T ∗. Then: Theorem 3.1. P is a densely defined closed self-adjoint operator, (P u, u) 0. Proof. We adopt the proof of Proposition V.5.7 in [25], where the statement is proved in case T = ∂ w , the ∂-operator in the sense of distributions. The proof in [25] is more or less taken from [7, Proposition 1.3.8], and is essentially due to Gaffney [12]. It is easy to see that P is a densely defined closed operator with (P u, u) 0 for all u ∈ Dom P , we will show that P is self-adjoint by checking that Dom P = Dom P ∗ . We need the following lemma of J. von Neumann as it is presented in [25, Lemma V.5.10]: Lemma 3.2. Let A : V → H be a closed densely defined operator on a Hilbert space H . Set R = id + A∗ A, S = id + AA∗ , Dom(R) = x ∈ Dom A: Ax ∈ Dom A∗ , Dom(S) = x ∈ Dom A∗ : A∗ x ∈ Dom A . Then R : Dom(R) → H , S : Dom(S) → H are linear bijective maps and R −1 , S −1 : H → H are continuous self-adjoint operators. Set F = id + P . So, F is a densely defined closed operator with 2 F uu (F u, u) = u2 + T u2 + T ∗ u for all u ∈ Dom F = Dom P . Hence ker F = 0 and R(F ) is closed. By Lemma 3.2, −1 id + T T ∗

and

−1 id + T ∗ T

are bounded self-adjoint operators. So, −1 −1 S := id + T T ∗ + id + T ∗ T − id is also bounded and self-adjoint.


3377

We will now show that F is surjective and S = F −1 . This implies that F and P are selfadjoint. Consider −1 −1 −1

id + T T ∗ − id = id − id + T T ∗ id + T T ∗ = −T T ∗ id + T T ∗ , −1

−1 −1 − id = id − id + T ∗ T id + T ∗ T = −T ∗ T id + T ∗ T . id + T ∗ T

(10) (11)

Therefore −1 ⊂ Dom T T ∗ , R id + T T ∗ −1 ⊂ Dom T ∗ T , R id + T ∗ T and (10) yields −1 −1 S = id + T ∗ T − T T ∗ id + T T ∗ . This implies with (13) and T 2 = 0 that R(S) ⊂ Dom T ∗ T and −1 T ∗ T S = T ∗ T id + T ∗ T . By symmetry, (11), (12) and (T ∗ )2 = 0 give R(S) ⊂ Dom T T ∗ and −1 T T ∗ S = T T ∗ id + T T ∗ . Summing up, R(S) ⊂ Dom F and F S = S + T ∗T S + T T ∗S −1 −1 −1 −1 = id + T T ∗ + id + T ∗ T − id + T ∗ T id + T ∗ T + T T ∗ id + T T ∗ −1 −1 = id + T ∗ T id + T ∗ T + id + T T ∗ id + T T ∗ − id = id on H . So, R(F ) = H and S = F −1 .

2

Corollary 3.3. P induces the orthogonal decomposition L2∗ (X, ϕ) = ker P ⊕ R(P ) = ker T ∩ ker T ∗ ⊕ R(T ) ⊕ R T ∗ .

(12) (13)

3378


Proof. The first equality follows from the self-adjointness of P , ker P = ker T ∩ ker T ∗ follows from (P u, u) = T u2 + T ∗ u2 . Clearly, R(P ) ⊂ R(T ) + R(T ∗ ) and R(T ) ⊥ R(T ∗ ) yield R(P ) ⊂ R(T ) ⊕ R(T ∗ ). On the other hand, assume that f ⊥ R(T ) ⊕ R(T ∗ ). Then (f, T g) = 0, (f, T ∗ h) = 0 for all g ∈ Dom T , h ∈ Dom T ∗ . Thus f ∈ Dom T ∗ ∩ Dom T and (T ∗ f, g) = (Tf, h) = 0 for all g, h ∈ L2∗ (X, ϕ) since T and T ∗ are densely defined. Hence, f ∈ (ker T ∩ ker T ∗ ). 2 Lemma 3.4. Let V1 , V2 , V3 ⊂ L2∗ (X, ϕ) be closed subspaces such that R(T |V1 ) ⊂ V2 ,

R(T |V2 ) ⊂ V3 ,

(14)

these ranges are closed, and T |∗V1 = T ∗ |V2 , T |∗V2 = T ∗ |V3 . It follows that the densely defined closed restricted operator Q = P |V2 : V2 → V2 is self-adjoint and has closed range. Hence, V2 = ker Q ⊕ R(Q) = ker T |V2 ∩ ker T ∗ |V2 ⊕ R(T |V1 ) ⊕ R T ∗ |V3 ,

(15) (16)

and there exists a constant c > 0 such that 2 cu2 T u2 + T ∗ u ,

for u ∈ Dom(T ) ∩ Dom T ∗ ∩ R(Q),

cu Qu for u ∈ Dom(Q) ∩ R(Q).

(17)

Proof. By assumption, R T ∗ |V2 = R T |∗V1 ⊂ V1 ,

R T ∗ |V3 = R T |∗V2 ⊂ V2 ,

(18)

and all these ranges are closed (see e.g. [21, Proposition A.1.2]). (14) and (18) together imply that R(P |V2 ) ⊂ V2 . It is clear that Q = P |V2 : V2 → V2 is closed and densely defined. Let V2⊥ be the orthogonal complement of V2 in L2∗ (X, ϕ) and u ∈ V2⊥ ∩ Dom P . Then (P u, v) = (u, P v) = 0 for all v ∈ V2 ∩ Dom P . As Dom P is dense in V2 , this yields R(P |V ⊥ ) ⊂ V2⊥ . 2

(19)


3379

It follows with (19) that P = P |V2 ⊕ P |V ⊥ : V2 ⊕ V2⊥ → V2 ⊕ V2⊥ . 2

Since P is self-adjoint, it follows that both operators Q = P |V2 and P |V ⊥ are self-adjoint. 2 Since R(T |V1 ) is a closed subspace of V2 , we get the orthogonal decomposition V2 = ker T |∗V1 ⊕ R(T |V1 ). On the other hand, R T |∗V2 = R T ∗ |V3 ⊂ ker T ∗ |V2 = ker T |∗V1 implies the orthogonal decomposition ker TV∗1 = ker TV∗1 ∩ ker T |V2 ⊕ R T |∗V2 . Together, we obtain (16). Since Q is self-adjoint, we also have the orthogonal decomposition V2 = ker Q ⊕ R(Q). Hence R(Q) = R(T |V1 ) ⊕ R T ∗ |V3 . To show that the range of Q is closed, let u ∈ R(T |V1 ). Then u = Tf with f ∈ (ker T |V1 )⊥ = R(T ∗ |V2 ). So, f = T ∗ g with g ∈ (ker T ∗ |V2 )⊥ = R(T |V1 ). Hence, g ∈ Dom Q and u = T T ∗ g = Qg. Analogously, if u ∈ R(T ∗ |V3 ), then there exists g ∈ Dom Q such that u = T ∗ T g = Qg. This shows that Q has closed range and (15) holds. To prove the two estimates, we follow [25, Theorem V.6.2]. First, we construct bounded solution operators for T and T ∗ . We elaborate that for T , the case of T ∗ is analogous. Let L = {u ∈ Dom T |V2 : u ⊥ ker T } be the Banach space with the norm u2L := u2 + T u2 . So, the mapping A : L → R(T |V2 ), is a bounded linear isomorphism. Therefore,

u → T u,

3380


A−1 : R(T |V2 ) → L is an L2 -bounded solution operator for T . Analogously, let B −1 : R T ∗ |V2 → u ∈ Dom T ∗ |V2 : u ⊥ ker T ∗ be the corresponding L2 -bounded solution operator for T ∗ . So, let u ∈ Dom T ∩ Dom T ∗ ∩ R(Q). Then u = u1 + u2 ∈ R(T |V1 ) ⊕ R T ∗ |V3 . Clearly, u1 ∈ Dom T ∗ ,

u2 ∈ Dom T ,

T ∗ u1 = T ∗ u,

T u2 = T u.

By our previous considerations, u1 = B −1 T ∗ u and u2 = A−1 T u. The continuity of A−1 and B −1 yields that 2 u2 = u1 2 + u2 2 C T ∗ u + T u2 with a constant C > 0 independent of u. For u ∈ Dom Q ∩ R(Q), the second inequality follows easily: 2 C −1 u2 T ∗ u + T u2 = (Qu, u) Quu.

2

Under the assumptions of Lemma 3.4 we can construct the Green operator to Q analogously to the construction of the solution operators for T and T ∗ in the proof of Lemma 3.4. Let L = {u ∈ Dom Q: u ⊥ ker Q} = u ∈ Dom Q: u ∈ R(Q) be the Banach space with the norm u2L := u2 + Qu2 . Note that uL Qu by (17). So, the mapping Q|L : L → R(Q) is a bounded linear isomorphism. Hence, Q|−1 L : R(Q) → L ⊂ Dom(Q) is an L2 -bounded solution operator for Q. We extend Q|−1 L to an operator


Q−1 : V2 → Dom(Q)

3381

(20)

by setting Q−1 u = 0 for u ∈ ker Q. Q−1 is called the Green operator associated to Q. The main objective of the present section is to study necessary and sufficient conditions for compactness of Q−1 . We need another useful representation of Q−1 which goes back to E. Straube in case of the complex Green operator (i.e. the ∂-Neumann operator) on pseudoconvex domains in Cn (see [42, Theorem 2.9]). From now on, let D := u ∈ V2 : u ∈ Dom T ∩ Dom T ∗ ∩ R(Q) be the Hilbert space with (u, v)D := (T u, T v)L2 + T ∗ u, T ∗ v L2 , and j : D → V2 the injection into V2 which is bounded by (17). Let j ∗ : V2 → D be the adjoint operator. Then Q−1 = j ◦ j ∗ as we will show now. Let u = u1 + u2 ∈ V2 = ker Q ⊕ R(Q). Then: (u, v)L2 = (u, j v)L2 = j ∗ u, v D for all v ∈ D. On the other hand, (u, v)L2 = (u2 , v)L2 = QQ−1 u2 , v L2 = Q−1 u2 , v D = Q−1 u, v D for all v ∈ D. Hence, if Q−1 is interpreted as an operator to D, then Q−1 = j ∗ . It follows that Q−1 = j ◦ j ∗ : V2 → V2 .

(21)

We will now characterize compactness of Q−1 under the assumption that T ⊕ T ∗ is elliptic in the interior of X in the sense that the Gårding inequality holds on relatively compact subsets of X: for each bounded open subset Ω ⊂⊂ X there exists a constant CΩ > 0 such that u2

W∗1,2 (Ω,ϕ)

2 CΩ u2L2 (Ω,ϕ) + T u2L2 (Ω,ϕ) + T ∗ uL2 (Ω,ϕ) ∗

∗

∗

∞ (Ω). The Sobolev-norm W 1,2 is well defined on Ω for Ω ⊂⊂ X. for all u ∈ C∗,cpt

(22)

3382


Natural choices for T are closed extensions of the operators ∂ cpt , ∂cpt or dcpt acting on smooth forms with compact support in X, for example T = ∂ w , the ∂-operator in the sense of distributions (the maximal closed L2 -extension of ∂ cpt ), or T = ∂ s , the ∂-operator in the sense of approximation by smooth forms with compact support (the minimal closed L2 -extension of ∂ cpt ). In both cases, it is well known that the Gårding inequality (22) holds (see e.g. [7, Theorem 2.2.1]). One important step in the characterization of compactness of the Green operator is the following observation which we present separately for later use: Lemma 3.5. Let V1 , V2 , V3 be as in Lemma 3.4, assume that the Gårding inequality (22) is satis∞ (Ω) ∩ V , and that V is closed under multiplication fied on open subsets Ω ⊂⊂ X for all u ∈ Ccpt 2 2 with smooth compactly supported functions. Let k > 0, 2 u2G = u2L2 + T u2L2 + T ∗ uL2 and K = u ∈ V2 : u ∈ Dom(T ) ∩ Dom T ∗ , u2G < k . Then: for all > 0 and all Ω ⊂⊂ X, there exists δ > 0 such that ∗ Φ u − u

L2

0 and Ω ⊂⊂ X. Fix Ω ⊂⊂ Ω1 ⊂⊂ Ω2 ⊂⊂ X and χ ∈ Ccpt 2 a cut-off function such that χ ≡ 1 on Ω1 . Then

∗ Φ u − u

L2

= Φ ∗ (χu) − χuL2

(23)

since Φ|X−Ω is just the identity mapping for all Φ ∈ Defδ (Ω, X). Since multiplication with χ preserves V2 and the domains of T and T ∗ , χu ∈ Dom(T ) ∩ Dom T ∗ ∩ V2 and there exists a constant Cχ > 0 such that 2 2 χu2L2 + T (χu)L2 + T ∗ (χu)L2 = χu2G Cχ k

(24)

for all u ∈ K. By the same argument, we can use a partition of unity subordinate to a finite covering of Ω2 by coordinate charts to achieve that the χu are supported in coordinate charts. So, we can assume that Ω2 is a bounded domain in Cn (taking Lemma 2.2 into account). Since χu has compact support in Ω2 , it can be approximated by smooth compactly supported forms in the L2 -sense such that arbitrary partial derivatives (up to a certain order) converge as well in the L2 -sense (making the · G -norm converge). So, we can assume that the χu are smooth with compact support in Ω2 because on the other hand


∗ Φ u − Φ ∗ u˜

L2

3383

2u − u ˜ L2

if only δ < δ0 (Ω) for some fix δ0 (Ω) > 0 (see (3) in the proof of Lemma 2.4). To simplify the notation, let v = χu. As | ∂t∂ Φt | 3δ for all Φ ∈ Defδ (Ω, X) by Definition 2.1, ∗ Φ v(z) − v(z) = v Φ1 (z) − v Φ0 (z) 1

∂ |v|1 Φt (z) Φt (z) dt ∂t

0

1 3δ

|v|1 Φt (z) dt,

0

where |v|1 denotes the pointwise norm of all derivatives of first order of all coefficients of v. Since (Φt )∗ ∞ , (Φt−1 )∗ ∞ < 1 + 3δ, it follows as in the proof of Lemma 2.4 that ∗ Φ v − v 2 2 9δ 2 L

1 Ω

2 e−ϕ(z) dVX (z)

0

1 9δ 2

Φt∗ |v|21 e−ϕ dVX dt

Ω

0

1 9δ 2

|v|1 Φt (z) dt

−1 (1 + 3δ)3n sup eϕ(z)−ϕ(Φt (z)) z∈Ω

0

|v|21 e−ϕ dVX dt.

Ω

Since ϕ ∈ C 0 (X) is uniformly continuous on compact subsets of X, there exists a constant −1 Cϕ = sup sup eϕ(z)−ϕ(Φt (z)) < ∞. t∈[0,1] z∈Ω2

So, we get ∗ Φ v − v 2 2 9δ 2 (1 + 3δ)3n Cϕ v2 1,2 . L W (Ω,ϕ) It follows with (23) that there exists a constant C(Ω, χ, ϕ) > 0 such that ∗ Φ u − u2 2 = Φ ∗ (χu) − χu2 2 C(Ω, χ, ϕ)δ 2 χu2 1,2 L L W (Ω

2 ,ϕ)

for all u ∈ K and all Φ ∈ Defδ (Ω, X). This clearly is now the place to use the assumption that T ⊕ T ∗ is elliptic in the sense of (22). Recall that we can assume that χu is smooth with compact support in Ω2 . Hence there exists a constant CΩ2 > 0 such that

3384


χu2W 1,2 (Ω

2 ,ϕ)

CΩ2 χu2L2 (Ω ∗

2 ,ϕ)

2 + T (χu)L2 (Ω ∗

2 ,ϕ)

2 + T ∗ (χu)L2 (Ω ∗

2 ,ϕ)

= CΩ2 χu2G for all u ∈ K. With (24), we arrive finally at ∗ Φ u − u2 2 δ 2 C(Ω, χ, ϕ)CΩ Cχ k. 2 L Therefore, Φ ∗ u − u2L2 < for all u ∈ K and all Φ ∈ Defδ (Ω, X) if δ is small enough.

2

It is now easy to give a necessary and sufficient condition for compactness of the Green operator. The criterion is inspired by the work of Gansberger [13] who treats domains in Cn . Part of his criterion goes back to an earlier work of Haslinger (see [15]). We restrict our attention to a Hermitian submanifold X of a Hermitian complex space Z in order to get an easy treatable notion of the boundary bX of X. Theorem 3.6. Let Z be a Hermitian complex space, X ⊂ Z an open Hermitian submanifold in Z, and T a linear partial differential operator acting on Dom(T ) ⊂ L2∗ (X, ϕ) → L2∗ (X, ϕ) which is densely defined, closed, elliptic in the interior of X and satisfies T 2 = 0. By ellipticity, we understand that the Gårding inequality (22) holds on each relatively compact subset of X. Let P = T ∗T + T T ∗. Assume that there are closed subspaces V1 , V2 , V3 ⊂ L2∗ (X, ϕ) such that the assumptions of Lemma 3.4 are satisfied, hence Q = P |V2 : Dom Q ⊂ V2 → V2 is self-adjoint with closed range, and let D = u ∈ V2 : u ∈ Dom(T ) ∩ Dom T ∗ ∩ R(Q) . Assume that V2 is closed under multiplication with smooth compactly supported functions. Then the following conditions are equivalent: (i) The Green operator Q−1 : V2 → V2 is compact. (ii) The injection j of D equipped with the graph norm u2D = T u2L2 + T ∗ u2L2 into L2∗ (X, ϕ) is compact. (iii) For all > 0, there exists Ω ⊂⊂ X such that uL2∗ (X−Ω,ϕ) < for all u ∈ L = {u ∈ D: uD < 1}. (iv) There exists a smooth function ψ ∈ C ∞ (X, R), ψ > 0, such that ψ(z) → ∞ as z → bX, and (Qu, u)L2 ψ|u|2 e−ϕ dVX for all u ∈ Dom(Q) ∩ R(Q). X


3385

Proof. First, we observe that (i) is equivalent to (ii). Since Q−1 = j ◦ j ∗ : V2 → V2 by (21), the assertion descends to the fact that a bounded operator S is compact exactly if S ∗ is compact (see [34, Theorem 4.19]), and SS ∗ is compact exactly if S and S ∗ are compact (use (S ∗ Sx, x) = (Sx, Sx)). We will now show that (ii) ⇒ (iii) ⇒ (iv) ⇒ (ii). Assume that j : D → L2∗ (X, ϕ) is compact. Then j (L) is precompact in L2∗ (X, ϕ) and (iii) holds by Theorem 2.5. If (iii) holds, it follows by linearity of T ⊕ T ∗ that for all > 0 there exists a domain Ω ⊂⊂ X such that u2L2 (X−Ω ∗

,ϕ)

2 u2D

(25)

for all u ∈ D. For such u, we have

|u|2 e−ϕ dVX c−1 u2D

(26)

X

by (17), and for k ∈ N, k 1:

2k |u|2 e−ϕ dVX 2k · 2−2k u2D = 2−k u2D

(27)

X−Ω2−k

by (25). So, let ψ ∈ C ∞ (X, R) be a real-valued smooth function such that ψ 2k ψ 2k−1

on Ω2−(k+1) − Ω2−k , k 0,

where we set Ω1 = ∅. It follows with (26) and (27) that

ψ |u|2 e−ϕ dVX c−1 + 1 u2D = c−1 + 1 (Qu, u)L2

X

for all u ∈ Dom(Q) ∩ R(Q). So, (iv) is satisfied with ψ = (c−1 + 1)−1 ψ . It remains to show (iv) ⇒ (ii). Assume that (iv) holds. It is enough to show that j (L) is precompact in L2∗ (X, ϕ). This will be done by checking the two conditions in Theorem 2.5 for j (L). The second condition in Theorem 2.5 is obvious: Let > 0. Choose Ω ⊂⊂ X such that ψ 1/ 2 on X − Ω . Then −2

X−Ω

|u|2 e−ϕ dVX

X

ψ|u|2 e−ϕ dVX u2D 1

3386


for all u ∈ L ∩ Dom(Q). That proves the second condition as Dom(Q) is dense in D. By Lemma 3.4 (17), we have 2 u2L2 c−1 T u2L2 + T ∗ uL2 = c−1 u2D < c−1

(28)

for all u ∈ L. Hence, L is a subset of K in Lemma 3.5 with k = 1 + c−1 , and so Lemma 3.5 yields the first condition in Theorem 2.5. 2 If T is a closed extension of the ∂-operator and G−1 the ∂-Neumann operator associated to this ∂-operator, Theorem 3.6 reads as: Theorem 3.7. Let Z be a Hermitian complex space of pure dimension n, X ⊂ Z an open Hermitian submanifold and ∂ a closed L2 (X, ϕ)-extension of the ∂ cpt -operator on smooth forms with compact support in X, for example ∂ = ∂ w the ∂-operator in the sense of distributions. Let 0 p, q n. Assume that ∂ has closed range in L2p,q (X, ϕ) and in L2p,q+1 (X, ϕ). Then 2p,q = ∂∂ ∗ + ∂ ∗ ∂ p,q has closed range and the following conditions are equivalent: (i) The ∂-Neumann operator Np,q = 2−1 p,q is compact. (ii) For all > 0, there exists Ω ⊂⊂ X such that uL2p,q (X−Ω,ϕ) < for all 2 u ∈ u ∈ Dom(∂) ∩ Dom ∂ ∗ ∩ R(2p,q ): ∂u2L2 + ∂ ∗ uL2 < 1 . (iii) There exists a smooth function ψ ∈ C ∞ (X, R), ψ > 0, such that ψ(z) → ∞ as z → bX, and (2p,q u, u)L2 ψ|u|2 e−ϕ dVX for all u ∈ Dom(2p,q ) ∩ R(2p,q ). X

Proof. The assumptions of Theorem 3.6 are satisfied for Vj = L2p,q−2+j (X, ϕ), We use V1 = {0} if q = 0, and V3 = {0} if q = n.

j ∈ {1, 2, 3}.

2

4. Compactness of the ∂-Neumann operator on complex spaces with isolated singularities 4.1. Compact solution operators for the ∂ w -equation In this section, we use some L2 -regularity results for the ∂ w -equation at isolated singularities due to Fornæss, Øvrelid and Vassiliadou (see [10]) to construct compact solution operators for the ∂ w -equation.


3387

Let X be a connected Hermitian complex space of pure dimension n with only isolated singularities and Ω ⊂⊂ X a relatively compact domain. Assume that either X is compact and Ω = X, or that X is Stein and Ω has smooth strongly pseudoconvex boundary which does not contain singularities, bΩ ∩ Sing X = ∅. Let Ω ∗ = Ω − Sing X and A = Ω ∩ Sing X = {a1 , . . . , am } the set of isolated singularities in Ω. For z ∈ X, we denote by dA (z) the distance distX (z, A) of the point z to the singular set A in X. Here, distX (x, y) is the infimum of the length of piecewise smooth curves connecting two points x, y in X. Theorem 4.1. Let X, Ω, A be as above and p +q < n, q 1. Then there exists a closed subspace H of finite codimension in ker ∂ w : L2p,q Ω ∗ → L2p,q+1 Ω ∗ and a constant C > 0 such that for each f ∈ H there exists u ∈ L2p,q−1 (Ω ∗ ) with ∂ w u = f satisfying

|u|2 dA−2 log−4 1 + dA−1 dVX

Ω∗

C

|f |2 dVX .

(29)

Ω∗

For p + q > n, there exist constants a > 0, Ca > 0 and a closed subspace L of finite codimension in ker ∂ w : L2p,q Ω ∗ → L2p,q+1 Ω ∗ such that for each f ∈ L there exists u ∈ L2p,q−1 (Ω ∗ ) with ∂ w u = f satisfying Ω∗

|u|2 dA−2a dVX Ca

|f |2 dVX .

(30)

Ω∗

If X is Stein and Ω has smooth strongly pseudoconvex boundary which does not contain singularities, then the ∂ w -equation is solvable with the estimate (30) if p +q > n, i.e. L = ker ∂ w , and a > 0 can be chosen arbitrarily in (0, 1). If Ω = X is compact, we have to assume in the second case (i.e. in the case p + q > n) that either p + q > n + 1 or that p = n and q = 1. Proof. We will first treat the case that X is Stein and Ω ⊂⊂ X has smooth strongly pseudoconvex boundary that does not contain singularities. We observe that there exists a strictly plurisubharmonic exhaustion function ρ of X which takes the value ρ = −∞ exactly on the singular set of X and which is real-analytic outside. This follows from [5, Theorem 1.2], and the observation that M is a 1-convex space with exceptional set π −1 (Sing X) if π : M → X is a resolution of singularities. We will explain desingularization

in more details below. Let ρ = eρ . After restricting ρ to a neighborhood of Ω, there exists an arbitrarily small regular value c > 0 of ρ such that {ρ < c} ⊂⊂ Ω. Since Ω has smooth strongly pseudoconvex boundary, it can be

3388


exhausted by an increasing sequence of smoothly bounded pseudoconvex domains. So, all the assumptions of Proposition 5.8 and Theorem 5.9 in [10] are fulfilled. The statement for p + q < n follows from the combination of Theorems 5.8 and 1.1 in [10] by the following observation: Let aj ∈ A be an isolated singularity. Then there exists a small neighborhood Uj of aj which can be embedded holomorphically in a complex number space CL such that aj = 0 ∈ CL and z dA (z), because the Euclidean distance of a point z to the origin is less or equal to the length of curves connecting z to the origin in X, if the length of a curve is measured with respect to the Euclidean metric. But the restriction of the Euclidean metric to X is isometric to the original Hermitian metric of X. So, if the equation ∂ w u = f is solvable on Uj − {aj } according to Theorem 1.1 from [10], then: −4 |u|2 dA−2 log−4 1 + dA−1 dVX |u|2 z−2 − log z2 dVX Uj −{aj }

Uj −{aj }

|f |2 dVX .

Uj −{aj }

By [10, Theorem 1.1], there are only finitely many obstructions to the equation ∂ w u = f on Uj − {aj } with that estimate. So, [10, Proposition 5.8] yields that there are only finitely many obstructions to solving the equation ∂ w u = f on Ω ∗ with the estimate (29). If p + q > n, then the statement of our theorem is just Theorem 5.9 in [10], L = ker ∂ w , and a > 0 can be chosen arbitrarily in (0, 1). It remains to treat the case that X is compact and Ω = X. This can be done by the use of a desingularization. Let π :M →X be a resolution of singularities (which exists due to Hironaka [18]), i.e. a proper holomorphic surjection such that π|M−E : M − E → X − Sing X is biholomorphic, where E = π −1 (Sing X) is the exceptional set. We can assume that E is a divisor with only normal crossings, i.e. the irreducible components of E are regular and meet complex transversely. For the topic of desingularization, we refer to [3,4,16]. Let γ := π ∗ h be the pullback of the Hermitian metric h of X to M. γ is positive semi-definite (a pseudo-metric) with degeneracy locus E.


3389

We give M the structure of a Hermitian manifold with a freely chosen (positive definite) metric σ . Then γ σ and γ ∼ σ on compact subsets of M − E. p,q Let Lσ be the sheaf of germs of (p, q)-forms which are locally square-integrable with respect to the metric σ and which have a ∂-derivate in the sense of distributions which is also square-integrable. Let I be the sheaf of ideals of the exceptional set E. Let k ∈ Z. If E is given in a point x ∈ M as the zero set of a (germ of a) holomorphic function f , then k p,q I Lσ x = u: f −k u ∈ Lp,q . σ x We have to use the weighted ∂-operator in the sense of distributions (∂ k )x ux := f k ∂ w f −k ux , which coincides with the usual ∂ w -operator if k 0. We obtain fine resolutions ∂k

p

∂k

∂k

k p,1 k p,n 0 → I k ΩM → I k Lp,0 σ −→ I Lσ −→ · · · −→ I Lσ → 0, p

where ΩM is the sheaf of germs of holomorphic p-forms on M. By the abstract theorem of de Rham, this implies p,q

p,q Hσ,k (U ) :=

p,q+1

ker(∂ k : I k Lσ (U ) → I k Lσ

(U )) ∼ q p = H U, I k ΩM

p,q−1 p,q Im(∂ k : I k Lσ (U ) → I k Lσ (U ))

for open sets U ⊂ M. We will use the well-known fact that p,q p dim Hσ,k (M) = dim H q M, I k ΩM < ∞

(31)

for all 0 p, q n since M is compact. By [35, Lemma 2.1], or [9, Lemma 3.1], respectively, there exists an integer N 0 depending on π : M → X such that N p,q Lp,q γ ⊂ I Lσ p,q

for all 0 p, q n, where Lγ is defined with respect to the pseudo-metric γ analogously to p,q Lσ . Note that the ∂ N -equation extends over the exceptional set by the ∂-extension Theorem 3.2 in [35]. Let ker(∂ w : L2p,q (Ω ∗ ) → L2p,q+1 (Ω ∗ )) Hwp,q Ω ∗ := . Im(∂ w : L2p,q−1 (Ω ∗ ) → L2p,q (Ω ∗ )) We can now define a map p,q Ψ = π ∗ : Hwp,q Ω ∗ → Hσ,N (M)

(32)

3390


by the following observation: Let u ∈ L2p,q−1 (Ω ∗ ) and g ∈ L2p,q (Ω ∗ ) such that ∂ w u = g on Ω ∗ . Then (M) ⊂ I N Lp,q−1 (M), π ∗ u ∈ Lp,q−1 γ σ N p,q π ∗ g ∈ Lp,q γ (M) ⊂ I Lσ (M),

where we extend π ∗ u and π ∗ g trivially over the exceptional set E. It is clear that ∂ w π ∗ u = π ∗ g on M − E, and it follows by the use of the ∂-extension Theorem 3.2 in [35] that ∂ N π ∗u = π ∗g on M. So, Ψ = π ∗ is a well-defined map on cohomology classes. We will now show that Ψ = π ∗ p,q is injective if p + q < n. Let [f ] ∈ Hw (Ω ∗ ) and assume that π ∗ [f ] = 0 ∈ Hσ,N (M), p,q

p,q−1

i.e. there exists a form u ∈ I N Lσ

(M) such that ∂ N u = π ∗ f.

∞ (X ∗ ), 0 χ 1, be a cut-off function such that 1 − χ is supported only in small Let χ ∈ Ccpt neighborhoods {U1 , . . . , Um } of the isolated singularities. Let U = Uj and K = X − U . Then u := (π ∗ χ)u has compact support in M − E, ∂ N u = ∂ w u on M, and

∂ w u = π ∗ f on π −1 (K) since χ ≡ 1 in a neighborhood of K. Set ∗ ∗ 2 v := π|−1 M−E u ∈ Lp,q−1 Ω . Then ∂wv = f on K and ∂ w v ≡ 0 in a neighborhood of the isolated singularities. Consider f := f − ∂ w v ∈ L2p,q Ω ∗ . This form is ∂ w -closed and has support in U = Uj . If p + q < n, then the equation ∂ w v = f is solvable in U in the category of L2 -forms with compact support in U such that the estimate

2 −2 −4 v d log 1 + d −1 dVX A

Uj −{aj }

A

2 f dVX

Uj −{aj }

holds by [10, Proposition 3.1], if we assume that U has been chosen appropriately. Hence

(33)


3391

f = ∂ w v + v , which shows that Ψ = π ∗ is injective if p + q < n, so that dim Hwp,q Ω ∗ < ∞ by the use of (31) and (32). The L2 -norms of v, ∂ w v and f can be dominated by the L2 -norm of f . Then v satisfies the estimate (29), and that is also clear for the form v which has support in a fixed compact set with positive distance to the singular set A. That proves the theorem if Ω = X is compact and p + q < n. Let us finally consider the case that Ω = X is compact and p + q > n. Here, we must distinguish between the cases q = 1 and p + q > n + 1. Let q = 1 which implies that p = n. We need an observation about the behavior of (n, 0) and (n, 1)-forms under the resolution of singularities π : M → X. Since σ is positive definite and γ is positive semi-definite, there exists a continuous function g ∈ C 0 (M, R) such that dVγ = g 2 dVσ . This yields |g||ω|γ = |ω|σ if ω is an (n, 0)-form, and |ω|σ |g||ω|γ if ω is an (n, q)-form, 0 q n. So, for an (n, q) form ω on M:

|ω|2σ

dVσ

g

2

|ω|2γ g −2 dVγ

=

|ω|2γ dVγ .

(34)

Hence, there exists a natural mapping Ψ = π ∗ : Hwn,q Ω ∗ → Hσn,q (M), which is an isomorphism by [37, Theorem 1.5]. That shows that especially Hwn,1 (Ω ∗ ) is finitedimensional, but we need some additional considerations to obtain also the estimate (30). As above, let U = Uj be a neighborhood of the isolated singularities such that the ∂ w equation is solvable for (n, 1)-forms on U with the estimate (30), and let χ1 ∈ C ∞ (U ), 0 χ1 1, be a cut-off function which is identically 1 in a smaller neighborhood of the singular set A. For [f ] ∈ Hwn,1 (Ω ∗ ), let u ∈ L2n,0 (U ∗ ) be a solution on U ∗ and set f := f − ∂(χ1 u) ∈ [f ] ∈ Hwn,1 Ω ∗ , where we extend χ1 u trivially to Ω ∗ . Now, if [f ] = 0 in Hwn,1 (Ω ∗ ) which is equivalent to [π ∗ f ] = 0 in Hσn,1 (M), then there exists g ∈ Ln,0 σ (M) such that ∂ w g = π ∗f

3392


on M. But π ∗ f vanishes identically in a fixed neighborhood of the exceptional set E. Hence g is a holomorphic n-form in a fixed neighborhood of the exceptional set. There, it is smooth and bounded and the sup-norm is bounded by the L2 -norm of f , which in turn is bounded by the L2 -norm of f . Let ϕ be a fixed weight function that vanishes exactly of order 1 along the exceptional set E. Then ϕ −(1−) g is square-integrable for a small > 0 and its L2 -norm can be estimated uniformly ∗ ∼ 2 by the L2 -norm of f . Since Ln,0 σ (M) = Ln,0 (Ω ), it follows by Lemma 3.1 in [9] that there exists an exponent a > 0 such that ∗ ∗ 2 dA−a π|−1 M−E g ∈ Ln,0 Ω . Hence, −1 ∗ π|M−E g + χ1 u is the desired solution of the equation ∂ w u = f which satisfies the estimate (30) with that exponent a > 0. Finally, let p + q > n + 1 which implies that q 2. Here, we proceed similar to the case p + q < n. First, we define a map p,q Ψ : Hwp,q Ω ∗ → Hσ,N (M) where N 1 is an integer such that p,q I N −1 Lp,q σ ⊂ Lγ

(35)

for all 0 p, q n, which is true if only N 1 is big enough by [35, Lemma 2.1], or [9, Lemma 3.1], respectively. p,q We can define the map Ψ as follows. Let [f ] ∈ Hw (Ω ∗ ). By solving the ∂-equation on the neighborhood U of the singular set as above, we can switch to the representative f = f − ∂(χ1 u) ∈ [f ]. Since f has compact support away from the singular set, π ∗ f ∈ I N Lσ (M), and we can define p,q

p,q Ψ [f ] := π ∗ f ∈ Hσ,N (M). We need to show that this assignment is well defined as a map on cohomology classes. So, assume that ∂wg = f on Ω ∗ . Let g = g − χ1 u. Then ∂ w g ≡ 0 on the neighborhood W of A where χ1 ≡ 1. By shrinking W appropriately, the ∂ w -equation is solvable on W in the L2 -sense for (p, q − 1) if p + q > n + 1. Hence, let v ∈ L2p,q−2 (W ∗ ) such that ∂v = g = g − χ1 u,


3393

∞ (W ), 0 χ 1, which is identically 1 in a smaller and choose a cut-off function χ2 ∈ Ccpt 2 neighborhood of the singular set A. Let

g

= g − ∂ w (χ2 v) = g − χ1 u − ∂ w (χ2 v) ∈ L2p,q−1 Ω ∗ . Then ∂ w g

= ∂ w g − ∂ w (χ1 u) = f and g

has compact support away from the singular set. Hence ∂ w π ∗ g

= π ∗ f , so that [π ∗ f ] = 0 in Hσ,N (M) and Ψ is actually well defined. It is clear that Ψ is injective because of the assumption (35), and we have arranged the index N 1 such that a solution ∂h = π ∗ f on M satisfies p,q

ϕ −(1−) h ∈ Lp,q−1 (M) γ as above. Hence, again ∗ ∗ 2 dA−a π|−1 M−E h ∈ Lp,q−1 Ω ∗ with the exponent a > 0 from above, and (π|−1 M−E ) h+χ1 u is the desired solution of the equation ∂ w u = f which satisfies the estimate (30) with that exponent. 2

Corollary 4.2. Let X, Ω and p, q be as in Theorem 4.1. Then the ∂-operator in the sense of distributions ∂ w : L2p,q−1 (Ω ∗ ) → L2p,q (Ω ∗ ) has closed range. We are now in the position to construct compact solution operators for the ∂ w -equation. Let ϕ be the weight ϕ = − log dA−2 log−4 1 + dA−1 if p + q < n or ϕ = − log dA−2a if p + q > n. For p + q = n, q 1, and q = 1 if Ω is compact and p + q = n + 1, let T1 : L2p,q−2 Ω ∗ → L2p,q−1 Ω ∗ , ϕ and T2 : L2p,q−1 Ω ∗ , ϕ → L2p,q Ω ∗ be the ∂-operators in the sense of distributions (ignore T1 if q = 1).

3394


T1 and T2 are closed densely defined operators, T2 ◦ T1 = 0 and T2 has closed range by Theorem 4.1. So, the adjoint operators T1∗ and T2∗ are closed densely defined operators with T1∗ ◦ T2∗ = 0 and T2∗ has closed range. We can use the orthogonal decomposition L2p,q−1 Ω ∗ , ϕ = ker T2 ⊕ R T2∗

(36)

to define a bounded solution operator for the ∂ w -equation as in Lemma 3.4. Let H ⊂ L2p,q (Ω ∗ ) be the closed subspace from Theorem 4.1 if p + q < n or H = L ⊂ L2p,q (Ω ∗ ) if p + q > n. For f ∈ H let Sf be the uniquely defined element u ⊥ ker T2 such that ∂ w u = f . So, S : H ⊂ L2p,q Ω ∗ → R T2∗ ⊂ L2p,q−1 Ω ∗ , ϕ

(37)

is a bounded solution operator for the ∂ w -equation and it satisfies T1∗ ◦ S = 0.

(38)

Since L2p,q−1 (Ω ∗ , ϕ) is contained in L2p,q−1 (Ω ∗ ), we can show by the use of the criterion for precompactness Theorem 2.5 as in the proof of Theorem 3.6 that S is compact as an operator to the latter space. Theorem 4.3. Let p + q = n. For q 2, and p + q = n + 1 if Ω is compact, the ∂ w -solution operator S is compact as an operator S : Dom S = H ⊂ L2p,q Ω ∗ → L2p,q−1 Ω ∗ . For q = 1, there exists a bounded operator P0 : H → L2p,0 (Ω ∗ ) such that S − P0 is a compact ∂ w -solution operator S − P0 : Dom S = H ⊂ L2p,1 Ω ∗ → L2p,0 Ω ∗ . Proof. We will only treat the case that Ω is Stein with smooth strongly pseudoconvex boundary. The compact case follows by the same arguments but is much easier because there is no boundary to consider. Let L = f ∈ H : f L2p,q (Ω ∗ ) < 1 . We will show that S(L) is precompact in L2p,q−1 (Ω ∗ ) if q 2. To do this, we have to treat ∞ (Ω), the singular set A and the strongly pseudoconvex boundary bΩ separately. So let χ ∈ Ccpt 0 χ 1, be a smooth cut-off function with compact support in Ω such that χ ≡ 1 in a neighborhood of the singular set A. Let us first show that L1 := χS(f ): f ∈ L is precompact in L2p,q−1 (Ω ∗ ) by the use of the criterion Theorem 2.5 with X = Ω ∗ .


3395

Since S is bounded as an operator to L2p,q−1 (Ω ∗ , ϕ), there exists a constant CS > 0 such that uL2

p,q−1 (Ω

CS

∗ ,ϕ)

for all u ∈ S(L). Let K = supp χ , K ∗ = K − A. Now then, let > 0. Choose Ω ⊂⊂ Ω ∗ such that e−ϕ 1/ 2 on K ∗ − Ω . This is possible because K ∗ − Ω is a neighborhood of A if Ω is big enough and e−ϕ(z) → +∞ as z approaches the singular set A. Then

−2

|χu| dVX = 2

−2

Ω ∗ −Ω

|χu|2 dVX

K ∗ −Ω

|χu|2 e−ϕ dVX

K ∗ −Ω

|u|2 e−ϕ dVX CS2

Ω∗

for all u ∈ S(L). Hence vL2

p,q−1 (Ω

∗ −Ω

)

CS

for all v ∈ L1 . That proves the second condition in Theorem 2.5, it remains to show the first condition. We can use Lemma 3.5 with X = Ω ∗ and T = T1 : V1 = L2p,q−2 Ω ∗ → V2 = L2p,q−1 Ω ∗ , ϕ , T = T2 : V2 = L2p,q−1 Ω ∗ , ϕ → V3 = L2p,q Ω ∗ , because we can use different weight functions for forms of different degree in all our considerations above. For u ∈ S(L), we have u2L2

p,q−1 (Ω

∗ ,ϕ)

+ T2 u2L2

p,q (Ω

∗)

< CS2 + 1

and T1∗ u = 0. Since χ is constant outside a compact subset of Ω ∗ , there exists a constant Cχ > 0 such that χu2L2

p,q−1

(Ω ∗ ,ϕ)

2 + T2 (χu)L2

p,q

(Ω ∗ )

2 + T1∗ (χu)L2

p,q−2 (Ω

∗)

< Cχ CS2 + 1 .

(39)

So, we can use Lemma 3.5 with k = Cχ (CS2 + 1) yielding L1 ⊂ K. Hence, L1 satisfies also the first condition in Theorem 2.5.

3396


The second step is to show that L2 = (1 − χ)S(f ): f ∈ L is precompact in L2p,q−1 (Ω ∗ ). But this follows from well-known results since Ω has smooth strongly pseudoconvex boundary and (1 − χ) has support away from the singular set A. Let V be an open neighborhood of A in Ω such that N = V ⊂⊂ z ∈ Ω: χ(z) = 1 ⊂⊂ Ω, and let π :M →X be a resolution of singularities as in the proof of Theorem 4.1. Set Ω = π −1 (Ω) and N = π −1 (N ). Again, let γ := π ∗ h be the pullback of the Hermitian metric h of X to M which is positive semi-definite with degeneracy locus E. As above, give M the structure of a Hermitian manifold with a freely chosen (positive definite) metric σ . Then γ σ on a neighborhood of Ω and γ ∼σ on Ω − N since the degeneracy locus E of γ is compactly contained in π −1 (V ). Recall that there exists a continuous function g ∈ C 0 (M, R) such that dVγ = g 2 dVσ . This yields |g||ω|γ = |ω|σ if ω is an (n, 0)-form, and |g||ω|γ |ω|σ if ω is a (p, 0)-form, 0 p n. So, for a (p, 0) form ω on M:

|ω|2γ dVγ

g −2 |ω|2σ g 2 dVσ =

|ω|2σ dVσ .

(40)

We can now show that L2 is precompact by the use of the resolution π : M → X and wellknown results about strictly pseudoconvex manifolds.


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Since γ = π ∗ h ∼ σ on Ω − N , we have 2,σ L2p,q−1 (Ω − N ) ∼ = Lp,q−1 Ω − N . But the forms in L2 have support in Ω − N . So, it is enough to show that π ∗ L2 is precompact in

L2,σ p,q−1 (Ω ). As in (39), there exists a constant Cχ > 0 such that v2L2

∗ p,q−1 (Ω ,ϕ)

+ T2 v2L2

p,q (Ω

∗)

2 + T1∗ v L2

p,q−2 (Ω

∗)

< Cχ

for all v ∈ L2 . We can ignore the weight ϕ since the forms in L2 have support away from the singular set A and get a constant Cχ

> 0 such that v2L2

∗ p,q−1 (Ω )

+ ∂ w v2L2

∗ p,q (Ω )

2 + ∂ ∗w v L2

p,q−2 (Ω

∗)

< Cχ

for all v ∈ L2 . For the same reason we obtain ∗ 2 π v 2,σ

Lp,q−1 (Ω )

2 2 + ∂ w π ∗ v L2,σ (Ω ) + ∂ ∗w π ∗ v L2,σ

p,q−2 (Ω

p,q

)

< Cχ

.

(41)

Since Ω is a relatively compact subset of M with a smooth strongly pseudoconvex boundary, Kohn’s basic estimate yields ∗ 2 π v

1/2,2,σ

Wp,q−1 (Ω )

< C1

for all v ∈ L2 with some constant C1 > 0 if q 2. In this setting, the embedding 1/2,2,σ Wp,q−1 Ω → L2,σ p,q−1 Ω is compact by the Sobolev embedding theorem, and this shows that π ∗ L2 is a precompact subset of L2p,q−1 (Ω ) if q 2. It remains to consider the case q = 1. Let 2,σ Π0 : L2,σ p,0 Ω → ker ∂ w ⊂ Lp,0 Ω be the Bergman projection (the orthogonal projection onto ker ∂ w ). We can now define the operator P0 : H → ker ∂ w ⊂ L2p,0 Ω ∗ as ∗ ∗ P0 (f ) := π|−1 Ω −E ◦ Π0 ◦ π (1 − χ)S(f ) .

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Since π : Ω − E → Ω ∗ is biholomorphic, it is clear that ∂ w P0 (f ) = 0, so that S − P0 remains a solution operator for the ∂ w -equation. Since (1 − χ) ≡ 0 on N , it is clear that f → Π0 ◦ π ∗ (1 − χ)S(f )

is a bounded map H → ker ∂ w ⊂ L2,σ p,0 (Ω ). On the other hand, (40) yields (because E is thin):

−1 π|

Ω −E

∗ ωL2

p,0 (Ω

∗)

= ωL2,γ (Ω ) ωL2,σ (Ω ) . p,0

p,0

Hence −1 ∗ 2,σ π|Ω −E : Lp,0 Ω → L2p,0 Ω ∗

(42)

is bounded, and we see that P0 is a bounded linear map. It is now easy to see by Kohn’s basic estimates that (1 − χ)S − P0 is compact. Because of (42), it is enough to show that π ∗ L 2 − Π0 π ∗ L 2

(43)

is precompact in L2,σ p,0 (Ω ). But (41) implies that there exists a constant C2 > 0 such that

∗ π v − Π 0 π ∗ v

1/2,2,σ

Wp,0

(Ω )

< C2

for all v ∈ L2 since Ω is a domain with smooth strongly pseudoconvex boundary. Hence, (43) follows by the Sobolev embedding theorem. 2 4.2. Compactness of the ∂ w -Neumann operator We can now study the ∂ w -Neumann operator on spaces with isolated singularities. Let X, Ω, A be as in Theorem 4.1. Then ∂ cpt : C∗∞ Ω ∗ → C∗∞ Ω ∗ is a densely defined operator on L2∗ (Ω ∗ ). In this section, we study the maximal closed extension, i.e. the ∂-operator in the sense of distributions, which we denote by ∂ w . Let 2 = ∂ w ∂ ∗w + ∂ ∗w ∂ w . By Theorem 3.1, 2 is a densely defined closed self-adjoint operator with (2u, u)L2 0.


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By Corollary 4.2, the ∂-operator in the sense of distributions has closed range in L2p,q (Ω ∗ ) if p + q = n. If Ω = X is compact, we have to assume in addition that q = 1 if p + q = n + 1. So, if p + q = n − 1, n (and q = 1 if p + q = n + 1 and Ω is compact), then 2p,q = 2|L2p,q : L2p,q Ω ∗ → L2p,q Ω ∗ has closed range and we have the orthogonal decomposition L2p,q Ω ∗ = ker 2p,q ⊕ R(2p,q ) by Lemma 3.4. Hence, the associated Green operator ∗ ∗ 2 2 Np,q = 2−1 p,q : Lp,q Ω → Dom(2) ⊂ Lp,q Ω is well defined as in (20). Np,q is called the ∂ w -Neumann operator. We will now observe that Np,q is a compact operator (if p + q = n − 1, n). This is the case exactly if the equivalent conditions of Theorem 3.6 are satisfied. However, we do not use Theorem 3.6 to verify compactness, but a classical argument due to Hefer and Lieb relying on the existence of compact solution operators (see [17]). Theorem 4.4. Let X be a Hermitian complex space of pure dimension n with only isolated singularities, and Ω ⊂⊂ X a relatively compact open subset such that either Ω = X is compact, or X is Stein and Ω has smooth strongly pseudoconvex boundary that does not contain singularities. Let p + q = n − 1, n and q 1. If Ω = X is compact and p + q = n + 1, let q = 1. Then the ∂-operator in the sense of distributions ∂ w has closed range in L2p,q (Ω ∗ ) and L2p,q+1 (Ω ∗ ) so that the ∂ w -Neumann operator −1 ∗ ∗ ∗ ∗ 2 2 Np,q = 2−1 p,q = ∂ w ∂ w + ∂ w ∂ w p,q : Lp,q Ω → Dom 2p,q ⊂ Lp,q Ω is well defined as in (20). Np,q is compact. Proof. Only compactness remains to show. By Theorems 4.1 and 4.3, there exist closed subspaces of finite codimension Hq ⊂ ker ∂ w ⊂ L2p,q Ω ∗ , Hq+1 ⊂ ker ∂ w ⊂ L2p,q+1 Ω ∗ , and compact linear operators Sq : Hq → Dom ∂ w ⊂ L2p,q−1 Ω ∗ , Sq+1 : Hq+1 → Dom ∂ w ⊂ L2p,q Ω ∗ , such that ∂ w Sq u = u and ∂ w Sq+1 u = u. Now then Np,q is compact by [17, Theorem 3.1]. We may outline the short and elegant proof for convenience of the reader. Let Uq and Uq+1 be the orthogonal complements of Hq and Hq+1

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in R(∂ w ) in L2p,q and L2p,q+1 with basis f1,q , . . . , frq ,q and f1,q+1 , . . . , frq+1 ,q+1 , respectively. Choose uj,k with ∂ w uj,k = fj,k and define the operators Tq and Tq+1 on R(∂ w ) in L2p,q and L2p,q+1 by Tk (f ) = Tk

rk

αj fj,k + g :=

j =1

rk

αj uj,k + Sq (g)

for g ∈ Hk .

j =1

Then Tk , k = q, q + 1, are compact linear solution operators for the ∂ w -operator on R(∂ w ). Extend these operators to be zero on R(∂ w )⊥ . For k ∈ {q, q + 1}, let Pk : L2p,k−1 (Ω ∗ ) → (ker ∂ w )⊥ and Qk : L2p,k (Ω ∗ ) → R(∂ w ) be the orthogonal projections on these closed subspaces, and define Kk := Pk Tk Qk

for k = q, q + 1.

∗ , and that yields compactness of N Hefer and Lieb show that Np,q = Kq∗ Kq + Kq+1 Kq+1 p,q by compactness of Tq , Tq+1 . 2

4.3. Compactness of the ∂ s -Neumann operator Another important operator is the minimal closed extension of ∂ cpt : C∗∞ (Ω ∗ ) → C∗∞ (Ω ∗ ), i.e. the closure of the graph in L2∗ (Ω ∗ ) × L2∗ (Ω ∗ ), which we denote by ∂ s . A form f ∈ L2p,q (Ω ∗ ) ∞ (Ω ∗ ) is in the domain of ∂ s iff it is in the domain of ∂ w and there exists a sequence {fj } ⊂ Cp,q such that fj → f in L2p,q (Ω ∗ ) and ∂fj → ∂ w f in L2p,q+1 (Ω ∗ ). The ∂ s -operator is dual to the ∂ w -operator in a sense we will elaborate now. Note that ∂ s = ∂ ∗∗ cpt since it is the closure of the graph. Let ∗ be the Hodge-∗-operator on Ω ∗ (mapping (p, q) to (n − q, n − p)-forms). Then ϑcpt = −∗∂ cpt ∗ is the formal adjoint of the ∂-operator (acting on smooth forms with compact support). By definition, ∗ ∂ w = ϑcpt .

We also obtain the ϑ -operator in the sense of distributions ϑw = ∂ ∗cpt = −∗∂ w ∗ and the minimal closed extension ∗∗ = −∗∂ s ∗. ϑs = ϑcpt


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Thus, we obtain the duality relations ∗∗ ∂ ∗w = ϑcpt = ϑs = −∗∂ s ∗

and ∂ ∗s = ∂ ∗cpt = ϑw = −∗∂ w ∗. Hence the ∂ s -Laplacian is related to the ∂ w -Laplacian as 2s = ∂ s ∂ ∗s + ∂ ∗s ∂ s = ∗2∗, and the ∂ s -Neumann operator N s is well defined on (n − p, n − q)-forms exactly if the ∂ w Neumann operator is well defined on (p, q)-forms, and in that case: −1 s = 2sn−p,n−q = ∗2−1 Nn−p.n−q p,q ∗ = ∗Np,q ∗. So, a direct consequence of Theorem 4.4 is: Theorem 4.5. Let X, Ω, p and q be as in Theorem 4.4, and a = n − p, b = n − q. Then the minimal closed extension ∂ s of the ∂-operator has closed range in L2a,b (Ω ∗ ) and L2a,b+1 (Ω ∗ ) so that the ∂ s -Neumann operator −1 ∗ −1 s Na,b = 2sa,b = ∂ s ∂ s + ∂ ∗s ∂ s a,b : L2a,b Ω ∗ → Dom 2a,b ⊂ L2a,b Ω ∗ s is compact. is well defined as in (20). Na,b

Acknowledgments The author thanks Klaus Gansberger and Dariush Ehsani for interesting and helpful discussions on the topic. The author also thanks Nils Øvrelid for pointing out that the restriction to the case q = 1 if X is compact and p + q = n + 1 in Theorems 1.1 and 1.2 is superfluous because Theorem 4.1 is valid on domains in Hermitian complex spaces with isolated singularities for all p + q = n if only the boundary of the domain is nice. We will elaborate that in a supplement to the present paper. References [1] H.W. Alt, Lineare Funktionalanalysis, Springer-Verlag, Berlin, 1992. [2] A. Andreotti, E. Vesentini, Carleman estimates for the Laplace Beltrami equation on complex manifolds, Publ. Math. Inst. Hautes Etudes Sci. 25 (1965) 81–130. [3] J.M. Aroca, H. Hironaka, J.L. Vicente, Desingularization Theorems, Mem. Math. Inst. Jorge Juan, vol. 30, 1977. [4] E. Bierstone, P. Milman, Canonical desingularization in characteristic zero by blowing-up the maximum strata of a local invariant, Invent. Math. 128 (2) (1997) 207–302. [5] M. Coltoiu, N. Mihalache, Strongly plurisubharmonic exhaustion functions on 1-convex spaces, Math. Ann. 270 (1985) 63–68.

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[6] K. Diederich, J.E. Fornæss, S. Vassiliadou, Local L2 results for ∂ on a singular surface, Math. Scand. 92 (2003) 269–294. [7] G.B. Folland, J.J. Kohn, The Neumann Problem for the Cauchy–Riemann Complex, Ann. of Math. Stud., vol. 75, Princeton University Press, Princeton, 1972. [8] J.E. Fornæss, L2 results for ∂ in a conic, in: International Symposium, Complex Analysis and Related Topics, Cuernavaca, in: Oper. Theory Adv. Appl., Birkhäuser, 1999. [9] J.E. Fornæss, N. Øvrelid, S. Vassiliadou, Semiglobal results for ∂ on a complex space with arbitrary singularities, Proc. Amer. Math. Soc. 133 (8) (2005) 2377–2386. [10] J.E. Fornæss, N. Øvrelid, S. Vassiliadou, Local L2 results for ∂: the isolated singularities case, Internat. J. Math. 16 (4) (2005) 387–418. [11] J. Fox, P. Haskell, Hodge decompositions and Dolbeault complexes on complex surfaces, Trans. Amer. Math. Soc. 343 (1994) 765–778. [12] M.P. Gaffney, Hilbert space methods in the theory of harmonic integrals, Trans. Amer. Math. Soc. 78 (1955) 426– 444. [13] K. Gansberger, On the resolvent of the Dirac operator in R2 , arXiv:1003.5124. [14] P. Haskell, L2 -Dolbeault complexes on singular curves and surfaces, Proc. Amer. Math. Soc. 107 (2) (1989) 517– 526. [15] F. Haslinger, Compactness for the ∂-Neumann problem – a functional analysis approach, ESI-Preprint 2208, arXiv:0912.4406. [16] H. Hauser, The Hironaka theorem on resolution of singularities, Bull. Amer. Math. Soc. (N.S.) 40 (3) (2003) 323– 403. [17] T. Hefer, I. Lieb, On the compactness of the ∂-Neumann operator, Ann. Fac. Sci. Toulouse Math. (6) 9 (3) (2000) 415–432. [18] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Ann. of Math. (2) 79 (1964) 109–326. [19] L. Hörmander, L2 -estimates and existence theorems for the ∂-operator, Acta Math. 113 (1965) 89–152. [20] L. Hörmander, An Introduction to Complex Analysis in Several Variables, Van Nostrand, Princeton, 1966. [21] M. Kashiwara, T. Kawai, The Poincaré lemma for varieties of polarized Hodge structure, Publ. Res. Inst. Math. Sci. 23 (1987) 345–407. [22] J.J. Kohn, Harmonic integrals on strongly pseudoconvex manifolds I, Ann. of Math. 78 (1963) 112–148. [23] J.J. Kohn, Harmonic integrals on strongly pseudoconvex manifolds II, Ann. of Math. 79 (1964) 450–472. [24] J.J. Kohn, L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965) 443–492. [25] I. Lieb, J. Michel, The Cauchy–Riemann Complex, Integral Formulae and Neumann Problem, Vieweg, Braunschweig/Wiesbaden, 2002. [26] M. Nagase, Remarks on the L2 -Dolbeault cohomology groups of singular algebraic surfaces and curves, Publ. Res. Inst. Math. Sci. 26 (5) (1990) 867–883. [27] T. Ohsawa, Hodge spectral sequence on compact Kähler spaces, Publ. Res. Inst. Math. Sci. 23 (1987) 265–274. [28] T. Ohsawa, K. Takegoshi, On the extension of L2 -holomorphic functions, Math. Z. 195 (2) (1987) 197–204. [29] N. Øvrelid, S. Vassiliadou, Solving ∂ on product singularities, Complex Var. Elliptic Equ. 51 (3) (2006) 225–237. [30] N. Øvrelid, S. Vassiliadou, Some L2 results for ∂ on projective varieties with general singularities, Amer. J. Math. 131 (2009) 129–151. [31] W. Pardon, The L2 -∂-cohomology of an algebraic surface, Topology 28 (2) (1989) 171–195. [32] W. Pardon, M. Stern, L2 -∂-cohomology of complex projective varieties, J. Amer. Math. Soc. 4 (3) (1991) 603–621. [33] W. Pardon, M. Stern, Pure Hodge structure on the L2 -cohomology of varieties with isolated singularities, J. Reine Angew. Math. 533 (2001) 55–80. [34] W. Rudin, Functional Analysis, International Series, Pure Appl. Math., McGraw–Hill, New York, 1991. [35] J. Ruppenthal, About the ∂-equation at isolated singularities with regular exceptional set, Internat. J. Math. 20 (4) (2009) 459–489. [36] J. Ruppenthal, The ∂-equation on homogeneous varieties with an isolated singularity, Math. Z. 263 (2009) 447–472. [37] J. Ruppenthal, L2 -theory for the ∂-operator on compact complex spaces, ESI-Preprint 2202, submitted for publication, arXiv:1004.0396. [38] J. Ruppenthal, E.S. Zeron, An explicit ∂-integration formula for weighted homogeneous varieties, Michigan Math. J. 58 (2009) 321–337. [39] J. Ruppenthal, E.S. Zeron, An explicit ∂-integration formula for weighted homogeneous varieties II, forms of higher degree, Michigan Math. J. 59 (4) (2010) 283–295.


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A classification of finite rank dimension groups by their representations in ordered real vector spaces ✩ Gregory R. Maloney a , Aaron Tikuisis b,∗ a Department of Mathematics, University of Massachusetts Boston, Boston, MA 02125, United States b Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

Received 4 October 2010; accepted 21 December 2010 Available online 30 December 2010 Communicated by Alain Connes

Abstract This paper systematically studies finite rank dimension groups, as well as finite-dimensional ordered real vector spaces with Riesz interpolation. We provide an explicit description and classification of finite rank dimension groups, in the following sense. We show that for each n, there are (up to isomorphism) finitely many ordered real vector spaces of dimension n that have Riesz interpolation, and we give an explicit model for each of them in terms of combinatorial data. We show that every finite rank dimension group can be realized as a subgroup of a finite-dimensional ordered real vector space with Riesz interpolation via a canonical embedding. We then characterize which of the subgroups of a finite-dimensional ordered real vector space have Riesz interpolation (and are therefore dimension groups). © 2010 Elsevier Inc. All rights reserved. Keywords: Ordered abelian groups; Riesz interpolation; Ordered vector spaces

1. Introduction Dimension groups are interesting algebraically, being only slightly more general than latticeordered abelian groups (in fact, the class of dimension groups comprises exactly the groups obtained by taking inductive limits of lattice-ordered abelian groups [14, Theorem 3.21]). More✩

The research of the author was supported by a scholarship from the Natural Sciences and Engineering Research Council of Canada. * Corresponding author. E-mail addresses: [email protected] (G.R. Maloney), [email protected] (A. Tikuisis). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.12.026

G.R. Maloney, A. Tikuisis / Journal of Functional Analysis 260 (2011) 3404–3428

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over, dimension groups are particularly relevant in operator algebras, as they often appear as the K0 -group of a C∗ -algebra, and thus constitute a crucial component in the range of the Elliott invariant [7–9,18]. Dimension groups also appear in the study of topological Markov chains [17,1], coding theory [10], and have examples in number theory [3, Proposition 4.4]. Systematic study of dimension groups has been undertaken previously. Most famously, Effros, Handelman and Shen proved in [3, Theorem 2.2] an equivalence between an abstract characterization of dimension groups (as ordered groups that are unperforated and have Riesz interpolation) and a concrete one (as inductive limits of ordered abelian groups of the form (Zr , Nr )). A more detailed exploration of the structure of finite rank dimension groups is contained in works by Effros and Shen [4,6]; Theorem 1.4 of the latter classifies all finite rank dimension groups under the restriction of being simple. Goodearl generalized this classification to all simple dimension groups in [14, Theorem 14.16]. Dimension groups that are also ordered real vector spaces (which we will refer to as ordered real vector spaces with Riesz interpolation) were studied systematically by Fuchs in [11,13]. Fuchs primarily focuses on describing the topological and order-theoretic structure of such vector spaces under the additional assumption of being antilattices. Additionally, he shows that any ordered real vector space with interpolation can be embedded into a cartesian product of antilattices. However, even with this result, there is much more to be understood about the structure of ordered real vector spaces with interpolation. In particular, the results contained here do not seem to follow from Fuchs’, and indeed, our techniques are quite different from his. This paper concerns the question of describing all finite rank dimension groups and all finitedimensional ordered real vector spaces with Riesz interpolation. Our ultimate results are explicit descriptions and classifications of finite rank dimension groups and finite-dimensional ordered real vector spaces with Riesz interpolation. For the finite-dimensional ordered real vector spaces with Riesz interpolation, Theorem 5.2, shows that for each n, there are (up to isomorphism) finitely many ordered real vector spaces of dimension n that have Riesz interpolation, and we give an explicit model for each of them in terms of combinatorial data. In Corollary 6.1 and the following remarks, we see that every finite rank dimension group can be realized as a subgroup of a finite-dimensional ordered real vector space with Riesz interpolation, via a canonical embedding with the property that every isomorphism of the dimension groups lifts to an isomorphism of the real vector spaces. Theorem 4.2 characterizes which of the subgroups of a finite-dimensional ordered real vector space have interpolation (and are therefore dimension groups). Putting these together yields an explicit description of all finite rank dimension groups (up to isomorphism), in terms of subgroups of Rn and combinatorial data describing the positive cone; additionally, an explicit description of when such dimension groups are isomorphic yields a complete classification of these dimension groups; altogether, this classification is the content of Corollary 6.2. To give an idea of the nature of the results, consider an n-dimensional ordered real vector space V with Riesz interpolation that has n extreme states (i.e. the positive functionals on V separate its points). In this case, each ideal I of V is determined by which of the extreme states are nonzero on I . This induces a lattice isomorphism between the ideals of V and a sublattice of the subsets of extreme states on V (or, by labelling the states with numbers 1 through n, a sublattice of 2{1,...,n} ); V is determined, as an ordered real vector space, by this sublattice of 2{1,...,n} , up to a permutation of {1, . . . , n}. Note that the simplifying assumptions of the last example assure that the extreme states on V already give a natural representation of V into Rn , and this representation provides a nice presentation of the ideal structure and positive cone of V . In the case that V has fewer than n

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extreme states, Theorem 3.2 is a non-trivial tool providing a representation of V that still gives a nice presentation of the ideal structure and the positive cone. Theorem 3.2 also applies to finite rank dimension groups (and indeed, to more general ordered groups than these two disjoint classes). The classification of finite rank dimension groups, however, is more complicated than that of finite-dimensional ordered real vector spaces with Riesz interpolation, because certain obstructions to having Riesz interpolation are automatically avoided by ordered real vector spaces. Section 2 contains the definitions of the basic concepts related to dimension groups and some standard background results. In Section 3, we explore the representation of dimension groups into Rn , leading the way to the main result, Theorem 3.2. This result gives a canonical representation into Rn of many dimension groups (including those that have finite rank and those that are finite-dimensional ordered real vector spaces), and describes the positive cone as a pull-back of a reasonably nice cone in Rn . Theorem 3.2 paves the way for the canonical embedding used in the classification results. In Section 4 we solve the problem of which subgroups of Rn with the reasonably nice cones as in Theorem 3.2 are in fact dimension groups. The solution of this problem is Theorem 4.2, which gives necessary and sufficient conditions for such an ordered group to have Riesz interpolation. Sections 5 and 6 contain the classifications of finite-dimensional ordered real vector spaces with interpolation and finite rank dimension groups respectively; these classifications are given simply by collecting and specializing the results in Theorems 3.2 and 4.2. 2. Preliminaries Definition 2.1. An ordered abelian group (or simply ordered group) is an abelian group G that is equipped with a partial order, , satisfying the following: (i) Compatibility with addition: for all g, h, x ∈ G, if g h then g + x h + x; and (ii) Directedness: for all g, h ∈ G, there exists y ∈ G such that g y. h An ordered vector space over a field F ⊆ R is a vector space V with an ordering such that (V , ) is an ordered group, and additionally satisfying: (iii) Compatibility with scalar multiplication: for all g, h ∈ G and r ∈ F , if g h and r 0 then rg rh. One associates to an ordered abelian group its positive cone, G+ := {g ∈ G: g 0}. This is a cone in G, meaning that it is closed under taking sums. It is a strict cone, meaning G+ ∩ −G+ = {0} (a consequence of being an order instead of a pre-order), and directedness of (G, ) amounts to G+ − G+ = G. Moreover, given any strict cone C ⊆ G such that C − C = G, it makes G into an ordered abelian group via the order g C h iff h − g ∈ C. Since the cone encapsulates all of the order information about G, there is a one-to-one correspondence between ordered abelian groups and groups with a strict cone C satisfying C − C = G. Hence, it is


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common practice (that will be taken here) to call (G, G+ ) an ordered abelian group (instead of (G, )). In the case that V is a vector space over F that is an ordered group with positive cone V + , being an ordered F -vector space is equivalent to V + being closed under scalar multiplication by positive elements of F . We call a cone C in a real vector space V a real cone if it is closed under scalar multiplication by positive real numbers. Every ordered group is torsion-free; this is an easy consequence of the existence of a strict cone that generates the group. Whenever we have an ordered group G and a subgroup H , we will treat H as an ordered group via the induced order, that is, H + = G+ ∩ H . Definition 2.2. Let G be an ordered abelian group. An ideal of G is a subgroup H satisfying the following: (i) (Order-)convexity: if g, h ∈ H and x ∈ G satisfies g x h then x ∈ H . (ii) (H, H + ) is itself an ordered group. (Since other conditions of an ordered group are automatic, this amounts to H being a directed subset, or H = H + − H + .) The ordered group G is said to be simple if it contains no ideals other than {0} and G. When V is an ordered vector space, note that every ideal is automatically a subspace. To see this, note that if I is an ideal and x ∈ I + , then for any scalar r ∈ [0, 1] we have 0 rx x, and thus rx ∈ I . It is then easy to see that rx ∈ I for any scalar r and x ∈ I + , and by directedness, this holds in fact for any x ∈ I . Definition 2.3. An order unit of an ordered group G is an element u ∈ G+ such that, for all g ∈ G, there exists n ∈ N such that −nu g nu. This is equivalent to saying that u generates G as an ideal. Definition 2.4. A positive functional on an ordered group G is a group homomorphism φ : G → R that is positive, meaning that if g ∈ G+ , then φ(g) 0. If u is an order unit for G then a positive functional φ is a state (with respect to u) if φ(u) = 1. Let us denote by S(G, u) the set of all states on G. When u is understood, let us write this simply as S(G). The set S(G, u) is a convex subset of the vector space of functionals on G (see [14, Proposition 6.2]). Let us denote by ∂e S(G, u) the set of all extreme points of S(G, u). In an important sense, the extreme points of S(G, u) do not depend on the choice of order unit u. Namely, if u, u are different order units then, of course, for every φ ∈ S(G, u), there is a unique real number kφ such that kφ φ ∈ S(G, u ). The map φ → kφ φ is of course a bijection, and although it may not be affine, it sends ∂e S(G, u) to ∂e S(G, u ) [14, Proposition 6.17]. The following result shows that we have order units (and therefore states) for the ordered groups that we are mostly concerned with in this paper. Proposition 2.5. Let (G, G+ ) be an ordered group. The ideals of G satisfy the ascending chain condition if and only if every ideal of G has an order unit. Proof. Note that for each element x ∈ G+ , x is an order unit in the ideal generated by x.

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⇒: Suppose that the ideals of G satisfy the ascending chain condition. Let H be an ideal of G, and let x1 ∈ H + . If x1 is not an order unit for H , then let H1 be the ideal generated by x1 , so that H1 H , and there exists x2 ∈ H + \H1 . Continuing this way, we either arrive at an order unit for H or get an infinite, strictly increasing chain of ideals. ⇐: Suppose that every ideal of G has an order unit. Let H1 ⊆ H2 ⊆ · · · be an increasing chain of ideals. Then the union H = Hi is an ideal, and therefore it has an order unit x. Therefore, x ∈ (Hi )+ for some i, from which it follows that H ⊆ Hi . 2 Definition 2.6. Let G be an ordered group. (i) G is unperforated if, for any g ∈ G and any positive integer n, if ng 0 then it must be the case that g 0. (ii) G has Riesz interpolation (or just interpolation, for short) if, for all a1 , a2 , b1 , b2 ∈ G satisfying a1 b 1 a2 b2 there exists some z ∈ G, called an interpolant, such that a1 b z 1. a2 b2 (iii) G is a dimension group if it is unperforated and has Riesz interpolation. We will use (often implicitly) some important facts proven in [12] about the ideals of a dimension group. Proposition 2.7. Let G be a dimension group. (i) [12, Theorem 5.6] The ideals of G form a distributive lattice, with operations (I, J ) → I + J and (I, J ) → I ∩ J . (ii) [12, Proposition 5.8] For ideals I1 , I2 of G, we have (I1 + I2 )+ = I1+ + I2+ . Let G be a dimension group and let I be an ideal of G. Consider the intersection of all of the kernels of positive functionals φ defined on an ideal J containing I for which φ(I ) = 0, that is KI =

ker φ: φ : J → R is positive, J is an ideal, I J and φ(I ) = 0 .

(2.1)

Evidently, KI contains I ; however, when I is not the proper intersection of two ideals, they may not be equal. Their inequality represents a sort of degeneracy in the positive cone, as is best illustrated in the case that G is a finite-dimensional real ordered vector space V that is simple (so that 0 is not a proper intersection of two ideals). In this case, the number of extreme states on V is equal to the dimension of V exactly when the closure of V + contains no non-trivial subspace (in fact, the difference between dim V and


3409

|∂e S(V )| is exactly the dimension of the largest subspace contained in V + ). The largest subspace of the closure of V + can in fact be found by taking the intersection of the kernels of the extreme states on V (or equivalently, the kernels of all positive functionals on V ); that is, it is equal to K{0} . In the non-simple case, there may be positive functionals on some ideal that do not extend as positive functionals on the entire ordered group, which is why we use positive functionals on ideals J containing I in (2.1). We define the degeneracy quotient of I as DegenI := KI /I. For an ordered group G and a field F contained in R, we may form the tensor product G ⊗Z F , into which G naturally embeds. Using the smallest (vector space) cone in G ⊗Z F that contains the image of G+ makes G ⊗Z F an ordered vector space, and the embedding G → G ⊗Z F is an order embedding precisely if G is unperforated. Proposition 2.8. Let G be an unperforated ordered group. (i) There is a one-to-one correspondence between the ideals of G and those of G ⊗Z Q given by I → I ⊗Z Q. (ii) If u ∈ G is an order unit then u ⊗ 1 is an order unit for G ⊗Z Q and S(G, u) = S(G ⊗Z Q, u ⊗ 1). (iii) If G has Riesz interpolation then so does G ⊗Z Q. Proof. (i) The proof of the last part of the statement of [6, Lemma 2.1] works here; (ii) and (iii) are easy. 2 3. Representation using functionals The main result of this section is Theorem 3.2, which describes how to represent certain dimension groups in Rn , in such a way that the ideals and the positive cone of the dimension group are described using the embedding and some combinatorial data. The next example shows that, to find such a representation, it is not enough simply to tensor a dimension group G with R. Example 3.1. Let θ ∈ R be irrational and let G = Z2 equipped with the positive cone G+ := {0} ∪ (a, b) ∈ G: a + θ b > 0 . Then (G, G+ ) is a dimension group [5]. Tensoring G with R yields a two-dimensional order real vector space, namely R2 with positive cone {0} ∪ (a, b) ∈ R2 : a + θ b > 0 . The map G → R given by (a, b) → a + θ b is a positive embedding of G into R, and is, in fact, the embedding that we will produce in Theorem 3.2. The embedding of G into R2 is undesirable

3410


because it introduces degeneracy into R2 that was not present in the original group G. In particular, the trivial ideal {0} of G is the kernel of the unique state on G, but within R2 the kernel of the unique state is non-trivial. When degeneracy is present in the group G (i.e., when DegenI = 0 for some I ), additional data (an embedding of DegenI into a real vector space) is requested by the following theorem to produce the embedding of G. Theorem 3.2. Let G be a dimension group that has finitely many ideals, finitely many extreme states on each ideal, and such that, for every ideal I , there exists an embedding ψI : DegenI → RkI (for some kI ), for which ψI (DegenI ) spans RkI . Then there exists a map φ = (φ1 , . . . , φn ) : G → Rn , a sublattice S of 2{1,...,n} and, for each S ∈ S, a subset PS> of S such that: (i) φ is one-to-one and φ(G) spans Rn . (ii) The positive cone is given by G+ =

g ∈ G: φi (x) > 0, ∀i ∈ PS> , φi (x) = 0, ∀i ∈ /S . S∈S

(iii) There is a lattice isomorphism between S and the ideals of G, given by S → IS where for S ∈ S, IS = g ∈ G: φi (g) = 0, ∀i ∈ /S . (iv) The order units of the ideal IS are exactly all g ∈ IS for which φi (g) > 0,

∀i ∈ PS> .

(v) For each S ∈ S, φ(IS ) spans (x1 , . . . , xn ) ∈ Rn : xi = 0, ∀i ∈ /S and the real cone generated by the image under φ of the order units of IS is

(x1 , . . . , xn ) ∈ Rn : xi = 0, ∀i ∈ / S, xi > 0, ∀i ∈ PS> .

(vi) For each ideal I of G which is not the proper intersection of two ideals, let S ∈ S be such that I = IS , and set DS :=

PT> ∪ S.

T ⊃S

Then |DS | = n − kI , and upon identifying RkI with (x1 , . . . , xn ): xi = 0, ∀i ∈ DS


3411

(by sending the ith coordinate to the ith nonzero coordinate), the following commutes DegenI

G/I α

ψI

RkI

=

{(x1 , . . . , xn ): xi = 0, ∀i ∈ DS }

where α(g) = (x1 , . . . , xn ) with xi = 0 if i ∈ DS and xi = φi (g) otherwise. Moreover, the number n, the lattice S, and the subsets PS> are determined (up to a permutation of the indices {1, . . . , n}) by the lattice of ideals of G, the space of positive functionals on each ideal, the restriction maps between the spaces of positive functionals, and the value of ki . The embedding φ is determined (up to a permutation of the indices {1, . . . , n} ) by the ordered group G and the embeddings ψI . Remark 3.3. This last result allows us to represent in Rn any countable dimension group with finitely many ideals and finitely many extreme states on each ideal—since in that case, the rank of DegenI is at most ℵ0 , so that we can in fact find an embedding ψI : DegenI → R. However, if ψI is allowed to be chosen in such an arbitrary manner, then the last theorem can produce different representations φ, φ : G → Rn such that for no vector space isomorphism α : Rn → Rn does the following commute φ

G = G

Rn α

φ

Rn .

In the following situations, we can pick ψI canonically, so that different representations given by Theorem 3.2 do lift to vector space isomorphisms of Rn . (i) If G is a finite-dimensional ordered vector space over R then DegenI is itself a finitedimensional vector space, so we can let ψI be a vector space isomorphism. (ii) If G is a finite-dimensional ordered vector space over Q then DegenI is itself a finitedimensional vector space over Q, so we can let ψI be the embedding DegenI → DegenI ⊗ R. (iii) More generally, if DegenI has finite rank then DegenI ⊗ R is a finite-dimensional real vector space, and again we let ψI be the embedding DegenI → DegenI ⊗ R. A proof of Theorem 3.2 comes by close examination of the states on the ideals of G. For ideals I ⊆ J , we have a restriction map r from the positive functionals on J to those on I . This restriction map sends each state on I either to zero or to a scalar multiple of a unique state on J . Modulo this adjustment by a scalar multiple, the upcoming results may be summarised as follows. • Lemma 3.4: r sends ∂e S(I ) to ∂e S(J ) ∪ {0}. • Lemma 3.5: r is one-to-one on the set of extreme states that do not map to zero.

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• Lemma 3.6 can be summarized as the existence of a pull-back state in the following diagram (where every map is a restriction map) ∃τ ∈ ∂e S(I1 + I2 )

τ1 ∈ ∂e S(I1 )

∂e S(I2 ) τ2

τ12 ∈ ∂e S(I1 ∩ I2 ) Lemma 3.4. Let G be an ordered group with Riesz interpolation and I an ideal. Suppose that G, I have order units u, v respectively. Then for τ ∈ ∂e S(G, u), τ |I is a scalar multiple (possibly 0) of some τ ∈ ∂e S(I, v). Proof. Replacing G by G ⊗ Q, by Proposition 2.8, we may assume that G is an ordered Q-vector space with Riesz interpolation. Suppose for a contradiction that τ |I = f1 + f2 where f1 , f2 are linearly independent positive functionals on I . So, there exist x, y ∈ I + such that f1 (x) < f2 (x)

and f1 (y) > f2 (y).

Let > 0 be such that f1 (x) < f2 (x) −

and f2 (y) < f1 (y) − .

By [16, Proposition I.9.1], let b ∈ G be such that min τ (x), τ (y) − < τ (b) and τ (b)
c/2. Notice that if 0 z x then z ∈ I + and so τ1 (z) = cτ2 (z). Since τ1 = τ2 , we can use [2] and [16, Theorem I.9.1] to find y ∈ G such that τ1 (y) ∈ (0, 1/2) and τ2 (y) > c/2, while τ (y) > 0 for all τ ∈ ∂e S(G, u). Likewise, we can find b ∈ G such that τ2 (b) ∈ (c/2, τ2 (y)) and τ (b)
0. Then f is a positive functional on G. To show that f is a scalar multiple of an extreme state, suppose that f =g+h where g, h are positive functionals on G, and let us show that g, h are linearly dependent. Since f |I1 ∩I2 = 0, WLOG, we have g|I1 ∩I2 = 0. Then, d1 τ1 = f |I1 = g|I1 + h|I1 , and since t1 is an extreme state, this implies that g|I1 , h|I1 are linearly dependent, so let h|I1 = Kg|I1 for some scalar K. Likewise, we see that g|I2 , h|I2 are linearly dependent, and since h|I1 ∩I2 = Kg|I1 ∩I2 and g|I1 ∩I2 = 0, we must have that h|I2 = Kg|I2 . Thus, h = Kg, so g, h are linearly dependent, as required.

2

Corollary 3.7. Let G be an ordered group with Riesz interpolation, such that every ideal I of G has an order unit v and ∂e S(I, v) is finite. For any ideal I with order unit v and any τ ∈ ∂e S(I, v), there exists some ideal I˜ with order unit w and some τ˜ ∈ ∂e S(I˜, w) such that τ is a scalar multiple of τ˜ |I , and I˜ is maximal in the sense that for any ideal J satisfying I ⊂ J I˜, if f : J → R satisfies f |I = τ then f is not a positive functional on J . Proof. Let I˜ be the sum of all ideals J such that τ extends to J (as a scalar multiple of an extreme state on J ). If τ extends to a scalar multiple of an extreme state on the ideal J then, by Lemma 3.5, this extension is unique. By Lemma 3.6, we can see that if I1 and I2 are both ideals, and τ extends to a scalar multiple of an extreme state on each of I1 and I2 then τ extends to a scalar multiple of an extreme state on I1 + I2 . Thus, we see that τ extends to a scalar multiple of an extreme state on all of I˜. Obviously, I˜ is maximal in the sense that it contains every ideal upon which τ has an extension that is a scalar multiple of an extreme state. However, if τ extends to a positive functional f on an ideal J ⊃ I then since ∂e S(J ) is finite, f can be written as a linear combination (with positive coefficients) of extreme traces on J , and it follows that one of the extreme traces appearing in the linear combination must restrict to a nonzero scalar multiple of τ . Thus, I˜ is also maximal in the sense stated. 2


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From Corollary 3.7, we that when G is a dimension group as in Theorem 3.2, we can define functionals τ1 , . . . , τk : G → R such that for every ideal I ⊆ G and every τ ∈ ∂e S(I ), there exists a unique i such that τ is a scalar multiple of τi . For each I , τi |I is positive if and only if τi is a scalar multiple of an extreme state on I . For each i, there is a maximal ideal upon which τi is positive; we denote this ideal by Iτi . In order to achieve condition (iii) of Theorem 3.2, we may need to include more functionals than τ1 , . . . , τk in the list φ1 , . . . , φn . For example, consider the case that G = R2 with the order given by G+ = (0, 0) ∪ (x, y) ∈ R2 : x > 0 . In this case, the ordered group (an ordered real vector space) is simple, and it has only one extreme state (namely, the functional x). For each ideal I of G, DegenI represents exactly the difference between the ideal I and what we would get if we used only τ1 , . . . , τk to pick out the ideal. Thus, we shall use ψI to give us more functionals which exactly recover each ideal. If an ideal I can be written as the proper intersection of two ideals J1 and J2 , and if we can write each of J1 and J2 as an intersection of kernels of some of the functionals φ1 , . . . , φn then of course we can do the same for I . This is why we only need to use the embeddings ψI for ideals I which are not the proper intersection of two ideals. The next result characterizes such ideals. Lemma 3.8. Suppose that the ideals of G satisfy the descending chain condition. Let I be a proper ideal of G. Then either I is the proper intersection of two ideals (i.e. I = J1 ∩ J2 , J1 = I = J2 ) or there exists a unique ideal MSI(I ) that properly contains I and that is minimal with respect to this condition (i.e. if I J then MSI(I ) ⊆ J ). Proof. Let MSI(I ) be the intersection of all the ideals that properly contain I . If MSI(I ) = I then it is clearly the minimum ideal that properly contains I . Otherwise, MSI(I ) = I and since the ideals satisfy the descending chain condition, I is the intersection of finitely many ideals which properly contain I . This implies that I is the proper intersection of two ideals. 2 Whenever we have the set up of the last lemma, and the ideal I is not the proper intersection of two ideals, we will continue to use the notation MSI(I ). Lemma 3.9. Suppose that the ideals of G satisfy the descending chain condition. Let I be a proper ideal of G that is not the proper intersection of two ideals. Let f : G → R be a functional satisfying f (I ) = 0 and f (MSI(I )) = 0. If J ⊂ G is another ideal satisfying f (J ) = 0 then J ⊆ I. Proof. If J I then I I + J , and so MSI(I ) ⊆ I + J . However, if f (J ) = 0 then f (I + J ) = 0, in contradiction to f (MSI(I )) = 0. 2 Lemma 3.10. Suppose that the ideals of G satisfy the descending chain condition. Let J be a proper ideal of G that is not the proper intersection of two ideals. Let f : G → R be a functional

3416


such that f (J ) = 0 and f (MSI(J )) = 0. Let I be an ideal such that f (I ) = 0 but f (K) = 0 for all K I . Then: (i) I has a unique maximal subideal, namely I ∩ J . (ii) Either I = MSI(J ) or we have I ∩ J J and I + J = MSI(J ). (iii) If I = MSI(J ) then every proper subideal of MSI(J ) is a subideal of either I or J . Proof. (i) Let K I . Then f (K) = 0 so by the last lemma, K ⊆ J . Hence every ideal of I is contained in I ∩ J . Moreover, I ∩ J = I , since f vanishes on I ∩ J but not on I . (ii) We have J I + J and so MSI(J ) ⊆ I + J . Consider the case that I = MSI(J ). If I ∩ J = J then J ⊆ I so MSI(J ) I . But then f does not vanish on the proper ideal MSI(J ) of I , a contradiction, so that I ∩ J J . If I + J = MSI(J ) then I MSI(J ) and so I ∩ MSI(J ) I . It follows from (i) that I ∩ MSI(J ) = I ∩ J , and by using distributivity, J = MSI(J ) ∩ J + (I ∩ J ) = MSI(J ) ∩ J + MSI(J ) ∩ I = MSI(J ) ∩ (J + I ) = MSI(J ), a contradiction. (iii) Suppose that I = MSI(J ) and K is a proper subideal of MSI(J ) such that K I and K J . We have first that J J + K and so J + K = MSI(J ). Secondly, K ∩ I I and so I ∩ K ⊆ I ∩ J by (i). Using distributivity, we have I ∩ J = (I ∩ J ) + (I ∩ K) = I ∩ (J + K) = I ∩ MSI(J ), and by (ii) we know that I ∩ MSI(J ) = I ∩ (I + J ) = I , which yields I ∩ J = I , a contradiction to (ii). 2 We have already described how to get functionals τ1 , . . . , τk that induce every extreme state on every ideal. Let us pick out the ideals of G that are not the proper intersection of two ideals, and label them J 1 , . . . , J . For s = 1, . . . , , we use ψJs : DegenJs → Ras (where as = kJs ) to define functionals fs,1 , . . . , fs,as from G to R. Therefore, we have Js =

ker τj : τj |MSI(Js ) 0 and τj (Js ) = 0 ∩ ker fs,1 ∩ · · · ∩ ker fs,as .

Proposition 3.11. The functionals τ1 , . . . , τk , f1,1 , . . . , f1,a1 , . . . , f ,1 , . . . , f ,a are linearly independent over R.

(3.2)


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Proof. Suppose that c1 τ1 + · · · + ck τk + d1,1 f1,1 + · · · + d ,a f ,a = 0

(3.3)

for some scalars c1 , . . . , ck , d1,1 , . . . , d ,a ∈ R. We will show, by induction, that for the ideal I ⊆ G, we have τi |I = 0

⇒

ci = 0,

fs,t |I = 0

⇒

ds,t = 0.

and

Since τi |G = 0 and fs,t |G = 0 for all i, s, t, this inductive argument will prove linear independence. We need to be careful about the order in which we enumerate the ideals for the induction process. We want to enumerate them such that when doing the step for an ideal I , we may assume the induction hypothesis for all proper subideals of I ; additionally, if I = Js and another ideal J is the proper intersection of two ideals and is maximal in MSI(Js ), then we may assume that the induction hypothesis holds for J . Some explanation is required for why this is possible. Let ≺ denote the relation on the ideals of G given by I ≺ Js if I is the proper intersection of two ideals and I is a maximal subideal of MSI(Js ); ≺ is clearly a (strict) pre-order. Let ≺ denote the (strict) pre-order relation generated by ≺ and . We want to enumerate in a non-decreasing order with respect to ≺, and to show that this is possible, we need the following. Claim. The pre-order ≺ is antisymmetric, and therefore a partial order. Proof. Notice that if I ≺ J and J K then since MSI(J ) is the minimum superideal of J , we must have MSI(J ) ⊆ K. Since I ≺ J implies that I MSI(J ), we must have I K. It follows that if I ≺ J then either I J or I ⊆ K ≺ J for some ideal K. In either case, it cannot happen that I = J . Now, if I J and J I then either I = J or else I ≺ J ≺ I , in which case I ≺ I , which contradicts what was just shown. 2 Let us now do the inductive step, where we may assume that for any ideal J ≺ I , the inductive hypothesis holds for J . For each i, if τi |I = 0 then there is nothing to prove; if τi |K = 0 for some K I then ci = 0 by the induction hypothesis. The only case of interest, then, is that τi |I = 0 but τi |K = 0 for all K I . Recall that Iτi is the maximal ideal upon which τi is positive. If I Iτi then I ∩ Iτi I and so τi |Iτi ∩I = 0 by induction. Hence, there exists g : G → R such that g|I = 0 but g|Iτi = τi |Iτi . Then Iτi = I + Iτi but g 0 on I + Iτi (since g = 0 on I and g 0 on Iτi ), which contradicts Corollary 3.7. Hence, we must have I ⊆ Iτi , and thus τi |I is a (nonzero) scalar multiple of some τ ∈ ∂e S(I ). Likewise, we have a trichotomy for each fs,t : the only case where there is something to prove is the case that fs,t |I = 0 but fs,t |K = 0 for all K I . For this case, we look to Lemma 3.10:

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by (i), I ∩ Js is the unique maximal subideal of I . By (ii), we can break our analysis into two cases, and proceed to show in each case that ds,t = 0 for all t = 1, . . . , as . Case 1. I = MSI(Js ). In this case, we cannot have that fs ,t |I = 0 but fs ,t |K = 0 for all K I , for some s = s. Certainly, if we did have this, then Js ∩ I would be the unique maximal subideal of I , and so Js = Js ∩ I , expressing Js as the proper intersection of ideals. Restricting (3.3) to {ker τi : τi (Js ) = 0} gives (ds,1 fs,1 + · · · + ds,as fs,as )| {ker τi : τi |MSI(Js ) 0 and τi (Ji )=0} = 0. If the scalars ds,t are not all nonzero, then this amounts to a linear dependence of the components of ψJs . However, since the range of ψJs spans Ras , the components are linearly dependent, so ds,t = 0 for all t. Case 2. I = MSI(Js ). In this case, by Lemma 3.10(iii), it follows that either Js ≺ I or else I = Js for some s . Let us first handle the case that I = Js for some s . Assume (by reordering indices) that τ1 , . . . , τb are nonzero on Js , τb+1 , . . . , τb are nonzero on Js , τ1 , . . . , τb are all zero on Js ∩ Js (and therefore on any proper subideal of either Js or Js ), and for i = b + 1, . . . , k, either τi |MSI(Js ) = 0 or ci = 0. Let us first show that τ1 |Js = 0. As argued above, since τ1 |K = 0 for all K Js , we have τ1 |Js 0. Likewise, we have τ1 |Js 0. Consequently, τ1 |MSI(Js ) 0, yet it is nonzero, so it is a scalar multiple of an extreme state. We may define f : MSI(Js ) → R by f |Js = 0 and f |Js = τ1 |Js . Likewise, we may define g : MSI(Js ) → R by g|Js = τ1 |Js and g|Js = 0. Since MSI(Js ) = Js + Js , we have f + g = τ1 |MSI(Js ) . If τ1 |Js = 0 then this would contradict the fact that τ1 |MSI(Js ) is a scalar multiple of an extreme state. Therefore, τ1 |Js = 0. Likewise, τi |Js = 0 for i = 1, . . . , b and τi |Js = 0 for i = b + 1, . . . , b . Eq. (3.3), restricted to MSI(Js ), becomes (c1 τ1 + · · · + cb τb + ds ,1 fs ,1 + · · · + ds ,as fs ,as )|MSI(Js ) = −(cb+1 τb+1 + · · · + cb τb + ds,1 fs,1 + · · · + ds,as fs,as )|MSI(Js ) . Let g denote the functional on Js that is given by each side of the above equation. Notice that g|Js = 0 from the right-hand side and g|Js = 0 from the left-hand side. Therefore, g = 0. From this, as in Case 1, we can show that ds,t = 0 for all t. Going back to the case that I is not of the form Js , so that the induction hypothesis holds for Js , in this situation, we can apply the same argument to MSI(Js ) to show that ds,t = 0 for all t. Finally, knowing that ds,t = 0 for all t, when we now restrict (3.3) to I , we get (c1 τ1 + · · · + ck τk )|I = 0,


3419

where for each i, either τi |I = 0 or ci = 0 or τi |I is a scalar multiple of an extreme state on I . Since the extreme states of I are linearly independent, it follows that ci = 0 whenever τi |I = 0. 2 We are now set to prove Theorem 3.2. Proof of Theorem 3.2. This is achieved by relabelling the functionals τ1 , . . . , τk , f1,1 , . . . , f1,a1 , . . . , f ,1 , . . . , f ,a as φ1 , . . . , φn . We already know that φ is one-to-one, and φ1 , . . . , φn are linearly independent. For each ideal I of G, set SI := i = 1, . . . , n: φi (I ) = 0 , so that, by (3.2), I = ISI . Let S consist of each SI given by an ideal I . Set

PSI := {i = 1, . . . , n: φi |I is a positive functional}.

Then PSI ⊇ SIc , and PS>I = PSI ∩ SI consists of each τi that is a nonzero positive functional on I . In light of the linear independence of φ1 , . . . , φn and the choice of τi ’s, this means that the extreme states on I are, up to positive scalar multiples, exactly the φi for which i ∈ PS>I , and so (iv) follows by [3, Theorem 1.4]. Item (ii) follows from (iv), since every positive element of G is an order unit in the ideal it generates. To see (v), note that the first part is a consequence of the fact that {φ1 , . . . , φn } is linearly independent, and the second part follows from the fact that the order units of the ideal I separate the extreme states on I . Finally, for two ideals I and J of G, we have SI +J = i: φi (I + J ) = 0 = i: φi (I ) = 0 or φi (J ) = 0 = SI ∪ SJ , and SI ∩J = i: φi (I ∩ J ) = 0 ⊆ SI ∩ SJ , but also, since I ∩ J = ISI ∩ ISJ (and by (v)), we see that we must have SI ∩J ⊇ SI ∩ SJ . This shows that the map I → SI is a lattice isomorphism, and that S is a lattice. 2 4. Necessary and sufficient conditions for interpolation Theorem 3.2 shows that, if a dimension group has finitely many ideals and finitely many extreme states on each ideal, and if, for each ideal I , DegenI has countable rank, then the dimension group is isomorphic to a subgroup G of Rn , and the order can be described using some combinatorial data, which we shall give a name for now.

3420


Definition 4.1. A standard triple is a triple (S, (PS> )S∈S , G) where S is a sublattice of 2{1,...,n} containing ∅ and {1, . . . , n}, PS> is a subset of S, and such that {g ∈ G: gi = 0, ∀i ∈ / S} spans (x1 , . . . , xn ) ∈ Rn : xi = 0, ∀i ∈ /S for each S ∈ S. Associated to the standard triple (S, (PS> )S∈S , G) is the positive cone G+ :=

g ∈ G: gi > 0, ∀i ∈ PS> , gi = 0, ∀i ∈ /S ,

S∈S

making G an ordered group. Note that the cone on (S, (PS> )S∈S , Rn ) is x ∈ Rn : xi > 0, ∀i ∈ PS> , xi = 0, ∀i ∈ /S , S∈S

which is the cone generated by G+ . Also, with this cone on Rn , the inclusion G ⊆ Rn is an order embedding. In this section, we will describe necessary and sufficient conditions on a standard triple (S, (PS> )S∈S , G) in order for (G, G+ ) to have Riesz interpolation. The main result of this section is the following. Theorem 4.2. Let (S, (PS> )S∈S , G) be a standard triple. Then (G, G+ ) is an unperforated ordered group, the ideals of which are / S} IS := {g ∈ G: gi = 0, ∀i ∈ for S ∈ S. (G, G+ ) has Riesz interpolation if and only if the following conditions hold.

(i) Letting PS = PS> ∪ S c for all S ∈ S, we have, for S1 , S2 ∈ S,

PS1 ∪S2 = PS1 ∩ PS2 . (ii) For S1 , S2 ∈ S, if S1 S2 then PS>2 S1 . (iii) For every pair S1 , S2 ∈ S, we have IS1 + IS2 = IS1 ∪S2 . (iv) For every pair S1 , S2 ∈ S for which S1 is a maximal proper subset of S2 , we either have:


3421

(a) If S2 \S1 = {i1 , . . . , i } then (xi1 , . . . , xi ): (x1 , . . . , xn ) ∈ IS2 is dense in R . / PT> or T \{k} ∈ S. (b) S2 = S1 {k}, and for any T ∈ S which contains S2 , either k ∈ Proof of the first part. Let us first prove that (G, G+ ) is an unperforated ordered group, the ideals of which are as described above. We can then discuss separately the necessity and sufficiency of conditions (i)–(iv). That G is an unperforated ordered group is clear from the definition of G+ . It is also clear from the definition of G+ that for S ∈ S the set IS is convex. To see that IS is directed, let g (1) , . . . , g (k) ∈ IS be an R-basis for {(x1 , . . . , xn ) ∈ Rn : xi = 0, ∀i ∈ / S}, and pick h(1) , h(2) ∈ IS . Let g (1) , . . . , g (k) denote the Euclidean norms of (1) / S} such that, if g , . . . , g (k) . Then we can find some c ∈ {(x1 , . . . , xn ) ∈ Rn : xi = 0, ∀i ∈ (t) x − c < g (1) + · · · + g (k) , then xi > hi for t = 1, 2 and for all i ∈ PS> . The element c can be written c = c1 g (1) + · · · + ck g (k) for c1 , . . . , ck ∈ R. Taking nearest integers c1 , . . . , ck , we obtain an element h := c1 g (1) + · · · + ck g (k) ∈ IS such that hi − hi > 0 for t = 1, 2 and for all i ∈ PS> . Therefore h − h(1) , h − h(2) ∈ G+ , and so IS is directed. Finally, let us pick an arbitrary convex and directed subgroup K of G and show that it equals IS for some S ∈ S. / S. Then certainly Let S be the largest set in S such that, for all k ∈ K, ki = 0 for all i ∈ K ⊆ IS . (i) For each i ∈ S, there is some k (i) ∈ K such that ki = 0. Then we can find some integer combination k of {k (i) : i ∈ S} such that ki = 0 for i ∈ S. Then k ∈ K because K is a group, and because K is directed, there is some k + ∈ K ∩ G+ such that k + k . By considering the definition of G+ , we see that we must have ki+ > 0 for all i ∈ PS> . But then, because K is convex, we see that IS ⊆ K as well. Therefore K = IS as required. 2 (t)

Now let us consider the necessity of the conditions (i)–(iv). The necessity of condition (iii) follows from [14, Proposition 2.4], which says that, in a dimension group, the sum of two ideals is also an ideal. The necessity of condition (i) then follows easily, because IS+1 ∪S2 = (IS1 + IS2 )+ = IS+1 + IS+2 . To show the necessity of condition (ii), suppose for a contradiction that PS>2 ⊆ S1 . By (i), this implies that PS>1 = PS>2 , so that if g is an order unit of IS1 then g is an order unit of IS2 . This contradicts the hypothesis that S1 S2 . The next proposition, combined with the fact that the quotient of a dimension group by an ideal is once again a dimension group, shows that if condition (iv)(a) does not hold then IS2 /IS1 must be cyclic.

3422


Proposition 4.3. Let G be a simple dimension group with order unit and finitely many extreme states τ1 , . . . , τk . Then either (G, G+ ) ∼ = (Z, N) or τ1 (g), . . . , τk (g) : g ∈ G is dense in Rk . Proof. This is simply the combination of [14, Proposition 14.3] and [15, Theorem 4.8].

2

Now knowing that if condition (iv)(a) fails then IS2 /IS1 is cyclic, the necessity of condition (iv)(b) comes from the following dichotomy, which generalizes [14, Proposition 17.4]. Proposition 4.4. Let G be a dimension group with an order unit and suppose that I is a cyclic ideal of G. If the unique state τ on I extends to a positive functional on G then I is a direct summand of G, as an ordered group; that is, G = I ⊕ J for some ideal J . Proof. By Lemmas 3.4 and 3.5, we see that there is a unique extreme state τ on G, the restriction of which is a positive scalar multiple of τ , and every other extreme state on G is zero on I . Let h be the positive generator of I , and let u be an order unit of G. Then, by considering separately the case that τ (u) is rational or irrational, we see that there exist m > 0 and n 0 such that the element g := −mu + nh satisfies τ (g) ∈ [0, τ (h)/2). Let us show that 0 h . g h−g Certainly, we already know that 0 h. The other inequalities amount to showing that both h − g and h − 2g are non-negative. For any state s on G, we have that either s = τ or s(I ) = 0. In the first case, τ (h − g) > τ (h) − τ (h)/2 > 0, and likewise, τ (h − 2g) > τ (h) − 2τ (h)/2 > 0. In the second case, s(h − g) = s(−g) = ms(u) > 0, and likewise, s(h − 2g) > 0. By [3, Theorem 1.4], we see that h − g and h − 2g are in fact order units in G. Since G has Riesz interpolation, there must be some interpolant z satisfying 0 h z . g h−g However, since h is the generator of the ideal I and 0 z h, we must have either z = 0 or z = h. Since g z h − g, in either case, it follows that g 0 and τ (g) = 0. Let J be the ideal


3423

generated by −g; by the definition of g, we see that G is the ideal generated by I and J . Yet, / J and thus I ∩ J = {0}. Hence, G = I ⊕ J as required. 2 since τ (−g) = 0, it follows that h ∈ Quotients of dimension groups by ideals are once again dimension groups, so the result of Proposition 4.4 applies in particular to the cyclic subideal IS{k} /IS of IT /IS . To say that the / PT> . To say that IS{k} /IS is a unique state on IS{k} /IS does not extend to IT /IS means that k ∈ direct summand in IT /IS means that T \{k} ∈ S as well. Therefore all of the conditions (i)–(iv) are necessary for G to have Riesz interpolation. Let us now prove that they are also sufficient. Proof of sufficiency of (i)–(iv). To show that (G, G+ ) has Riesz interpolation, choose elements a (1) , a (2) , b(1) , b(2) ∈ G satisfying a (1) b(1) (2) . (2) a b (t)

(t)

(t)

(t)

Say that a (t) = (a1 , . . . , an ) and b(t) = (b1 , . . . , bn ). For each pair s, t ∈ {1, 2}, define Ts,t := (s)

(s) (t) S ∈ S: ai = bi , ∀i ∈ /S .

(t)

Clearly, Ts,t ∈ S and ai = bi for all i ∈ / Ts,t . (t) (s) Furthermore, we must have bi − ai > 0 for all i ∈ PT>s,t . Indeed, to see this, suppose other-

/S wise. Then, because b(t) − a (s) ∈ G+ , there is some set S ∈ S such that ai = bi for all i ∈ (t) (s) (t) (s) > (so that Ts,t ⊆ S), and bi − ai > 0 for all i ∈ PS (so that Ts,t S). But, since bi − ai = 0 for all i ∈ S\Ts,t , we must have PS> ⊆ Ts,t , contradicting condition (ii). Next let us show that (s)

(t)

T1,1 ∪ T2,2 = T1,2 ∪ T2,1 .

(4.1)

Indeed, if we let (b(1) + b(2) − a (1) − a (2) )i denote the ith entry of b(1) + b(2) − a (1) − a (2) and / T , then both T denote the smallest set in S such that (b(1) + b(2) − a (1) − a (2) )i = 0 for all i ∈ sides of (4.1) are equal to T . By symmetry, it suffices to show that T1,1 ∪ T2,2 = T . (1) (1) (2) (2) / T1,1 and bi − ai = 0 for all i ∈ / T2,2 , so (b(1) + b(2) − We have bi − ai = 0 for all i ∈ (1) (1) (2) (2) a (1) − a (2) )i = (bi − ai ) + (bi − ai ) = 0 for all i ∈ / T1,1 ∪ T2,2 . Moreover, T1,1 ∪ T2,2 ∈ S, so by the minimality of T , we must have T ⊆ T1,1 ∪ T2,2 . (s) (s) >, Suppose for a contradiction that T T1,1 ∪ T2,2 . Since bi − ai > 0 for i ∈ Ps,s (1) b + b(2) − a (1) − a (2) i > 0

c ) ∪ (P > ∩ T c ) ∪ (P > ∩ P > ). But this set is exactly (P for i ∈ (PT>1,1 ∩ T2,2 T2,2 T1,1 T2,2 T1,1 ∩ PT2,2 )\ 1,1

c ∩ T c ). By condition (i), (P (1) + b(2) − a (1) − a (2) ) > 0 (T1,1 i T1,1 ∩ PT2,2 ) = PT1,1 ∪T2,2 , so (b 2,2

c ∩ Tc ) = P> (1) + b(2) − a (1) − a (2) ) = 0 for all i ∈ / T, for all i ∈ PT1,1 ∪T2,2 \(T1,1 i T1,1 ∪T2,2 . But (b 2,2 > which means that PT1,1 ∪T2,2 ⊆ T , and this contradicts condition (ii). Therefore T = T1,1 ∪ T2,2 .

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For i ∈ / (T1,1 ∩ T1,2 ) ∪ (T2,1 ∩ T2,2 ), this means that for some t, t we have (1)

(t)

ai

= bi

(t )

(2)

(2)

and ai

(t )

= bi .

If t = t , then ai = bi = bi = ai . If t = t , then i ∈ (T1,t ∪ T2,t )c , which, by (4.1), is equal to (T1,t ∪ T2,t )c . Then ei ther ai(1) = bi(t ) or ai(2) = bi(t) ; in either case, we have ai(1) = ai(2) . Therefore a (1) − a (2) ∈ I(T1,1 ∩T1,2 )∪(T2,1 ∩T2,2 ) . By condition (iii), there exist g (1) ∈ IT1,1 ∩T1,2 and g (2) ∈ IT2,1 ∩T2,2 such that (1)

(t)

a (1) − a (2) = g (2) − g (1) . Set c := a (1) + g (1) = a (2) + g (2) , and let ci denote the ith entry of c. The ith entry of g (1) is 0 for i ∈ / T1,1 ∩ T1,2 , so (1)

ci = ai (1)

(t)

(2)

(2)

(t)

whenever ai = bi for some t. Likewise, we have ci = ai whenever ai = bi for some t. Let R denote the set T1,1 ∩ T1,2 ∩ T2,1 ∩ T2,2 ∈ S. Suppose that we are in the situation of condition (iv)(b), and R plays the role of T ; that is, there exist a set S and index k such that S {k} ∈ S, S {k} ⊆ R, and IS{k} /IS is cyclic. Suppose further that k ∈ PR> . Then, by condition (iv)(b), R\{k} is also in S. We can repeat this procedure now with the set R2 := R\{k}. If there exist S2 and k2 ∈ / S2 such that S2 , S2 {k2 } ∈ S, S2 {k2 } ⊆ R2 , IS2 {k2 } /IS2 is cyclic, and k2 ∈ PR>2 , then R3 := R2 \{k2 } is also in S. Continuing in this fashion, we will eventually arrive at a set Rˆ ∈ S such that, if S, S {k} fall ˆ then k ∈ . into condition (iv)(b) and S {k} ⊆ R, / P> Rˆ = {i1 , . . . , i } Using an induction argument and condition (iv)(a), we can show that, if P > Rˆ then (gi1 , . . . , gi ): g ∈ IRˆ is dense in R . Note that for i ∈ P > , we have ˆ R

ai(1) bi(1) (2) < (2) . ai bi Therefore we can find m ∈ IRˆ such that the element z := c + m satisfies (1) (2) (1) (2) max ai , ai < zi < min bi , bi for i ∈ P > . By our choice of c, we have ˆ R


3425

(1) (2) zi = max ai , ai for all i satisfying (1) (2) (1) (2) max ai , ai = min bi , bi

ˆ or i ∈ R\R.

Therefore, we see that for t = 1, 2 we have z − a (t) ∈ I +ˆ and b(t) − z ∈ IR+ . Therefore R

b(1) a (1) z (2) , (2) a b as required.

2

5. Classification of finite-dimensional ordered real vector spaces with Riesz interpolation Certain of the conditions in Theorem 4.2 are automatically satisfied in the case that the group G is all of Rn . We have eliminated these conditions to give the following corollary. Corollary 5.1. Let (S, (PS )S∈S , Rn ) be a standard triple and use V + to denote the associated positive cone. Then (Rn , V + ) is an ordered real vector space, and it has Riesz interpolation if and only if the following two conditions hold:

(i) Letting PS = PS> ∪ S c for all S ∈ S, we have, for S1 , S2 ∈ S,

PS1 ∪S2 = PS1 ∩ PS2 ;

and

(ii) For S1 , S2 ∈ S, if S1 S2 then PS>2 S1 . By Theorem 3.2 and the following remark, every finite-dimensional ordered real vector space with interpolation occurs as one in the corollary just described, where (S, (PS> )S∈S ) is determined up to permutation of the indices 1, . . . , n. This yields the following classification.

Corollary 5.2. For n ∈ N, let Dn denote the set of all pairs (S, (PS )S∈S ) where 1. S is a sublattice of 2{1,...,n} that contains ∅ and {1, . . . , n}; 2. PS is a subset of {1, . . . , n} that properly contains S c ; 3. For S1 , S2 ∈ S,

PS1 ∪S2 = PS1 ∩ PS2 ; 4. For S1 , S2 ∈ S, if S1 S2 then PS>2 S1 .

and

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Define the equivalence relation ≡ on Dn by (S, (PS )S∈S ) ≡ (S , (QS )S∈S ) if there exists a permutation σ on {1, . . . , n} such that S = σ (S): S ∈ S , and for each S ∈ S, Qσ (S) = σ PS . Then the (ordered real vector space) isomorphism classes of n-dimensional ordered real vector spaces with Riesz interpolation are in one-to-one correspondence with the equivalence classes in Dn . Remark 5.3. If V , V are two finite-dimensional ordered real vector spaces corresponding to two elements d, d of Dn , and V ∼ = V then the isomorphism gives rise to a permutation σ that induces the equivalence d ≡ d . While a permutation σ that induces an equivalence d ≡ d lifts to an isomorphism V → V , this isomorphism need not be unique since there is choice in the maps DegenIS → DegenIσ (S) . 6. Classification of finite rank dimension groups In light of Theorem 4.2, we see that for any dimension group G satisfying the hypotheses of Theorem 3.2 and any representation φ : G → Rn given by that theorem, if V + denotes the real cone in Rn generated by φ(G+ ) (that is, V + is closed under addition and multiplication by positive scalars) then (Rn , V + ) has Riesz interpolation. In particular, we have the following. Corollary 6.1. Let G be a dimension group that has finitely many ideals, finitely many extreme states on each ideal, and such that, for every ideal I , the rank of DegenI is at most 2ℵ0 . Then G arises as a subgroup of an ordered real vector space with interpolation (with the induced ordering). In fact, the ordered real vector space can be chosen to have finite dimension. If DegenI has finite rank for every ideal I (in particular, if G has finite rank) then, according to the remark following Theorem 3.2, we can find a canonical representation, i.e. we can impose a restriction on the representation φ that ensures that two embeddings φ and φ of G into Rn give rise to an isomorphism of Rn (in fact, it is an isomorphism of ordered real vector spaces, when we equip Rn with the cone generated by φ(G+ ) and that generated by φ (G+ )). Here are the conditions on the embedding φ : G → Rn that make it canonical; throughout, we let V + denote the real cone generated by φ(G+ ), and use V to denote the ordered real vector space Rn , the positive cone of which is V + . (i) The ordered real vector space (V , V + ) has Riesz interpolation; (ii) The ideals of G correspond to the ideals of V via the map I → span φ(I ); and (iii) For every pair of ideals I and J , we have span φ(I + J ) = span φ(I ) + span φ(J ). (iv) For each ideal I of G, the rank of DegenI equals the dimension of Degenφ(I ) .


3427

A consequence of these conditions (which is explicit in the proof of Theorem 3.2) is that the positive functionals on each ideal I correspond exactly to the positive functionals on φ(I ), so that if we pick an order unit for I and use its image as an order unit for φ(I ) then this produces an identification of the state space of I and of φ(I ). Corollary 6.2. Let C denote the set of all standard triples (S, (PS> )S∈S , G) that satisfy conditions (i)–(iv) of Theorem 4.2 as well as: (v) For every S ∈ S, letting {i1 , . . . , ik } = i = 1, . . . , n: i ∈ / S and i ∈ / PT> , ∀T ⊃ S , then the rank of (xi1 , . . . , xik ): (x1 , . . . , xn ) ∈ G and xi = 0, ∀i ∈ / {i1 , . . . , ik } ∪ S (as an abelian group, i.e., regardless of the embedding into Rk ) is equal to k. Define the equivalence relation ≡ on C by (S, (PS> )S∈S , G) ≡ (S , (Q> S )S∈S , H ) if S and S are sublattices of 2{1,...,n} for the same n, and there exists a permutation σ on {1, . . . , n} and a vector space isomorphism φ : Rn → Rn satisfying the following.

(i) S = {σ (S): S ∈ S}; > (ii) For each S ∈ S, Q> σ (S) = σ (PS ); and (iii) For each S ∈ S, φ (x1 , . . . , xn ) ∈ Rn : xi = 0, ∀i ∈ / S and xi > 0, ∀i ∈ PS> = (x1 , . . . , xn ) ∈ Rn : xi = 0, ∀i ∈ / σ (S) and xi > 0, ∀i ∈ σ PS> ;

and

(iv) φ(G) = H . Then the equivalence classes in C are in one-to-one correspondence with the (ordered group) isomorphism classes of dimension groups which have finitely many ideals, finitely many extreme traces on each ideal, and for which DegenI has finite rank for each I . Remark 6.3. The classification of finite rank dimension groups follows: these correspond exactly to the equivalence classes in C for which the group G has finite rank. Proof of Corollary 6.2. This is an immediate consequence of Theorems 3.2 and 4.2, and the remark following Theorem 3.2. 2 Corollary 6.4. Let (G, G+ ) be a finite rank ordered group that is unperforated. Then G is a dimension group if and only if G ⊗ Q is a dimension group and, for every triple I, J, K of ideals of G, such that I ⊂ J ⊂ K and J /I is cyclic, either the unique state on J /I does not extend to K/I or else J /I is a direct summand in K/I .

3428


Question 6.5. Although our methods only allow us to prove Corollaries 6.1 and 6.4 for dimension groups with finite-rank type restrictions, there is no obvious reason why these results would not hold for general dimension groups (the generalization of Corollary 6.1 would not use a finitedimensional ordered real vector space). This leads one to ask: do these results of them hold for general dimension groups? Acknowledgments The research presented here grew out of meetings at the Operator Algebras seminar at the Fields Institute in Toronto. The authors would like to thank this institute for its hospitality during the course of this research. The authors would also like to thank Chris Phillips for helpful comments. References [1] J. Cuntz, W. Krieger, A class of C ∗ -algebras and topological Markov chains, Invent. Math. 56 (1980) 251–268. [2] D.A. Edwards, Séparation des fonctions réelles définies sur un simplexe de Choquet, C. R. Acad. Sci. Paris 261 (1965) 2798–2800. [3] E.G. Effros, D.E. Handelman, C.L. Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (1980) 385–407. [4] E.G. Effros, C.L. Shen, Dimension groups and finite difference equations, J. Operator Theory 2 (1979) 215–231. [5] E.G. Effros, C.L. Shen, Approximately finite C ∗ -algebras and continued fractions, Indiana Univ. Math. J. 29 (1980) 191–204. [6] E.G. Effros, C.L. Shen, The geometry of finite rank dimension groups, Illinois J. Math. 25 (1981) 27–38. [7] G.A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976) 29–44. [8] G.A. Elliott, On the classification of C ∗ -algebras of real rank zero, J. Reine Angew. Math. 443 (1993) 179–219. [9] G.A. Elliott, A classification of certain simple C ∗ -algebras. II, J. Ramanujan Math. Soc. 12 (1997) 97–134. [10] L. Farhane, Coding of the dimension group, Adv. Math. 206 (2006) 455–465. [11] L. Fuchs, On partially ordered vector spaces with the Riesz interpolation property, Publ. Math. Debrecen 12 (1965) 335–343. [12] L. Fuchs, Riesz groups, Ann. Sc. Norm. Super. Pisa (3) 19 (1965) 1–34. [13] L. Fuchs, Riesz Vector Spaces and Riesz Algebras, Queen’s Papers in Pure and Appl. Math., vol. 1, Queen’s University, Kingston, ON, 1966. [14] K.R. Goodearl, Partially Ordered Abelian Groups with Interpolation, Math. Surveys Monogr., vol. 20, Amer. Math. Soc., Providence, RI, 1986. [15] K.R. Goodearl, D.E. Handelman, Metric completions of partially ordered abelian groups, Indiana Univ. Math. J. 29 (1980) 861–895. [16] K.R. Goodearl, D.E. Handelman, J.W. Lawrence, Affine representations of Grothendieck groups and applications to Rickart C ∗ -algebras and ℵ0 -continuous regular rings, Mem. Amer. Math. Soc. 26 (1980) vii+163. [17] W. Krieger, On dimension functions and topological Markov chains, Invent. Math. 56 (1980) 239–250. [18] J. Villadsen, The range of the Elliott invariant of the simple AH-algebras with slow dimension growth, K-Theory 15 (1998) 1–12.


Broken translation invariance in quasifree fermionic correlations out of equilibrium Walter H. Aschbacher Ecole Polytechnique, Centre de Mathématiques Appliquées, UMR CNRS – 7641, 91128 Palaiseau Cedex, France Received 2 July 2010; accepted 23 February 2011

Communicated by L. Gross

Abstract Using the C ∗ algebraic scattering approach to study quasifree fermionic systems out of equilibrium in quantum statistical mechanics, we construct the nonequilibrium steady state in the isotropic XY chain whose translation invariance has been broken by a local magnetization and analyze the asymptotic behavior of the expectation value for a class of spatial correlation observables in this state. The effect of the breaking of translation invariance is twofold. Mathematically, the finite rank perturbation not only regularizes the scalar symbol of the invertible Toeplitz operator generating the leading order exponential decay but also gives rise to an additional trace class Hankel operator in the correlation determinant. Physically, in its decay rate, the nonequilibrium steady state exhibits a left mover–right mover structure affected by the scattering at the impurity. © 2011 Elsevier Inc. All rights reserved. Keywords: Nonequilibrium quantum statistical mechanics; Quasifree fermions; C ∗ algebraic scattering; Toeplitz theory

1. Introduction In the mathematical study of open quantum systems, the role played by quasifree fermionic systems is an important one. Within the framework of algebraic quantum statistical mechanics, they not only allow for a powerful description by means of scattering theory on the one-particle Hilbert space over which the fermionic algebra of observables is built, being thus ideally suited for rigorous analysis on many levels, but they also represent a class of systems which are indeed realized in nature, see, for example, Culvahouse et al. [16], D’Iorio et al. [17], and Sologubenko E-mail address: [email protected]. 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.02.021

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W.H. Aschbacher / Journal of Functional Analysis 260 (2011) 3429–3456

et al. [24]. A special instance of this class is the finite XY spin chain introduced by Lieb et al. [20] and extended to the infinite two-sided discrete line by Araki [5] in the framework of C ∗ dynamical systems. As a matter of fact, this spin model can be mapped, in some precise sense, onto a gas of free fermions with the help of the Araki–Jordan–Wigner transformation. In order to study the effect of the breaking of translation invariance in this system, we choose the physically interesting and computationally convenient emptiness formation correlation observable. The socalled emptiness formation probability (EFP), i.e. the expectation value of this observable in a given state, describes, in the spin picture, the probability that all spins in a string of a given length point downwards. However, we would like to underline that the analysis is not limited to this observable but can rather be carried out for a broad class of spatial correlations. The asymptotic behavior of the EFP in the XY chain for large string length has already been analyzed for the cases where the state is a ground state or a thermal equilibrium state at positive temperature. In both cases, the EFP can be written as the determinant of the section of a Toeplitz operator with scalar symbol. Since the higher order asymptotics of a Toeplitz determinant is highly sensitive to the regularity of the symbol of the Toeplitz operator, the asymptotic behavior of the ground state EFP is qualitatively different in the so-called critical and noncritical regimes corresponding to certain values of the anisotropy and the exterior magnetic field of the XY chain, i.e., in (19) below, the parameters γ and λ, respectively. It has been found that the EFP decays like a Gaussian in one of the critical regimes (with some additional explicit numerical prefactor and some power law prefactor), see Shiroishi et al. [23] and references therein. In a second critical regime and in all noncritical regimes, the EFP decays exponentially (in contrast to the noncritical regimes, there is an additional power law prefactor in the second critical regime whose exponent differs from the one in the first critical regime), see Abanov and Franchini [1,18]. These results have been derived by using powerful theorems of Szeg˝o, Widom, and Fisher–Hartwig, and the yet unproven Basor–Tracy conjecture and some of its extensions, see Widom [26] and Böttcher and Silbermann [14,15]. Furthermore, in thermal equilibrium at positive temperature, the EFP can again be shown to decay exponentially by using a theorem of Szeg˝o, see, for example, Shiroishi et al. [23] and Franchini and Abanov [18]. In contrast, out of equilibrium, the situation is more subtle. The typical open system consists of a confined sample which is coupled to extended ideal reservoirs at different temperatures. Using this paradigm, a translation invariant nonequilibrium steady state (NESS) has been constructed in Aschbacher and Pillet [13] for the XY chain using the scattering approach to algebraic quantum statistical mechanics developed by Ruelle [22] (for γ = λ = 0, this NESS has also been found by Araki and Ho [6] using a different method; moreover, using the latter approach, the magnetization profile at intermediate but large times has been studied by Ogata [21]). In this NESS, the EFP can still be recast into the form of a Toeplitz determinant, but now, the symbol is, in general, no longer scalar and regular. Due to the lack of control of higher order determinant asymptotics in Toeplitz theory with nontrivial irregular block symbols, we started off by studying bounds on the leading asymptotic order for a class of general block Toeplitz determinants in Aschbacher [7]. There, it turned out that suitable basic spectral information on the density of the state is sufficient to derive a bound on the rate of the exponential decay of the EFP in general translation invariant fermionic quasifree states. This bound proved to be exact not only for the decay rates of the ground states and the equilibrium states at positive temperature treated in Abanov and Franchini [1,18] and Shiroishi et al. [23] but also for the translation invariant NESS in the isotropic XY chain analyzed in Aschbacher [9]. In the present paper, new results are obtained for the asymptotic behavior of a class of spatial correlations, and in particular, for the asymptotic behavior of the EFP, in a NESS of the isotropic


3431

XY chain whose translation invariance has been broken by a local magnetization in the form of a finite rank perturbation. Although such a spatial correlation can again be transformed from its initial Paffian form into a scalar Toeplitz determinant, the effect of the breaking of translation invariance manifests itself in a regularization of the Toeplitz symbol and the appearance of an additional Hankel operator whose symbol is smooth. Hence, due to Peller’s theorem, this operator is of trace class, and the spatial asymptotics is governed by an exponential decay due to the invertibility of the Toeplitz operator at nonvanishing temperature. Moreover, the decay rate, determined by the Toeplitz symbol, exhibits the underlying left mover–right mover structure affected by the scattering at the impurity (see also Aschbacher [8] and Aschbacher and Barbaroux [10] for the left mover–right mover structure of the NESS expectation of several other types of correlation observables). The paper is organized as follows. In Section 2, we set the stage for the nonequilibrium XY chain with impurity, construct its NESS, and derive the basic expression for the NESS EFP. Section 3 then contains the asymptotic analysis of the NESS EFP. Several ingredients of the proofs have been transferred to Appendices A–E, as, for example, the construction of the wave operators by means of stationary scattering theory or the summary of the spectral properties of the so-called magnetic Hamiltonian. 2. Nonequilibrium setting In this section, we will shortly summarize the setting for the system out of equilibrium used in Aschbacher and Pillet [13]. In contradistinction to the presentation there, we skip the formulation of the two-sided XY chain as a spin system and rather focus directly on the underlying C ∗ dynamical system structure in terms of Bogoliubov automorphisms on a selfdual CAR algebra as in Araki [5]. A C ∗ -dynamical system is a pair (A, τ ), where A is a C ∗ algebra and R t → τ t ∈ Aut(A) a strongly continuous group of ∗-automorphism of A. For more information on the algebraic approach to open quantum systems, see, for example, Aschbacher et al. [11]. For some given N ∈ N ∪ {0}, the nonequilibrium configuration is set up by cutting the finite piece ZS := {x ∈ Z | −N x N }

(1)

out of the two-sided discrete line Z. This piece will play the role of the confined sample whereas the remaining parts, (2) ZL := x ∈ Z x −(N + 1) , ZR := {x ∈ Z | x N + 1},

(3)

will act as infinitely extended thermal reservoirs, eventually carrying different temperatures, see Fig. 1. The observables of the system are specified by the following selfdual CAR algebra over the wave functions on the chain. Definition 1 (Observables). Let F(h) denote the fermionic Fock space over the one-particle Hilbert space of wave functions on the discrete line, h := 2 (Z).

(4)

3432


Fig. 1. The nonequilibrium setting for the XY chain.

With the help of the creation and annihilation operators a ∗ (f ), a(f ) ∈ L(F(h)) with f ∈ h (where L(H) denotes the bounded linear operators on the Hilbert space H), the complex linear mapping B : h⊕2 → L(F(h)) is defined, for F := [f1 , f2 ] ∈ h⊕2 , by B(F ) := a ∗ (f1 ) + a(f¯2 ).

(5)

The observables are described by the selfdual CAR algebra over h⊕2 with antiunitary involution J generated by the operators B(F ) ∈ L(F(h)) for all F ∈ h⊕2 , i.e. we have, for all F, G ∈ h⊕2 , ∗ B (F ), B(G) = (F, G), ∗

B (F ) = B(J F ),

(6) (7)

where J F := [f¯2 , f¯1 ] for all F := [f1 , f2 ] ∈ h⊕2 , the anticommutator of A, B ∈ L(H) is {A, B} := AB + BA, and the scalar product in h⊕2 is written as the one in h. We denote this algebra by A := A(h⊕2 , J ). Remark 2. The concept of selfdual CAR algebras has been introduced and developed in Araki [3,4]. Here, it is just a convenient way of working with the linear combination (5). Also in view of future generalizations of the present paper, for example to the case of the truly anisotropic XY chain and other classes of correlations, we will stick to this notation in the present context. We next specify the Bogoliubov ∗-automorphisms on the selfdual CAR algebra which describe the time evolutions used for the construction of the NESS. Definition 3 (Dynamics). Let the coupling strength be κ > 0, and let u ∈ L(h) be the translation given by (uf )(x) := f (x − 1) for all f ∈ h and all x ∈ Z. The XY, the decoupled, and the magnetic one-particle Hamiltonians h, h0 , hB ∈ L(h), respectively, are defined by h := Re(u), h0 := h − (vL + vR ), hB := h + κv,

(8) (9) (10)

where the decoupling operators vL , vR ∈ L0 (h) (with L0 (H) the finite rank operators on H) and the operator v ∈ L0 (h) which breaks translation invariance have the form vL := Re u−(N +1) p0 uN , vR := Re uN p0 u−(N +1) , v := p0 .

(11) (12) (13)


3433

Here, the projection p0 ∈ L0 (h) is given by p0 := (δ0 , ·)δ0 , where δx ∈ h for x ∈ Z denotes the Kronecker function (moreover, the real part of A ∈ L(H) is given by Re(A) := (A + A∗ )/2). For all t ∈ R, the XY, the decoupled, and the magnetic time evolutions are the Bogoliubov ∗automorphisms τ t , τ0t , τBt ∈ Aut(A) defined on the generators B(F ) ∈ A with F ∈ h⊕2 by τ t B(F ) := B eitH F , τ0t B(F ) := B eitH0 F , τBt B(F ) := B eitHB F ,

(14) (15) (16)

where we set H := h ⊕ −h, H0 := h0 ⊕ −h0 , and HB := hB ⊕ −hB . Remark 4. For the sake of an easy exposition, we restrict the analysis to the case κ > 0, the case κ < 0 being strictly analogous. Remark 5. The magnetic Hamiltonian HB ∈ L(h⊕2 ) breaks translation invariance in the sense that the commutator [HB , u ⊕ u] = [hB , u] ⊕ −[hB , u] is nonvanishing (where [A, B] := AB − BA is the commutator of A, B ∈ L(H)), i.e., for all f ∈ h, it holds [hB , u]f = κ f (−1)δ0 − f (0)δ1 .

(17)

Remark 6. Since HB ∈ L(h⊕2 ) anticommutes with the antiunitary involution J , the magnetic Hamiltonian HB generates a Bogoliubov transformation in the sense of Araki [3,4], i.e. that, for all t ∈ R, we have itH e B , J = 0.

(18)

The same also holds for the XY and the decoupled Hamiltonian H, H0 ∈ L(h⊕2 ). Remark 7. As mentioned at the beginning of this section, this model has its origin in the XY spin chain whose formal Hamiltonian is given by H =−

1 (x) (x+1) (x) (x+1) (x) (1 + γ )σ1 σ1 + (1 − γ )σ2 σ2 + 2λσ3 , 4

(19)

x∈Z

where γ ∈ (−1, 1) denotes the anisotropy, λ ∈ R the external magnetic field, and the Pauli basis of C2×2 reads

σ0 =

1 0 , 0 1

σ1 =

0 1 , 1 0

σ2 =

0 i

−i , 0

σ3 =

1 0 . 0 −1

(20)

The Hamiltonian h from (8) corresponds to the case of the isotropic XY chain without external magnetic field, i.e. to the case where γ = 0 and λ = 0.

3434


The left and right reservoirs carry the inverse temperatures βL and βR , respectively. Pour fixer les idées, we assume, w.l.o.g., that they satisfy 0 < βL βR < ∞.

(21)

Moreover, for later use, we set β := (βR + βL )/2 and δ := (βR − βL )/2. We next specify the state in which the system is prepared initially. It consists of a KMS state at the corresponding temperature for each reservoir, and, w.l.o.g., of the chaotic state for the sample. For the definition of fermionic quasifree states, see Appendix A. Definition 8 (Initial state). The initial state ω0 ∈ Q(A) is the quasifree state specified by the density S0 ∈ L(h⊕2 ) of the form S0 := s0,− ⊕ s0,+ ,

(22)

where the operators s0,± ∈ L(h) are defined by −1 s0,± := 1 + e±k0 ,

(23)

and k0 ∈ L(h hL ⊕ hS ⊕ hR ) is given by k0 := βL hL ⊕ 0 ⊕ βR hR .

(24)

Here, for α = L, S, R, we used the definitions hα := 2 (Zα ) and hα := iα∗ hiα ∈ L(hα ), where iα : hα → h is the natural injection defined, for any f ∈ hα , by iα f (x) := f (x) if x ∈ Zα , and zero otherwise. Remark 9. Note that S0 ∈ L(h⊕2 ) is well defined, and that it satisfies the properties of a density given in Definition 32 of Appendix A. Remark 10. The one-particle Hilbert space h over Z = ZL ∪ ZS ∪ ZR decomposes as h

hL ⊕ hS ⊕ hR . It follows from (9) in Definition 3 that, w.r.t. this decomposition, the decoupled Hamiltonian h0 does not couple the different subsystems to each other, indeed, i.e. we have h0 = hL ⊕ hS ⊕ hR . As discussed in the Introduction, we pick the EFP correlation observable in order to study the effect of the breaking of translation invariance on nonequilibrium expectation values. This observable is defined as follows. Definition 11 (EFP). Let x0 ∈ Z and n ∈ N. The EFP observable An ∈ A is defined by An :=

2n

B(Fi ),

i=1

where, for all i ∈ N, the form factors Fi ∈ h⊕2 are given by

(25)


3435

F2i−1 := ui ⊕ ui G1 ,

(26)

F2i := ui ⊕ ui G2 ,

(27)

and the initial form factors G1 , G2 ∈ h⊕2 look like G1 := J G2 := [0, δx0 −1 ].

(28)

Moreover, the expectation value P : N → [0, 1] of the EFP observable An ∈ A in the NESS ωB ∈ E(A) constructed in Theorem 17 below is denoted by P(n) := ωB (An ). Remark 12. As for the name EFP, note that An =

x0 +n−1 ax , we have, for any state ω ∈ E(A), x=x0

(29)

x0 +n−1 x=x0

ax ax∗ , and that, with Bn :=

x0 +n−1 0 ω(An ) = ω Bn Bn∗ Bn 2 δx 2 = 1.

(30)

x=x0

Remark 13. The analysis of this paper can also be carried out for different form factors. If we choose the initial form factors Gi =: [gi,1 , gi,2 ] ∈ h⊕2 for i = 1, 2 to be of the completely localized form gi,l = ail δxil for ail ∈ C and xil ∈ Z with l = 1, 2, we cover the case G1 = [−δ−1 , δ−1 ] (0) (n) and G2 = [δ0 , δ0 ]. This choice describes the prominent spin-spin correlations σ1 σ1 , see, for example, Aschbacher and Barbaroux [10]. The following definition from Ruelle [22] introduces the concept of nonequilibrium steady state (NESS) in the framework of C ∗ -dynamical systems. For the situation at hand, the C ∗ dynamical system is given in terms of the magnetic Bogoliubov ∗-automorphism group τB on the selfdual CAR algebra A. Definition 14 (NESS). A NESS associated with the C ∗ -dynamical system (A, τB ) and the initial state ω0 ∈ E(A) is a weak-∗ limit point for T → ∞ of the net

1 T

T dtω0 ◦ τBt

T >0 .

(31)

0

Next, we define the time dependent correlation matrix of the EFP observable An ∈ A w.r.t. the initial state ω0 ∈ E(A) and the magnetic dynamics τBt ∈ Aut(A). Definition 15 (Correlation matrix). Let Fi ∈ h⊕2 for i ∈ N be the form factors of Definition 11. For all t ∈ R, the skew-symmetric correlation matrix Ωn (t) ∈ C2n×2n := {A ∈ C2n×2n | At = a t −A} (where A is the transpose of A) is defined, for all i, j = 1, . . . , 2n, by its entries

3436


⎧ ∗ itH itH ⎨ ω0 (B (e B J Fi )B(e B Fj )), Ωij (t) := 0, ⎩ −Ωj i (t), ap

pa

if i < j, if i = j, if i > j.

(32)

pp

Moreover, the matrices Ωnaa (t), Ωn (t), Ωn (t), Ωn (t) ∈ C2n×2n are defined, for i, j = 1, . . . , 2n a and i < j , by Ωijaa (t) := ω0 B ∗ eitHB 1ac (HB )J Fi B eitHB 1ac (HB )Fj , ap Ωij (t) := ω0 B ∗ eitHB 1ac (HB )J Fi B eitHB 1pp (HB )Fj , pa Ωij (t) := ω0 B ∗ eitHB 1pp (HB )J Fi B eitHB 1ac (HB )Fj , pp Ωij (t) := ω0 B ∗ eitHB 1pp (HB )J Fi B eitHB 1pp (HB )Fj ,

(33) (34) (35) (36)

and are to be completed as in (32) for i j . Here, 1ac (HB ), 1pp (HB ) ∈ L(h⊕2 ) are the spectral projections onto the absolutely continuous and the pure point subspaces of HB , respectively. The contributions which will play a role in the large time limit are defined as follows. Definition 16 (Asymptotic correlation matrix). Let Fi ∈ h⊕2 for i ∈ N be the form factors of pp Definition 11. The matrices Ωnaa , Ωn ∈ C2n×2n are defined, for i, j = 1, . . . , 2n and i < j , by a Ωijaa := ω0 B ∗ W (H0 , HB )J Fi B W (H0 , HB )Fj , pp Ωij := ω0 B ∗ 1e (HB )J Fi B 1e (HB )Fj ,

(37) (38)

e∈specpp (HB )

and are to be completed as in (32) for i j . Here, 1e (HB ) ∈ L(h⊕2 ) denotes the spectral projection onto the eigenspace associated with the eigenvalue e in the set of eigenvalues specpp (HB ) of HB , and the wave operator W (H0 , HB ) ∈ L(h⊕2 ) is defined by W (H0 , HB ) := s-lim e−itH0 eitHB 1ac (HB ). t→∞

(39)

The following theorem establishes the existence and uniqueness of the NESS and yields an expression for the EFP in this NESS. From now on, whenever an entry of a skew-symmetric matrix is written down, we always assume that the row index is strictly smaller than the column index. Moreover, for the definition of the Pfaffian, see Appendix A. Theorem 17 (NESS and NESS EFP). There exists a unique quasifree NESS ωB ∈ Q(A) associated with the C ∗ -dynamical system (A, τB ) and the initial state ω0 ∈ E(A) whose density SB ∈ L(h⊕2 ) has the form SB = W ∗ (H0 , HB )S0 W (H0 , HB ) +

e∈specpp (HB )

1e (HB )S0 1e (HB ).

(40)


3437

Moreover, the expectation value of the EFP observable in this NESS is given by pp P(n) = pf Ωnaa + Ωn .

(41)

Proof. We proceed similarly to the proof of Theorem 3.2 in Aschbacher et al. [12]. To this end, we note that the expectation value in the quasifree initial state ω0 ∈ Q(A) of the correlation observable An ∈ A propagated in time with the magnetic dynamics τBt ∈ Aut(A) can be written, for all t ∈ R, as the Pfaffian of the correlation matrix Ωn (t) ∈ C2n×2n from Definition 15, a ω0 τBt (An ) = pf Ωn (t) ,

(42)

where we used (18) in Remark 6 to commute the antiunitary involution J across the unitary group generated by HB ∈ L(h⊕2 ). In order to treat the argument of the Pfaffian, we make use of assertion (a) in Theorem 36 of Appendix B which states that, for the singular continuous spectrum, we have specsc (HB ) = ∅.

(43)

Hence, injecting 1ac (HB ) + 1pp (HB ) = 1 ∈ L(h⊕2 ) to the left of J Fi and Fj in the correlation matrix entry Ωij (t) = ω0 B ∗ eitHB J Fi B eitHB Fj = eitHB J Fi , S0 eitHB Fj ,

(44)

the correlation matrix can be decomposed as ap

pa

pp

Ωn (t) = Ωnaa (t) + Ωn (t) + Ωn (t) + Ωn (t),

(45)

where the matrices on the r.h.s. of (45) are given in Definition 15. Since the NESS is constructed in the large time limit, we separately study this limit for all the terms in (45). So, using that the initial state is invariant under the decoupled time evolution, i.e. [H0 , S0 ] = 0, the first term can be written as Ωijaa (t) = e−itH0 eitHB 1ac (HB )J Fi , S0 e−itH0 eitHB 1ac (HB )Fj .

(46)

Thus, with the help of the Kato–Rosenblum theorem from scattering theory for perturbations of trace class type (see, for example, Yafaev [27]), we find lim Ωnaa (t) = Ωnaa ,

t→∞

(47)

where we used that H0 − HB ∈ L0 (h⊕2 ), and the r.h.s. is given in Definition 16. For the second term on the r.h.s. of (45), we have the bound ap Ω (t) 1pp (HB )S0 eitHB 1ac (HB )J Fi Fj . ij

(48)

3438


Since the pure point spectrum of HB consists of the two simple eigenvalues ±eB , where eB is given in assertion (c) of Theorem 36 in Appendix B, we have 1pp (HB ) ∈ L0 (h⊕2 ), and, hence, it follows that ap

lim Ωn (t) = 0.

t→∞

(49)

pa

The same holds for Ωn (t), of course. For the last term on the r.h.s. of (45), using 1pp (HB ) = 1eB (HB ) + 1−eB (HB ), we get pp

Ωij (t) =

e−it (e −e) 1e (HB )J Fi , S0 1e (HB )Fj .

(50)

e,e ∈{±eB }

Moreover, since assertion (c) of Theorem 36 also states that ran(1eB (HB )) ⊂ h ⊕ 0 and ran(1−eB (HB )) ⊂ 0 ⊕ h, and since the density of the initial state S0 ∈ L(h⊕2 ) has the block diagonal form given in (22) of Definition 8, the terms in (50) for different energies vanish, and, hence, the time dependence drops out of (50). This leads to pp

pp

Ωn (t) = Ωn

(51)

→C for all t ∈ R, where the r.h.s. is given in Definition 16. Finally, since the Pfaffian pf : C2n×2n a is a continuous mapping, we get 1 P(n) = lim T →∞ T

T

dtω0 τBt (An )

0

1 = lim T →∞ T

T

dtpf Ωn (t)

0

pp = pf Ωnaa + Ωn .

(52)

Note that we didn’t make use of the specific structure of the form factors Fi . Hence, since the algebra of observables A is generated by the operators B(F ) for F ∈ h⊕2 , and since the mapping A A → ω0 (τBt (A)) ∈ C is continuous uniformly in t ∈ R, the relation (52) defines the unique NESS ωB ∈ Q(A). The form (40) of the density SB follows from (37) and (38). Moreover, due to the completeness of the wave operator and Remark 6, SB has the defining properties of a density given in Definition 32 of Appendix A. This is the assertion. 2 3. NESS correlation asymptotics In order to approach the asymptotic behavior of the NESS EFP from Theorem 17, we start off by studying more closely the two pieces of the asymptotic correlation matrix given in Definition 16. For this purpose, besides the position space, we will use the momentum space and the energy space defined before Theorem 35 of Appendix A and in Definition 39 of Appendix C, respectively.


3439

Lemma 18 (ac-structure). The asymptotic correlation matrix Ωnaa ∈ C2n×2n has the decomposia tion Ωnaa,σ , (53) Ωnaa = σ =± aa,± where the matrices Ωnaa,± ∈ C2n×2n are defined, for all i, j = 1, . . . , n, by Ω2i−1 a 2j −1 := aa,± aa,− aa,+ Ω2i 2j := Ω2i 2j −1 := Ω2i−1 2j = 0, and the nonvanishing entries are given by

aa,− Ω2i−1 2j := w− (h, hB )δi+x0 −1 , s− w− (h, hB )δj +x0 −1 , aa,+ Ω2i 2j −1 := w+ (h, hB )δi+x0 −1 , s+ w+ (h, hB )δj +x0 −1 .

(54) (55)

Here, s± ∈ L(h) are the density components of the translation invariant XY NESS given in Theorem 35 of Appendix A, and the wave operators w± (h, hB ) ∈ L(h) are defined by w± (h, hB ) := s-lim eith e−ithB 1ac (hB ), t→±∞

(56)

where, from now on, all the spectral projections of hB are denoted as the ones for HB given in the Definitions 15 and 16 with HB replaced by hB . Proof. In order to rewrite the absolutely continuous contribution to the asymptotic correlation matrix from Definition 16, we want to take advantage of the fact that the operator S = W ∗ (H0 , H )S0 W (H0 , H )

(57)

is the known density of the translation invariant XY NESS (i.e. the NESS for κ = 0) given in Theorem 35 of Appendix A. For this purpose, we use the chain rule W (H0 , HB ) = W (H0 , H )W (H, HB ) which is permissible since H − H0 , HB − H ∈ L0 (H) (the wave operators W (H0 , H ), W (H, HB ) ∈ L(h⊕2 ) are defined as in (39) with the appropriate replacements). Hence, the absolutely continuous contribution becomes (58) Ωijaa = W (H, HB )J Fi , SW (H, HB )Fj . Using the block diagonal structure of the operators H, HB , S ∈ L(h⊕2 ) and plugging the explicit form of the form factors from Definition 11 into (58) leads to the assertion. 2 In order to evaluate the nonvanishing entries (54) and (55) from Lemma 18, we determine the action of the wave operators on completely localized wave functions. The main computations are carried out in Appendix C. Proposition 19 (Wave operators). Let x ∈ Z be any site. Then, in momentum space hˆ = L2 (T), the wave operators w± (h, hB ) ∈ L(h) act on the completely localized wave function δx ∈ h as wˆ ± (h, hB )ex (k) = ex (k) ∓ iκ

e|x| (∓|k|) , sin(|k|) ± iκ

where we set ex (k) := δˆx (k) = eikx for all x ∈ Z and for all k ∈ (−π, π].

(59)

3440


Proof. Plugging (141) into (131) in Appendix C and applying ˜f : hˆ → h˜ from (124) to ex , the action of the wave operator is expressed in energy space h˜ as −1/4 x x w˜ ± (h, hB )δ˜x (e) = (2π)−1/2 1 − e2 e + i 1 − e2 , e − i 1 − e2 √ (e ∓ i 1 − e2 )|x| [1, 1] . ∓ iκ √ 1 − e2 ± iκ

(60)

Applying ˜f∗ : h˜ → hˆ from (126) in Appendix C to (60) yields the assertion.

2

Remark 20. The action (59) relates to the action of the wave operator for the one-center δinteraction on the continuous line by replacing sin(|k|) by |k|, see, for example, Albeverio et al. [2]. We next turn to the pure point contribution. pp

Lemma 21 (pp-structure). The asymptotic correlation matrix Ωn ∈ C2n×2n has the decomposia tion pp

Ωn =

pp,σ

Ωn

(61)

,

σ =± pp,±

where the matrices Ωn pp,±

pp,−

pp,+

pp,±

∈ C2n×2n are defined, for all i, j = 1, . . . , n, by Ω2i−1 2j −1 := a

Ω2i 2j := Ω2i 2j −1 := Ω2i−1 2j := 0, and the nonvanishing entries are given by pp,− Ω2i−1 2j := 1pp (hB )δi+x0 −1 , s0,− 1pp (hB )δj +x0 −1 , pp,+ Ω2i 2j −1 := 1pp (hB )δi+x0 −1 , s0,+ 1pp (hB )δj +x0 −1 .

(62) (63)

Here, s0,± ∈ L(h) are the density components of the initial state given in Definition 8. Proof. Using the block diagonal structures of HB , S0 ∈ L(h⊕2 ) and plugging the explicit form of the form factors from Definition 11 into (38) leads to the assertion. 2 In order to evaluate the nonvanishing entries (62) and (63) from Lemma 21, we determine the form of the projections onto the pure point subspaces of the magnetic Hamiltonian. A summary of its spectral properties is given in Appendix B. Lemma 22 (Pure point projection). The projection onto the pure point subspace of the magnetic Hamiltonian hB satisfies dim ran 1pp (hB ) = 1,

(64)

and its range is spanned by an exponentially localized eigenfunction fB ∈ h of hB ∈ L(h) with eigenvalue eB > 1.


3441

Fig. 2. The symbol a(k) > 0 with k ∈ (−π, π ] for βR = 2, βL = 12 , and κ = 0 to the left and κ = 15 to the right. The nonvanishing magnetic field regularizes the symbol.

Proof. See Theorem 36 in Appendix B.

2

Collecting the properties of the absolutely continuous and the pure point contributions to the asymptotic correlation matrix from Lemma 18 to Lemma 22, we get the following structural assertion. For the ingredients from Toeplitz theory referred to in the remainder of the present section, see, for example, Böttcher and Silbermann [14,15]. Moreover, we denote by χA : R → {0, 1} the characteristic function of the set A ⊂ R. Proposition 23 (Determinantal structure). The NESS EFP is the determinant of the finite section of a Toeplitz operator, a Hankel operator, and an operator of finite rank. The symbol a ∈ L∞ (T) of the Toeplitz operator reads a = ϕB sˆ−,L + (1 − ϕB )ˆs−,R ,

(65)

where the functions ϕB , sˆ±,α ∈ L∞ (T) with α = L, R, are defined, for k ∈ (−π, π], by 1 1 ± tanh 12 βα cos(k) , 2 sin2 (k) , ϕB (k) := χ[0,π] (k) 2 sin (k) + κ 2

sˆ±,α (k) :=

(66) (67)

see Fig. 2. Moreover, the symbol of the Hankel operator is smooth. Remark 24. In the limit κ → 0, we recover the symbol derived in Aschbacher [7,9] for the translation invariant case. Remark 25. Note that for nonvanishing coupling, the characteristic function in (67) is smoothed out. This will play an essential role in the asymptotic analysis of the corresponding Toeplitz determinant.

3442

W.H. Aschbacher / Journal of Functional Analysis 260 (2011) 3429–3456 pp

Proof. The total asymptotic correlation matrix, defined by Ωn := Ωnaa + Ωn ∈ C2n×2n , has the a 2 × 2 block substructure Ωn = [Aij ]i,j =1,...,n , where the matrices Aij ∈ C2×2 are defined, for i, j = 1, . . . , n, by ⎧ 0 bij ⎪ , if i < j, ⎪ c 0 ⎪ ⎨ ij 0 bii (68) Aij := −bii 0 , if i = j, ⎪ ⎪ ⎪ ⎩ −At , if i > j, ji and the entries are given by pp,−

if i j,

(69)

pp,+

if i < j.

(70)

aa,− bij := Ω2i−1 2j + Ω2i−1 2j , aa,+ cij := Ω2i 2j −1 + Ω2i 2j −1 ,

In order to rewrite the argument of the Pfaffian in P(n) = pf(Ωn ) from Theorem 17 in a form more suited for the subsequent analysis, we want to apply a similarity transformation to Ωn . To this end, for any i, j = 1, . . . , 2n with i < j , we denote by R [ij ] ∈ O(2n) the elementary matrix whose left multiplication with any matrix A ∈ C2n×2n exchanges the ith and j th row of A (where O(n) stands for the orthogonal matrices in Rn×n ). Then, using the matrix R ∈ O(2n) defined by n−1 k−1 [2(n−k)+l,2(n−k)+l+1] , we can transform Ωn into off-diagonal block form, R := k=1 l=0 R

0 Θn , (71) R t Ωn R = −Θnt 0 where the matrix Θn ∈ Cn×n , called the reduced correlation matrix, is defined by its entries Θij := θij , and, for all i, j ∈ N, the numbers θij are given by if i j , bij , θij := (72) −cj i , if i > j . Hence, using assertions (a) and (b) from Lemma 34 of Appendix A, we get P(n) = pf(Ωn ) = (−1)

n(n−1) 2

pf

0 −Θnt

Θn 0

= det(Θn ).

(73)

Let us next analyze the structure of Θn . In order to do so, we subdivide the discussion into the following two cases w.r.t. the starting site x0 ∈ Z of the EFP string. Case 1: x0 0. With the help of Lemma 44 of Appendix D, we make the decomposition Θn = ΘT ,n + ΘH,n , where the matrix ΘT ,n ∈ Cn×n has the entries ΘT ,ij := θT ,ij given by θT ,ij := bT ,ij if i j , and θT ,ij := −cT ,j i if i > j . Here, for all i, j ∈ N, we define bT ,ij := (ei−j , sˆ− e0 ) + (ei−j , a− e0 ),

if i j ,

(74)

−cT ,j i := (ej −i , sˆ+ e0 ) + (ei−j , a+ e0 ),

if i > j ,

(75)


3443

where sˆ± , a± ∈ L∞ (T) are given in Theorem 35 of Appendix A and Lemma 44 of Appendix D, respectively. Similarly, the matrix ΘH,n ∈ Cn×n has the entries ΘH,ij := θH,ij given by θH,ij := bH,ij if i j , and θH,ij := −cH,j i if i > j . Here, for all i, j ∈ N, we define bH,ij := (ei+j , b− e0 ),

if i j ,

(76)

−cH,j i := (ei+j , b+ e0 ),

if i > j ,

(77)

where b± ∈ L∞ (T) are given in Lemma 44 of Appendix D. Using (66) and (67) in (74), and sˆ+,R − sˆ+,L = −(ˆs−,R − sˆ−,L ) and (ei−j , sˆ+ e0 ) = −(ej −i , sˆ− e0 ) in (75), where the last identity is due to Definition 32 of the density in Appendix A, we have bT ,ij = (ei−j , ae0 ) for i j and −cT ,j i = (ei−j , ae0 ) for i > j . Hence, ΘT ,n is the finite section of the Toeplitz operator T [a] ∈ L(2 (N)) generated by the symbol a ∈ L∞ (T), i.e. ΘT ,n = Tn [a].

(78)

Moreover, as for (76) and (77), using (119) in Remark 37 of Appendix B, we find that bH,ij = (ei+j −1 , be0 ) for i j and −cH,j i = (ei+j −1 , be0 ) for i > j , where the function b ∈ C ∞ (T) is defined by b(k) := iκ

e−ik(2x0 −1) (fB , s0,− fB ) − s ˆ (k) . −,R sin(k) + iκ eB2

(79)

Hence, ΘH,n is the finite section of the Hankel operator H [b] ∈ L(2 (N)) generated by the symbol b ∈ C ∞ (T), i.e. ΘH,n = Hn [b].

(80)

Therefore, it follows from (73) that the NESS EFP is the determinant of the finite section of the sum of a Toeplitz and a Hankel operator, P(n) = det Tn [a] + Hn [b] ,

(81)

where, in this case, the finite rank operator from the formulation of the assertion vanishes. Let us now turn to the case where the EFP string starts to the left of the origin. Case 2: x0 < 0. For n 1 + n0 , where we set n0 := −x0 , we again have from Lemma 44, that, for all i, j = 1, . . . , n − n0 , Θn,i+n0 j +n0 = Tn−n0 ,ij [a] + Hn−n0 ,ij [c],

(82)

where we set c := e−(2n0 +1) b. Defining the operator Θ : Cn0 ⊕ 2 (N) → Cn0 ⊕ 2 (N) on [ξ, f ] ∈ Cn0 ⊕ 2 (N) by the matrix multiplication with the infinite matrix θij from (72), we have M := Θ − 0 ⊕ T [a] + H [c] ∈ L0 Cn0 ⊕ 2 (N) .

(83)

3444


Since the reduced correlation matrix satisfies Θn = Rn ΘRn ran(Rn ) with Rn := 1 ⊕ Pn ∈ L(Cn0 ⊕ 2 (N)), it follows from (73) that P(n) = det 0 ⊕ Tn−n0 [a] + Hn−n0 [c] + Mn , where Mn := Rn MRn ran(Rn ) . Hence, we arrive at the assertion.

(84)

2

We are now ready to formulate our main result on the behavior of the NESS EFP for large string lengths. Theorem 26 (Exponential decay). For n → ∞, the NESS EFP P : N → [0, 1] has an exponentially decaying bound, P(n) = O e−Γ n .

(85)

The decay rate Γ := ΓR + ΓB > 0 contains the two parts 1 ΓR := − 2 ΓB := −

1 2

π −π

π −π

dk log sˆ−,R (k) , 2π

(86)

dk log σB (k)ˆs−,L (k) + 1 − σB (k) sˆ−,R (k) , 2π

(87)

where the function σB ∈ L∞ (T) is given by σB (k) :=

sin2 (k) sin2 (k) + κ 2

.

(88)

Remark 27. Note that Theorem 26 holds for any coupling strength. In the small coupling limit, we recover the exact decay rate from Aschbacher [9], and the bound derived for general quasifree systems in Aschbacher [7]. Remark 28. As can be seen in (87), the NESS EFP decay rate displays a left mover – right mover structure. It is composed of a left mover carrying temperature βR and coming from +∞, a left mover carrying temperature βR having been reflected at the perturbation at the origin, and a right mover carrying temperature βL having been transmitted through the origin. This left mover–right mover structure has already been observed in translation invariant systems for several types of correlation functions, see Aschbacher [8,9] and Aschbacher and Barbaroux [10]. Remark 29. Defining ΓL analogously to (86), we have the rewritings, for α = L, R, π

2 Γα = − − π2

dk log 14 1 − tanh2 12 βα cos(k) , 2π

(89)


3445

Fig. 3. For βR = 2 and βL = 12 , the integrand of ΓL (left, thin line) and ΓR (right, thin line) compared to ΓB with κ = 15 (left and right, thick line). π

2 ΓB = − − π2

dk log 14 1 − 1 − σB (k) tanh 12 βR cos(k) 2π

+ σB (k) tanh

1

2 .

2 βL cos(k)

(90)

From (89) and (90), it immediately follows that, if the system is truly out of equilibrium, i.e. if δ > 0, the decay rates are ordered as 0 < ΓL < ΓB < ΓR ,

(91)

see also Fig. 3. Remark 30. It follows from assertion (a) in Proposition 45 of Appendix E that the nonvanishing coupling regularizes the underlying Toeplitz theory in the sense that the symbol which determines the decay rate is smoother than in the case κ = 0, see Fig. 2. Namely, the latter case requires Fisher–Hartwig theory and, if δ > 0, leads to a strictly positive power law subleading order as given in Aschbacher [9]. Proof. Since the Toeplitz symbol a ∈ L∞ (T) from (65) is real-valued, we can make use of the Hartman–Wintner theorem in order to control the spectrum of the selfadjoint Toeplitz operator T [a] ∈ L(2 (N)). Moreover, due to Proposition 45 of Appendix E, we have a ∈ C(T),

(92)

and, hence, spec(T [a]) = ran(a) = [ˆs−,R (0), sˆ+,R (0)], where 0 < sˆ−,R (0) < sˆ+,R (0) < 1 in the temperature range 0 < βL βR < ∞. Therefore, T [a] ∈ L(2 (N)) is invertible, 0∈ / spec T [a] ,

(93)

3446


and the spectrum is independent of the coupling strength. Moreover, since (92) and (93) hold, the Gohberg–Feldman theorem implies that the sequence {Tn [a]}n∈N is stable, (94) lim supTn−1 [a] < ∞. n→∞

For the following analysis, as in the proof of Proposition 23, we discuss the cases x0 0 and x0 < 0 separately. For convenience of exposition, we start with the second case. Case 2: x0 < 0. Since we want to analyze the asymptotic behavior for large n with the help of Szeg˝o’s strong limit theorem, we write, using (94), det(Tn−n0 [a]) det(Tn [a]) P(n) P(n) = , n G(a) det(Tn−n0 [a]) det(Tn [a]) G(a)n

(95)

where G(a) is the exponential of the 0th Fourier coefficient of log(a), and n0 := −x0 as before. Due to (84) and (94), the first factor on the r.h.s. of (95) can be written as P(n) −1 = det 1 + 1 ⊕ Tn−n [a] (−1) ⊕ Hn−n0 [c] + Mn . 0 det(Tn−n0 [a])

(96)

Moreover, since we know from Proposition 23 that c ∈ C ∞ (T), we also have c ∈ L∞ (T) ∩ B11 (T),

(97)

where Bpα (T) are the usual Besov spaces. Therefore, Peller’s theorem allows us to conclude that (L1 (H) are the trace class operators on the Hilbert space H) H [b] ∈ L1 2 (N) . (98) Due to (83), (84), and (98), the r.h.s. of (96) converges to the constant K(a, b) := det(1 + 1 ⊕ T −1 [a](−1 ⊕ H [b] + M)). In order to treat the second factor on the r.h.s. of (95), we apply Szeg˝o’s first limit theorem which is applicable due to (92) and (93). Hence, if we factorize the quotient as 0 det(Tn−i [a]) det(Tn−n0 [a]) = , det(Tn [a]) det(Tn+1−i [a])

n

(99)

i=1

each factor on the r.h.s. of (99) converges to 1/G(a). In order to treat the third factor on the r.h.s. of (95), we make use of Szeg˝o’s strong limit theorem. This theorem states that, since a(t) > 0 for all t ∈ T s.t. ind(a) = 0, and since 1/2

a ∈ W (T) ∩ B2 (T),

(100)

which follows from a ∈ C 1 (T) ∩ P C ∞ (T) in Proposition 45 of Appendix E (W (T) is the Wiener algebra and P C ∞ (T) are the piecewise smooth functions), the quotient converges to a constant usually denoted by E(a). Plugging the foregoing three limits into the r.h.s. of (95), we get P(n) = K(a, b)E(a)G(a)x0 . n→∞ G(a)n lim

(101)


3447

Hence, in order to determine an exponential bound on the asymptotic decay, we are left with the computation of the constant G(a). Decomposing the integral in the 0th Fourier coefficient of log(a) w.r.t. positive and negative momenta and using the fact that sˆ−,α for α = L, R and σB are even functions in k ∈ (−π, π], we arrive at the expressions (86) and (87) for the decay rate of the bound on the exponential decay. The case where the EFP string starts at nonnegative sites is simpler and is treated analogously as follows. Case 1: x0 0. Writing (81) as in (95), where, in this case, the second factor is absent, we can proceed as for Case 2. In particular, the determinant of the Toeplitz contribution can again be separated due to (94), (98) holds for the symbol b satisfying (97), and the constant now reads K(a, c) = det(1 + T −1 [a]H [c]). Finally, the last factor in (101) is absent. Hence, we arrive at the assertion. 2 Remark 31. The study of the present problem for the anisotropic XY model, i.e. for the case where γ = 0 in (19) of Remark 7, is more complicated. Not only the Pfaffian structure of the correlation cannot be preserved in the present form, but also one has to cope with Toeplitz theory for operators with nonscalar symbols. We will study this set of problems for general quasifree systems elsewhere. Acknowledgments We would like to thank the editor and the referee for their constructive remarks. Moreover, the support provided by the German Research Foundation (DFG) is gratefully acknowledged. Appendix A. Fermionic quasifree states Let A be the selfdual CAR algebra from Definition 1. We denote by E(A) the set of states, i.e. the normalized positive linear functionals on the C ∗ algebra A. Definition 32 (Density). The density of a state ω ∈ E(A) is defined to be the operator S ∈ L(h⊕2 ) with 0 S ∗ = S 1 and J SJ = 1 − S satisfying, for all F, G ∈ h⊕2 , ω B ∗ (F )B(G) = (F, SG).

(102)

An important class are the quasifree states. Definition 33 (Quasifree state). A state ω ∈ E(A) is called quasifree if it vanishes on the odd polynomials in the generators and if it is a Pfaffian on the even polynomials in the generators, i.e. if, for all F1 , . . . , F2n ∈ h⊕2 and for any n ∈ N, we have ω B(F1 ) . . . B(F2n ) = pf(Ωn ),

(103)

where the skew-symmetric matrix Ωn ∈ C2n×2n = {A ∈ C2n×2n | At = −A} is defined, for i, j = a 1, . . . , 2n, by

3448


Fig. 4. Some of the pairings for n = 3. The total number of intersections I relates to the signature as sign(π ) = (−1)I .

⎧ ⎨ ω(B(Fi )B(Fj )), Ωij := 0, ⎩ −ω(B(Fj )B(Fi )),

if i < j , if i = j , if i > j .

(104)

Aπ(2j −1),π(2j ) ,

(105)

→ C is given by Here, the Pfaffian pf : C2n×2n a pf(A) :=

sign(π)

n j =1

π

where the sum is running over all pairings of the set {1, 2, . . . , 2n}, i.e. over all the (2n)!/(2n n!) permutations π in the permutation group of 2n elements which satisfy π(2j − 1) < π(2j + 1) and π(2j − 1) < π(2j ), see Fig. 4. The set of quasifree states is denoted by Q(A). The following lemma has been used in Section 3. Lemma 34 (Pfaffian). The Pfaffian has the following properties. (a) Let X, Y ∈ C2n×2n with Y t = −Y . Then, pf XY X t = det(X) pf(Y ).

(106)

(b) Let X ∈ Cn×n . Then,

pf

0 −X t

X 0

Proof. See, for example, Stembridge [25].

= (−1)

n(n−1) 2

det(X).

(107)

2

Next, we state the properties of the NESS for the translation invariant case κ = 0, the so-called XY NESS. To this end, let f : h → hˆ := L2 (T) (with unitcircle T) be the Fourier transformation defined with the sign convention fˆ(k) := (ff )(k) := x∈Z f (x)eikx . Moreover, for any a ∈ L(h), we use the notation aˆ := faf∗ . We then have the following. Theorem 35 (XY NESS). There exists a unique quasifree NESS ω ∈ Q(A) associated with the C ∗ -dynamical system (A, τ ) and the initial state ω0 ∈ Q(A) whose density S ∈ L(h⊕2 ) has the form S = s− ⊕ s+ , where the operators s± ∈ L(h) act in momentum space hˆ as multiplication by

(108)


sˆ± (k) :=

1 1 ± ± (k) , 2

3449

(109)

and the functions ± : T → (−1, 1) are defined by ± (k) := tanh 12 β ± sign sin(k) δ cos(k) .

(110)

2

Proof. See Aschbacher and Pillet [13]. Appendix B. Magnetic Hamiltonian

In this section, we summarize the spectral theory of HB ∈ L(h⊕2 ) needed above. To this end, we denote by specsc (A), specac (A), and specpp (A) the singular continuous, the absolutely continuous, and the pure point spectrum of the operator A, respectively. Theorem 36 (Magnetic spectrum). The magnetic Hamiltonian HB ∈ L(h⊕2 ) has the following properties. (a) specsc (HB ) = ∅; (b) specac (HB ) = [−1, 1]; (c) specpp (HB ) = {±eB } with eB > 1. The eigenvalues ±eB are simple, and eB =

1 + κ 2.

(111)

The normalized eigenfunction of HB with eigenvalue eB is given by fB ⊕ 0 ∈ h⊕2 , where fB is exponentially localized, i.e. for all x ∈ Z, it has the form fB (x) :=

1 −λB |x| e , nB

(112)

and the decay rate and the normalization constant are given by λB := log(κ + eB ), eB nB := . κ

(113) (114)

Moreover, the eigenfunction of HB with eigenvalue −eB reads 0 ⊕ fB ∈ h⊕2 . Proof. Assertions (a) and (b) are proven in a more general context in Hume and Robinson [19] (which also contains the case of the truly anisotropic XY model without magnetic field, and more general perturbations). The fact that there is no eigenvalue embedded in the continuum specac (HB ) = specac (H ) = spec(H ) = [−1, 1] also follows from [19]. Hence, in order to derive assertion (c), we compute eigenfunctions of the operator HB by looking for solutions of the eigenvalue equation hB f = ef for e ∈ R with |e| > 1 and not identically vanishing f ∈ h. Since such e lie in the resolvent set of h, we can write f = −(h − e)−1 vf = −κf (0)(h − e)−1 δ0 . By taking the scalar product of this equation with δx for any x ∈ Z, we have

3450


f (x) = −κf (0) δx , (h − e)−1 δ0 ,

(115)

implying that f (0) = 0. Plugging x = 0 into (115), we get the eigenvalue equation 1 + κ δ0 , (h − e)−1 δ0 = 0.

(116)

Switching to the momentum space representation and using Cauchy’s residue theorem, we get, for all e ∈ R with |e| > 1 and all x ∈ Z, √ (e − sign(e) e2 − 1)|x| . δx , (h − e)−1 δ0 = − sign(e) √ e2 − 1

(117)

If we plug x = 0 into (117) and use the assumption κ > 0, we see that the equation (116) can be satisfied for e > 1 only. Solving (116) for this case leads to (111). Next, plugging (111) into (117), we get f (x) = f (0)(κ + eB )−|x| from (115). Choosing f (0) > 0, we arrive at (112) with (113) and (114). 2 Remark 37. The Fourier transformation of fB ∈ h is given, for all k ∈ (−π, π], by κ 1 . fˆB (k) = nB eB − cos(k)

(118)

Cauchy’s residue theorem yields that, for x ∈ Z with x 0, we also have

e

−λB x

π = ieB −π

e−ikx dk . 2π sin(k) + iκ

(119)

The integrand on the r.h.s. of (119) is used for the extraction of the Hankel symbol in the proof of Proposition 23. Appendix C. Wave operators In this section, we use the stationary approach to scattering theory in order to compute the wave operators w± (h, hB ) ∈ L(h) appearing in the ac-contribution to the asymptotic correlation matrix from Lemma 18. To this end, we first need to express the resolvent of the magnetic Hamiltonian by the resolvent of the XY Hamiltonian. This is done in the following lemma. For any operator a ∈ L(h) and any z ∈ C in the resolvent set of a, we denote by rz (a) := (a − z)−1 ∈ L(h) the resolvent of a at the point z. Lemma 38 (Magnetic resolvent). Let e ∈ R and ε > 0. Then, at the points e ± iε, the resolvent of hB ∈ L(h) can be expressed in terms of the resolvent of h ∈ L(h) as re±iε (hB ) = re±iε (h) −

κ re∓iε (h)δ0 , · re±iε (h)δ0 . 1 + κ(δ0 , re±iε (h)δ0 )

(120)


3451

Proof. In order to simplify notation, we drop the index e ± iε of the resolvents. With the help of the resolvent identity r(hB ) = r(h) − κr(h)vr(hB ), we can write, for all f ∈ h, r(hB )f + κ δ0 , r(hB )f r(h)δ0 = r(h)f.

(121)

Taking the scalar product of (121) from the left with δ0 , we get

1 + κ δ0 , r(h)δ0 δ0 , r(hB )f = r(h)∗ δ0 , f .

(122)

Since, due to (1 + κ(δ0 , r(h)δ0 ))(1 − κ(δ0 , r(hB )δ0 )) = 1, the first factor on the l.h.s. of (122) is nonvanishing, we can solve (122) for (δ0 , r(hB )f ). Plugging the resulting expression into (121) yields the assertion. 2 In the next definition, we introduce the energy space being the direct integral decomposition of the absolutely continuous subspace of the XY Hamiltonian h ∈ L(h) w.r.t. which h is diagonal. Definition 39 (Energy space). Let the direct integral over specac (h) with fiber C2 be denoted by h˜ := L2 [−1, 1], C2 ,

(123)

ˆ and let us call h˜ the energy space of h. Moreover, the mapping ˜f : hˆ → h˜ is defined, for all ϕ ∈ h, by −1/4 ϕ arccos(e) , ϕ − arccos(e) . (˜fϕ)(e) := (2π)−1/2 1 − e2

(124)

We will use the notation f˜ := ˜fff for all f ∈ h, and a˜ := ˜ffaf∗ ˜f∗ for all a ∈ L(h), where the Fourier transform f : h → hˆ is defined in Appendix A. Moreover, the Euclidean scalar product in C2 will be denoted as ·,·. We then have the following lemma. ˆ h) ˜ is unitary, and the XY Hamiltonian h ∈ Lemma 40 (Diagonalization). The mapping ˜f ∈ L(h, ˜ L(h) acts, on any η ∈ h, as the multiplication by the energy variable e, ˜ (hη)(e) = eη(e).

(125)

Proof. A simple computation shows that ˜f is a surjective isometry with ˜f−1 = ˜f∗ : h˜ → hˆ acting on all η =: [η1 , η2 ] ∈ h˜ as ∗ ˜f η (k) = (2π)1/2 1 − cos2 (k) 1/4 χ[0,π] (k) η1 cos(k) + χ[−π,0] (k)η2 cos(k) . (126) Equality (125) then follows immediately.

2

We introduce the following abbreviations.

3452


Definition 41 (Boundary values). Let e ∈ R and ε > 0. For all f, g ∈ h, we define f,g (e ± iε) := f, re±iε (h)g , γf,g (e, ε) :=

1 f,g (e + iε) − f,g (e − iε) . 2πi

(127) (128)

Moreover, if the limits exist, we write f,g (e ± i0) := lim f,g (e ± iε),

(129)

ε→0+

γf,g (e) := lim γf,g (e, ε).

(130)

ε→0+

The wave operators then have the following form. ˜ the action of the wave operators Proposition 42 (Wave operators). In energy space h, w± (h, hB ) ∈ L(h) on any f ∈ h has the form w˜ ± (h, hB )f˜(e) = f˜(e) −

κδ0 ,f (e ± i0) δ˜0 (e). 1 + κδ0 ,δ0 (e ± i0)

(131)

Proof. In order to compute the wave operators w± (h, hB ) ∈ L(h) with the help of the stationary scheme in scattering theory (see, for example, Yafaev [27]), we write them in the weak abelian form ∞ w± (h, hB ) = w-lim 2ε ε→0+

dt e−2εt 1ac (h)e±ith e∓ithB 1ac (hB ).

(132)

0

Applying Parseval’s identity to (132) and using that re±iε (h) = ±i write, for all f, g ∈ h, ε f, w± (h, hB )g = lim ε→0+ π

∞

∞ 0

dt e∓it (h−(e±iε)) , we can

de re±iε 1ac (h)f, re±iε (hB )1ac (hB )g .

(133)

−∞

Moreover, if the limits ε → 0+ of (re±iε (h)f, re±iε (hB )g) exist for all f, g ∈ h and almost all e ∈ R (the set of full measure depending on f and g), we get f, w± (h, hB )g =

1

−1

de lim

ε→0+

ε re±iε (h)f, re±iε (hB )g π

(134)

because 1ac (h) = 1 and spec(h) = [−1, 1]. In order to compute the limit in (134), we express the resolvents re±iε (h) of the magnetic Hamiltonian in terms of the resolvents re±iε (h) of the XY Hamiltonian. Plugging (120) from Lemma 38 into the scalar product on the r.h.s. of (134) and using (127) and (128) from Definition 41, we have κ ε re±iε (h)f, re±iε (hB )g = γf,g (e, ε) − γf,δ0 (e, ε)δ0 ,g (e ± iε). π 1 + κδ0 ,δ0 (e ± iε)

(135)


3453

Here, we made use of the fact that, due to the resolvent identity, we have the equality γf,g (e, ε) = (f, πε re±iε (h)re∓iε (h)g). Now, we know that, for any f, g ∈ h and almost all e ∈ [−1, 1], the following limits exist, d(f, ρ(e)g) f,g (e ± i0) = ±πi + p.v. de

1

de

−1

e

1 d(f, ρ(e )g) , −e de

(136)

where the p.v.-integral denotes Cauchy’s principle value, the mapping ρ : B(R) → L(h) with B(R) the Borel sets on R is the projection-valued spectral measure of the XY Hamiltonian h, and we used that d(f, ρ(e)g) de. d f, ρ(e)g = χ[−1,1] (e) de

(137)

Moreover, we get from (128) and (136), γf,g (e) =

d(f, ρ(e)g) . de

(138)

Therefore, plugging (135), (136), and (138) into (134), we can write f, w± (h, hB )g = (f, g) − κ

1

−1

de

γf,δ0 (e)δ0 ,g (e ± i0) , 1 + κδ0 ,δ0 (e ± i0)

(139)

1 where, in the first term on the r.h.s., we used −1 deγf,g (e) = (f, 1ac (h)g) = (f, g). In order to write the derivatives in (138) entering (139) more explicitly, we switch to the energy space representation from Definition 39. Using the diagonalization (125), we have, for all f, g ∈ h, that d(f, ρ(e)g) ˜ = f (e), g(e) ˜ , de

(140)

where we recall from Definition 39 that ·,· denotes the scalar product in the fiber C2 of the direct integral h˜ = L2 ([−1, 1], C2 ), and f˜ = ˜fff for all f ∈ h. Hence, plugging (138) and (140) into (139), we arrive at the assertion. 2 Finally, since the wave operators w± (h, hB ) ∈ L(h) appearing in the ac-contribution to the asymptotic correlation matrix act on completely localized wave functions δx ∈ h with x ∈ Z, we compute the terms δ0 ,δx (e ± i0) on the r.h.s. of (131) in Proposition 42. Lemma 43 (Boundary values). Let x ∈ Z and e ∈ (−1, 1). Then, we have √ (e ∓ i 1 − e2 )|x| δ0 ,δx (e ± i0) = ±i . √ 1 − e2

(141)

3454


Proof. Let x ∈ Z with x 0, e ∈ (−1, 1), and ε > 0 sufficiently small. Writing δ0 ,δx (e − iε) in the momentum space representation, using Cauchy’s residue theorem, and taking the limit ε → 0+ , we get the expression (141) for δ0 ,δx (e − i0). Moreover, using (127), the translation and parity invariance of h, i.e. [h, u] = 0 and [h, θ ] = 0, respectively, where θ : h → h is defined, for all f ∈ h, by (θf )(x) := f (−x), we have, for all x ∈ Z with x 0, that δ0 ,δx (e + iε) = δ0 ,δ−x (e − iε),

(142)

δ0 ,δ−x (e − iε) = δ0 ,δx (e − iε). This yields the assertion.

(143)

2

Appendix D. Asymptotic correlation matrix In this section, we compute the nonvanishing entries of the total asymptotic correlation matrix used above. Lemma 44 (Structure). Let x0 0 and i, j 1, or x0 < 0 and i, j 1 − x0 . Then, the entries of the asymptotic correlation matrix have the structure pp,−

if i j ,

(144)

pp,+

if i > j ,

(145)

aa,− Ω2i−1 2j + Ω2i−1 2j = (ei−j , sˆ− e0 ) + (ei−j , a− e0 ) + (ei+j , b− e0 ), aa,+ Ω2j 2i−1 + Ω2j 2i−1 = (ej −i , sˆ+ e0 ) + (ei−j , a+ e0 ) + (ei+j , b+ e0 ),

where the functions a± , b± ∈ L∞ (T) are defined by a± (k) := κ 2 χ[0,π] (k) b± (k) = (−iκ)

sˆ±,R (k) − sˆ±,L (k) sin2 (k) + κ 2

,

sˆ±,R (k) −2ik(x0 −1) κ e−2λB (x0 −1) e . + 2 (fB , s0,± fB ) sin(k) + iκ eB − cos(k) nB

(146) (147)

Proof. For i j , plugging the wave operator (59) into ac-contribution (54) yields aa,− Ω2i−1 2j = (δi+x0 −1 , s− δj +x0 −1 )

π + iκ −π

π − iκ −π

π +κ

2 −π

dk e−i(k(i+x0 −1)−|k||j +x0 −1|) sˆ− (k) 2π sin(|k|) − iκ dk e−i(|k||i+x0 −1|−k(j +x0 −1)) sˆ− (k) 2π sin(|k|) + iκ dk e−i|k|(|i+x0 −1|−|j +x0 −1|) sˆ− (k) . 2π sin2 (k) + κ 2

(148)


3455

Moreover, using Lemma 22, we have pp,−

Ω2i−1 2j =

1 (fB , s0,− fB )e−λB (|i+x0 −1|+|j +x0 −1|) . n2B

(149)

The expressions for i > j are analogous. Then, if x0 0 and i, j 1, or x0 < 0 and i, j 1 − x0 , we use the translation invariance of s± , resolve the absolute values, and decompose the integrals w.r.t. the sign of the momentum in order to get rid of the sign function in the density sˆ± . This leads to (144) and (145). 2 Appendix E. Toeplitz symbol regularity The following proposition is used in Theorem 26. Proposition 45 (Regularity). The Toeplitz symbol a ∈ L∞ (T) of Proposition 23 has the following properties. (a) a ∈ C 1 (T) ∩ P C ∞ (T); (b) The left and right derivatives D± a (k) exist for all k ∈ (−π, π], but, for k+ := 0 and k− := π , we have D+ a (k± ) − D− a (k± ) = ±

1 sinh[ 12 (βR − βL )] . κ 2 cosh[ 12 βR ] cosh[ 12 βL ]

(150)

Proof. From the very form of the symbol given in (65)–(67), we get a ∈ P C ∞ (T) with jumps at k± , and since ϕB ∈ C(T) for nonvanishing coupling, we also have a ∈ C(T). Moreover, the one-sided limits yield a (k± + 0) = a (k± − 0) = 0 which is assertion (a), and analogously for assertion (b). 2 References [1] G.A. Abanov, F. Franchini, Emptiness formation probability for the anisotropic XY spin chain in a magnetic field, Phys. Lett. A 316 (2003) 342–349. [2] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, Amer. Math. Soc., Providence, 2000. [3] H. Araki, On the diagonalization of a bilinear Hamiltonian by a Bogoliubov transformation, Publ. RIMS Kyoto Univ. 6 (1968) 385–442. [4] H. Araki, On quasifree states of CAR and Bogoliubov automorphisms, Publ. RIMS Kyoto Univ. 6 (1971) 385–442. [5] H. Araki, On the XY-model on two-sided infinite chain, Publ. RIMS Kyoto Univ. 20 (1984) 277–296. [6] H. Araki, T.G. Ho, Asymptotic time evolution of a partitioned infinite two-sided isotropic XY-chain, Proc. Steklov Inst. Math. 228 (2000) 191–204. [7] W.H. Aschbacher, On the emptiness formation probability in quasi-free states, Contemp. Math. 447 (2007) 1–16. [8] W.H. Aschbacher, Non-zero entropy density in the XY chain out of equilibrium, Lett. Math. Phys. 79 (2007) 1–16. [9] W.H. Aschbacher, A remark on the subleading order in the asymptotics of the nonequilibrium emptiness formation probability, Confluentes Math. 2 (2010) 293–311, arXiv:1009.1584. [10] W.H. Aschbacher, J.M. Barbaroux, Exponential spatial decay of spin–spin correlations in translation invariant quasifree states, J. Math. Phys. 48 (2007) 113302-1–113302-14. [11] W.H. Aschbacher, V. Jakšić, Y. Pautrat, C.A. Pillet, Topics in Non-Equilibrium Quantum Statistical Mechanics, Lecture Notes in Math., vol. 1882, Springer, New York, 2006, pp. 1–66.

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[12] W.H. Aschbacher, V. Jakšić, Y. Pautrat, C.A. Pillet, Transport properties of quasi-free fermions, J. Math. Phys. 48 (2007) 032101-1–032101-28. [13] W.H. Aschbacher, C.A. Pillet, Non-equilibrium steady states of the XY chain, J. Stat. Phys. 112 (2003) 1153–1175. [14] A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Springer, New York, 1999. [15] A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, Springer, Berlin, 2006. [16] J.W. Culvahouse, D.P. Schinke, L.G. Pfortmiller, Spin-spin interaction constants from the hyperfine structure of coupled ions, Phys. Rev. 177 (1969) 454–464. [17] M. D’Iorio, R.L. Armstrong, D.R. Taylor, Longitudinal and transverse spin dynamics of a one-dimensional XY system studied by chlorine nuclear relaxation in Pr Cl3 , Phys. Rev. B 27 (1983) 1664–1673. [18] F. Franchini, A.G. Abanov, Asymptotics of Toeplitz determinants and the emptiness formation probability for the XY spin chain, J. Phys. A: Math. Gen. 38 (2005) 5069–5095. [19] L. Hume, D.W. Robinson, Return to equilibrium in the XY model, J. Stat. Phys. 44 (1986) 829–848. [20] E. Lieb, T. Schultz, D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. 16 (1961) 407–466. [21] Y. Ogata, Diffusion of the magnetization profile in the XX model, Phys. Rev. E 66 (2002) 066123-1–066123-7. [22] D. Ruelle, Natural nonequilibrium states in quantum statistical mechanics, J. Stat. Phys. 98 (2000) 57–75. [23] M. Shiroishi, M. Takahashi, Y. Nishiyama, Emptiness formation probability for the one-dimensional isotropic XY model, J. Phys. Soc. Japan 70 (2001) 3535–3543. [24] A.V. Sologubenko, K. Giannò, H.R. Ott, A. Vietkine, A. Revcolevschi, Heat transport by lattice and spin excitations in the spin-chain compounds SrCuO2 and Sr2 CuO3 , Phys. Rev. B 64 (2001) 054412-1–054412-11. [25] J.R. Stembridge, Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math. 83 (1990) 96–131. [26] H. Widom, The strong Szeg˝o limit theorem for circular arcs, Indiana Univ. Math. J. 21 (1971) 277–283. [27] D.R. Yafaev, Mathematical Scattering Theory: General Theory, Amer. Math. Soc., Providence, RI, 1992.


Lévy–Ornstein–Uhlenbeck transition semigroup as second quantized operator S. Peszat ∗ Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland Received 16 August 2010; accepted 3 March 2011

Communicated by L. Gross

Abstract Let μ be an invariant measure for the transition semigroup (Pt ) of the Markov family defined by the Ornstein–Uhlenbeck type equation dX = AX dt + dL on a Hilbert space E, driven by a Lévy process L. It is shown that for any t 0, Pt considered on L2 (μ) is a second quantized operator on a Poisson Fock space of eAt . From this representation it follows that the transition semigroup corresponding to the equation on E = R, driven by an α-stable noise L, α ∈ (0, 2), is neither compact nor symmetric. © 2011 Elsevier Inc. All rights reserved. Keywords: Lévy–Ornstein–Uhlenbeck processes; Poisson chaos decomposition

1. Introduction Let μ be an invariant measure for a Markov family X = (X x ) on a measurable space (E, B), with the transition semigroup Pt ψ(x) = Eψ X x (t) ,

t 0, x ∈ E.

* Fax: +48 126173165.


3458

S. Peszat / Journal of Functional Analysis 260 (2011) 3457–3473

Then (Pt ) is a semigroup of contractions on any Lp (μ) := Lp (E, B(E), μ)-space, p ∈ [1, +∞]. From a probabilistic and an analytic point of view, there is motivation to analyze (Pt ) and μ as μ can be treated as a reference measure (especially in infinite-dimensional case) and the generator of (Pt ) as a reference operator. In the present paper X is defined by the Ornstein–Uhlenbeck type equation dX = AX dt + dL,

X(0) = x ∈ E,

(1)

where (A, D(A)) generates a C0 -semigroup eAt , t 0, on a Hilbert space (E, ·,·E ). The main result covers the general case of the so-called cylindrical process; that is the case where L is a Lévy process taking values in a Hilbert space E˜ ← E, for more details see Section 5. Analytic properties of the transition semigroup (Pt ) corresponding to the equation on a finite or infinite-dimensional state space, driven by a Wiener process have been studied for many years and now they are rather well understood. For example, there are if and only if conditions for its compactness, self-adjointness, analyticity, and hypercontractivity (see e.g. [7–11,14,13,23–26]). In particular, in the case of E = R, (Pt ) is symmetric, compact (even nuclear) and hypercontractiv. It will be shown in Theorem 7.3 of the present paper, that in the case of an α-stable noise, α = 2, (Pt ) in neither symmetric nor compact. The fact that the transition semigroup (Pt ) is not symmetric has been shown using different methods, by Albeverio, Rüdiger, and Wu [1] for α-stable processes, and by Applebaum and Goldys [3] for general noise, whereas the fact that (Pt ) is not hyperbounded; that is Pt L(Lp (μ),Lq (μ)) = ∞ for any t > 0 and 1 < p < q < ∞, was shown by Röckner and Wang [30]. Ornstein–Uhlenbeck equations with Lévy noise have been studied for over twenty years (see e.g. [2,5,4,6,15,20,21,28–30]) and now they are a subject of intensive studies (see e.g. [5,4,28]). However, even in the case of E = R, our knowledge on properties of their transition semigroups is rather limited. Namely, apart of the Albeverio, Rüdiger, and Wu, and the Applebaum and Goldys results on the lack of symmetry of (Pt ), Lescot and Röckner [20,21] identified the generator of (Pt ) as a pseudo-differential operator with an explicit symbol, and obtained an explicit formula for the square field operator of (Pt ), next Röckner and Wang [30] established Poincaré and Harnack type inequalities and showed that generally (Pt ) is not hyperbounded (for related results see [16]). In the case of a Wiener noise; Chojnowska-Michalik and Goldys [8] following Simon [32] and Feyel and de La Pradelle [12] showed that for each t 0, Pt is equal to the second quantized operator Γ (S0∗ (t)) of the adjoint semigroup S0∗ (t), t 0, where S0 (t) is the original semigroup eAt , t 0, “regarded” on the Reproducing Hilbert Kernel Space of μ. This representation is very useful for study properties of the transition semigroup (see Section 2 where the ChojnowskaMichalik and Goldys results will be sketched). The goal of the present paper is to formulate an analogous result for the Markov family defined by equation with Lévy noise (see Section 5). We will use the fact that μ is the distribution of ∞ Y∞ =

eAt dL(t),

(2)

0

and that the transition semigroup (Pt ) is given by the generalized Mehler formula (see e.g. [2,15])


f eAt x + y μt (dy),

Pt f (x) =

x ∈ E, t 0,

3459

(3)

E

where μt is the distribution of t Yt :=

eA(t−s) dL(s)

(4)

0

or equivalently of

t 0

eAs dL(s).

2. Gaussian case In this section we are dealing with (1) driven by a Wiener process. Namely, we assume that L(t) = BW (t), where W is a cylindrical Wiener process on E, see e.g. [8,11,28], B is a bounded linear operator on E, and eAt , t 0, is an exponentially stable semigroup on E satisfying ∞ At 2 e B

L(H S) (E,E)

dt < ∞.

0

Note that this estimate is in fact if and only if condition for the existence of an invariant measure μ. Moreover, due to the stability of the semigroup eAt , t 0, μ is unique. Finally μ is the distribution of Y∞ given by (2), and μ is mean-zero, Gaussian with the covariance operator ∞ Q∞ :=

∗

eAs BB ∗ eA s ds.

0

To simplify the exposition we assume that Ker Q∞ = {0}. 1/2 Let us recall that the Reproducing Hilbert Kernel Space of μ is the space E0 := Range Q∞ , equipped with the scalar product

1/2

1/2

Q∞ u, Q∞ v

E0

= u, vE ,

u, v ∈ E.

2.1. Second quantization −1/2

Given h ∈ E0 define a linear functional ψh (x) := Q∞ h, xE , x ∈ E. Then

−1/2 −1/2 ψh (x)ψu (x) μ(dx) = Q∞ Q∞ h, Q∞ u E = h, uE ,

h, u ∈ E0 .

(5)

E

Since E0 is dense in E, for any h ∈ E there is a sequence (hn ) ⊂ E0 converging in E to h. By (5),

3460


(ψhn ) converges in L2 (μ). The limit will be denoted by ψh . Clearly, ψh (x)ψu (x) μ(dx) = h, uE ,

∀h, u ∈ E.

E

Let Pn be the closed subspace of L2 (μ) spanned by p(ψh1 , . . . , ψhk ), k ∈ N, h1 , . . . , hk ∈ E, and p is a polynomial of order n. Let H0 be the space of all constant functions, and let Hn , n ∈ N, be the orthogonal complement of Hn−1 in Pn . The Itô–Wiener chaos decomposition says that L2 (μ) =

∞

Hn .

n=0

Let Prn be the orthogonal projection of L2 (μ) into Hn . Given an R ∈ L(E, E) define Γn (R) : Hn → Hn , n = 0, 1 . . . , by Γn (R)Prn (ψh1 . . . ψhn ) := Prn (ψRh1 . . . ψRhn ),

h1 , . . . , hn ∈ E.

One can show that Γn (R) is well defined and Γn (R) L(Hn ,Hn ) = R nL(E,E) . Hence for any linear contraction R on E, Γ (R) :=

∞

Γn (R)Prn

n=0

defines a contraction on L2 (μ). We call Γ (R) the second quantized operator of R, and Γ the second quantization operator. In the Gaussian case the action of the second quantization operator is well understood. In fact, the following lemma gathers some basic properties of Γ . For its proof we refer the reader to Lemma 2 and Proposition 2 from [8], and to Chapter 1 of [32]. Theorem 2.1. Assume that R, R1 , R2 are contractions on E. Then: (a) Γ (IE ) = IL2 (μ) , where IE and IL2 (μ) are the identity operators. (b) Γ (R1 R2 ) = Γ (R1 )Γ (R2 ), Γ (R ∗ ) = Γ (R)∗ . (c) Γ (R)1 = 1, and Γ (R) is positivity preserving; that is if f 0, μ-a.s., then Γ (R)f 0, μ-a.s. (d) The operator Γ (R) has an extension (restriction) to a positive contraction on every Lp (μ) for p 1. (e) For any p 1 and q0 = 1 +

p−1 , R 2L(E,E)

we have Γ (R) L(Lp (μ),Lq0 (μ)) = 1 and if q > q0 , then Γ (R) L(Lp (μ),Lq (μ)) = ∞.


3461

(f) If R is self-adjoint with a complete set of eigenvectors (vk ), then Γ (R) is also self-adjoint with the complete orthogonal set of eigenvectors ∞

a Praj ψvjj : (aj ) ⊂ N ∪ {0} and

j =1

aj < ∞.

j

(g) Let p, q 1 and R = 0. Then Γ (R) : Lp (μ) → Lq (μ) is compact if and only if R is a compact strict contraction and q < q0 , q0 is given in (e). (h) The operator Γ (R) is Hilbert–Schmidt on L2 (μ) if and only if R is a strict Hilbert–Schmidt contraction. Moreover, Γ (R)

L(H S) (L2 (μ),L2 (μ))

=√

1 . det(I − R ∗ R)

2.2. Second quantization of Mehler semigroup The following lemma and theorem were formulated and proven in [8], see Lemma 4, and Theorems 1, 2, 3. Let μt be the distribution of the random variable Yt given by (4). Clearly, μt is mean-zero Gaussian with the covariance t Qt :=

∗

eAs BB ∗ eA s ds.

0

It is convenient to formulate the following condition 1/2

Range Qt

1/2

= Range Q∞ = E0 . −1/2

(6) 1/2

Lemma 2.2. For any t 0, eAt E0 ⊂ E0 , and S0 (t) = Q∞ eAt Q∞ , t 0, is a C0 -semigroup of contractions on E. Moreover, S0 (t) L(E,E) < 1 if and only if (6) holds. Theorem 2.3. For any t 0, Pt = Γ (S0∗ (t)) and Pt∗ = Γ (S0 (t)). Moreover, the following statements hold: (a) Let t 0. If (6) holds, then for any p, q 1, Pt L(Lp (μ),Lq (μ)) = 1 if and only if q 1+

p−1 . S0 (t) 2L(E,E)

Otherwise, Pt L(Lp (μ),Lq (μ)) = ∞. (b) For p, q 1 and t 0, the operator Pt is compact from Lp (μ) into Lq (μ) if and only if (6) holds, S0 (t) is compact and q 0, the semigroup eAt has an extension to a bounded linear map, denoted also by eAt , from E˜ into E, and that eAt , t 0, is stable on E; that is |eAt x|E → 0 as t ↑ +∞ for any x ∈ E. (H.3) L is a pure jump process; that is E eix,L(t)E˜ = e−tΨ (x) ,

˜ x ∈ E,

where the co-called Lévy exponent Ψ (x) :=

1 − eix,yE˜ + ix, yE˜ χ{|y|E˜ 1} ν(dy),

E˜

and ν, called the Lévy measure of L, is a non-negative measure on E˜ satisfying 2 ∧ 1) ν(dx) < ∞. (|x| E˜ E˜ ∞ As e y |χ{|y| 1} − χ As (H.4) {|e y|E 1} | ν(dy) ds < ∞. E E˜ 0 E˜

∞

(H.5)

As 2 e y ∧ 1 ν(dy) ds < ∞. E

0 E˜

For t ∈ [0, +∞], define t mt :=

eAs y(χ{|y|E˜ 1} − χ{|eAs y|E 1} ) ν(dy) ds,

(10)

0 E˜

t νt := 0

−1 ν ◦ eAs ds.

(11)


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Clearly, by (H.4) and (H.5), for each t ∈ [0, +∞], the integral appearing in (10) converges, mt ∈ E, and νt is a measure on E satisfying

2 |y|E ∧ 1 νt (dy) < ∞.

E

Let Ψt (x) := ix, mt E +

1 − eix,yE + ix, yE χ{|y|E 1} νt (dy).

(12)

E

Proposition 5.1. Eq. (1) defines a Markov family (X x , x ∈ E) on E, with a unique invariant measure μ. Moreover, X x (t) = eAt x + Yt , where Yt is given by (4), and its distribution μt is infinitely divisible with the Lévy exponent Ψt and Lévy measure νt . Finally μ is the distribution of Y∞ defined by (2), it is infinitely divisible with the Lévy exponent Ψ∞ and Lévy measure ν∞ . Proof. This result is a rather standard generalization of [6]. Namely, the identity X x (t) = eAt x + Yt is the so-called mild formula for the solution. The question is only, if the process X x or equivalently Y takes values in E. To see this note that the law of Y is infinitely divisible ˜ with the jump measure νt . Then, by (H.5), νt is a jump measure of an infinitely-divisible in E, law on E. For more details see [6]. 2 ˜ and that the semigroup eAt , t 0, is exponentially stable; that Remark 5.2. Assume that E = E, is there are M, ω > 0 such that eAt L(E,E) Me−ωt , t 0. Then (H.4) and (H.5) hold if and only if log |x|E ν(dx) < ∞, {|x|E 1}

see [31] for finite-dimensional case and [6] for infinite-dimensional case. From now on μ = μ∞ , λ = ν∞ , and Π is a Poisson random measure on E with the intensity measure λ. Given a ξ ∈ Z+ (E) write ξ (dx) = ξ(dx)χ{|x|E >1} + ξ(dx) − λ(dx) χ{|x|E 1} . Recall that by the Lévy–Khinchin decomposition theorem there is a vector m ∈ E such that Y∞ = m +

x Π(dx). E

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6. Main result The main result of the present paper can be illustrated by the following diagram. Pt

L2 (μ) ⏐ ⏐ j

−−−−→

L2 (PΠ ) τ

−−−−→

L2 (μ) ⏐ ⏐ j

Γ (eAt )

∞

∞

2 n n n=0 L(s) (E , λ )

n n=0 ρeAt

−−−−−→

L2 (PΠ ) τ

(13)

∞

2 n n n=0 L(s) (E , λ ).

More precisely, we have the following theorem: Theorem 6.1. Under assumptions (H.1) to (H.5), for any t > 0, ρeAt is a contraction on L2 (E, λ) and (13) holds with

(jf )(ξ ) := f m + x ξ (dx) ,

f ∈ L2 (μ), ξ ∈ Z+ (E),

E

and τ :=

∞ 1 √ T n. n! n=0

Proof. Let us fix a t > 0. The contractivity of ρeAt on L2 (E, λ) follows from the fact that λ = ν∞ is given by (11). We have

ρ

e

2 At f (x) λ(dx) =

∞

E

E

At 2 f e x ν ◦ eAs −1 (dx) ds E

0 E

∞ =

A(t+s) 2 f e x E ν(dx) ds

0 E

∞

As 2 f e x ν(dx) ds = E

0 E

E

To see (13) it is enough to show that for all n ∈ N and f ∈ L2 (μ), T n (j Pt f ) = ρenAt T n (jf ).

f (x)2 λ(dx). E


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To do this fix an f ∈ L2 (μ). Given a t ∈ [0, +∞) denote by Y˜t a copy of Yt , independent of Ys , s ∈ [0, +∞], and by E˜ the expectation with respect to Y˜t , t ∈ [0, +∞). Note that for any t, the random variable eAt Y∞ + Y˜t has the law μ. Finally, by Proposition 5.1, the Mehler formula (3) reads ˜ eAt x + Y˜t . (14) Pt f (x) = Ef If n = 0, then, by (14), ˜ eAt Y∞ + Y˜t = Ef (Y∞ ) T 0 (j Pt f ) = E(j Pt f )(Π) = EPt f (Y∞ ) = EEf = Ejf (Π) = T 0 (jf ). Assume now that n 1. Then by (7), T n (j Pt f )(y1 , . . . , yn ) = EDyn1 ,...,yn j Pt f (Π)

n−|I | =E (−1) Pt f m + y δyi + Π (dy) I

=E

(−1)n−|I | Pt f

I

i∈I

E

yi + Y∞ .

i∈I

Hence, again by (14), T n (j Pt f )(y1 , . . . , yn ) =

˜ (−1)n−|I | EEf eAt yi + eAt Y∞ + Y˜t I

i∈I

n−|I | At = (−1) Ef e yi + Y∞ I

=

I

=

i∈I

(−1)n−|I | Ef m + y δeAt yi + Π (dy)

E

(−1)n−|I | Ejf

i∈I

δeAt yi + Π

I

i∈I n = EDeAt y ,...,eAt y jf (Π) n 1

= T n (jf ) eAt y1 , . . . , eAt yn .

2

As a direct consequence of Theorem 6.1 we have the following decomposition formula. Corollary 6.2. For all t 0 and f ∈ L2 (μ), Pt f (Y∞ ) =

∞

1 n n In ρeAt T (jf ) , n! n=0

where the series converges in L2 (Ω, F, P).

(15)

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7. One-dimensional case with density Assume that E = R, and that μ is absolutely continuous with respect to Lebesgue measure. Let q := dμ/dx. We start this section with two simple general observations. We will use them to the study of the equation driven by an α-stable noise. Note that κ : L2 (R, dx) f → f q −1/2 ∈ L2 (μ) is a linear isometry. Let T˜ n := T n ◦ j ◦ κ : L2 (R, dx) → L2(s) (Rn , λn ). By Theorem 6.1 we have the following fact. Lemma 7.1. Let t > 0. If Pt is a compact operator on L2 (μ), then for any n, ρenAt ◦ T˜ n is a compact operator from L2 (dx) to L2(s) (Rn , λn ). For f : R → R write ∇yn1 ,...,yn f (x) =

n−|I |

(−1)

f x− yi ,

I ⊂{1,...,n}

n ∈ N.

i∈I

Lemma 7.2. For any n, f : R → R and y = (y1 , . . . , yn ) ∈ Rn , n ˜ T f (y) = Gn (x, y)f (x) dx, R

where Gn (x, y) := q −1/2 (x)∇yn q(x). Proof. We have T˜ n f (y1 , . . . , yn ) = EDyn1 ,...,yn j f q −1/2 (Π)

n−|I | −1/2 = z+ (−1) f z+ yi q yi q(z) dz I ⊂{1,...,n}

=

i∈I

R

Gn (x, y)f (x) dx.

i∈I

2

R

7.1. α-Stable case Assume that L is a symmetric α-stable process taking values in E = R. Then its Lévy exponent is Ψ (x) = |x|α and the Lévy measure is λ(dx) = cα |x|−1−α dx. Given γ > 0, consider the following one-dimensional Lévy–Ornstein–Uhlenbeck equation dX = −γ X dt + (αγ )1/α dL.

(16)


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Note that the invariant measure μ for the Markov family given by (16) is symmetric α-stable. Our main result of this section is the following theorem. Theorem 7.3. Let α ∈ (0, 2). Then for any t > 0, Pt is neither compact nor self-adjoint. Moreover, the adjoint semigroup is given by Pt∗ g(Y∞ ) =

∞ −nγ αt

e n=0

n!

In ρenγ t T n (jg) ,

∀t 0, ∀g ∈ L2 (μ),

(17)

and (Pt ) satisfies the following spectral gap property 2 Pt f (x) − Pt f (z) μ(dz) μ(dx) e−γ αt |f |2 2 L (μ) R

(18)

R

for all t 0 and f ∈ L2 (μ). Proof. Note for any z ∈ R \ {0}, ρz is a bounded bijection on L2 (R, λ). Therefore, as far as compactness is concerned, then, taking into account Lemma 7.1, it is enough to show that T˜ := T˜ 1 is not compact from L2 (dx) into L2 (R, λ). By Lemma 7.2, T˜ is given by the kernel G1 (x, y) =

q(x − y) − q(x) , √ q(x)

x, y ∈ R,

where q is the density of the α-stable law μ. We will use the fact, see e.g. [17], that q ∈ C 1 (R), q(x) > 0, and q(x) decreases like |x|−1−α as |x| → ∞. Let fn (x) = χ[n,n+1] (x), n ∈ N, x ∈ R. Then

gn (y) := T˜ fn (y) =

1 0

q(x + n − y) − q(x + n) dx, √ q(x + n)

y ∈ R, n ∈ N.

Hence there are constants C0 , C1 > 0 and n0 ∈ N such that for m n0 , 1 gm (y) C0

q(x + m)−1/2 dx C1 m(1+α)/2 ,

∀y ∈ [m, m + 1].

0

Next note that for any n there is an mn such that for all m mn , q(x + n − y) q(x + n), Therefore, for all n ∈ N and m mn ,

∀x ∈ [0, 1], ∀y ∈ [m, m + 1].

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S. Peszat / Journal of Functional Analysis 260 (2011) 3457–3473 m+1

|gm − gn |2L2 (R,λ)

m

gm (y) − gn (y)2 cα dy |y|1+α

m+1

C12 m1+α m

cα dy |y|1+α

αC12 cα m1+α m−α − (m + 1)−α , and consequently there is a constant C > 0 such that |gm − gn |2L2 (R,λ) C,

∀n n0 , ∀m mn .

Thus the sequence (gn ) is not relatively compact in L2 (R, λ), and hence T˜ is not compact. We will show that for t > 0, Pt is not symmetric. By Theorems 3.1 and 6.1, for f, g ∈ L2 (μ), Pt f, gL2 (μ) = EPt f (Y∞ )g(Y∞ ) = E(j Pt f )(Π)(jg)(Π) =

∞

1 n E ρe−γ t T n (jf ), T n (jg) L2 (R,λn ) . n! n=0

Note that for the operator ρen−γ t considered on L2 (Rn , λn ) we have

ρen−γ t

∗

h(y1 , . . . , yn ) = e−nγ αt h eγ t y1 , . . . , eγ t yn ,

and hence (17) holds. To show that Pt is not symmetric it is enough to find a g ∈ L2 (μ) such that I1 ρe−γ t T (jg) = e−γ αt I1 ρeγ t T (jg) . Since T (jg)(y) = E g(Y∞ + y) − g(Y∞ ) =

g(x + y) − g(x) μ(dx),

R

(19) can be written in the equivalent form KL :=

g x + e−γ t y − g(x) μ(dx) Π(dy)

R R

= KR := e

−γ αt

R R

g x + eγ t y − g(x) μ(dx) Π(dy).

(19)


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Take g(x) = eix . Then KL = e−1

ie−γ t y e − 1 Π(dy),

KR = e−γ αt−1

R

ieγ t y e − 1 Π(dy).

R

Thus for any z ∈ R, Ee

izeKL

ie−γ t y e = E exp iz − 1 Π(dy) R

ie−γ t y −1) = exp − 1 − eiz(e λ(dy) R

and ieγ t y e − 1 Π(dy) EeizeKR = E exp ize−γ αt R

γt ize−γ αt (eie y −1) 1−e λ(dy) . = exp − R

To see the spectral gap property note that by Theorems 3.1, 6.1, and 3.3, for f ∈ L2 (μ), 2 Pt f (x) − Pt f (z) μ(dz) μ(dx) = EPt f (Y∞ ) − EPt f (Y∞ )2 R

R

=

∞

n ρ −γ t T n (jf )2 2 e

L (Rn ,λn )

.

n=1

By direct calculation for any z > 0 and any h ∈ L2 (Rn , λn ), we have |ρzn h|L2 (Rn ,λn ) = znα/2 |h|L2 (Rn ,λn ) . Hence n ρ

z L(L2 (Rn ,λn );L2 (Rn ,λn ))

and consequently (18) holds.

= znα/2 ,

2

Acknowledgments The work has been supported by Polish Ministry of Science and Higher Education Grant “Stochastic Equations in Infinite-Dimensional Spaces” Nr N N201 419039. The author would like to thank the anonymous referee for his valuable comments and remarks which helped to improve the presentation of the results.

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The Cuntz semigroup and comparison of open projections ✩ Eduard Ortega a , Mikael Rørdam b,∗ , Hannes Thiel b a Department of Mathematical Sciences, NTNU, NO-7491 Trondheim, Norway b Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100,

Copenhagen Ø, Denmark Received 24 August 2010; accepted 15 February 2011 Available online 3 March 2011 Communicated by S. Vaes

Abstract We show that a number of naturally occurring comparison relations on positive elements in a C ∗ -algebra are equivalent to natural comparison properties of their corresponding open projections in the bidual of the C ∗ -algebra. In particular we show that Cuntz comparison of positive elements corresponds to a comparison relation on open projections, that we call Cuntz comparison, and which is defined in terms of—and is weaker than—a comparison notion defined by Peligrad and Zsidó. The latter corresponds to a well-known comparison relation on positive elements defined by Blackadar. We show that Murray–von Neumann comparison of open projections corresponds to tracial comparison of the corresponding positive elements of the C ∗ -algebra. We use these findings to give a new picture of the Cuntz semigroup. © 2011 Elsevier Inc. All rights reserved. Keywords: C ∗ -algebras; Cuntz semigroup; von Neumann algebras; Open projections

✩ This research was supported by the NordForsk Research Network “Operator Algebras and Dynamics” (grant #11580). The first named author was partially supported by the Research Council of Norway (project 191195/V30), by MECDGESIC (Spain) through Project MTM2008-06201-C02-01/MTM, by the Consolider Ingenio “Mathematica” project CSD2006-32 by the MEC, and by 2009 SGR 1389 grant of the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. The second named author was supported by grants from the Danish National Research Foundation and the Danish Natural Science Research Council (FNU). The third named author was partially supported by the Marie Curie Research Training Network EU-NCG and by the Danish National Research Foundation. * Corresponding author. E-mail addresses: [email protected] (E. Ortega), [email protected] (M. Rørdam), [email protected] (H. Thiel).


E. Ortega et al. / Journal of Functional Analysis 260 (2011) 3474–3493

3475

1. Introduction There is a well-known bijective correspondence between hereditary sub-C ∗ -algebras of a C ∗ algebra and open projections in its bidual. Thus to every positive element a in a C ∗ -algebra A one can associate the open projection pa in A∗∗ corresponding to the hereditary sub-C ∗ -algebra Aa = aAa. Any comparison relation between positive elements in a C ∗ -algebra that is invariant under the relation a ∼ = b, defined by a ∼ = b ⇔ Aa = Ab , can in this way be translated into a comparison relation between open projections in the bidual. Vice versa, any comparison relation between open projections corresponds to a comparison relation (which respects ∼ =) on positive elements of the underlying C ∗ -algebra. Peligrad and Zsidó defined in [19] an equivalence relation (and also a sub-equivalence relation) on open projections in the bidual of a C ∗ -algebra as Murray–von Neumann equivalence with the extra assumption that the partial isometry that implements the equivalence gives an isomorphism between the corresponding hereditary sub-C ∗ -algebras of the given C ∗ -algebra. Very recently, Lin [17], noted that the Peligrad–Zsidó (sub-)equivalence of open projections corresponds to a comparison relation of positive elements considered by Blackadar in [6]. The Blackadar comparison relation of positive elements is stronger than the Cuntz comparison relation of positive elements that is used to define the Cuntz semigroup of a C ∗ -algebra. The Cuntz semigroup has recently come to play an influential role in the classification of C ∗ algebras. We show that Cuntz comparison of positive elements corresponds to a natural relation on open projections, that we also call Cuntz comparison. It is defined in terms of—and is weaker than—the Peligrad–Zsidó comparison. It follows from results of Coward, Elliott, and Ivanescu [10], and from our results, that the Blackadar comparison relation is equivalent to Cuntz comparison of positive elements when the C ∗ -algebra is separable and has stable rank one, and consequently that Peligrad–Zsidó comparison is equivalent to our notion of Cuntz comparison of open projections in this case. The best known and most natural comparison relation for projections in a von Neumann algebra is the one introduced by Murray and von Neumann. It is weaker than the Cuntz and the Peligrad–Zsidó comparison relations. We show that Murray–von Neumann (sub-)equivalence of open projections in the bidual in the separable case is equivalent to tracial comparison of the corresponding positive elements of the C ∗ -algebra. Tracial comparison is defined in terms of dimension functions arising from lower semicontinuous tracial weights on the C ∗ -algebra. The proof of this equivalence builds on two results on von Neumann algebras that may have independent interest, and which probably are known to experts: One says that Murray–von Neumann comparison of projections in any von Neumann algebra which is not too big (in the sense of Tomiyama—see Section 5 for details) is completely determined by normal tracial weights on the von Neumann algebra. The other result states that every lower semicontinuous tracial weight on a C ∗ -algebra extends (not necessarily uniquely) to a normal tracial weight on the bidual of the C ∗ -algebra. We use results of Elliott, Robert, and Santiago [11], to show that tracial comparison of positive elements in a C ∗ -algebra is equivalent to Cuntz comparison if the C ∗ -algebra is separable and exact, its Cuntz semigroup is weakly unperforated, and the involved positive elements are purely non-compact. We also relate comparison of positive elements and of open projections to comparison of the associated right Hilbert A-modules. The Hilbert A-module corresponding to a positive element a in A is the right ideal aA. We show that Blackadar equivalence of positive elements is equivalent to isomorphism of the corresponding Hilbert A-modules, and we recall that Cuntz comparison of

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positive elements is equivalent to the notion of Cuntz comparison of the corresponding Hilbert A-modules introduced in [10]. 2. Comparison of positive elements in a C ∗ -algebra We remind the reader about some, mostly well-known, notions of comparison of positive elements in a C ∗ -algebra. If a is a positive element in a C ∗ -algebra A, then let Aa denote the hereditary sub-C ∗ -algebra generated by a, i.e., Aa = aAa. The Pedersen equivalence relation on positive elements in a C ∗ -algebra A is defined by a ∼ b if a = x ∗ x and b = xx ∗ for some x ∈ A, where a, b ∈ A+ , and it was shown by Pedersen, that this indeed defines an equivalence relation. Write a ∼ = b if Aa = Ab . The equivalence relation generated by these two relations was considered by Blackadar in [5, Definition 6.1.2]: Definition 2.1 (Blackadar comparison). Let a and b be positive elements in a C ∗ -algebra A. Write a ∼s b if there exists x ∈ A such that a ∼ = x ∗ x and b ∼ = xx ∗ , and write a s b if there exists + a ∈ Ab with a ∼s a . (It follows from Lemma 4.2 below that ∼s is an equivalence relation.) Note that s is not an order relation on A+ /∼s since in general a s b s a does not imply a ∼s b (see [16, Theorem 9]). If p and q are projections, then p ∼s q agrees with the usual notion of equivalence of projections defined by Murray and von Neumann, denoted by p ∼ q. The relation defining the Cuntz semigroup that currently is of importance in the classification program for C ∗ -algebras is defined as follows: Definition 2.2 (Cuntz comparison of positive elements). Let a and b be positive elements in a C ∗ -algebra A. Write a b if there exists a sequence {xn } in A such that xn∗ bxn → a. Write a ≈ b if a b and b a. 2.3 (The Cuntz semigroup). Let us briefly remind the reader about the ordered Cuntz semigroup + W (A) associated to a C ∗ -algebra A. Let M∞ (A)+ denote the disjoint union ∞ n=1 Mn (A) . For + + + a ∈ Mn (A) and b ∈ Mm (A) set a ⊕ b = diag(a, b) ∈ Mn+m (A) , and write a b if there exists a sequence {xk } in Mm,n (A) such that xk∗ bxk → a. Write a ≈ b if a b and b a. Put W (A) = M∞ (A)+ /≈, and let a ∈ W (A) be the equivalence class containing a. Let us denote by Cu(A) the completion of W (A) with respect to countable suprema, i.e., Cu(A) := W (A ⊗ K). Lastly we define comparison by traces. We shall here denote by T (A) the set of (norm) lower semicontinuous tracial weights on a C ∗ -algebra A. We remind the reader that a tracial weight on A is an additive function τ : A+ → [0, ∞] satisfying τ (λa) = λτ (a) and τ (x ∗ x) = τ (xx ∗ ) for all a ∈ A+ , x ∈ A, and λ ∈ R+ . That τ is lower semicontinuous means that τ (a) = lim τ (ai ) whenever {ai } is a norm-convergent increasing sequence (or net) with limit a. Each τ ∈ T (A) gives rise to a lower semicontinuous dimension function dτ : A+ → [0, ∞] given by dτ (a) = supε>0 τ (fε (a)), where fε : R+ → R+ is the continuous function that is 0 on 0, 1 on [ε, ∞), and linear on [0, ε]. Any dimension function gives rise to an additive order preserving state on the Cuntz semigroup, and in particular it preserves the Cuntz relation . Definition 2.4 (Comparison by traces). Let a and b be positive elements in a C ∗ -algebra A. Write a ∼tr b and a tr b if dτ (a) = dτ (b), respectively, dτ (a) dτ (b), for all τ ∈ T (A).


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Remark 2.5. Observe that a s b

⇒

ab

⇒

a tr b,

a ∼s b

⇒

a≈b

⇒

a ∼tr b

for all positive elements a and b in any C ∗ -algebra A. In Section 6 we discuss under which conditions these implications can be reversed. 3. Open projections The bidual A∗∗ of a C ∗ -algebra A can be identified with the von Neumann algebra arising as the weak closure of the image of A under the universal representation πu : A → B(Hu ) of A. Following Akemann [1, Definition II.1], and Pedersen [18, Proposition 3.11.9, p. 77], a projection p in A∗∗ is said to be open if it is the strong limit of an increasing sequence of positive elements from A, or, equivalently, if it belongs to the strong closure of the hereditary sub-C ∗ -algebra pA∗∗ p ∩ A of A. We shall denote this hereditary sub-C ∗ -algebra of A by Ap . (This agrees with the previous definition of Ap if p is a projection in A.) The map p → Ap furnishes a bijective correspondence between open projections in A∗∗ and hereditary sub-C ∗ -algebras of A. The open projection corresponding to a hereditary sub-C ∗ -algebra B of A is the projection onto the closure of the subspace πu (B)Hu of Hu . Let Po (A∗∗ ) denote the set of open projections in A∗∗ . A projection in A∗∗ is closed if its complement is open. For each positive element a in A we let pa denote the open projection in A∗∗ corresponding to the hereditary sub-C ∗ -algebra Aa of A. Equivalently, pa is equal to the range projection of πu (a), and if a is a contraction, then pa is equal to the strong limit of the increasing sequence {a 1/n }. Notice that pa = pb if and only if Aa = Ab if and only if a ∼ = b. If A is separable, then each hereditary sub-C ∗ -algebra of A contains a strictly positive element and hence is of the form Aa for some a. It follows that every open projection in A∗∗ is of the form pa for some positive element a in A, whence there is a bijective correspondence between open projections in A∗∗ and positive elements in A modulo the equivalence relation ∼ =. 3.1 (Closure of a projection). If K ⊆ Po (A∗∗ ) is a family of open projections, then their supre mum K is again open. Dually, the infimum of a family of closed projections is again closed. Therefore, if we are given any projection p, then we can define its closure p as p :=

q ∈ P A∗∗ : q is closed, p q .

We shall consider various notions of comparisons and equivalences of open projections in A∗∗ that, via the correspondence a → pa , match the notions of comparison and equivalences of positive elements in a C ∗ -algebra considered in the previous section. First of all we have Murray– von Neumann equivalence ∼ and subequivalence of projections in any von Neumann algebra. We shall show in Section 5 that they correspond to tracial comparison. Peligrad and Zsidó made the following definition: Definition 3.2 (PZ-equivalence). (See [19, Definition 1.1].) Let A be a C ∗ -algebra, and let p and q be open projections in A∗∗ . Then p, q are equivalent in the sense of Peligrad and Zsidó (PZ-equivalent, for short), denoted by p ∼PZ q, if there exists a partial isometry v ∈ A∗∗ such

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that p = v ∗ v,

q = vv ∗ ,

vAp ⊆ A,

v ∗ Aq ⊆ A.

Say that p PZ q if there exists p ∈ Po (A∗∗ ) such that p ∼PZ p q. PZ-equivalence is stronger than Murray–von Neumann equivalence. We will see in Section 6 that it is in general strictly stronger, but the two equivalences do agree for some C ∗ -algebras and for some classes of projections. We will now turn to the question of PZ-equivalence of left and right support projections. Peligrad and Zsidó proved in [19, Theorem 1.4] that pxx ∗ ∼PZ px ∗ x for every x ∈ A (and even for every x in the multiplier algebra of A). One can ask whether the converse is true. The following result gives a satisfactory answer. Proposition 3.3. Let p, q ∈ Po (A∗∗ ) be two open projections with p ∼PZ q. If p is the support projection of some element in A, then so is q, and in this case p = pxx ∗ and q = px ∗ x for some x ∈ A. Proof. There is a partial isometry v in A∗∗ with p = v ∗ v, vv ∗ = q, and vAp ⊆ A. This implies that vAp v ∗ ⊆ A, so the map x → vxv ∗ defines a ∗ -isomorphism from Ap onto Aq . By assumption, p = pa for some positive element a in A. Upon replacing a by a−1 a we can assume that a is a contraction. Put b := vav ∗ ∈ A+ . Then 1/n pb = sup vav ∗ = sup va 1/n v ∗ = vpa v ∗ = q. n

n

Hence q is a support projection, and moreover for x := va 1/2 ∈ A we have a = x ∗ x and xx ∗ = b. 2 Remark 3.4. As noted above, every open projection in the bidual of a separable C ∗ -algebra is realized as a support projection, so that PZ-equivalence of two open projections means precisely that they are the left and right support projections of some element in A. 3.5 (Compact and closed projections). We define below an equivalence relation and an order relation on open projections that we shall show to match Cuntz comparison of positive elements (under the correspondence a → pa ). To this end we need to define the concept of compact containment, which is inspired by the notion of a compact (and closed) projection developed by Akemann. The idea first appeared in [1], although it was not given a name there, and it was later termed in the slightly different context of the atomic enveloping von Neumann algebra in [2, Definition II.1]. Later again, it was studied by Akemann, Anderson, and Pedersen in the context of the universal enveloping von Neumann algebra (see [3, after Lemma 2.4]). A closed projection p ∈ A∗∗ is called compact if there exists a ∈ A+ of norm one such that pa = p. See [3, Lemma 2.4] for equivalent conditions. Note that a compact, closed projection p ∈ A∗∗ must be dominated by some positive element of A (since pa = p implies p = apa a 2 ∈ A). The converse also holds (this follows from the result [2, Theorem II.5] transferred to the context of the universal enveloping von Neumann algebra).


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Definition 3.6 (Compact containment). Let A be a C ∗ -algebra, and let p, q ∈ Po (A∗∗ ) be open projections. We say that p is compactly contained in q (denoted p q) if p is a compact projection in Aq , i.e., if there exists a positive element a in Aq with a = 1 and pa = p. Further, let us say that an open projection p is compact if it is compactly contained in itself, i.e., if p p. Proposition 3.7. An open projection in A∗∗ is compact if and only if it belongs to A. Proof. Every projection in A is clearly compact. If p is open and compact, then by definition there exists a ∈ (Ap )+ such that pa = p. This implies that p p a p, whence p = a ∈ A. 2 Remark 3.8. Note that compactness was originally defined only for closed projections in A∗∗ (see 3.5). In Definition 3.6 above we also defined a notion of compactness for open projections in A∗∗ by assuming it to be compactly contained in itself. This should cause no confusion since, by Proposition 3.7, a compact, open projection is automatically closed as well as compact in the sense defined for closed projections in 3.5. Now we can give a definition of (sub-)equivalence for open projections that we term Cuntz (sub-)equivalence, and which in the next section will be shown to agree with Cuntz (sub-)equivalence for positive elements and Hilbert modules in a C ∗ -algebra. We warn the reader that our definition of Cuntz equivalence (below) does not agree with the notion carrying the same name defined by Lin in [17]. The latter was the one already studied by Peligrad and Zsidó that we (in Definition 3.2) have chosen to call Peligrad–Zsidó equivalence (or PZ-equivalence). Our definition below of Cuntz equivalence for open projections turns out to match the notion of Cuntz equivalence for positive elements, also when the C ∗ -algebra does not have stable rank one. Definition 3.9 (Cuntz comparison of open projections). Let A be a C ∗ -algebra, and let p and q be open projections in A∗∗ . We say that p is Cuntz subequivalent to q, written p Cu q, if for every open projection p p there exists an open projection q with p ∼PZ q q. If p Cu q and q Cu p hold, then we say that p and q are Cuntz equivalent, which we write as p ∼Cu q. 4. Comparison of positive elements and the corresponding relation on open projections We show in this section that the Cuntz comparison relation on positive elements corresponds to the Cuntz relation on the corresponding open projections. We also show that the Blackadar relation on positive elements, the Peligrad–Zsidó relation on their corresponding open projections, and isometric isomorphism of the corresponding Hilbert modules are equivalent. 4.1 (Hilbert modules). See [4] for a good introduction to Hilbert A-modules. Throughout this note all Hilbert modules are assumed to be right modules and countably generated. Let A be a general C ∗ -algebra. We will denote by H(A) the set of isomorphism classes of Hilbert Amodules. Every closed, right ideal in A is in a natural way a Hilbert A-module. In particular, Ea := aA is a Hilbert A-module for every element a in A. The assignment a → Ea defines a natural map from the set of positive elements of A to H(A).

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If E and F are Hilbert A-modules, then E is said to be compactly contained in F , written E F , if there exists a positive element x in K(F ), the compact operators of L(F ), such that xe = e for all e ∈ E. For two Hilbert A-modules E, F we say that E Cu F (E is Cuntz subequivalent to F ) if for every Hilbert A-submodule E E there exists F F with E ∼ = F (isometric isomorphism). Further declare E ≈ F (Cuntz equivalence) if E Cu F and F Cu E. Before relating the Blackadar relation with the Peligrad–Zsidó relation we prove the following lemma restating the Blackadar relation: Lemma 4.2. Let A be a C ∗ -algebra, and let a and b be positive elements in A. The following conditions are equivalent: (i) (ii) (iii) (iv) (v)

a ∼s b, there exist a , b ∈ A+ with a ∼ = a ∼ b ∼ = b, there exists x ∈ A such that Aa = Ax ∗ x and Ab = Axx ∗ , there exists b ∈ A+ with a ∼ b ∼ = b, there exists a ∈ A+ with a ∼ = a ∼ b.

Proof. (ii) is just a reformulation of (i), and (iii) is a reformulation of (ii) keeping in mind that Ac = Ad if and only if c ∼ = d. (iv) ⇒ (ii) and (v) ⇒ (ii) are trivial. (iii) ⇒ (v): Take x ∈ A such that Aa = Ax ∗ x and Ab = Axx ∗ . Let x = v|x| be the polar decomposition for x (with v a partial isometry in A∗∗ ). Then c → v ∗ cv defines an isomorphism from Axx ∗ = Ab onto Ax ∗ x = Aa . This isomorphism maps the strictly positive element b of Ab onto a strictly positive element a = v ∗ bv of Aa . Hence b ∼ a ∼ = a as desired. The proof of (iii) ⇒ (iv) is similar. 2 The equivalence of (i) and (iv) in the proposition below was noted to hold in Lin’s recent paper [17]. We include a short proof of this equivalence for completeness. Proposition 4.3. Let A be a C ∗ -algebra, and let a and b be positive elements in A. The following conditions are equivalent: (i) (ii) (iii) (iv)

a ∼s b, Ea and Eb are isomorphic as Hilbert A-modules, there exists x ∈ A such that Ea = Ex ∗ x and Eb = Exx ∗ , pa ∼PZ pb .

Proof. (i) ⇒ (iv): As remarked earlier, it was shown in [19, Theorem 1.4] that px ∗ x ∼PZ pxx ∗ for all x ∈ A. In other words, a ∼ b implies pa ∼PZ pb . Recall also that pa = pb when a ∼ = b. These facts prove the implication. (iv) ⇒ (i): If pa ∼PZ pb , then by Proposition 3.3, there exist positive elements a and b in A such that pa = pa , pb = pb , and a ∼ b . Now, pa = pa and pb = pb imply that a ∼ = a and ∼ b = b , whence (i) follows (see also Lemma 4.2). (ii) ⇒ (iii): Let Φ : Ea → Eb be an isomorphism of Hilbert A-modules, i.e., a bijective Alinear map preserving the inner product. Set x := Φ(a) ∈ Eb . Then


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xA = Φ(a)A = Φ(aA) = Eb , whence Eb = Ex = Exx ∗ . Since Φ preserves the inner product,

a 2 = a, a Ea = Φ(a), Φ(a) E = x ∗ x. b

Hence Ea = Ea 2 = Ex ∗ x and Eb = Exx ∗ . (iii) ⇒ (ii): Let x = v|x| be the polar decomposition of x in A∗∗ . Note that E|x| = Ex ∗ x and Exx ∗ = E|x ∗ | . Define an isomorphism E|x| → E|x ∗ | by z → vz. (i) ⇔ (iii): This follows from the one-to-one correspondence between hereditary sub-C ∗ algebras and right ideals: A hereditary sub-C ∗ -algebra B corresponds to the right ideal BA, and, conversely, a right ideal R corresponds the hereditary algebra R ∗ R. In particular, Ea = Aa A and Aa = Ea∗ Ea . If (i) holds, then, by Lemma 4.2, Aa = Ax ∗ x and Axx ∗ = Ab for some x ∈ A. This shows that Ea = Aa A = Ax ∗ x A = Ex ∗ x and, similarly, Eb = Exx ∗ . In the other direction, if Ea = Ex ∗ x and Exx ∗ = Eb for some x ∈ A, then Aa = Ea∗ Ea = ∗ Ex ∗ x Ex ∗ x = Ax ∗ x and, similarly, Ab = Axx ∗ , whence a ∼s b. 2 4.4. It follows from the proof of (ii) ⇒ (iii) of the proposition above that if a is a positive element in a C ∗ -algebra A and if F is a Hilbert A-module such that Ea ∼ = F , then F = Eb for some positive element b in A. In fact, if Φ : Ea → F is an isometric isomorphism, then we can take b to be Φ(a) as in the before mentioned proof. 4.5. For any pair of positive elements a and b in a C ∗ -algebra A we have the following equivalences: a ∈ Ab

⇔

Aa ⊆ Ab

⇔

Ea ⊆ Eb

⇔

p a pb ,

as well as the following equivalences: a ∈ Ab

and b ∈ Aa

⇔

a∼ =b

⇔

Aa = Ab

⇔

Ea = Eb

⇔

pa = pb .

As a consequence of Proposition 4.3, Lemma 4.2, and the remark above we obtain the following proposition: Proposition 4.6. Let A be a C ∗ -algebra, and let a and b be positive elements in A. The following conditions are equivalent: (i) (ii) (iii) (iv)

a s b, there exists a Hilbert A-module E such that Ea ∼ = E ⊆ Eb , there exists x ∈ A with Ea = Ex ∗ x and Exx ∗ ⊆ Eb , pa PZ pb .

Lemma 4.7. Let a and e be positive elements in a C ∗ -algebra A and assume that e is a contraction. Then the following equivalences hold ae = a

⇔

pa e = pa

⇔

pa e = pa .

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Proof. The two “⇐”-implications are trivial. Suppose that ae = a. Let χ be indicator function for the singleton {1}, and put q = χ(e) ∈ A∗∗ . Then qe = q and q is the largest projection in A∗∗ with this property. Moreover, q is the projection onto the kernel of 1 − e, hence 1 − q is the projection onto the range of 1 − e, i.e., 1 − q = p1−e . This shows that q is a closed projection. As a and 1−e are orthogonal so are their range projections pa and p1−e , whence pa 1−p1−e = q. Thus pa q. This shows that pa e = pa . 2 Lemma 4.8. Let A be a C ∗ -algebra, and let e and a be positive elements in A. If ae = a, then p a pe . Proof. Upon replacing e with f (e), where f : R+ → R+ is given by f (t) = max{t, 1}, we may assume that e is a contraction. If ae = a, then pa e = pa by Lemma 4.7, and this implies that p a pe . 2 We show below that the two previously defined notions of compact containment agree. To do so we introduce a third notion of compact containment: Definition 4.9. Let a and b be positive elements in a C ∗ -algebra. Then a is said to be compactly contained in b, written a b, if and only if there exists a positive element e in Ab such that ea = a. Following the proof of Lemma 4.8, the element e above can be assumed to be a contraction. Proposition 4.10. Let A be a C ∗ -algebra, let b be a positive element in A, and let a be a positive element in Ab . Then the following statements are equivalent: (i) (ii) (iii) (iv)

Ea Eb , a b, pa pb , pa pb and pa is compact in A.

Proof. (i) ⇔ (ii): By definition, (i) holds if and only if there exists a positive element e in K(Eb ), such that e acts as the identity on Ea . We can identify K(Eb ) with Ab , as elements of the latter act on Eb by left-multiplication. Thus (i) is equivalent to the existence of a positive element e in Ab such that ex = x for all x ∈ Ea = aA. The latter condition is fulfilled precisely if ea = a. (ii) ⇔ (iii): (iii) holds if and only if there exists a positive element e in Ab such that pa e = pa ; and (ii) holds if and only if there exists a positive element e in Ab such that ae = a. In both cases e can be taken to be a contraction, cf. the proof of Lemma 4.8. The bi-implication now follows from Lemma 4.7. (ii) and (iii) ⇒ (iv): If a b, then there is a positive contraction e in Ab such that ae = a. By Lemma 4.8 this implies that pa pe pb . From (iii) we have that pa is compact in Ab which entails that p a also is compact in A. (iv) ⇒ (iii): This is [3, Lemma 2.5]. 2 Remark 4.11. In many cases it is automatic that p is compact, and then p q is equivalent to the condition p q. For example, if A is unital, then all closed projections in A∗∗ are compact. More generally, if a ∈ A+ sits in some corner qAq for a projection q ∈ A, then pa is compact.


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Lemma 4.12. Let a be a positive element in a C ∗ -algebra A. (i) If E is a Hilbert A-module that is compactly contained in Ea , then E ⊆ E(e−ε)+ for some positive element e ∈ Aa and some ε > 0. (ii) If q, q are open projections in A∗∗ such that q is compactly contained in q, then q p(e−ε)+ for some positive element e ∈ Aq and some ε > 0. Proof. (i): By definition there is a positive element e in K(Ea ) = Aa such that ex = x for all x ∈ E . This implies that (e − 1/2)+ x = 12 x for all x ∈ E , whence E ⊆ E(e−1/2)+ . (ii): If q is compactly contained in q, then there is a positive element e in Aq such that q e = q (in fact such that q e = q ). It follows that q (e − 1/2)+ = 12 q , and hence that q p(e−1/2)+ . 2 Proposition 4.13. Let a and b be positive elements in a C ∗ -algebra A. Then the following statements are equivalent: (i) a b. (ii) Ea Cu Eb . (iii) pa Cu pb . Proof. The equivalence of (i) and (ii) was first shown in [10, Appendix], see also [4, Theorem 4.33]. (ii) ⇒ (iii): Suppose that Ea Cu Eb , and let p be an arbitrary open projection in A∗∗ which is compactly contained in pa . Then, by Lemma 4.12, p p(e−ε)+ for some positive element e in Aa and some ε > 0. Notice that (e − ε)+ a. It follows from Proposition 4.10 that E(e−ε)+ is compactly contained in Ea . Accordingly, E(e−ε)+ ∼ = F for some Hilbert A-module F that is compactly contained in Eb . By 4.4, F = Ec for some positive element c in A. It now follows from Proposition 4.10 and from Proposition 4.3 that p p(e−ε)+ ∼PZ pc pb . This shows that pa Cu pb . (iii) ⇒ (ii): Suppose that pa Cu pb , and let E be an arbitrary Hilbert A-module which is compactly contained in Ea . Then, by Lemma 4.12, E ⊆ E(e−ε)+ for some positive element e in Aa and some ε > 0. It follows from Proposition 4.10 that p(e−ε)+ is compactly contained in pa . Accordingly, p(e−ε)+ ∼PZ q for some open projection q in A∗∗ that is compactly contained in pb . By Proposition 3.3, q = pc for some positive element c in A. It now follows from Proposition 4.10 and from Proposition 4.3 that E ⊆ E(e−ε)+ ∼ = Ec Eb . This shows that Ea Cu Eb .

2

By the definition of Cuntz equivalence of positive elements, Hilbert A-modules, and of open projections, the proposition above immediately implies the following:

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Corollary 4.14. For every pair of positive elements a and b in a C ∗ -algebra A we have the following equivalences: a≈b

⇔

Ea ≈ Eb

⇔

pa ∼Cu pb .

We conclude this section by remarking that the pre-order PZ on the open projections is not algebraic (unlike the situation for Murray–von Neumann subequivalence). Indeed, if p and q are open projections A∗∗ with p q, then q − p need not be an open projection. For the same reason, Cu is not an algebraic order. However, Cuntz comparison is approximately algebraic in the following sense. Proposition 4.15. Let A be a C ∗ -algebra, and let p, p , q ∈ A∗∗ be open projections with p p Cu q. Then there exists an open projection r ∈ A∗∗ such that p ⊕ r Cu q Cu p ⊕ r. Proof. By Lemma 4.12 (ii) there exists an open projection p with p p p (take p to be p(a−ε/2)+ in that lemma). By the definition of Cuntz sub-equivalence there exists an open projection q such that p ∼PZ q q. Since p ∼PZ q implies p ∼Cu q , there exists an open projection q with p ∼PZ q q . Then r := q − q is an open projection. Since q q implies q q , and q q , we get p ⊕ r ∼PZ q ⊕ r PZ q = q + r Cu q ⊕ r ∼PZ p ⊕ r p ⊕ r as desired.

2

Translated, this result says that for positive elements a , a, b in A with a a b there exists a positive element c such that a ⊕ c b a ⊕ c. To formulate the result in the ordered Cuntz semigroup, we recall that an element α ∈ Cu(A) is called way-below β ∈ Cu(A), denoted α β, if for every increasing sequence {βk } in Cu(A) with β supk βk there exists l ∈ N such that already α βl . Consequently, in the Cuntz semigroup we get the following almost algebraic order: Corollary 4.16 (Almost algebraic order in the Cuntz semigroup). Let A be a C ∗ -algebra, and let α , α, β in Cu(A) be such that α α β. Then there exists γ ∈ Cu(A) such that α + γ β α +γ. 5. Comparison of projections by traces In this section we show that Murray–von Neumann (sub-)equivalence of open projections in the bidual of a separable C ∗ -algebra is equivalent to tracial comparison of the corresponding positive elements of the C ∗ -algebra. For the proof we need to show that every lower semicontinuous tracial weight on a C ∗ -algebra extends (not necessarily uniquely) to a normal tracial weight on its bidual and that Murray–von Neumann comparison of projections in any von Neumann algebra “that is not too big” is determined by tracial weights. We expect those two results to be known to experts, but in lack of a reference and for completeness we have included their proofs. Recall that a weight ϕ on a C ∗ -algebra A is an additive map ϕ : A+ → [0, ∞] satisfying ϕ(λa) = λϕ(a) for all a ∈ A+ and all λ ∈ R+ . We say that ϕ is densely defined if the set


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{a ∈ A+ : ϕ(a) < ∞} is dense in A+ . Recall from Section 2 that the set of (norm) lower semicontinuous tracial weights on A in this paper is denoted by T (A). If M is a von Neumann algebra, then let W (M) denote the set of normal weights on M, and let Wtr (M) denote the set of normal tracial weights on M, i.e., weights ϕ for which ϕ(x ∗ x) = ϕ(xx ∗ ) for all x ∈ M. The standard trace on B(H ) is an example of a normal tracial weight. For the extension of weights on a C ∗ -algebra to its universal enveloping von Neumann algebra, we use the result below from [9, Proposition 4.1 and Proposition 4.4]. For every f in the dual A∗ of a C ∗ -algebra A, let f denote the unique normal extension of f to A∗∗ . (One can equivalently obtain f via the natural pairing: f(z) = f, z for z ∈ A∗∗ .) Proposition 5.1. (See Combes [9].) Let A be a C ∗ -algebra, let ϕ : A+ → [0, ∞] be a densely defined lower semicontinuous weight. Define a map ϕ : (A∗∗ )+ → [0, ∞] by ϕ (z) := sup f(z): f ∈ A∗ , 0 f ϕ ,

+ z ∈ A∗∗ .

Then ϕ is a normal weight on A∗∗ extending ϕ. Moreover, if ϕ is tracial, then ϕ is the unique extension of ϕ to a normal weight on A∗∗ . Combes did not address the question whether the (unique) normal weight on A∗∗ that extends a densely defined lower semicontinuous tracial weight on A is itself a trace. An affirmative answer to this question is included in the proposition below. Proposition 5.2. Let A be a C ∗ -algebra, and let ϕ be a lower semicontinuous tracial weight on A. Then there exists a normal, tracial weight on A∗∗ that extends ϕ. Proof. The closure of the linear span of the set {a ∈ A+ : ϕ(a) < ∞} is a closed two-sided ideal in A. Denote it by Iϕ . The restriction of ϕ to Iϕ is a densely defined tracial weight, which therefore, by Combes’ extension result (Proposition 5.1), extends (uniquely) to a normal weight ϕ on Iϕ∗∗ . The ideal Iϕ corresponds to an open central projection p in A∗∗ via the identification Iϕ = A∗∗ p ∩ A, and Iϕ∗∗ = A∗∗ p. In other words, Iϕ∗∗ is a central summand in A∗∗ . Extend ϕ to a normal weight ϕ on the positive elements in A∗∗ by the formula

ϕ (z) =

ϕ (z), ∞,

if z ∈ Iϕ∗∗ , otherwise.

It is easily checked that ϕ is a normal weight that extends ϕ, and that ϕ is tracial if we knew that ϕ is tracial. To show the latter, upon replacing A with Iϕ , we can assume that ϕ is densely defined. We proceed to show that ϕ is tracial under the assumption that ϕ is densely defined. To this end it suffices to show that ϕ is unitarily invariant, i.e., that ϕ (uzu∗ ) = ϕ (z) for all unitaries u in A∗∗ ∗∗ the unitization and all positive elements z in A . We first check this when the unitary u lies in A, ∗ ∗∗ of A, which we view as a unital sub-C -algebra of A , and for an arbitrary positive element z in A∗∗ . For each f in A∗ let u.f denote the functional in A∗ given by (u.f )(a) = f (uau∗ ) for a ∈ A. By the trace property of ϕ we see that if f ∈ A∗ is such that 0 f ϕ, then also 0 u.f ϕ, and vice versa since f = u∗ .(u.f ). It follows that

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(z): f ∈ A∗ , 0 f ϕ ϕ uzu∗ = sup f uzu∗ : f ∈ A∗ , 0 f ϕ = sup u.f ϕ (z). = sup f(z): f ∈ A∗ , 0 f ϕ = For the general case we use Kaplansky’s density theorem (see [18, Theorem 2.3.3, p. 25]), is σ -strongly dense in U (A∗∗ ). Thus, given u in U (A∗∗ ) which says that the unitary group U (A) converging σ -strongly to u. It follows that (uλ zu∗ ) converges we can find a net (uλ ) in U (A) λ σ -strongly (and hence σ -weakly) to uzu∗ . As ϕ is σ -weakly lower semicontinuous (see [6, III.2.2.18, p. 253]), we get ϕ uzu∗ = ϕ lim uλ zu∗λ lim ϕ (z). ϕ uλ zu∗λ = λ

λ

The same argument shows that ϕ (z) = ϕ (u∗ (uzu∗ )u) ϕ (uzu∗ ). This proves that ϕ (uzu∗ ) = ϕ (z) as desired. 2 The extension ϕ in Proposition 5.2 need not be unique if ϕ is not densely defined. Take for example the trivial trace ϕ on the Cuntz algebra O2 (that is zero on zero and infinite elsewhere). Then every normal tracial weight on O2∗∗ that is infinite on every (non-zero) properly infinite element is an extension of ϕ, and there are many such normal tracial weights arising from the type I∞ and type II∞ representations of O2 . On the other hand, every densely defined lower semicontinuous tracial weight on a C ∗ -algebra extends uniquely to a normal tracial weight on its bidual by Combes’ result (Proposition 5.1) and by Proposition 5.2. Remark 5.3. Given a C ∗ -algebra A equipped with a lower semicontinuous tracial weight τ and a positive element a in A. Then we can associate to τ a dimension function dτ on A (as above Definition 2.4). Let τ be (any) extension of τ to a normal tracial weight on A∗∗ (cf. Proposition 5.2). Then dτ (a) = τ (pa ). To see this, assume without loss of generality that a is a contraction. Then pa is the strong operator limit of the increasing sequence {a 1/n }, whence τ (pa ) τ a 1/n = dτ (a) = lim τ a 1/n = lim n→∞

n→∞

by normality of τ. Corollary 5.4. Let a and b be positive elements in a C ∗ -algebra A. If pa pb in A∗∗ , then a tr b in A; and if pa ∼ pb in A∗∗ , then a ∼tr b in A. Proof. Suppose that pa pb in A∗∗ . Then ω(pa ) ω(pb ) for every tracial weight ω on A∗∗ . Now let τ ∈ T (A) be any lower semicontinuous tracial weight, and let dτ be the corresponding dimension function. By Proposition 5.2, τ extends to a tracial, normal weight τ on A∗∗ . Using τ (pa ) τ (pb ) = dτ (b). This proves that a tr b. the remark above, it follows that dτ (a) = The second statement in the corollary follows from the first statement. 2 We will now show that the converse of Corollary 5.4 is true for separable C ∗ -algebras. First we need to recall some facts about the dimension theory of (projections in) von Neumann algebras. A good reference is the recent paper [23] of David Sherman.


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Definition 5.5. (Tomiyama [24, Definition 1], see also [23, Definition 2.3].) Let M be a von Neumann algebra, p ∈ P (M) a non-zero projection, and κ a cardinal. Say that p is κ-homogeneous if p is the sum of κ mutually equivalent projections, each of which is the sum of centrally orthogonal σ -finite projections. Set κM := sup{κ: M contains a κ-homogeneous element}. A projection can be κ-homogeneous for at most one κ ℵ0 ; and if κ ℵ0 , then two κhomogeneous projections are equivalent if they have identical central support (see [24,23]). We shall use these facts in the proof of Proposition 5.7. But first we show that the enveloping von Neumann algebra A∗∗ of a separable C ∗ -algebra A has κA∗∗ ℵ0 , a property that has various equivalent formulations and consequences (see [23, Propositions 3.8 and 5.1]). This property is useful, since it means that there are no issues about different “infinities”. For instance, the set of projections up to Murray–von Neumann equivalence in an arbitrary II∞ factor M (not necessarily with separable predual) can be identified with [0, ∞) ∪ {κ: ℵ0 κ κM }, see [23, Corollary 2.8]. Thus, tracial weights on M need not separate projections up to equivalence. However, if κM ℵ0 , then normal, tracial weights on M do in fact separate projections up to Murray–von Neumann equivalence. Lemma 5.6. Let A be a separable C ∗ -algebra. Then κA∗∗ ℵ0 . Proof. We show the stronger statement that whenever {pi }i∈I is a family of non-zero pairwise equivalent and orthogonal projections in A∗∗ , then card(I ) ℵ0 . The universal representation πu of A is given as πu = ϕ∈S(A) πϕ , where S(A) denotes the set of states on A, and where πϕ : A → B(Hϕ ) denotes the GNS-representation corresponding to the state ϕ. It follows that

A∗∗ = πu (A) ⊆

B(Hϕ ).

ϕ∈S(A)

The projections {pi }i∈I are non-zero in at least one summand B(Hϕ ); but then I must be countable because each Hϕ is separable. 2 Proposition 5.7. Let M be a von Neumann algebra with κM ℵ0 , and let p, q ∈ P (M) be two projections. Then p q if and only if ω(p) ω(q) for all normal tracial weights ω on M. Proof. The “only if” part is obvious. We prove the “if” part and assume accordingly that ω(p) ω(q) for all normal tracial weights ω on M, and we must show that p q. We show first that it suffices to consider the case where q p. There is a central projection z in M such that zp zq and (1 − z)p (1 − z)q. We are done if we can show that (1 − z)p (1 − z)q. Every normal tracial weight on (1 − z)M extends to a normal tracial weight on M (for example by setting it equal to zero on zM), whence our assumptions imply that ω((1 − z)p) ω((1 − z)q) for all tracial weights ω on (1 − z)M. Upon replacing M by (1 − z)M, and p and q by (1 − z)p and (1 − z)q, respectively, we can assume that p q, i.e., that q ∼ q p for some projection q in M. Upon replacing q by q we can further assume that q p as desired.

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There is a central projection z in M such that zq is finite and (1 − z)q is properly infinite (see [14, 6.3.7, p. 414]). Arguing as above it therefore suffices to consider the two cases where q is finite and where q is properly infinite. Assume first that q is finite. We show that p = q. Suppose, to reach a contradiction, that p − q = 0. Then there would be a normal tracial weight ω on M such that ω(q) = 1 and ω(p − q) > 0. But that would entail that ω(p) > ω(q) in contradiction with our assumptions. To see that ω exists, consider first the case where q and p − q are not centrally orthogonal, i.e., that cq cp−q = 0. Then there are non-zero projections e q and f p − q such that e ∼ f . Choose a normal tracial state τ on the finite von Neumann algebra qMq such that τ (e) > 0. Then τ extends uniquely to a normal tracial weight ω0 on Mcq and further to a normal tracial weight ω on M by the recipe ω(x) = ω0 (xcq ). Then ω(q) = τ (q) = 1 and ω(p − q) ω0 (f ) = ω0 (e) = τ (e) > 0. In the case where q and p − q are centrally orthogonal, take a normal tracial weight ω0 (for example as above) such that ω0 (q) = 1 and extend ω0 to a normal tracial weight ω on M by the recipe ω(x) = ω0 (x) for all positive elements x ∈ Mcq and ω(x) = ∞ whenever x is a positive element in M that does not belong to Mcq . Then ω(q) = 1 and ω(p − q) = ∞. Assume next that q is properly infinite. Every properly infinite projection can uniquely be written as a central sum of homogeneous projections (see [24, Theorem 1], see also [23, Theorem 2.5] and the references cited there). By the assumption that κM ℵ0 we get that every properly infinite projection is ℵ0 -homogeneous. Therefore q is ℵ0 -homogeneous and hence equivalent to its central support projection cq . Let ω be the normal tracial weight on M which is zero on Mcq and equal to ∞ on every positive element that does not lie in Mcq . Then ω(p) ω(q) = 0, which shows that p ∈ Mcq , and hence cp cq . It now follows that p cp cq ∼ q, and so p q as desired. 2 We can now show that Murray–von Neumann (sub-)equivalence of open projections in the bidual of a C ∗ -algebra is equivalent to tracial (sub-)equivalence of the corresponding positive elements in the C ∗ -algebra. Theorem 5.8. Let a and b be positive elements in a separable C ∗ -algebra A. Then pa pb in A∗∗ if and only if a tr b in A; and pa ∼ pb in A∗∗ if and only if a ∼tr b in A. Proof. The “only if parts” have already been proved in Corollary 5.4. Suppose that a tr b. Let ω be a normal tracial weight on A∗∗ , and denote by ω0 its restriction to A. Then ω0 is a norm lower semicontinuous tracial weight on A, whence ω(pa ) = dω0 (a) dω0 (b) = ω(pb ), cf. Remark 5.3. As ω was arbitrary we can now conclude from Lemma 5.6 and Proposition 5.7 that pa pb . The second part of the theorem follows easily from the first part. 2 Corollary 5.9. Let A be a separable C ∗ -algebra, and p and q be two open projections in A∗∗ . Then p PZ q

⇒

p Cu q

⇒

p q,

p ∼PZ q

⇒

p ∼Cu q

⇒

p ∼ q.

The first implication in each of the two strings holds without assuming A to be separable.


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Proof. Since A is separable there are positive elements a and b such that p = pa and q = pb . The corollary now follows from Remark 2.5, Proposition 4.3, Proposition 4.13, and Theorem 5.8. 2 It should be remarked, that one can prove the corollary above more directly without invoking Remark 2.5. Remark 5.10. There is a certain similarity of our main results with the following result recently obtained by Robert in [21, Theorem 1]: If a, b are positive elements of a C ∗ -algebra A, then the following are equivalent: (i) τ (a) = τ (b) for all norm lower semicontinuous tracial weights on A, (ii) a and i.e., there exists a sequence {xk } in A such that b are Cuntz–Pedersen ∞ equivalent, ∗ and b = ∗ x (the sums are norm-convergent). a= ∞ x x x k k k=1 k=1 k k It is known that Cuntz–Pedersen equivalence and Murray–von Neumann equivalence agree for projections in a von Neumann algebra (see [13, Theorem 4.1]), but they are different for projections in a C ∗ -algebra. 6. Summary and applications In the previous sections we have established equivalences and implications between different types of comparison of positive elements and their corresponding open projections and Hilbert modules. The results we have obtained can be summarized as follows. Given two positive elements a and b in a (separable) C ∗ -algebra A with corresponding open projections pa and pb in A∗∗ and Hilbert A-modules Ea and Eb , then:

(∗)

a s b

pa PZ pb

a ∼s b

pa ∼PZ pb

Ea ∼ = Eb

ab

pa Cu pb

a≈b

pa ∼Cu pb

Ea ∼Cu Eb

a tr b

pa pb

a ∼tr b

pa ∼ pb

We shall discuss below to what extend the reverse (upwards) implications hold. First we remark how the middle bi-implications yield an isomorphism between the Cuntz semigroup and a semigroup of open projections modulo Cuntz equivalence. 6.1 (The semigroup of open projections). Given a C ∗ -algebra A. We wish to show that its Cuntz semigroup Cu(A) can be identified with an ordered semigroup of open projections in (A ⊗ K)∗∗ . More specifically, we show Po ((A ⊗ K)∗∗ )/∼Cu is an ordered abelian semigroup which is isomorphic to Cu(A).

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First we note how addition is defined on the set Po ((A ⊗ K)∗∗ )/∼Cu . Note that A ⊗ B 2 ⊆ M(A ⊗ K) ⊆ (A ⊗ K)∗∗ . Choose two isometries s1 and s2 in B(2 ) satisfying the Cuntz relation 1 = s1 s1∗ + s2 s2∗ , and consider the isometries t1 = 1 ⊗ s1 and t2 = 1 ⊗ s2 in M(A ⊗ K) ⊆ (A ⊗ K)∗∗ . For every positive element a in A ⊗ K and for every isometry t in M(A ⊗ K) we have a ∼s tat ∗ in A ⊗ K and pa ∼PZ tpa t ∗ = ptat ∗ in (A ⊗ K)∗∗ . We can therefore define addition in Po ((A ⊗ K)∗∗ )/∼Cu by (∗∗)

[p]Cu + [q]Cu := t1 pt1∗ + t2 qt2∗ Cu ,

p, q ∈ Po (A ⊗ K)∗∗ .

The relation Cu yields an order relation on Po ((A⊗K)∗∗ )/∼Cu , which thus becomes an ordered abelian semigroup. Proposition 4.13 and Corollary 4.14 applied to the C ∗ -algebra A ⊗ K yield that the mapping a → [pa ]Cu , for a ∈ (A ⊗ K)+ , defines an isomorphism Cu(A) ∼ = Po (A ⊗ K)∗∗ /∼Cu of ordered abelian semigroups whenever A is a separable C ∗ -algebra. In more detail, Proposition 4.13 and Corollary 4.14 imply that the map a → [pa ]Cu is well defined, injective, and order preserving. Surjectivity follows from the assumption that A (and hence A ⊗ K) are separable, whence all open projections in (A ⊗ K)∗∗ are of the form pa for some positive element a ∈ A ⊗ K. Additivity of the map follows from the definition of addition defined in (∗∗) above and the fact that a + b = t1 at1∗ + t2 bt2∗ in Cu(A). 6.2 (The stable rank one case). It was shown by Coward, Elliott, and Ivanescu in [10, Theorem 3] that in the case when A is a separable C ∗ -algebra with stable rank one, then two Hilbert Amodules are isometrically isomorphic if and only if they are Cuntz equivalent, and that the order structure given by Cuntz subequivalence is equivalent to the one generated by inclusion of Hilbert modules together with isometric isomorphism (see also [4, Theorem 4.29]). Combining those results with Proposition 4.3, Proposition 4.6, Proposition 4.13 and Corollary 4.14 shows that the following holds for all a, b ∈ A+ and for all p, q ∈ Po (A∗∗ ): (1) (1) (2) (2)

a b ⇔ a s b, and a ≈ b ⇔ a ∼s b. p Cu q ⇔ p PZ q, and p ∼Cu q ⇔ p ∼PZ q. If a s b and b s a, then a ∼s b. If p PZ q and q PZ p, then p ∼PZ q.

Hence the vertical implications between the first and the second row of (∗) can be reversed when A is separable and of stable rank one. The right-implications in (1) and (2) (and hence in (1) and (2) ) above do not hold in general. Counterexamples were given by Lin in [16, Theorem 9], by Perera in [20, before Corollary 2.4], and by Brown and Ciuperca in [8, Section 4]. For one such example take non-zero projections p and q in a simple, purely infinite C ∗ -algebra. Then, automatically, p q, p s q, q s p, and p ≈ q; but p ∼ q and p ∼s q hold (if and) only if p and q define the same K0 -class (which they do not always do).


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It is unknown whether (1)–(2) hold for residually stably finite C ∗ -algebras, and in particular whether they hold for stably finite simple C ∗ -algebras. 6.3 (Almost unperforated Cuntz semigroup). We discuss here when the vertical implications between the second and the third row of (∗) can be reversed. This requires both a rather restrictive assumption on the C ∗ -algebra A, and also an assumption on the positive elements a and b. To define the latter, we remind the reader of the notion of purely non-compact elements from [11, before Proposition 6.4]: The quotient map πI : A → A/I induces a morphism Cu(A) → Cu(A/I ) whenever I is an ideal in A. An element a in Cu(A) is purely non-compact if whenever πI (a) is compact for some ideal I , it is properly infinite, i.e., 2πI (a) = πI (a) in Cu(A/I ). Recall that an element α in the Cuntz semigroup Cu(B) of a C ∗ -algebra B is called compact if it is way-below itself, i.e., α α (see the end of Section 4 for the definition). It is shown in [11, Theorem 6.6] that if Cu(A) is almost unperforated and if a and b are implies b positive elements in A ⊗ K such that a is purely non-compact in Cu(A), then a means b that a b in Cu(A). In the notation of [11], and using [11, Proposition 4.2], a that dτ (a) dτ (b) for every (lower semicontinuous, possibly unbounded) 2-quasitrace on A. In the case where A is exact it is known that all such 2-quasitraces are traces by Haagerup’s theorem [12] (extended to the non-unital case by Kirchberg [15], and Blanchard and Kirchberg if and only if a tr b. We can thus rephrase [11, b [7, Remark 2.29(i)]) so it follows that a Theorem 6.6] (see also [22, Corollary 4.6 and Corollary 4.7]) as follows: Suppose that A is an exact, separable C ∗ -algebra with Cu(A) almost unperforated. Then the following holds for all positive elements a, b in A ⊗ K: (3) If a ∈ Cu(A) is purely non-compact, then a tr b ⇔ a b. (4) If a , b ∈ Cu(A) are purely non-compact, then a ∼tr b ⇔ a ≈ b. We wish to rephrase (3) and (4) above for open projections. We must first deal with the problem of choosing which kind of compactness of open projection to be invoked. Compactness of an open projection p ∈ A∗∗ as in Definition 3.6 means that p ∈ A (see Proposition 3.7). On the other hand, compactness for an element of the Cuntz semigroup Cu(A) is defined in terms of its ordering. Compactness of pa implies compactness of a ∈ Cu(A) for every positive element a in A ⊗ K. Brown and Ciuperca have shown that the converse holds in stably finite C ∗ -algebras [8, Corollary 3.3]. Recall that a C ∗ -algebra is called stably finite if its stabilization contains no infinite projections. From now on, we restrict our attention to the residually stably finite case, which means that all quotients of the C ∗ -algebra are stably finite. We define an open projection p in A∗∗ to be residually non-compact if there is no closed, central projection z ∈ A∗∗ such that pz is a nonzero, compact (open) projection in A∗∗ z. Here, we identify A∗∗ z with the bidual of the quotient A/I , where I is the ideal corresponding to the open, central projection 1 − z, i.e., I = A1−z = (1 − z)A∗∗ (1 − z) ∩ A. It follows from Proposition 3.7 that an open projection p ∈ A∗∗ is residually non-compact if and only if there is no closed, central projection z ∈ A∗∗ such that pz is non-zero and belongs to Az. Applying [8, Corollary 3.3] to each quotient of A, we get that a ∈ Cu(A) is purely non-compact if and only if pa is residually non-compact whenever a is a positive element in A ⊗ K. Thus, for open projections p, q in the bidual of a separable, exact, residually stably finite C ∗ -algebra A with Cu(A) almost unperforated, the following hold:

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(3) If p is residually non-compact, then p q ⇔ p Cu q. (4) If p and q are residually non-compact, then p ∼ q ⇔ p ∼Cu q. If, in addition, A is assumed to be simple, then an open projection p in A∗∗ is residually non-compact if and only if it is not compact, i.e., if and only if p ∈ / A, thus: / A, then p q ⇔ p Cu q. (3) If p ∈ / A, then p ∼ q ⇔ p ∼Cu q. (4) If p, q ∈ If A is stably finite, and p, q are two Cuntz equivalent open projections in A∗∗ , then p is compact if and only if q is compact (see [8, Corollary 3.4]). Together with (3) and (4) this gives the following new picture of the Cuntz semigroup: Let A be a separable, simple, exact, stably finite C ∗ -algebra with Cu(A) almost unperforated. Then Cu(A) = V (A) Po (A ⊗ K)∗∗ \P (A ⊗ K) /∼ . In other words, the Cuntz semigroup can be decomposed into the monoid V (A) (of Murray– von Neumann equivalence classes of projections in A⊗K) and the non-compact open projections modulo Murray–von Neumann equivalence in (A ⊗ K)∗∗ . In conclusion, let us note that the vertical implications between the second and the third row of (∗) cannot be reversed in general. Actually, these implications will fail whenever Cu(A) is not almost unperforated, which tends to happen when A has “high dimension”. These implications can also fail for projections in very nice C ∗ -algebras. Indeed, if p and q are projections, then p ∼tr q simply means that τ (p) = τ (q) for all traces τ . It is well known that the latter does not imply Murray–von Neumann or Cuntz equivalence even for simple AF-algebras, if their K0 groups have non-zero infinitesimal elements. Acknowledgment We thank Uffe Haagerup for his valuable comments on von Neumann algebras that helped us to shorten and improve some of the proofs in Section 5. References [1] C.A. Akemann, The general Stone–Weierstrass problem, J. Funct. Anal. 4 (1969) 277–294. [2] C.A. Akemann, A Gelfand representation theory for C ∗ -algebras, Pacific J. Math. 39 (1971) 1–11. [3] C.A. Akemann, J. Anderson, G.K. Pedersen, Approaching infinity in C ∗ -algebras, J. Operator Theory 21 (2) (1989) 255–271. [4] P. Ara, F. Perera, A.S. Toms, K-theory for operator algebras. Classification of C ∗ -algebras, arXiv:0902.3381, 2009. [5] B. Blackadar, Comparison theory for simple C ∗ -algebras, in: Operator Algebras and Applications, vol. 1, in: London Math. Soc. Lecture Note Ser., vol. 135, Cambridge Univ. Press, Cambridge, 1988, pp. 21–54. [6] B. Blackadar, Operator algebras, in: Theory of C ∗ -Algebras and von Neumann Algebras, Operator Algebras and Non-Commutative Geometry, III, in: Encyclopaedia Math. Sci., vol. 122, Springer-Verlag, Berlin, 2006. [7] E. Blanchard, E. Kirchberg, Non-simple purely infinite C ∗ -algebras: The Hausdorff case, J. Funct. Anal. 207 (2004) 461–513. [8] N.P. Brown, A. Ciuperca, Isomorphism of Hilbert modules over stably finite C ∗ -algebras, J. Funct. Anal. 257 (1) (2009) 332–339. [9] F. Combes, Poids sur une C ∗ -algèbre, J. Math. Pures Appl. (9) 47 (1968) 57–100. [10] K.T. Coward, G.A. Elliott, C. Ivanescu, The Cuntz semigroup as an invariant for C ∗ -algebras, J. Reine Angew. Math. 623 (2008) 161–193.


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[11] G.A. Elliott, L. Robert, L. Santiago, The cone of lower semicontinuous traces on a C ∗ -algebras, Amer. J. Math., in press. [12] U. Haagerup, Quasitraces on exact C ∗ -algebras are traces, preprint, 1992. [13] R.V. Kadison, G.K. Pedersen, Equivalence in operator algebras, Math. Scand. 27 (1970) 205–222. [14] R.V. Kadison, J.R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, in: Pure Appl. Math., vol. 100-2, Academic Press/Harcourt Brace Jovanovich, Orlando, 1986, pp. 399–1074, XIV. [15] E. Kirchberg, On the existence of traces on exact stably projectionless simple C ∗ -algebras, in: Operator Algebras and Their Applications, Waterloo, ON, 1994/1995, in: Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 171–172. [16] H. Lin, Equivalent open projections and corresponding hereditary C ∗ -subalgebras, J. Lond. Math. Soc. (2) 41 (2) (1990) 295–301. [17] H. Lin, Cuntz semigroups of C ∗ -algebras of stable rank one and projective Hilbert modules, arXiv:1001.4558, 2010. [18] G.K. Pedersen, C ∗ -Algebras and Their Automorphism Groups, London Math. Soc. Monogr. Ser., vol. 14, Academic Press, London/New York/San Francisco, 1979, X+416 pp. [19] C. Peligrad, L. Zsidó, Open projections of C ∗ -algebras: comparison and regularity, in: Operator Theoretical Methods, Timi¸soara, 1998, Theta Found., Bucharest, 2000, pp. 285–300. [20] F. Perera, The structure of positive elements for C ∗ -algebras with real rank zero, Int. J. Math. 8 (3) (1997) 383–405. [21] L. Robert, On the comparison of positive elements of a C ∗ -algebra by lower semicontinuous traces, Indiana Univ. Math. J. 58 (6) (2009) 2509–2515. [22] M. Rørdam, The stable and the real rank of Z-absorbing C ∗ -algebras, Int. J. Math. 15 (10) (2004) 1065–1084. [23] D. Sherman, On the dimension theory of von Neumann algebras, Math. Scand. 101 (1) (2007) 123–147. [24] J. Tomiyama, Generalized dimension function for W ∗ -algebras of infinite type, Tohoku Math. J. (2) 10 (1958) 121–129.


A Monge–Kantorovich mass transport problem for a discrete distance N. Igbida a , J.M. Mazón b,∗ , J.D. Rossi c , J. Toledo b a Institut de recherche XLIM, UMR–CNRS 6172, Faculté des sciences et techniques, Université de Limoges, France b Departament d’Anàlisi Matemàtica, Universitat de València, València, Spain c Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina

Received 7 September 2010; accepted 28 February 2011

Communicated by C. Villani To the memory of Fuensanta Andreu, our friend and colleague

Abstract This paper is concerned with a Monge–Kantorovich mass transport problem in which in the transport cost we replace the Euclidean distance with a discrete distance. We fix the length of a step and the distance that measures the cost of the transport depends of the number of steps that is needed to transport the involved mass from its origin to its destination. For this problem we construct special Kantorovich potentials, and optimal transport plans via a nonlocal version of the PDE formulation given by Evans and Gangbo for the classical case with the Euclidean distance. We also study how these problems, when rescaling the step distance, approximate the classical problem. In particular we obtain, taking limits in the rescaled nonlocal formulation, the PDE formulation given by Evans–Gangbo for the classical problem. © 2011 Elsevier Inc. All rights reserved. Keywords: Mass transport; Nonlocal problems; Monge–Kantorovich problems

1. Introduction and preliminaries The Monge mass transport problem, as proposed by Monge in 1781, deals with the optimal way of moving points from one mass distribution to another so that the total work done is min* Corresponding author.

E-mail addresses: [email protected] (N. Igbida), [email protected] (J.M. Mazón), [email protected] (J.D. Rossi), [email protected] (J. Toledo). 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.02.023

N. Igbida et al. / Journal of Functional Analysis 260 (2011) 3494–3534

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imized. In general, the total work is proportional to some cost function. In the classical Monge problem the cost function is the Euclidean distance, and this problem has been intensively studied and generalized in different directions that correspond to different classes of cost functions. We refer to the surveys and books [1,3,10,17,19,20] for further discussion of Monge’s problem, its history, and applications. However, even being the case of discontinuous cost functions very interesting for concrete situations and applications, it seems not to be well covered in the literature, maybe for the lack of convexity of the associated cost functions, which, nevertheless, enhance the interest of the problem. For instance, assume that you want to transport an amount of sand located somewhere to a hole at other place, then you count the number of steps that you have to move each part of sand to its final destination in the hole and try to move the total amount of sand making as less as possible steps. This amounts to the classical Monge–Kantorovich problem for the discrete distance: ⎧ 0 if x = y, ⎪ ⎪ ⎨ 1 if 0 < |x − y| 1, d1 (x, y) = 2 if 1 < |x − y| 2, ⎪ ⎪ ⎩. .. that count the number of steps. This transport problem also appears naturally when one considers, in a simplified way, a transport problem between cities in which the cost is measured by the toll in the road (that is a discrete function of the number of kilometers). We want to mention that our first motivation for the study of this problem comes from an interpretation of a nonlocal model for sandpiles studied in [5] (which is a nonlocal version of the sandpile model of Aronsson– Evans–Wu [6], see also [14]); in this model the height u of a sandpile evolves following the equation:

f (t, ·) − ut (t, ·) ∈ ∂IKd

(RN ) 1

u(t, ·)

a.e. t ∈ (0, T ),

u(x, 0) = u0 (x), where Kd1 (RN ) is the set of 1-Lipschitz L2 -functions w.r.t. d1 and f is a source. The interpretation reads as follows (it is similar to the one given in [10] for the sandpile model of Aronsson–Evans–Wu with the Euclidean distance): at each moment of time, the height function u(t, ·) of the sandpile is deemed also to be the potential generating the Monge–Kantorovich reallocation of μ+ = f (t, ·) dx to μ− = ut (t, ·) dy when the cost distance considered is d1 . In other words, the mass μ+ is instantly and optimally transported downhill by the potential u(t, ·) into the mass μ− . The aim of this paper is a detailed study of the mass transport problem for the discrete cost function d1 . It is clear that our problem falls into the scope of lower semi-continuous metric cost functions, so that standard results, like the existence of a solution for the relaxed problem, the so called Monge–Kantorovich problem, or the Kantorovich duality, stated in terms of the Kantorovich potentials, remain true for d1 . Nevertheless the above standard results rely on a general theory and our interest resides in giving concrete characterizations: since d1 is discrete, the characterization of the potentials, the Evans–Gangbo approach [11], as well as concrete computations of optimal transport plans and/or maps are not covered in the literature; in particular, the potentials cannot be characterized in a standard way, i.e., by using standard differentiation. It is also

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worth to mention that, adapting an example of [16], it is easy to see that the Monge infimum and the Monge–Kantorovich minimum does not coincide in general. We find a special class of Kantorovich potentials and perform a detailed study of the onedimensional case with concrete examples that illustrate the obstructions to the existence of optimal transport maps; we show that the Monge problem is, in fact, ill-posed. In any dimension, we give an equation for the Kantorovich potentials, in the way of Evans–Gangbo, obtained as a limit of nonlocal p-Laplacian problems, and, what is quite important, we use it to construct optimal transport plans. We want to remark that all these developments can be done in the same way for the discrete distance with steps of size ε,

dε (x, y) =

⎧ 0 ⎪ ⎪ ⎨ε 2ε ⎪ ⎪ ⎩. ..

if x = y, if 0 < |x − y| ε, if ε < |x − y| 2ε,

Then, finally, we give the connection between the Monge–Kantorovich problem with the discrete distance dε and the classical Monge–Kantorovich problem with the Euclidean distance, proving that, when the length of the step tends to zero, these discrete/nonlocal problems give an approximation to the classical one; in particular, we recover the PDE formulation given by Evans–Gangbo in [11]. Whenever T is a map from a measure space (X, μ) to an arbitrary space Y , we denote by T # μ the pushforward measure of μ by T . Explicitly, (T # μ)[B] = μ[T −1 (B)]. When we write T # f = g, where f and g are non-negative functions, this means that the measure having density f is pushed-forward to the measure having density g. The general framework in which we will move is in a bounded convex domain Ω in RN . The Monge problem for the cost function d1 . Take two non-negative Borel function f + , f − ∈ L1 (Ω) satisfying the mass balance condition

f + (x) dx =

Ω

f − (y) dy.

(1.1)

Ω

Let A(f + , f − ) be the set of transport maps pushing f + to f − , that is, the set of Borel maps T : Ω → Ω such that T # f + = f − . The Monge problem consists in finding a map T ∗ ∈ A(f + , f − ) which minimizes the cost functional Fd1 (T ) :=

d1 x, T (x) f + (x) dx

Ω

in the set A(f + , f − ). T ∗ is called an optimal transport map pushing f + to f − . The original problem studied by Monge corresponds to the cost function d|·| (x, y) := |x − y| the Euclidean distance. In general, the Monge problem is ill-posed. To overcome the difficulties of the Monge problem, L.V. Kantorovich (1942) [15] proposed to study a relaxed version of the Monge problem and, what is more relevant here, introduced a dual variational principle.


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We will use the usual convention of denoting by πi : RN × RN the projections, π1 (x, y) := x, π2 (x, y) := y. Given a Radon measure μ in Ω × Ω, its marginals are defined by projx (μ) := π1 # μ, projy (μ) := π2 # μ. The Monge–Kantorovich relaxed problem for d1 . Fix f + and f − satisfying (1.1). Let π (f + , f − ) the set of transport plans between f + and f − , that is the set of non-negative Radon measures μ in Ω × Ω such that projx (μ) = f + (x) dx and projy (μ) = f − (y) dy. The Monge– Kantorovich problem is to find a measure μ∗ ∈ π (f + , f − ) which minimizes the cost functional d1 (x, y) dμ(x, y), Kd1 (μ) := Ω×Ω

in the set π (f + , f − ). A minimizer μ∗ is called an optimal transport plan between f + and f − . Remark that we say plans between f + and f − since this problem is reversible, which is not true in general for the Monge problem. As a consequence of [1, Propostion 2.1], we have

inf Kd1 (μ): μ ∈ π f + , f − inf Fd1 (T ): T ∈ A f + , f − . On the other hand, since d1 is a lower semi-continuous cost function, it is well known the existence of an optimal transport plan (see [1,16] and the references therein). Therefore we have the following result. Proposition 1.1. Let f + , f − ∈ L1 (Ω) be two non-negative Borel functions satisfying the mass balance condition (1.1). Then, there exists an optimal transport plan μ∗ ∈ π (f + , f − ) solving the Monge–Kantorovich problem Kd1 (μ∗ ) = min{Kd1 (μ): μ ∈ π (f + , f − )}. The Kantorovich dual problem for d1 . Since the cost function d1 is a lower semi-continuous metric, we have the following result (see for instance [19, Theorem 1.14]). Theorem 1.2 (Kantorovich–Rubinstein Theorem). Let f + , f − ∈ L1 (Ω) be two non-negative Borel functions satisfying the mass balance condition (1.1). Then,

min Kd1 (μ): μ ∈ π f + , f − = sup Pf + ,f − (u): u ∈ Kd1 (Ω) ,

(1.2)

where Pf + ,f − (u) :=

u(x) f + (x) − f − (x) dx,

Ω

and Kd1 (Ω) is the set of 1-Lipschitz functions w.r.t. d1 ,

Kd1 (Ω) := u ∈ L2 (Ω): u(x) − u(y) d1 (x, y) for all x, y ∈ Ω . The maximizers u∗ of the right-hand side of (1.2) are called Kantorovich (transport) potentials.

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The Kantorovich dual problem consists in finding this Kantorovich potentials. Although it can be studied for masses being Borel measures, we will restrict ourselves to Lebesgue integrable functions in order to avoid more technicalities. If we denote by IKd1 (Ω) to the indicator function of Kd1 (Ω),

IKd1 (Ω) (u) :=

0 if u ∈ Kd1 (Ω), +∞ if u ∈ / Kd1 (Ω),

we have that the Euler–Lagrange equation associated with the variational problem

sup Pf + ,f − (u): u ∈ Kd1 (Ω) is the equation f + − f − ∈ ∂IKd1 (Ω) (u).

(1.3)

That is, the Kantorovich potentials of (1.2) are solutions of (1.3). In the particular case of the Euclidean distance d|·| (x, y) and for adequate masses f + and − f , Evans and Gangbo in [11] find a solution of the related equation (1.3) as a limit, as p → ∞, of solutions to the local p-Laplace equation with Dirichlet boundary conditions in a sufficiently large ball BR (0):

−p up = f + − f − , up = 0,

BR (0), ∂BR (0).

Moreover, they characterize the solutions to the limit equation (1.3) by means of a PDE. Theorem 1.3 (Evans–Gangbo Theorem). Let f + , f − ∈ L1 (Ω) be two non-negative Borel functions satisfying the mass balance condition (1.1). Assume additionally that f + and f − are Lipschitz continuous functions with compact support such that supp(f + ) ∩ supp(f − ) = ∅. Then, there exists u∗ ∈ Lip1 (Ω, d|·| ) such that Ω

u∗ (x) f + (x) − f − (x) dx = max u(x) f + (x) − f − (x) dx: u ∈ Lip1 (Ω, d|·| ) ; Ω

and there exists 0 a ∈ L∞ (Ω) (the transport density) such that f + − f − = −div a∇u∗ in D (Ω).

(1.4)

Furthermore |∇u∗ | = 1 a.e. on the set {a > 0}. The function a that appear in the previous result is the Lagrange multiplier corresponding to the constraint |∇u∗ | 1, and it is called the transport density. Moreover, what is very important from the point of view of mass transport, Evans and Gangbo use this PDE to find a proof of the existence of an optimal transport map for the classical Monge problem, different to the first one given by Sudakov in 1979 by means of probability methods ([18], see also [1] and [3]).


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One of our main aims will be to perform such program for the discrete distance. Before starting with it, we want to remark that, as it is known (see [16]), the equality between Monge’s infimum and Kantorovich’s minimum is not true in general if the cost function is not continuous. The example given by Pratelli in [16] can be adapted to get a counterexample also for the case of the cost function given by the metric d1 . Example 1.4. Consider R, S and T the parallel segments in R2 given by R := {(−1, y): y ∈ [−1, 1]}, S := {(0, y): y ∈ [−1, 1]} and Q := {(1, y): y ∈ [−1, 1]}. Let f + := 2H1 S and f − := H1 R + H1 Q. It is not difficult to see that min{Kd1 (μ): μ ∈ π (f + , f − )} = 2 and the minimum is achieved by the transport plan splitting the central segment S in two parts and translating them on the left and on the right. On the other hand, we claim that

inf Fd1 (T ): T ∈ A f + , f − 4.

(1.5)

To prove (1.5), fix T ∈ A(f + , f − ) and consider I (T ) := {x ∈ S: d1 (x, T (x)) = 1}. If we see that f + I (T ) = 0,

(1.6)

then Fd1 (T ) =

d1 x, T (x) df + (x) 2

S

dH1 (x) = 4,

S\I (T )

and (1.5) follows. Finally, let us see that (1.6) holds. If we define

I (T )R := x ∈ I (T ): T (x) ∈ R

and I (T )Q := x ∈ I (T ): T (x) ∈ Q ,

we have I (T ) = I (T )R ∪ I (T )Q and I (T )R ∩ I (T )Q = ∅, and by the definition of I (T ), if E = T (I (T )), it is easy to see that H1 (E) = H1 (E ∩ R) + H1 (E ∩ Q) = H1 I (T )R + H1 I (T )R = H1 I (T ) . Therefore, f + (I (T )) = 2f − (E). But since T ∈ A(f + , f − ) one has f − (E) = f + (T −1 (E)) f + (I (T )) = 2f − (E), that implies f + (I (T )) = 0 and (1.6) is proved. 2. Kantorovich potentials The aim of this section is the study of the Kantorovich potentials that maximize

sup Pf + ,f − (u): u ∈ K1 , where K1 := Kd1 (Ω) for shortness. Following ideas from [11], we first show that it is possible to construct Kantorovich potentials for the cost function d1 taking limit, as p goes to ∞, in some p-Laplacian problems but of nonlocal nature. Afterwards, we prove the existence of Kantorovich potentials with a finite number

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of jumps of size one (a specially interesting result for searching/constructing optimal transport maps and plans). Let ⎧ N radial function with ⎪ ⎨ J : R → R be a non-negative continuous (2.1) J (x) dx = 1. supp(J ) = B1 (0), J (0) > 0 and ⎪ ⎩ RN

We will use the following Poincaré type inequality from [4]. Proposition 2.1. (See [4].) Given p 1, J and Ω, there exists βp = β(J, Ω, p) > 0 such that βp

p p u − 1 1 u J (x − y) u(y) − u(x) dy dx |Ω| 2

Ω

Ω

∀u ∈ Lp (Ω).

Ω Ω

Proposition 2.2. Let f ∈ L2 (Ω) and p > 2. Then the functional p 1 Fp (u) = J (x − y) u(y) − u(x) dy dx − f (x)u(x) dx 2p Ω Ω

has a unique minimizer up in Sp := {u ∈ Lp (Ω):

Ω

Ω

u(x) dx = 0}.

Proof. Let un be a minimizing sequence. Hence, Fp (un ) C, that is 1 2p

p J (x − y) un (y) − un (x) dy dx −

Ω Ω

f (x)un (x) dx C. Ω

Then, 1 2p

p J (x − y) un (y) − un (x) dy dx

Ω Ω

f (x)un (x) dx + C. Ω

From the Poincaré inequality (2.2) and Hölder’s inequality, we get 1 2p

p J (x − y) un (y) − un (x) dy dx

Ω Ω

f L2 (Ω) un L2 (Ω) + C 1 2 2 1 f L2 (Ω) J (x − y) u(y) − u(x) dy dx +C 2β2 Ω Ω

1/p 2−p 2p p J (x − y) un (y) − un (x) dy dx J (x − y) + C. C(f ) Ω Ω

Ω Ω

(2.2)


Therefore, we have that

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p J (x − y) un (y) − un (x) dy dx C.

Ω Ω

Then, applying again Poincaré’s inequality (2.2), we have {un : n ∈ N} is bounded in Lp (Ω). p Hence, we can extract a subsequence that converges weakly in L (Ω) to some u (that clearly has to verify Ω u = 0) and we obtain p p 1 1 J (x − y) un (y) − un (x) dy dx J (x − y) u(y) − u(x) dy dx lim inf n→+∞ 2p 2p Ω Ω

Ω Ω

and

f (x)un (x) dx =

lim

n→+∞ Ω

f (x)u(x) dx. Ω

Therefore, u is a minimizer of Fp . Uniqueness is a direct consequence of the fact that Fp is strictly convex. 2 Lemma 2.3. Given u ∈ L1 (Ω) such that

E := (x, y) ∈ Ω × Ω: u(x) − u(y) > d1 (x, y) is a null set of Ω × Ω, there exists uˆ ∈ K1 such that u = uˆ a.e. in Ω.

(2.3)

Proof. We can assume that u is defined everywhere in Ω and bounded. Indeed, let A be the null set in Ω such that for all x ∈ Ω \ A, Ex = {y ∈ Ω: (x, y) ∈ E} is null and u(x) is finite. Take x ∈ Ω \ A, then, for all y ∈ Ω \ Ex , u(x) − d1 (x, y) u(y) u(x) + d1 (x, y), and therefore u(y) is a.e. bounded by M := |u(x)| + supz∈Ω d1 (x, z). Take now B the null set in Ω where |u| > M and define u(x) ˜ := u(x) in Ω \ B, u(x) ˜ := 0 in B. Then u˜ = u a.e. and u(x) ˜ − u(y) ˜ d1 (x, y) ∀(x, y) ∈ Ω × Ω \ E ∪ (B × Ω) ∪ (Ω × B) . Let us consider 1 uε (x) = |Bε (x)|

u(z) dz, Bε (x)

where u is extended by 0 to RN \ Ω. Then, for any x ∈ Ω, we define u(x) ˆ := lim sup uε (x). ε→0

It is clear that uˆ = u a.e. in Ω.

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Let x, y ∈ Ω be such that |x − y| = i for any i = 0, 1, 2, . . . . Then, there exists i ∈ N such that i − 1 < |x − y| 0 such that Bε0 (x), Bε0 (y) ⊂ Ω and i − 1 < |z1 − z2 | < i,

for any (z1 , z2 ) ∈ Bε0 (x) × Bε0 (y).

This implies that, for any 0 < ε ε0 , we have uε (x) − uε (y) =

=

1 |Bε (x)|

u(z) dz − Bε (x)

1 |Bε (0)|2

1 |Bε (0)|2

1 |Bε (y)|

u(z) dz Bε (y)

u(z1 ) − u(z2 ) dz1 dz2

Bε (x)×Bε (y)

d1 (z1 , z2 ) dz1 dz2 Bε (x)×Bε (y)

= d1 (x, y). Then, letting ε → 0, we deduce that u(x) ˆ d1 (x, y) + u(y) ˆ

for any (x, y) ∈ Ω × Ω, |x − y| = i, i = 1, 2, . . . .

(2.4)

Now, assume that x, y ∈ Ω, |x − y| = i, for some i ∈ N. And let ε0 be such that Bε0 (x), B2ε0 (y) ⊂ Ω. Let yn ∈ Ω be such that yn → y, Bε0 (yn ) ⊂ Ω and i − 1 < |x − yn | < i. Using the continuity of uε and (2.4) we see that, for any 0 < ε ε0 , uε (x) − uε (y) = lim

n→∞

= lim

n→∞

1 |Bε (x)|

1 |Bε (0)|

u(z) ˆ dz − Bε (x)

1 |Bε (yn )|

u(z) ˆ dz Bε (yn )

u(x ˆ + z) − u(y ˆ n + z) dz

Bε (0)

lim d1 (x, yn ) = i = d1 (x, y). n→∞

Letting ε → 0, we obtain that u(x) ˆ d1 (x, y) + u(y). ˆ The proof is finished.

2

Now we show that the limit as p goes to ∞ of the sequence up of minimizers of Fp in Sp gives a Kantorovich potential. Theorem 2.4. Let f + , f − ∈ L2 (Ω) be two non-negative Borel functions satisfying the mass balance condition (1.1). Let up be the minimizer in Proposition 2.2 for f = f + − f − , p > 2.


3503

Then, there exists a subsequence {upn }n∈N having as weak limit a Kantorovich potential u for f ± and the metric cost function d1 , that is, u(x) f + (x) − f − (x) dx = max v(x) f + (x) − f − (x) dx. v∈K1

Ω

Ω

Proof. For 1 q, we set |||u|||q :=

q J (x − y) u(y) − u(x) dx dy

1 q

.

Ω Ω

By Hölder’s inequality, for r q: |||u|||q

r J (x − y) u(y) − u(x) dx dy

1 r

Ω Ω

r−q J (x − y) dx dy

rq

,

Ω Ω

that is, for (r, q), r q, r−q

|||u|||q |||u|||r

J (x − y) dx dy

rq

(2.5)

.

Ω Ω

Since Fp (up ) Fp (0) = 0 and Poincaré’s inequality (2.2),

p

|||up |||p 2p

f (x)up (x) dx 2p f 2 up 2 Ω

2p f 2 |||up |||2 . (2β2 )1/2

Then, for 2 q < p, using (2.5) twice (for (p, q) and for (q, 2)), p |||up |||q

p |||up |||p

p−q J (x − y) dx dy

q

Ω Ω

2p f 2 |||up |||2 (2β2 )1/2 2p f 2 |||up |||q (2β2 )1/2

p−q q

J (x − y) dx dy Ω Ω

J (x − y) dx dy

p−q + q−2 q

2q

.

Ω Ω

Consequently, |||up |||q

2p f 2 (2β2 )1/2

1 p−1

J (x − y) dx dy Ω Ω

1− q

1 2(p−1)

.

(2.6)

3504


Then, {|||up |||q : p > q} is bounded. Hence, by Poincaré’s inequality (2.2), we have that {up : p > q} is bounded in Lq (Ω). Therefore, we can assume that up u weakly in Lq (Ω). By a diagonal process, we have that there is a sequence pn → ∞, such that upn u weakly in Lm (Ω), as n → +∞, for all m ∈ N. Thus, u ∈ L∞ (Ω). Since the functional v → |||v|||q is weakly lower semi-continuous, having in mind (2.6), we have |||u|||q

1 J (x − y) dx dy

q

.

Ω Ω

Therefore, limq→+∞ |||u|||q 1, from where it follows that |u(x) − u(y)| d1 (x, y) a.e. in Ω × Ω. Now, thanks to Lemma 2.3 we can suppose, that u ∈ K1 . Let us see that u is a Kantorovich potential associated with the metric d1 . Fix v ∈ K1 . Then,

1 f up 2p

− Ω

p J (x − y) up (y) − up (x) dx dy −

Ω Ω

Ω

p J (x − y) v(y) − v(x) dx dy −

Ω Ω

1 2p

J (x − y) dx dy − Ω Ω

f (x)up (x) dx Ω

1 = Fp (up ) Fp v − v |Ω| 1 = 2p

f (x)v(x) dx Ω

f (x)v(x) dx, Ω

where we have used Ω f = 0 for the second equality and the fact that v ∈ K1 for the last inequality. Hence, taking limit as p → ∞, we obtain that + − u(x) f (x) − f (x) dx v(x) f + (x) − f − (x) dx. 2 Ω

Ω

Let us now study a special class of Kantorovich potentials. We begin with the following lemma. Lemma 2.5. Assume that v ∈ K1 takes a finite number of values. Then, there exists u ∈ K1 that also takes a finite number of values but with jumps of length 1, the number of points in its image is less or equal than the number of points in the image of v and improves in the maximization problem, that is, + − u(x) f (x) − f (x) dx v(x) f + (x) − f − (x) dx. Ω

Ω

Proof. The proof runs by induction in the number of nonempty level sets of v. Take f := f + − f − and suppose that v ∈ K1 is given by, without loss of generality, v(x) = a0 χ A0 + a1 χ A1 + · · · + ak χ Ak , a0 = 0, |Ai | > 0, Ai ∩ Aj = ∅ for any i = j .


3505

Set s := Sign( A0 f ), where

Sign(r) =

if r 0, if r < 0,

1 −1

and consider t0 = max{t 0: ut := (a0 + st)χ A0 + a1 χ A1 + · · · + ak χ Ak ∈ K1 }. So, t0 is such that ∃i = 0, dist(Ai , A0 ) 1 and |a0 + st0 − ai | = 1 and f (x)v(x) dx f (x)ut (x) dx. Ω

Ω

Hence, replacing v by ut0 , we can assume that Ai are disjoint sets, dist(A0 , A1 ) 1 and |u0 − u1 | = 1. Now, we set s := Sign( A0 ∪A1 f ) and we consider

t0 = max t 0; ut := (a0 + st)χ A0 + (a1 + st)χ A1 + a2 χ A2 + · · · + ak χ Ak ∈ K1 . So, t0 is such that ∃i ∈ {0, 1} and ∃ji ∈ / {0, 1} such that dist(Ai , Aji ) 1, |ai + st0 − aji | = 1 and f (x)v(x) dx f (x)ut (x) dx. Ω

Ω

Hence, replacing v by ut0 , we can assume that Ai are disjoint sets and |ui − uj | ∈ {0, 1, 2}, for any i, j ∈ {0, 1, 2}. Now, by induction assume that we have u = a0 χ A0 + · · · + al χ Al + · · · + ak χ Ak , where Ai are disjoint sets, and |ai − aj | ∈ N, for any i, j = 0, 1, . . . , l, and let us prove that we can assume that Ai are disjoint compact sets, and |ai − aj | ∈ N, for any i, j ∈ {0, 1, . . . , l + 1}. We set s := Sign

f ,

A0 ∪···∪Al

and we consider

t0 = max t 0; ut := (a0 + st)χ A0 + · · · + (al + st)χ Al + al+1 χ Al+1 + · · · + ak χ Ak ∈ K1 . So, t0 is such that ∃i ∈ {0, 1, . . . , l} and ∃ji ∈ / {0, 1, . . . , l} for which |ui + st0 − uji | = 1 and f (x)u(x) dx f (x)ut (x) dx. dist(Ai , Aji ) 1, Ω

Ω

Hence, replacing u by ut0 , we can assume that the sets Ai are disjoint and |ai − aj | ∈ N, for any i, j ∈ {0, 1, . . . , l + 1}. Finally, by induction, we deduce that we can assume that Ai are disjoint compact sets, and |ai − aj | ∈ N, for any i, j ∈ {0, 1, . . . , k}. 2 Now we find the special Kantorovich potentials.

3506


Theorem 2.6. Let f + , f − ∈ L∞ (Ω) be two non-negative Borel functions satisfying the mass balance condition (1.1) and such that supp(f + ) ∩ supp(f − ) is a null set. Then there exists a Kantorovich potential u∗ for f ± , associated with the metric d1 , such that u∗ (Ω) ⊂ Z and takes a finite number of values. Proof. Take f := f + − f − . By density, we have that there exists a maximizing sequence vn ∈ K1 such that vn takes a finite number of values and

vn f → max

wf.

w∈K1

Ω

Ω

Thanks to the previous lemma, there exists un ∈ K1 , un = 0χ C0n + 1χ C1n + · · · + kn χ Ckn , kn ∈ N ∪ {0}, n n n n C > 0, Ci ∩ Cj = ∅, if i = j, i a new maximizing sequence, that is,

un f → max

Ω

(2.7)

wf.

w∈K1 Ω

Notice now that the sequence {kn } is uniformly bounded by a constant that only depends on Ω. Indeed, if u ∈ K1 is of the form u(x) = 0χ C0 + 1χ C1 + · · · + k χ Ck , with |Ci | > 0, Ci ∩ Cj = ∅ for i = j , then |x − y| > 1 for every (x, y) ∈ (Ci−1 × Ci+1 ) for all i, otherwise u ∈ / K1 . Therefore, since Ω has finite diameter, this provides a bound m0 ∈ N for the number of possible sets k, and consequently, 0 kn m0 for all n ∈ N. By Fatou’s Lemma and having in mind (2.7), we get

wf

max

w∈K1 Ω

lim sup(un f ). n→∞

Ω

Now, since supp(f + ) ∩ supp(f − ) is a null set and having in mind that un (x) ∈ {0, 1, . . . , m0 } for all n ∈ N, it is easy to see that lim sup(un f ) f + lim sup un − f − lim inf un = f + n→∞

n→∞

n→∞

m0

i χ Ai − f −

i=0

m0

i χ Bi = f

i=0

∪ (Bi ∩ {f − (x) > 0}) for i > 0 and C0 = Ω \ where Ci = (Ai ∩ {f + (x) > 0}) m0 Therefore, setting u∗ = i=0 i χ Ci , we have

wf

max

w∈K1 Ω

Ω

f u∗ .

m0

i χ Ci ,

i=0

m0

i=0 Ci .


3507

To finish the proof let us see that u∗ ∈ K1 . Take x, y ∈ Ω. Let us suppose that

x ∈ Ai ∩ f + > 0

and y ∈ Bj ∩ f − > 0

(the other cases being similar), then we have ∗ u (x) − u∗ (y) = |i − j | d1 (x, y). If not, that is, if |i − j | > d1 (x, y), assuming for instance that i < j , we have that there exists 0 < < 1 such that i < i + < j − < j and there exists n ∈ N such that un (x) ∈ [i, i + ], and un (y) ∈ [j − , j ], that is, un (x) = i and un (y) = j , which contradicts that |un (x) − un (y)| d1 (x, y). 2 Remark 2.7. Let us remark that the results we have obtained are also true if in the definition of the metric d1 we change the Euclidean norm by any norm · of RN . Especially interesting is the case in which we consider the · ∞ norm since in this case it counts the maximum of steps moving parallel to the coordinate axes. That is, in this case we measure the distance cost as the number of blocks that the taxi has to cover going from x to y in a city. Remark 2.8. If we assume that u∗ takes only the values {j, j + 1, j + 2, . . . , j + k}, j ∈ Z, that is, u∗ = j χ A0 + (j + 1)χ A1 + (j + 2)χ A2 + · · · . + (j + k)χ Ak , then, Ak ∩ supp f − = 0 and A0 ∩ supp f + = 0.

(2.8)

In fact, if not, just redefine u∗ to be ∗

u˜ (x) =

j + k − 1 in Ak ∩ supp(f− ), otherwise, u∗ (x)

and we get that u˜ ∗ ∈ K1 with

u∗ f
x − 1 in K. Then, it is easy to see that there exists θ ∈ (0, 1) such that |T −1 ((−1, θ − 1))| < |A1 ∩ (0, θ )|. Therefore, since T −1 ((−i, θ − i)) ⊂ Ai ∩ (0, θ ) for all i, we have


3513

L L 1 −1 θ = (−i, θ − i) = T (−i, θ − i) L i=1

i=1

L L −1 = (−i, θ − i) < Ai ∩ (0, θ ) = θ, T i=1

i=1

and we arrive to a contradiction. With a similar proof it can be proved that there is no transport map T between f + = Lχ [0,1] and f − = χ [−L,0] with L ∈ N if one considers the distance d1/k with k ∈ N. Remark 3.2. Observe that it is easy to construct an optimal transport plan μ∗ ∈ π (f + , f − ) solving the Monge–Kantorovich problem. Indeed, if define the measure μ∗ in Ω × Ω by 1 1 1 μ∗ (x, y) := Lχ [0,1] (x) δ[y=−1+x] + δ[y=−2+x] + · · · + δ[y=−L+x] , L L L then μ∗ ∈ π (f + , f − ) and, moreover, since Kd1 μ∗ =

d1 (x, y) dμ∗ (x, y)

Ω×Ω

1 =L

1 1 1 d1 (x, −1 + x) + d1 (x, −2 + x) + · · · + d1 (x, −L + x) dx L L L

0

L(L + 1) 2

= sup P(u): u ∈ K1

= min K1 (μ): μ ∈ π f + , f − ,

=

we have that μ∗ is an optimal plan. 3.1.3. A precise construction of optimal transport plans Let us now see that in one dimension we can give, in a quite easy way, a construction of optimal transport plans by using the special Kantorovich potentials obtained in Section 3.1.1. This is independent of the general construction given afterward. We will construct an optimal transport plan under the assumptions (3.1); Remark 3.1 says how to work in a more general situation. Let u∗ = θα be the Kantorovich potential given from (3.2) and construct a new configuration of equal masses as follows: f0+ (x) =

L−1 j =0

f + (x + j ) χ ]0,1[ (x),

f0− (x) =

L−1 j =0

f − (x − j ) χ ]−1,0[ (x).

3514


For these masses, the same u∗ is a Kantorovich potential. Moreover, L

u∗ (x) f + (x) − f − (x) dx

−L

1 =

L−1 u (x) f0+ (x) − f0− (x) dx + ∗

1

+

jf (x + j ) dx +

j =0 0

−1

L−1

0

jf − (x − j ) dx.

j =0 −1

By (2.9) there exists β ∈ [α, 1] such that β

f0+

0

f0− .

= −1+α

α

Consider the smallest of such β. Take also the smallest γ ∈ [−1, −1 + α] such that 1

f0+

γ =

f0− .

−1

β

For x ∈ (0, 1), we define T0 by y x ⎧ sup{y ∈ R : −1+α f0− = α f0+ } ⎪ ⎪ ⎨ y − x + T0 (x) = sup{y ∈ R : −1 f0 = β f0 } ⎪ ⎪ x y ⎩ sup{y ∈ R : γ f0− = 0 f0+ }

0

0

α

β

1

if x ∈ (α, β), if x ∈ (β, 1), if x ∈ (0, α).

x

γ

−1

y The straight lines are only illustrative.

It is easy to see that T0 ∈ A(f + , f − ) and that d1 x, T0 (x) = u∗ (x) − u∗ T0 (x)

a.e. x ∈ supp f + .


3515

Then, by Lemma 2.9 (or a direct computation), μ00 (x, y) = f0+ (x)δ[y=T0 (x)] is an optimal transport plan between f0+ and f0− for the cost function d1 . Once we have the above construction, it is also easy to see that μ0 (x, y) =

L−1

f + (x)χ (j,j +1) (x)δ[y=T0 (x−j )]

j =0

is an optimal transport plan between f + and f0− for the cost function d1 . A remarkable observation is that these μ00 and μ0 are induced by transport maps and that for the above configurations the Monge infimum and the Monge–Kantorovich minimum coincide. By splitting the mass L−1

+

f (x)χ (j,j +1) (x) =

gi,j (x),

j = 0, 1, . . . , L − 1,

(3.6)

if x ∈ (0, β),

(3.7)

if x ∈ (β, 1),

(3.8)

i=0

is such a way that, for i = 0, 1, . . . , L − 1, x+j L−1

T0 (x)−i

j =0 j

γ −i

x+j L−1

T0 (x)−i

f−

gi,j =

and f−

gi,j =

j =0β+j

−1−i

we can finally see that μ(x, y) =

L−1 L−1

gi,j (x)χ (j,j +1) (x)δ[y=−i+T0 (x−j )]

i=0 j =0

is a transport plan between f + and f − for the cost function d1 : taking x = β in (3.7), and x = 1 in (3.8), respectively, we get β+j L−1

−i

gi,j =

j =0 j

f

−

and

1+j L−1

γ −i gi,j =

j =0β+j

γ −i

f −.

−1−i

Adding the last two equalities, we obtain 1+j L−1

−i

gi,j (x) dx =

j =0 j

−1−i

−

0

f (x) dx = −1

f − (x − i) dx.

3516


Hence, L

u f+ −f− = ∗

d1 (x, y)μ0 (x, y) +

0 L−1

j f − (x − j ) dx

j =0−1

−L

=

j +1 L−1

L−1 d1 x, T0 (x − j ) f + (x) + i

j =0 j

=

f − (x − i) dx

i=0 −1

j +1 L−1 L−1 L−1 d1 x, T0 (x − j ) gi,j (x) dx + i gi,j (x) dx

j +1 L−1 j =0 j

=

0

i=0

j =0 j

i=0

j +1 L−1 L−1

d1 x, T0 (x − j ) + i gi,j (x) dx

i=0 j =0 j

=

j +1 L−1 L−1

d1 x, −i + T0 (x − j ) gi,j (x) dx

i=0 j =0 j

=

d1 (x, y)μ(x, y).

Ω×Ω

In the following example, μ(x, y) = f + (x)δ[y=T1∗ (x)] illustrates the above construction. Example 3.3. Set f − = 14 χ ]−1,0[ and f + = χ ] 7 ,2[ . Then M = − 14 χ ]0, 3 [ + 34 χ ] 3 ,1[ and therefore 4 4 4 u∗ (x) = θ 3 is (up to adding a constant) the unique Kantorovich potential associated with the 4 metric d1 for f + and f − , moreover, u∗ (f + − f − ) = 11 16 . Nevertheless, there exist infinitely many optimal transport maps. For example, the following two are optimal transport maps, ⎧ ⎪ ⎨ 4x − ∗ T1 (x) = 4x − ⎪ ⎩ x

29 4 33 4

if if

28 16 29 16

<x
2(3−L) we have to distribute the mass f + in three equiweighted parts, in this case, L set the rectangles with corner coordinates, lower-left, lli = (xi , yi ),

lower-right, lri = (xi+1 , yi ),

upper-left, uli = (xi , yi+1 ),

upper-right, uri = (xi+1 , yi+1 ),

i = 1, 2, . . . , where now x1 = xi+1 = yi + 1,

2(3 − L) , L yi+1 = yi +

y1 = 2 − L, L L (xi+1 − xi ) = (xi+1 − 1) 3 3

(observe that lri ∈ [y = x − 1] and lli , uri ∈ [y = L3 (x − 1)]); in each rectangle we can trace three parallel segments of slope L defined by the lines y = L(x − xi ) + yi ,

y = L(x − xî ) + yi ,

xî = xi +

xi+1 − xi , 3

and y = L(x − x˜i ) + yi ,

x˜i = xi + 2

xi+1 − xi ; 3

then Ti (x) = f + (x)χ (xi ,xî ) (x)δ[y=L(x−xi )+yi ] + f + (x)χ (xî ,x˜i ) (x)δ[y=L(x−xî )+yi −1] + f + (x)χ (x˜i ,xi+1 ) (x)δ[y=L(x−x˜i )+yi −2] push in an optimal way f + χ (xi ,xi+1 ) to f − χ (yi ,yi+1 )∪(yi −1,yi+1 −1)∪(yi −2,yi+1 −2) , for i = 1, 2, . . . .


3519

3.2. Characterizing the Euler–Lagrange equation: A nonlocal version of the Evans–Gangbo approach Our first objective is to characterize the Euler–Lagrange equation associated with the variational problem sup{Pf + ,f − (u): u ∈ Kd1 (Ω)}, that is, characterize f + − f − ∈ ∂IK1 (u), where, as above, we denote for simplicity K1 := Kd1 (Ω). Let Mab (Ω × Ω) := {bounded antisymmetric Radon measures in Ω × Ω}. And define the multivalued operator B1 in L2 (Ω) as follows: (u, v) ∈ B1 if and only if u ∈ K1 , v ∈ L2 (Ω), and there exists σ ∈ Mab (Ω × Ω) such that

(x, y) ∈ Ω × Ω: |x − y| 1 , ξ(x) dσ (x, y) = ξ(x)v(x) dx, ∀ξ ∈ Cc (Ω), σ =σ

Ω×Ω

Ω

and |σ |(Ω × Ω) 2

v(x)u(x) dx. Ω

Theorem 3.5. The following characterization holds: ∂IK1 = B1 . Proof. Let us first see that B1 ⊂ ∂IK1 . Let (u, v) ∈ B1 , to see that (u, v) ∈ ∂IK1 we need to prove that 0 v(x) u(x) − ξ(x) dx, ∀ξ ∈ K1 . Ω

Using an approximation procedure, we can assume that ξ ∈ K1 is continuous. Then,

1 v(x) u(x) − ξ(x) dx |σ |(Ω × Ω) − 2

Ω

v(x)ξ(x) dx Ω

1 = |σ |(Ω × Ω) − 2

ξ(x) dσ (x, y)

Ω×Ω

1 1 = |σ |(Ω × Ω) − 2 2

ξ(x) − ξ(y) dσ (x, y) 0,

Ω×Ω

where in the last equality we have used the antisymmetry of σ . Therefore, we have B1 ⊂ ∂IK1 . Since ∂IK1 is a maximal monotone operator, to see that the operators are equal we only need to show that for every f ∈ L2 (Ω) there exists u ∈ K1 such that u + B1 (u) f.

(3.9)

3520


Let J : RN → R as in (2.1). By results in [5], given p > N and f ∈ L2 (Ω) there exists a unique solution up ∈ L∞ (Ω) of the nonlocal p-Laplacian problem

p−2 up (y) − up (x) dy = Tp (f )(x) J (x − y) up (y) − up (x)

up (x) −

∀x ∈ Ω,

(3.10)

Ω

where Tk (r) := max{min{k, r}, −r}. And we also know, using again Lemma 2.3, that there exists u ∈ K1 such that up → u

in L2 (Ω) as p → +∞,

with u + ∂IK1 (u) f , from where it follows that f (x) − u(x) w(x) − u(x) dx 0,

(3.11)

∀w ∈ K1 ,

Ω

and consequently, u = PK1 (f ). Multiplying (3.10) by up and integrating, we get

1 Tp (f )(x) − up (x) up (x) dx = 2

Ω

p J (x − y) up (y) − up (x) dx dy,

(3.12)

Ω×Ω

from where it follows that p 2 J (x − y) up (y) − up (x) dx dy + up (x) dx f 2L2 (Ω) . Ω×Ω

(3.13)

Ω

If we set σp (x, y) := J (x − y)|up (y) − up (x)|p−2 (up (y) − up (x)), by Hölder’s inequality,

σp (x, y) dx dy

Ω×Ω

=

p−1 J (x − y) up (y) − up (x) dx dy

Ω×Ω

p J (x − y) up (y) − up (x) dx dy

p−1 p

Ω×Ω

=

1 J (x − y) dx dy

Ω×Ω


p−1 p

.

Ω×Ω

Now, by (3.13), we have Ω×Ω

p−1 p . σp (x, y) dx dy f 2 2 L (Ω)

p


3521

Hence, {σp : p 2} is bounded in L1 (Ω × Ω), and consequently we can assume that weakly∗ in Mb (Ω × Ω).

σp (.,.) σ

(3.14)

Obviously, since each σp is antisymmetric, σ ∈ Mab (Ω × Ω). Moreover, since supp(J ) = B1 (0), we have σ = σ {(x, y) ∈ Ω × Ω: |x − y| 1}. On the other hand, given ξ ∈ Cc (Ω), by (3.10), (3.11) and (3.14), we get

ξ(x) dσ (x, y) = lim

p→+∞ Ω×Ω

Ω×Ω

= lim

p→+∞ Ω×Ω

= lim

p→+∞

ξ(x)σp (x, y) dx dy p−2 up (y) − up (x) ξ(x) dx dy J (x − y) up (y) − up (x)

Tp (f )(x) − up (x) ξ(x) dx

Ω

f (x) − u(x) ξ(x) dx.

= Ω

Then, to prove (3.9), we only need to show that |σ |(Ω × Ω) 2 fact, by (3.14), we have |σ |(Ω × Ω) lim inf

p→+∞

Ω (f (x)

− u(x))u(x) dx. In

σp (x, y) dx dy.

Ω Ω

Now, by (3.12),

σp (x, y) dx dy

Ω×Ω


p−1 p

Ω×Ω

p−1 p Tp (f )(x) − up (x) up (x) dx = 2 Ω

=2

p−1 p

Tp (f )(x) − up (x) up (x) dx

p−1 p

.

Ω

Therefore |σ |(Ω × Ω) 2

Ω (f (x) − u(x))u(x) dx.

2

We can rewrite the operator B1 as follows. Corollary 3.6. (u, v) ∈ B1 if and only if u ∈ K1 , v ∈ L2 (Ω), and there exists σ ∈ Mab (Ω × Ω) such that

3522


(x, y) ∈ Ω × Ω: |x − y| 1, u(x) − u(y) = 1 ,

σ − = σ − (x, y) ∈ Ω × Ω: |x − y| 1, u(y) − u(x) = 1 , ξ(x) dσ (x, y) = ξ(x)v(x) dx, ∀ξ ∈ Cc (Ω),

σ+ = σ+

Ω×Ω

Ω

and |σ |(Ω × Ω) = 2

v(x)u(x) dx. Ω

Proof. Let (u, v) ∈ B1 , then

ξ(x) dσ (x, y) =

∀ξ ∈ Cc (Ω).

ξ(x)v(x) dx,

Ω×Ω

(3.15)

Ω

Hence, by approximation, we can take ξ ∈ L2 (Ω) in (3.15) and Ω Ω ξ(x) dσ (x, y) has this sense. Taking ξ = u in (3.15) and using the antisymmetric of σ and the previous result we get

u(x) − u(y) dσ (x, y)

|σ |(Ω × Ω) Ω×Ω

=2

u(x) dσ (x, y)

Ω×Ω

=2

u(x)v(x) dx Ω

|σ |(Ω × Ω).

2

As consequence of the above results, we have that u∗ ∈ K1 is a Kantorovich potential for d1 , f − , if and only if

f +,

f + − f − ∈ B1 u∗ ,

(3.16)

that is, if u∗ ∈ K1 and there exists σ ∗ ∈ Mab (Ω × Ω), such that ⎧ ∗ + ∗ +

⎪ σ (x, y) ∈ Ω × Ω: u∗ (x) − u∗ (y) = 1, |x − y| 1 , = σ ⎪ ⎪ ⎪ ⎪ ⎨ σ ∗ − = σ ∗ − (x, y) ∈ Ω × Ω: u∗ (y) − u∗ (x) = 1, |x − y| 1, ⎪ ⎪ ∗ ⎪ ξ(x) dσ (x, y) = ξ(x) f + (x) − f − (x) dx, ⎪ ⎪ ⎩ Ω×Ω

Ω

(3.17)


3523

and ∗ σ (Ω × Ω) = 2

+ f (x) − f − (x) u∗ (x) dx = 2P u∗ .

(3.18)

Ω

We want to highlight that (3.16) plays the role of (1.4). Moreover, we will see in the next subsection that we can construct optimal transport plans from it, more precisely, we shall see that the potential u∗1 and the measure σ1∗ encode all the information that we need to construct an optimal transport plan associated with the problem. 3.3. Constructing optimal transport plans We will use a gluing lemma (see Lemma 7.6 in [19]), which permits to glue together two transport plans in an adequate way. As remarked in [19], it is possible to state the gluing lemma in the following way (we present it for the distance d1 ). Lemma 3.7. Let f1 , f2 , g be three positive measures in Ω. If μ1 ∈ π (f1 , g) and μ2 ∈ π (g, f2 ), there exists a measure G(μ1 , μ2 ) ∈ π (f1 , f2 ) such that Kd1 G(μ1 , μ2 ) Kd1 (μ1 ) + Kd1 (μ2 ).

(3.19)

Let us now proceed with the general construction. Given f + , f − ∈ L∞ (Ω) two non-negative Borel functions satisfying the mass balance condition (1.1) and | supp(f + ) ∩ supp(f − )| = 0, by Theorems 1.2 and 2.6, there exists a Kantorovich potential u∗ taking a finite number of entire values such that

min Kd1 (μ): μ ∈ π f + , f − = u∗ (x) f + (x) − f − (x) dx. Ω

Then, by Corollary 3.6, there exists σ ∈ Mab (Ω × Ω) satisfying (3.17) and (3.18). We are going to give a method to obtain an optimal transport plan μ∗ from the measure σ . We divide the construction in two steps. We assume without loss of generality that u∗ = 0χ A0 + 1χ A1 + · · · + k χ Ak ,

with Ai = x ∈ Ω: u∗ (x) = i .

Step 1. How the measures σ + (Aj × Aj −1 ) work. Taking into account the antisymmetry of σ and (3.17), we have that projx (σ + ) − projy (σ + ) = f + − f − , which implies g := projx (σ + ) − f + = projy (σ + ) − f − . By (2.8), projx (σ + ) Ak = f + χ Ak and projx (σ + ) A0 = f + χ A0 = 0, then g

Ak = g

A0 = 0.

Moreover, we have projx (σ + (Aj ×Aj −1 )) = projx (σ + ) Aj and projy (σ + (Aj ×Aj −1 )) = projx (σ + ) Aj −1 , then projx (σ + (Aj × Aj −1 )) = f + χ Aj + g Aj and projy (σ + (Aj × Aj −1 )) = f − χ Aj −1 + g Aj −1 . Let us call μj := σ + (Aj × Aj −1 ). Let us briefly comment what these measures do. The first one, μk , transports f + χ Ak into f − χ Ak−1 plus something else,

3524


that is g Ak−1 . Afterwards, μj transports f + χ Aj +g Aj into f − χ Aj −1 again plus something else, that is g Aj −1 . The last one, μ1 , transports f + χ A1 + g A1 to f − χ A0 . Step 2. The gluing. Now, we would like to glue this transportations, and, in order to apply the gluing lemma, we consider the measures μlk (x, y) := μk (x, y) + f + (x)χ Ak−1 (x)δ[y=x] , and μrk−1 (x, y) := μk−1 (x, y) + f − (x)χ Ak−1 (x)δ[y=x] . It is easy to see that μlk ∈ π f + χ Ak + f + χ Ak−1 , f − χ Ak−1 + projx σ +

Ak−1

and μrk−1 ∈ π f − χ Ak−1 + projx σ +

Ak−1 , f − χ Ak−1 + f − χ Ak−2 + g

Ak−2 .

Therefore, by the gluing lemma, G μlk , μrk−1 ∈ π f + χ Ak + f + χ Ak−1 , f − χ Ak−1 + f − χ Ak−2 + g

Ak−2 .

Let us now consider the measures μlk−1 (x, y) := G μlk , μrk−1 (x, y) + f + (x)χ Ak−2 (x)δ[y=x] and μrk−2 (x, y) := μk−2 (x, y) + f − (x)χ Ak−1 (x) + f − (x)χ Ak−2 (x) δ[y=x] . Then we have μlk−1 ∈ π f + χ Ak + f + χ Ak−1 + f + χ Ak−2 , f − χ Ak−2 + f − χ Ak−1 + projx σ + and μrk−2 ∈ π f − χ Ak−2 + f − χ Ak−1 + projx σ + f

−χ

Ak−1

+f

−χ

Ak−2

+f

−χ

Ak−3

+g

Ak−2 , Ak−3 .

Consequently, G μlk−1 , μrk−2 ∈ π f + χ Ak + f + χ Ak−1 + f + χ Ak−2 , f − χ Ak−1 + f − χ Ak−2 + f − χ Ak−3 + g

Ak−3 .

Ak−2


3525

Proceeding in this way we arrive to the construction of μl2 (x, y) = G μl3 , μr2 (x, y) + f + (x)χ A1 (x)δ[y=x] , μr1 (x, y) = μ1 (x, y) +

k−1

f − (x)χ Ai (x) δ[y=x]

i=1

and μ∗ = G μl2 , μr1 ∈ π f + , f − , which is, in fact, an optimal transport plan since, by (3.19), Kd1 μ∗ = Kd1 G μl2 , μr1 Kd1 μl2 + Kd1 μr1 = Kd1 G μl3 , μr2 + Kd1 (μ1 ) Kd1 μl3 + Kd1 μr2 + Kd1 (μ1 ) k−1 = Kd1 G μl4 , μr3 + Kd1 (μ2 ) + Kd1 (μ1 ) . . . Kd1 μlk + Kd1 (μj ) j =1

=

k

Kd1 (μj ) =

j =1

k

dσ

+

(Aj × Aj −1 ) =

j =1Ω×Ω

dσ +

Ω×Ω

1 = |σ |(Ω × Ω) = min Kd1 (μ): μ ∈ π f + , f − . 2 We want to remark that a similar construction works for any Kantorovich potential u∗ , without assuming that u∗ (Ω) ⊂ Z, but the above one is simpler. 4. Convergence to the classical problem The task of this section is the connection between this discrete mass transport problem and the classical transport problem for the Euclidean distance. In particular we recover the PDE formulation (1.4) of Evans–Gangbo by means of this discrete approach. Let us begin by remarking that an equivalent result to Corollary 3.5 for dε gives us that (u∗ε , σε∗ ) is a solution of the Euler–Lagrange equation f + − f − ∈ ∂IKdε (Ω) (u), that corresponds to the maximization problem max Ω

+ − u(x) f (x) − f (x) dx: u ∈ Kdε (Ω) ,

(4.1)

3526


if and only if u∗ε ∈ Kdε (Ω) and σε∗ in Ω is an antisymmetric bounded Radon measure such that ∗ + ∗ +

σε = σε (x, y) ∈ Ω × Ω: u∗ε (x) − u∗ε (y) = ε, |x − y| ε , ∗ − ∗ −

σε = σε (x, y) ∈ Ω × Ω: u∗ε (y) − u∗ε (x) = ε, |x − y| ε , ξ(x) dσε∗ (x, y) = ξ(x) f + (x) − f − (x) dx, Ω×Ω

(4.2) (4.3)

Ω

and ∗ σ (Ω × Ω) = 2 ε ε

+ 2 f (x) − f − (x) u∗ε (x) dx = P u∗ε . ε

(4.4)

Ω

4.1. Convergence to the classical problem Let us fix f + , f − ∈ L2 (Ω) satisfying the mass balance condition (1.1). First of all, in the following result we state the convergence to the Monge–Kantorovich problems. We will denote Kε = Kdε (Ω) and Kd|·| = Kd|·| (Ω) for simplicity (recall that d|·| denotes the Euclidean distance), and

W := sup Pf + ,f − (u): u ∈ Kd|·| = min Kd|·| (μ): μ ∈ π f + , f −

= inf F (T ): T ∈ A f + , f − ,

Wε := sup Pf + ,f − (u): u ∈ Kε = min Kε (μ): μ ∈ π f + , f − . Proposition 4.1. For the costs Wε and W the following facts hold: Wε Wε for ε ε . 0 Wε − W ε f + (x) dx for any ε > 0.

(4.5)

Ω

For the primal problems, it also holds:

lim inf Fε (μ): μ ∈ π f + , f − = W.

(4.6)

dε (x, y) − ε d|·| (x, y) dε (x, y),

(4.7)

ε→0+

Proof. Since

given μ ∈ π (f + , f − ), we have Ω×Ω

dε (x, y) − ε dμ(x, y)

Ω×Ω

d|·| (x, y) dμ(x, y) Ω×Ω

dε (x, y) dμ(x, y).


3527

Then, taking the minimum over all μ ∈ π (f + , f − ), and having in mind that dμ(x, y) = f + (x) dx, Ω×Ω

Ω

we obtain (4.5). Moreover, since dε dε for ε ε , the sequence of costs {Wε }ε>0 is monotone nonincreasing as ε decreases to zero. Let us now prove (4.6), which, by Example 1.4, is not a trivial consequence of the above statement. Precisely, this previous statement gives:

lim Wε = inf F (T ): T ∈ A f + , f − .

ε→0+

(4.8)

Take now T a transport map. Thanks to (4.7),

lim sup inf Fε (T ): T ∈ A f + , f − ε→0

= lim sup inf ε→0

lim sup ε→0

+ + − dε x, T (x) f (x) dx: T ∈ A f , f

Ω

dε x, T (x) f + (x) dx =

Ω

x − T (x) f + (x) dx.

Ω

Therefore,

lim sup inf Fε (T ): T ∈ A f + , f − inf F (T ): T ∈ A f + , f − .

(4.9)

ε→0

On the other hand,

Wε = min Kε (μ): μ ∈ π f + , f − inf Fε (T ): T ∈ A f + , f − . Taking now the lim infε→0 in the above expression and taking into account (4.8) and (4.9) we obtain (4.6). 2 Let us now proceed with the approximation of optimal transport plans. Let us consider, for each ε > 0, an optimal transport plan με between f + and f − for dε , that is, με ∈ π (f + , f − ) such that

Kε (με ) = min Kε (μ): μ ∈ π f + , f − . Proposition 4.2. There exists a sequence εn → 0 as n → ∞ and μ∗ ∈ π (f + , f − ) such that μεn μ∗

as measures

and

K μ∗ = min K(μ): μ ∈ π f + , f − .

3528


Proof. To prove this we just observe that d|·| (x, y) = |x − y| dε (x, y) |x − y| + ε (note that this implies dε (x, y) → |x − y| uniformly as ε → 0). Hence,

|x − y| dμε (x, y) Ω×Ω

dε (x, y) dμε (x, y)

Ω×Ω

|x − y| + ε dμε (x, y).

Ω×Ω

On the other hand, by Prokhorov’s Theorem, we can assume that, there exists a sequence εn → 0 as n → ∞ such that μεn converges weakly∗ in the sense of measures to a limit μ∗ . Therefore, we conclude that |x − y| dμ∗ (x, y) = lim dεn (x, y) dμεn (x, y). n→+∞ Ω×Ω

Ω×Ω

Finally, by Proposition 4.1 we obtain that μ∗ is a minimizer for the usual Euclidean distance.

2

To illustrate these results, we present an example in one dimension that shows how one can recover the unique monotone nondecreasing optimal transport map for the Euclidean distance between f + and f − . Example 4.3. Let f + = 2χ [0,1] and f − = χ [−2,0] . Set Ω an interval containing [−2, 2]. As we set in Section 3.1.2, there is no transport map T between f + and f − if one considers the distance d1/k with k ∈ N. Nevertheless, for each n ∈ N, μn (x, y) = χ [ 2n −1 ,1] (x)δ[y=x−1] + 2n

n −1 2

m=1

χ [ 2n −m−1 , 2n −m+1 ] (x)δ[y=x−1− mn ] + χ [0, 2n

is an optimal transport plan between f + and f − for the distance d μn f + (x)δ[y=T (x)]

2

2n

1 2n

1 ] 2n

(x)δ[y=x−2]

such that

weakly∗ as measures,

where T (x) = 2x − 2 is the unique monotone nondecreasing optimal transport map for the Euclidean distance between f + and f − . Let us finish this subsection with a convergence result for Kantorovich potentials. Proposition 4.4. Let u∗ε be a Kantorovich potential for f + − f − associated with the metric dε . Then, there exists a sequence εn → 0 as n → ∞ such that u∗εn u∗

in L2 ,

where u∗ is a Kantorovich potential associated with the Euclidean metric d|·| .


3529

Proof. It is an obvious fact that {uε } is L∞ -bounded, then, there exists a sequence u∗εn v

in L2 .

Therefore, lim

n→+∞

u∗εn (x) f + (x) − f − (x) dx =

Ω

v(x) f + (x) − f − (x) dx.

Ω

Now, since

u∗εn (x) f + (x) − f − (x) dx = sup Pf + ,f − (u): u ∈ Kεn ,

Ω

by Proposition 4.1, we conclude that

v(x) f + (x) − f − (x) dx = sup Pf + ,f − (u): u ∈ Kd|·| . Ω

In order to have that the limit v is a maximizer u∗ we need to show that v ∈ Kd|·| , and this follows by the Mosco-convergence of IKε to IKd|·| (see [5]). 2 4.2. Approximating the Evans–Gangbo PDE The main task in this subsection is to show how from the solutions (u∗ε , σε∗ ) of the Euler– Lagrange equation f + − f − ∈ ∂IKdε (Ω) (u), that corresponds to the maximization problem

max u(x) f + (x) − f − (x) dx: u ∈ Kdε (Ω) , Ω

we can recover u∗ ∈ Kd|·| (Ω) such that

u∗ (x) f + (x) − f − (x) dx = max u(x) f + (x) − f − (x) dx: u ∈ Kd|·| (Ω) ,

Ω

Ω

and 0 a ∈ L∞ (Ω) such that f + − f − = −div a∇u∗ in D (Ω),

∗ ∇u = 1 a.e. on the set {a > 0}.

Remember that u∗ε ∈ Kdε (Ω) and σε∗ is an antisymmetric bounded Radon measure in Ω satisfying (4.2), (4.3) and (4.4). Moreover, by Proposition 4.4, after a subsequence,

3530


u∗ε u∗

in L2 (Ω) as ε → 0,

where u∗ is a Kantorovich potential associated with the metric d|·| . Let us now fix Ω Ω Ω

(4.10)

be such that |x − y| > r = diam(supp(f + − f − )) for any x ∈ supp(f + − f − ) and any y ∈ Ω \ Ω . By (4.3),

ξ(x) f + (x) − f − (x) dx =

Ω

ξ(x) dσε∗ (x, y),

∀ξ ∈ Cc (Ω).

(4.11)

Ω×Ω

Hence, for ξ ∈ Cc1 (Ω), by (4.11) and the antisymmetry of σε∗ , we have that

ξ(x) f + (x) − f − (x) dx =

Ω

ξ(x) dσε∗ (x, y) =

Ω×Ω

ε ∗ ξ(x) − ξ(y) d σε (x, y) , ε 2

Ω×Ω

and

ξ(x) f + (x) − f − (x) dx =

Ω

ξ(x) dσε∗ (x, y)

Ω×Ω

=

ξ(x) − ξ(y) ∗ + d ε σε (x, y) . ε

(4.12)

Ω×Ω

Now observe that for ϕ ∈ Cc (Ω × Ω), if φ(x, z) = ϕ(x, x + εz) and Tε (x, y) = Ω×Ω

+ ϕ(x, y) d σε∗ (x, y) =

y−x ε ,

then

+ φ (π1 , Tε )(x, y) d σε∗ (x, y)

Ω×Ω

=

+ φ(x, z) d (π1 , Tε ) # σε∗ (x, z)

Ω× Ω−Ω ε

=

+ ϕ(x, x + εz) d (π1 , Tε ) # σε∗ (x, z).

Ω× Ω−Ω ε

Also, since ∗ + ∗ +

εσε = εσε (x, y) ∈ Ω × Ω: u∗ε (x) − u∗ε (y) = ε, |x − y| ε , and (π1 , Tε ) is one to one and continuous, we have that, setting με := (π1 , Tε ) # [εσε∗ ]+ ,

με = με (π1 , Tε ) (x, y) ∈ Ω × Ω: u∗ε (x) − u∗ε (y) = ε, |x − y| ε ,


3531

that is, με = με

(x, z): x ∈ Ω, x + εz ∈ Ω, |z| 1, u∗ε (x) − u∗ε (x + εz) = ε .

Therefore, we can rewrite (4.12) as

ξ(x) f + (x) − f − (x) dx =

Ω

ξ(x) − ξ(x + εz) dμε (x, z). ε

(4.13)

Ω×B 1 (0)

On the other hand, by (4.4), με is bounded by a constant independent of ε. Therefore there exists a subsequence εn → 0 such that weakly as measures,

μεn ϑ

(4.14)

with ϑ =ϑ

(x, z): x ∈ Ω, |z| 1 .

Then, taking limit in (4.13), for ε = εn , as n goes to infinity, we obtain

ξ(x) f + (x) − f − (x) dx =

Ω

∇ξ(x) · (−z) dϑ(x, z).

(4.15)

Ω×B 1 (0)

Now, by disintegration of the measure ϑ (see [2]), ϑ = (ϑ)x ⊗ μ, with μ = π1 # ϑ, that is a non-negative measure. Moreover, if we define ν(x) := (−z) d(ϑ)x (z),

x ∈ Ω,

B 1 (0)

then, ν ∈ L1μ (Ω, RN ) and we can rewrite (4.15) as Ω

ξ(x) f + (x) − f − (x) dx =

∇ξ(x) · ν(x) dμ(x),

∀ξ ∈ Cc1 (Ω).

(4.16)

Ω

Let us see that supp(μ) Ω.

(4.17)

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The proof of (4.17) follows the argument of [1, Lemma 5.1] (we include this argument here for the sake of completeness). In fact, let x0 ∈ supp(f + − f − ) be a minimum point for the restriction of u∗ to supp(f + − f − ) and define

+ w(x) := min u∗ (x) − u∗ (x0 ) , dist x, Ω \ Ω , where Ω verifies (4.10). Then, w(x) = u∗ (x) − u∗ (x0 ) on supp(f + − f − ) and w ≡ 0 on Ω \ Ω . On the other hand, μ(Ω) = ϑ(Ω × RN ) lim inf με (Ω × RN ) lim inf ε[σε∗ ]+ (Ω × RN ) ε→0 ε→0 = lim inf u∗ε (x)(f + (x) − f − (x)) dx ε→0

=

Ω

u∗ (x)(f + (x) − f − (x)) dx,

(4.18)

Ω

and, for a regularizing sequence {ρ 1 }, on account of (4.16) and using that |ν(x)| 1, we have n

u (x) f + (x) − f − (x) dx = ∗

Ω

∗ u (x) − u∗ (x0 ) f + (x) − f − (x) dx

Ω

= lim n

(w ∗ ρ 1 )(x) f + (x) − f − (x) dx n

Ω

= lim n

∇(w ∗ ρ 1 )(x) · ν(x) dμ(x) μ Ω , n

Ω

where Ω verifies (4.10). So, μ(Ω \ Ω ) = 0, and (4.17) is satisfied. Let us now recall some tangential calculus for measures (see [7,8]). We introduce the tangent space Tμ to the measure μ which is defined μ-a.e. by setting Tμ (x) := Nμ⊥ (x) where:

Nμ (x) = ξ(x): ξ ∈ Nμ

being

N ∗ ∞ Nμ = ξ ∈ L∞ μ Ω, R : ∃un smooth, un → 0 uniformly, ∇un ξ weakly in Lμ . In [7], given u ∈ D(Ω), for μ-a.e. x ∈ Ω, the tangential derivative ∇μ u(x) is defined as the projection of ∇u(x) on Tμ (x). Now, by [8, Proposition 3.2], there is an extension of the linear operator ∇μ to Lip1 (Ω, d|·| ) the set of Lipschitz continuous functions. Let us see that ν(x) ∈ Tμ (x),

μ-a.e. x ∈ Ω.

(4.19)

For that we need to show that ν(x) · ξ(x) dμ(x) = 0, Ω

∀ ξ ∈ Nμ .

(4.20)


3533

In fact, given ξ ∈ Nμ , there exists un smooth, un → 0 uniformly, ∇un ξ weakly∗ in L∞ μ . Then, taking ξ = un in (4.16), which is possible on account of (4.17), we obtain

un (x) f + (x) − f − (x) dx =

Ω

∇un (x) · ν(x) dμ(x), Ω

from here, taking limit as n → +∞, we get v(x)ξ(x) · ν(x) dμ(x) = 0,

∀v ∈ D(Ω),

Ω

from where (4.20) follows. Now, if we set Φ := νμ, by (4.16) we have −div(Φ) = f + − f −

in D (Ω).

Then, having in mind (4.19), by [8, Proposition 3.5], we get

u∗ (x) f + (x) − f − (x) dx =

Ω

ν(x)∇μ u∗ (x) dμ(x),

(4.21)

Ω

where ∇μ u∗ is the tangential derivative. Then, since |ν(x)| 1 and |∇μ u∗ (x)| 1 for μ-a.e. x ∈ Ω, from (4.21) and (4.18), we obtain that ν(x) = ∇μ u∗ (x) and |∇μ u∗ (x)| = 1, μ-a.e. x ∈ Ω. Therefore, we have

−div μ∇μ u∗ = f + − f − ∇μ u∗ (x) = 1

in D (Ω), μ-a.e. x ∈ Ω.

Now, by the regularity results given in [12] (see also [1] and [13]), since f + , f − ∈ L∞ (Ω), we have that the transport density μ ∈ L∞ (Ω). Consequently we conclude that the density transport of Evans–Gangbo is represented by a = π1 # ϑ for any ϑ obtained as in (4.14). Acknowledgments We thank the referee for a pertinent and insightful report, which led us to improve this work. Theorem 3.5 was obtained jointly with Professors B. Andreianov and F. Andreu in the bosom of an interesting working week at University of València. N.I., J.M.M. and J.T. have been partially supported by the Spanish MEC and FEDER, project MTM2008-03176. J.D.R. has been partially supported by UBA X196 and by CONICET, Argentina. References [1] L. Ambrosio, Lecture notes on optimal transport problems, in: Mathematical Aspects of Evolving Interfaces, Funchal, 2000, in: Lecture Notes in Math., vol. 1812, Springer-Verlag, Berlin, 2003, pp. 1–52. [2] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., 2000.

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[3] L. Ambrosio, A. Pratelli, Existence and stability results in the L1 theory of optimal transportation, in: Optimal Transportation and Applications, Martina Franca, 2001, in: Lecture Notes in Math., vol. 1813, Springer-Verlag, Berlin, 2003, pp. 123–160. [4] F. Andreu, J.M. Mazón, J.D. Rossi, J. Toledo, A nonlocal p-Laplacian evolution equation with Neumann boundary conditions, J. Math. Pures Appl. 90 (2008) 201–227. [5] F. Andreu, J.M. Mazón, J.D. Rossi, J. Toledo, The limit as p → ∞ in a nonlocal p-Laplacian evolution equation. A nonlocal approximation of a model for sandpiles, Calc. Var. Partial Differential Equations 35 (2009) 279–316. [6] G. Aronsson, L.C. Evans, Y. Wu, Fast/slow diffusion and growing sandpiles, J. Differential Equations 131 (1996) 304–335. [7] G. Bouchitté, G. Buttazzo, P. Seppecher, Energy with respect to measures and applications to low dimensional structures, Calc. Var. Partial Differential Equations 5 (1997) 37–54. [8] G. Bouchitté, T. Champion, C. Jiminez, Completion of the space of measures in the Kantorovich norm, Riv. Mat. Univ. Parma (7) (2005) 127–139. [9] L.A. Caffarelli, M. Feldman, R.J. McCann, Constructing optimal maps for Monge’s transport problem as limit of strictly convex costs, J. Amer. Math. Soc. 15 (2001) 1–26. [10] L.C. Evans, Partial differential equations and Monge–Kantorovich mass transfer, in: Current Developments in Mathematics, Cambridge, MA, 1997, International Press, Boston, MA, 1999, pp. 65–126. [11] L.C. Evans, W. Gangbo, Differential Equation methods in the Monge–Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (653) (1999) viii+66. [12] L. De Pascale, A. Pratelli, Regularity properties for Monge transport density and for solutions of some shape optimization problem, Calc. Var. Partial Differential Equations 14 (2002) 249–274. [13] L. De Pascale, A. Pratelli, Sharp summability for Monge transport density via interpolation, ESAIM Control Optim. Calc. Var. 10 (2004) 549–552. [14] N. Igbida, Partial integro-differential equation in granular matter and its connection with stochastic model, preprint. [15] L.V. Kantorovich, On the transfer of masses, Dokl. Nauk SSSR 37 (1942) 227–229. [16] A. Pratelli, On the equality between Monge’s infimum and Kantorovich’s minimum in optimal mass transportation, Ann. Inst. H. Poincaré Probab. Statist. 43 (2007) 1–13. [17] S.T. Rachev, L. Rüschendorf, Mass Transportation Problems, Springer-Verlag, 1998. [18] V.N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions, Proc. Steklov Inst. Math. 141 (1979) 1–178. [19] C. Villani, Topics in Optimal Transportation, Grad. Stud. Math., vol. 58, 2003. [20] C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss., vol. 338, 2008.


Geometric methods for nonlinear many-body quantum systems Mathieu Lewin CNRS and Laboratoire de Mathématiques (CNRS UMR 8088), Université de Cergy-Pontoise, 95 000 Cergy-Pontoise, France Received 17 October 2010; accepted 23 November 2010 Available online 15 December 2010 Communicated by C. Villani

Abstract Geometric techniques have played an important role in the seventies, for the study of the spectrum of many-body Schrödinger operators. In this paper we provide a formalism which also allows to study nonlinear systems. We start by defining a weak topology on many-body states, which appropriately describes the physical behavior of the system in the case of lack of compactness, that is when some particles are lost at infinity. We provide several important properties of this topology and use them to write a simple proof of the famous HVZ theorem in the repulsive case. In the second step we recall the method of geometric localization in Fock space as proposed by Dereziński and Gérard, and we relate this tool to our weak topology. We then provide several applications. We start by studying the so-called finite-rank approximation which consists in imposing that the many-body wavefunction can be expanded using finitely many one-body functions. We thereby emphasize geometric properties of Hartree–Fock states and prove nonlinear versions of the HVZ theorem, in the spirit of works of Friesecke. In the last section we study translation-invariant many-body systems comprising a nonlinear term, which effectively describes the interactions with a second system. As an example, we prove the existence of the multi-polaron in the Pekar–Tomasevich approximation, for certain values of the coupling constant. © 2010 Elsevier Inc. All rights reserved. Keywords: Geometric methods; Many-body Schrödinger operators; Many-body density matrices; Geometric localization; HVZ theorem; Hartree–Fock method; Multi-configuration theory; Pekar–Tomasevich model; Multi-polaron


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M. Lewin / Journal of Functional Analysis 260 (2011) 3535–3595

Contents 0. 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Spaces and algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Creation and annihilation operators . . . . . . . . . . . . . . . . . . 1.3. Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. States, density matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Geometric convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Compactness results . . . . . . . . . . . . . . . . . . . . . . 2.2. Application: HVZ theorem in the lower semi-continuous case 3. Geometric localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Convergence results . . . . . . . . . . . . . . . . . . . . . . 3.2. Application: HVZ theorem in the general case . . . . . . . . . . 4. Finite-rank approximation of many-body systems . . . . . . . . . . . . . 4.1. States living on a subspace of H, finite-rank states . . . . . . . . 4.1.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Geometric properties of finite-rank states . . . . . . . . 4.2. HVZ-type results for finite-rank many-body systems . . . . . . 4.2.1. A general result . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Two corollaries for fermions . . . . . . . . . . . . . . . . 4.2.3. Translation-invariant Hartree–Fock theory . . . . . . . 5. Many-body systems with effective nonlinear interactions . . . . . . . . 5.1. A general result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Application: the multi-polaron . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3536 3542 3542 3544 3545 3547 3551 3551 3551 3554 3556 3561 3561 3561 3563 3564 3568 3568 3568 3571 3573 3573 3575 3577 3580 3580 3588 3592 3593

0. Introduction A system of N (spinless) quantum particles is usually described by an energy functional Ψ → E(Ψ ) ∈ R where Ψ is a normalized function of the N -body space HN :=

N

N . L2 R d L2 R d

(1)

n=1

Here d is the dimension of the space in which the N particles evolve, that is, d = 3 in the physical case. If the particles are indistinguishable bosons (resp. fermions) it is additionally assumed that Ψ is symmetric (resp. antisymmetric) with respect to exchanges of variables (x1 , . . . , xN ) ∈ (Rd )N . In the simplest case the energy E is the quadratic form associated with a self-adjoint operator on HN . For nonrelativistic particles interacting with a two-body potential W and submitted to an external potential V , the corresponding N -body Hamiltonian reads


H V (N ) =

N xj − + V (xj ) + 2 j =1

W (xk − x ).

3537

(2)

1k N leads to multiconfiguration methods [19,30]. Despite the fact that these methods are essential tools of quantum physics and chemistry, their geometric properties have deserved little interest in the literature so far. In [19], Friesecke was, to our knowledge, the first to consider both the Hartree– Fock and the multiconfiguration theories as real N -body models and to use geometric techniques in order to derive nonlinear HVZ-type results. Our goal is to emphasize geometric properties of finite-rank states, that is, to find what can be said on geometric limit points of sequences or on geometric localization of such special states. For instance, we show in Section 4.1.2 that the geometric limit of a sequence of pure Hartree– Fock states is always a convex combination of pure Hartree–Fock states, see Example 16 below. Using such properties, we are able to provide a simple proof of Friesecke’s results, as well as to derive other theorems. For instance, in Theorem 22 below, we prove a nonlinear HVZ-type result for a translation-invariant Hartree–Fock theory, combining ideas of Lions [38,39] and geometric techniques. This result is in the same spirit as what was done for neutron stars in a recent collaboration with Lenzmann [29]. In Section 5 we study another kind of nonlinear models where all possible many-body states are considered but nonlinear effective terms are added to the quantum energy E in order to describe some specific physical effects. To be more precise we concentrate on translation-invariant models of the form

E(Ψ ) = Ψ, H 0 (N )Ψ + F (ρΨ )

(10)

where F is a concave nonlinear function of the charge density ρΨ , and H 0 (N ) is the N -body Hamiltonian (2) with V = 0. In practice the purpose of the nonlinear term F (ρΨ ) is to model the interaction of the N particles with a second complicated system. For instance, we consider in Section 5.2 the multi-polaron in the Pekar–Tomasevich approximation. This is a system of N nonrelativistic electrons with an effective nonlinear term ρΨ (x)ρΨ (y) α dx dy F (ρΨ ) = − 2 |x − y| R3 R3

modeling interactions with the phonons of a polar crystal in the regime of strong coupling. We show the existence of bound states for all α > τc (N ) where τc (N ) < 1, which covers the physical case. This complements recent results of [17,18].

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The paper is organized as follows. In Section 1 we provide necessary notations and some preliminary results. The reader at ease with the concepts of Fock space, creation and annihilation operators and density matrices may want to skip most of the material of Section 1. Of importance is Lemma 1 which provides crucial properties of density matrices for states on a truncated Fock space F N . Section 2 is devoted to the definition and the derivation of important properties of the geometric topology and convergence. This is followed by a proof of the HVZ theorem in the repulsive case. In Section 3 geometric localization is defined and the general HVZ theorem is proved. Sections 4 and 5 are respectively devoted to the study of the finite-rank approximation, and of nonlinear systems of the form (10). For the sake of clarity we usually do not state the most general results and rather favor some chosen applications. Many of our theorems can be generalized in several directions. 1. Notation and preliminaries We start by fixing some important notation and vocabulary, as well as by providing some preliminary results that will be useful throughout the paper. The reader acquainted with Fock spaces can jump to Section 1.4 where density matrices are defined and some of their important properties are derived. 1.1. Spaces and algebras For a (separable) Hilbert space H, we denote by B(H) and K(H) the algebras of, respectively, bounded and compact operators on H. The Schatten space Sp (H) ⊂ K(H) is defined [58] by requiring that 1/p < ∞, ASp (H) := Tr |A|p √ with |A| = A∗ A. Operators in S1 (H) have a well-defined trace Tr(A) = i fi , Afi (for any orthonormal basis {fi } of H). Operators in S2 (H) are called Hilbert–Schmidt. We recall that [58] K(H) = S1 (H)

and

S1 (H) = B(H).

(11)

Since K(H) is separable when H is separable, (11) means that any bounded sequence {Γn } in S1 (H) has a subsequence which converges weakly–∗ in the sense that limn→∞ TrH (Γn K) = TrH (Γ K) for all K ∈ K(H). The same holds for bounded sequences in B(H), with K(H) replaced by S1 (H). In the whole paper we fix as space for one quantum particle H = L2 (Rd ). We could as well work in a domain Ω with appropriate boundary conditions, use a discrete model, or even, for most of our results, take an abstract Hilbert space. These obvious generalizations are left to the reader for shortness. Similarly, for simplicity we almost always restrict ourselves to the case of quantum systems made of one kind of indistinguishable particles ( fermions or bosons) without spin. Most results can be easily generalized to the case of several kinds of particles having internal degrees of freedom. The space for N indistinguishable fermions is the antisymmetric tensor product HN a :=

N 1

N H = L2a Rd


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consisting of wavefunctions Ψ which are antisymmetric with respect to exchanges of variables: Ψ (x1 , . . . , xi , . . . , xj , . . . , xN ) = −Ψ (x1 , . . . , xj , . . . , xi , . . . , xN ), with xk ∈ Rd for k = 1, . . . , N . The space for N indistinguishable bosons is the symmetric tensor product HN s :=

N

N H = L2s Rd

1

consisting of wavefunctions Ψ which are symmetric with respect to exchanges of variables: Ψ (x1 , . . . , xi , . . . , xj , . . . , xN ) = Ψ (x1 , . . . , xj , . . . , xi , . . . , xN ), with xk ∈ Rd for k = 1, . . . , N . The corresponding fermionic or bosonic Fock space is denoted as Fa/s = C ⊕

HN a/s .

N 1

Saying differently, it is the space composed of sequences of the form Ψ = (ψ 0 , ψ 1 , ψ 2 , . . .) ∈ C × H × H2a/s × · · · satisfying the constraint that Ψ 2Fa/s :=

2 ψ n n < ∞. H a/s

n0

It is a Hilbert space when endowed with the scalar product

Ψ1 , Ψ2 Fa/s =

ψ1n , ψ2n

n0

Hna/s

.

The vacuum state is by convention defined as Ω := (1, 0, 0, . . .) ∈ Fa/s . As we consider N -body systems, we must always work in the ‘truncated’ Fock space N

Fa/s := C ⊕

N

Hna/s

(12)

n=1

which we identify to a closed subspace of Fa/s . Similarly, any N -body vector of HN a/s can be N

viewed as a vector of Fa/s or of Fa/s . As we explain later in Section 2, the ‘geometric’ limit of N

a sequence (ψn ) ⊂ HN a/s will always live in the truncated Fock space Fa/s .

N2 N1 +N2 1 For ψ1 ∈ HN a and ψ2 ∈ Ha , we define the antisymmetric tensor product ψ1 ∧ ψ2 ∈ Ha as follows:

ψ1 ∧ ψ2 (x1 , . . . , xN1 +N2 ) 1 := √ N1 !N2 !(N1 + N2 )! sgn(σ )ψ1 (xσ (1) , . . . , xσ (N1 ) ) ψ2 (xσ (N1 +1) , . . . , xσ (N1 +N2 ) ). × σ ∈SN1 +N2

(13)

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Here SN is the group of permutations of {1, . . . , N}. When {fi } is an orthonormal basis of H, then {fi1 ∧ · · · ∧ fiN }i1

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