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Journal of Functional Analysis 255 (2008) 1–12 www.elsevier.com/locate/jfa Estimates for Hilbertian Koszul homology Xia...

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Journal of Functional Analysis 255 (2008) 1–12 www.elsevier.com/locate/jfa

Estimates for Hilbertian Koszul homology Xiang Fang 1 Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA Received 21 March 2007; accepted 19 December 2007 Available online 14 April 2008 Communicated by G. Pisier

Abstract The objective of this paper is to give new kind of estimates for Hilbertian Koszul homology, inspired by commutative algebra, in multivariable Fredholm theory. © 2008 Elsevier Inc. All rights reserved. Keywords: Multivariable Fredholm theory; Koszul complex; Homology; Lech’s formula

0. Introduction The Fredholm index of a single operator admits a generalization to several variables via Koszul complexes over Hilbert spaces, which is, in general, difficult to calculate. In particular, in sharp contrast with rich results on Noetherian algebraic modules, over Hilbert modules currently there are essentially no systematic estimates for higher Koszul homology groups. In [13–15], we initiated a study of Fredholm theory through the asymptotic behavior of higher powers of a tuple T¯ . See also Eschmeier’s [12]. In this paper, the asymptotic methods lead to estimates for all powers of T¯ . Let T¯ = (T1 , . . . , Tn ) (n ∈ N) be a Fredholm tuple of commuting operators on a Hilbert space H . This means that the homology groups Hi (K(T1 , . . . , Tn )) (i = 0, 1, . . . , n) of the associated Koszul complex K(T1 , . . . , Tn ) of Ti over the Hilbert space H are all finite-dimensional. Let k = (k1 , . . . , kn ) ∈ Nn be a multi-index, and T¯ k = (T1k1 , . . . , Tnkn ). If T¯ is Fredholm, then so E-mail address: [email protected]. 1 Partially supported by National Science Foundation Grant DMS 0400509.

0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2007.12.016

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X. Fang / Journal of Functional Analysis 255 (2008) 1–12

is T¯ k . For convenience, let hi (k1 , . . . , kn ) = dim(Hi (K(T1k1 , . . . , Tnkn ))). The main result of this paper is Theorem 1. For any Fredholm tuple (T1 , . . . , Tn ), there exist e0 , e1 , . . . , en ∈ Z, and a constant C > 0 such that for all i = 0, 1, . . . , n, and k1 , k2 , . . . , kn ∈ N, k1 k2 · · · kn · ei hi (k1 , . . . , kn ) k1 k2 · · · kn

C ei + . min ki

A few remarks follow: • Considering the multi-index k is indeed useful, say, in [14], where n = 2, and supk hi (1, k) < ∞ implies ei = 0. h (T k ,...,T k )

• Clearly, our result implies that ei = limk→∞ i 1 k n n (see Corollary 2.3 in [12]) and that index index(T¯ ) = ni=0 (−1)i ei by the multiplicity formula index(T1k1 , . . . , Tnkn ) = k1 · · · kn index(T1 , . . . , Tn ). • When H is replaced by a finitely generated module over a Noetherian ring, the corresponding function hi is dominated by a polynomial of ki with degree n − i, hence ei = 0 except for exists. possibly e0 [26]. It is not clear whether limk→∞ hi (k,...,k) k n−i Two main ingredients in the proof of Theorem 1. Many arguments in this paper refine those of Eschmeier’s [12] in order to obtain quantitative results. The first set of techniques is sheaf theoretic. First touched upon by Markoe [22], sheaf theory for operators was systematically investigated later [25], and the primary reference is the monograph [11]. The second set is commutative algebra in nature, and is more closely related to our previous work. In particular, we own a deep intellectual debt to C. Lech [14,18,19], from which we borrow many ideas. Both sets of techniques are well known, and in fact easy, to experts in algebra and analysis, respectively. What we do here is to bring them together to yield estimates which appear of value in operator theory. 1. Background Definitions. In order to study the spectral theory of a tuple of commuting operators, instead of a single operator, J.L. Taylor, in 1970, introduced a seminal approach via Koszul complexes over Banach spaces [28,29]. For a commuting tuple T¯ = (T1 , . . . , Tn ) on a Banach space H , its Koszul complex K(T1 , . . . , Tn ) is K(T¯ ):

0→H ⊗

n

Cn → H ⊗

n−1

Cn → · · · → H ⊗

0

Cn → 0.

n Here n Cn is the kth exterior power of Cn . Let {e1 , . . . , en } be an orthonormal basis nfor C , and let ci be the creation operator associated with ei , that is, ci (ξ ) = ei ∧ ξ for ξ ∈ C . Then the boundary operator is B = T1 ⊗ c1∗ + · · · + Tn ⊗ cn∗ . The tuple (T1 , . . . , Tn ) is called invertible if the complex K(T¯ ) is exact.


3

Subsequently, a multivariable Fredholm theory is formulated: a tuple T¯ of commuting operators is Fredholm if K(T¯ ) has a finite-dimensional homology group at each stage, that is, dimC Hi K(T¯ ) < ∞ for all i = 0, 1, . . . , n [1,2,7,8,11,30]. We also write Hi (T¯ ) instead of Hi (K(T¯ )) for convenience. The n + 1 homology groups of K(T¯ ) are the multivariable analogs of the kernel ker(T ) and cokernel H /T H of a single operator T ∈ B(H ). When (T1 , . . . , Tn ) is Fredholm, define the multivariable Fredholm index by index(T1 , . . . , Tn ) =

n (−1)i dimC Hi K(T¯ ) , i=0

the Euler characteristic of K(T¯ ). The multivariable index index(T¯ ) is connected with a variety of problems in both classical analysis and algebraic topology [3,11,20,21]. Currently, however, there is essentially no effective computational tools, especially for higher homology groups Hi (·), that is, for those groups with i > 0. Most known examples are, or are reduced to, acyclic tuples: Hi (·) = 0 except for i = 0, hence index(·) = dim(H0 (·)). Consequently, there is a current need to get a better grasp on those higher homology groups. Motivation. Our approach to Hi (·) originates from an effort to generalize the following simple arguments from [14] to several variables: for a single Fredholm operator T acting on a separable Hilbert space H , by the definition of Fredholm index, and the multiplicity formula,

index(T ) = =

index(T k ) k dim(ker(T k )) dim(H /T k H ) − k k dim(ker(T k )) dim(H /T k H ) − lim . k→∞ k→∞ k k

= lim

Here both limits exist, and are in fact integers. This leads to links to commutative algebra through the Hilbert function k → dim(H /T k H ), [10], and a celebrated result of J.-P. Serre, relating the Euler characteristics of Koszul complexes to Samuel multiplicities [27]. 2. Correction modules C(M, L; J ) This section is purely commutative algebra. We introduce a notion of correction modules, which, simple as it is, seems not discussed explicitly in literature. For operator theorists wanting more algebraic references, see standard texts [4,10] for Samuel multiplicity, and see [14,18,19], and [26] for Lech’s formulas.

4


Definition 2. Let R be a ring, J ⊂ R be an ideal, and M ⊂ L be a submodule of an R-module L. Define the correction module of M in L with respect to an ideal J to be C(M, L; J ) =

M ∩ JL . JM

Remark. When R and L are Noetherian, the Artin–Rees lemma is useful for the study of the asymptotic behavior of C(M, L; J k ) when k → ∞. Lemma 3. Let R be a local Noetherian ring, I = (x1 , . . . , xn ) ⊂ R be its maximal ideal, and Ik = (x1k1 , . . . , xnkn ) for any k ∈ Nn . If L is a finitely generated R-module, and M ⊂ L is a submodule, then there exists a constant C such that for all k ∈ Nn , length C(M, L; Ik ) k1 k2 · · · kn ·

C . min kj

Proof. Let N = L/M be the quotient module. For any ideal J ⊂ R, applying the functor (·) ⊗R R/J , which is only right-exact, to a short exact sequence of R-modules 0 → M → L → N → 0,

(2.1)

→ M/J M → L/J L → N/J N → 0.

(2.2)

we get a right-exact sequence

By the definition of correction module, it follows 0 → C(M, L; J ) → M/J M → L/J L → N/J N → 0.

(2.3)

Consider J = Ik , and by the algebraic Lech’s formula (see Lemma 4), there exists a constant CE for the modules E = M, L, or N , such that CE . k1 · · · kn · e(E) length(E/Ik E) k1 · · · kn e(E) + min kj

(2.4)

Here length(E/I t E) t→∞ tn

e(E) = n! lim

is the Samuel multiplicity of E with respect to I . By the additivity of Samuel multiplicity over short exact sequence (2.1), we have e(L) = e(M) + e(N ). Now the proof is completed by observing


5

length C(M, L; Ik ) = length(M/Ik M) + length(N/Ik N ) − length(L/Ik L) k1 k2 · · · kn ·

C M + CN . min kj

2

Remarks. (1) We derive the name of C(M, L; J ) from (2.3). (2) For the proof of Theorem 1, the only case we need is R = O0 , the local ring of germs of holomorphic functions around the origin in Cn . For readers’ convenience we record the following. Lemma 4 (Lech’s inequality). Let J = (x1 , . . . , xn ) be an ideal of a local ring R, generated by xi , and let M be a Noetherian R-module such that length(M/J M) < ∞, then there exists a constant C such that k1 C kn , k1 · · · kn e(M, J ) length M/ x1 , . . . , xn M k1 · · · kn e(M, J ) + minj kj here e(M, J ) is the Samuel multiplicity of M with respect to J . The original proof of Lech is contained in the proof of Theorem 2 in [18], which in fact only covers the case M = R. The (Hilbert) module case is treated in [14]. Both proofs can be easily adopted to prove Lemma 4. 3. Difference between Hp (L• /J L• ) and Hp (L• )/J Hp (L• ) as correction modules This section is still purely algebraic. Let R be any commutative ring, and L• :

· · · → Lp → Lp−1 → · · ·

be a complex of R-modules, with Hp (L• ) denoting the homology group at the pth stage, p ∈ Z. For any ideal J ⊂ R, we represent the difference between Hp (L• /J L• ) and Hp (L• )/J Hp (L• ) as correction modules in this section. This is also considered in [12]. Here we refine the arguments in [12] and obtain more quantitative results. Since the difference between Hp (L• /J L• ) and Hp (L• )/J Hp (L• ) is often encountered, say in base change theorems, in algebraic geometry, our results here may be of interests to algebraists. Recall that for any R-module M, there exists a natural morphism (say, by [5]) Hp (L• ) ⊗R M → Hp (L• ⊗R M). Here we will consider M = J , and R/J . Let Zp ⊂ Lp be the set of closed elements, that is, Zp = ker(Lp → Lp−1 ), and let Bp ⊂ Lp be the set of boundary elements, that is, Bp = Image(Lp+1 → Lp ). Note that Hp (L• ) = Zp /Bp . Lemma 5. For the natural morphism j : Hp (L• )/J Hp (L• ) → Hp (L• /J L• ), the cokernel is isomorphic to coker(j ) ∼ = C(Bp−1 , Lp−1 ; J ).

6


The kernel ker(j ) is resolved by an exact sequence of correction modules 0 → C(Bp , Zp ; J ) → C(Bp , Lp ; J ) → C(Zp , Lp ; J ) → ker(j ) → 0. Proof. The standard strategy in algebra is to analyze the natural morphism j : Hp (L• )/J Hp (L• ) → Hp (L• /J L• ) by embedding it into a commutative diagram. To resolve Hp (L• /J L• ), we consider the long exact sequence associated with 0 → J L• → L• → L• /J L• → 0, and get the first row of the diagram (3.1). To resolve Hp (L• )/J Hp (L• ) we consider the straightforward short exact sequence which leads to the second row of the diagram (3.1). Together with the natural morphisms j1 , j2 = id, and j , we obtain a commutative diagram Hp (J L• ) j1

0

d1

Hp (L• )

d2

Hp (L• /J L• )

j2

J Hp (L• )

δ

Hp−1 (J L• )

d3

Hp−1 (L• ) (3.1)

j

Hp (L• )

Hp (L• )/J Hp (L• )

0.

The cokernel part is easier. By the second commutative square in the diagram (3.1), and the exactness of the first row in (3.1), coker(j ) = coker(d2 ) ∼ = Image(δ) = ker(d3 ). Let Z∗ (J L• ) (respectively B∗ (J L• )) denote the closed (respectively boundary) elements of the complex J L• . Then ker(d3 ) =

Zp−1 (J L• ) ∩ Bp−1 J Lp−1 ∩ Zp−1 ∩ Bp−1 J Lp−1 ∩ Bp−1 = = . Bp−1 (J L• ) J Bp−1 J Bp−1

Now consider ker(j ). By the second commutative square, and exactness of both rows in (3.1), ker(j ) =

Image(d1 ) ker(d2 ) = . J Hp (L• ) J Hp (L• )

Note that Image(d1 ) =

J Lp ∩ Zp + Bp Bp

and J Hp (L• ) =

J Zp + Bp . Bp

Hence we can resolve ker(j ) by C(Zp , Lp ; J ) 0→

J Lp ∩ Zp J Lp ∩ Zp + Bp (J Lp ∩ Zp ) ∩ (J Zp + Bp ) → → → 0. J Zp J Zp J Zp + Bp

(3.2)


7

Observe that (J Lp ∩ Zp ) ∩ (J Zp + Bp ) = (J Lp ∩ Zp ) ∩ Bp + J Zp = J Lp ∩ Bp + J Zp . Here the first equality is because if x ∈ J Zp , y ∈ Bp such that x + y ∈ J Lp ∩ Zp , then y ∈ J Lp ∩ Zp since x ∈ J Zp ⊂ J Lp ∩ Zp . Hence the left-hand side of (3.2) is isomorphic to J Lp ∩ Bp J Lp ∩ Bp J Lp ∩ Bp + J Zp ∼ = . = J Zp (J Lp ∩ Bp ) ∩ J Zp J Zp ∩ Bp But the last one is resolved by correction modules C(Bp , Zp ; J ) and C(Bp , Lp ; J ) 0→

J Zp ∩ Bp J Lp ∩ Bp J Lp ∩ Bp → → → 0. J Bp J Bp J Zp ∩ Bp

Now patching (3.2) and (3.3) completes the proof of the lemma.

(3.3)

2

4. Parametrized Koszul complexes In this section sheaf theory comes into the play. For more background interested readers should see [11], especially those arguments related to Lemma 2.1.5, Proposition 9.4.5, and Theorem 10.3.13. Here our approach is slightly more algebraic. It allows conceptual proofs and leads to conjectures for further development. We start with a connection between a Hilbert module H over a ring R, associated with an operator tuple T¯ = (T1 , . . . , Tn ), and its sheaf model [11,25], h = O(H )/(z − T¯ )O(H ), as well as its stalk at the origin h0 = O0 (H )/(z − T¯ )O0 (H ). Here R is any of the following three rings C[z1 , . . . , zn ],

O(Cn ),

and O(U ),

with U being a Stein neighborhood of the Taylor spectrum σ (T¯ ). In any case, and even for O0 , let I = (z1 , . . . , zn ) be the maximal ideal at the origin. We usually assume that dim(H /I H ) < ∞ and 0 ∈ σ (T¯ ). In an effort to relate H to h0 , Douglas and Yan showed in [9] that the Hilbert function of H , with respect to I , is greater than or equal to the Hilbert function of h0 . In [15] we showed that the inequality between the two Hilbert functions is in fact an equality. This plays a key role in the proof of the semi-continuity of Samuel multiplicity over Hilbert modules. The result from [15] can be reformulated as that the completions of H and h0 in the so-called I -adic topology [27] are isomorphic, Hˆ ∼ = hˆ 0 ,

(4.1)

which is better suited for generalization. In particular, an easy consequence of (4.1) is H /I H ∼ = h0 /I h0 .

(4.2)

8


Next we aim at the homological generalizations of (4.1) and (4.2). First, rewrite the completion Hˆ as an inverse limit H /I k H. Hˆ = lim ←− k→∞

Let Ik = (z1k , . . . , znk ) ⊂ R. Then, by basic facts on inverse limits H /Ik H. Hˆ = lim ←− k→∞

Observe that H /Ik H can be written as the 0th homology group of the Koszul complex K(T1k , . . . , Tnk ; H ) of (T1k , . . . , Tnk ) on H . On the other hand, the sheaf model h can be written as the 0th homology group H0 (z − T¯ , O(H )) of the Koszul complex of z − T¯ = (z1 − T1 , . . . , zn − Tn ) on O(H ). To generalize Eq. (4.1), observe that, for each i = 0, 1, . . . , n, we can form an inverse system of the Koszul homology groups, [6,16],

k Hi T1 , . . . , Tnk ; H , k = 1, 2, . . . . Definition 6. For each i = 0, 1, . . . , n, we define Hˆ i = lim Hi T1k , . . . , Tnk ; H . ←− k→∞

For the sheaf side, as generalization of the sheaf model h = h(0) , we call h(i) = Hi z − T¯ , O(H ) , i = 0, 1, . . . , n, the homological sheaf models of H . The modules Hˆ i are reminiscent of Grothendieck’s local cohomology modules in algebraic geometry [16]. According to Markoe [22], h(i) is in fact a coherent analytic sheaf around the origin for each i when T¯ is Fredholm. The significance of Hˆ i and h(i) is yet to be understood. As a first step, and as a generalization of (4.1), we offer the following conjecture. Let h(i),0 denote the stalk at the origin, and hˆ (i),0 its I -adic completion. Conjecture. For any Fredholm tuple T¯ and each i = 0, 1, . . . , n, we have a natural isomorphism Hˆ i ∼ = hˆ (i),0 of modules over the ring of power series C z1 , . . . , zn . As for Eq. (4.2), observe that h0 /I h0 can be written as the 0th homology group of the Koszul complex K(z − T¯ ; R/I ⊗ H ) of z − T¯ = (z1 − T1 , . . . , zn − Tn ) on O0 (H )/I O0 (H ) = R/I ⊗C H . Note that O0 /I O0 are isomorphic to R/I , as Artinian rings, for any of R = C[z1 , . . . , zn ], O(Cn ), and O(U ). For each of these three rings, we generalize (4.2) to Lemma 7. Let f = (f1 , . . . , fn ) be a regular sequence in R, and (f ) be the ideal generated by fj . Then Hi f1 (T¯ ), . . . , fn (T¯ ); H ∼ = Hi z − T¯ , R/(f ) ⊗C H .


9

Remarks. (1) Here f being regular means that the Koszul complex of f on R yields a free resolution of R/(f ) [10]. In particular, length(R/(f )) < ∞. (2) Under the condition f −1 (0) = 0, Lemma 7 is already covered in [12] which, in turn, is modeled after the proof of Theorem 10.3.13 in [11]. Modulo technical matters, what is new here is just the way it is presented. (3) Our proof is essentially only a series observations in homological algebra, which can establish the result for a larger category, and motivates a further conjecture—see the remark at the end of the paper. Proof. When i = 0, both sides are directly verified to be H /

fj (T¯ )H . In fact one has

H0 z − T¯ , R/(f ) ⊗C H ∼ = R/(f ) ⊗R H. The natural map from the left to the right is the class of x ⊗ y ∈ R/(f ) ⊗C H being sent to the class of x ⊗ y ∈ (R/(f )) ⊗ H . It is clearly surjective with kernel being the submodule generated by rx ⊗C y − x ⊗C ry, which is the same as that generated by (zi − Ti )(x ⊗C y) = zi x ⊗C y − x ⊗C Ti y [10]. For general i, since f is regular, the Koszul homology is also given by the derived functors TorR i (·, ·), Hi f1 (T¯ ), . . . , fn (T¯ ); H = TorR i R/(f ), H . Let Rw denote the ring R with variables written in w = (w1 , . . . , wn ), and consider H as a module over Rw . Then Hi (z − T¯ , R/(f ) ⊗C H ), viewed as a module over Rz ⊗ Rw , is naturally a module over Rw . Hence, the functors Fi : M → Hi (z − T¯ , R/(f ) ⊗C M) can be regarded as over the category of Rw (= R)-modules. To show the sequence of functors Fi : M → Hi (z − T¯ , R/(f ) ⊗C M) coincide with the derived functors M → TorR i (R/(f ), M) for any R-module M, we only need to show that F = (Fi ) is a universal δ-functor—here we use the machinery in homological algebra as explained in Section 3.1, [17]. Being a δ-functor is clear by definition. To show it is universal, it suffices to show that Fi is coeffaceable for i > 0. This can be verified by (1) the category of all R-modules have enough projectives, and (2) Fi (R) = 0 when i > 0. The first is algebraic folklore, and the second follows easily from the definition of regular sequence. Next we give more details for Hi z − w, O(U )/(f ) ⊗C O(U ) = 0 (i > 0) for readers’ convenience [10]. Because (z − w) forms a regular sequence in O(U ) ⊗C O(U ), the Koszul homology can be calculated via O(U )⊗C O(U )

Tori

O(U ) ⊗C O(U )/(z − w), O(U )/(f ) ⊗C O(U ) .

Since (f ) is regular by assumption, the Koszul complex K(f, O(U )) provides a free resolution of O(U )/(f ), hence K(f, O(U )) ⊗C O(U ) a free resolution of O(U )/(f ) ⊗C O(U ). Hence the above Tori can be calculated through the complex

10


K f, O(U ) ⊗C O(U ) ⊗O(U )⊗C O(U ) O(U ) ⊗C O(U )/(z − w) ∼ = K f, O(U ) ⊗C O(U )/(z − w). The last term, regarded as a complex of O(U )-modules in the variable z, is isomorphic to K(f, O(U )), which is acyclic, hence Hi (· · ·) = 0. 2 ∼ O0 (H )/J O0 (H ). If L• = K(z − T¯ , Let J = (f ) = (z1k1 , . . . , znkn ), then R/(f ) ⊗C H = ¯ O0 (H )) denotes the Koszul complex of z − T on O0 (H ), then, by Lemma 7, Hi T1k1 , . . . , Tnkn ∼ = Hi (L• /J L• ). Since O0 (H ) in L• is an infinitely generated O0 -module when dim(H ) = ∞, a standard strategy for parametrized complexes is to find a complex of finitely generated O0 -modules with isomorphic homology groups, which will allow us to apply results from Section 3. Lemma 8. If T¯ is Fredholm, then there exists a complex E• of finitely generated O0 -modules: · · · → Ei → Ei−1 → · · · , such that for J = 0, or any k = (k1 , . . . , kn ) ∈ Nn and J = (z1k1 , . . . , znkn ), Hi (L• /J L• ) ∼ = Hi (E• /J E• ),

i ∈ Z.

Proof. This is essentially due to [11] and [12]. The construction of E• is detailed in [11]. The isomorphism between homology groups is verified in [12]. 2 Proof of Theorem 1. Since the components Ei in E• are finitely generated O0 -modules, so are the homology groups Hi (E• ). By Lemma 4, the function φ(k) = dim Hi (E• )/J Hi (E• ) satisfies, for some constant C, φ(k) k1 · · · kn ei +

C , min kj

here ei = ei (T¯ ) is the Samuel multiplicity of Hi (E• ) = Hi (L• ) with respect to I . Now, the estimates on correction modules, together with the representation of the difference between Hi (E• )/J Hi (E• ) and Hi (E• /J E• ) as correction modules, completes the proof of the upper bound in Theorem 1. The lower bound is much easier, and is in fact part of Theorem 2.4 in [12]. Our treatment here is just slightly different. For fixed k, we claim that hi (k1 , . . . , kn ) = ei · k1 · · · kn when the tuple is T¯ − λ, where λ is in a small neighborhood of the origin except for a possibly thin subvariety. Because hi (k1 , . . . , kn ) is upper semi-continuous around the origin as a function of λ in T¯ − λ, we get the lower bound. Since the singularity set of the coherent sheaf Hi (L• ) is thin, we can choose small λ such that, with respect to the tuple T¯ − λ, Hi (L• ) is free, and, for any primary ideal J , the following are naturally isomorphic: Hi (L• /J L• ) ∼ = Hi (L• )/J Hi (L• ) (see Grauert’s comparison theorem [5,17]).


11

Here the latter has dimension ei · dim(O0 /J ) since Hi (L• ) is free. Choosing J = (z1k1 , . . . , znkn ) gives us the claim. 2 Because Hi (z − T¯ , O(H )) is coherent around the origin when T¯ is Fredholm [22], and ei is the Samuel multiplicity of the stalk of Hi (z − T¯ , O(H )) at the origin, a straightforward consequence is, by the invariance of Samuel multiplicity of stalks of a coherent analytic sheaf, that is Hi (z − T¯ , O(H )) in our case, the local constancy of ei (T¯ − λ). Corollary 9. If T¯ is Fredholm, then the function λ ∈ Cn → ei (T¯ − λ) is locally constant in a neighborhood of the origin. In other words, let Ω be a connected component of the Fredholm domain Cn \ σe (T¯ ), then ei (T¯ − λ) is a constant for λ ∈ Ω. Motivated by the base change formula for Fredholm index index(f1 (T¯ ), . . . , fn (T¯ )) [24], it is natural to ask whether similar formulas hold for Hi (f1 (T¯ ), . . . , fn (T¯ )) and ei (f1 (T¯ ), . . . , fn (T¯ )). In general, Hi (·) is too unstable to enjoy a nice base change formula. For ei , however, we observe that the proof of the base change formula in Theorem 10.3.16 in [11] goes, roughly, as follows. The key in reduction is that index(T¯ − λ) is locally constant in λ. For a neighborhood U ⊃ σ (T¯ ) of the Taylor spectrum σ (T ), and a map F = (f1 , . . . , fn ) : U → Cn with F (0) = 0, we can consider index(F (T¯ ) − λ) such that the fibre (F − λ)−1 (0) is simple, that is, a collection of k distinct points {p1 , . . . , pk }, here k being the mapping degree of f at 0. Then over each simple point pi , the contribution to index can be counted directly, hence leading to the base change formula. Now, based on Corollary 9, the whole proof in [11] carries over for ei (·). Corollary 10. Let T¯ be a Fredholm tuple, and F ∈ O(U )n be an n-tuple of analytic functions defined on an open neighborhood U of the Taylor spectrum σ (T¯ ). Assume that F (0) = 0 and 0∈ / σe (F (T¯ )), and let mz (F ) denote the multiplicity of F at z. Then, for each i = 0, 1, . . . , n,

ei F (T¯ ) =

mz (F )ei (T¯ − z).

z∈F −1 (0)∩σ (T¯ )

We end the paper with a remark when f = (f1 , . . . , fn ) in R = C[z1 , . . . , zn ], O(Cn ), or O(U ), is not necessarily a regular sequence. If we rewrite R/(f ) as the 0th Koszul homology of f on R, then Lemma 7 becomes Hi f1 (T¯ ), . . . , fn (T¯ ); H ∼ = Hi z − T¯ , H0 (f, R) ⊗C H . Hence it motivates Conjecture. For general f , there exists a spectral sequence, with E 2 page 2 ∼ Epq = Hp z − T¯ , Hq (f, R) ⊗C H , convergent to Hp+q (f1 (T¯ ), . . . , fn (T¯ ); H ).

12


Adopting this viewpoint, Lemma 7, that is when f is regular, actually follows immediately from Grothendieck’s spectral sequence of composition functors [23,31]. For the general case, we will address the conjecture by constructing spectral sequences directly from double complexes in a coming work. Acknowledgment The author thanks J. Eschmeier for sending several of his manuscripts and papers, and for communications which prompt this work. References [1] C.-G. Ambrozie, F.-H. Vasilescu, Banach Space Complexes, Math. Appl., vol. 334, Kluwer, Dordrecht, 1995. [2] W. Arveson, The Dirac operator of a commuting d-tuple, J. Funct. Anal. 189 (1) (2002) 53–79. [3] M. Atiyah, A survey of K-theory, in: K-Theory and Operator Algebras, Proc. Conf., Univ. Georgia, Athens, GA, 1975, in: Lecture Notes in Math., vol. 575, Springer-Verlag, Berlin, 1977, pp. 1–9. [4] S. Balcerzyk, T. Jozefiak, Commutative Rings. Dimension, Multiplicity and Homological Methods, Prentice–Hall, Englewood Cliffs, NJ, 1989. [5] C. Banica, O. Stanasila, Algebraic Methods in the Global Theory of Complex Spaces, Wiley, 1976 (translated from Rumanian). [6] M.P. Brodmann, R.Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Stud. Adv. Math., vol. 60, Cambridge Univ. Press, Cambridge, 1998. [7] R.E. Curto, Fredholm and invertible n-tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1) (1981) 129–159. [8] R.E. Curto, Applications of several complex variables to multiparameter spectral theory, in: Surveys of Some Recent Results in Operator Theory, vol. II, in: Pitman Res. Notes Math. Ser., vol. 192, Longman, Harlow, 1988, pp. 25–90. [9] R. Douglas, K. Yan, Hilbert–Samuel polynomials for Hilbert modules, Indiana Univ. Math. J. 42 (1993) 811–820. [10] D. Eisenbud, Commutative Algebra. With a View toward Algebraic Geometry, Grad. Texts in Math., vol. 150, Springer-Verlag, New York, 1995. [11] J. Eschmeier, M. Putinar, Spectral Decompositions and Analytic Sheaves, London Math. Soc. Monogr. (N.S.), vol. 10, Oxford Univ. Press, New York, 1996. [12] J. Eschmeier, Samuel multiplicity and Fredholm theory, preprint. [13] X. Fang, Samuel multiplicity and the structure of semi-Fredholm operators, Adv. Math. 186 (2) (2004) 411–437. [14] X. Fang, The Fredholm index of a pair of commuting operators, Geom. Funct. Anal. 16 (2) (2006) 367–402. [15] X. Fang, The Fredholm index of a pair of commuting operators, II, preprint. [16] R. Hartshorne, Local Cohomology, Lecture Notes in Math., vol. 41, Springer-Verlag, Berlin, New York, 1967. [17] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math., vol. 52, Springer-Verlag, New York, 1977. [18] C. Lech, On the associativity formula for multiplicities, Ark. Mat. 3 (1957) 301–314. [19] C. Lech, Notes on multiplicities of ideals, Ark. Mat. 4 (1959) 63–86. [20] R.N. Levy, The Riemann–Roch theorem for complex spaces, Acta Math. 158 (3–4) (1987) 149–188. [21] R.N. Levy, Algebraic and topological K-functors of commuting n-tuple of operators, J. Operator Theory 21 (2) (1989) 219–253. [22] A. Markoe, Analytic families of differential complexes, J. Funct. Anal. 9 (1972) 181–188. [23] J. McCleary, A User’s Guide to Spectral Sequences, second ed., Cambridge Stud. Adv. Math., vol. 58, 2000. [24] M. Putinar, Base change and the Fredholm index, Integral Equations Operator Theory 8 (5) (1985) 674–692. [25] M. Putinar, Spectral theory and sheaf theory, II, Math. Z. 192 (3) (1986) 473–490. [26] P. Roberts, Multiplicities and Chern Classes in Local Algebra, Cambridge Tracts in Math., vol. 133, Cambridge Univ. Press, Cambridge, 1998. [27] J.-P. Serre, Local Algebra, Springer Monogr. Math., Springer-Verlag, Berlin, 2000. [28] J.L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6 (1970) 172–191. [29] J.L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970) 1–38. [30] F.-H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, translated from the Romanian, Math. Appl. (East European Ser.), vol. 1, D. Reidel Publ. Co., Dordrecht, 1982 [Editura Academiei Republicii Socialiste Romania, Bucharest, 1982]. [31] C.A. Weibel, An Introduction to Homological Algebra, Cambridge Stud. Adv. Math., vol. 38, Cambridge Univ. Press, Cambridge, 1994.


Spectral properties of the canonical solution operator to ∂¯ Friedrich Haslinger 1 , Bernhard Lamel ∗,2 Universität Wien, Fakultät für Mathematik, Nordbergstrasse 15, A-1090 Wien, Österreich Received 25 July 2007; accepted 17 March 2008 Available online 28 April 2008 Communicated by Paul Malliavin

Abstract We study boundedness, compactness, and Schatten-class membership of the canonical solution operator ¯ restricted to (0, 1)-forms with holomorphic coefficients, on L2 (dμ) where μ is a measure with the to ∂, property that the monomials form an orthogonal family in L2 (dμ). The characterizations are formulated in terms of moment properties of μ. Our results generalize the results of the first author to several variables, contain some known results for several variables, and also cover new ground. © 2008 Elsevier Inc. All rights reserved. Keywords: dbar-Equation; Solution operator; Compactness

1. Introduction and statement of results In this paper, we study spectral properties of the canonical solution operator to ∂¯ acting on spaces of (0, 1)-forms with holomorphic coefficients in L2 (dμ) for measures μ with the property that the monomials zα , α ∈ Nn , are orthogonal in L2 (dμ). This situation covers a number of basic examples:

* Corresponding author.

E-mail addresses: [email protected] (F. Haslinger), [email protected] (B. Lamel). 1 Supported by the FWF, Projekt P19147. 2 Supported by the FWF, Projekt P17111.


14

F. Haslinger, B. Lamel / Journal of Functional Analysis 255 (2008) 13–24

• Lebesgue measure on bounded domains in Cn which are invariant under the torus action (θ1 , . . . , θn )(z1 , . . . , zn ) → eiθ1 z1 , . . . , eiθn zn (i.e. Reinhardt domains). • Weighted L2 spaces with radial-symmetric weights (e.g., generalized Fock spaces). • Weighted L2 spaces with decoupled radial weights, that is, dμ = e

j

ϕj (|zj |2 )

dV ,

where ϕj : R → R is a weight function. Sufficient conditions for the weight in order for the Fock space to be infinite-dimensional are known from the work of Shigekawa [12]. Some of these examples have been studied previously; our approach has the advantage of unifying these previous result as well as of being applicable in new situations as well. Our main focus in this paper is the case n > 1; indeed, we generalize results of the first author (see [4–6]) to this setting. The behaviour of the canonical solution operator S is interesting from many points of view. ¯ First, there is a close connection between properties of S and properties of the ∂-Neumann op∗ ¯ erator N ; indeed, S = ∂ N . In particular, noncompactness of S prohibits compactness of N . As is well known, S behaves quite nicely on spaces of (0, 1)-forms with holomorphic coefficients, and we shall exploit this connection. On the other hand, for convex domains, a result of Fu and Straube [2] shows that compactness of S on forms with holomorphic coefficients is also sufficient for compactness on all of L2 . There is also an intriguing connection between the canonical solution operator S and the theory of magnetic Schrödinger operators (see [3] and [7]); this connection has been exploited in the recent paper of the first author and Helffer [8] in order to study compactness of S on general (not rotation-invariant) weighted L2 -spaces on Cn . Let us introduce the notation used in this paper. We denote by A2 (dμ) = zα : α ∈ Nn , the closure of the monomials in L2 (dμ), and write mα = cα−1

=

α 2 z dμ.

We will give necessary and sufficient conditions in terms of these multimoments of the measure ¯ when restricted to (0, 1)-forms with coefficients in μ for the canonical solution operator to ∂, 2 A (dμ) to be bounded, compact, and to belong to the Schatten class Sp . This is accomplished by presenting a complete diagonalization of the solution operator by orthonormal bases with corresponding estimates. In the case of radial-symmetric measures our results specializes to the results of [10] applied to this specific case; we are also able to characterize membership in Sp for all positive p in some cases (a question left open in [10]).


15

As usual, for a given function space F , F(0,1) denotes the space of (0, 1)-forms with coefficients in F , that is, expressions of the form n

fj d z¯ j ,

fj ∈ F.

j =0

The ∂¯ operator is densely defined operator ¯ = ∂f

n ∂f d z¯ j . ∂ z¯ j j =1

The canonical solution operator S assigns to each ω ∈ L2(0,1) (dμ) the solution to the ∂¯ equation which is orthogonal to A2 (dμ); this solution need not exist, but if the ∂¯ equation for ω can be solved, then Sω is defined, and is given by the unique f ∈ L2 (dμ) which satisfies ¯ =ω ∂f

in the sense of distributions and f ⊥ A2 (dμ).

Our main interest in this paper is the spectral behaviour of the map S restricted to A2(0,1) (dμ). We first give a criterion for S to be a bounded operator. We will frequently encounter multiindices γ which might have one (but not more than one) entry equal to −1: in that case, we define cγ = 0. We will denote the set of these multi-indices by Γ . We let ej = (0, . . . , 1, . . . , 0) be the multi-index with a 1 in the j th spot and 0, elsewhere. Theorem 1. S : A2(0,1) (dμ) → L2 (dμ) is bounded if and only if there exists a constant C such that cγ +ep cγ +2ep

cγ 0. Then S : A2(0,1) (dμ) → A2 (dμ) is in the Schatten-p-class Sp if and only if cγ +ej γ ∈Γ

−

cγ +2ej

j

p/2

cγ

< ∞.

cγ +ej

(2)

The condition above is substantially easier to check if p = 2 (we will show that the sum is actually a telescoping sum then), i.e. for the case of the Hilbert–Schmidt class; we state this as a theorem. Theorem 4. The canonical solution operator S is in the Hilbert–Schmidt class if and only if

lim

k→∞

cγ < ∞. cγ +ep

γ ∈Nn , |γ |=k 1pn

(3)

1.1. Application in the case of decoupled weights Let us apply Theorem 1 to the case of decoupled weights, or more generally, of product measures dμ = dμ1 × · · · × dμn , where each dμj is a (circle-invariant) measure on C. Note that for such measures, there is definitely no compactness by Theorem 2. If we denote by

j

ck =

|z|2j dμk

−1 ,

C

we have that c(γ1 ,...,γn ) =

n

γ

ck k .

k=1

We thus obtain the following corollary. Corollary 5. For a product measure dμ = dμ1 × · · · × dμn as above, the canonical solution operator S : A2(0,1) (dμ) → L2 (dμ) is bounded if and only if there exists a constant C such that j +1

ck

j +2

ck

j

−

ck

j +1

0. Then the canonical solution operator to ∂¯ is in the Schatten class Sp , as an operator from A2(0,1) (dμ) to L2 (dμ) if and only if

∞

n + d − 2 (2n + d − 1)md+1 d=1

n−1

(n + d)md

md − md−1

p/2 < ∞.

(9)

18


In particular, Corollary 9 improves [10, Theorem C] in the sense that it also covers the case 0 < p < 2. We would like to note that our techniques can be adapted to the setting of [10] by considering the canonical solution operator on a Hilbert space H of holomorphic functions endowed with a norm which is comparable to the L2 -norm on each subspace generated by monomials of a fixed degree d, if in addition to the requirements in [10] we also assume that the monomials belong to H; this introduces the additional weights found by [10] in the formulas, as the reader can check. In our setting, the formulas are somewhat “cleaner” by working with A2 (dμ) (in particular, Corollary 8 only holds in this setting). 2. Monomial bases and diagonalization In what follows, we will denote by uα =

√ α cα z

the orthonormal basis of monomials for the space A2 (dμ), and by Uα,j = uα d z¯ j the corre¯ sponding basis of A2(0,1) (dμ). We first note that it is always possible to solve the ∂-equation for the elements of this basis; indeed, ∂¯ z¯ j uα = Uα,j . The canonical solution operator is also easily determined for forms with monomial coefficients: Lemma 10. The canonical solution Szα d z¯ j for monomial forms is given by Szα d z¯ j = z¯ j zα −

cα−ej cα

zα−ej ,

α ∈ Nn .

(10)

Proof. We have ¯zj zα , zβ = zα , zβ+ej ; so this expression is nonzero only if β = α − ej (in particular, if this implies (10) for multi-indices α with αj = 0; recall our convention that cγ = 0 if one of the entries of γ is negative). Thus Szα d z¯ j = z¯ j zα + czα−ej , and c is computed by −1 0 = z¯ j zα + czα−ej , zα−ej = cα−1 + ccα−e , j which gives c = −cα−ej /cα .

2

We are going to introduce an orthogonal decomposition A2(0,1) (dμ) =

Eγ

γ ∈Γ

of A2(0,1) (dμ) into at most n-dimensional subspaces Eγ indexed by multi-indices γ ∈ Γ (we will describe the index set below), and a corresponding sequence of mutually orthogonal finitedimensional subspaces Fγ ⊂ L2 (dμ) which diagonalizes S (by this we mean that SEγ = Fγ ). To motivate the definition of Eγ , note that

Sz d z¯ k , Sz d z¯ = α

β

0,

cα 1 cα cα+e

−

cα−ek cα+e −ek

,

β = α + e − ek , β = α + e − ek ,

(11)


19

so that Szα d z¯ k , Szβ d z¯ = 0 if and only if there exists a multi-index γ such that α = γ + ek and β = γ + e . We thus define Eγ = span{Uγ +ej ,j : 1 j n} = span zγ +ej d z¯ j : 1 j n , and likewise Fγ = SEγ . Recall that Γ is defined to be the set of all multi-indices whose entries are greater or equal to −1 and at most one negative entry. Note that Eγ is 1-dimensional if exactly one entry in γ equals −1, and n-dimensional otherwise. We have already observed that Fγ are mutually orthogonal subspaces of L2 (dμ). Whenever we use multi-indices γ and integers p ∈ {1, . . . , n} as indices, we use the convention that the p run over all p such that γ + ep 0; that is, for a fixed multi-index γ ∈ Γ , either the indices are either all p ∈ {1, . . . , n} or there is exactly one p such that γp = −1, in which case the index is exactly this one p. We next observe that we can find an orthonormal basis of Eγ and an orthonormal basis of Fγ such that in these bases Sγ = S|Eγ : Eγ → Fγ acts diagonally. First note that it is enough to do this if dim Eγ = n (since an operator between one-dimensional spaces is automatically diagonal). Fixing γ , the functions Uj := Uγ +ej ,j are an orthonormal basis of Eγ . The operator Sγ is clearly nonsingular on this space, so the functions SUj = Ψj constitute a basis of Fγ . For a basis B of j j vectors v j = (v1 , . . . , vn ), j = 1, . . . , n, of Cn we consider the new basis Vk =

n

j

v k Uj ;

j =1

since the basis given by the Uj is orthonormal, the basis given by the Vk is also orthonormal provided that the vectors vk = (vk1 , . . . , vkn ) constitute an orthonormal basis for Cn with the standard Hermitian product. Let us write Φk = SVk =

j

vk SUj .

j

The inner product Φp , Φq is then given by ⎛

Φ1 , Φ1 · · · ⎜ .. ⎝ .

Φn , Φ1 · · · ⎛ v1 · · · 1 ⎜ . = ⎝ .. vn1 · · ·

⎞

Φ1 , Φn ⎟ .. ⎠ .

Φn , Φn n v1 ⎞ ⎛ Ψ1 , Ψ1 .. .. ⎟ ⎜ . . ⎠⎝ n

Ψn , Ψ1 vn

j k j,k vp v¯ q SUj , SUk .

···

We therefore have

⎞ ⎛ v¯ 1

Ψ1 , Ψn 1 ⎟⎜ . .. ⎠ ⎝ .. .

· · · Ψn , Ψn

v¯1n

···

v¯n1 ⎞ .. ⎟ . . ⎠ v¯nn

(12)

Since the matrix ( Ψj , Ψk )j,k is Hermitian, we can unitarily diagonalize it; that is, we can choose an orthonormal basis B of Cn such that with this choice of B the vectors ϕγ ,k = Vk = j j vk Uγ +ej ,j of Eγ are orthonormal, and their images Φk = SVk are orthogonal in Fγ . Therefore, Φk / Φk is an orthonormal basis of Fγ such that Sγ : Eγ → Fγ is diagonal when expressed in terms of the bases {V1 , . . . , Vn } ⊂ Eγ and {Φ1 / Φ1 , . . . , Φn / Φn } ⊂ Fγ , with entries Φk .

20


Furthermore, the Φk are exactly the square roots of the eigenvalues of the matrix ( Ψp , Ψq ) which by (11) is given by

Ψp , Ψq = SUγ +ep ,p , SUγ +eq ,q = cγ +ep cγ +eq Szγ +ep d z¯ p , Szγ +eq d z¯ q

cγ +ep cγ 1 − = cγ +ep cγ +eq cγ +ep cγ +ep +eq cγ +eq =

cγ +ep cγ +eq − cγ cγ +ep +eq . √ cγ +ep +eq cγ +ep cγ +eq

(13)

Summarizing, we have the following proposition. Proposition 11. With μ as above, the canonical solution operator S : A2(0,1) (dμ) → L2(0,1) (dμ) admits a diagonalization by orthonormal bases. In fact, we have a decomposition A2(0,1) = γ Eγ into mutually orthogonal finite-dimensional subspaces Eγ , indexed by the multi-indices γ with at most one negative entry (equal to −1), which are of dimension 1 or n, and orthonormal bases ϕγ ,j of Eγ , such that Sϕγ ,j is a set of mutually orthogonal vectors in L2 (dμ). For fixed γ , the norms Sϕγ ,j are the square roots of the eigenvalues of the matrix Cγ = (Cγ ,p,q )p,q given by Cγ ,p,q =

cγ +ep cγ +eq − cγ cγ +ep +eq . √ cγ +ep +eq cγ +ep cγ +eq

(14)

In particular, we have that n

Sϕγ ,j = trace(Cγ ,p,q )p,q = 2

j =1

n

cγ +ep p=1

cγ +2ep

−

cγ cγ +ep

.

(15)

3. Boundedness: Proof of Theorem 1 In order to prove Theorem 1, we are using Proposition 11. We have seen that we have an orthonormal basis ϕγ ,j , γ ∈ Γ , j ∈ {1, . . . , n}, such that the images Sϕγ ,j are mutually orthogonal. Thus, S is bounded if and only if there exists a constant C such that

Sϕγ ,j 2 C for all γ ∈ Γ and j ∈ {1, . . . , dim Eγ }. If dim Eγ = 1, then γ has exactly one entry (say the j th √ one) equal to −1; in that case, let us write ϕγ = Uγ +ej d z¯ j . We have Sϕγ = cγ +ej z¯ j zγ +ej , and so

Sϕγ 2 =

cγ +ej cγ +2ej

.


21

On the other hand, if dim Eγ = n, we argue as follows: writing Sϕγ ,j 2 = λ2γ ,j with λγ ,j > 0, from (15) we find that n

λ2γ ,j

j =1

=

n

cγ +ej j =1

−

cγ +2ej

cγ cγ +ej

.

The last two equations complete the proof of Theorem 1. 4. Compactness In order to prove Theorem 2, we use the following elementary lemma (which is for example contained in [1]). Lemma 12. Let H1 and H2 be Hilbert spaces, and assume that S : H1 → H2 is a bounded linear operator. Then S is compact if and only if for every ε > 0 there exists a compact operator Tε : H1 → H2 such that the following inequality holds:

Sv 2H2 Tε v 2H2 + ε v 2H1 .

(16)

Proof of Theorem 2. We first show that (1) implies compactness. We will use the notation which was already used in the proof of Theorem 1; that is, we write Sϕγ ,j 2 = λ2γ ,j . Let ε > 0. There / Aε , exists a finite set Aε of multi-indices γ ∈ Γ such that for all γ ∈ n j =1

λ2γ ,j =

n

cγ +ej

cγ +2ej

j =1

−

cγ

cγ +ej

< ε.

Hence, if we consider the finite-dimensional (and thus, compact) operator Tε defined by aγ ,j ϕγ ,j = Tε aγ ,j Sϕγ ,j , γ ∈Aε

for any v =

aγ ,j ϕγ ,j ∈ A2(0,1) (dμ) we obtain 2

Sv 2 = Tε v 2 + S a ϕ γ ,j γ ,j γ ∈A / ε

= Tε v 2 +

|aγ ,j |2 Sϕγ ,j 2

γ ∈A / ε

= Tε v 2 +

|aγ ,j |2 λ2γ ,j

γ ∈A / ε

Tε v 2 + ε

|aγ ,j |2

γ ∈A / ε

Tε v + ε v 2 . 2

Hence, (16) holds and we have proved the first implication in Theorem 2.

22


We now turn to the other direction. Assume that (1) is not satisfied. Then there exists a K > 0 and an infinite family A of multi-indices γ such that for all γ ∈ A, n

λ2γ ,j =

j =1

n

cγ +ej j =1

cγ +2ej

−

cγ cγ +ej

> nK.

In particular, for each γ ∈ A, there exists a jγ such that λ2γ ,jγ > K. Thus, we have an infinite orthonormal family {ϕγ ,jγ : γ ∈ A} of vectors such that their images Sϕγ ,jγ are orthogonal and √ have norm bounded from below by K, which contradicts compactness. 2 5. Membership in the Schatten classes Sp and in the Hilbert–Schmidt class We keep the notation introduced in the previous sections. We will also need to introduce the usual grading on the index set Γ , that is, we write Γk = γ ∈ Γ : |γ | = k ,

k −1.

(17)

In order to study the membership in the Schatten class, we need the following elementary lemma. Lemma 13. Assume that p(x) and q(x) are continuous, real-valued functions on RN which are homogeneous of degree 1 (i.e. p(tx) = tp(x) and q(tx) = tq(x) for t ∈ R), and q(x) = 0 as well as p(x) = 0 implies x = 0. Then there exists a constant C such that 1 q(x) p(x) C q(x). C

(18)

Proof. Note that the set Bq = {x: q(x) = 1} is compact: it is closed since q is continuous, and since |q| is bounded from below on S N by some m > 0, it is necessarily contained in the closed ball of radius 1/m. Now, the function |p| is bounded on the compact set Bq ; say, by 1/C from below and C from above. Thus for all x ∈ RN ,

x 1 C, p C q(x) which proves (18).

2

Proof of Theorem 3. Note that S is in the Schatten class Sp if and only if

p

λγ ,j < ∞.

γ ∈Γ,j

We rewrite this sum as γ ∈Γ

j

p λγ ,j

=: M ∈ R ∪ {∞}.

(19)


23

Lemma 13 implies that there exists a constant C such that for every γ ∈ Γ , p/2

1 2 p/2 p λγ ,j λγ ,j C λ2γ ,j . C j

j

j

Hence, M < ∞ if and only if γ

p/2 < ∞,

λ2γ ,j

j

which after applying (15) becomes the condition (2) claimed in Theorem 3.

2

Proof of Theorem 4. S is in the Hilbert–Schmidt class if and only if

λ2γ ,j < ∞.

(20)

γ ∈Γ,j

We will prove that k

λ2γ ,j =

=−1 γ ∈Γ ,j

α∈Nn , |α|=k+1 1pn

cα , cα+ep

(21)

which immediately implies Theorem 4. The proof is by induction over k. For k = −1, the lefthand side of (21) is n j =1

λ2−ej ,j =

n

zj 2 c0 =

j =1

n c0 , cep j =1

which is equal to the right-hand side. Now assume that the (21) holds for k = K − 1; we will show that this implies it holds for k = K. We write K

λ2γ ,j =

=−1 γ ∈Γ ,j

cα

α∈Nn , |α|=K−1 1pn

=

α∈Nn , |α|=K 1pn

This finishes the proof of Theorem 4.

2

cα+ep cα

cα+ep

.

+

cγ +ej γ ∈ΓK ,j

cγ +2ej

−

cγ cγ +ej

24


References [1] J.P. D’Angelo, Inequalities from Complex Analysis, Carus Math. Monogr., vol. 28, Math. Assoc. America, Washington, DC, 2002. [2] S. Fu, E.J. Straube, Compactness of the ∂-Neumann problem on convex domains, J. Funct. Anal. 159 (2) (1998) 629–641. [3] S. Fu, E.J. Straube, Semi-classical analysis of Schrödinger operators and compactness in the ∂-Neumann problem, J. Math. Anal. Appl. 271 (1) (2002) 267–282. [4] F. Haslinger, The canonical solution operator to ∂ restricted to Bergman spaces, Proc. Amer. Math. Soc. 129 (11) (2001) 3321–3329 (electronic). [5] F. Haslinger, Compactness of the canonical solution operator to ∂ restricted to Bergman spaces, in: Functionalanalytic and Complex Methods, Their Interactions, and Applications to Partial Differential Equations, Graz, 2001, World Sci. Publ., River Edge, NJ, 2001, pp. 394–400. [6] F. Haslinger, The canonical solution operator to ∂ restricted to spaces of entire functions, Ann. Fac. Sci. Toulouse Math. (6) 11 (1) (2002) 57–70. [7] F. Haslinger, Magnetic Schrödinger operators and the ∂-equation, J. Math. Kyoto 46 (2) (2006) 249–257. [8] F. Haslinger, B. Helffer, Compactness of the solution operator to ∂ in weighted L2 -spaces, J. Funct. Anal. 243 (2007) 679–697. [9] W. Knirsch, G. Schneider, Continuity and Schatten–von Neumann p-class membership of Hankel operators with anti-holomorphic symbols on (generalized) Fock spaces, J. Math. Anal. Appl. 320 (1) (2006) 403–414. [10] S. Lovera, E.H. Youssfi, Spectral properties of the ∂-canonical solution operator, J. Funct. Anal. 208 (2) (2004) 360–376. [11] G. Schneider, Non-compactness of the solution operator to ∂ on the Fock-space in several dimensions, Math. Nachr. 278 (3) (2005) 312–317. [12] I. Shigekawa, Spectral properties of Schrödinger operators with magnetic fields for a spin 12 particle, J. Funct. Anal. 101 (2) (1991) 255–285.


Boundedness and compactness of Hankel operators on the sphere ✩ Jingbo Xia Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, USA Received 14 August 2007; accepted 23 March 2008 Available online 28 April 2008 Communicated by N. Kalton

Abstract We consider Hankel operators on the Hardy space of the unit sphere in Cn . We show that a large amount of information about the function f − Pf can be recovered from the Hankel operator Hf . For example, if Hf is compact, then the function f − Pf is necessarily in VMO. © 2008 Elsevier Inc. All rights reserved. Keywords: Hankel operator; Mean oscillation

1. Introduction Let S denote the unit sphere {z ∈ Cn : |z| = 1} in Cn . Let σ be the positive, regular Borel measure on S which is invariant under the orthogonal group O(2n), i.e., the group of isometries on Cn ∼ = R2n which fix 0. Furthermore we normalize σ such that σ (S) = 1. The Cauchy projection P is defined by the integral formula (Pf )(w) =

✩

f (ζ ) n dσ (ζ ), 1 − w, ζ

|w| < 1.

This work was supported by National Science Foundation grant DMS-0456448. E-mail address: [email protected].


26

J. Xia / Journal of Functional Analysis 255 (2008) 25–45

See [7, p. 39]. Recall that P is the orthogonal projection from L2 (S, dσ ) onto the Hardy space H 2 (S). For each z ∈ Cn with |z| < 1, we write kz (w) =

(1 − |z|2 )n/2 , (1 − w, z)n

|w| 1.

It is well known that the formula 1/2 d(ζ, ξ ) = 1 − ζ, ξ ,

ζ, ξ ∈ S,

(1.1)

defines metric a on S [7, p. 66]. Throughout the paper, we denote 1/2 B(ζ, r) = x ∈ S: 1 − x, ζ < r for ζ ∈ S and r > 0. There is a constant A0 ∈ (2−n , ∞) such that 2−n r 2n σ B(ζ, r) A0 r 2n

(1.2)

√ for all ζ ∈ S and 0 < r 2 [7, Proposition 5.1.4]. A function f ∈ L1 (S, dσ ) is said to have bounded mean oscillation if 1 f BMO = sup |f − fB(ζ,r) | dσ < ∞, ζ ∈S σ (B(ζ, r)) r>0

B(ζ,r)

where fB = B f dσ/σ (B), the average of f over B. A function f ∈ L1 (S, dσ ) is said to have vanishing mean oscillation if 1 lim sup |f − fB(ζ,r) | dσ = 0. δ↓0 ζ ∈S σ (B(ζ, r)) 0 0 and continuous if limτ (E)→0+ |||E||| = 0. The following theorem is proved in Section 11. Theorem G. Let ||| · ||| be a unitarily invariant norm on M and let T be the topology induced by ||| · ||| on M1,· = {T ∈ M: T 1}. If ||| · ||| is singular, then T is the operator norm topology on M1,· . If ||| · ||| is continuous, then T is the measure topology (in the sense of Nelson [14]) on M1,· . Let M be a type II1 factor and let ||| · ||| be a unitarily invariant norm on M. We denote by be the completion of M with respect M|||·||| the completion of M with respect to ||| · |||. Let M to the measure topology in the sense of Nelson [14]. In Section 12, we prove that there is an that extends the identity map from M onto M. An element injective map from M|||·||| into M in M can be identified with a closed, densely defined operator affiliated with M (see [14]). So generally speaking, an element in M|||·||| should be treated as an unbounded operator. We will consider the following two questions in Section 13: Question 1. Under what conditions is M|||·|||# the dual space of M|||·||| in the following sense: for every φ ∈ M|||·||| # , there is a unique X ∈ M|||·|||# such that φ(T ) = τ (T X),

∀T ∈ M|||·||| ,

and φ = |||T |||? Question 2. Under what conditions is M|||·||| a reflexive Banach space? 1 Let ||| · |||1 be the symmetric gauge norm on (L∞ [0, 1], 0 dx) corresponding to ||| · ||| on M as in Corollary 2. Then the same questions can be asked about L∞ [0, 1]|||·|||1 , the completion of L∞ [0, 1] with respect to ||| · |||1 . As further consequences of Theorem A, we prove the following theorems that answer the Questions 1 and 2, respectively. Theorem H. Let M be a type II1 factor, ||| · ||| be a unitarily invariant norm on M and ||| · |||# be the dual unitarily invariant norm on M. Let ||| · |||1 be the symmetric gauge norm on 1 (L∞ [0, 1], 0 dx) corresponding to ||| · ||| on M as in Corollary 2. Then the following conditions are equivalent: 1. 2. 3. 4.

M|||·|||# is the dual space of M|||·||| in the sense of Question 1; L∞ [0, 1]|||·|||# is the dual space of L∞ [0, 1]|||·|||1 in the sense of Question 1; 1 ||| · ||| is a continuous norm on M; ||| · |||1 is a continuous norm on L∞ [0, 1].

Theorem I. Let M be a type II1 factor, ||| · ||| be a unitarily invariant norm on M and let ||| · |||# be the dual unitarily invariant norm on M. Let ||| · |||1 be the symmetric gauge norm on 1 (L∞ [0, 1], 0 dx) corresponding to ||| · ||| on M as in Corollary 2. Then the following conditions are equivalent:

J. Fang et al. / Journal of Functional Analysis 255 (2008) 142–183

1. 2. 3. 4.

149

M|||·||| is a reflexive space; L∞ [0, 1]|||·|||1 is a reflexive space; both ||| · ||| and ||| · |||# are continuous norms on M; both ||| · |||1 and ||| · |||#1 are continuous norms on L∞ [0, 1].

A key step to proving Theorem H is based on the following fact: if ||| · ||| is a continuous unitarily invariant norm on M and φ ∈ M|||·||| # , then the restriction of φ to M is an ultraweakly continuous linear functional, i.e., φ is in the predual space of M. A significant advantage of our approach is that we develop a relatively complete theory of unitarily invariant norms on type II1 factors before handling unbounded operators. Indeed, unbounded operators are slightly involved only in the last two sections (Sections 12 and 13). Compared with the classical methods (e.g., [19]), which have to do a lot of subtle analysis on unbounded operators, our methods are much simpler. Let M be a finite factor. Recall that a norm ||| · ||| on M is called a normalized norm if |||1||| = 1. Let N(M) be the set of normalized unitarily invariant norms on M. Then N(M) is a convex compact set in the pointwise weak topology. Let Ne (M) be the set of extreme points of N(M). By the Krein–Milman theorem, N(M) is the closure of the convex hull of Ne (M) in the pointwise weak topology. So it is an interesting question of characterizing the set Ne (M). In Section 10, we prove the following theorems. Theorem J. Ne (M2 (C)) = {max{tT , T 1 }: 1/2 t 1}, where T 1 = τ2 (|T |). Theorem K. If M is a type II1 factor and t is a rational number such that 0 t 1, then the Ky Fan tth norm is an extreme point of N(M). This paper is almost self-contained and we do not assume any backgrounds on noncommutative Lp -theory. 2. Preliminaries 2.1. Nonincreasing rearrangements of functions Throughout this paper, we denote by m the Lebesgue measure on [0, 1]. In the following, a measurable function and a measurable set mean a Lebesgue measurable function and a Lebesgue measurable set, respectively. For two measurable sets A and B, A = B means m((A \ B) ∪ (B \ A)) = 0. Let f (x) be a real measurable function on [0, 1]. The non-increasing rearrangement function, f ∗ (x), of f (x) is defined by ∗

f (x) =

sup{y: m({f > y}) > x}, 0 x < 1; ess inf f, x = 1.

(2.1)

We summarize some useful properties of f ∗ (x) in the following proposition. Proposition 2.1. Let f (x), g(x) be real measurable functions on [0, 1]. Then we have the following:

150


1. f ∗ (x) is a non-increasing, right-continuous function on [0, 1] such that f ∗ (0) = ess sup f ; 1 1 2. if f (x) and g(x) are bounded functions and 0 f n (x) dx = 0 g n (x) dx for all n = 0, 1, 2, . . . , then f ∗ (x) = g ∗ (x); 1 1 3. f (x) and f ∗ (x) are equi-measurable and 0 f (x) dx = 0 f ∗ (x) dx when either integral is well defined. 2.2. Invertible measure-preserving transformations on [0, 1] Let G = {φ: φ(x) is an invertible measure-preserving transformation on [0, 1]}. It is well known that G acts on [0, 1] ergodically (see [6, pp. 3, 4], for instance), i.e., for a measurable subset A of [0, 1], φ(A) = A for all φ ∈ G implies that m(A) = 0 or m(A) = 1. Lemma 2.2. Let A, B be two measurable subsets of [0, 1] such that m(A) = m(B). Then there is a φ ∈ G such that φ(A) = B. Proof. We can assume that m(A) = m(B) > 0. Since G acts ergodically on [0, 1], there is a φ ∈ G such that m(φ(A)∩B) > 0. Let B1 = φ(A)∩B and A1 = φ −1 (B1 ). Then m(A1 ) = m(B1 ) and φ(A1 ) = B1 . By Zorn’s lemma and maximality arguments, we prove the lemma. 2 Corollary 2.3. Let A1 , . . . , An and B1 , . . . , Bn be disjoint measurable subsets of [0, 1] such that m(Ak ) = m(Bk ) for 1 k n. Then there is a φ ∈ G such that φ(Ak ) = Bk for 1 k n. Proof. We can assume that A1 ∪ · · · ∪ An = B1 ∪ · · · ∪ Bn = [0, 1]. By Lemma 2.2, there is a φk ∈ G such that φk (Ak ) = Bk , 1 k n. Define φ(x) = φk (x) for x ∈ Ak . Then φ ∈ G and φ(Ak ) = Bk for 1 k n. 2 1 For f (x) ∈ L∞ [0, 1], define τ (f ) = 0 f (x) dx. The following theorem is a version of the Dixmier’s averaging theorem (see [3] or [11]) and it has a similar proof. Theorem 2.4. Let f (x) ∈ L∞ [0, 1] be a real function. Then τ (f ) is in the L∞ -norm closure of the convex hull of {f · φ(x): φ ∈ G}. We end this subsection with the following proposition. Proposition 2.5. If φ(x) is an invertible measure-preserving transformation on [0, 1], then θ (f ) = f ◦ φ is a ∗-automorphism of L∞ [0, 1] preserving τ . Conversely, if θ is a ∗-automorphism of L∞ [0, 1] preserving τ , then there is an invertible measure-preserving transformation on [0, 1] such that θ (f ) = f ◦ φ for all f (x) ∈ L∞ [0, 1]. Proof. The first part of the proposition is easy to see. Suppose θ is a ∗-automorphism of L∞ [0, 1]. Let φ(x) = θ (f )(x), where f (x) ≡ x. Then it is easy to see the second part of the proposition. 2


151

2.3. s-Numbers of operators in type II1 factors In [5], Fack and Kosaki give a rather complete exposition of generalized s-numbers of τ -measurable operators affiliated with semi-finite von Neumann algebras. For the sake of reader’s convenience and our purpose, we provide sufficient details on s-numbers of bounded operators in finite von Neumann algebras in the following. We will define s-numbers of bounded operators in finite von Neumann algebras from the point of view of non-increasing rearrangement of functions. The following lemma is well known. The proof is an easy exercise. Lemma 2.6. Let (A, τ ) be a separable (i.e., with separable predual ) diffuse abelian von Neumann algebra with a faithful normal trace τ on A. Then there is a ∗-isomorphism α from (A, τ ) 1 1 onto (L∞ [0, 1], 0 dx) such that τ = 0 dx ◦ α. Let M be a type II1 factor and let τ be the unique trace on M. For T ∈ M, there is a separable diffuse abelian von Neumann subalgebra A of M containing |T |. By Lemma 2.6, there is a ∗-iso1 1 morphism α from (A, τ ) onto (L∞ ([0, 1], 0 dx) such that τ = 0 dx ◦ α. Let f (x) = α(|T |) and f ∗ (x) be the non-increasing rearrangement of f (x) (see (2.1)). Then the s-numbers of T , μs (T ), are defined as μs (T ) = f ∗ (s),

0 s 1.

Lemma 2.7. μs (T ) does not depend on A and α. Proof. Let A1 be another separable diffuse abelian von Neumann subalgebra of M containing 1 1 |T | and let β be a ∗-isomorphism from (A1 , τ ) onto (L∞ [0, 1], 0 dx) such that τ = 0 dx · β. 1 n 1 Let g(x) = β(|T |). For every number n = 0, 1, 2, . . . , 0 f (x) dx = τ (|T |n ) = 0 g n (x) dx. ∗ Since both f (x) and g(x) are bounded positive functions, by 2 of Proposition 2.1, f (x) = g ∗ (x) for all x ∈ [0, 1]. 2 Corollary 2.8. For T ∈ M and p 0, τ (|T |p ) =

1 0

μs (T )p ds.

The following lemma says that the above definition of s-numbers coincides with the definition of s-numbers given by Fack and Kosaki. Recall that P(M) is the set of projections in M. Lemma 2.9. For 0 s 1, μs (T ) = inf T E: E ∈ P(M), τ E ⊥ = s . Proof. By the polar decomposition and the definition of μs (T ), we may assume that T is positive. Let A be a separable diffuse abelian von Neumann subalgebra of M containing T and 1 1 let α be a ∗-isomorphism from (A, τ ) onto (L∞ [0, 1], 0 dx) such that τ = 0 dx · α. Let f (x) = α(T ) and let f ∗ (x) be the non-increasing rearrangement of f (x). Then μs (T ) = f ∗ (s). By the definition of f ∗ ,

1 m f ∗ > μs (T ) = lim m f ∗ > μs (T ) + s n→∞ n

152


and

∗ 1 ∗ m f μs (T ) lim m f > μs (T ) − s. n→∞ n Since f ∗ and f are equi-measurable, m({f > μs (T )}) s and m({f μs (T )}) s. Therefore, there is a measurable subset A of [0, 1], {f > μs (T )} ⊂ [0, 1] \ A ⊂ {f μs (T )}, such that m([0, 1] \ A) = s and f (x)χA (x)∞ = μs (T ) and f (x)χB (x)∞ μs (T ) for all B ⊂ [0, 1] \ A such that m(B) > 0. Let F = α −1 (χA ). Then τ (F ⊥ ) = s, T F = α −1 (f χA )∞ = μs (T ) and T F μs (T ) for all non-zero subprojections F of F ⊥ . This proves that μs (T ) inf{T E: E ∈ P(M), τ (E ⊥ ) = s}. Similarly, for every > 0, there is a projection F ∈ M such that τ (F⊥ ) = s + , T F = μs+ (T ) and T F μs+ (T ) for all non-zero subprojections F of F⊥ . Suppose E ∈ M is a projection such that τ (E ⊥ ) = s. Then τ (E ∧ F⊥ ) = τ (E) + τ (F⊥ ) − τ (E ∨ F⊥ ) = 1 + − τ (E ∨ F ⊥ ) > 0. Hence, T E T (E ∧ F⊥ ) μs+ (T ). This proves that inf{T E: E ∈ P(M), τ (E ⊥ ) = s} μs+ (T ). Since μs (T ) is rightcontinuous, μs (T ) inf{T E: E ∈ P(M), τ (E ⊥ ) = s}. 2 Corollary 2.10. Let S, T ∈ M. Then μs (ST ) Sμs (T ) for s ∈ [0, 1]. We refer to [4,5] for other interesting properties of s-numbers of operators in type II1 factors. 2.4. s-Numbers of operators in finite von Neumann algebras Throughout this paper, a finite von Neumann algebra (M, τ ) means a finite von Neumann algebra M with a faithful normal tracial state τ . An embedding of a finite von Neumann algebra (M, τ ) into another finite von Neumann algebra (M1 , τ1 ) means a ∗-isomorphism α from M to M1 such that τ = τ1 ◦ α. Let (L(F2 ), τ ) be the free group factor with the faithful normal trace τ . Then the reduced free product von Neumann algebra M1 = (M, τ ) ∗ (L(F2 ), τ ) is a type II1 factor with a (unique) faithful normal trace τ1 such that the restriction of τ1 to M is τ . So every finite von Neumann algebra can be embedded into a type II1 factor. Definition 2.11. Let (M, τ ) be a finite von Neumann algebra and T ∈ M. If α is an embedding of (M, τ ) into a type II1 factor (M1 , τ1 ), then the s-numbers of T are defined as μs (T ) = μs α(T ) . Similar to the proof of Lemma 2.7, we can see that μs (T ) is well defined, i.e., does not depend on the choice of α and M1 . Let T ∈ (Mn (C), τn ), where τn is the normalized trace on Mn (C). Then |T | is unitarily equivalent to a diagonal matrix with diagonal elements s1 (T ) · · · sn (T ) 0. In the classical matrices theory [1,7], s1 (T ), . . . , sn (T ) are also called s-numbers of T . It is easy to see that the relation between μs (T ) and s1 (T ), . . . , sn (T ) is the following μs (T ) = s1 (T )χ[0,1/n) (s) + s2 (T )χ[1/n,2/n) (s) + · · · + sn (T )χ[n−1/n,1] (s).

(2.2)

Since no confusions will arise, we will use both s-numbers for matrices in Mn (C). We refer to [1,7] for other interesting properties of s-numbers of matrices. We end this section by the following definition.


153

Definition 2.12. Positive operators S and T in a finite von Neumann algebra (M, τ ) are equimeasurable if μs (S) = μs (T ) for 0 s 1. By 2 of Proposition 2.1 and Corollary 2.8, positive operators S and T in a finite von Neumann algebra (M, τ ) are equi-measurable if and only if τ (S n ) = τ (T n ) for all n = 0, 1, 2, . . . . 3. Tracial gauge semi-norms on finite von Neumann algebras satisfying the weak Dixmier property 3.1. Gauge semi-norms Definition 3.1. Let (M, τ ) be a finite von Neumann algebra. A semi-norm ||| · ||| on M is called gauge invariant if for every T ∈ M, |||T ||| = |T |. Lemma 3.2. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a semi-norm on M. Then the following conditions are equivalent: 1. ||| · ||| is gauge invariant; 2. ||| · ||| is left unitarily invariant, i.e., for every unitary operator U ∈ M and operator T ∈ M, |||U T ||| = |||T |||; 3. for operators A, T ∈ M, |||AT ||| A · |||T |||. Proof. “3 ⇒ 2” and “2 ⇒ 1” are easy to see. We only prove “1 ⇒ 3.” We need to prove that if A < 1, then |||AT ||| |||T |||. Since A < 1, there are unitary operators U1 , . . . , Uk such k kT that A = U1 +···+U (see [10,16]). Since |U1 T | = · · · = |Uk T | = |T |, |||AT ||| = ||| U1 T +···+U ||| k k |||U1 T |||+···+|||Uk T ||| |||T |||. 2 k Corollary 3.3. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a gauge invariant semi-norm on M such that |||T V ||| = |||T ||| for every unitary operator V ∈ M and operator T ∈ M. If 0 S T , then |||S||| |||T |||. Proof. Since 0 S T , there is an operator A ∈ M such that S = AT A∗ and A 1. Similar to the proof of Lemma 3.2, |||S||| = |||AT A∗ ||| A · |||T ||| · A∗ |||T |||. 2 Definition 3.4. A normalized semi-norm on a finite von Neumann algebra (M, τ ) is a semi-norm ||| · ||| such that |||1||| = 1. By Lemma 3.2, we have the following corollary. Corollary 3.5. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a normalized gauge semi-norm on M. Then for every T ∈ M, |||T ||| T . A simple operator in a finite von Neumann algebra (M, τ ) is an operator T = a1 E1 + · · · + an En , where E1 , . . . , En are projections in M such that E1 + · · · + En = 1.

154


Corollary 3.6. Let (M, τ ) be a finite von Neumann algebra, ||| · |||1 and ||| · |||2 be two gauge invariant semi-norms on M. Then ||| · |||1 = ||| · |||2 on M if |||T |||1 = |||T |||2 for all positive simple operators T ∈ M. Proof. Without loss of generality, assume |||1|||1 = |||1|||2 = 1. Let T ∈ M be a positive operator. By the spectral decomposition theorem, there is a sequence of positive simple operators Tn ∈ M such that limn→∞ T − Tn = 0. By Corollary 3.5, limn→∞ |||T − Tn |||1 = limn→∞ |||T − Tn |||2 = 0. By the assumption of the corollary, |||Tn |||1 = |||Tn |||2 . Hence, |||T |||1 = |||T |||2 . Since both ||| · |||1 and ||| · |||2 are gauge invariant, ||| · |||1 = ||| · |||2 . 2 3.2. Tracial gauge semi-norms Definition 3.7. Let (M, τ ) be a finite von Neumann algebra. A semi-norm ||| · ||| on M is called tracial if |||S||| = |||T ||| for every two equi-measurable positive operators S, T in M. A semi-norm ||| · ||| on M is called a tracial gauge semi-norm if it is both tracial and gauge invariant. Since for a positive operator T in a finite von Neumann algebra (M, τ ), T = 1 limn→∞ (τ (T n )) n , the operator norm · is a tracial gauge norm on (M, τ ). Another less obvious example of a tracial gauge norm on (M, τ ) is the non-commutative L1 -norm: T 1 = 1 τ (|T |) = 0 μs (T ) ds. The less obvious part is to show that · 1 satisfies the triangle inequality. The following lemma overcomes this difficulty. Lemma 3.8. A1 = sup{|τ (U A)|: U ∈ U(M)}, where U(M) is the set of unitary operators in M. Proof. By the polar decomposition theorem, there is a unitary operator V ∈ M such that A = V |A|. By the Schwarz inequality, |τ (U A)| = |τ (U V |A|)| = |τ (U V |A|1/2 |A|1/2 )| τ (|A|)1/2 · τ (|A|)1/2 = τ (|A|). Hence A1 sup{|τ (U A)|: U ∈ U(M)}. Let U = V ∗ , we obtain A1 sup{|τ (U A)|: U ∈ U(M)}. 2 Corollary 3.9. A + B1 A1 + B1 . Lemma 3.10. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a gauge invariant semi-norm on M. Then ||| · ||| is tracial if |||S||| = |||T ||| for every two equi-measurable positive simple operators S, T in M. Proof. We can assume that |||1||| = 1. Let A, B be two equi-measurable positive operators in M. By the spectral decomposition theorem, there are two sequences of positive simple operators An , Bn in M such that An and Bn are equi-measurable and limn→∞ A − An = limn→∞ B − Bn = 0. By Corollary 3.5, limn→∞ |||A − An ||| = limn→∞ |||B − Bn ||| = 0. By the assumption of the lemma, |||An ||| = |||Bn |||. Hence, |||A||| = |||B|||. 2 3.3. Symmetric gauge semi-norms Definition 3.11. Let (M, τ ) be a finite von Neumann algebra and let Aut(M, τ ) be the set of ∗-automorphisms on M preserving τ . A semi-norm ||| · ||| on M is called symmetric if θ (T ) = |||T |||, ∀T ∈ M, θ ∈ Aut(M, τ ).


155

A semi-norm ||| · ||| on M is called a symmetric gauge semi-norm if it is both symmetric and gauge invariant. n Example 3.12. Let M = Cn and τ (T ) = x1 +···+x , where T = (x1 , . . . , xn ) ∈ Cn . Then n Aut(M, τ ) is the set of permutations on {1, . . . , n}. So a semi-norm ||| · ||| on M is a symmetric gauge semi-norm if and only if for every (x1 , . . . , xn ) ∈ Cn and a permutation π on {1, . . . , n},

(x1 , . . . , xn ) = |x1 |, . . . , |xn | , and (x1 , . . . , xn ) = (xπ(1) , . . . , xπ(n) ). Lemma 3.13. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a semi-norm on M. If ||| · ||| is tracial gauge invariant, then ||| · ||| is symmetric gauge invariant. Proof. Let θ ∈ Aut(M, τ ) and T ∈ M. We need to prove that |||θ (T )||| = |||T |||. Since |θ (T )| = θ (|T |) and ||| · ||| is gauge invariant, we can assume that T is positive. Since θ ∈ Aut(M, τ ), T and θ (T ) are equi-measurable. Hence, |||T ||| = |||θ (T )|||. 2 Example 3.14. Let M = C ⊕ M2 (C) and τ (a ⊕ B) = a2 + τ2 (B) 2 , where τ2 is the normalized trace on M2 (C). Define |||a ⊕ B||| = |a|/2 + τ2 (|B|). Then ||| · ||| is a symmetric gauge norm but not a tracial gauge norm. Note that 1 ⊕ 0 and 0 ⊕ 1 are equi-measurable, but 1/2 = |||1 ⊕ 0||| = |||0 ⊕ 1||| = 1. Aut(M, τ ) acts on M ergodically if θ (T ) = T for all θ ∈ Aut(M, τ ) implies T = λ1. Lemma 3.15. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a semi-norm on M. If Aut(M, τ ) acts on M ergodically, then the following are equivalent: 1. ||| · ||| is a tracial gauge semi-norm; 2. ||| · ||| is a symmetric gauge semi-norm. Proof. “1 ⇒ 2” by Lemma 3.13. We need to prove “2 ⇒ 1.” By Corollary 3.6, we need to prove |||S||| = |||T ||| for two equi-measurable simple operators S, T in M. Similar to the proof of Corollary 2.3, there is a θ ∈ Aut(M, τ ) such that S = θ (T ). Hence |||S||| = |||T |||. 2 1 Corollary 3.16. A semi-norm on (L∞ [0, 1], 0 dx) or (Cn , τ ) is a tracial gauge norm if and n only if it is a symmetric gauge norm, where τ ((x1 , . . . , xn )) = x1 +···+x . n 3.4. Unitarily invariant semi-norms Definition 3.17. Let (M, τ ) be a von Neumann algebra. A semi-norm ||| · ||| on M is unitarily invariant if |||U T V ||| = |||T ||| for all T ∈ M and unitary operators U, V ∈ M.

156


Proposition 3.18. Let ||| · ||| be a semi-norm on M. Then the following statements are equivalent: 1. ||| · ||| is unitarily invariant; 2. ||| · ||| is gauge invariant and unitarily conjugate invariant, i.e., |||U T U ∗ ||| = |||T ||| for all T ∈ M and unitary operators U ∈ M; 3. ||| · ||| is left-unitarily invariant and |||T ||| = |||T ∗ ||| for every T ∈ M; 4. for all operators T , A, B ∈ M, |||AT B||| A · |||T ||| · B. Proof. “1 ⇒ 4” is similar to the proof of Lemma 3.2. “4 ⇒ 3,” “3 ⇒ 2,” and “2 ⇒ 1” are routine. 2 For a unitary operator U ∈ M, let θ (T ) = U T U ∗ . Then θ ∈ Aut(M, τ ). By Proposition 3.18, we have the following. Corollary 3.19. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a symmetric, gauge invariant semi-norm on M. Then ||| · ||| is a unitarily invariant semi-norm on M. n . Define |||(x1 , . . . , xn )||| = Example 3.20. Let M = Cn , n 2 and τ ((x1 , . . . , xn )) = x1 +···+x n |x1 |. Then ||| · ||| is a unitarily invariant semi-norm but not a symmetric gauge semi-norm on M.

Lemma 3.21. Let (M, τ ) be a finite factor and ||| · ||| be a semi-norm on M. Then the following conditions are equivalent: 1. ||| · ||| is a tracial gauge semi-norm; 2. ||| · ||| is a symmetric gauge semi-norm; 3. ||| · ||| is a unitarily invariant semi-norm. Proof. “1 ⇒ 2” by Lemma 3.13 and “2 ⇒ 3” by Corollary 3.19. We need to prove “3 ⇒ 1.” By Corollary 3.6, we need to prove |||S||| = |||T ||| for two equi-measurable positive simple operators S, T ∈ M. Suppose S = a1 E1 + · · · + an En and T = a1 F1 + · · · + an Fn , where E1 + · · · + En = 1 and F1 + · · · + Fn = 1 and τ (Ek ) = τ (Fk ) for 1 k n. Since M is a factor, there is a unitary operator U ∈ M such that Ek = U Fk U ∗ for 1 k n. Therefore, S = U T U ∗ and |||S||| = |||T |||. 2 3.5. Weak Dixmier property Definition 3.22. A finite von Neumann algebra (M, τ ) satisfies the weak Dixmier property if for every positive operator T ∈ M, τ (T ) is in the operator norm closure of the convex hull of {S ∈ M: S and T are equi-measurable}. A finite factor (M, τ ) satisfies the Dixmier property (see [2,11]): for every operator T ∈ M, τ (T ) is in the operator norm closure of the convex hull of {U T U ∗ : U ∈ U(M)}. Hence finite factors satisfy the weak Dixmier property. In the following, we will characterize finite von Neumann algebras satisfying the weak Dixmier property. There is a central projection P in a finite von Neumann algebra (M, τ ) such that P M is of type I and (1 − P )M is of type II. A type II von Neumann algebra is diffuse, i.e, there are no non-zero minimal projections in the von Neumann algebra. Furthermore, there are central


157

projections P1 , . . . , Pn , . . . in M, such that P1 + · · · + Pn + · · · = P and Pn M = An ⊗ Mn (C), An is abelian. We can decompose An into an atomic part Aan and a diffuse part Acn , i.e., there is a projection Qn in An , Aan = Qn An , such that Qn = En1 + En2 + · · · , where Enk is a minimal projection in Aan and τ (Enk ) > 0, and Acn = (1 − Qn )An is diffuse. Let Ma = ⊕ Aan ⊗ Mn (C) and Mc = ⊕ Acn ⊗ Mn (C) ⊕ (1 − P )M. Then M = Ma ⊕ Mc . We call Ma the atomic part of M and Mc the diffuse part of M. A finite von Neumann algebra (M, τ ) is atomic if M = Ma and is diffuse if M = Mc . Lemma 3.23. Let (M, τ ) be a finite-dimensional von Neumann algebra such that for every two non-zero minimal projections E, F ∈ M, τ (E) = τ (F ). Then (M, τ ) satisfies the weak Dixmier property. Proof. Since M is finite-dimensional, M ∼ = Mk1 (C) ⊕ · · · ⊕ Mkr (C). Since τ (E) = τ (F ) for every two non-zero minimal projections E, F ∈ M, (M, τ ) can be embedded into (Mn (C), τn ), where n = k1 + · · · + kr . So we can assume that (M, τ ) is a von Neumann subalgebra of (Mn (C), τn ) such that M contains all diagonal matrices a1 E1 + · · · + an En . Now for every positive operator T ∈ M, there is a unitary operator U ∈ M such that U T U ∗ = a1 E1 + · · · + an En , (a E +···+aπ(n) En ) n . Then τ (T ) = π π(1) 1 n! . 2 a1 , . . . , an 0 and τ (T ) = a1 +···+a n Lemma 3.24. Let (M, τ ) be a diffuse finite von Neumann algebra. Then (M, τ ) satisfies the weak Dixmier property. Proof. Let A be a separable diffuse abelian von Neumann subalgebra of M. By Lemma 2.6, 1 1 there is a ∗-isomorphism α from (A, τ ) onto (L∞ [0, 1], 0 dx) such that 0 dx · α = τ . For a positive operator T ∈ M, there is an operator S ∈ A such that α(S) = μs (T ). Hence 1 τ (T ) = τ (S) = 0 μs (T ) ds. By Theorem 2.4, for any > 0, there are S1 , . . . , Sn in A such that n < . Hence (M, τ ) satisfies the weak S, S1 , . . . , Sn are equi-measurable and τ (S) − S1 +···+S n Dixmier property. 2 Lemma 3.25. Let (M, τ ) be an atomic finite von Neumann algebra with two minimal projections E and F in M such that τ (E) = τ (F ). Then (M, τ ) does not satisfy the weak Dixmier property. Proof. Since (M, τ ) is an atomic finite von Neumann algebra, M ∼ = Mk1 (C) ⊕ Mk2 (C) ⊕ · · · . Let Eij be minimal projections in Mki such that Eij = 1. Without loss of generality, assume that τ (E11 ) > τ (E21 ) τ (E31 ) · · · . Let ⎛ ⎜ T =⎝

⎞

1 ..

⎟ ⎠ ⊕ A,

. 1

k1

where ⎛ ⎜ A=⎝

1 2

⎞ ..

⎛

⎟ ⎜ ⎠⊕⎝

. ( 12 )k2

( 12 )k2 +1

⎞ ..

⎟ ⎠ ⊕ ···.

. ( 12 )k2 +k3

158


If T1 ∈ M and T are equi-measurable, then ⎛ 1 ⎜ .. T1 = ⎝ .

⎞ ⎟ ⎠ ⊕ A1 , 1

k1

where A and A1 are equi-measurable. Hence, if τ (T ) is in the operator norm closure of the convex hull of {S ∈ M: S and T are equi-measurable}, then τ (T ) = 1. This is a contradiction. 2 Let (M, τ ) be a finite von Neumann algebra and let E ∈ M be a non-zero projection. The induced finite von Neumann algebra (ME , τE ) is the von Neumann algebra ME = EME with E) a faithful normal trace τE (ET E) = τ (ET τ (E) . The proof of the following lemma is similar to the proof of Lemma 3.25. Lemma 3.26. Let (M, τ ) be a finite von Neumann algebra such that Ma = 0 and Mc = 0. Then M does not satisfy the weak Dixmier property. Proof. Let P be the central projection such that Ma = P M and Mc = (1 − P )M. Let A be a separable diffuse abelian von Neumann subalgebra of (Mc , τ1−P ). By Lemma 2.6, there is a positive operator A in Mc such that μs (A) = 1−s 2 with respect to (Mc , τ1−P ). Consider T = P + A(1 − P ). Then 1, 0 s < τ (P ); μs (T ) = 1−s 1 2τ (1−P ) 2 , τ (P ) s 1 with respect to (M, τ ). If T1 ∈ M and T are equi-measurable, then T1 = P + A1 such that A1 and A are equi-measurable. Hence, if τ (T ) is in the operator norm closure of the convex hull of {S ∈ M: S and T are equi-measurable}, then τ (T ) = 1. This is a contradiction. 2 Summarizing Lemmas 3.23–3.26, we can characterize finite von Neumann algebras satisfying the weak Dixmier property as the following theorem. Theorem 3.27. Let (M, τ ) be a finite von Neumann algebra. Then M satisfies the weak Dixmier property if and only if M satisfies one of the following conditions: 1. M is finite-dimensional (hence atomic) and for every two non-zero minimal projections E, F ∈ M, τ (E) = τ (F ), or equivalently, (M, τ ) can be identified as a von Neumann subalgebra of (Mn (C), τn ) that contains all diagonal matrices; 2. M is diffuse. Corollary 3.28. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and E ∈ M be a non-zero projection. Then (ME , τE ) also satisfies the weak Dixmier property. The following example shows that we cannot replace the weak Dixmier property by the following condition: τ (T ) is in the operator norm closure of the convex hull of {θ (T ): θ ∈ Aut(M, τ )}.


159

Example 3.29. (C ⊕ M2 (C), τ ), τ (a ⊕ B) = 13 a + 23 τ2 (B), satisfies the weak Dixmier property. On the other hand, let T = 1 ⊕ 2 ∈ C ⊕ M2 (C). Then for every θ ∈ Aut(M, τ ), θ (T ) = T . Hence, τ (T ) is not in the operator norm closure of the convex hull of {θ (T ): θ ∈ Aut(M, τ )}. 3.6. A comparison theorem The following theorem is the main result of this section. Theorem 3.30. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property. If ||| · ||| is a normalized tracial gauge semi-norm on M, then for all T ∈ M, T 1 |||T ||| T . In particular, every tracial gauge semi-norm on M is a norm. Proof. By Corollary 3.5, |||T ||| T for every T ∈ M. To prove T 1 |||T |||, we can assume T 0. Let > 0. Since (M, τ ) satisfies the weak Dixmier property, there are S1 , . . . , Sk in k M such that T , S1 , . . . , Sk are equi-measurable and τ (T ) − S1 +···+S < . By Corollary 3.5, k S1 +···+Sk S1 +···+Sk k |||τ (T ) − ||| τ (T ) − < . Hence T = |τ (T )| ||| S1 +···+S ||| + 1 k k k |||S1 |||+···+|||Sk ||| + = |||T ||| + . 2 k By Theorem 3.30 and Lemma 3.21, we have the following corollary. Corollary 3.31. Let (M, τ ) be a finite factor and let ||| · ||| be a normalized unitarily invariant norm on M. Then T 1 |||T ||| T ,

∀T ∈ M.

In particular, every unitarily invariant semi-norm on a finite factor is a norm. By Theorem 3.30 and Lemma 3.15, we have the following corollary. Corollary 3.32. Let ||| · ||| be a normalized symmetric gauge semi-norm on (L∞ [0, 1], n (or (Cn , τ ), where τ ((x1 , . . . , xn )) = x1 +···+x ). Then n T 1 |||T ||| T ,

1 0

dx)

∀T ∈ L∞ [0, 1] or Cn .

In particular, every symmetric gauge semi-norm on (L∞ [0, 1], is a norm.

1 0

dx) (or (Cn , τ ), respectively)

4. Proof of Theorem B To prove Theorem B, we need the following lemmas. Lemma 4.1. Let E1 , . . . , En be projections in M such that E1 + · · · + En = 1 and T ∈ M. Then S = E1 T E1 + · · · + En T En is in the convex hull of {U T U ∗ : U ∈ U(M)}.

160


Proof. Let T = (Tij ) be the matrix with respect to the decomposition 1 = E1 + · · · + En . Let U = −E1 + E2 + · · · + En . Then simple computation shows that ⎛

T11 ⎜ 0 1 ⎜ UT U∗ + T = ⎜ . ⎝ .. 2 0

0 T22 .. .

··· ··· .. .

⎞ 0 T2n ⎟ ⎟ .. ⎟ = E1 T E1 + (1 − E1 )T (1 − E1 ). . ⎠

Tn2

···

Tnn

By induction, S = E1 T E1 + · · · + En T En is in the convex hull of {U T U ∗ : U ∈ U(M)}.

2

Corollary 4.2. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a unitarily invariant norm on M. Let E1 , . . . , En be projections in M such that E1 + · · · + En = 1 and T ∈ M and S = E1 T E1 + · · · + En T En . Then |||S||| |||T |||. Recall that for a (non-zero) finite projection E in M, τE (ET E) = on ME = EME.

τ (ET E) τ (E)

is the induced trace

Lemma 4.3. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and ||| · ||| be a tracial gauge norm on M. Suppose T , E1 , . . . , En ∈ M, T 0, E1 + · · · + En = 1. Then |||T ||| |||τE1 (E1 T E1 )E1 + · · · + τEn (En T En )En |||. Proof. We may assume that |||1||| = 1. Since M satisfies the weak Dixmier property, by Corollary 3.28, (MEi , τEi ) also satisfies the weak Dixmier property, 1 i n. Let > 0. There are operators Si1 , . . . , Sik in MEi such that Ei T Ei , Si1 , . . . , Sik are equi-measurable and Si1 + · · · + Sik − τ (E T E )E Ei i i i < . k Let Sj = S1j E1 + · · · + Snj En , 1 j k. Then T , S1 , . . . , Sn are equi-measurable and S1 + · · · + Sk − τE1 (E1 T E1 )E1 + · · · + τEn (En T En )En < . k By Corollary 3.5, |||

S1 + · · · + Sk − τE1 (E1 T E1 )E1 + · · · + τEn (En T En )En ||| < . k

Hence, |||τE1 (E1 T E1 )E1 + · · ·+ τEn (En T En )En ||| |||T ||| + . Since > 0 is arbitrary, we obtain the lemma. 2 Corollary 4.4. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. If A is a finite-dimensional abelian von Neumann subalgebra of M and EA is the normal conditional expectation from M onto A preserving τ , then for every T ∈ M, |||EA (T )||| |||T |||.


161

Proof. Let A = {E1 , . . . , En } such that E1 + · · · + En = 1. Then for every T ∈ M, EA (T ) = τE1 (E1 T E1 )E1 + · · · + τEn (En T En )En . By Corollary 4.2 and Lemma 4.3, |||EA (T )||| |||T |||.

2

Proof of Theorem B. By Lemma 3.13 and Corollary 3.19, ||| · ||| is unitarily invariant. Suppose Tα is a net in M1,|||·||| such that limα Tα = T in the weak operator topology. Let T = U |T | be the polar decomposition of T . Then limα U ∗ Tα = |T | in the weak operator topology. Since ||| · ||| is unitarily invariant, |||U Tα ||| 1 and ||||T |||| = |||T |||. So we may assume that T 0 and Tα = Tα∗ . By the spectral decomposition theorem and Corollary 3.5, to prove |||T ||| 1, we need to prove |||S||| 1 for every positive simple operator S such that S T . Let S = a1 E1 + · · · + an En and > 0. Since limα Tα = T S, limα Ei Tα Ei = Ei T Ei ai Ei for 1 i n. Hence, limα τEi (Ei (Tα + )Ei ) ai + > ai . So there is a β such that τE1 (E1 (Tβ + )E1 )E1 + · · · + τEn (En (Tβ + )En )En S. By Lemma 4.3 and Corollary 3.3, 1 + |||Tβ + ||| |||τ (E1 (Tβ + )E1 )E1 + · · · + τ (En (Tβ + )En )En ||| |||S|||. Since > 0 is arbitrary, |||S||| 1. 2 Proof of Corollary 1. Since A is a separable abelian von Neumann algebra, there is a sequence of finite-dimensionalabelian von Neumann subalgebras An such that A1 ⊂ A2 ⊂ · · · ⊂ A and A is the closure of n An in the strong operator topology. Let EAn be the normal conditional expectation from M onto An preserving τ . Then for every T ∈ M, EA (T ) = limn→∞ EAn (T ) in the strong operator topology. By Theorem B and Corollary 4.4, |||EA (T )||| |||T |||. 2 In the following we give some other useful corollaries of Theorem B. Corollary 4.5. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. Suppose 0 T1 T2 · · · T in M such that limn→∞ Tn = T in the weak operator topology. Then limn→∞ |||Tn ||| = |||T |||. Proof. By Corollary 3.3, |||T1 ||| |||T2 ||| · · · |||T |||. Hence, limn→∞ |||Tn ||| |||T |||. By Theorem B, limn→∞ |||Tn ||| |||T |||. 2 Corollary 4.6. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and ||| · |||1 and ||| · |||2 be two tracial gauge norms on M. Then ||| · |||1 = ||| · |||2 on M if |||T |||1 = |||T |||2 for every operator T = a1 E1 + · · · + an En in M such that a1 , . . . , an 0 and τ (E1 ) = · · · = τ (En ) = n1 , n = 1, 2, . . . . Proof. We need only to prove |||T |||1 = |||T |||2 for every positive operator T in M. By Theorem 3.27, M is either a finite-dimensional von Neumann algebra such that τ (E) = τ (F ) for arbitrary two non-zero minimal projections in M or M is diffuse. If M is a finite-dimensional von Neumann algebra such that τ (E) = τ (F ) for arbitrary two non-zero minimal projections in M, then the corollary is obvious. If M is diffuse, by the spectral decomposition theorem, there is a sequence of operators Tn ∈ M satisfying the following conditions: 1. 0 T1 T2 · · · T ,

162


2. Tn = an1 En1 + · · · + ann Enn , an1 , . . . , ann 0 and τ (En1 ) = · · · = τ (Enn ) = n1 , 3. limn→∞ Tn = T in the weak operator topology. By the assumption of the corollary, |||Tn |||1 = |||Tn |||2 . By Corollary 4.5, |||T |||1 = |||T |||2 .

2

Corollary 4.7. Let M be a type II1 factor and ||| · |||1 and ||| · |||2 be two unitarily invariant norms on M. Then ||| · |||1 = ||| · |||2 on M if ||| · |||1 = ||| · |||2 on all type In subfactors of M, n = 1, 2, . . . . 5. Ky Fan norms on finite von Neumann algebras Let (M, τ ) be a finite von Neumann algebra and 0 t 1. For T ∈ M, define the Ky Fan tth norm by

|||T |||(t)

T , 1 t t 0 μs (T ) ds,

t = 0; 0 < t 1.

Let M1 = (M, τ ) ∗ (LF2 , τ ) be the reduced free product von Neumann algebra of M and the free group factor LF2 . Then M1 is a type II1 factor with a faithful normal trace τ1 such that the restriction of τ1 to M is τ . Recall that U(M1 ) is the set of unitary operators in M1 and P(M1 ) is the set of projections in M1 . Lemma 5.1. For 0 < t 1, t|||T |||(t) = sup{|τ1 (U T E)|: U ∈ U(M1 ), E ∈ P(M1 ), τ1 (E) = t}. Proof. We may assume that T is a positive operator. Let A be a separable diffuse abelian von Neumann subalgebra of M1 containing T and let α be a ∗-isomorphism from (A, τ1 ) onto 1 1 (L∞ [0, 1], 0 dx) such that τ1 = 0 dx · α. Let f (x) = α(T ) and let f ∗ (x) be the non-increasing rearrangement of f (x). Then μs (T ) = f ∗ (s). By the definition of f ∗ (see (2.1)),

1 t m f ∗ > f ∗ (t) = lim m f ∗ > f ∗ (t) + n→∞ n and

∗ 1 ∗ ∗ ∗ m f f (t) lim m f > f (t) − t. n→∞ n Since f ∗ and f are equi-measurable, m({f > f ∗ (t)}) t and m({f f ∗ (t)}) t. Therefore, there is a measurable subset A of [0, 1], {f > f ∗ (t)} ⊂ A ⊂ {f f ∗ (t)}, such that m(A) = t. t Since f (x) and f ∗ (x) are equimeasurable, A f (s) ds = 0 f ∗ (s) ds. Let E = α −1 (χA ). t ∗ Then τ1 (E ) = t and τ1 (T E ) = A f (s) ds = 0 f (s) ds = t|||T |||(t) . Hence, t|||T |||(t) sup{|τ1 (U T E)|: U ∈ U(M), E ∈ P(M1 ), τ1 (E) = t}. We need to prove that if E is a projection in M1 , τ1 (E) = t, and U ∈ U(M1 ), then t|||T |||(t) |τ1 (U T E)|. By the Schwarz inequality, |τ1 (U T E)| = τ1 (EU T 1/2 T 1/2 E) τ1 (U ∗ EU T )1/2 × 1 τ1 (ET )1/2 . By Corollary 2.8, τ1 (ET ) = 0 μs (ET ) ds. By Corollary 2.10, μs (ET ) t min{μs (T ), μs (E)T }. Note that μs (E) = 0 for s τ1 (E) = t. Hence, τ1 (ET ) 0 μs (T ) ds = t|||T |||t . Similarly, τ1 (U ∗ EU T ) t|||T |||t . So |τ1 (U T E)| t|||T |||t . This proves that t|||T |||(t) sup{|τ1 (U T E)|: U ∈ U(M1 ), E ∈ P(M1 ), τ1 (E) = t}. 2


163

Theorem 5.2. For 0 t 1, ||| · |||(t) is a normalized tracial gauge norm on (M, τ ). Proof. We only prove the triangle inequality, since the other parts are obvious. We may assume that 0 < t 1. Let S, T ∈ M. By Lemma 5.1, t|||S + T |||(t) = sup{|τ1 (U (S + T )E)|: U ∈ U(M1 ), E ∈ P(M1 ), τ1 (E) = t} sup{|τ1 (U SE)|: U ∈ U(M1 ), E ∈ P(M1 ), τ1 (E) = t} + sup{|τ1 (U T E)|: U ∈ U(M1 ), E ∈ P(M1 ), τ1 (E) = t} = t|||S|||(t) + t|||T |||(t) . 2 Proposition 5.3. Let (M, τ ) be a finite von Neumann algebra and T ∈ (M, τ ). Then |||T |||(t) is a non-increasing continuous function on [0, 1]. Proof. Let 0 < t1 < t2 1. 1 |||T |||(t1 ) − |||T |||(t2 ) = t1 =

1 t1

t1 0

1 μs (T ) ds − t2

t1 0

μs (T ) ds −

t2 μs (T ) ds 0

t2

1 t2 −t1

t1

μs (T ) ds

t2 (t2 − t1 )

0.

Since μs (T ) is right-continuous, |||T |||(t) is a non-increasing continuous function on [0, 1].

2

Example 5.4. The Ky Fan nk th norm of a matrix T ∈ (Mn (C), τn ) is |||T ||| k = n

s1 (T ) + · · · + sk (T ) , k

1 k n.

6. Dual norms of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property 6.1. Dual norms Let ||| · ||| be a norm on a finite von Neumann algebra (M, τ ). For T ∈ M, define |||T |||#M = sup τ (T X): X ∈ M, |||X||| 1 . When no confusion arises, we simply write ||| · |||# instead of ||| · |||#M . Lemma 6.1. ||| · |||# is a norm on M. Proof. If T = 0, |||T |||# τ (T T ∗ )/|||T ∗ ||| > 0. It is easy to see that |||λT |||# = |λ| · |||T |||# and |||T1 + T2 |||# |||T1 |||# + |||T2 |||# . 2 Definition 6.2. ||| · |||# is called the dual norm of ||| · ||| on M with respect to τ . The next lemma follows directly from the definition of dual norm.

164


Lemma 6.3. Let ||| · ||| be a norm on a finite von Neumann algebra (M, τ ) and let ||| · |||# be the dual norm on M. Then for S, T ∈ M, |τ (ST )| |||S||| · |||T |||# . The following corollary is a generalization of Hölder’s inequality for bounded operators in finite von Neumann algebras. Corollary 6.4. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a gauge norm on M. Then for S, T ∈ M, ST 1 |||S||| · |||T |||# . Proof. By Lemma 3.8, ST 1 = sup{|τ (U ST )|: U ∈ U(M)}. By Lemmas 6.3 and 3.2, |τ (U ST )| |||U S||| · |||T |||# = |||S||| · |||T |||# . 2 Proposition 6.5. If ||| · ||| is a unitarily invariant norm on a finite von Neumann algebra (M, τ ), then ||| · |||# is also a unitarily invariant norm on M. Proof. Let U be a unitary operator. Then |||U T |||# = sup{|τ (U T X)|: X ∈ M, |||X||| 1} = sup{|τ (T XU )|: X ∈ M, |||X||| 1} = sup{|τ (T X)|: X ∈ M, |||X||| 1} = |||T ||| and |||T U |||# = sup{|τ (T U X)|: X ∈ M, |||X||| 1} = sup{|τ (T X)|: X ∈ M, |||X||| 1} = |||T |||. 2 Proposition 6.6. If ||| · ||| is a symmetric gauge norm on a finite von Neumann algebra (M, τ ), then ||| · |||# is also a symmetric gauge norm on (M, τ ). Proof. Let θ ∈ Aut(M, τ ). Then |||θ (T )|||# = sup{|τ (θ (T )X)|: X ∈ M, |||X||| 1} = sup{|τ (θ (T θ −1 (X)))|: X ∈ M, |||X||| 1} = sup{|τ (T θ −1 (X))|: X ∈ M, |||X||| 1} = sup{|τ (T X)|: X ∈ M, |||X||| 1} = |||T |||. 2 Lemma 6.7. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. If T ∈ M is a positive operator, then |||T |||# = sup τ (T X): X ∈ M, X 0, XT = T X, |||X||| 1 . Proof. Let A be a separable abelian von Neumann subalgebra of M containing T and let EA be the normal conditional expectation from M onto A preserving τ . For every Y ∈ M such that |||Y ||| 1, let X = EA (Y ). By Corollary 1, ||||X|||| = |||X||| |||Y ||| 1. Furthermore, |τ (T Y )| = |τ (EA (T Y ))| = |τ (T EA (Y ))| = |τ (T X)| τ (T |X|). Hence, |||T |||# = sup τ (T X): X ∈ M, X 0, XT = T X, |||X||| 1 .

2

Lemma 6.8. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. Suppose T = a1 E1 + · · · + an En is a positive simple operator in M. Then |||T ||| = sup τ (T X): X = b1 E1 + · · · + bn En 0 and |||X|||# 1 n # = sup ak bk τ (Ek ): X = b1 E1 + · · · + bn En 0 and |||X||| 1 . k=1


165

Proof. By Lemma 6.7, |||T |||# = sup{|τ (T X)|: X ∈ M, X 0, XT = T X, |||X||| 1}. Let A = {E1 , . . . , En } and let EA be the normal conditional expectation from M onto A preserving τ . Then S = EA (X) = τE1 (E1 XE1 )E1 + · · · + τEn (En XEn )En is a positive operator, τ (T X) = τ (EA (T X)) = τ (T EA (X)) = τ (T S), and |||S||| |||X||| by Corollary 4.4. Combining the definition of dual norm, this proves the lemma. 2 Corollary 6.9. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. Suppose S, T are equi-measurable, positive simple operators in M. Then |||S|||# = |||T |||# . Theorem 6.10. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. Then ||| · |||# is also a tracial gauge norm on M. Furthermore, if |||1||| = 1, then |||1|||# = 1. Proof. By Lemma 3.13, ||| · ||| is a symmetric gauge norm on M. By Proposition 6.6, Corollary 6.9 and Lemma 3.10, ||| · |||# is a tracial gauge norm on M. Note that |||1||| = 1, hence, |||1|||# τ (1 · 1) = 1. On the other hand, by Theorem 3.30, |||1|||# = sup{|τ (X)|: X ∈ M, |||X||| 1} sup{|||X|||: X ∈ M, |||X||| 1} 1. 2 Corollary 6.11. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. If N is a von Neumann subalgebra of M satisfying the weak Dixmier property, then ||| · |||#N is the restriction of ||| · |||#M to N . Proof. Let ||| · |||1 = ||| · |||#N and let ||| · |||2 be the restriction of ||| · |||#M to N . By Theorem 6.10, both ||| · |||1 and ||| · |||2 are tracial gauge norms on N . By Lemma 3.6, to prove ||| · |||1 = ||| · |||2 , we need to prove |||T |||1 = |||T |||2 for every positive simple operator T ∈ N . Let A be a finitedimensional abelian von Neumann subalgebra of N containing T . By Lemma 6.8, |||T |||#M = |||T |||#N = |||T |||#A . So |||T |||1 = |||T |||2 . 2 6.2. Dual norms of Ky Fan norms For (x1 , . . . , xn ) ∈ Cn , τ (x) =

x1 +···+xn n

defines a trace on Cn . For 1 k n, the Ky Fan x ∗ +···+x ∗

(Cn , τ ) is |||(x1 , . . . , xn )|||( k ) = 1 k k , where (x1∗ , . . . , xn∗ ) is the decreasing rearn rangement of (|x1 |, . . . , |xn |). Let = {(x1 , . . . , xn ) ∈ Cn : x1 x2 xk = xk+1 = · · · = xn 0, x1 +···+xk 1} and E be the set of extreme points of . k The proof of the following lemma is an easy exercise. k n th norm on

k k Lemma 6.12. E consists of k + 1 points: (k, 0, . . .), ( k2 , k2 , 0, . . .), . . . , ( k−1 , . . . , k−1 , 0, . . .), (1, 1, . . . , 1) and (0, 0, . . . , 0).

The following lemma is well known. For a proof we refer to [8, 10.2]. Lemma 6.13. Let s1 s2 · · · sn 0 and t1 , . . . , tn 0. If t1∗ t2∗ · · · tn∗ is the decreasing rearrangement of t1 , . . . , tn , then s1 t1∗ + · · · + sn tn∗ s1 t1 + · · · + sn tn .

166


Lemma 6.14. For T ∈ (Mn (C), τn ),

k |||T |||#( k ) = max T , T 1 . n n Proof. Let |||T |||1 = |||T |||# k and |||T |||2 = max{ nk T , T 1 }. Then both ||| · |||1 and ||| · |||2 are (n)

unitarily invariant norms on Mn (C). To prove ||| · |||1 = ||| · |||2 , we need only to prove |||T |||1 = |||T |||2 for every positive matrix T in Mn (C). We can assume that ⎞ ⎛ s1 ⎟ ⎜ .. T =⎝ ⎠, . sn where s1 , . . . , sn are s-numbers of T such that s1 s2 · · · sn . By Lemmas 6.8 and 6.13,

n

n i=1 si ti i=1 si ti : (t1 , . . . , tn ) ∈ = sup : (t1 , . . . , tn ) ∈ E . |||T |||1 = sup n n Note that T = s1 s2 · · · sn 0. By Lemma 6.12 and simple computations, |||T |||1 = max{ nk T , T 1 } = |||T |||2 . 2 The next lemma simply follows from the definition of dual norms. Lemma 6.15. Let (M, τ ) be a finite von Neumann algebra and ||| · |||, ||| · |||1 , ||| · |||2 be norms on M such that |||T |||1 |||T ||| |||T |||2 ,

∀T ∈ M.

|||T |||#2 |||T |||# |||T |||#1 ,

∀T ∈ M.

Then

Corollary 6.16. Let (M, τ ) be a finite von Neumann algebra and ||| · |||1 , ||| · |||2 be equivalent norms on M. Then ||| · |||#1 and ||| · |||#2 are equivalent norms on M. Theorem 6.17. Let M be a type II1 factor and 0 t 1. Then |||T |||#(t) = max{tT , T 1 },

∀T ∈ M.

Proof. Firstly, we assume t = nk is a rational number. Let Nr be a type Irn subfactor of M. Then the restriction of ||| · |||(t) to Nr is ||| · |||( rk ) . By Lemma 6.14 and Corollary 6.11, |||T |||#(t) = rn

max{tT , T 1 } for T ∈ Nr . By Corollary 4.7, |||T |||#(t) = max{tT , T 1 } for all T ∈ M. Now assume t is an irrational number. Let t1 , t2 be two rational numbers such that t1 < t < t2 . By Lemma 6.15, for every T ∈ M, max t2 T , T 1 |||T |||#(t) max t1 T , T 1 . Since t1 t t2 are arbitrary, |||T |||#(t) = max{tT , T 1 }.

2


167

6.3. Proof of Theorem C Lemma 6.18. Let n ∈ N and τ be an arbitrary faithful state on Cn . If ||| · ||| is a norm on (Cn , τ ) and ||| · |||# is the dual norm with respect to τ , then ||| · |||## = ||| · |||. Proof. By Lemma 6.3, |||T |||## = sup{|τ (T X)|: X ∈ Cn , |||X|||# 1} |||T |||. We need to prove |||T ||| |||T |||## . By the Hahn–Banach theorem, there is a continuous linear functional φ on Cn with respect to the topology induced by ||| · ||| on Cn such that |||T ||| = φ(T ) and φ = 1. Since all norms on Cn induce the same topology, there is an element Y ∈ Cn such that φ(S) = τ (SY ) for all S ∈ Cn . By the definition of dual norm, |||Y |||# = φ = 1. By Lemma 6.3, |||T ||| = φ(T ) = τ (T Y ) |||T |||## . 2 Proof of Theorem C. By Theorem 6.10, both ||| · |||## and ||| · ||| are tracial gauge norms on M. By Corollary 3.6, to prove ||| · |||## = ||| · |||, we need to prove that |||T ||| = |||T |||## for every positive simple operator T ∈ M. Let A be the abelian von Neumann subalgebra generated by T . ## By Corollary 6.11 and Lemma 6.18, |||T |||## M = |||T |||A = |||T |||. 2 7. Proof of Theorem A Let (M, τ ) be a finite von Neumann algebra. Lemma 7.1. Let n ∈ N, a1 a2 · · · an an+1 = 0 and f (x) = a1 χ[0, 1 ) (x) + a2 χ[ 1 , 2 ) (x) + n n n · · · + an χ[ n−1 ,1] (x). For T ∈ M, define n

1 |||T |||f =

(7.1)

f (s)μs (T ) ds. 0

Then |||T |||f =

n k(ak − ak+1 )

n

k=1

Proof. Since t|||T |||(t) =

t 0

1

=

2

n f (s)μs (T ) dt = a1

0 n k=1

(7.2)

n

μs (T ) ds, summation by parts shows that

1 |||T |||f =

|||T ||| k .

n μs (T ) ds + a2

0

k(ak − ak+1 ) |||T ||| k . n n

1 μs (T ) ds + · · · + an

1 n

μs (T ) ds n−1 n

2

Corollary 7.2. The norm ||| · |||f defined as above is a tracial gauge norm on M and |||1|||f = 1 a1 +···+an . 0 f (x) dx = n Lemma 7.3. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let {||| · |||α } be a set of tracial gauge norms on (M, τ ) such that |||1|||α 1 for all α. For

168


every T ∈ M, define |||T ||| = sup |||T |||α . α

Then ||| · |||

α ||| · |||α

is also a tracial gauge norm on (M, τ ).

Proof. By Corollary 3.5, |||T ||| T is well defined. It is easy to check that ||| · ||| is a tracial gauge norm on (M, τ ). 2 Proof of Theorem A. Let

F = μs (X): X ∈ M, |||X|||# 1, X = b1 F1 + · · · + bk Fk 0, 1 where F1 + · · · + Fk = 1 and τ (F1 ) = · · · = τ (Fk ) = , k = 1, 2, . . . . k 1 For every positive operator X ∈ M such that |||X|||# 1, 0 μs (X) ds = τ (X) = X1 |||X|||# 1 by Theorem 3.30. Hence F ⊂ F and μs (1) = χ[0,1] (s) ∈ F by Theorem 6.10. For T ∈ M, define |||T ||| = sup |||T |||f : f ∈ F . By Corollary 7.2, |||·||| is a tracial gauge norm on M. To prove that |||·||| = |||·|||, by Corollary 4.6, we need prove that |||T ||| = |||T ||| for every positive operator T ∈ M such that T = a1 E1 + · · · + an En and τ (E1 ) = · · · = τ (En ) = n1 . By Lemma 6.8 and Theorem C, n 1 # |||T ||| = sup ak bk : X = b1 E1 + · · · + bn En 0 and |||X||| 1 . n k=1

Note that if X = b1 E1 + · · · + bn En is a positive simple operator in M and |||X|||# 1, 1 then μs (X) ∈ F and |||T |||μs (X) = 0 μs (X)μs (T ) ds = n1 nk=1 ak∗ bk∗ , where {ak∗ } and {bk∗ } are non-increasing rearrangements of {ak } and {bk }, respectively. By Lemma 6.13, |||T ||| sup{|||T |||f : f ∈ F } = |||T ||| . We need to prove |||T ||| |||T ||| . Let X = b1 F1 + · · · + bk Fk be a positive operator in M such that F1 + · · · + Fk = 1, τ (F1 ) = · · · = τ (Fk ) = 1k and |||X|||# 1. We need only prove that |||T ||| |||T |||μs (X) . Since (M, τ ) satisfies the weak Dixmier property, by Theorem 3.27, (M, τ ) is either a von Neumann subalgebra of (Mn (C), τn ) that contains all diagonal matrices or M is a diffuse von Neumann algebra. In either case, we may assume that T = a˜ 1 E˜ 1 + · · · + a˜ r E˜ r and X = b˜1 F˜1 +· · ·+ b˜r F˜r , where E˜ 1 +· · ·+ E˜ r = F˜1 +· · ·+ F˜r = 1 and τ (E˜ i ) = τ (F˜i ) = 1r for 1 i r, a˜ 1 · · · a˜ r 0 and b˜1 · · · b˜r 0. Let Y = b˜1 E˜ 1 +· · ·+ b˜r E˜ r . Then X and Y are two equimeasurable operators in M and μs (X) = μs (Y ). By Theorem 6.10, |||Y |||# 1. By Lemma 6.3, 1 ˜ |||T ||| τ (T Y ) = a˜ i bi = r r

i=1

1

1 μs (Y )μs (T ) ds =

0

μs (X)μs (T ) ds = |||T |||μs (X) . 0

2


169

Combining Theorem A and Lemma 3.21, we obtain the following corollary. Corollary 7.4. Let (M, τ ) be a finite factor and let ||| · ||| be a normalized unitarily invariant norm on M. Then there is a subset F of F containing the constant 1 function on [0, 1] such that for all T ∈ M, |||T ||| = sup{|||T |||f : f ∈ F }. Combining Theorem A and Lemma 3.15 we obtain the following corollary. 1 Corollary 7.5. Let ||| · ||| be a normalized symmetric gauge norm on (L∞ [0, 1], 0 dx). Then there is a subset F of F containing the constant 1 function on [0, 1] such that for all T ∈ L∞ [0, 1], |||T ||| = sup{|||T |||f : f ∈ F }. 8. Proof of Theorems D and E 1 Lemma 8.1. Let θ1 , θ2 be two embeddings from (L∞ [0, 1], 0 dx) into a finite von Neumann algebra (M, τ ). If ||| · ||| is a tracial gauge norm on M, then |||θ1 (f )||| = |||θ2 (f )||| for every f ∈ L∞ [0, 1]. Proof. If f ∈ L∞ [0, 1] is a positive function, then θ1 (f ) and θ2 (f ) are equi-measurable operators in M. Hence |||θ1 (f )||| = |||θ2 (f )|||. 2 Proof of Theorem D. We prove Theorem D for diffuse finite von Neumann algebras. The proof of the atomic case is similar. We may assume that the norms on M or L∞ [0, 1] are normalized. By the definition of Ky Fan norms, there is a one-to-one correspondence between Ky Fan tth 1 norms on (M, τ ) and Ky Fan tth norms on (L∞ [0, 1], 0 dx) as in Theorem D. By Lemma 7.1, Theorems 3.27 and A, there is a one-to-one correspondence between normalized tracial norms 1 on (M, τ ) and normalized symmetric gauge norms on (L∞ [0, 1], 0 dx) as in Theorem D. 2 Example 8.2. For 1 p ∞, the Lp -norm on L∞ [0, 1] defined by f (x) = p

1 ( 0 |f (x)|p dx)1/p , ess sup |f |,

1 p < ∞; p=∞

1 is a normalized symmetric gauge norm on (L∞ [0, 1], 0 dx). By Corollaries 2 and 2.8, the induced norm 1 (τ (|T |p ))1/p = ( 0 |μs (T )|p ds)1/p , 1 p < ∞; T p = T , p=∞ is a normalized unitarily invariant norm on a type II1 factor M. The norms { · p : 1 p ∞} are called Lp -norms on M. Corollary 8.3. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on (M, τ ). If (M, τ ) can be embedded into a finite factor (M1 , τ1 ), then there is a unitarily invariant norm ||| · |||1 on (M1 , τ1 ) such that ||| · ||| is the restriction of ||| · |||1 to (M, τ ).

170


The following example shows that without the weak Dixmier property, Corollary 8.3 may fail. Example 8.4. On (C2 , τ ), τ ((x, y)) = 13 x + 23 y, let |||(x, y)||| = 23 |x| + 13 |y|. It is easy to see that ||| · ||| is a tracial gauge norm on (C2 , τ ). Let M1 be the reduced free product of (C2 , τ ) with the free group factor L(F2 ). Then M1 is a type II1 factor with a faithful normal trace τ1 such that the restriction of τ1 to C2 is τ . Suppose ||| · |||1 is a unitarily invariant norm on M1 such that the restriction of ||| · |||1 to C2 is ||| · |||. Let E = (1, 0) and F = (0, 1) in C2 . Then τ1 (E) = τ (E) < τ (F ) = τ1 (F ). So there is a unitary operator U in M1 such that U EU ∗ F . By Corollary 3.3, 23 = |||E||| = |||E|||1 = |||U EU ∗ |||1 |||F |||1 = |||F ||| = 13 . This is a contradiction. Proof of Theorem E. Let ||| · |||2 be the tracial gauge norm on M corresponding to the symmetric 1 gauge norm ||| · |||#1 on (L∞ [0, 1], 0 dx) as in Theorem D. By Lemma 4.6, to prove ||| · |||2 = ||| · |||# on M, we need to prove |||T |||2 = |||T |||# for every positive simple operator T = a1 E1 + · · · + an En in M such that τ (E1 ) = · · · = τ (En ) = n1 . We may assume that a1 · · · an 0. Then μs (T ) = a1 χ[0, 1 ) (s) + · · · + an χ[ n−1 ,1] (s). By Lemma 6.8, n

n

n 1 # |||T ||| = sup ak bk : X = b1 E1 + · · · + bn En 0 and |||X||| 1 . n #

k=1

By Lemma 6.13,

n 1 |||T ||| = sup ak bk : X = b1 E1 + · · · + bn En 0, b1 · · · bn 0, |||X||| 1 . n #

k=1

By Theorem D and Lemma 6.8, n # 1 |||T |||2 = μs (T ) = sup ak bk : g(s) = b1 χ[0, 1 ) (s) + · · · + bn χ[ n−1 ,1] (s) 0, n n n k=1 g(s) 1 . By Lemma 6.13, n # 1 |||T |||2 = μs (T ) = sup ak bk : g(s) = b1 χ[0, 1 ) (s) + · · · + bn χ[ n−1 ,1] (s) 0, n n n k=1 b1 · · · bn 0, g(s) 1 . Note that if b1 · · · bn 0, then μs (b1 E1 + · · · + bn En ) = b1 χ[0, 1 ) (s) + · · · + bn χ[ n−1 ,1] (s). n n Since ||| · ||| is the tracial gauge norm on (M, τ ) corresponding to the symmetric gauge norm ||| · |||1


on (L∞ [0, 1],

1 0

171

dx) as in Theorem D, |||b1 E1 + · · · + bn En ||| 1 if and only if |||b1 χ[0, 1 ) (s) +

· · · + bn χ[ n−1 ,1] (s)|||1 1. Therefore, |||T |||2 = |||T |||# . n

2

n

p Example 8.5. If p = 1, let q = ∞. If 1 < p < ∞, let q = p−1 . Then the Lq -norm on L∞ [0, 1] is p ∞ the dual norm of the L -norm on L [0, 1]. By Theorem E, the Lq -norm on a type II1 factor M is the dual norm of the Lp -norm on M.

9. Proof of Theorem F Proof of Theorem F. Let ||| · ||| be a tracial gauge norm on M. By Lemma 7.1, |||S|||f |||T |||f for every f ∈ F . By Theorem A, |||S||| |||T |||. 2 Corollary 9.1. Let M be a type II1 factor and S, T ∈ M. If |||S|||(t) |||T |||(t) for all Ky Fan tth norms, 0 t 1, then |||S||| |||T ||| for all unitarily invariant norms ||| · ||| on M. By Corollary 9.1, we obtain Ky Fan’s dominance theorem [13]. Ky Fan’s dominance theorem. If S, T ∈ Mn (C) and |||S|||(k/n) |||T |||(k/n) , i.e., ki=1 si (S) k i=1 si (T ) for 1 k n, then |||S||| |||T ||| for all unitarily invariant norms ||| · ||| on Mn (C). 10. Extreme points of normalized unitarily invariant norms on finite factors In this section, we assume that M is a finite factor with the unique tracial state τ . 10.1. N(M) Let N(M) be the set of normalized unitarily invariant norms on M. It is easy to see that N(M) is a convex set. Let F(M) be the set of complex functions defined on M. Then F(M) is a locally convex space such that a neighborhood of f ∈ F(M) is N (f, T1 , . . . , Tn , ) = g ∈ F(M): g(Ti ) − f (Ti ) < . In this topology, fα → f means limα fα (T ) = f (T ) for every T ∈ M. We call this topology the pointwise weak topology. Lemma 10.1. N(M) ⊆ F(M) is a compact convex subset in the pointwise weak topology. Proof. It is clear that N(M) is a convex subset of F(M). Suppose ||| · |||α ∈ F(M) and f (T ) = limα |||T |||α for every T ∈ M. It is easy to check that f (T ) defines a unitarily invariant semi-norm on M such that f (1) = 1. By Corollary 3.31, f (T ) is a norm and f ∈ N(M). 2 Let Ne (M) be the subset of extreme points of N(M). By the Krein–Milman theorem, the closure of the convex hull of Ne (M) is N(M) in the pointwise weak topology. It is an interesting question of characterizing Ne (M). In the following, we will provide some results on Ne (M).

172


10.2. Ne (Mn (C)) For n 2, let 1 ⊕ s2 ⊕ · · · ⊕ sn be the matrix ⎛ 1 ⎜ s2 ⎜ ⎜ .. ⎝ .

⎞ ⎟ ⎟ ⎟ ∈ Mn (C). ⎠ sn

Let ||| · ||| be a normalized unitarily invariant norm on Mn (C). For 0 sn · · · s2 1, define f (s2 , . . . , sn ) = f|||·||| (s2 , . . . , sn ) = |||1 ⊕ s2 ⊕ · · · ⊕ sn |||.

(10.1)

In the following, let Ωn−1 = {(s2 , . . . , sn ): 0 sn · · · s2 1}. By [7, Lemma 3.2] and Corollary 3.31, we have the following lemma. Lemma 10.2. Let f (s2 , . . . , sn ) be a function defined on Ωn−1 . In order that f (s2 , . . . , sn ) = f|||·||| (s2 , . . . , sn ) for some ||| · ||| ∈ N(Mn (C)), it is necessary and sufficient that the following conditions are satisfied: 1. f (s2 , . . . , sn ) > 0 for all (s2 , . . . , sn ) ∈ Ωn−1 and f (1, . . . , 1) = 1; 2. f (s2 , . . . , sn ) is a convex function on Ωn−1 ; 3. for 0 sn sn−1 · · · s1 , 0 tn tn−1 · · · t1 , if ki=1 si ki=1 ti for 1 k n, then s1 · f ( ss21 , . . . , ssn1 ) t1 · f ( tt21 , . . . , ttn1 ). If f (s2 , . . . , sn ) satisfies the above conditions, then f satisfies 1 + s2 + · · · + sn f (s2 , . . . , sn ) 1 n for all (s2 , . . . , sn ) ∈ Ωn−1 . Let ||| · |||1 , ||| · |||2 ∈ N(Mn (C)). If |||S|||1 = |||S|||2 for all S = 1 ⊕ s2 ⊕ · · · ⊕ sn , (s2 , . . . , sn ) ∈ Ωn−1 , then |||T |||1 = |||T |||2 for every matrix T ∈ Mn (C). This implies the following lemma. Lemma 10.3. Let |||·|||1 , |||·|||2 ∈ N(Mn (C)). Then |||·|||1 = |||·|||2 if and only if f|||·|||1 (s2 , . . . , sn ) = f|||·|||2 (s2 , . . . , sn ) for all (s2 , . . . , sn ) ∈ Ωn−1 . Let 1 m n. Suppose ||| · ||| is a normalized unitarily invariant norm on Mm (C) and g(s2 , . . . , sm ) = g|||·||| (s2 , . . . , sm ) is the function on Ωm−1 induced by ||| · ||| (see (10.1)). Define f (s2 , . . . , sn ) on Ωn−1 by f (s2 , . . . , sn ) = g(s2 , . . . , sm ),

(s2 , . . . , sn ) ∈ Ωn−1 .

It is easy to check that f (s2 , . . . , sn ) is a function on Ωn−1 satisfying Lemma 10.2. By Lemmas 10.2 and 10.3, there is a unique normalized unitarily invariant norm ||| · |||1 ∈ N(Mn (C)) such that f (s2 , . . . , sn ) = f|||·|||1 (s2 , . . . , sn ) = g(s2 , . . . , sm ) for all (s1 , . . . , sn ) ∈ Ωn−1 . (This fact can also be obtained by Corollary 7.4 and Lemma 10.3.) ||| · |||1 is called the induced norm of ||| · |||.


173

Conversely, suppose ||| · |||1 is a normalized unitarily invariant norm on Mn (C) and f (s2 , . . . , sn ) = f|||·|||1 (s2 , . . . , sn ) is the function on Ωn−1 induced by ||| · |||1 . If f (s2 , . . . , sn ) = g(s2 , . . . , sm ) for all (s2 , . . . , sn ) ∈ Ωn−1 , then g(s2 , . . . , sm ) satisfies Lemma 10.2. Hence, there is a unique normalized unitarily invariant norm ||| · ||| on Mm (C) such that g(s2 , . . . , sm ) = g|||·||| (s2 , . . . , sm ) for all (s2 , . . . , sm ) ∈ Ωm−1 . ||| · ||| is called the reduced norm of ||| · |||1 . Proposition 10.4. For 1 k n, the Ky Fan extreme point of N(Mn (C)).

k n th

norm (see Example 5.4) on Mn (C) is an

Proof. Suppose 0 < α < 1 and ||| · |||1 , ||| · |||2 ∈ N(Mn (C)) satisfy ||| · |||( k ) = α||| · |||1 + n (1 − α)||| · |||2 . Let f (s2 , . . . , sn ) = f|||·||| k (s2 , . . . , sn ), f1 (s2 , . . . , sn ) = f|||·|||1 (s2 , . . . , sn ) and (n)

f2 (s2 , . . . , sn ) = f|||·|||2 (s2 , . . . , sn ) for (s2 , . . . , sn ) ∈ Ωn−1 . Then f (s2 , . . . , sn−1 ) = αf1 (s2 , . . . , sn−1 ) + (1 − α)f2 (s2 , . . . , sn−1 ). ∂f k , ∂s∂fk+1 = · · · = ∂s = 0. Since f1 (s2 , . . . , sn ), f2 (s2 , . . . , sn ) Since f (s2 , . . . , sn ) = 1+s2 +···+s k n are convex functions on Ωn−1 ,

∂fi ∂sj

0 for i = 1, 2 and k +1 j n. Since f = αf1 +(1−α)f2 ,

∂fi ∂sj

= 0 for i = 1, 2 and k + 1 j n. This implies that fi (s2 , . . . , sn ) = gi (s2 , . . . , sk ) for all (s2 , . . . , sn ) ∈ Ωn−1 and i = 1, 2. By the discussions above the proposition, there are normalized unitarily invariant norms ||| · |||1 , ||| · |||2 on Mk (C) such that gi (s2 , . . . , sk ) = (gi )|||·|||i (s2 , . . . , sk ) for all (s2 , . . . , sk ) ∈ Ωk−1 and i = 1, 2. By Lemma 10.2, gi (s2 , . . . , sk )

1 + s2 + · · · + sk k

for all (s2 , . . . , sk ) ∈ Ωk−1 and i = 1, 2. Since f = αf1 + (1 − α)f2 , 1 + s2 + · · · + sk = αg1 (s2 , . . . , sk ) + (1 − α)g2 (s2 , . . . , sk ). k This implies that g1 (s2 , . . . , sk ) = g2 (s2 , . . . , sk ) =

1+s2 +···+sk . k

So f = f1 = f2 .

2

The proof of the following proposition is similar to that of Proposition 10.4. Proposition 10.5. Let 1 m n and ||| · ||| be a normalized unitarily invariant norm on Mm (C). If ||| · ||| is an extreme point of N(Mm (C)), then the induced norm ||| · |||1 on Mn (C) is also an extreme point of N(Mn (C)). Question. For n 3, find all extreme points of N(Mn (C)). 10.3. Ne (M2 (C)) In this subsection, we will prove Theorem J. We need the following auxiliary results. The following lemma is a corollary of Lemma 10.2 in the case n = 2.

174


Lemma 10.6. Let f (s) be a function on [0, 1]. If there is a normalized unitarily invariant norm ||| · ||| on M2 (C) such that f (s) = f|||·||| (s) = |||1 ⊕ s|||, then f (s) is an increasing convex function on [0, 1] satisfying 1+s f (s) 1, 2

∀s ∈ [0, 1].

Corollary 10.7. For 0 a b 1, we have 1 0 f (a−) f (a+) f (b−) f (b+) f (1−) . 2 Proof. Since f (s) is an increasing convex function, 0 f (a−) f (a+) f (b−) f (b+) f (1−). By Lemma 10.6, f (1) − f (1 − h) 1 − (2 − h)/2 1 lim = . h→0+ h→0+ h h 2

f (1−) = lim For

1 2

2

t 1, define ||| · |||t = max{tT , T 1 }.

Lemma 10.8. For 1/2 t 1, ||| · |||t is an extreme point of N(M2 (C)). Proof. Suppose 0 < α < 1 and ||| · |||1 , ||| · |||2 ∈ N(M2 (C)) such that ||| · |||t = α||| · |||1 + (1 − α)||| · |||2 . Let f (s) = f|||·|||t (s), f1 (s) = f|||·|||1 (s) and f2 (s) = f|||·|||2 (s). Then f (s) = αf1 (s) + (1 − α)f2 (s). Note that t 0 s 2t−1 2 ; f (s) = s+1 2t−1 2 2 s 1. 1 2t−1 Hence, f (s) = 0 if 0 s < 2t−1 2 and f (s) = 2 if 2 < s 1. By Corollary 10.7, f1 (s) = 2t−1 1 2t−1 f2 (s) = 0 if 0 s < 2 and f1 (s) = f2 (s) = 2 if 2 < s 1. Since f (s), f1 (s), f2 (s) are convex functions and hence continuous and f (1) = f1 (1) = f2 (1) = 1, f (s) = f1 (s) = f2 (s) for all 0 s 1. This implies that ||| · |||t = ||| · |||1 = ||| · |||2 . 2

Lemma 10.9. The mapping: t → ||| · |||t is continuous with respect to the usual topology on [1/2, 1] and the pointwise weak topology on N(M2 (C)). In particular, {||| · |||t : 1/2 t 1} is compact in the pointwise weak topology. Proof. For every 0 s 1, |||1 ⊕ s|||t = max{t, 1+s 2 } is a continuous function on [0, 1]. Hence, the mapping: t → ||| · |||t is continuous with respect to the usual topology on [1/2, 1] and the pointwise weak topology on N(M2 (C)). 2 Lemma 10.10. The set

1

S = ||| · |||: ||| · ||| =

||| · |||t dμ(t), μ is a regular Borel probability measure on [1/2, 1]

1/2

is a convex compact subset of N(M2 (C)) in the pointwise weak topology.


175

Proof. Suppose {||| · |||α } is a net in S such that ||| · |||α → ||| · ||| ∈ N(M2 (C)) in the pointwise weak topology. Let μα be the regular Borel probability measure on [1/2, 1] corresponding to ||| · |||α . Then there is a subnet of μα that converges weakly to a regular Borel probability measure μ on [1/2, 1], i.e., for every continuous function φ(t) on [1/2, 1], 1

1 φ(t) dμαβ (t) =

lim α

1/2

φ(t) dμ(t).

1/2

In particular, for every T ∈ M2 (C), we have 1 |||T ||| = lim |||T |||αβ = lim αβ

αβ

1 |||T |||t dμαβ (t) =

1/2

Hence ||| · ||| ∈ S.

|||T |||t dμ(t).

1/2

2

Lemma 10.11. Let f (s) be a convex, increasing function on [0, 1] such that 1+s f (s) 1, 2

∀s ∈ [0, 1].

Then there is an element ||| · ||| ∈ S such that f (s) = |||1 ⊕ s|||. Proof. We can approximate f uniformly by piecewise linear functions satisfying the conditions of the lemma. By Lemma 10.10, we may assume that f (s) is a piecewise linear function. Furthermore, we may assume that 0 = a0 < a1 < a2 < · · · < an = 1 and f (s) is linear on [ai , ai+1 ] for 0 i n − 1. Let f (s) = αi /2 on [ai , ai+1 ]. By Corollary 10.7, 0 = α0 α1 · · · αn−1 1. Let g(s) = (1 − αn−1 )1 ⊕ s + (αn−1 − αn−2 )|||1 ⊕ s|||αn−1 + · · · + (α1 − α0 )|||1 ⊕ s|||α1 + α0 1 ⊕ s1 . Then g(1) = f (1) = 1 and g (s) = αi /2 on [ai , ai+1 ]. So g (s) = f (s) except s = αi for 1 i n. Hence f (s) = g(s) for all 0 s 1. 2 Proof of Theorem J. By Lemma 10.8, {||| · |||t : 1/2 t 1} are extreme points of N(M). By Lemmas 10.10, 10.11 and 10.3, the closure of the convex hull of {||| · |||t : 1/2 t 1} in the pointwise weak topology is N(M2 (C)). By Lemmas 10.8, 10.9 and by [11, Theorem 1.4.5], Ne (M2 (C)) = {||| · |||t : 1/2 t 1}. 2 Corollary 10.12. Let f (s) be a function on [0, 1]. Then the following conditions are equivalent: 1. f (s) = f|||·||| (s) = |||1 ⊕ s||| for some normalized unitarily invariant norm ||| · ||| on M2 (C); 2. f (s) is an increasing convex function on [0, 1] such that 1+s 2 f (s) 1 for all s ∈ [0, 1]; 3. f (s) is an increasing convex function on [0, 1] such that f (1) = 1 and f (1−) 12 . In the following, we will show how to write the Lp -norms on M2 (C) in terms of extreme points of N(M2 (C)). Recall that for 1 p < ∞, the Lp -norm of 1 ⊕ s is

1 ⊕ sp =

1 + sp 2

1/p .

176

J. Fang et al. / Journal of Functional Analysis 255 (2008) 142–183 p

1/p , 0 s 1. Then f (1) = 1 and Let fp (s) = f·p (s) = ( 1+s p 2 )

fp (s) =

s p−1 1 + s p 1/p−1 , 2 2

fp (0) = 0, fp (1) = 12 . Lemma 10.13. For 1 < p < ∞ and 0 s 1, 1

|||1 ⊕ s|||t 4fp (2t − 1) dt.

fp (s) = 1/2

Proof. 1

|||1 ⊕ s|||t 4fp (2t − 1) dt

1/2

1 =

|||1 ⊕ s||| x+1 2fp (x) dx 2

0

s =

1 + s 2fp (x) dx + 2

1

1 + x 2fp (x) dx 2

s

0

1

= (1 + s)f (s) − (1 + s)f (0) + 2f (1) − (1 + s)f (s) −

fp (x) dx

s

= 1 − fp (1) + fp (s) = fp (s).

2

Corollary 10.14. For 1 < p < ∞ and T ∈ M2 (C), 1 T p =

|||T |||t 4fp (2t − 1) dt.

1/2

10.4. Proof of Theorem K Lemma 10.15. Let M be a type II1 factor and let ||| · ||| be a normalized unitarily invariant norm on M. Suppose N1 ⊂ N2 ⊂ · · · is a sequence of type Inr subfactors of M such that Nr ∼ = Mnr (C) and limr→∞ nr = ∞. If the restriction of ||| · ||| to Nr is an extreme point of N(Nr ) for all r = 1, 2, . . . , then ||| · ||| is an extreme point of N(M). Proof. Suppose 0 < α < 1 and ||| · |||1 , ||| · |||2 ∈ N(M) such that ||| · ||| = α||| · |||1 + (1 − α)||| · |||2 on M. Then for every r = 1, 2, . . . , ||| · ||| = α||| · |||1 + (1 − α)||| · |||2 on Nr . By the assumption of


177

the lemma, ||| · ||| = ||| · |||1 = ||| · |||2 on Nr . By Corollary 4.7, ||| · ||| = ||| · |||1 = ||| · |||2 on M. So ||| · ||| is an extreme point of N(M). 2 Proof of Theorem K. By the assumption of the theorem, t = nk is a rational number. Then we can construct a sequence of type Irn subfactor Mrn of M such that Mn ⊆ M2n ⊆ · · · . Then the restriction of || · |||(t) on Mrn is || · |||( rk ) . By Proposition 10.4, the restriction of || · |||(t) on Mrn is rn an extreme point of N(Mrn (C)). By Lemma 10.15, || · |||(t) is an extreme point of N(M). 2 Remark 10.16. Here we point out other interesting examples of extreme points of N(M). For 0 t 1, recall that || · |||(t) is the tth Ky Fan norm on M. For any non-negative function c(t) on [0, 1] such that c(t)∞ = 1 and T ∈ M, define T |||[c(t)] = c(t)T |||(t) ∞ . Then it is easy to see that || · |||[c(t)] is a normalized unitarily invariant norm on M. It can be proved that if c(t) is a simple function or if tc(t) is a simple function, then|| · |||[c(t)] is an extreme point of N(M). 11. Proof of Theorem G In this section, we assume that M is a type II1 factor with the unique tracial state τ and ||| · ||| is a unitarily invariant norm on M. For two projections E, F in M, τ (E) τ (F ) if and only if there is a unitary operator U ∈ M such that U EU ∗ F . By Corollary 3.3, if τ (E) τ (F ), |||E||| |||F |||. So we can define r ||| · ||| =

lim

τ (E)→0+

|||E|||.

Definition 11.1. A unitarily invariant norm ||| · ||| on M is singular if r(||| · |||) > 0 and continuous if r(||| · |||) = 0. Example 11.2. The operator norm is singular since r( · ) = limτ (E)→0+ E = 1. If 0 < t 1, the Ky Fan tth norm ||| · |||(t) is continuous since r(||| · |||(1) ) = r( · 1 ) = limτ (E)→0+ τ (E) = 0 and r(||| · |||(t) ) 1t · r(||| · |||(1) ) = 0. If 1 p < ∞, it is easy to see that the Lp -norm on M is also continuous. Lemma 11.3. If ||| · ||| is singular, then ||| · ||| is equivalent to the operator norm · . Indeed, for every T ∈ M, we have r ||| · ||| T |||T ||| |||1||| · T . Proof. By Lemma 3.2, |||T ||| |||1||| · T . We need to prove r(||| · |||)T |||T |||. We may assume that T > 0. For any > 0, let E = χ[T −,T ] (T ) > 0. Then T (T − )E. By Corollary 3.3 and Lemma 3.2, |||T ||| |||(T − )E||| (T − ) · |||E||| (T − )r(||| · |||). Since > 0 is arbitrary, r(||| · |||)T |||T |||. 2

178


Recall that a neighborhood N (, δ) of 0 ∈ M in the measure topology (see [14]) is N(, δ) = T ∈ M, there is a projection E ∈ M such that τ (E) < δ and T E ⊥ < Proof of Theorem G. By Lemma 11.3, if ||| · ||| is singular, then T is the operator topology on M1,· . Suppose ||| · ||| is continuous. For , δ > 0 and T ∈ M such that T 1 and |||T ||| < δ, by Corollary 3.31, τ (χ[,1] (|T |) T1 |||T ||| < δ and T · χ[0,) (|T |) < . This implies that {T ∈ M1,· : |||T ||| < δ} ⊆ N (, δ). Conversely, let ω > 0. Since r(||| · |||) = 0, there is an , 0 < < ω/2, such that if τ (E) < then |||E||| < ω/2. For every T ∈ N (, ω/2) and T 1, choose E ∈ M such that τ (E) < and T E ⊥ < ω/2. By Proposition 3.18 and Corollary 3.5, |||T ||| |||T E||| + |||T E ⊥ ||| < T · |||E||| + T E ⊥ < ω/2 + ω/2 = ω. Hence {T ∈ N(, ω/2): T 1} ⊆ {T ∈ M: |||T ||| < ω}. 2 Corollary 11.4. Topologies induced by the Lp -norms, 1 p < ∞, on the unit ball of a type II1 factor are the same. 12. Completion of type II1 factors with respect to unitarily invariant norms In this section, we assume that M is a type II1 factor with the unique tracial state τ and ||| · ||| is a unitarily invariant norm on M. The completion of M with respect to ||| · ||| is denoted by M|||·||| . We will use the traditional notation Lp (M, τ ) to denote the completion of M with respect to be the completion the Lp -norm defined as in Example 11.2. Note that L∞ (M, τ ) = M. Let M of M in the measure topology in the sense of [14]. 12.1. Embedding of M|||·||| into M Lemma 12.1. Let ||| · ||| be a continuous unitarily invariant norm on M and T ∈ M. For every > 0, there is a δ > 0 such that if τ (E) < δ, then |||T E||| < . Proof. Since ||| · ||| is continuous, limτ (E)→0 |||E||| = 0. Hence, for every > 0, there is a δ > 0 such that if τ (E) < δ, then |||E||| < 1+T . By Proposition 3.18, |||T E||| T · |||E||| < . 2 Lemma 12.2. Let ||| · ||| be a continuous unitarily invariant norm on M and let {Tn } in M be a Cauchy sequence with respect to ||| · |||. For every > 0, there is a δ > 0 such that if τ (E) < δ, then |||Tn E||| < for all n. Proof. Since {Tn } is a Cauchy sequence with respect to ||| · |||, there is an N such that for all n N , |||Tn − TN ||| < /2. By Lemma 12.1, there is a δ1 such that if τ (E) < δ1 then |||TN E||| < /2. By Proposition 3.18, for n N , |||Tn E||| |||(Tn − TN )E||| + |||TN E||| < |||(Tn − TN )||| · E + /2 < . A simple argument shows that we can choose 0 < δ < δ1 such that if τ (E) < δ then |||Tn E||| < for all n. 2 The following proposition generalizes Theorem 5 of [14]. Proposition 12.3. Let M be a type II1 factor and let ||| · ||| be a unitarily invariant norm on M. that extends the identity map from M to M. There is an injective map from M|||·||| to M


179

Proof. If ||| · ||| is singular, by Lemma 11.3, M|||·||| = M. So we will assume that ||| · ||| is continuous. If {Tn } in M is a Cauchy sequence with respect to ||| · |||, then {Tn } is a Cauchy sequence |) in the L1 -norm by Corollary 3.31. For every δ > 0 and T ∈ M, τ (χ(δ,∞) (|T |) τ (|T δ . Hence, 1 if {Tn } is a Cauchy sequence in M in the L -norm, then {Tn } is a Cauchy sequence in the that extends the identity map measure topology. So there is a natural map Φ from M|||·||| to M from M to M. To prove that Φ is injective, we need to prove that if {Tn } in M is a Cauchy sequence with respect to ||| · ||| and Tn → 0 in the measure topology, then limn→∞ |||Tn ||| = 0. Let > 0. By Lemma 12.2, there is a δ > 0 such that if τ (E) < δ then |||Tn E||| < /2 for all n. Since Tn → 0 in the measure topology, there are N and δ1 , 0 < δ1 < δ, such that for all n N , there is a projection En such that τ (En ) < δ1 and Tn En⊥ < /2. By Corollary 3.31, |||Tn ||| |||Tn En⊥ ||| + |||Tn En ||| < Tn En⊥ + /2 < . This proves that limn→∞ |||Tn ||| = 0 and that extends the identity map from M to M. 2 hence Φ is an injective map from M|||·||| to M By the proof of Proposition 12.3, we have the following. Corollary 12.4. There is an injective map from M|||·||| to L1 (M, τ ) that extends the identity map from M to M. The following corollary is very By Proposition 12.3, we will consider M|||·||| as a subset of M. useful. Corollary 12.5. Let M be a type II1 factor and let ||| · ||| be a unitarily invariant norm on M. If {Tn } ⊂ M is a Cauchy sequence with respect to ||| · ||| and limn→∞ Tn = T in the measure topology, then T ∈ M|||·||| and limn→∞ Tn = T in the topology induced by ||| · |||. satisfying the following conditions: Corollary 12.6. M|||·||| is a linear subspace of M 1. if T ∈ M|||·||| , then T ∗ ∈ M|||·||| ; 2. T ∈ M|||·||| if and only if |T | ∈ M|||·||| ; 3. if T ∈ M|||·||| and A, B ∈ M, then AT B ∈ M|||·||| and |||AT B||| A · |||T ||| · B. In particular, ||| · ||| can be extended to a unitarily invariant norm, also denoted by ||| · |||, on M|||·||| . and L1 (M, τ ) 12.2. M The following theorem is due to Nelson [14]. is a ∗-algebra and T ∈ M if and only if T is a closed, Theorem 12.7. (See Nelson [14].) M is a positive operator, then densely defined operator affiliated with M. Furthermore, if T ∈ M limn→∞ χ[0,n] (T ) = T in the measure topology. as in [5]. In the following, we define s-numbers for unbounded operators in M and 0 s 1, define the sth numbers of T by Definition 12.8. For T ∈ M μs (T ) = inf T E: E ∈ M is a projection such that τ E ⊥ = s .

180


such Theorem 12.9. (See Fack and Kosaki [5].) Let T and Tn be a sequence of operators in M that limn→∞ Tn = T in the measure topology. Then for almost all s ∈ [0, 1], limn→∞ μs (Tn ) = μs (T ). Let {Tn } be a sequence of operators in M such that T = limn→∞ Tn in the L1 -norm. By Lemma 3.8, {τ (Tn )} is a Cauchy sequence in C. Define τ (T ) = limn→∞ τ (Tn ). It is obvious that τ (T ) does not depend on the sequence {Tn }. In this way, τ is extended to a linear functional on L1 (M, τ ). Lemma 12.10. Let ||| · ||| be a normalized unitarily invariant norm on a type II1 factor M. If T ∈ M|||·||| and X ∈ M, then T X ∈ L1 (M, τ ). Proof. By the proof of Proposition 12.3, limn→∞ Tn = T in the measure topology. Hence limn→∞ Tn X = T X in the measure topology (see [14, Theorem 1]). By Corollary 6.4, Tn X − Tm X1 |||Tn − Tm ||| · |||X|||# . So {Tn X} is a Cauchy sequence in the L1 -norm. By Corollary 12.5, T X ∈ L1 (M, τ ) and limn→∞ Tn X = T X in the L1 -norm. 2 12.3. Elements in M|||·||| Lemma 12.11. For all T ∈ M|||·||| , |||T ||| = sup τ (T X): X ∈ M, |||X|||# 1 . Proof. Let {Tn } be a sequence of operators in M such that limn→∞ Tn = T with respect to ||| · |||. By Corollary 6.4, if X ∈ M and |||X|||# 1, then |τ (T X)| = limn→∞ |τ (Tn X)| limn→∞ |||Tn ||| = |||T |||. Therefore, |||T ||| sup{|τ (T X)|: X ∈ M, |||X|||# 1}. We need to prove that |||T ||| sup{|τ (T X)|: X ∈ M, |||X|||# 1}. Let > 0. Since limn→∞ Tn = T with respect to ||| · |||, there is an N such that |||T − TN ||| < /3. For TN , there is an X ∈ M, |||X|||# 1, such that |||TN ||| |τ (TN X)| + /3. By the proof of Lemma 12.10 and Corollary 6.4, τ (T X) − τ (TN X) = lim τ (Tn X) − τ (TN X) n→∞

lim |||Tn − TN ||| · |||X|||# |||T − TN ||| < /3. n→∞

So |τ (T X)| |τ (TN X)| − |τ ((TN − T )X)| |||TN ||| − /3 − /3 |||T ||| − . Therefore, |||T ||| sup{|τ (T X)|: X ∈ M, |||X|||# 1}. 2 The following theorem generalizes Theorem A. Its proof is based on Lemma 12.11 and is similar to the proof of Theorem A. So we omit the proof. Theorem 12.12. If ||| · ||| is a unitarily invariant norm on a type II1 factor M, then there is a subset F of F containing the constant 1 function on [0, 1] such that for all T ∈ M|||·||| , |||T ||| = sup |||T |||f : f ∈ F ,


181

where |||T |||f is defined in Lemma 7.1 by Eq. (3) or by Eq. (4) and F = {f (x) = a1 χ[0, 1 ) (x) + a2 χ[ 1 , 2 ) (x) + · · · + an χ[ n−1 ,1] (x): a1 a2 · · · an 0, n n

n

a1 +···+an n

1, n = 1, 2, . . .}.

n

Combining Theorems 12.12 and 12.9, we have the following corollary. Corollary 12.13. Let ||| · ||| be a unitarily invariant norm on a type II1 factor M and let ||| · ||| 1 be the corresponding symmetric gauge norm on (L∞ [0, 1], 0 dx) as in Corollary 2. If T ∈ M, then T ∈ M|||·||| if and only if μs (T ) ∈ L∞ [0, ∞)|||·||| . In this case, |||T ||| = |||μs (T )||| . and 1 p ∞. Then T ∈ Lp (M, τ ) if and only if μs (T ) ∈ Example 12.14. Let T ∈ M 1 ∞ p L ([0, 1]). In this case, T p = ( 0 μs (T )p ds)1/p = ( 0 λp dμ|T | (λ))1/p . 12.4. A generalization of Hölder’s inequality Lemma 12.15. Let ||| · ||| be a unitarily invariant norm on a finite factor M and let T ∈ M|||·||| be a positive operator. Then limn→∞ χ[0,n] (T ) = T with respect to ||| · |||. Proof. If ||| · ||| is singular, then T ∈ M by Lemma 11.3 and the lemma is obvious. We may assume that ||| · ||| is continuous. Let Tn = χ[0,n] (T )) and > 0. By Lemma 12.2, there is a δ > 0 such that if τ (E) < δ then |||T E||| < . There is an N such that μs ([N, ∞)) < δ. So for m > n N , |||Tm − Tn ||| = |||T · χ(m,n] (T )||| < . This implies that {Tn } is a Cauchy sequence of M with respect to ||| · |||. Since limn→∞ Tn = T in the measure topology, by Corollary 12.5, limn→∞ Tn = T in the topology induced by ||| · |||. 2 The following theorem is a generalization of Hölder’s inequality. Theorem 12.16. Let ||| · ||| be a normalized unitarily invariant norm on a finite factor M. If T ∈ M|||·||| and S ∈ M|||·|||# , then T S ∈ L1 (M, τ ) and T S1 |||T ||| · |||S|||# . Proof. By the polar decomposition and Corollary 12.6, we may assume that S and T are positive operators. Let Tn = χ[0,n] (T ) and Sn = χ[0,n] (S). By Lemma 12.15, limn→∞ |||T − Tn ||| = limn→∞ |||S − Sn |||# = 0. Let K be a positive number such that |||Tn ||| K and |||Sn |||# K for all n and > 0. Then there is an N such that for all m > n N , |||Tm − Tn ||| < /(2K) and |||Sm − Sn |||# < /(2K). By Corollary 6.4, Tm Sm − Tn Sn 1 (Tm − Tn )Sm 1 + Tn (Sm − Sn )1 |||Tm − Tn ||| · |||Sm |||# + |||Tn ||| · |||Sm − Sn |||# < . This implies that {Tn Sn } is a Cauchy sequence in M with respect to · 1 . Since limn→∞ Tn Sn = T S in the measure topology, by Proposition 12.3, limn→∞ Tn Sn = T S in · 1 . By Corollary 6.4, Tn Sn 1 |||Tn ||| · |||Sn |||# for every n. Hence, T S1 |||T ||| · |||S|||# . 2 Combining Example 8.5 and Theorem 12.16, we obtain the non-commutative Hölder’s inequality. Corollary 12.17. Let M be a finite factor with the faithful normal tracial state τ . If T ∈ Lp (M, τ ) and S ∈ Lq (M, τ ), then T S ∈ L1 (M, τ ) and T S1 T p · Sq , where 1 p, q ∞ and

1 p

+

1 q

= 1.

182


13. Proof of Theorems H and I In this section, we assume that M is a type II1 factor with the unique tracial state τ , ||| · ||| is a unitarily invariant norm on M and ||| · |||# is the dual unitarily invariant norm on M (see 1 Definition 6.2). Let ||| · |||1 be the corresponding symmetric gauge norm on (L∞ [0, 1], 0 dx) as 1 in Theorem D and ||| · |||#1 be the dual norm on (L∞ [0, 1], 0 dx). Lemma 13.1. If M|||·|||# is the dual space of M|||·||| in the sense of Question 1, then L∞ [0, 1]|||·|||# 1

is the dual space of L∞ [0, 1]|||·|||1 in the sense of Question 1.

Proof. By Corollary 2 and Lemma 2.6, there is a separable diffuse abelian von Neumann sub1 algebra A of M and a ∗-isomorphism α from A onto L∞ [0, 1] such that τ = 0 dx ◦ α and |||α(T )|||1 = |||T ||| for each T ∈ A. By Theorem E, |||α(T )|||#1 = |||T |||# for each T ∈ A. So we need only prove that A|||·|||# is the dual space of A|||·||| in the sense of Question 1. Let φ ∈ A|||·||| # . By the Hahn–Banach extension theorem, φ can be extended to a bounded linear functional ψ on M|||·||| such that ψ = φ. By the assumption of the lemma, there is an operator X ∈ M#|||·||| such that ψ(S) = τ (SX) for all S ∈ M|||·||| and ψ = |||X|||# . Let X = U |X| be the polar decomposition of X and Xn = U · χ[0,n] (|X|). By Lemma 12.15, limn→∞ Xn = X with respect to the norm ||| · |||# . Let Yn = EA (Xn ) for n = 1, 2, . . . . By Corollary 1, {Yn } is a Cauchy sequence in A with respect to the norm ||| · |||# and |||Yn |||# |||Xn |||# . Let Y = limn→∞ Yn with respect to the norm ||| · |||# . Then Y ∈ A#|||·||| and |||Y |||# |||X|||# = ψ = φ. For T ∈ A|||·||| , φ(T ) = ψ(T ) = τ (T X) = limn→∞ τ (T Xn ) = limn→∞ τ (EA (T Xn )) = limn→∞ τ (T Yn ) = τ (T Y ). By Lemma 12.11, φ = |||Y |||# . 2 Recall that ||| · ||| is a singular norm on M if limτ (E)→0+ |||E||| > 0 and is a continuous norm on M if limτ (E)→0+ |||E||| = 0 (see Section 11). Corollary 13.2. If ||| · ||| is a singular unitarily invariant norm on M, then M|||·|||# is not the dual space of M|||·||| in the sense of Question 1. Proof. Since ||| · ||| is a singular norm on M, by Lemma 11.3, ||| · ||| is equivalent to the operator norm on M and M|||·||| = M. By Corollary 6.16 and Theorem 6.17, ||| · |||# is equivalent to the L1 -norm on M. So ||| · |||1 is equivalent to the L∞ -norm on L∞ [0, 1] and ||| · |||#1 is equivalent to the L1 -norm on L∞ [0, 1] by Theorem E. Note that L∞ [0, 1]|||·|||1 = L∞ [0, 1] is not separable with respect to ||| · |||1 but L∞ [0, 1]|||·|||# is separable with respect to ||| · |||#1 . So L∞ [0, 1]|||·|||# is not 1

1

the dual space of L∞ [0, 1]|||·|||1 in the sense of Question 1. By Lemma 13.1, M|||·|||# is not the dual space of M|||·||| in the sense of Question 1. 2 Lemma 13.3. If ||| · ||| is a continuous unitarily invariant norm on M, then M|||·|||# is the dual space of M|||·||| in the sense of Question 1. Proof. We may assume that |||1||| = 1. By Theorem 12.16, M|||·|||# is a subspace of the dual space of M|||·||| in the sense of Question 1. Let φ be a linear functional in the dual space of M|||·||| . Then for every T ∈ M|||·||| , |φ(T )| φ · |||T |||. By Corollary 3.31, for every T ∈ M,


183

|φ(T )| φ · T . So φ is a bounded linear functional on M. Since ||| · ||| is a continuous norm on M, limτ (E)→0 |||E||| = 0. Hence, limτ (E)→0 φ(E) = 0. This implies that φ is an ultraweakly continuous linear functional on M and hence in the predual space of M. So there is an operator X ∈ L1 (M, τ ) such that for all T ∈ M, φ(T ) = τ (T X). By Lemma 12.11, |||X|||# = φ < ∞. This implies that X ∈ M|||·|||# . So φ(T ) = τ (T X) for all T ∈ M|||·||| and φ = |||X|||# . This proves the lemma. 2 Proof of Theorems H and I. Combining Lemmas 13.1, 13.3 and Theorem A gives the proof of Theorems H and I. 2 Example 13.4. If 1 p < ∞ and p1 + L1 (M, τ ) is not the dual space of M.

1 q

= 1, then Lq (M, τ ) is the dual space of Lp (M, τ ).

Example 13.5. For 1 < p < ∞, Lp (M, τ ) is a reflexive space. L1 (M, τ ) and M are not reflexive spaces. By Theorem 6.17, for 0 t 1, M|||·|||(t) is not a reflexive space. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997. J. Dixmier, Les anneaux d’operateurs de classe finie, An. Sci. École. Norm. Sup. Paris 66 (1949) 209–261. J. Dixmier, Von Neumann Algebras, North-Holland, Amsterdam, 1981. T. Fack, Sur la notion de valeur caractéristique, J. Operator Theory 7 (2) (1982) 307–333. T. Fack, H. Kosaki, Generalized s-numbers of τ -measurable operators, Pacific J. Math. 123 (2) (1986) 269–300. N.A. Friedman, Introduction to Ergodic Theory, Van Nostrand/Reinhold, 1970. I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., vol. 18, Amer. Math. Soc., Providence, RI, 1969. G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, second ed., Cambridge Math. Library, Cambridge Univ. Press, Cambridge, 2001. E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vol. 2, Springer-Verlag, Berlin, 1970. R.V. Kadison, G.K. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (2) (1985) 249–266. R. Kadison, J. Ringrose, Fundamentals of the Theory of Operator Algebras, vols. 1, 2, Academic Press, New York, 1986. R.A. Kunze, Lp Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc. 89 (1958) 519–540. K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760–766. E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974) 103–116. G. Pisier, Q. Xu, Non-commutative Lp -spaces, in: Handbook of the Geometry of Banach Spaces, vol. 2, NorthHolland, Amsterdam, 2003, pp. 1459–1517. B. Russo, H.A. Dye, A note on unitary operators in C ∗ -algebras, Duke Math. J. 33 (1966) 413–416. R. Schatten, A Theory of Cross-spaces, Ann. Math. Studies, vol. 26, Princeton Univ. Press, Princeton, NJ, 1950. R. Schatten, Norm Ideals of Completely Continuous Operators, Springer-Verlag, Berlin, 1960. I.E. Segal, A non-commutative extension of abstract integration, Ann. of Math. (2) 57 (1953) 401–457. B. Simon, Trace Ideals and Their Applications, second ed., Amer. Math. Soc., Providence, RI, 2005. J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286–300.


Dual pair correspondences for non-linear covers of orthogonal groups Hung Yean Loke a,∗ , Gordan Savin b a Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore b Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA

Received 1 November 2007; accepted 9 April 2008 Available online 12 May 2008 Communicated by P. Delorme

Abstract In this paper we study compact dual pair correspondences arising from smallest representations of nonlinear covers of odd orthogonal groups. We identify representations appearing in these correspondences with subquotients of cohomologically induced representations. © 2008 Elsevier Inc. All rights reserved. Keywords: Lie groups; Representations; Dual pairs

1. Introduction Let p be an odd positive integer and let q be an even positive integer. Let SO0 (p, q) be the identity component of the Lie group SO(p, q) and let G be the central extension of SO0 (p, q) with a maximal compact subgroup Spin(p) × SO(q) if p < q, K0 = SO(p) × Spin(q) if q < p. The group G is not a linear group. In [9], we investigated the smallest representations of G that do not factor through the linear quotient SO0 (p, q). (Such representations are called genuine.) We described the corresponding Harish-Chandra modules: one such module V if p < q and two * Corresponding author.

E-mail addresses: [email protected] (H.Y. Loke), [email protected] (G. Savin). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.009

H.Y. Loke, G. Savin / Journal of Functional Analysis 255 (2008) 184–199

185

modules V + and V − if p > q. These representations are interesting for a variety of reasons. For example, if G is split then V (in the case p + 1 = q) or V + and V − (in the case p − 1 = q) lift to a trivial representation (of an appropriate algebraic group) via the local Shimura correspondence [1]. Let g be the complexified Lie algebra of G. (Lie algebras in this paper are complex unless specified otherwise.) Let W be the Harish-Chandra module of one of the smallest representations above. We showed in [9] that W is a (g, K)-module where K ⊇ K 0 is obtained by replacing the SO-factor of K 0 by the corresponding full orthogonal group. This extension is important for investigation of dual pair correspondences arising from W . More precisely, let K2 = O(s). Consider a standard embedding of K2 into the O-factor of K. Note that, by Witt’s lemma, this embedding is unique up to a conjugation. Let g1 be the centralizer of K2 in g. Then g1 =

so(p, r), r = q − s, so(r, q), r = p − s,

if p < q, if p > q.

Let G1 be a connected subgroup of G corresponding to the Lie algebra g1 and let K10 = G1 ∩ K 0 . Then W , when restricted to g1 × K2 , decomposes discretely W=

Θ(τ ) ⊗ τ

τ

where the sum is taken over all irreducible finite-dimensional representations of K2 , and Θ(τ ) is naturally a (g1 , K10 )-module. In [9], we obtained some partial results about Θ(τ ), such as irreducibility of Θ(τ ), which were necessary to established a correspondence of infinitesimal characters. Our objective in this paper is to give a more thorough investigation of the correspondence. Let q m = p−1 2 and m = 2 . Consider a θ -stable maximal parabolic subgroup q1 = l1 + n1 in g1 whose Levi component corresponds to a subgroup L1 =

U(m) × SO0 (1, r) SO(r, 0) × U(m )

if p < q, if p > q

U(m) ⊆ Spin(p) is a two-fold cover of U(m), which is given as a pull-back of in G1 . Here U(m) ⊆ SO(p). We identify Θ(τ ) with subquotients of modules which are cohomologically induced from irreducible representations of L1 which are trivial on the SO-factor and genuine on the U-factor. In particular this implies that these cohomologically induced subquotients are unitarizable and we have a detailed information about their K1 -types, since the types of Θ(τ ) could be computed by the usual branching rules of orthogonal groups. One can consider representations cohomologically induced from representations of L1 which are trivial on the SO-factor and not genuine on the U-factor. It is interesting to note that these representations (of the linear quotient of G1 ) appear as double lifts from compact orthogonal groups in the Howe correspondence [8] and [10]. In Section 6 we highlight a special case. Assume that r > q is an odd integer. Knapp [5] introduced a family πs of (so(r, q), SO(r) × Spin(q))-modules, s = 0, 1, 2 . . . . The module πs is a Harish-Chandra module of a genuine representation of G1 if and only if s is even. If s is even then p = r + s is odd. We show that πs is isomorphic to our Θ(0) where 0 denotes the trivial representation of O(s). These results, therefore, complement the results of Paul and Trapa [11].

186


It is shown there that πs for s odd appear as double lifts of trivial representations of compact groups in the Howe correspondence [8,10]. The study of our compact dual pairs unfortunately requires use of disconnected groups for technical reasons. In order to avoid the complications of treating covers of disconnected Lie groups, we will work exclusively with Harish-Chandra modules in this paper. The main results and proofs for V and V ± are similar but each requires slightly different set of notations. Hence we will divide the paper into two parts. The first part consists of Sections 2–4 where we concentrate on one family of dual pairs for the smallest representation V . The main purpose is to explain the main ideas quickly and clearly without being buried by the notations. In Section 5, we will state but without proofs the corresponding results for V ± . 2. The smallest representation In Sections 2–4, we will assume that p < q. Let V be the Harish-Chandra module of the smallest representation of G as in [9]. The module V is unitarizable and it extends to an irreducible (g, K)-module for K = Spin(p) × O(q). We need some notation in order to describe the K-types of V . 2.1. Notation The following convention will be used throughout the paper. Given a multiple of numbers λ = (λ1 , . . . , λr , 0, . . . , 0) then, by adding or removing 0’s at the tail, λ can be considered an s-tuple for every s r. Let 1k := (1, . . . , 1) and 0k := (0, . . . , 0) where there are k copies of 1’s and 0’s respectively. We set εi = (0, . . . , 0, 1, 0, . . . , 0) where 1 appears at the ith position. Given β = (β1 , . . . , βr ) and γ = (γ1 , . . . , γs ), we will denote (β1 , . . . , βr , γ1 , . . . , γs ) by (β, γ ) if there is no fear of confusion. Let Λ(n) denote the set of highest weights λ = (λ1 , . . . , λ[n/2] ) of so(n). For e = 0, 12 , let Λ(n, e) denote the subset of Λ(n) consisting of λ = (λ1 , . . . , λ[n/2] ) where λi ∈ Z + e. Hence Λ(n) = Λ(n, 0) ∪ Λ(n, 12 ). Let τnλ denote the finite-dimensional irreducible representations of so(n) with the highest weight λ. If λ is in Λ(n, 0) then τnλ is an irreducible representation of the compact group SO(n). Otherwise it is an irreducible representation of Spin(n) which does not descend to SO(n). The trivial representation may be denoted by CSO(n) . Let n−4 ρn = ( n−2 2 , 2 , . . .) ∈ Λ(n) denote the half sum of positive roots of so(n). Next we discuss irreducible representations of O(n). Let Λ(O(n)) denote the subset of elements in Zn such of the form (λ1 , . . . , λk , 0n−k )

or

(λ1 , . . . , λk , 1n−2k , 0k )

(1)

where λi are positive integers, and k n2 . Irreducible representations of O(n) are parameterized by Λ(O(n)) (see [2] and [4]). We will call an element λ of Λ(O(n)) a highest weight of O(n). λ Let τO(n) denote the corresponding irreducible finite-dimensional representation of O(n). The trivial representation of O(n) is sometimes denoted by CO(n) . λ λ Finally we recall a branching rule. Suppose n > s, then τO(n) contains τO(s) if and only if λi λi λi+n−s for all 1 i s.


187

With this notation in hand, we can now describe the K-types of V . Recall that m = restriction of V to K = Spin(p) × O(q) is

V=

λ+ q−p 2 1m

τp

p−1 2 .

The

λ ⊗ τO(q) .

λ∈Λ(p,0) λ Here λ in τO(q) is considered as an element of Λ(O(q)) by adding 0’s at the tail. In particular, q−p

the minimal K-type of V is τp 2 μp,q

1m

⊗ CO(q) . The infinitesimal character of V is

q −1 p−1 1 3 , , ,..., . = 1, 2, . . . , 2 2 2 2

We now consider the restriction of V to g1 × K2 , where g1 = so(p, r) and K2 = O(s) for some integers r and s such that r + s = q. We obtain a direct sum V=

λ Θ(λ ) ⊗ τO(s) .

(2)

λ ∈Λ(O(s))

Note that every Θ(λ ) is a (g1 , K1 )-module, where K1 = Spin(p) × O(r). Since V is admissible with respect to Spin(p) ⊆ K1 , it follows that each Θ(λ ) is an admissible (g1 , K1 )-module. 2.2. The K1 -types of Θ(τ ) Let λ be in Λ(O(s)). Write λ = (λ1 , . . . , λt , 0, . . . , 0). We will now describe the K1 -types of λ . Θ(λ ) = Θ τO(s) λ+ q−p 2 1m

Let δ1 be a K1 -type of Θ(λ ). Obviously, δ1 must be isomorphic to τp

λ+ q−p 2 1m

some λ = (λ1 , . . . , λm ) in Λ(p, 0), and it has to lie in the K-type δ = τp Furthermore, the multiplicity of δ1 in Θ(λ ) is given by

μ

⊗ τO(r) for

λ ⊗ τO(q) of V .

μ λ λ dimC HomK1 ×K2 δ1 ⊗ τKλ 2 , δ = dimC HomO(r)×O(s) τO(r) ⊗ τO(s) . , τO(q)

(3)

By the branching rule stated after (1), the right-hand side is nonzero only if λi λi for all i m, and λi = 0 for all i > m. In particular Θ(λ ) is nonzero if and only if the number of nonzero integers in λ is not greater than (p − 1)/2, that is, t m. (If that is the case then λ can be viewed as a highest weight for so(p).) Moreover, the branching rule implies that λ + q−p 2 1m

W (λ ) = τp

⊗ CO(r)

appears in Θ(λ ) with multiplicity one and it is the (unique) minimal K1 -type of Θ(λ ). Let K10 = Spin(p) × SO(r) be the identity component of K1 . We can view Θ(λ ) as a (g1 , K10 )-module. The minimal K1 -type restricts irreducibly to K10 , and it is not hard to see that it becomes the unique minimal K10 -type of Θ(λ ).

188


We will now state [9, Theorem 9.1]. The use of disconnected K2 is crucial here. (Note that we have just proved the second part.) Theorem 2.1. Recall that g1 = so(p, r), K2 = O(s) and K10 = Spin(p) × SO(r). Let Θ(τ ) be the lift of an irreducible representation τ of K2 . Then (i) The (g1 , K10 )-module Θ(τ ) is either zero or irreducible. (ii) Suppose τ and τ are non-isomorphic irreducible representations of K2 , and suppose Θ(τ ) and Θ(τ ) are nonzero. Then the minimal K10 -types of Θ(τ ) and Θ(τ ) are non-isomorphic. In particular, Θ(τ ) and Θ(τ ) are non-isomorphic (g1 , K10 )-modules. 3. Cohomological induction The purpose of this section is to introduce cohomological induction and realize V in terms of the cohomological induction. 3.1. Notation We recall some basic definitions and notation from [6] and [14]. We use a subscript 0 to denote a real Lie algebra. Those without are complex Lie algebras. Consider a connected Lie group G. Let K 0 be a maximal compact subgroup. Let g0 and k0 be the Lie algebras of G and K 0 , respectively. Let θ be the Cartan involution of g0 fixing k0 . Let q = l + n be a θ -stable parabolic subalgebra of g. Let q denote its opposite parabolic subalgebra. Let L denote the corresponding 0 connected Lie subgroup top of G with Lie algebra l0 . If Z is an irreducible (l, L ∩ K )-module, then

n and we put Z = Z ⊗ g

indq¯ Z = U(g) ⊗q¯ Z. We will write ind Z if it is clear what g and q are. If Z has infinitesimal character λZ , then ind Z has infinitesimal character λZ + ρ(n). Let Li (Z) = Πi ind Z g,K 0

where Πi = (Πg,L∩K 0 )i is the ith derived functor of the Bernstein functor. If Z = Cλ is the onedimensional character of (l, L ∩ K 0 ), then we denote Aq (λ) = Ls0 (Cλ ) and it has infinitesimal character λ + ρ(g). Given a (g, K 0 )-module W , we set W h to be the subspace of K 0 -finite vectors in the conjugate linear dual vector space of W . Let s0 := dim(n ∩ k) and let Γ s0 be the s0 th derived functor of the Zuckerman functor of taking K 0 -finite vectors. By [6, Eq. (6.25)], Ls0 (Z)h = Γ s0 ((ind Z )h ). By [14, Theorem 6.3.5], there is a non-degenerate sesquilinear pairing between Γ s0 ((ind Z )h ) and Γ s0 (ind Z ). Hence if Γ s0 (ind Z ) is K 0 -admissible then Ls0 (Z) = Γ s0 ind Z . In this paper, we find it more convenient to work with Γ s0 (ind Z ) and ignore Ls0 (Z) completely. However we will state all final results in Ls0 (Z) because it is a more widely accepted definition.


189

3.2. A positive root system We now specialize to g = so(p, q) and K 0 = Spin(p) × SO(q). Recall that m = p−1 2 and = q2 . Let g0 and k0 be the real Lie algebras of G and K 0 , respectively. Choose a compact Cartan subalgebra h0 ⊆ k0 of g0 and positive root system Φ + with respect to h0 such that the simple roots εi −εi+1 for 1 i m−1 belong to so(p), and εi −εi+1 for m+1 i m+m −1 belong to so(q). The non-compact simple roots are εm − εm+1 and εm+m . √ Let λ0 = (1m , 0m ) ∈ −1h∗0 . Let q = l + n be the maximal parabolic subalgebra in g where l is spanned by roots perpendicular to λ0 . Then q is θ -stable. The Levi factor l corresponds to the subgroup m

L= U(m) × SO0 (1, q) in G. Here U(m) ⊆ Spin(p) is a two-fold cover of U(m) ⊆ SO(p). We note that the weights of finite-dimensional representations of U(m) which do not descend to U(m) can be identified with m-tuples of half-integers. The one-dimensional representation with the weight ( 12 , . . . , 12 ) is 1/2 denoted by detu(m) . Under the adjoint action of L, the radical n decomposes as n = Cm ⊗ C1+q ⊕

2 Cm

representation of U(m) and C1+q is the standard representation of where Cm is the standard 0 SO (1, q). The summand 2 (Cm ) is spanned by long roots εi + εj for 1 i < j m. These long roots and short roots εi for 1 i m are precisely all compact roots contained in n. It follows that s0 = dim(n ∩ k) =

m(m + 1) p 2 − 1 = , 2 8

and this number is independent of q. A maximal compact subgroup of L is L∩K 0 = U(m)×SO(q). However, since our considerations involve a disconnected group, we also need to consider a slightly larger group U(m) × O(q). We view C1+q , in the decomposition of n above, as a natural (so(1, q), O(q))-module. Then, as (l, U(m) × O(q))-modules,

top

n∼ = det u(m) ⊗ det m O(q) . q+m

The action of so(1, q) is, of course, trivial. Recall that if Z is an (l, U(m) × O(q))-module then, using the cohomological induction, Z gives rise to a (g, K 0 )-module Γ s0 (ind Z ). There are two important observations to be made here. First, since ind Z is already SO(q)-finite, the functor Γ s0 is simply the s0 th derived functor of the Zuckerman functor of taking Spin(p)-finite vectors. Using the definition and the treatment of Γ s0 in [14, Chapter 6], Γ s0 (ind Z ) can be computed by considering ind Z as an (so(p), U(m))-module. Furthermore since ind Z is an O(q)-module, and the action of so(p) commutes with the action of O(q), Γ s0 (ind Z ) is naturally an O(q)module. In other words, Γ s0 (ind Z ) extends to a (g, K)-module.

190


Let Z0 be a one-dimensional (l, U(m) × O(q))-module such that the action of so(1, q) ⊆ l is trivial and, as U(m) × O(q)-modules, − p+q

Z0 ∼ = det u(m)2 ⊗ det m O(q) .

(4)

We set M0 = Γ s0 (ind Z0 ). One easily checks that the infinitesimal character of M0 is μp,q , the infinitesimal character of V . Lemma 3.1. The (g, K)-module M0 is Spin(p)-admissible so M0 = Ls0 (Z0 ). It contains the Kq−p

1m

⊗ CO(q) with multiplicity one. The K-type W0 is also the minimal K 0 -type type W0 = τp 2 of M0 . In particular, M0 is nonzero. We will derive this lemma as a corollary of the proof of Theorem 4.2 in Section 4. Alternatively the lemma also follows from the Blattner formula (see [6, Theorem 5.64]). Since the K-type W0 appears in Ls0 (Z0 ) with multiplicity one, we define Ls0 (Z0 ) to be the unique irreducible (g, K)-subquotient of Ls0 (Z0 ) containing W0 . Proposition 3.2. The irreducible (g, K)-modules V and Ls0 (Z0 ) are isomorphic. Proof. Both representations have the same infinitesimal character μp,q and the minimal K 0 type W0 . We showed in [9] that V is the unique irreducible (g, K 0 )-module with infinitesimal q−p

1m

character μp,q and minimal K 0 -type τp 2 ⊗ CSO(q) . Hence the two modules are isomorphic (g, K 0 )-modules. There are two ways to extend V from a (g, K 0 )-module to a (g, K)-module. One differs from the other by the determinant character of O(q). Hence V and Ls0 (Z0 ) are the same because they have the same minimal K-type W0 . 2 4. Identifying Θ(λ ) Let r and s be two integers such that r + s = q. Choose a standard embedding of O(s) into O(q), the second factor of K. Let g1 ∼ = so(p, r) be the centralizer of O(s) in g. Note that g1 is θ -invariant. In this section we consider the restriction of V to (g1 , K10 ) × K2 where K10 = Spin(p) × SO(r) and K2 = O(s). Suppose λ = (λ1 , . . . , λs ) is in Λ(O(s)) such that Θ(λ ) in (2) is nonzero. Then by (3), λi = 0 if i > m = p−1 2 . In particular, λ can be considered in element in Λ(p, 0) by adding or removing some 0’s at the tail. The irreducible (g1 , K10 )-module Θ(λ ) has a unique minimal K10 -type λ + q−p 2 1m

W (λ ) = τp

⊗ CSO(r) .

Using the θ -stable parabolic q = l + n in g, we define q1 = q ∩ g1 . Write q1 = l1 + n1 . Then l1 corresponds to a subgroup L1 = U(m) × SO0 (1, r)


191

in G1 . For every λ such that Θ(λ ) = 0 (or equivalently λi = 0 for i > m) let Z(λ ) be an irreducible L1 -module such that the action of SO0 (1, r) is trivial and

q−p−2r 1m 2

λ+ Z(λ ) ∼ = τu(m)

(5)

as U(m)-modules. Set M(λ ) := Γ s0 (ind Z(λ ) ). We have explained in the previous section that we may take Γ s0 to be the s0 th derived functor of the Zuckerman functor of taking Spin(p)-finite vectors. Lemma 4.1. The (g1 , K10 )-module M(λ ) is Spin(p)-admissible so M(λ ) = Ls0 (Z(λ )). Any of its Spin(p)-type is isomorphic to λ + q−p 2 1m +κ

τp

where κ is an m-tuple of non-negative integers. The module M(λ ) contains the K10 -type W (λ ) with multiplicity one and it is the minimal K10 -type. We will prove Lemma 4.1 together with Theorem 4.2 below. One could also verify this lemma directly using the Blattner’s formula. Let Ls0 (Z(λ )) denote the unique irreducible subquotient of M(λ ) = Ls0 (Z(λ )) containing the minimal K10 -type W (λ ). We can now state the main result of this section. Theorem 4.2. The irreducible (g1 , K10 )-modules Θ(λ ) and Ls0 (Z(λ )) are isomorphic. In particular Ls0 (Z(λ )) is unitarizable and it has K10 -types given by the branching (3). Remarks. It is interesting to note that Ls0 (Z(λ )) is not always in the good or weakly good range (see [6, Definition 0.49]). Hence it may be reducible. It is of separate interest that the image of the bottom layer map induces an unitarizable subquotient. The infinitesimal character of L(Z(λ )) is

q − p − 2r λ + 1m , 0[ r+1 ] + ρp+r . 2 2

Hence Theorem 4.2 gives an alternative proof of the correspondence of infinitesimal characters of so(p, r) and so(s) [9, Theorem 1.2]. The rest of this section contains the proofs of Lemmas 3.1, 4.1 and Theorem 4.2. It is inspired by the work of [3], [15] and [16]. Recall that n1 ⊆ n. We have a decomposition n = n1 + n2 such that n2 = Cm ⊗ Cs is a tensor product of standard representations of U(m) and O(s), while the group SO0 (1, r) acts trivially on it. We extend n2 to a representation of U(m) × U(s). It is well known that (see [2] and [4]) Symn n2 =

μ

μ

μ

τU(m) ⊗ τU(s)

(6)

192


where the sum is taken over all partitions μ of n of length not longer than min(m, s). (So every such partition can be viewed as a highest weight for both U(m) and U(s).) We further restrict the μ summand τU(s) to O(s) μ

τU(s) =

λ τO(s) .

(7)

λ ↑μ

λ The notation λ ↑ μ simply means that τO(s) is a subrepresentation of τU(s) , and the sum is taken μ μ with multiplicities. Note that τO(s) appears in the restriction from τU(s) with multiplicity one. Using this notation, we get

Symn n2 =

μ

μ λ ↑μ

μ

λ τU(m) ⊗ τO(s)

(8)

as a sum of irreducible representations of U(m) × O(s).

We now recall the definition of Z0 from (4). One easily sees that the restriction of Z0 to L1 × O(s) is given by q−1

2 Z0 = det u(m) ⊗ Cso(1,r) ⊗ CO(s) .

Let symm : Sym(g) → U(g) denote the symmetrization map (see [14, §0.4.2 ]). By the Poincaré– Birkhoff–Witt theorem,

ind Z0 = U(n1 ) ⊗ symm Sym(n2 ) ⊗ Z0

(9)

as L1 × O(s)-modules. Let Sn (n2 ) = ni=0 Symi (n2 ). We define Fn to be the (g1 , L1 ∩ K10 )

submodule of ind Z0 generated by 1 ⊗ symm(Sn (n2 )) ⊗ Z0 . Hence {Fn : n = 0, 1, 2, . . .} forms

an exhaustive increasing filtration of g1 × O(s)-submodules of ind Z0 . We will now state a special case of a known fact which is used in proof of the Blattner formula in [6]. Lemma 4.3. For every positive integer n, we have an isomorphism of g1 × O(s)-modules Fn /Fn−1 =

μ λ ↑μ

q−1 )1m g μ+( λ indq¯ 11 τu(m) 2 ⊗ Cso(1,r) ⊗ τO(s)

λ where μ is any partition of n of length not more than min(m, s) and τO(s) is counted with multiμ plicity with which it appears in the restriction of τU(s) .

We shall use the filtration Fn to compute Γ s0 (ind Z0 ). Case 1. We first consider the filtration Fn in the case r = 0 and s = q. In particular, g1 = so(p). Put q−1 )1m g μ+( . V (μ) = indq¯ 11 τu(m) 2


193

μ+ q−p 1m

The infinitesimal character of V (μ) is the same as the infinitesimal of τp 2 . In particular, these infinitesimal characters are pairwise different for different partitions μ. It follows that the filtration Fn splits:

λ V (μ) ⊗ τO(q) . (10) ind Z0 = μ λ ↑μ

Here the first sum is taken over all partitions μ of length no more than m = μ counted with multiplicity with which it appears in τU(q) .

p−1 2 ,

λ and τO(q) is

Lemma 4.4. Let μ be a partition of length not more than m. Then V (μ) is an irreducible so(p)module. Proof. Since V (μ) is u(m)-finite generalized Verma module, any proper submodule of V (μ) must be a quotient of some V (μ ) where μ = μ . Note that the lowest u(m)-type τ of V (μ ) is a nonzero u(m)-type of V (μ). Let hm denote the maximal Cartan subalgebra of u(m). We claim that the highest weight of τ is of the form μ + ( q−1 2 )1m + κ where κ is sum of roots of n1 restricted to hm . Indeed by (9), μ+( q−1 2 )1m

V (μ) = Sym(n1 ) ⊗ τu(m)

as a u(m)-module. By [13, Proposition 3.2.12], an irreducible u(m)-module (in particular τ ) on the right-hand side of the above equation has highest weight μ + ( q−1 2 )1m + κ where κ is a hm -weight of Sym(n1 ). This proves our claim. The roots of n1 are of the form εi or εi + εj so κ is a m-tuple of non-negative integers. Since V (μ ) is proper, κ is nonzero. The infinitesimal characters of V (μ) and V (μ ) correspond to the q−p weights μ + q−p 2 1m + ρp and μ + 2 1m + κ + ρp , respectively, under the Harish-Chandra homomorphism. These two weights correspond to partitions of different lengths because the entries of μ, κ, ρp are non-negative, q > p and κ is nonzero. Hence V (μ) and V (μ ) do not have the same infinitesimal character, and V (μ ) cannot map to V (μ). The lemma is proved. 2 We recall the previous section that the Zuckerman functor Γ j is computed in the category of (so(p), U(m))-modules. If we apply Γ j to both sides of (10) then

λ Γ j ind Z0 = Γ j V (μ) ⊗ τO(q) . (11) μ λ ↑μ

Since s0 = dim(n1 ∩ k1 ) =

m(m+1) 2

=

p 2 −1 8 , by the Borel–Weil–Bott–Kostant theorem, μ+ q−p 1m τp 2 . The reader may recognize that we have

Γ j (V (μ)) = 0 if j = s0 and Γ s0 (V (μ)) = essentially followed the proof of the Blattner formula in [6] to compute K-types of Ls0 (Z0 ). Now we have the following conclusions:

μ+ q−p 2 1m

(A) Spin(p)-type of Γ s0 (ind Z0 ) is of the form τp

μ

with multiplicity given by dim τU(q) .

Therefore Γ s0 (ind Z0 ) is admissible with respect to Spin(p). This also follows from a very

general criterion in [7]. We now have Γ s0 (ind Z0 ) = Ls0 (Z0 ).

194

H.Y. Loke, G. Savin / Journal of Functional Analysis 255 (2008) 184–199 μ+ q−p 2 1m

(B) A K-type of Ls0 (Z0 ) is of the form τp

λ , where τ λ ⊗ τO(q) O(q) appears in the restriction q−p

μ

1m

⊗ CO(q) which occurs with from τU(q) . In particular, the minimal K-type is W0 = τp 2 multiplicity one. It is also the image of the bottom layer map. With this, we have proven Lemma 3.1. Case 2. Now we return to the general r for g1 = so(p, r). Consider the filtration Fn in this situation. We recall (5) and we abbreviate q−1 )1m g g μ+( L(μ) = indq¯ 11 Z(μ) = indq¯ 11 τu(m) 2 ⊗ Cso(1,r) . Then, Fn /Fn−1 is a direct sum of L(μ) where μ is a partition of n of length not more than min(m, s). By (10) and Lemma 4.4, L(μ) is a direct sum of various V (μ ), and L(μ) is a U(m))(so(p), U(m))-submodule of (10). Since Γ s0 is computed in the category of (so(p),

modules, Γ s0 (L(μ)) is a Spin(p)-submodule of Γ s0 (ind Z0 ). By Conclusion (A) in Case 1,

Γ s0 (ind Z0 ) is Spin(p)-admissible so Γ s0 (L(μ)) is Spin(p)-admissible and Γ s0 (L(μ)) = Ls0 (Z(μ)). This proves the first assertion of Lemma 4.1. In order to understand Spin(p)-types of Γ s0 (L(μ)), we must describe μ such that V (μ ) ⊆ L(μ). Lemma 4.5. If V (μ ) ⊆ L(μ), then μ = μ + κ where κ is an m-tuple of non-negative integers. Proof. The proof is similar to part of the proof of Lemma 4.4. Let hm denote the Cartan subalgebra of u(m). Let τ be the lowest u(m)-type of V (μ ). It has highest hm -weight μ + ( q−1 2 )1m . μ+( q−1 2 )1m

As a u(m)-module L(μ) = Sym(n2 ) ⊗ τu(m)

. Since τ is a u(m)-type in L(μ), by [13,

Proposition 3.2.12], the highest hm -weight of τ is of the form μ + ( q−1 2 )1m + κ where κ is a hm -weight of Sym(n1 ), i.e. sum of roots of n1 . Since the roots of n1 when restricted to hm are of the form εi or εi + εj , κ is an m-tuple of non-negative integers. 2 In addition, L(μ) contains a unique copy of V (μ) and SO(r) acts trivially on it. By the above μ+ q−p 2 1m +κ

lemma, the Spin(p)-types of Γ s0 (L(μ)) are τp integers, and the Lemma 4.1.

K10 -type

μ+ q−p 1m W (μ) = τp 2

, here κ is an m-tuple of non-negative

⊗ CSO(r) occurs with multiplicity one. This proves

Lemma 4.6. Let Fn be the filtration as in Lemma 4.3. Then Γ j (Fn ) = 0 if j = s0 . Furthermore we have an exact sequence 0 → Γ s0 (Fn−1 ) → Γ s0 (Fn ) → Γ s0 (Fn /Fn−1 ) → 0. Proof. Let F¯ n := Fn /Fn−1 . Then Fn and F¯ n are direct sums of V (μ)’s in (10). As explained in U(m))-modules. Hence the previous section, we may compute Γ j F¯ n in the category of (so(p), Γ j (Fn ) and Γ j (F¯ n ) are direct sums of Γ j (V (μ)) and we have shown that these are zeros if j = s0 . Finally we apply the functor Γ to the exact sequence 0 → Fn−1 → Fn → F¯ n → 0 to get the long exact sequence. The exact sequence in the lemma follows immediately.

2


195

Lemma 4.7. In the category of (g1 , K10 )-modules, Γ s0 (Fn ) is an exhaustive increasing filtration of Ls0 (Z0 ) and Γ s0 (Fn )/Γ s0 (Fn−1 ) = Γ s0 (Fn /Fn−1 ) =

λ Γ s0 L(μ) ⊗ τO(s) .

μ λ ↑μ

λ is Here the first sum is taken over all partitions of length no more than min(m, s) and τO(s) μ counted with multiplicity with which it appears in τU(s) .

Proof. This follows from Lemmas 4.6 and 4.3.

2

We are finally ready to prove Theorem 4.2, that is, compute Θ(λ ) where λ is in Λ(O(s)) of length not more than min(m, s). We define S(λ ) as the set of all partitions μ of length not more μ λ . Since V is an irreducible subquotient of L (Z ), than min(m, s) such that τU(s) contains τO(s) s0 0 it follows that Θ(λ ) is an irreducible subquotient of Ls0 (Z0 ), considered as (g1 , K10 )-module. It follows that Θ(λ ) is an irreducible subquotient of Γ s0 (L(μ)) for some μ in S(λ ). We now need the following lemma. λ + q−p 2 1m

Lemma 4.8. Let μ be in S(λ ). The K10 -type W (λ ) = τp if and only if μ = λ .

⊗ CSO(r) occurs in Γ s0 (L(μ))

Proof. We check Spin(p)-types. If W (λ ) is contained in Γ s0 (L(μ)) for some then, as we have μ λ , this just seen, λ = μ + (κ1 , . . . , κm ) where κi 0. On the other hand, since τU(s) contains τO(s) is possible only if μ = λ as desired. 2 Since Θ(λ ) contains W (λ ) the lemma implies that Θ(λ ) is an irreducible subquotient of s0 (Z(λ )). This proves Theorem 4.2.

Γ s0 (L(λ )) = L

5. The smallest representation V + In this section, we will extend Theorems 2.1 and 4.2 to representations V + and V − . Since the proofs are almost identical to those in the previous sections, we will only state the main results. q Let g = so(p, q) and K = O(p) × Spin(q). Recall that m = p−1 2 and m = 2 . Let g0 and k0 be the real Lie algebras of G and K 0 , respectively. Choose a compact Cartan subalgebra h0 ⊆ k0 of g0 and positive root system Φ + such that the simple roots εi − εi+1 for 1 i m − 1 belong to so(q) and, εi − εi+1 for m + 1 i m + m − 1 and εm +m belong to so(p). The non-compact simple root is εm − εm +1 . We refer√ to the notation on cohomological induction introduced in Section 3. We set λ0 = (1m , 0m ) ∈ −1h∗0 and we let q = l + n be the corresponding parabolic subalgebra. The algebra l corresponds to the subgroup L = SO(p) × U(m ) in G. We have s0 =

m (m −1) 2

=

q(q−2) . 8

Let Z0 be a one-dimensional O(p) × U(m )-module

−( p+q )

2 Z0 = det m O(p) ⊗ det u(m ) .

196


We consider the (g, K)-module Ls0 (Z0 ). It is equal to Aq (λ) where λ = − p+q 2 λ0 . The following is essentially a result of [5] and [12]. The only difference is that we consider K and not K 0 . See Section 6 for more details. Theorem 5.1. Recall that p > q and K = O(p) × Spin(q). p−q

1

(i) The minimal K-type of Ls0 (Z0 ) is W0 = CO(p) ⊗ τq 2 m and it occurs in Ls0 (Z0 ) with multiplicity 1. (ii) Let V + = Ls0 (Z0 ) denote the irreducible subquotient of Ls0 (Z0 ) generated by W0 . Then V + is an unitarizable (g, K)-module.

Remark. As in the case of V in Section 4, we work with Γ s0 (ind Z0 ) instead of Ls0 (Z0 ). Part of

the proof involves establishing the fact that Γ s0 (ind Z0 ) is K 0 -admissible so that Γ s0 (ind Z0 ) = Ls0 (Z0 ). The same applies to Ls0 (Z(λ )) in Theorem 5.3 below. The restriction of V + to K = O(p) × Spin(q) is

V+ =

λ+ p−q 2 1m

λ τO(p) ⊗ τq

.

λ∈Λ(q+1,0) p−2 p−4 1 + remains irreIts infinitesimal character is ( q2 , q−2 2 , . . . , 1, 2 , 2 , . . . , 2 ). The module V ducible as a (g, K 0 )-module. In [9] we call V + a smallest representation of the non-linear cover of SO(p, q), and there is also an outline of a construction of V + using Gelfand–Zetlin bases.

Remark. We note that by an outer automorphism action of the pair (so(p, q), K) on V + , we get another smallest representation V − . All the results in this paper on V + would immediately give corresponding results for V − via this outer automorphism. Therefore we will only work with V + . Choose a standard embedding of K2 = O(s) into O(p), the first factor of K. Let g1 ∼ = so(r, q) be the centralizer of O(s) in g. Note that g1 is θ -invariant. In this section we consider the restriction of V + to K2 × (g1 , K1 ) where K1 = O(r) × Spin(q). V+ =

λ τO(s) ⊗ Θ(λ ).

λ ∈Λ(O(s))

Since O(s) is compact, the right-hand side is a direct sum. Furthermore V + is admissible with respect to Spin(q), so Θ(λ ) is an admissible (g1 , K1 )-module. The K1 -types of Θ(λ ) can be computed using branching rules similar to (3). More precisely, μ

λ+ q−p 2 1m

suppose δ1 = τO(r) ⊗ τq λ+ q−p 2 1m

λ ⊗ τq τO(p)

is a K1 -type of Θ(λ ). Then δ1 has to lie in the K-type δ =

of V + . The multiplicity of δ in Θ(λ ) is given by

μ λ λ λ dimC HomK1 ×O(s) δ1 ⊗ τO(s) . , δ = dimC HomO(r)×O(s) τO(r) ⊗ τO(s) , τO(p)

(12)


197

By the right-hand side of (12), Θ(λ ) is nonzero if and only if nonzero entries of λ is not greater than q2 . The minimal K10 -type of Θ(λ ) is λ + p−q 2 1m

W (λ ) = CSO(r) ⊗ τq

.

(13)

We compare the next theorem with Theorem 2.1. Theorem 5.2. Recall that g1 = so(r, q), K2 = O(s) and K10 = SO(r) × Spin(q). Let Θ(τ ) be the lift of an irreducible representation τ of K2 . Then: (i) The (g1 , K10 )-module Θ(λ ) is either zero or irreducible. (ii) Suppose Θ(λ ) and Θ(η ) are nonzero. Then Θ(λ ) and Θ(η ) are isomorphic (g1 , K10 )modules if and only if λ = η . Part (i) follows the same argument as that of [9, Theorem 9.1]. We will omit the proof. Part (ii) is a consequence of (13) because if λ = η , then Θ(λ ) and Θ(η ) have distinct minimal K10 types. 5.1. Cohomological induction We would like to identify Θ(λ ) as a subquotient of a cohomological induced module. Suppose Θ(λ ) is nonzero. Then the number of nonzero entries in λ is not greater than m . Let q1 = q ∩ g1 be a theta-stable parabolic subalgebra of g1 . Its Levi subalgebra l1 corresponds to a subgroup U(m ) L1 = SO(r) × in G1 . Let Z(λ ) be an irreducible L1 -module which is trivial on SO(r) and such that the restriction to U(m ) is λ + p−q−2r 1m 2

Z(λ ) ∼ = τu(m )

.

We consider the cohomologically induced representation Ls0 (Z(λ )). Its minimal K10 -type is W (λ ) in (13) and it occurs with multiplicity one. Let Ls0 (Z(λ )) denote the unique irreducible (g1 , K10 )-subquotient of Ls0 (Z(λ )) containing W (λ ). The next theorem is proved in the same way as Theorem 4.2. Theorem 5.3. The irreducible (g1 , K10 )-modules Θ(λ ) and Ls0 (Z(λ )) are isomorphic. In particular, Ls0 (Z(λ )) is nonzero and unitarizable. 6. On results of Knapp and Trapa The aim of this section is to relate our results to some results of Knapp and Trapa. Assume that r is an integer and r q. For every non-negative integer s, Knapp [5] defined

198


an (so(r, q), K10 )-module πs as a certain (naturally unitarizable) subquotient of Aq (λ) where q = l + n, l = u(m ) + so(r) and λ=

s −r −q 1m , 0[ 2r ] . 2

The module πs contains the minimal K10 -type of Aq (λ). Trapa showed in [12] that πs is irreducible. We now focus our attention to non-negative integral values of s so that Aq (λ) is a faithful representation of K10 . This implies that s−r−q ∈ Z + 12 , that is, r + s is odd. 2 + Consider W = V and the dual pair (g1 , K1 ) × O(s) where g1 = so(r, q), K1 = O(r) × Spin(q) and p = r + s. Let Θ(0) denote the theta lift of the trivial representation of O(s). Then Θ(0) is an (so(r, q), K10 )-module. The next theorem follows from Theorem 5.3. Theorem 6.1. Let r and s be two positive integers such that r q and p = r + s is odd. Then the (so(r, q), K10 )-module Θ(0) is isomorphic to πs . We note that Knapp computed K10 -types of πs . His computation shows that K10 -types of πs coincide with K10 -types of Θ(0). Hence this paper gives an independent proof of the fact that πs is irreducible (see [12]). An interesting way to formulate the above result for odd r is as follows. Let π0 , π2 , . . . be ∼ Θ(0) where Θ(0) is the theta lift of the Knapp’s family for so(p, q) , where p > q. Then π2a = appear in trivial representation of O(2a). Again, we note that Paul and Trapa studied how π2a+1 the Howe correspondence [11]. Acknowledgments The first author would like to thank the hospitality of the Mathematics Department at University of Utah while the initial idea for this paper was first conceived. Likewise, the second author would like to thank the hospitality of the Mathematics Department at National University of Singapore during 2007 when this paper was completed. We would also like to thank Peter Trapa for some very insightful discussions, and the referee for pointing out a few gaps in a previous version. The first author was supported by an NUS grant R-146-000-085-112. The second author was supported by an NSF grant DMS 0551846. References [1] J. Adams, D. Barbasch, A. Paul, P. Trapa, D. Vogan, Shimura correspondences for split real groups, J. Amer. Math. Soc. 20 (2007) 701–751. [2] Roe Goodman, Nolan R. Wallach, Representations and Invariants of the Classical Groups, Cambridge Univ. Press, Cambridge, 1998, third corrected printing, 2003. [3] Benedict H. Gross, Nolan R. Wallach, On quaternionic discrete series representations, and their continuations, J. Reine Angew. Math. 481 (1996) 73–123. [4] Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, in: The Schur Lectures, 1992, Tel Aviv, in: Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. [5] Anthony Knapp, Nilpotent orbits and some unitary representations of indefinite orthogonal groups, J. Funct. Anal. 209 (2004) 36–100. [6] Anthony Knapp, David Vogan, Cohomological Induction and Unitary Representations, Princeton Univ. Press, Princeton, NJ, 1995.


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[7] T. Kobayashi, Discrete decomposability of the restriction of Aq (λ) with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (2) (1998) 229–256. [8] H.Y. Loke, Howe quotients of unitary characters and unitary lowest weight modules, Represent. Theory 10 (2006) 21–47. [9] H.Y. Loke, G. Savin, The smallest representations of non-linear covers of odd orthogonal groups, Amer. J. Math., in press. [10] K. Nishiyama, C.B. Zhu, Theta lifting of unitary lowest weight modules and their associated cycles, Duke Math. J. 125 (3) (2004) 415–465. [11] Annegret Paul, Peter E. Trapa, Some small unipotent representations of indefinite orthogonal groups and the theta correspondence, in: Univ. Aarhus Publ. Ser., vol. 48, Aarhus Univ., Aarhus, 2007, pp. 103–125. [12] Peter E. Trapa, Some small unipotent representations of indefinite orthogonal groups, J. Funct. Anal. 213 (2004) 290–320. [13] David Vogan, Representations of Real Reductive Lie Groups, Birkhäuser Boston, Boston, MA, 1981. [14] Nolan R. Wallach, Real Reductive Groups I, Academic Press, New York, 1988. [15] Nolan R. Wallach, Transfer of unitary representations between real forms, in: Representation Theory and Analysis on Homogeneous Spaces, New Brunswick, NJ, 1993, in: Contemp. Math., vol. 177, Amer. Math. Soc., Providence, RI, 1994, pp. 181–216. [16] Nolan R. Wallach, C.-B. Zhu, Transfer of unitary representations, in: Special Issue in Memory of A. Borel, Asian J. Math. 8 (4) (2004) 861–880.


On non-equilibrium stochastic dynamics for interacting particle systems in continuum Yuri Kondratiev a,b , Oleksandr Kutoviy a,∗ , Robert Minlos c a Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33615 Bielefeld, Germany b Research Center BiBoS, Universität Bielefeld, 33615 Bielefeld, Germany c IITP, Russian Academy of Science, Moscow, Russia

Received 26 November 2007; accepted 11 December 2007 Available online 1 February 2008 Communicated by Paul Malliavin

Abstract We propose a general scheme for construction of Markov stochastic dynamics on configuration spaces in continuum. An application to the Glauber-type dynamics with competitions is considered. © 2007 Elsevier Inc. All rights reserved. Keywords: Configuration space; Glauber dynamics; Non-equilibrium Markov process

1. Introduction Interacting particle systems (IPS) is a large and growing area of probability theory and infinitedimensional analysis which is devoted to the study of certain models that arise in statistical physics, biology, economics, etc. Most of the results in the theory of IPS are related to the study of the so-called lattice systems and their Markov stochastic evolutions. In such systems the spatial structure of the considered model is presented by a lattice (or an infinite graph). Considered processes are usually specified by transition rates and associated Markov generators. The existence problem for the corresponding Markov process on the lattice configuration space can be solved positively under quite general assumptions about the transition rates, see e.g. [22]. * Corresponding author.

E-mail addresses: [email protected] (Y. Kondratiev), [email protected] (O. Kutoviy), [email protected] (R. Minlos). 0022-1236/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2007.12.006

Y. Kondratiev et al. / Journal of Functional Analysis 255 (2008) 200–227

201

Comparing with the lattice case, the situation with Markov stochastic dynamics for IPS in continuum is essentially different. In particular, it is true for an important class of birth-and-death processes in continuum (or so-called spatial birth-and-death processes). To this class belongs the Glauber type dynamics in continuum which are under active consideration, see [3,29]. Another class of interesting stochastic processes is formed by the Kawasaki type dynamics in continuum [19] and gradient diffusions [1,17]. Most of the results we have up to now for these processes are related to the equilibrium case (via the Dirichlet forms approach) [13,18] or to the processes in bounded domains (see, e.g., [8,24]). The situation with the non-equilibrium case is much pure. In particular, non-equilibrium spatial birth-and-death processes were constructed recently by Garcia and Kurtz for a special class of transition rates using techniques of stochastic differential equations [6] and a graphical construction was applied in [5]. Note that in both mentioned papers the death rate was considered to be a constant and the latter plays an essential technical role. A continuous version of the lattice contact model was analyzed in [14]. In contrast to the lattice case, constructions of the stochastic dynamics in continuum show essential difference between the Markov processes and Markov functions concepts. The latter notion (due to E. Dynkin) concerns the case of the processes with given initial distributions contrary to the more usual initial points framework. This weaker notion of the Markov function is not so essential in the lattice models because corresponding Markov processes can be constructed (typically) under very general assumptions. A principal role of dynamics with given classes of initial distributions was clarified at first for (deterministic) Hamiltonian dynamics in continuum, see e.g. [4]. In the present paper the role of the Markov functions approach is clarified for an infinite particle stochastic dynamics in continuum. This approach is based on the study of the corresponding (dual) Kolmogorov equation on measures. Such equation can be transported to an equation for corresponding correlation functions. Typically, this correlation functions equation does not admit a direct perturbation theory approach. In fact, the main technical observation made in the paper is related to the consideration of its dual time evolution on the so-called quasi-observables. This approach appeared for the first time in the literature on stochastic IPS in continuum in our paper [16] in the particular case of a Glauber-type dynamics. The idea to move the dynamics to a proper quasi-observables space (as well as the notion of quasi-observables itself) follows naturally from the concepts of harmonic analysis on configuration spaces, see e.g. [11]. We apply a perturbation technique to this dynamics in proper weighted L1 -spaces of functions on finite configurations and produce time evolutions of correlation functions as a dual object. The choice of corresponding weights gives precise description of the class of admissible initial distributions for our processes. One should emphasize also another principal moment of the paper. Namely, even if we have constructed a time evolution of the correlation functions, we need to show that they correspond to a time evolution of measures. In fact, this point is hidden in several works in statistical physics concerning BBGKY-hierarchy, etc. A rigorous mathematical analysis of this problem is based on a proper concept of positive definiteness of the correlation functions which was developed in [2,11]. The power of the described general scheme we illustrate by the application to a particular model of Glauber-type stochastic dynamics in continuum. In this process the birth of points is independent and uniformly distributed in space. Without a death part, the density of the system will grow to the infinity with the time. To prevent such unbounded growth we can introduce a self-regulation in the model. The latter can be done in several ways and one of them is to introduce the competition between points via a proper death rate. This competition (via density

202


dependent mortality in terminology of the spatial ecology), we choose in such a way that a Gibbs measure on the configuration space becomes a symmetrizing measure for the considered generator. Note that such type of stochastic dynamics with competition may be also realized as a proper framework for individual based models of complex socio-economic systems. Considered Glauber dynamics has unbounded death rate. Therefore, all known results in this field cannot be applied for the construction of the corresponding Markov process. In the present work, we have constructed a family of Markov functions for the Glauber dynamics with a competition corresponding to a class of initial distributions explicitly defined in the paper and depending on the interaction potential. This place needs an additional explanation: for the continuous IPS it would be too much to expect the existence of the stochastic dynamics for arbitrary initial distribution. The latter is wrong even for the systems without interactions, see e.g. [18]. The class of admissible initial distributions shows “how far” from the a priori reversible state an initial distribution can be chosen to be able to prevent an explosion in the dynamics. Note that the right scale of the deviations from the equilibrium state is an important technical problem for several models of continuous infinite particle dynamics (stochastic or deterministic ones). 2. Foundations We consider the Euclidean space Rd . By B(Rd ) we denote the family of all Borel sets in Rd . Bb (Rd ) denotes the system of all sets in B(Rd ) which are bounded. The space of n-point configuration is (n)

Γ0

(n) = Γ0,Rd := η ⊂ Rd |η| = n ,

n ∈ N0 := N ∪ {0},

where |A| denotes the cardinality of the set A. (n) (n) (n) The space ΓΛ = Γ0,Λ for Λ ∈ Bb (Rd ) is defined analogously to the space Γ0 . As a set, Γ0(n) is equivalent to the symmetrization of n n Rd = (x1 , . . . , xn ) ∈ Rd xk = xl if k = l , d )n /S , where S is the permutation group of {1, . . . , n}. Hence, one can introduce i.e. to the (R n n the corresponding topology and Borel σ -algebra, which we denote by O(Γ0(n) ) and B(Γ0(n) ), respectively. The space of finite configurations Γ0 :=

(n)

Γ0

n∈N0

is equipped with the topology O(Γ0 ) of disjoint union. Let B(Γ0 ) denotes the corresponding Borel σ -algebra. A set B ∈ B(Γ0 ) is called bounded if there exists Λ ∈ Bb (Rd ) and N ∈ N such that (n) B⊂ N n=0 ΓΛ . The configuration space Γ := γ ⊂ Rd |γ ∩ Λ| < ∞, for all Λ ∈ Bb Rd


203

is equipped with the vague topology O(Γ ). It is a Polish space (see e.g. [12]). B(Γ ) denotes the corresponding Borel σ -algebra. The filtration on Γ with a base set Λ ∈ Bb (Rd ) is given by BΛ (Γ ) := σ NΛ Λ ∈ Bb Rd , Λ ⊂ Λ , where NΛ : Γ0 → N0 is such that NΛ (η) := |η ∩ Λ|. For short we write ηΛ := η ∩ Λ. (n) For every Λ ∈ Bb (Rd ) the projection pΛ : Γ → ΓΛ := n0 ΓΛ is defined as pΛ (γ ) := γΛ . One can show that Γ is the projective limit of the spaces {ΓΛ }Λ∈Bb (Rd ) with respect to this projections. In the sequel we will use the following classes of function on Γ0 : • L0 (Γ0 )—the set of all measurable functions on Γ0 ; • L0ls (Γ0 )—the set of measurable functions with local support, i.e. G ∈ L0ls (Γ0 ) if there exists Λ ∈ Bb (Rd ) such that G Γ0 \ΓΛ = 0; • L0bs (Γ0 )—the set of measurable functions with bounded support, i.e. G ∈ L0bs (Γ0 ) if there exists Λ ∈ Bb (Rd ) and N ∈ N such that G Γ \ N Γ (n) = 0; 0 n=0 Λ • B(Γ0 )—the set of bounded measurable functions; • Bbs (Γ0 )—the set of bounded functions with bounded support; Λ (Γ ), Λ ∈ B (Rd )—the set of function from B (Γ ), whose support is a subset of Λ; • Bbs 0 b bs 0 Λ (Γ )—the set of continuous functions from B • CBΛ 0 bs bs (Γ0 ). On Γ we consider the set of cylinder functions FL0 (Γ ), i.e. the set of all measurable functions G ∈ L0 (Γ ) which are measurable with respect to BΛ (Γ ) for some Λ ∈ Bb (Rd ). These functions are characterized by the following relation: F (γ ) = F ΓΛ (γΛ ). Those cylinder functions which are measurable with respect to BΛ (Γ ) for fixed Λ ∈ Bb (Rd ) we will denote by FL0 (Γ, BΛ (Γ )). Next we would like to describe some facts from the harmonic analysis on the configuration space based on [11]. The following mapping between functions on Γ0 and functions on Γ plays the key role in our further considerations: KG(γ ) :=

G(ξ ),

G ∈ L0ls (Γ0 ), γ ∈ Γ,

ξ γ

see e.g. [20,21]. The summation in the latter expression is taken over all finite subconfigurations of γ , which is denoted by symbol ξ γ . K-transform is linear, positivity preserving, and invertible, with K −1 F (η) :=

(−1)|η\ξ | F (ξ ), ξ ⊂η

F ∈ FL0 (Γ ), η ∈ Γ0 .

(1)

204


It is easy to see that for any Λ ∈ Bb (Rd ) and arbitrary F ∈ FL0 (Γ, BΛ (Γ )), K −1 F (η) = 1ΓΛ (η)K −1 F (η),

∀η ∈ Γ0 .

(2)

The map K, as well as the map K −1 , can be extended to more wide classes of functions. For details and further properties of the map K see, e.g., [11]. One can introduce a convolution : L0 (Γ0 ) × L0 (Γ0 ) → L0 (Γ0 ), G1 (ξ1 ∪ ξ2 )G2 (ξ2 ∪ ξ3 ), (G1 , G2 ) → (G1 G2 )(η) :=

(3)

(ξ1 ,ξ2 ,ξ3 )∈P∅3 (η)

where P∅3 (η) denotes the set of all partitions (ξ1 , ξ2 , ξ3 ) of η in 3 parts, i.e., all triples (ξ1 , ξ2 , ξ3 ) with ξi ⊂ η, ξi ∩ ξj = ∅ if i = j , and ξ1 ∪ ξ2 ∪ ξ3 = η. It has the property that for G1 , G2 ∈ L0ls (Γ0 ) K(G1 G2 ) = KG1 · KG2 . Due to this convolution we can interpret the K-transform as the Fourier transform in configuration space analysis, see also [2]. Let M1fm (Γ ) be the set of all probability measures μ which have finite local moments of all orders, i.e.

|γΛ |n μ(dγ ) < +∞ Γ

for all Λ ∈ Bb (Rd ) and n ∈ N0 . A measure ρ on Γ0 is called locally finite if ρ(A) < ∞ for all bounded sets A from B(Γ0 ). The set of such measures is denoted by Mlf (Γ0 ). A measure ρ ∈ Mlf (Γ0 ) is called positive definite if

(G G)(η)ρ(dη) 0,

∀G ∈ Bbs (Γ0 ),

Γ0

where G is a complex conjugate of G. A measure ρ is called normalized if and only if ρ({∅}) = 1. One can define a transform K ∗ : M1fm (Γ ) → Mlf (Γ0 ), which is dual to the K-transform, i.e., for every μ ∈ M1fm (Γ ), G ∈ Bbs (Γ0 ) we have

KG(γ )μ(dγ ) =

Γ

Γ0

G(η)(K ∗ μ)(dη).


205

The measure ρμ := K ∗ μ is called the correlation measure of μ. As shown in [11] for μ ∈ M1fm (Γ ) and any G ∈ L1 (Γ0 , ρμ ) the series

KG(γ ) :=

G(η),

(4)

ηγ

is μ-a.s. absolutely convergent. Furthermore, KG ∈ L1 (Γ, μ) and

G(η)ρμ (dη) =

Γ0

(KG)(γ )μ(dγ ).

(5)

Γ

Fix a non-atomic and locally finite measure σ on (Rd , B(Rd )). For any n ∈ N the product d )n and hence on Γ (n) . The measure σ ⊗n can be considered by restriction as a measure on (R 0 (n) measure on Γ0 we denote by σ (n) . The Lebesgue–Poisson measure λzσ on Γ0 is defined as λzσ :=

∞ n z n=0

n!

σ (n) .

Here z > 0 is the so-called activity parameter. The restriction of λzσ to ΓΛ will be also denoted by λzσ . We write λz instead of λzσ , if the measure σ is considered to be fixed. The Poisson measure πzσ on (Γ, B(Γ )) is given as the projective limit of the family of meaΛ} Λ Λ −zσ (Λ) λ . sures {πzσ zσ Λ∈Bb (Rd ) , where πzσ is the measure on ΓΛ defined by πzσ := e A measure μ ∈ M1fm (Γ ) is called locally absolutely continuous with respect to πzσ iff μΛ := −1 Λ = π ◦ p −1 for all Λ ∈ B (Rd ). In this case, μ ◦ pΛ is absolutely continuous with respect to πzσ zσ b Λ ∗ ρμ := K μ is absolutely continuous with respect to λzσ . Let kμ : Γ0 → R+ be the corresponding Radon–Nikodym derivative, i.e. kμ (η) :=

dρμ (η), dλzσ

η ∈ Γ0 .

Remark 2.1. The functions

kμ(n) (x1 , . . . , xn ) :=

kμ(n) : (Rd )n → R+ , kμ ({x1 , . . . , xn }), 0,

d )n , if (x1 , . . . , xn ) ∈ (R otherwise

(6)

are well-known correlation functions in statistical physics, see e.g. [27, 28]. Next, we recall the theorem about characterization of correlation measures (or correlation functions).

206


Theorem 2.1. (Cf. [2,11].) Let ρ ∈ Mlf (Γ0 ) be given. Assume that ρ is positive definite, normalized and that for each bounded open Λ ⊂ Rd , for every C > 0 there exists DΛ,C > 0 such that (n) ρ ΓΛ DΛ,C C n ,

n ∈ N0 .

Then, there exists a unique measure μ ∈ M1fm (Γ ) with ρ = K μ. Remark 2.2. A sufficient condition for the bound in the theorem has the following form: for each bounded open Λ ⊂ Rd there exist εΛ > 0 and CΛ > 0 such that (n) ρ ΓΛ (n!)−εΛ (CΛ )n .

(7)

For the technical purposes we also recall the following result. Lemma 2.1. Let n ∈ N, n 2, and z > 0 be given. Then

... Γ0

G(η1 ∪ · · · ∪ ηn )H (η1 , . . . , ηn ) dλzσ (η1 ) . . . dλzσ (ηn )

Γ0

=

G(η) Γ0

H (η1 , . . . , ηn ) dλzσ (η)

(η1 ,...,ηn )∈Pn (η)

for all measurable functions G : Γ0 → R and H : Γ0 × · · · × Γ0 → R with respect to which both sides of the equality make sense. Here Pn (η) denotes the set of all ordered partitions of η in n parts, which may be empty. This lemma is known in the literature as Minlos lemma (cf. [15,23]) and it will be crucial for calculations in many places proposed in the next sections. 3. General approach to the construction of non-equilibrium dynamics for interacting particle systems (IPS) In this section we investigate the existence problem for non-equilibrium Markov processes of IPS in continuum. The mechanism of an evolution of IPS on Γ , which we would like to study, is formally described by the heuristically given generator L, defined on some proper domain of functions on Γ . The problem of construction of the corresponding process in Γ , in mathematically rigorous sense, is related to the problem of construction of a semigroup associated with L on a functional space over Γ . The latter problem in its turn concerns the possibility to find a solution to the Kolmogorov equation, which corresponds to the generator of this semigroup. Formally (only in the sense of action of operator), it has the form dFt = LFt , dt Ft |t=0 = F0 .


207

In the most interesting cases the construction of the corresponding semigroup on functional spaces on Γ seems to be a very difficult question. This difficulty is mostly related to the complex structure of infinite-dimensional space Γ . In this section we propose an alternative way for the construction of the corresponding dynamic which uses deeply the harmonic analysis technique described in the previous section. Let := K −1 LK L be the formal K-transform image of L or symbol of the operator L. This object we consider as the starting point on the way to the mathematically rigorous description of the model. Let : Γ0 → R+ be an arbitrary and fixed positive function, such that

(η) C |η| , for some C > 0. We consider : D(L) ⊂ L( ) → L( ) L in the Banach space L = L( ) := L1 (Γ0 , dλ1 ), where λ1 is the Lebesgue–Poisson measure with parameters z = 1 and σ is the Lebesgue measure on Rd . One should emphasize, that the Banach space L has a Fock space structure: ∞

(n) L1 Γ0 , (n) σ (n) ,

n=0 (n)

where (n) is the nth component of the function on Γ0 . plays the crucial role in our technique. The following condition on the operator L D(L)) is a generator of a C0 -semigroup in L( ), which will be denoted Assumption 3.1. (L, t , t 0. by U t , t 0, gives the solution Gt = U t G0 to the following evolutional Remark 3.1. The semigroup U equation for the operator L in the Banach space L( ): dGt = LGt , dt Gt |t=0 = G0 .

208


t , t 0, gives the possibilThe functional evolution on L(ρ) constructed via the semigroup U ity to construct the corresponding evolution of locally finite measures on Γ0 . In order to do this we consider the dual space K( ) := k : Γ0 → R k · −1 ∈ L∞ (Γ0 , λ1 ) to the Banach space L( ). The duality is given by the following expression

G, k := G · k dλ1 , G ∈ L( ).

(8)

Γ0

It is clear that K( ) is the Banach space with the norm k := k −1 L∞ (Γ

0 ,λ1 )

.

Note also, that k · −1 ∈ L∞ (Γ0 , λ1 ) means that the function k satisfies the bound k(η) const (η), λ1 -a.e. t , t 0, is constructed in the following way: The evolution on K( ), which corresponds to U t G, k. G, kt := U We denote t k := kt . U t , t 0, is a semigroup on the Banach space K( ). But it is not necessarily Remark 3.2. U a C0 -semigroup. The continuity in 0 of a L∞ -semigroups implies the boundness of the corresponding generators, which is not necessarily the case in our situation. Let k ∈ K( ) be a correlation function of some measure μ ∈ M1 (Γ ), where M1 (Γ ) denotes the class of all probability measures on Γ . Let t k, kt := U

t 0,

be the corresponding evolution of the function k in time. In order to say that there exists the corresponding evolution of probability measures on Γ we assume Assumption 3.2. For any t 0, kt ∈ K(ρ) is a positive definite, normalized function. We set M1ρ (Γ ) := μ ∈ M1 (Γ ) kμ const · ρ, λ1 -a.e. . Under Assumption 3.2, due to Theorem 2.1 about the characterization of correlation measures, one can easily construct a time evolution of the measures on M1ρ :


209

k → μ, t k → μt , kt = U

t 0,

Ut μ := μt ∈ M1ρ . Remark 3.3. It is not difficult to see that Ut , t 0, is a semigroup. But, of course, not necessarily a C0 -semigroup. Remark 3.4. Suppose that the operator L is a generator of a semigroup on some functional space on Γ . Suppose also that it is possible to define the adjoint operator L to the operator L on M1 (Γ ). Then, the constructed above semigroup Ut , t 0, determines the solution μt := Ut μ0 to the dual Kolmogorov equation for the operator L : ∂μt = L μ t , ∂t μt |t=0 = μ0 ∈ M1ρ . Theorem 3.1. Suppose that Assumptions 3.1, 3.2 are satisfied. Then, for any μ ∈ M1ρ , there μ exists a Markov process (Xt )t0 on the configuration space Γ with the initial distribution μ associated with the generator L. Proof. Let n ∈ N, A1 , . . . , An ∈ B(Γ ) and the moments of time 0 t1 · · · tn be arbitrary and fixed. Then there exists a process, defined on some probability space (Ω, F, P ), the finitedimensional distribution of which is given by the following formula: μ μ P Xt1 ∈ A1 , . . . , Xtn ∈ An =

1An Utn −tn−1 . . . Ut2 −t1 1A1 Ut1 μ (dγ ),

Γ

where for A ∈ B(Γ ) and t 0 the measure 1A Ut μ on Γ is defined by

1A Ut μ(S) :=

1A (γ )Ut μ(dγ ),

S ∈ B(Γ ).

S

Moreover, 1A Ut μ ∈ M1ρ since the indicator function of each A ∈ B(Γ ) is bounded by 1. Eventually, we have constructed the non-equilibrium Markov process. 2 4. Application to the Glauber dynamics with competition The approach proposed in the previous section was successfully applied to a special class of Glauber dynamics on Γ with the birth rate equal to a constant (see [16]). Below we study the model with the death rate equal to some unbounded function, that makes it impossible to apply the approaches developed by [5] and [6]. In applications this death rate may be considered as reflection of the competition between the particles of the system.

210


4.1. Potential and Gibbs measures on configuration spaces A pair potential is a Borel, even function φ : Rd → R ∪ {+∞}. Below we list some standard conditions on φ, known from statistical physics: (S) (Stability) There exists B > 0 such that, for any η ∈ Γ0

E(η) :=

φ(x − y) −B|η|.

{x,y}⊂η

Notice that the stability condition implies that the potential φ is semi-bounded from below. (I) (Integrability) For any β > 0,

1 − exp −βφ(x) dx < ∞.

C(β) := Rd

(SI) (Strong integrability) For any β > 0,

Cst (β) :=

1 − exp βφ(x) dx < ∞.

Rd

(P) (Positivity) φ(x) 0 for all x ∈ Rd . For γ ∈ Γ and x ∈ Rd \ γ we define the relative energy of interaction as follows: E(x, γ ) :=

y∈γ

φ(x − y),

+∞,

if y∈γ |φ(x − y)| < ∞, otherwise.

The energy of the configuration η ∈ Γ0 , or the Hamiltonian E φ : Γ0 → R ∪ {+∞}, which corresponds to the potential φ, is defined by E φ (η) =

φ(x − y),

η ∈ Γ0 , |η| 2.

{x,y}⊂η φ

The Hamiltonian EΛ : ΓΛ → R for Λ ∈ Bb (Rd ), which corresponds to the potential φ, is defined by φ

EΛ (η) =

φ(x − y),

η ∈ ΓΛ , |η| 2.

{x,y}⊂η φ

For fixed φ we will write for short E = E φ and EΛ = EΛ . For given γ¯ ∈ Γ we define the interaction energy between η ∈ ΓΛ and γ¯Λc = γ¯ ∩ Λc , Λc = d R \ Λ: WΛ (η|γ¯ ) =

x∈η, y∈γ¯Λc

φ(x − y).


211

The interaction energy is said to be well defined if for any Λ ∈ Bb (Rd ), η ∈ ΓΛ and γ¯ ∈ Γ it is finite or +∞. For β > 0 we define EΛ (η|γ¯ ) = EΛ (η) + WΛ (η|γ¯ ) and

ZΛ (γ¯ ) :=

exp −βEΛ (η|γ¯ ) λz (dη)

ΓΛ

the so-called partition function. Let Λ ∈ Bb (Rd ), β > 0, be arbitrary, and let γ¯ ∈ Γ . The finite volume Gibbs measure on the space ΓΛ with the boundary configuration γ¯ is defined by PΛ,γ¯ (dη) =

exp {−βEΛ (η|γ¯ )} λz (dη). ZΛ (γ¯ )

Let {πΛ } denote the specification associated with z and the Hamiltonian E (see [25]) which is defined by

πΛ,γ¯ (A) = PΛ,γ¯ (dη) A

where A = {η ∈ ΓΛ : η ∪ (γ¯Λc ) ∈ A}, A ∈ B(Γ ) and γ¯ ∈ Γ . A probability measure μ on Γ is called a Gibbs measure for E and z if μ πΛ,γ¯ (A) = μ(A) for every A ∈ B(Γ ) and every Λ ∈ Bb (Rd ). This relation is the well-known (DLR)-equation (Dobrushin–Lanford–Ruelle equation), see [7] for more details. The set of all Gibbs measures, which correspond to the potential φ, activity parameter z > 0, and inverse temperature β > 0, will be denoted by G(φ, z, β). For a fixed potential φ we will write G(z, β) instead of G(φ, z, β). 4.2. Glauber type dynamics. Generator and the corresponding symbol on the space of finite configurations According to the general scheme the mechanism of an evolution of configurations in Γ should be specified by some formally given generator. The action of such generator in the case of Glauber type dynamics has the following form:

− d(x, γ \ x)Dx F (γ ) + b(x, γ )Dx+ F (γ ) dx, (LF )(γ ) := (Lb,d )F (γ ) = x∈γ

Rd

where Dx− F (γ ) = F (γ \ x) − F (γ ) and Dx+ F (γ ) = F (γ ∪ x) − F (γ ).

212


It is known that the Gibbs measure μ ∈ G(z, β) is reversible with respect to the Markov process associated with L (i.e. the operator L is symmetrical in L2 (Γ, μ)) if and only if the following condition on coefficients b and d (birth and death rates) is fulfilled: b(x, γ ) = ze−βE(x,γ ) d(x, γ ).

(9)

In the sequel we will be interested only in the models with birth and death rates of the form • Glauber dynamics (G+ ): b(x, γ ) = ze−βE(x,γ ) ,

d(x, γ ) = 1.

Such model was investigated by many authors, see e.g. [13,15,16]. As it was mentioned before, in the first paper the authors used the particular realization of the general approach proposed in the present work. Under conditions (I) and (P), the non-equilibrium Glauber type dynamics on the configuration space Γ was constructed. In the present paper we consider another example of the Glauber type dynamics • Glauber dynamics (G− ): b(x, γ ) = z,

d(x, γ ) = eβE(x,γ ) .

The generator which corresponds to (G− ) we denote by the same symbol L. Remark 4.1. In the case, when E(x, γ ) is given via a potential with a positive part, the death rate of the operator L will be unbounded. In the considered model, the death rate reflects a competition between points in the configuration. In the spatial ecology models such a case is related to a density dependent mortality notion. For the technical reasons we will be also interested in the model with the birth and death rates localized in some volume Λ ∈ Bb (Rd ): bΛ (x, γ ) = z1Λ (x),

dΛ (x, γ ) = 1Λ (x)eβE(x,γΛ ) .

The corresponding operator we denote by LΛ . 4.3. Symbol of the operator L Let us consider the operator L on functions FL0 (Γ, BΛ (Γ )). One can easily check that this operator has the Markov property (it satisfies the maximum principle for the generators of Markov semigroups). Therefore, one may think about this operator as about Markov pregenerator. Proposition 4.1. The image of L under the K-transform (or symbol of L) on functions G ∈ Bbs (Γ0 ) is given by


213

(LG)(η) := K −1 LKG (η) = −G(η) eβφ(x−y) x∈η y∈η\x

−

G(ξ )

ξ ⊂η, ξ =η

eβφ(x−y)

x∈ξ y∈ξ \x

eβφ(x−y) − 1 + G(η ∪ x)z dx.

y∈η\ξ

Rd

we have Proof. According to the definition of the operator L (LG)(η) = K −1

eβE(x,·\x) Dx− KG(·) + z

x∈·

=

(−1)|η\ξ |

ξ ⊂η

+z

eβE(x,ξ \x) Dx− KG(ξ )

|η\ξ |

(−1)

ξ ⊂η

Dx+ KG(·) dx (η)

Rd

x∈ξ

Dx+ KG(ξ ) dx.

(10)

Rd

At the beginning we transform the first expression in the sum (10) I1 G(η) :=

(−1)|η\ξ |

ξ ⊂η

=

eβE(x,ξ \x) Dx− KG(ξ )

x∈ξ

(−1)|η\ξ |

ξ ⊂η

eβE(x,ξ \x)

G(ρ) −

ρ⊂ξ \x

x∈ξ

G(ρ)

ρ⊂ξ

=− (−1)|η\ξ | eβE(x,ξ \x) G(ρ ∪ x). ξ ⊂η

ρ⊂ξ \x

x∈ξ

Using the definitions of the K-transform and its inverse mapping we obtain I1 G(η) = −

(−1)|η\ξ | eβE(x,ξ \x) K G(· ∪ x) (ξ \ x) ξ ⊂η

=−

x∈ξ

(−1)|η\(ξ ∪x)| eβE(x,ξ ) K G(· ∪ x) (ξ )

x∈η ξ ⊂η\x

=−

|(η\x)\ξ |

(−1)

βφ(x−y) e K − 1 (ξ )K G(· ∪ x) (ξ )

x∈η ξ ⊂η\x

y∈·\x

βφ(x−y) −1 K =− e K − 1 K G(· ∪ x) (η \ x). x∈η

y∈·\x

For any measurable function f on Rd we denote eλ (f, η) :=

x∈η

f (x),

η ∈ Γ0 .

214


A direct application of the definition of the convolution yields eλ eβφ(x−·) − 1 G(· ∪ x) (η \ x)

I1 G(η) = −

x∈η

=−

eλ eβφ(x−·) − 1 (ξ1 ∪ ξ2 )G(ξ2 ∪ ξ3 ∪ x)

x∈η (ξ1 ,ξ2 ,ξ3 )∈P3 (η\x)

=−

eλ eβφ(x−·) − 1 (ξ1 ∪ ξ2 )G(ξ2 ∪ ξ3 ∪ x).

x∈η ξ1 ⊂η\x (ξ2 ,ξ3 )∈P2 (η\(ξ1 ∪x))

Picking out from the last expression the item which corresponds to ξ1 = ∅ we get I1 G(η) = −

G(η)eλ eβφ(x−·) − 1 (ξ2 )

x∈η (ξ2 ,ξ3 )∈P2 (η\x)

−

x∈η

ξ1 ⊂η\x, ξ1 =∅

eλ eβφ(x−·) − 1 (ξ1 ∪ ξ2 )G(ξ2 ∪ ξ3 ∪ x). (11)

(ξ2 ,ξ3 )∈P2 (η\(ξ1 ∪x))

Changing the summation in (11) we have I1 G(η) = −G(η) −

eλ eβφ(x−·) − 1 (ξ )

x∈η ξ ⊂η\x

x∈η

eλ eβφ(x−·) − 1 (η \ x) \ (ξ2 ∪ ξ3 ) ∪ ξ2 G(ξ2 ∪ ξ3 ∪ x).

(ξ2 ,ξ3 )∈P2 (η\x), ξ1 ∪ξ2 =η\x

Using the fact that Keλ eβφ(x−·) − 1 (η) = eλ eβφ(x−·) (η),

η ∈ Γ0 ,

we obtain I1 G(η) = − G(η)

eβφ(x−y)

x∈η y∈η\x

−

eλ eβφ(x−·) − 1 (η \ x) \ ξ ∪ ρ G(ξ ∪ x).

x∈η ξ ⊂η\x, ξ =η\x ρ⊂ξ

Finally, changing the summation in the last term and using (12) we get I1 G(η) = −G(η)

x∈η y∈η\x

−

ξ ⊂η, ξ =η

G(ξ )

eβφ(x−y) x∈ξ y∈ξ \x

Now, we transform the second item of (10):

eβφ(x−y)

eβφ(x−y) − 1 . y∈η\ξ

(12)


I2 G(η) := z

|η\ξ |

(−1)

ξ ⊂η

=z

Dx+ KG(ξ ) dx

Rd

(−1)|η\ξ |

ξ ⊂η

Rd

215

G(ρ) −

ρ⊂ξ ∪x

G(ρ) dx.

ρ⊂ξ

A direct use of the definitions of the K-transform and its inverse yields

I2 G(η) = z (−1)|η\ξ | G(ρ ∪ x) dx ξ ⊂η

Rd

ρ⊂ξ

|η\ξ | =z (−1) K G(· ∪ x) (ξ ) dx = z G(η ∪ x) dx. Rd

ξ ⊂η

2

Rd

4.4. Verification of Assumption 3.1 In the following subsections, the Lebesgue–Poisson measure λ1 , defined for the general approach, will be denoted for simplicity by λ. We assume also that the potential φ satisfies conditions (S) and (SI). in the Banach space For arbitrary and fixed C > 0, we consider the operator L (13) LC := L1 Γ0 , C |η| λ(dη) . s

Symbol · stands for the norm of the space (13) and symbol → denotes the strong convergence of operators in LC . Remark 4.2. According to the general scheme (η) := C |η| , η ∈ Γ0 . For any Λ ∈ Bb (Rd ) we set LΛ C := {G ∈ LC | G Γ0 \ΓΛ = 0}.

(14)

Λ It is not difficult to show that LΛ C is a closed linear subset in (LC , · ). Therefore, (LC , · ) is a subspace of (LC , · ). For any ω > 0 we introduce a set H(ω, 0) of all densely defined closed operators T on LC , the resolvent set ρ(T ) of which contains the following sector π π + ω := ζ ∈ C |arg ζ | < + ω , ω > 0, Sect 2 2

and for any ε > 0 (T − ζ 1)−1 Mε , |ζ |

|arg ζ |

π + ω − ε, 2

where Mε does not depend on ζ . Let H(ω, θ ), θ ∈ R denotes the set of all operators of the form T = T0 + θ with T0 ∈ H(ω, 0).

216


Remark 4.3. It is well known (see e.g., [10]), that any T ∈ H(ω, θ ) is a generator of a holomorphic semigroup U (t) in the sector |arg t| < ω. The function U (t) is not necessarily uniformly bounded, but it is quasi-bounded, i.e. U (t) consteθt in any sector of the form |arg t| ω − ε. Proposition 4.2. For any C > 0, the operator (L0 G)(η) = (L0,β G)(η) := −G(η) D(L0 ) = G ∈ LC

eβφ(x−y) ,

x∈η y∈η\x

βφ(x−y) e G(η) ∈ LC x∈η y∈η\x

is a generator of a contraction semigroup on LC . Moreover, L0 ∈ H(ω, 0) for all ω ∈ (0, π2 ). Proof. It is not difficult to show that the operator L0 is densely defined and closed. Let 0 < ω < π 2 be arbitrary and fixed. Since the potential V satisfies (S), for all η ∈ Γ0 , |η| > 1, we have

1

eβφ(x−y) |η|e |η|

2β

{x,y}∈η φ(x−y)

|η|e−2Bβ .

(15)

x∈η y∈η\x

This inequality implies that for all ζ ∈ Sect( π2 + ω) βφ(x−y) > 0, e + ζ

η ∈ Γ0 .

x∈η y∈η\x

Therefore, for any ζ ∈ Sect( π2 + ω) the inverse operator (L0 − ζ 1)−1 , the action of which is given by (L0 − ζ 1)−1 G (η) = −

x∈η

1

y∈η\x

eβφ(x−y) + ζ

G(η),

(16)

is well defined on the whole space LC . Moreover, it is a bounded operator in this space and (L0 − ζ 1)−1

1 |ζ | , M |ζ | ,

if Re ζ 0, if Re ζ < 0,

(17)

where the constant M does not depend on ζ . Indeed, the case Re ζ 0 is a direct consequence of (16) and the inequality x∈η y∈η\x

eβφ(x−y) + Re ζ |η|e−2Bβ + Re ζ Re ζ 0.


217

We prove now the bound (17) for the case Re ζ < 0. Using (16), we have 1 (L0 − ζ 1)−1 G = G(·) | βφ(x−y) + ζ | e x∈· y∈·\x |ζ | 1 = G(·) . |ζ | | x∈· y∈·\x eβφ(x−y) + ζ | Since ζ ∈ Sect( π2 + ω), π + ω = |ζ | cos ω. | Im ζ | |ζ |sin 2 Hence, |

x∈η

|ζ | 1 |ζ | =: M βφ(x−y) + ζ | | Im ζ | cos ω y∈η\x e

and (17) is fulfilled. The rest statement of the lemma follows now directly from the theorem of Hille–Yosida (see e.g., [10]). 2 An additional parameter of the model. Since the intensity z of the Lebesgue–Poisson measure in the definition of the Banach space LC is equal to 1, let z, which was involved in the structure of the birth rate of L, now plays the role of an additional parameter := z. We set now (L1 G)(η) := (L1,β G)(η) eβφ(x−y) − 1 , G(ξ ) eβφ(x−y) =− ξ ⊂η, ξ =η

x∈ξ y∈ξ \x

y∈η\ξ

D(L1 ) := D(L0 ) and

(L2, G)(η) =

G(η ∪ x) dx, Rd

D(L2 ) := D(L0 ). The well-definiteness of these operators will be clear from the lemma below. We will sometimes to emphasize the dependence on and β. use the notation L ,β instead of L Lemma 4.1. For any δ > 0 there exist 0 > 0 and β0 > 0 such that for all 0 and β β0 (L1,β + L2, )G aL0 G + bG, G ∈ D(L0 ), (18) with a = a(, β) < δ, b = b(, β) < δ.

218


Proof. It is not difficult to show that βE(x,ξ \x) |η| βφ(x−y) G(ξ ) 1−e C λ(dη) e

L1 G

Γ0 ξ ⊂η, ξ =η

x∈ξ

y∈η\ξ

βE(x,ξ \x) |η| βφ(x−y) G(ξ ) 1−e C λ(dη) e = Γ0 ξ ⊂η

−

x∈ξ

y∈η\ξ

βE(x,η\x) |η| G(η) e C λ(dη).

(19)

x∈η

Γ0

The application of the Minlos lemma to (19) gives us

L1 G

βE(x,η\x) |η| G(η) e C λ(dη) x∈η

Γ0

= eCst (β)C − 1 L0 G.

1 − eβφ(y) C |ξ | λ(dξ ) − L0 G

Γ0 y∈ξ

We estimate the norm L2 G using the Minlos lemma and the bound (15). Namely,

G(η ∪ x) dx C |η| λ(dη)

L2 G Γ0 Rd

|η|G(η)C |η|−1 λ(dη)

Γ0

e2Bβ

Γ0

eβE(x,η\x) G(η)C |η|−1 λ(dη) = e2Bβ C −1 L0 G.

x∈η

Therefore, (L1 + L2 )G eCst (β)C + e2Bβ C −1 − 1 L0 G. And hence the assertion of the lemma is fulfilled with the coefficients a := eCst (β)C + e2Bβ C −1 − 1,

b := 0

which can be taken less then δ for the appropriate choice of and β.

2

Theorem 4.1. For any C > 0, and for all , β > 0 which satisfy 2eCst (β)C + 2e2Bβ C −1 < 3 the operator L ,β is a generator of a holomorphic semigroup in LC .

(20)


219

Proof. It follows from the theorem about perturbation of holomorphic semigroup (see e.g., [10]). For the reader’s convenience, below we give its formulation: For any T ∈ H(ω, θ ) and for any ε > 0 there exist positive constants , δ such that if the operator A satisfies Au aT u + bu,

u ∈ D(T ) ⊂ D(A),

with a < δ, b < δ, then T + A ∈ H(ω − ε, ). In particular, if θ = 0 and b = 0, then T + A ∈ H(ω − ε, 0). 2 Remark 4.4. Applying the proof of the theorem about perturbation of the generator of a holomorphic semigroup (see, e.g. [10]) to our case and taking into account the fact that L0 ∈ H(ω, 0), for any ω ∈ (0, π2 ), one can conclude that δ in this theorem can be chosen to be 12 . For our further purposes we have to show that the holomorphic semigroup constructed in Theorem 4.1 can be approximated by semigroups localized in bounded volumes. Let Λ ∈ Bb (Rd ) be arbitrary and fixed. Then all results proved in this subsection hold true for the operator L Λ G(η) := −

eβφ(x−y) G(η)

x∈ηΛ y∈ηΛ \x

−

G(ξ )

ξ ⊂η, ξ =η

eβφ(x−y)1Λ (y) − 1 + G(η ∪ x) dx

eβφ(x−y)

x∈ξΛ y∈ξΛ \x

y∈η\ξ

Λ

acting in the functional space LΛ C with the domain D(L Λ ) := G ∈ LC

eβφ(x−y) G(η) ∈ LΛ C .


Namely, the main result can be formulated as follows. Theorem 4.2. For any Λ ∈ Bb (Rd ), and any triple of constants C, > 0, and β > 0 which satisfy 2eCst (β)C + 2e2Bβ C −1 < 3 Λ the operator L Λ is a generator of a holomorphic semigroup in LC .

Remark 4.5. The arguments, analogous to those which were proposed in the proof of Lemma 4.1, imply that (18) holds for the operators L 0,Λ G(η) := −

eβφ(x−y) G(η),


L 1,Λ G(η) := −

ξ ⊂η, ξ =η

G(ξ )

x∈ξΛ y∈ξΛ \x

eβφ(x−y)

eβφ(x−y)1Λ (y) − 1 , y∈η\ξ

220


and

(L2,,Λ G)(η) =

G(η ∪ x) dx, Λ

with βE(x,ηΛ \x) Λ D(L0,Λ ) = D(L1,Λ ) = D(L2,Λ ) := G ∈ LC e G(η) ∈ LC . x∈ηΛ

Moreover, the bound (18) in this case will be uniform with respect to Λ ∈ Bb (Rd ), i.e. the coefficients a > 0 and b > 0 in (18) can be chosen independent of Λ. Fix any triple of positive constants C, and β which satisfies (20) and any Λ ∈ Bb (Rd ). Λ Remark 4.6. Let U t (C, , β) be a holomorphic semigroup generated by the operator (LΛ , D(LΛ )) Λ Λ on LC . Then Ut (C, , β)PΛ , t 0, where PΛ G(η) := 1ΓΛ (η)G(η),

G ∈ LC ,

is a semigroup on LC generated by the operator L Λ PΛ with the domain D(LΛ PΛ ) := G ∈ LC

e

βφ(x−y)

1ΓΛ (η)G(η) ∈ LC .


Remark 4.7. The theorem about perturbation of the generator of a holomorphic semigroup, mentioned before in this subsection (see also [10]), implies that for any Λ ∈ Bb (Rd ) and ε > 0 there exists > 0 and a constant M > 0 which does not depend on Λ such that for any ζ from the half-plane Re ζ > the following bound holds: −1 (L Λ PΛ − ζ )

Mε , |ζ − |

arg (ζ − ) π + ω − ε. 2

be a sequence of bounded Borel sets such that Λn ⊂ Λn+1 , for all n ∈ N, and Let {Λn }n1 d . Below, we formulate the following approximation theorem. Λ = R n1 n Λn t (C, , β) and {U Theorem 4.3. Let U t (C, , β), n 1} be holomorphic semigroups generated Λn and {L by L Λn , , n 1} in the spaces LC and LC , respectively. Then, s t (C, , β), UtΛn (C, , β)PΛn → U

uniformly on any finite interval of t 0.

n → ∞,


221

Proof. Using the approximation theorem for quasi-bounded semigroups (see e.g. [10]), it is enough to show that −1 s (L → (L − ζ )−1 Λn , PΛn − ζ )

for some ζ ∈ C such that Re ζ > θ . Let ζ ∈ C, Re ζ > θ be arbitrary and fixed. For any G ∈ LC it holds −1 (L − ζ )−1 G Λn , PΛn − ζ ) G − (L −1 − ζ )−1 G. = (L Λn , PΛn − ζ ) [L −L Λn , PΛn ](L ) = D(L0 ) For any G ∈ D(L − L [L Λn , PΛn ]G(η) =− eβφ(x−y) 1 − 1ΓΛn (η) G(η) x∈η y∈η\x

−

G(ξ )

ξ ⊂η, ξ =η

+

eβφ(x−y) − 1 1 − 1ΓΛn (ξ )1ΓΛn (η \ ξ )

eβφ(x−y)

x∈ξ y∈ξ \x

y∈η\ξ

1 − 1Λn (η ∪ x) G(η ∪ x) dx +

G(η ∪ x) dx,

Λcn

Λn

where Λcn = Rd \ Λn . Using the simple inequality 1 − 1Γ (ξ )1Γ (η) 1 − 1Γ (ξ ) + 1 − 1Γ (η), Λn Λn Λn Λn

ξ, η ∈ Γ0 ,

and the estimates analogous to those which were proposed in Lemma 4.1 we obtain [L − L Λn , PΛn ]G(η)

C (β)C βφ(x−y) 2Bβ −1 st e + e C 1 − 1ΓΛn (·) e G(·)

−1

x∈· y∈·\x

|G(·)

| ·Λcn + e C

βφ(x−y) 1 − 1Γ (η)K(η)C |η| λ(dη), + e G(·) Λn 2Bβ

x∈· y∈·\x

ΓΛn

where K(η) :=

1 − eβφ(x) ,

η ∈ Γ0 .

x∈η

All of terms in the right-hand side of the last inequality tends to zero, when n → ∞.

(21)

222


Using Remark 4.7 and (21) we easily conclude that the difference in (21) also tends to zero when n → ∞. 2 4.5. Verification of Assumption 3.2 t (C, , β) be a Fix any triple of positive constants C, and β which satisfies (20). Let U and let holomorphic semigroup generated by L ,β KC := k : Γ0 → R k(·)C −|·| ∈ L∞ (Γ0 , λ) be the dual space to the space LC with respect to the following duality:

G, k := Gk dλ.

(22)

(23)

Γ0

It is also called the space of “so-called correlation functions.” Analogously to the general scheme, KC is a Banach space. Note, also that k(·)C −|·| ∈ L∞ (Γ0 , λ) means that the function k satisfies the following bound k(η) const C |η| , a.a. η ∈ Γ0 with respect to λ, (24) which is known as the classical Ruelle bound, see e.g. [27]. (C, , β) be a semigroup on KC determined by According to the general scheme, let U t t (C, , β) via the duality (23). U Next, we solve the following problem. Suppose that k0 ∈ KC is a correlation function, i.e., there exists a probability measure μ0 ∈ M1fm (Γ ), locally absolutely continuous with respect to the Poisson measure, whose correlation function is exactly k0 . We would like to investigate now (C, , β) preserves the property whether the evolution of k0 in time given by the semigroup U t (C, , β)k0 , at any moment of time t > 0, is a correlation described above. Namely, whether U t function or not? In order to answer this question, one can apply, for example, the theorem about characterization of correlation functions, proposed in [11]. The conditions of this theorem, which must be checked for our particular model are the following: (C, , β)k0 0, for any t 0: G G, U t

∀G ∈ Bbs (Γ0 ).

Further explanations will be devoted to the verification of the latter condition. Let μ ∈ G(β, z) and {πΛ,∅ }Λ∈Bb (Rd ) denotes the specification with empty boundary conditions corresponding to the Gibbs measure μ. We define

E(F, G) := Dx+ F (γ )Dx+ G(γ ) πΛ (dγ , ∅), F, G ∈ KCBΛ bs (Γ0 ), Γ

x∈γ

Λ where KCBΛ bs (Γ0 ) is K-image of CBbs (Γ0 ). Now we would like to list some facts the proofs of which are completely analogous to those proposed in [13,17].


223

2 d Lemma 4.2. The set KCBΛ bs (Γ0 ) is dense in L (Γ, πΛ,∅ ) for any Λ ∈ Bb (R ).

Lemma 4.3. Let Λ ∈ Bb (Rd ) be arbitrary and fixed. Then (E, KCBΛ bs (Γ0 )) is a well-defined bilinear form on L2 (Γ, πΛ,∅ ). Lemma 4.4. Let Λ ∈ Bb (Rd ) be arbitrary and fixed. Suppose that conditions (S) and (SI) are satΛ isfied. Then (LΛ , KCBΛ bs (Γ0 )) is an operator associated with the bilinear form (E, KCBbs (Γ0 )) 2 in L (Γ, πΛ,∅ ), i.e.

E(F, G) = LΛ F (γ )G(γ ) πΛ,∅ (dγ ), F, G ∈ KCBΛ bs (Γ0 ). Γ

Lemma 4.5. Let Λ ∈ Bb (Rd ) be arbitrary and fixed. Suppose that conditions (S) and (SI) are satisfied and μ ∈ G(z, β). Then there exists a self-adjoint positive Friedrichs’ extension Λ 2 (L Λ , D(L Λ )) of the operator (LΛ , KCBbs (Γ0 )) in L (Γ, πΛ,∅ ). Moreover, (L Λ , D(L Λ )) is a 2 generator of a contraction semigroup which preserves 1 in L (Γ, πΛ,∅ ), associated with some Markov process. Remark 4.8. It is well known (see e.g. [26]) that under the condition of Lemma 4.5 the semi1 d group generated by (L Λ , D(L Λ )) can be extended to L (Γ, πΛ,∅ ). For any Λ ∈ Bb (R ), the extension of this semigroup in L1 (Γ, πΛ,∅ ) we will denote by (UtΛ )t0 . For the generator of this semigroup we will use the notation (L Λ , D1 (L Λ )), where D1 (L Λ ) ⊃ D(L Λ ) is the domain of 1 L Λ in L (Γ, πΛ,∅ ). Now, we introduce one of the crucial lemmas about the evolution of the “so-called correlation functions.” Lemma 4.6. Let the positive constants C, and β which satisfy (20) be arbitrary and fixed. The (C, , β) on KC preserves positive semi-definiteness, i.e. for any t 0 semigroup U t (C, , β)k 0, ∀G ∈ Bbs (Γ0 ), G G, U t iff G G, k 0,

(25)

for any G ∈ Bbs (Γ0 ). Remark 4.9. Let MC stands for the set of all probability measures on Γ , locally absolutely continuous with respect to the Poisson measure, with locally finite moments, whose correlation functions satisfy the bound (24). As it was pointed out at the beginning of this section, the condition (25) on the function k ∈ KC insures the existence of a unique measure μ ∈ MC whose correlation function is k, see [11]. Proof of Lemma 4.6. Under the assumptions of the lemma we have to show that for any t 0 t (C, , β)(G G), k 0, ∀G ∈ Bbs (Γ0 ). U (26)

224


But G G ∈ Bbs (Γ0 ) for any G ∈ Bbs (Γ0 ). Therefore, due to Theorem 4.3 it is enough to show that for any t 0 and any G ∈ Bbs (Γ0 ) there exists Λ ∈ Bb (Rd ) such that for all Λ ∈ Bb (Rd ), Λ ⊃ Λ UtΛ (C, , β)PΛ (G G), k 0.

(27)

Let Λ ∈ Bb (Rd ) be arbitrary and fixed. We set −1 Λ UtΛ := K U t (C, , β)K ,

t 0.

(UtΛ )t0 is a semigroup on −1 Λ L1 := KLΛ C , · 1 := K · L C

which is a Banach space. Moreover, it is not difficult to show that the generator of this semigroup coincides with (LΛ , KD(L Λ )). 1 Proposition 4.3. For any F ∈ LΛ 1 ⊂ L (Γ, πΛ,∅ ),

UtΛ F = UtΛ F,

t 0, in L1 (Γ, πΛ,∅ ),

where (UtΛ )t0 is defined in Remark 4.8. Λ Λ Proof. The fact that (LΛ , KD(L Λ )) is the generator of (Ut )t0 in (L1 , · 1 ) implies the following (see e.g. [9])

−n Λ U F − 1 − t L Λ F → 0, t n 1

n → ∞, for all F ∈ LΛ 1.

Since · 1 · L1 (Γ,πΛ,∅ ) , the latter fact gives −n Λ U F − 1 − t L Λ F → 0, t 1 n L (Γ,πΛ,∅ )


(28)

Λ Analogously, the fact that (L Λ , D1 (L Λ )) is the generator of (Ut )t0 gives us

−n Λ Ut F − 1 − t L F → 0, Λ 1 n L (Γ,πΛ,∅ )


(29)

As was shown before, there exists > 0 such that for any real ζ > −1 −1 −1 −1 (L Λ − ζ 1) F − (LΛ − ζ 1) F = (L Λ − ζ 1) [LΛ − L Λ ](LΛ − ζ 1) F.

The function Fζ := (LΛ − ζ 1)−1 F ∈ KD(L Λ ). Hence, [LΛ − L Λ ]Fζ = 0. The latter fact means that


−n Λ Ut F − 1 − t L F Λ 1 n L (Γ,πΛ,∅ ) −n Λ t = F → 0, Ut F − 1 − n LΛ 1 L (Γ,πΛ,∅ )


The convergences (28) and (29) imply the assertion of the proposition.

225

(30)

2

Corollary 4.1. Lemma 4.5 implies that for any moment of time t 0 UtΛ F 0,

for all non-negative F ∈ KBbs (Γ0 ).

(31)

Let t 0 and G ∈ Bbs (Γ0 ) be arbitrary and fixed. Suppose that N ∈ N and Λ ∈ Bb (Rd ) are such that G G Γ

N

(n) n=0 ΓΛ

0\

= 0.

d 2 2 Then, K(G G) = |KG|2 ∈ LΛ 1 for all Λ ∈ Bb (R ), Λ ⊃ Λ . Moreover, PΛ |KG| = |KG| . d Hence, the left-hand side of (27) for any Λ ∈ Bb (R ), Λ ⊃ Λ , is equal to the following expression:

UtΛ (C, , β)PΛ (G G), k =

Λ KU t (G G)(γ )μ (dγ )

Γ

UtΛ K(G G)(γ )μ (dγ ) =

= Γ

UtΛ |KG|2 (γ )μΛ (dγ ),

ΓΛ

where μΛ is a projection of μ on ΓΛ (see Remark 4.9). Let us mention that the measure μ is locally absolutely continuous with respect to the Poisson measure π . Therefore, UtΛ (C, , β)PΛ (G G), k =

UtΛ |KG|2 (γ )

ΓΛ

dμΛ (γ )πΛ (dη). dπΛ

Corollary 4.1 implies that there exists a set S ⊂ Γ, πΛ,∅ (S) = 0, such that for all γ ∈ Γ \ S: UtΛ |KG|2 (γ ) 0. But πΛ,∅ is absolutely continuous with respect to πΛ . Furthermore, the corresponding Radon– Nikodym derivative is positive almost surely with respect to πΛ . Hence, πΛ (SΛ ) = 0, where SΛ is a projection of the set S to ΓΛ , and UtΛ (C, , β)PΛ (G G), k =

UtΛ |KG|2 (γ )

ΓΛ \SΛ

The latter proves the assertion of Lemma 4.6.

2

dμΛ (γ )πΛ (dη) 0. dπΛ

226


The result obtained in Lemma 4.6 and the characterization of correlation functions in [11] imply the following corollary. Corollary 4.2. Let positive constants C, and β which satisfy (20) be arbitrary and fixed. Let k ∈ KC be such that G G, k 0, for any G ∈ Bbs (Γ0 ). Then for any t 0 there exists a (C, , β)k. unique measure μt ∈ MC whose correlation function is U t In Corollary 4.2, we denote the evolution of the measure μ in time by Ut (C, , β)μ := μt . According to the general scheme (Ut (C, , β))t0 is a semigroup on MC,β . This leads us directly to the construction of a non-equilibrium Markov process (or rather a Markov function) on Γ . Theorem 4.4. Suppose that conditions (S) and (SI) are satisfied. For any triple of positive conμ stants C, and β which satisfies (20) and any μ ∈ MC there exists a Markov process Xt ∈ Γ with initial distribution μ associated with the generator L . Proof. The proof is a direct consequence of Theorem 3.1.

2

Acknowledgments The financial support of the DFG through the SFB 701 (Bielefeld University) and German– Ukrainian Project 436 UKR 113/80 is gratefully acknowledged. R. Minlos gratefully acknowledges the financial support of RFFI No. 06-01-00449 and CRDF research funds N RM1-2085. References [1] S. Albeverio, Yu. Kondratiev, M. Röckner, Analysis and geometry on configuration spaces, J. Funct. Anal. 154 (1998) 444–500. [2] Yu.M. Berezansky, Yu.G. Kondratiev, T. Kuna, E. Lytvynov, On a spectral representation for correlation measures in configuration space analysis, Methods Funct. Anal. Topology 5 (4) (1999) 87–100. [3] L. Bertini, N. Cancrini, F. Cesi, The spectral gap for a Glauber-type dynamics in a continuous gas, Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 91–108. [4] R.L. Dobrushin, Ya.G. Sinai, Yu.M. Suhov, Dynamical system of the statistical mechanics, in: Sovrem. Mat. Fundam. Napravl., vol. 2, VINITI, 1985, pp. 235–284. [5] R. Fernandez, P. Ferrari, G. Guerberoff, Spatial birth-and-death processes in random environment, Math. Phys. Electron. J. 11 (2005) 3–52. [6] N. Garcia, T. Kurtz, Spatial birth and death processes as solutions of stochastic equations, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 281–303. [7] H.O. Georgii, Gibbs Measures and Phase Transitions, de Gruyter, 1982. [8] R.A. Holley, D.W. Stroock, Nearest neighbor birth and death processes on the real line, Acta Math. 140 (1978) 103–154. [9] K. Ito, F. Kappel, Evolution Equations and Approximations, World Scientific, 2002. [10] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966. [11] Yu.G. Kondratiev, T. Kuna, Harmonic analysis on configuration space I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2) (2002) 201–233. [12] Yu.G. Kondratiev, O.V. Kutoviy, On the metrical properties of the configuration space, Math. Nachr. 279 (7) (2006) 774–783. [13] Yu.G. Kondratiev, E. Lytvynov, Glauber dynamics of continuous particle systems, Ann. Inst. H. Poincaré Probab. Statist. 41 (4) (2005) 685–702. [14] Yu.G. Kondratiev, A. Skorokhod, On contact process in continuum, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2) (2006) 187–198.


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[15] Yu.G. Kondratiev, R. Minlos, E. Zhizhina, One-particle subspaces of the generator of Glauber dynamics of continuous particle systems, Rev. Math. Phys. 16 (9) (2004) 1–42. [16] Yu.G. Kondratiev, O.V. Kutoviy, E. Zhizhina, Nonequilibrium Glauber-type dynamics in continuum, J. Math. Phys. 47 (11) (2006) 17 pp. [17] Yu.G. Kondratiev, E. Lytvynov, M. Röckner, Infinite interacting diffusion particles. I. Equilibrium process and its scaling limit, Forum Math. 18 (1) (2006) 9–43. [18] Yu.G. Kondratiev, E. Lytvynov, M. Röckner, Equilibrium Kawasaki dynamics of continuous particle systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2) (2007) 185–209. [19] Yu. G. Kondratiev, T. Kuna, M.J. Oliveira, J.L. da Silva, L. Streit, Hydrodynamic limits for the free Kawasaki dynamics of continuous particle systems SFB-701, preprint, University of Bielefeld, Bielefeld, Germany, 2007. [20] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I, Arch. Ration. Mech. Anal. 59 (1975) 219–239. [21] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. II, Arch. Ration. Mech. Anal. 59 (1975) 241–256. [22] T.M. Liggett, Interacting Particle Systems, Springer-Verlag, 1985. [23] R.A. Minlos, Lectures on statistical physics, Uspekhi Mat. Nauk 23 (1) (1968) 133–190 (in Russian). [24] C. Preston, Spatial birth-and-death processes, in: Proceedings of the 40th Session of the International Statistical Institute, Warsaw, 1975, vol. 2, Bull. Inst. Internat. Statist. 46 (1975) 371–391. [25] C. Preston, Random Fields, Lecture Notes in Math., vol. 534, Springer-Verlag, 1976. [26] M. Reed, B. Simon, Methods of Modern Mathematical Physics, 2. Fourier Analysis, Self-Adjointness, Academic Press, 1972. [27] D. Ruelle, Statistical Mechanics, Benjamin, 1969. [28] D. Ruelle, Superstable interactions in classical statistical mechanics, Comm. Math. Phys. 18 (1970) 127–159. [29] L. Wu, Estimate of spectral gap for continuous gas, Ann. Inst. H. Poincaré Probab. Statist. 40 (4) (2004) 387–409.


Matrix Riemann–Hilbert problems and factorization on Riemann surfaces M.C. Câmara a,1 , A.F. dos Santos a,∗,1 , Pedro F. dos Santos b,2 a Centro de Análise Funcional e Aplicações, Departamento de Matemática, Instituto Superior Técnico, Portugal b Centro de Análise, Geometria e Sistemas Dinâmicos, Departamento de Matemática,

Instituto Superior Técnico, Portugal Received 7 December 2007; accepted 10 January 2008 Available online 29 February 2008 Communicated by Paul Malliavin

Abstract The Wiener–Hopf factorization of 2 × 2 matrix functions and its close relation to scalar Riemann–Hilbert problems on Riemann surfaces is investigated. A family of function classes denoted C(Q1 , Q2 ) is defined. To each class C(Q1 , Q2 ) a Riemann surface Σ is associated, so that the factorization of the elements of C(Q1 , Q2 ) is reduced to solving a scalar Riemann–Hilbert problem on Σ. For the solution of this problem, a notion of Σ-factorization is introduced and a factorization theorem is presented. An example of the factorization of a function belonging to the group of exponentials of rational functions is studied. This example may be seen as typical of applications of the results of this paper to finite-dimensional integrable systems. © 2008 Elsevier Inc. All rights reserved. Keywords: Riemann–Hilbert problem; Factorization; Riemann surfaces; Integrable systems


E-mail addresses: [email protected] (M.C. Câmara), [email protected] (A.F. dos Santos), [email protected] (P.F. dos Santos). 1 Partially supported by Fundação para a Ciência e a Tecnologia through Program POCI 2010/FEDER and FCT Project PTDC/MAT/81385/2006. 2 Partially supported by Fundação para a Ciência e a Tecnologia through Program POCI 2010/FEDER. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.01.008

M.C. Câmara et al. / Journal of Functional Analysis 255 (2008) 228–254

229

1. Introduction The central object of this paper is the interplay between the problem of Wiener–Hopf factorization of 2 × 2 matrix functions and scalar Riemann–Hilbert problems on Riemann surfaces. The first part of the paper deals with the definition of certain classes of matrix functions that play a fundamental role in the study of the factorization of 2 × 2 matrix functions and the second part concentrates on the associated Riemann–Hilbert problems on Riemann surfaces. Before we give an overview of the main results of the paper we recall the definition of Wiener– Hopf factorization of a 2×2 Hölder continuous matrix function G on a piecewise smooth contour ΓC which divides C into disjoint regions ΩC+ , ΩC− such that C = ΩC+ ∪ ΓC ∪ ΩC− (Definition 3.1 below). This is a representation of the form G = G− DG+ ,

(1.1)

where G± and their inverses are analytic in ΩC± and continuous in ΩC± and D = diag(zk1 , zk2 ) with ki ∈ Z and k1 k2 . In the paper we define a family of classes C(Q1 , Q2 ) of all the G ∈ Cμ (ΓC ) satisfying GT Q1 G = hQ2 ,

(1.2)

where the upperscript T denotes transposition and h is an invertible scalar function on ΓC . Denoting by Cμ± (ΓC ) the spaces of functions of Cμ (ΓC ) that have analytic extensions into ΩC± , we assume in (1.2) that Q1 ∈ Cμ− (ΓC ) + R and Q2 ∈ Cμ+ (ΓC ) + R, where R denotes the space of rational functions with poles off ΓC . The first important result of the paper is that all functions that have a factorization of the form (1.1) belong to some class C(Q1 , Q2 ) with the additional condition that det Q1 = det Q2 (Theorems 2.4 and 2.7). This fact means that we may associate with each class C(Q1 , Q2 ) a Riemann surface (which may be the Riemann sphere or even trivial) defined by an algebraic curve of the form μ2 = det Q1 = det Q2 . This, in turn, provides us with a tool that enables us to study the solvability of the factorization problem for important classes of matrix functions and obtain formulas for the factors when the factorization exists. In the paper a general representation for the elements of each class C(Q1 , Q2 ) is derived (Theorem 2.12) and this leads to a technique to obtain a related scalar Riemann–Hilbert problem on the above-mentioned Riemann surface. In Theorem 2.20 we state an important result that gives an alternative characterization of the classes C(Q1 , Q2 ) in terms of an equivalence of multiplication operators on [Cμ (ΓC )]2 and scalar multiplication operators on the preimage of the contour ΓC under the standard projections from the Riemann surface to the complex plane. The results of Section 2 give, for the first time, a general framework for the study of the factorizations of 2 × 2 matrix functions, framework that goes significantly beyond the dispersive results that can be found in the specialized literature. In Section 3 we introduce the concept of Σ -factorization relative to a contour Γ on a Riemann surface Σ (Definition 3.1), which we recall here. A function f ∈ Cμ (Γ ), invertible on the contour Γ is said to have a Σ -factorization if it can be represented in the form f = f− rf+

(1.3)

230


with r a rational function on Σ and f+±1 ∈ Cμ+ (Γ ), f−±1 ∈ Cμ− (Γ ) where Cμ± (Γ ) are the subspaces of Cμ (Γ ) of functions that have analytic extensions into the regions that are preimages of Ω ± , continuous in Ω ± . If r is a constant the factorization is called special. It is shown in Theorem 3.4 that this case occurs if and only if we have 1 2πi

(log f )ω ∈ Zg + BZg , Γ

where ω is the vector of normalized holomorphic differentials and Zg + BZg is the lattice of periods of ω. Definition (1.3) is crucial in the development of a technique for solving scalar Riemann– Hilbert problems on a Riemann surface in a rigorous and elegant way. This is done through a factorization theorem (Theorem 3.4) which in general terms states that if f ∈ Cμ (Γ ) is such that log f ∈ Cμ (Γ ) then f possesses a Σ-factorization with factors given by f+ = exp PΓ+ log f h+ , f− = exp PΓ− log f − log h h− ,

(1.4)

where h is a function depending on log f (see Theorem 3.4) whose factorization h = h− rh+ is given in Proposition 3.10. In (1.4) PΓ± are the complementary projections defined by the meromorphic analog of the Cauchy kernel (see Appendix A). The above result has the convenient feature of avoiding the use of the so called discontinuous analog of the Cauchy kernel for Riemann surfaces [17]. The method applies to surfaces with genus greater than 1 although the calculations become considerably more difficult in the general case. Section 4 deals with an example that belongs to the group of exponentials of rational matrix functions which, besides illustrating the techniques developed in Sections 2 and 3, has the additional interest of being typical of problems appearing in the study of finite-dimensional integrable systems [4,13]. To end this introduction we make some brief remarks on references related to the problems that are dealt with in this paper. The study of the class C(Q1 , Q2 ) was initiated in [3], with emphasis on the case Q1 = Q2 . However the theory expounded in the present paper is much more general and the connection with Riemann–Hilbert problems on Riemann surfaces was not touched in [3]. A reduction of the factorization problem of 2 × 2 Daniele–Khrapkov matrix functions to a scalar Riemann–Hilbert problem on a Riemann surface was studied for the first time in [10]. However the treatment followed in [10] is unnecessarily complicated and appears incomplete from the point of view of the relation between the dimension of the spaces of solutions of the two problems. Also the solvability of the resulting Riemann–Hilbert problem on the associated Riemann surface is not studied in that paper. General references on Riemann–Hilbert problems on Riemann surfaces are [9,14,16,17]. In [14] the solvability conditions are presented but no usable formulas are given. [17] is a useful general reference on the Riemann–Hilbert problem on Riemann surfaces, including the question of the analogs of the Cauchy kernel. The paper is almost entirely written in a classical complex analysis perspective.


231

The notions of Σ-factorization and the factorization Theorem 3.4 are presented here for the first time and cannot be found in any of the above references. 2. Matrix factorization and the class C(Q1 , Q2 ) In what follows we restrict the study of matrix factorization to 2 × 2 matrix-functions because of its interest in applications and its relation to factorization on a Riemann surface to be developed later. However many results of this section can be generalized to n × n matrix functions at the expense of greater computational complexity. Before we define the class C(Q1 , Q2 ) we introduce some terminology. By H + (∂D) and − H (∂D) we denote, respectively, the Hardy spaces of bounded analytic functions in D (the unit disc in C) and C \ D. These spaces will be identified with closed subspaces of L∞ (∂D). By R(∂D) we denote the space of rational functions on C with poles off ∂D. If A is an algebra, we denote by G(A) the group of invertible elements in the algebra A. Definition 2.1. Let G ∈ G([L∞ (∂D)]2×2 ). G is said to possess a bounded Wiener–Hopf factorization if it can be represented in the form G = G− DG+ ,

(2.1)

where G± ∈ G([H ± (∂D)]2×2 ) and D = diag(zk1 , zk2 ) with ki ∈ Z and k1 k2 . The factorization (2.1) is said to be canonical if k1 = k2 = 0. Remark 2.2. In the above definition we used the expression “Wiener–Hopf factorization” to denote the factorization (2.1). This is the standard designation in the area of singular operator theory where most of the results concerning this concept can be found. However, in other areas of mathematics, in particular in integrable systems, the designation Riemann–Hilbert factorization is commonly used [7]. The expression Birkhoff factorization is also adopted in some areas where the above notion appears. It is, for example, the case of the classification of holomorphic vector bundles over the Riemann sphere [8, Chapter 2]. The concept of factorization presented in Definition 2.1 is a particular case of the concept of generalized factorization, but is sufficient and appropriate for our purpose (for the theory of generalized factorization, see e.g. [5] and [2]). Definition 2.3. Let Q1 , Q2 be symmetric matrix functions such that 2×2 Q1 ∈ G H − (∂D) + R(∂D) ,

2×2 Q2 ∈ G H + (∂D) + R(∂D) .

We denote by C(Q1 , Q2 ) the set of all matrix functions G ∈ G([L∞ (R)]2×2 ) satisfying the relation GT Q1 G = hQ2 ,

(2.2)

232


where the upperscript T stands for matrix transposition, and det Q1 h = det G , det Q2 where the square-root is assumed to have positive real part. In the case Q1 = Q2 = Q we have Q ∈ R2×2 and the class C(Q1 , Q2 ) is denoted by C(Q). In the following theorem we show that every factorizable 2 × 2 matrix function belongs to some class in the family of classes C(Q1 , Q2 ). Theorem 2.4. If G ∈ G([L∞ (∂D)]2×2 ) admits a bounded factorization then there exist Q1 , Q2 such that G ∈ C(Q1 , Q2 ). Proof. Let G = G− DG+ be a factorization of G. Take −1 Q1 = GT− G−1 − ,

Q2 = GT+ D 2 G+ .

Obviously Q1 , Q2 are symmetric and Q1 ∈ G[H − (∂D) + R(∂D)]2×2 , Q2 ∈ G[H + (∂D) + R(∂D)]2×2 . A straightforward calculation shows that GT Q1 G = Q2 , i.e., h = 1 in this case. 2 We will assume from now on that Q1 , Q2 are of the form [qij ], where qij ∈ Cμ± (∂D) + R(∂D) and either q11 = 0 or q11 ∈ G(Cμ± (∂D) + R(∂D)), taking the upperscripts + and − as corresponding to Q2 and Q1 , respectively. We also assume that det Q1 and det Q2 admit a square-root in Cμ (∂D). The set of such pairs (Q1 , Q2 ) will be denoted by Q. Moreover we focus our attention in matrix functions G ∈ (Cμ (∂D))2×2 . Indeed, several classes that are relevant from the point of view of applications are of this type and belong to some class C(Q1 , Q2 ) for an appropriate (Q1 , Q2 ) ∈ Q [4,12,13]. In this case, since it is clear that Q1 , Q2 in Theorem 2.4 are not unique, we are able to make a specially interesting choice for Q1 , Q2 . To show this we give next some auxiliary results. Lemma 2.5. Let Q ∈ [Cμ (∂D)]2×2 be a symmetric matrix function of the form q1 q2 . Q= q2 q3 Then (1) If q1 is invertible, then 1 Q = q1 S T J S, 2

(2.3)

where

0 1 J= , 1 0 with 2 = − det Q.

S=

1 (q2 + )q1−1

1 (q2 + )q1−1


233

(2) If q1 = 0, 1 Q = S T J S, 2

(2.4)

where S=

0 2q2

1 . q3

Proof. A straightforward calculation shows that (2.3) and (2.4) hold.

2

Corollary 2.6. Let (Q1 , Q2 ) ∈ Q. Then C(Q1 , Q2 ) = ∅ and there is G0 ∈ C(Q1 , Q2 ) ∩ [Cμ (∂D)]2×2 admitting a bounded factorization and such that the function h in (2.2) admits a bounded factorization. Proof. Taking into account our previous assumptions on Q1 , Q2 , we can assume without loss of generality that the element in the first row and first column of both matrices is either 0 or 1. Let 1 Q1 = S1T J S1 , 2

1 Q2 = S2T J S2 , 2

according to Lemma 2.5, and let G0 = S1−1 S2 . We have G0 ∈ C(Q1 , Q2 ) ∩ [Cμ (∂D)]2×2 and det G0 ∈ GCμ (∂D), so that G0 admits a bounded factorization. Moreover, G0 satisfies (2.2) with h = 1. 2

1 , Q

2 ) ∈ Q such that Theorem 2.7. Let (Q1 , Q2 ) ∈ Q. Then there is a pair (Q

1 = det Q

2 = p, det Q where p is a monic polynomial admitting, at most, simple zeros and

1 , Q

2 ). C(Q1 , Q2 ) = C(Q Proof. Let G0 ∈ C(Q1 , Q2 ) admit a bounded factorization and satisfy GT0 Q1 G0 = hQ2 , with h = h− zμ h+ ,

μ ∈ Z, h± ∈ GH ± (∂D),

and let moreover g = det G0 = g− zk g+ , Define

k ∈ Z, g± ∈ GH ± (∂D).

(2.5)

234


1 = g− h−1 Q − Q1 , −1 μ−k

2 = h+ g+ z Q2 . Q

(2.6)

2 −2

1 = g− det Q h− det Q1 .

(2.7)

Then

From (2.5) we have (det G0 )2 det Q1 = h2 det Q2 . Substituting this result in (2.7), and using the expression for the factorization of det G0 (det G0 = g− zk g+ ), we get −2 2(μ−k) 2 −2 −2 2

1 = g−

2 . det Q h− g h det Q2 = h2+ g+ z = det Q

(2.8)

Now, since

1 ∈ H − (∂D) + R(∂D) 2×2 , Q

2 ∈ H + (∂D) + R(∂D) 2×2 Q

we have, from the equality (2.8),

1 , det Q

2 ∈ R(∂D). det Q

1 , Q

2 are obtained from Q1 , Q2 through multiplication by scalar functions Noting that Q (cf. (2.6)) it follows that

1 , Q

2 ). C(Q1 , Q2 ) = C(Q

(2.9)

1 , Q

2 ) = C(r Q

1 , r Q

2 ) for any r ∈ G(R(∂D)), we see that (2.9) holds with Q

1 , Q

2 Since C(Q

1 , det Q

2 are monic polynomials admitting at most simple zeros, as we set to such that det Q prove. 2 Theorem 2.7 means that we can associate with each class C(Q1 , Q2 ) a certain polynomial function (which may be a constant). We shall see now that this enables us to associate in a unique way an algebraic curve to each class C(Q1 , Q2 ). We start by considering the case where (Q1 , Q2 ) ∈ Q and Q1 = Q2 = Q, in which case we will say that Q ∈ Q. We will also use the notation

J = αI α ∈ GCμ (∂D) . It is clear that J ⊂ C(Q) for any Q ∈ Q. Theorem 2.8. Let Q, Q∗ ∈ Q. If C(Q) ∩ C(Q∗ ) = J then Q = βQ∗ for some β ∈ R(∂D) and C(Q) = C(Q∗ ).


235

Proof. Let Q be given by (2.3) or (2.4), according to Lemma 2.5, and assume that there is G∈ / J , G ∈ C(Q) ∩ C(Q∗ ). Define

= SGS −1 . G Since G ∈ C(Q) and noting that S T J S = q1 Q, we have

= q1 S −1 T GT QGS −1 = hJ,

T J G G

Hence, from the proof of Theorem 2.12 below, it follows that G

is a where h = det G = det G. diagonal matrix,

= diag(a, d) with a = d. G

∈ C(J ∗ ) with J ∗ = (S −1 )T Q∗ S −1 . It folOn the other hand, since G ∈ C(Q∗ ), we have G ∗ ∗ ∗ lows that J = β J with β ∈ G(Cμ (∂D)), which is equivalent to Q∗ = βQ with β ∈ R(∂D) and, consequently C(Q) = C(Q∗ ). 2 Corollary 2.9. If C(Q) = C(Q∗ ), for Q, Q∗ ∈ Q, then Q∗ = βQ with β ∈ R. Proof. We have J Q ∈ C(Q) = C(Q∗ ), for J =

0 1 −1 0

and J Q ∈ / J , so that the result follows from Theorem 2.8.

2

Now we consider the classes C(Q1 , Q2 ) in general. Theorem 2.10. Let (Q1 , Q2 ) ∈ Q. If C(Q1 , Q2 ) = C(Q∗1 , Q∗2 ), then Qi = βi Q∗i with βi ∈ GCμ (∂D) for i = 1, 2. Proof. We recall from [3, Theorem 3.5] that C(Q1 , Q2 ) = C(Q1 )H = C Q∗1 H = C Q∗1 , Q∗2 , for some H ∈ C(Q1 , Q2 ) = C(Q∗1 , Q∗2 ). Thus C(Q1 ) = C(Q∗1 ) and it will follow from Corollary 2.9 that Q1 = β1 Q∗1 . Analogously, C(Q1 , Q2 ) = H C(Q2 ) = C Q∗2 H and Q2 = β2 Q∗2 .

2

As an immediate consequence of Theorems 2.10 and 2.7, we have the following result.

236


Theorem 2.11. For every (Q1 , Q2 ) ∈ Q, there is a unique monic polynomial p admitting, at most, simple zeros such that

2

1 = det Q p = det Q

1 , Q

2 ) satisfying the conditions of Theorem 2.7. for any pair (Q Next we give an important result on the a representation of any G belonging to C(Q1 , Q2 ). Theorem 2.12. Let G ∈ C(Q1 , Q2 ) with det Q1 = det Q2 . Then G = S1−1 DS2 , where S1 , S2 correspond to the matrix S of the representation of Q1 , Q2 (cf. Lemma 2.5) and D is diagonal or anti-diagonal. Proof. For G ∈ C(Q1 , Q2 ) we have, for some h, GT Q1 G = hQ2 .

(2.10)

Since S1 and S2 are invertible we may write G = S1 DS2−1 , for some matrix D. Substituting this expression in (2.10) gives T −1 T T S2 D S1 Q1 S1 DS2−1 = hQ2 , from which it follows that D T S1T Q1 S1 D = hS2T QS2 and, in view of Lemma 2.5, DT J D =

hJ,

(2.11)

for same scalar function

h. It is worth noting that (2.11) defines a group as can be easily checked. To obtain all the elements of this group we write D in the general form a b D= . c d Substitution of this matrix in (2.11) leads to the equations ac = 0,

bd = 0

which can only have non-trivial solutions if a = d = 0 or

b = c = 0.


237

In this first case D anti-diagonal and in the second it is diagonal (the group defined by (2.11) has two connected components). The other two equations that come from (2.11) simply give

h = bc = − det D,

h = ad = det D,

2

respectively for the first and second cases.

Corollary 2.13. With the same assumptions of Theorem 2.12 the Riemann–Hilbert problem Gφ + = φ − ,

2 φ ± ∈ H ± (∂D) ,

(2.12)

is equivalent to D S2 φ + = S1 φ − ,

2 φ ± ∈ H ± (∂D) .

(2.13)

Remark 2.14. It is clear that, from the point of view of solving the Riemann–Hilbert (2.13), the case where D is anti-diagonal is entirely analogous to the case where D is diagonal. Thus, unless otherwise stated, we shall assume that D is diagonal. Before we leave the results of Theorems 2.12 and 2.4 it is useful to write Eq. (2.13) in system form: d1 q21 φ1+ + q22 φ2+ + φ2+ = q11 φ1− + q12 φ2− + φ2− , d2 q21 φ1+ + q22 φ2+ − φ2+ = q11 φ1− + q12 φ2− − φ2− ,

(2.14)

where the first subscript in the q corresponds to Q1 or Q2 . It will be shown later that the system (2.14) is equivalent to a scalar Riemann–Hilbert problem on the hyperelliptic Riemann surface defined by the equation μ2 = det Q1 = det Q2 = 2

(2.15)

assuming that det Qi is not a constant. This fact yields a powerful tool for solving Eqs. (2.14). In what follows we shall denote by Σ the Riemann surface obtained by the compactification of the above algebraic curve which henceforth we write in the form μ2 = p(λ), where p(λ) is assumed to be a polynomial of degree 2(g + 1) (g 0) with simple roots. Thus, Σ is a hyperelliptic Riemann surface of genus g. It is convenient to view it as a branched cover of C via the meromorphic function λ : Σ → C induced by (λ, μ) → λ. The meromorphic function induced by (λ, μ) → μ will be denoted by μ. We shall assume that p(λ) has an even number 2(g + 1) (with g −1) of zeros inside D and no zeros on ∂D. This implies √ that there is a continuous branch of log p on ∂D. We shall denote by : ∂D → C the branch of p(λ) for which Re > 0. Note also that the contour Γ = λ−1 (∂D) consists of two disjoint closed paths which divide Σ into two disjoint regions.

238


Definition 2.15. We denote by Γ1 the component of Γ = λ−1 (∂D) where Re μ > 0. The other component is denoted by Γ2 . The contour Γ = Γ1 Γ2 divides Σ into two disjoint regions defined by Ω + = λ−1 (D) and Ω − = Σ \ Ω + . In the process of reducing our Riemann–Hilbert problem on ∂D to a scalar Riemann–Hilbert on a contour in Σ we shall obtain another characterization of the class C(Q1 , Q2 ). Firstly we define a transformation TΣ from the space of Hölder continuous functions on ∂D to the space of Hölder continuous functions on the contour Γ in Σ . Definition 2.16. Let TΣ : [Cμ (∂D)]2 → Cμ (Γ ) be the linear transformation defined by TΣ (φ1 , φ2 )|Γi = φi . The following proposition gives the main properties of TΣ . Proposition 2.17. Let TΣ be as in Definition 2.16. Then the following assertions hold: (1) TΣ maps (φ1 + φ2 , φ1 − φ2 ) into φ1 + μφ2 (here φ1 , φ2 ∈ Cμ (∂D) and, on Γ , we use φi to abbreviate λ∗ φi = φi ◦ λ); (2) TΣ is invertible with inverse given by 2 TΣ−1 : Cμ (Γ ) → Cμ (∂D) ,

TΣ−1 Ψ = (ψ|Γ1 , ψ|Γ2 ).

Proof. Follows straightforwardly from Definition 2.16.

2

For the next result we recall Eqs. (2.14) which we write again in the form (2.13) (renumbered (2.16) for convenience), D S2 φ + = S1 φ + ,

(2.16)

where Si =

1 1

qi − qi +

(i = 1, 2).

(2.17)

Before we derive a Riemann–Hilbert problem on the Riemann surface defined by (2.15) we need some definitions and notation concerning function spaces on Σ . Definition 2.18. By Cμ+ (Γ ) we will denote the subspace of Cμ (Γ ) whose elements are boundary values of analytic functions in Ω + . Analogously for Cμ− (Γ ). We are now in a position to state the result of the following proposition. Proposition 2.19. Eq. (2.13) is equivalent to the scalar Riemann–Hilbert problem on Γ , d φ1+ + q2 φ2+ + μφ2+ = φ1− + q1 φ2− + μφ2− , where d = TΣ (d1 , d2 ) and φ1± , φ2± ∈ Cμ± (Γ ).

(2.18)


239

Proof. Applying TΣ to both sides of (2.16) and using Definition 2.16 gives (i) for the right-hand side, TΣ S1 φ − = TΣ φ1− + q1 φ2− − φ2− , φ1− + q1 φ2− + φ2− = φ1− + q1 φ2− − μφ2− ,

(2.19)

where φi− ∈ Cμ− (Γ ); (ii) for the left-hand side: TΣ DS2 φ = dTΣ S2 φ = d φ1+ + q2 φ2+ + μφ2+ ,

(2.20)

where we have used the fact that TΣ Dψ = dTΣ ψ . 2

From (2.20) and (2.19) we obtain (2.18) as desired.

We are now in a position to state an alternative characterization of the classes C(Q1 , Q2 ) in terms of multiplication operators on [Cμ (∂D)]2 and Cμ (Γ ). In the following we denote by mG the operator of multiplication by G on [Cμ (∂D)]2 and md the operator of multiplication by d on Cμ (Γ ). Theorem 2.20. The matrix valued function G in [G(L∞ (∂D))]2×2 belongs to C(Q1 , Q2 ) where det Q1 = det Q2 = p(λ) ∈ C[λ], if and only if there exists an operator md : Cμ (Γ ) → Cμ (Γ ) such that md = TΣ mS1 mG mS −1 TΣ−1

(2.21)

md = τ ∗ TΣ mS1 mG mS −1 TΣ−1 ,

(2.22)

2

or 2

where Σ is the Riemann surface defined by the equation μ2 = p(λ), S1 , S2 are the matrices given in Eq. (2.17), and τ ∗ : Cμ (Γ ) → Cμ (Γ ) denotes the composition with the hyperelliptic involution τ : Σ → Σ . Proof. Suppose G ∈ C(Q1 , Q2 ). Then, by Theorem 2.12, the matrix D = S1 GS2−1 is either diagonal or anti-diagonal. A direct calculation gives TΣ mD TΣ = md , if D = d01 d02 , and TΣ mD TΣ = τ ∗ md , if D = d02 d01 . Hence either (2.21) or (2.22) holds. Conversely, if (2.21) or (2.22) hold, then the matrix D = S1 GS2−1 is either diagonal or antidiagonal and it follows from Theorem 2.12 that G ∈ C(Q1 , Q2 ). 2 3. Σ-factorization In this section we define a factorization for scalar functions belonging to Cμ (Γ ) where Γ is a contour in a hyperelliptic Riemann surface Σ . This factorization allows us to study scalar

240


Riemann–Hilbert problems such as those arising from vector Riemann–Hilbert problems in C as shown in Section 2. Recall from Section 2 that Σ is the Riemann surface associated to the equation μ2 = p(λ), where p(λ) is a polynomial of degree 2g + 2 without multiple roots. Hence Σ is a hyperelliptic Riemann surface of genus g and is obtained from the plane algebraic curve

Σ0 = (λ, μ) ∈ C2 μ2 = p(λ) by adding two points “at infinity,” ∞1 , ∞2 , such that ς = λ−1 is a local parameter at these points. There are two natural meromorphic functions on Σ : those induced by the projections (λ, μ) → λ and (λ, μ) → μ. They will be denoted respectively by λ and μ. The field of rational functions on Σ will be denoted by R(Σ). As usual, it is convenient to view Σ as a 2-branched cover of P(C2 ) under the map λ : Σ → P(C2 ) = C ∪ {∞}. The two points in λ−1 (0) will be denoted by 01 and 02 . Definition 3.1. Let Γ be a contour in Σ and let f ∈ GCμ (Γ ), 0 < μ < 1; f is said to possess a Σ-factorization relative to Γ if it has a representation of the form f = f− rf+ ,

(3.1)

where (f+ )±1 ∈ Cμ+ (Γ ), (f− )±1 ∈ Cμ− (Γ ) (see Definition 2.18) and r ∈ R(Σ). If r is constant (3.1) is called a special Σ -factorization. It is easily seen that, if f = f− f+ and f = f − f + are two special Σ -factorizations for f , then f + = cf+ ,

f − = cf− ,

where c is a constant. Indeed we have f + f+−1 = f− f−−1 = c. Keeping in mind the application to vector valued Riemann–Hilbert problems in C, we consider only the case where Γ is the (oriented) boundary of a region Ω + ⊂ Σ defined in Definition 3.2 below. For a general reference on Riemann surfaces see, for example, [11] or [15]. Definition 3.2. We will denote by Ω + the inverse image under λ of the unit disk D ⊂ C. We assume that D contains 2(g + 1) zeros of p(λ) so that, if g = −1, Ω + is a union of two disjoint disks; if g 0, Ω + is a Riemann surface of genus g with two closed disks removed. We will also consider the open set Ω − = Σ \Ω + . Note that Ω − is a Riemann surface of genus g = g −g −1 with two disks removed. Assumption 3.3. Henceforth we assume Γ = ∂Ω + with the orientation induced from Ω + .


241

Next we show that every function f ∈ GCμ (Γ ) such that log f ∈ Cμ (Γ ) has a Σ-factorization. Theorem 3.4. Let f ∈ Cμ (Γ ) be such that log f ∈ Cμ (Γ ) and let h be the function given by μ μ h = exp α1 + · · · + αg g λ λ with αk =

1 4πi

log f

λk−1 dλ μ

(k = 1, . . . , g).

Γ

Then f has a Σ-factorization f = f− rf+

(3.2)

with f+ = exp PΓ+ log f h+ , g μ − αk k h− , f− = exp PΓ log f − λ

(3.3) (3.4)

k=1

where PΓ± are the bounded projections on Cμ (Γ ) defined in Appendix A, and h+ , h− , r are the factors of the Σ -factorization h = h+ rh− given in Proposition 3.10 below. Proof. Putting φ = log f and denoting by PΓ± the bounded projections on Cμ (Γ ) defined in Appendix A, we have

Γ φ + φ = PΓ+ φ + PΓ− φ = PΓ+ φ + P

g

αk

k=1

μ , λk

(3.5)

Γ ∈ Cμ− (Γ ) (cf. proof of Proposition A.1) and where PΓ+ φ ∈ Cμ+ (Γ ), P αk =

1 4πi

ξ k−1 φ(ξ, τ ) dξ. τ

Γ

The result now follows from (3.5) and Proposition 3.10 below.

2

Theorem 3.4 reduces the problem of computing a Σ -factorization g of f to the computation of a Σ-factorization for the exponential h = exp() where = k=1 αk μ/λk . To obtain this factorization we will need some information about the periods of the differential d. We start by fixing bases for the first homology group H1 (Σ; Z) and the space of holomorphic differentials Ω 1 (Σ).

242


Let {a1 , . . . , ag , b1 , . . . , bg } be a canonical basis of H1 (Σ; Z) (this means that ai · aj = bi · bj = 0 and ai · bj = δij ) whose elements do not pass through 0j , ∞j . We denote by ζ = (ζ1 , . . . , ζg ) the vector of differentials given by ζk =

λk−1 dλ. μ

It is well known that {ζ1 , . . . , ζg } is a basis of holomorphic differentials [15]. However, it is usually more convenient to use a basis ω = (ω1 , . . . , ωg ) which is dual to the homology basis {aj , bj }, i.e., it satisfies ai ωj = δij . The bases ζ , ω are related by ω = Cζ , where C = (cij ) is the inverse of the matrix A = (aij ) of a-periods of ζ : aij = ζj . ai

We will also need the matrix of b-periods of ω, denoted by B = (bij ): bij = ωj . bi

Notation 3.5. For convenience we introduce the following notation concerning integrals and residues of vectors of differentials. (1) Given a differential ϕ and a vector of 1-cycles c = (c1 . . . , cg ) we denote by c ϕ the vector of c-periods of ϕ: c ϕ = ( c1 ϕ, . . . , cg ϕ). (2) Given a vector of meromorphic differentials ϕ = (ϕ1 , . . . , ϕg ) we denote by Resp (ϕ) the vector of residues (Resp (ϕ1 ), . . . , Resp (ϕg )). (3) Given a vector ϕ = (ϕ1 , . . . , ϕg ) and a 1-cycle c we denote by c ϕ the vector of differentials of periods ( c ϕ1 , . . . , c ϕg ). g Lemma 3.6. Given α = (α1 , . . . , αg ) ∈ Cg consider the function = k=1 αk λμk . Then d = γ∞ − γ0 where γ∞ and γ0 are differentials of the second kind satisfying: (i) γ∞ is holomorphic in Σ \ {∞1 , ∞2 }; in Σ \ {01 , 02 }; (ii) γ0 is holomorphic (iii) aj γ∞ = aj γ0 = 0; = (b1 , . . . , bg ) and C is the matrix defined above (ω = (iv) b γ∞ = b γ0 = 4πiCα, where b Cζ ). Proof. The meromorphic differential d is holomorphic in Σ \ {0j , ∞j | j = 1, 2} and all its residues are zero. That is, d is a differential of the second kind with singularities at 0j , ∞j . It follows from the properties of the differentials of the second kind [15, Chapter 8] that there exists a meromorphic differential γ∞ with the same principal part as d at ∞1 , ∞2 and holomorphic elsewhere. Setting γ0 := γ∞ − d, all the required properties in the statement are satisfied except possibly (iii) and (iv). Changing γ0 , γ∞ by adding an appropriate linear combination of the differentials ωj gives (iii).


243

It remains to prove (iv). Let P be a standard polygon for Σ and let P + , denote the preimage of Ω + under the identification map P → Σ . Since P is simply connected there is a holomorphic function f = (f1 , . . . , fg ) : P → Cg such that df = ω. Applying the bilinear relations and the properties of γ0 , γ∞ , we obtain γ0 = 2πi

Resp (γ0 f) = 2πi

p∈P

b

=−

Resp (γ0 f) =

p∈P +

ω =

∂P+

γ0 f

∂P+

(d)f =

∂P+

ω.

∂Ω +

Now, for each j , we have ωj = ∂Ω +

αr

r

= 4πi

∂Ω +

μ ω = αr cj s j λr r,s

λs−r−1 dλ ∂Ω +

cj r αr = 4πi(Cα)j .

r

Hence the second equality in (iv) holds. Since γ∞ − γ0 is exact, the first equality also holds.

2

Definition 3.7. Fix a point p0 ∈ Σ \ {∞j , 0j }. Define p A(p) =

ω ∈ Cg . p0

Of course, the value of this integral depends on the choice of a path between p0 and p. Therefore the expressions involving A(p) are, in general, multivalued. Given a constant v ∈ Cg and p ∈ Σ, we define F (p | v) = θ A(p) − v − K, B ∈ C, where θ is the Riemann theta function, θ (z, B) = n∈Zg exp(2πi( 12 nT Bn + nT z)), B is the matrix of b-periods of ω and K is the vector of Riemann constants [6]. Remark 3.8. The function F (p | v) is multivalued for its definition involves A(p). The effect of changing the integration path between p0 and p is determined by the quasi-periodicity properties of the theta function θ (z, B): 1 t T θ (z + n + Bm, B) = exp −2πi m Bm + m z θ (z, B). 2 We can now obtain the Σ -factorization for the function h that is referred to in Theorem 3.4. The method used to obtain this factorization is closely related to the construction of a Baker– Akhiezer function in [6]. In order to state the result we need one more definition.

244


− + − Definition 3.9. Let D+ = p+ 1 + · · · + pg and D− = p1 + · · · + pg be nonspecial divisors + − + − (see [11]) such that pi ∈ Ω and pi ∈ Ω .

Proposition 3.10. Let h : Σ → C be given by μ μ h = exp α1 + · · · + αg g λ λ

(3.6)

and let u = 2Cα (cf. Lemma 3.6). Then for all N ∈ N large enough, h has a Σ-factorization h = h+ rh−

(3.7)

with p h+ (p) = exp

γ∞

p0

F (p | A(D− ) + Nu )N , F (p | A(D− ))N

(3.8)

p F (p | A(D+ ) − Nu )N h− (p) = exp − γ0 , F (p | A(D+ ))N

(3.9)

p0

where the same path from p0 to p is used to evaluate the integrals of γ0 , γ∞ and A(p). The function r is given by F (p | A(D+ ) − Nu ) F (p | A(D− ) + r = h(p0 ) F (p | A(D+ )) F (p | A(D− ))

u −N N)

.

(3.10)

In particular r is a rational functional. − + − Proof. Since D+ = p+ 1 + · · · + pg and D− = p1 + · · · + pg are nonspecial divisors such that + − + − pi ∈ Ω and pi ∈ Ω , it follows [6, Chapter II] that the functions

F p A(D+ ) ,

F p A(D− )

− + − have exactly g zeros at the points p+ 1 , . . . , pg and p1 , . . . , pg , respectively. It also follows that we can choose N ∈ N large enough so that the functions F (p | A(D+ ) + Nu ), F (p | A(D− ) − Nu ) − + − are not identically zero. Their zeros are points q+ 1 , . . . , qg and q1 , . . . , qg such that

u + A q+ 1 + · · · + qg = A(D+ ) + N ,

u − A q− 1 + · · · + qg = A(D− ) − N .

− + − We assume N is large so that q+ i ∈ Ω and qi ∈ Ω . Consider the functions h+ and h− defined in (3.8) and (3.9). To show that h+ is independent of the path of integration we consider the effect of adding to it a cycle homologous to i (ni ai +


245

mi bi ). Set n = (n1 , . . . , ng ) and m = (m1 , . . . , mg ). By Lemma 3.6(iv) the first factor in the formula for h+ transforms as follows: p exp

γ∞

p γ∞ , exp 2πi u m exp →

T

p0

p0

while the second factor is multiplied by exp 2πiN[− 12 mT Bm − mT (A(p) + A(D− ) +

u N

− K)]

exp 2πiN [− 12 mT Bm − mT (A(p) + A(D− ) − K)]

= exp 2πi −uT m ,

hence the value of h+ remains unchanged. The verification that h− is well defined is similar. It is clear from the properties of γ∞ , γ0 and the preceding remarks that h±1 + is holomorphic ±1 + − in Ω and h− is holomorphic in Ω . Since d = γ∞ − γ0 (see Lemma 3.6) and h = exp it follows that F (p | A(D+ ) − Nu ) F (p | A(D− ) + h = h+ h− h(p0 ) F (p | A(D+ )) F (p | A(D− ))

u −N N)

.

2

Corollary 3.11. Let h and u be as in Proposition 3.10. If u ∈ Zg + BZg then h has a special Σ -factorization. Proof. Let u = n1 + Bm1 ∈ Zg + BZg . Then taking N = 1 in (3.8) and (3.9) we get well-defined elements of GCμ± (Γ ) given (up to multiplicative constants) by the following expressions: p h+ (p) = exp

γ∞ exp −mT1 A(p) ,

p0

p h− (p) = exp − γ0 exp mT1 A(p) . p0

Since h/(h+ h− ) = h(p0 ) we conclude that h = h+ h− h(p0 ) is a special Σ -factorization.

2

Theorem 3.12. Let f ∈ Cμ (Γ ) be such that log f ∈ Cμ (Γ ). Then f has a special Σ -factoriza 1 g g tion with respect to Γ if and only if 2πi Γ (log f )ω ∈ Z + BZ . Proof. From Theorem 3.4 it follows that f has a special Σ -factorization iff the same is true for function h = exp(α1 μ/λ + · · · + αg μ/λg ) where αk =

1 4πi

(log f )ζk Γ

246


and ζk = (λk−1 /μ) dλ. Denoting by ζ the vector of holomorphic differentials (ζ1 , . . . , ζg ) we have 1 1 C (log f )ζ = 2Cα. (log f )ω = 2πi 2πi Γ

Γ

By Corollary 3.11, u = 2Cα ∈ Zg + BZg is a sufficient condition for h to have a special Σfactorization. Hence it remains to show that this condition is necessary. Let P be a standard polygon for Σ and let P + , P − , respectively, denote the preimage of Ω + and Ω − under the identification map P → Σ . Recall from Definition 3.2 that D contains 2(g + 1) branch points (with g −1). If g > 0, P + is simply connected. If D contains no branch points (g = −1) then P + is a disjoint union of two simply connected sets. In either case we can define in P + a continuous branch of log f+ , which is holomorphic in the interior of P + . Hence we get 1 2πi

(log f+ )ω = 0.

(3.11)

∂P+

Proceeding similarly for P − , we obtain a continuous branch of log f− : P − → C such that 1 2πi

(log f− )ω = 0.

(3.12)

∂P−

Denoting by Γ ± the preimage of Γ in P ± (with the orientation induced from ∂P ± ) we obtain

(log f )ω =

(log f )ω =

Γ+

Γ

(log f+ )ω −

Γ+

(log f− )ω

Γ−

for f = f+ f− on Γ and Γ + , Γ − have opposite orientations. Now, if p1 , p2 ∈ P ± are two points with the same image under the map P → Σ then log f± (p1 ) = log f± (p2 ) + 2πi n± for some n± ∈ Z. From this and Eqs. (3.11) and (3.12) we conclude that 1 2πi

(log f ) ≡ Γ

1 2πi

(log f+ )ω −

∂P+

(log f− )ω

mod Zg + BZg

∂P−

≡ 0 mod Z + BZ . g

g

2

Next we illustrate how the factorization Theorem 3.4 can be used to solve a Riemann–Hilbert problem on the Riemann surface Σ. We restrict our study to a homogeneous problem f ψ+ = ψ− ,

(3.13)

where f ∈ Cμ (Γ ) and ψ+ , ψ− are assumed to be holomorphic respectively in Ω + , Ω − and continuous in Ω + , Ω − .


247

Let f = f+ rf− be a Σ -factorization of f then f ψ+ = ψ−

⇔

rf+ ψ+ = f−−1 ψ− .

Hence, if (ψ+ , ψ− ) is a solution of the Riemann–Hilbert problem (3.13), there exists R ∈ R(Σ) such that rf+ ψ+ in Ω + , R= f−−1 ψ− in Ω − . Let D = div(r|Ω + ) (the divisor of r|Ω + .) Then we have R ∈ L(−D) where we set

L(−D) := g ∈ R(Σ) div(R) − D 0 , following standard notation [11]. Conversely, if g ∈ L(−D) then the pair (ψ+ , ψ− ) = (gr −1 f+−1 , gf− ) is a solution problem (3.13). This proves: Proposition 3.13. Let D = div(r|Ω + ), then the space of solutions of (3.13) is isomorphic to L(−D). The computation of the dimension of the space L(−D) is a classical problem whose answer is given by the Riemann–Roch theorem [11,15]. In the special case where log f ∈ Cμ (Γ ) we obtain the following result. Proposition 3.14. Under the condition log f ∈ Cμ (Γ ), the Riemann–Hilbert problem (3.13) has a non-trivial solution if and only if f has a special Σ -factorization. In this case the dimension of the space of solutions is 1. Proof. If f has a special Σ -factorization it is clear that the space of solutions of problem (3.13) has dimension one. Conversely, assume the space of solutions of (3.13) has dimension one. We start by computing the degree of the divisor D. Clearly deg D = IndΓ (r). Since f±±1 is holomorphic in Ω ± it follows that IndΓ (f± ) = 0 and so IndΓ (r) = IndΓ (f ). The condition log f ∈ Cμ (Γ ) gives IndΓ (f ) = 0, hence deg D = 0. Since deg D = 0 we have dim L(−D) 1 and the equality occurs iff there is a rational function r− ∈ R(Σ) such that div(r− ) = D [15]. In this case, we set r+ = r/r− and r has a special Σ -factorization r = r+ r− . Therefore f = (f+ r+ )(r− f− ) is a special factorization for f . 2 For the example discussed in the next Section it will be convenient to have to the following generalization of Proposition 3.14. Proposition 3.15. Let D be a divisor on Σ. Then the space of solutions (ψ+ , ψ− ) of (3.13) satisfying div(ψ+ ) + div(ψ− ) D is isomorphic to the space of rational functions L(−D − D), where D = div(r|Ω + ).

(3.14)

248


Proof. Let f = f+ rf− be a Σ -factorization of f . As in the proof of Proposition 3.14 we obtain a rational function rf+ ψ+ in Ω + , R= f−−1 ψ− in Ω − . From (3.14) it follows that div(R) − D D , i.e., R ∈ L(−D − D). Conversely, if R ∈ L(−D − D) then the pair (Rr −1 f+−1 , Rf− ) is a solution of (3.13) satisfying (3.14). 2 4. Example In this section we illustrate the results of the previous sections by solving a Riemann–Hilbert problem corresponding to a 2 × 2 matrix symbol that belongs to a family of exponentials of rational matrices. Symbols of this form appear for example in the study of finite-dimensional integrable systems (cf. [4,13]). Specifically, let G = exp(tL),

(4.1)

where L is a rational 2 × 2 matrix function and t ∈ R. The symbol (4.1) belongs to the class

with Q

= J L, where C(Q) J =

0 1 . −1 0

Indeed we have

= exp tLT J L exp(tL) = exp tLT J exp(tL)L. GT QG

(4.2)

But for any 2 × 2 matrix A we have AT J A = (det A)J

which, introduced in (4.2), gives

= (det G)J L = (det G)Q,

GT QG

i.e., G ∈ C(Q). For our example we take

v L(λ) = −ku

u , −v

(4.3)

where u and v are Laurent polynomials in λ given by u = aλ − xλ−1 , with a, x, k positive real constants.

v = xλ−1 ,

λ ∈ ∂D,

(4.4)


249

The Riemann–Hilbert problem we have to solve is Gφ + = φ − ,

(4.5)

where G is given by (4.1). Recall from Section 2 that the Riemann surface associated with problem (4.5) is given by the equation

μ2 = − det Q.

by Q = λQ

we obtain C(Q) = C(Q)

with det Q a polynomial in λ. Hence the Replacing Q associated Riemann surface Σ has the form considered in Section 3, i.e., it is the compactification of the algebraic curve μ2 = p(λ),

(4.6)

where p is the polynomial p(λ) = −ka 2 λ4 + 2akxλ2 + x 2 (1 − k). The zeros of p(λ) are given by 1

1 x ± xk − 2 x 1 ± k− 2 . λ = = 2a 2a

2

For k > 1 all zeros of p(λ) are real and symmetric in pairs. We shall consider three distinct cases: (i) x < (ii)

a , 1+k −1/2

a 1+k −1/2

(iii) x >

which leads to all zeros of p(λ) inside D;

<x
0, x ∈ R3 , and denotes the Laplacian on R3 . The functions σ and b are Lipschitz continuous, and the process F˙ is the formal derivative of a Gaussian random field, white in time and correlated in space defined as follows. Let f (x) = ϕ(x)kβ (x),

(2)

where ϕ is a smooth positive function and kβ denotes the Riesz kernel kβ (x) = |x|−β , β ∈ ]0, 2[. We shall assume that f defines a tempered measure and R3

μ(dξ ) < ∞, 1 + |ξ |2

(3)

where μ = F −1 (f ) and F denotes the Fourier transform operator. This condition is satisfied for instance for densities of the form (2) with ϕ(x) = exp(−σ 2 |x|2 /2) and β ∈ ]0, 2[ (see [5]). Let D(R4 ) be the space of Schwartz test functions (see [16]). Then, on some probability space, there exists a Gaussian process F = (F (ϕ), ϕ ∈ D(R4 )) with mean zero and covariance functional defined by E F (ϕ)F (ψ) =

R+

ds

˜ dx f (x) ϕ(s) ∗ ψ(s) (x),

(4)

R3

˜ x) = ψ(s, −x). where ψ(s, Riesz kernels are a class of singular correlation functions which have already appeared in several papers related with the stochastic heat and wave equations, for instance in [2,3,7–9]. 2 We recall that the fundamental solution G(t) associated to the wave operator L = ∂t∂ 2 − in 1 dimension three is given by G(t) = 4πt σt , t > 0, where σt denotes the uniform surface measure on the sphere of radius t ∈ [0, T ], hence with total mass 4πt 2 . The properties of G(t) together with the particular form of the covariance of the noise play a crucial role in giving a rigorous formulation to the initial value problem (1). Here, we shall follow the same formulation as in [5] which for the purpose of existence and uniqueness of solution of (1) introduces a localization of the SPDE by means of a set related with the past light cone, as follows. Let D be a bounded domain in R3 . Set KaD (t) = y ∈ R3 : d(y, D) a(T − t) ,

t ∈ [0, T ],

(5)

where a 1 and d denotes the Euclidean distance. Then, a solution to the SPDE (1) in D is an adapted, mean-square continuous stochastic process (u(t)1KaD (t) , t ∈ [0, T ]) with values in L2 (R3 ), satisfying

M. Sanz-Solé / Journal of Functional Analysis 255 (2008) 255–281

257

d G(t) ∗ v0 + G(t) ∗ v˜0 (·) u(t, ·)1KaD (t) (·) = 1KaD (t) (·) dt t + 1KaD (t) (·)

G(t − s, · − y)σ u(s, y) 1KaD (s) (y)M(ds, dy)

0 R3

t + 1KaD (t) (·)

ds G(t − s) ∗ b u(s, ·) 1KaD (s) (·) ,

(6)

0

almost surely, for any t ∈ [0, T ], where we consider the stochastic integral defined in [4] and M denotes the martingale measure derived from F (see [3]). The following result is a quotation of [5, Theorem 4.11] and will be invoked repeatedly in this paper. Theorem 1.1. Assume that: (a) the covariance density is of the form (2) with the covariance factor ϕ bounded and positive, ϕ ∈ C 1 (Rd ) and ∇ϕ ∈ Cbδ (Rd ), for some δ ∈ ]0, 1]; (b) the initial values v0 , v˜0 are such that v0 ∈ C 2 (R3 ), and v0 and v˜0 are Hölder continuous with orders γ1 , γ2 ∈ ]0, 1], respectively; (c) the coefficients σ and b are Lipschitz. 1+δ Then for any q ∈ [2, ∞[ and α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [, there is C > 0 such that for (t, x), (t¯, y) ∈ [0, T ] × D, q αq E u(t, x) − u(t¯, y) C |t − t¯| + |x − y| . (7)

In particular, almost surely, the stochastic process (u(t, x), (t, x) ∈ [0, T ] × D) solution of (6) has α-Hölder continuous sample paths, jointly in (t, x), and q sup (8) E u(t, x) < ∞ (t,x)∈[0,T ]×D

for any q ∈ [1, ∞[. In this paper we are interested in studying the properties of the density of the solution of (6) as a function of (t, x) ∈ ]0, T ] × D, where D is a bounded subset of R3 . We shall denote this density by pt,x (y). We shall prove that (t, x) → pt,x (y) is jointly Hölder continuous, uniformly in y on compact sets. This question is trivial in the very particular case where the initial conditions v0 , v˜0 and the coefficient b vanish, and the coefficient σ is a constant function. In fact, with these assumptions and σ = 1 the solution to Eq. (6) is a Gaussian process, centered, stationary in the space variable, and with σt2

2 := E u(t, x) =

t ds 0

μ(dξ ) R3

sin2 (s|ξ |) . |ξ |2

258


From the expression pt,x (y) =

√1 2π σt

exp(− |y|2 ) it is not difficult to prove that 2

2σt

pt,x (y) − pt¯,x (y) C(t0 , t1 , D)|t − t¯|, for 0 < t0 t < t¯ t1 , D ⊂ R3 . However, in the general situation that we are considering in this article, the problem becomes much more involved. Suppose that v0 , v˜0 are null functions, assume also that the covariance of the process F is given by (4) with dx f (x) replaced by Γ (dx), where Γ is a non-negative, tempered, non-negative definite measure. Set μ = F −1 (Γ ). We introduce an assumption on μ, denoted by (Hη ), saying

μ(dξ ) < ∞, for some value of η ∈ ]0, 1]. Assume that the coefficients σ and b are of that R3 (1+|ξ |2 )η class C 1 with bounded derivatives and that (Hη ) holds for some η ∈ (0, 1). Then, the existence of the density pt,x at any fixed point (t, x) ∈ ]0, T ] × D has been established in [13]. Moreover, assuming that σ and b are C ∞ functions with bounded derivatives of order greater or equal to one, and that (Hη ) holds for some η ∈ (0, 12 ), it is proved in [14] that y → pt,x (y) is a C ∞ function. We refer the reader to [15] for results on applications of Malliavin calculus to the analysis of probability laws of SPDEs. In [12], it is shown that the extension of Walsh’s integral introduced in [2] does not require for the integrands any stationary property in the spatial variable. As a consequence of this fact, the results of [2,13,14] and [15] concerning the stochastic wave equation can be formulated with non-null deterministic initial conditions. In addition, the solution of the equation in this setting coincides with the solution to (6). Furthermore, in the particular case of absolutely continuous covariance measures Γ (dx) = f (x) dx satisfying (3) the existence and smoothness of the density pt,x are proved in [12] under the weaker assumption (H1 ). Hence, on the basis of the above mentioned references and remarks, we can write the next statement, which together with Theorem 1.1 are the starting point of our work.

Theorem 1.2. Assume assumptions (a) and (b) of Theorem 1.1. Suppose also that σ and b are C ∞ functions with bounded derivatives of order greater or equal to one, and inf{|σ (z)|, z ∈ R} σ0 > 0. Then, for any fixed (t, x) ∈ ]0, T ] × D, the law of the real valued random variable u(t, x) solution to (6) has a density pt,x ∈ C ∞ . The main purpose of this paper is to prove that with the assumptions of this theorem, for any y ∈ R, the mapping (t, x) ∈ ]0, T ] × R3 → pt,x (y) 1+δ is jointly α-Hölder continuous with α ∈ ]0, inf(γ1 , γ2 , 2−β 2 , 2 )[ (see Theorem 2.1 in Section 1). For stochastic differential equations and some finite-dimensional stochastic evolution systems with an underlying semigroup structure one can find results of this type for instance in [6]. For SPDEs the problem has not been yet very much explored. To the best of our knowledge, this issue has only been studied for the stochastic heat equation in spatial dimension d = 1 in [10] and for the wave equation with d = 2 in [9] (see [1] and [7] for the existence and regularity of the density for these two types of SPDEs). It is worthy noticing that in these two references, the Hölder degree regularity of pt,x (y) in (t, x) is better than for the sample paths of the solution process u(t, x), while in the equation under consideration we obtain the same order. As it will


259

become clear from the proof, the reason is the rather degenerate character of the fundamental solution of the wave equation in dimension three. The method of our proof is based on the integration by parts formula of Malliavin calculus, as in the above mentioned references. We next give the main ideas and steps of the proof. Fix y ∈ R and let (gn,y , n 1) be a sequence of smooth functions converging pointwise to the Dirac delta function δ{y} . Fix t, t¯ ∈ ]0, T ], x, x¯ ∈ D and assume that we can prove sup E gn,y u(t, x) − gn,y u(t¯, x) ¯ C |t − t¯|β1 + |x − x| ¯ β2 ,

(9)

n∈N,y∈K

for some β1 , β2 > 0, where K ⊂ R. Then, since pt,x (y) = E[δ{y} (u(t, x))] (see [17, Theorem 1.12] for a rigorous meaning of this identity), by passing to the limit as n → ∞, we will have joint Hölder continuity of the mapping (t, x) ∈ ]0, T ] × D → pt,x (y) ∈ R with degree β1 in t and β2 in x, uniformly in y ∈ K. An estimate like (9) is obtained by the following procedure. For simplicity we write g instead ¯ around u(t, x) up to a certain order r0 of gn,y . We first consider a Taylor expansion of g(u(t¯, x)) chosen in such a way to obtain optimal values of β1 and β2 . Then for any r r0 , we estimate terms of the type (r) r , E g u(t, x) u(t, x) − u(t¯, x) ¯ and the term corresponding to the rest in the Taylor expansion, whose structure is similar. For this we use the version of the integration by parts formula for one-dimensional random variables given in [17, Lemma 2, p. 54] (see also [11, Eqs. (2.29)–(2.31)]) which we now quote as a lemma. Lemma 1.3. On an abstract Wiener space (Ω, H, P ), we consider two real-valued random varip ∞ r ables ξ and Z such that ξ ∈ D∞ , Dξ −1 p2 L (Ω), Z ∈ D . Let g be a function in C , H ∈ for some r 1. Denote by g˜ the antiderivative of g. Then, the following formula holds: ˜ )Hr+1 (Z, ξ ) , E g (r) (ξ )Z = E g(ξ

(10)

where Hr , r 1, is defined recursively by ZDξ , H1 (Z, ξ ) = δ Dξ 2H ZDξ , Hr+1 (Z, ξ ) = δ Hr (Z, ξ ) Dξ 2H

r 1.

In this lemma, δ stands for the adjoint operator of the Malliavin derivative, also termed divergence operator or Skorohod integral and we have used the notations of [11] and [15], as we shall do throughout the paper when referring to notions and results of Malliavin calculus. The abstract Wiener space that we shall consider here is the one associated with the Gaussian process F restricted to the time interval [0, T ], as is described in [15, Section 6.1]. For the sake of completeness and its further use, we recall that H := HT , HT = L2 ([0, T ]; H) and that H is the completion of the inner product space consisting of test functions endowed with the inner product

260


ϕ, ψ H =

˜ dx f (x)(ϕ ∗ ψ)(x).

R3

Assume that the function g˜ in Lemma 1.3 is bounded. From (10) it clearly follows that (r) E g (ξ )Z g ˜ ∞ Hr+1 (Z, ξ )L1 (Ω) . (11) Furthermore, as a consequence of the continuity property of the Skorohod integral and the assumptions on ξ , for any r 1, k 1 and p ∈ (1, ∞), Hr (Z, ξ ) CZk+r,4r p k,p (see [10, Corollary 4.1]). Consequently, under the previous assumptions from (11) we obtain (r) E g (ξ )Z Cg ˜ ∞ Zr+1,4r+1 . (12) Let us recall that for a natural number k and a real number p ∈ [1, ∞[, Zk,p = ZLp (Ω) +

k r D Z r=1

Lp (Ω;HT⊗r )

.

We shall apply (12) mainly to ξ := u(t, x) and Z := (u(t, x) − u(t¯, x)) ¯ r , for natural values of r. Under the hypotheses of Theorem 2.1 the assumptions of Lemma 1.3 are satisfied (see [14] and [15, Chapters 7 and 8]). Thus we face the problem of giving upper bounds for (u(t, x) − u(t¯, x)) ¯ r r+1,4r+1 . Malliavin derivatives of the solution of (1) satisfy evolution equations (see [15, Theorem 7.1] k u(t, x), (t, x) ∈ [0, T ] × D) is a and [14]). Indeed, for x ∈ D, and a natural number k 1, (D·,∗ ⊗k HT -valued process satisfying Dτ,∗ u(t, x) = 0 for τ > t, and for τ t it is the solution of the evolution equation t k k u(t, x) = Zτ,∗ (t, x) + Dτ,∗

G(t − s, x − y) 0 R3

k × Γ k σ, u(s, y) + σ u(s, y) Dτ,∗ u(s, y) M(ds, dy) +

t ds 0

k × Γ k b, u(s, y) + b u(t − s, x − y) Dτ,∗ u(t − s, x − y) .

G(s, dy) R3

(13)

In this equation, (Z k (t, x), (t, x) ∈ [0, T ] × D) is a HT⊗k -valued stochastic process and for a given function g ∈ C k and a random variable X ∈ Dk,2 , Γ k (g, X) = D k (g(X)) − g (X)D k X. The solution of (13) satisfies u(s, y) < +∞ sup (14) k,p (t,x)∈[0,T ]×D

for any p ∈ [1, ∞[ (see [15, Theorem 7.1]).


261

In the next section, we shall make use of the explicit form of (13) for k = 1. In this case 1 Zτ,∗ (t, x) = G(t − τ, x − ∗)σ u(τ, ∗)

(15)

and Γ 1 (g, X) = 0. With some effort, using the tools on stochastic integration of Hilbert-valued processes developed in [15] it can be proved that the conclusions of Theorem 1.1 also apply to the HT⊗k valued stochastic process solution to (13). More precisely, for any k 1, q ∈ [2, ∞[ and 1+δ ¯ α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [, there is C > 0 such that for (t, x), (t , y) ∈ [0, T ] × D, u(t, x) − u(t¯, y) C |t − t¯| + |x − y| α . k,q

(16)

Hence, with the Hölder continuity property on u(t, x) and its Malliavin derivatives we may be able to prove (9) for specific values of β1 , β2 . We shall fix what is the top order r0 in the Taylor expansion of g(u(t¯.x)). ¯ Clearly, the lower exponents βi should come from the first order term. However, in the examples studied so far, terms of first and second order give the same exponent. For Eq. (6) the situation is different. Already at the first order level of the expansion, we shall see that the contribution of the pathwise integral involving the coefficient b is of the same order than the Hölder continuity exponent given in Theorem 1.1. Clearly, the second order term would provide twice the Hölder continuity degree. Therefore, a Taylor expansion of first order gives the best possible result. However, to conclude whether the regularity of the density pt,x in (t, x) is the same as that of the sample paths of u(t, x), we have to check that the contribution to the first order term in the Taylor expansion of the stochastic integral is not worse than that of the pathwise integral. This explains the strategy of the proof of the main result in the next section. 2. Main result Throughout this section D denotes a fixed bounded domain of R3 and C will be any positive finite constant. We assume that (3) holds. Our purpose is to prove the following theorem. Theorem 2.1. Assume that: (a) the covariance density is of the form (2) and the covariance density factor ϕ is bounded and positive, ϕ ∈ C 1 (Rd ) and ∇ϕ ∈ Cbδ (Rd ) for some δ ∈ ]0, 1]; (b) the initial values v0 , v˜0 are such that v0 ∈ C 2 (R3 ), and v0 and v˜0 are Hölder continuous with orders γ1 , γ2 ∈ ]0, 1], respectively; (c) the coefficients σ and b are C ∞ functions with bounded derivatives of order greater or equal to one, and there exists σ0 > 0 such that inf{|σ (z)|, z ∈ R} σ0 . Then the mapping (t, x) ∈ ]0, T ] × D → pt,x (y) 1+δ is α-Hölder continuous jointly in (t, x) with α ∈ ]0, inf(γ1 , γ2 , 2−β 2 , 2 )[, uniformly in y ∈ R.

262


Proof. Fix y ∈ R and let (gn,y , n 1) be a sequence of regular functions converging pointwise to δ{y} as n → ∞; for example, a sequence of Gaussian kernels with mean y and variances converging to zero. We may assume that the corresponding antiderivatives g˜ n,y are uniformly bounded by 1. To simplify the notation, we shall write g instead of gn,y . Step 1 (Time increments). For (t, x) ∈ [0, T ] × D we consider the Taylor expansion E g u(t + h, x) − g u(t, x) = E g u(t, x) u(t + h, x) − u(t, x) 2 ˜ x, h) u(t + h, x) − u(t, x) , + E g u(t,

(17)

where h > 0 and u(t, ˜ x, h) denotes a random variable lying on the segment determined by u(t + h, x) and u(t, x). First order term. Set T1 (t, x, h) = E g u(t, x) u(t + h, x) − u(t, x) . We aim to prove that sup

(t,x)∈[0,T ]×D

T1 (t, x, h) Chα ,

1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. By using Eq. (6), we write T1 (t, x, h) 3i=1 T1,i (t, x, h), with

d G(t + h) ∗ v0 + G(t + h) ∗ v˜0 (x) T1,1 (t, x, h) = E g u(t, x) dt d − G(t) ∗ v0 + G(t) ∗ v˜0 (x) , dt t+h T1,2 (t, x, h) = E g u(t, x) ds G(t + h − s, dz)b u(s, x − z) 0

t −

ds 0

R3

G(t − s, dz)b u(s, x − z)

R3

,

t+h T1,3 (t, x, h) = E g u(t, x) G(t + h − s, x − z)σ u(s, x − z) M(ds, dz) 0 R3

t − 0 R3

G(t − s, x − z)σ u(s, x − z) M(ds, dz) .

In fact, by our choice of (t, x) all the indicator functions in (6) take the value 1.

(18)


263

We shall apply repeatedly the inequality (12) with r = 1, ξ := u(t, x) and different choices of Z. To begin with, we take Z :=

d d G(t + h) ∗ v0 + G(t + h) ∗ v˜0 (x) − G(t) ∗ v0 + G(t) ∗ v˜0 (x). dt dt

Since Z is deterministic, Zk,p = |Z|, for any k and p. Then, applying (12) and [5, Lemma 4.9] yields sup

(t,x)∈]0,T ]×D

T1,1 (t, x, h) Chγ1 ∧γ2 .

(19)

We next study the term T1,2 (t, x, h). Let t+h T1,2,1 (t, x, h) = E g u(t, x) ds G(t + h − s, dz)b u(s, x − z) . t

We apply (12) to Z :=

t+h t

ds

R3

R3

G(t + h − s, dz)b(u(s, x − z)) and consider the measure on

given by dsG(t + h − s, dz) with total mass [t, t applying Minkowski’s inequality, we obtain + h] × R3

h2 2

and an arbitrary p ∈ [1, ∞[. By

t+h ds G(t + h − s, dz)b u(s, x − z) t

R3

2,p

t+h ds G(t + h − s, dz)b u(s, x − z) 2,p . t

R3

By the chain rule of Malliavin calculus, b u(s, y)

2,p

2 C 1 + u(s, y)2,p + u(s, y)2,2p .

Consequently, sup (s,y)∈[0,T ]×D2T

b u(s, y)

2,p

0, we finally ob(1) tain I1 , is bounded uniformly in t, h ∈ [0, T ]. (2) Similar but simpler arguments show that the same property hold for I1 . η For any λ ∈ [0, 1] set ψ(λ) = k3−α˜ (w + λ t−r ). It is easy to check that |ψ (λ)| Ck4−α˜ (w + η λ t−r ). Moreover, for |w| 2, and |η| = t − r, w + λ η |w| − λ η |w| − 1 |w| , t −r t − r 2 by the triangular inequality.

274


Thus, (3) I1

t C

dr 0

t +h−r t −r

G(t − r, dξ )G(t − r, dη) Bt−r (0) Bt−r (0)

1

×

kα+β (η − ξ − hw) dw ˜

(B2

(0))c

t 2 dr FG(r)(ξ ) k3−(α+β) dw k4−α˜ (w) dξ . ˜ r

C (B2

0

η dλ k4−α˜ w + λ t −r

(0))c

0

(37)

R3

For α˜ ∈ ]0, 1[ the integral (B2 (0))c dw k4−α˜ (w) is finite. Moreover, if α˜ + β ∈ ]0, 2[ the last integral in (37) is also finite, owing to [5, Lemma 2.3] applied to the value b = 1. This lead us to conclude that I1(3) is bounded uniformly in t, h ∈ [0, T ]. Summarizing, as a consequence of (35), (36) and the preceding discussion, we have proved that t

μh2 (dr; dξ, dη) Chα˜ ,

sup t∈[0,t]

(38)

0 Bt−r (0) Bt−r (0)

with α˜ ∈ ]0, 2−β 2 [. We can now apply Hölder’s inequality with respect to the measure μh2 (dr; dξ, dη). By virtue of (38), the linear growth of σ and (8) we obtain sup

(t,x)∈[0,T ]×D

T1,3,2,2,2 Chαp ,

with α ∈ ]0, 2−β 2 [. Finally, the estimates (31), (33) and (39) imply that h B (t, x) Chα , sup 1 p (t,x)∈[0,T ]×D

(39)

(40)

and a fortiori sup

(t,x)∈[0,T ]×D

α D·,∗ u(t, x), B h (t, x) ·,∗ H p Ch , T

(41)

1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. This finishes the analysis of the · p contribution to the left-hand side of (27). h (t, x) We next consider the Lp (Ω, HT )-norm of D D·,∗ u(t, x), B·,∗ HT . As in the previous h h ˜ with arbitrary t˜ ∈ [0, T ], x˜ ∈ R3 . By virtue of (26) step, we shall replace B·,∗ (t, x) by B·,∗ (t˜, x) and (41) it suffices to study the Lp (Ω, HT )-norm of DBih (t, x; t˜, x) ˜ for i = 1, 2. We start with ˜ the analysis of B2h (t, x; t˜, x).


275

By applying the differential rules of Malliavin calculus we have D.∗ B2h (t, x; t˜, x) ˜ ˜ = G(t − ·, x − ∗)σ u(.∗) D(s, y; t˜, x) t +

G(t − s, x − y) σ u(s, y) D.∗ u(s, y)D(s, y; t˜, x) ˜

0 R3

+ σ u(s, y) D.∗ D(s, y; t˜, x) ˜ M(ds, dy). Applying Hölder’s inequality and using that σ is bounded, we obtain, as in (28), p E G(t − ·, x − ∗)σ u(.∗) D(s, y; t˜, x) ˜ H

T

t C

p ˜ . E D(s, y; t˜, x)

ds sup

(42)

y∈KaD (s)

0

For fixed t˜, x˜ we consider the HT -valued process defined by K(s, y; t˜, x) ˜ = σ u(s, y) D.∗ u(s, y)D(s, y; t˜, x) ˜ + σ u(s, y) D.∗ D(s, y; t˜, x), ˜

(43)

(s, y) ∈ [0, T ] × R3 , for which we have p 1 2p 1 p 2 E Du(s, y)2p 2 ˜ H σ ∞ E D(s, y; t˜, x) E K(s, y; t˜, x) ˜ HT T p p ˜ . + σ ∞ E DD(s, y; t˜, x)

(44)

HT

We can apply the Lp -estimates for stochastic integrals with respect to the Gaussian process M of Hilbert-valued integrands (see [15, Eq. (6.8) of Theorem 6.1] and [12, p. 289]) yielding t p G(t − s, x − y)K(s, y; t˜, x)M(ds, ˜ dy) p

L (Ω,HT )

0 R3

t C

ds sup y∈KaD (s)

0

p 2p 1 2 + DD(s, y; t˜, x) ˜ Lp (Ω;H ) . E D(s, y; t˜, x) ˜

(45)

T

By taking t˜ = t and x˜ = x and considering the inequalities (42), (45), we obtain DB h (t, x)p p 2 L (Ω;H

T)

t C

ds sup 0

y∈KaD (s)

p 2p 1 h h (t, x) H + E D·,∗ u(s, y), B·,∗ (t, x) H 2 E D·,∗ u(s, y), B·,∗ T

p h + D D·,∗ u(s, y), B·,∗ (t, x) H Lp (Ω;H ) . T

T

T

(46)

276


Then, (41) and Gronwall’s lemma yield p h sup D D·,∗ u(t, x), B·,∗ (t, x) H Lp (Ω;H

sup

t∈[0,T ] x∈KaD (t)

C

sup

T)

T

p sup DB1h (t, x)Lp (Ω;H

T

t∈[0,T ] x∈KaD (t)

αp , + h )

(47)

1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. The last step of the proof consist of checking that for an arbitrary bounded set D ⊂ R3 , DB h (t, x) p sup Chα , (48) 1 L (Ω;H ) (t,x)∈[0,T ]×D

T

1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. The proof of this fact can be done following the same lines as for (40). We apply the results on the densities ν(dr; dξ, dη), μh1 (dr; dξ, dη), μh2 (dr; dξ, dη), defined in (30), (32) respectively, proved so far. Instead of the process {σ (u(s, y)), (s, y) ∈ [0, T ] × R3 } and the Lp (Ω)-norm, we shall deal here with the HT -valued process {D(σ (u(s, y))), (s, y) ∈ [0, T ] × R3 } and the Lp (Ω; HT )-norm. In addition to (7), we should also apply (16) and (14). We leave the details to the reader. Together with (19) and (24) this proves (18) and concludes the proof of the first step of the proof.

Remark 2.2. Applying first (12) and then estimates for the · 2,p -norm of the stochastic integral leads to t σ u(s, y) G(t + h − s, x − y) − G(t − s.x − y) M(ds, dy) T1,3,2 C 0 R3

2,43

α

Ch 2 . Thus, we loose accuracy. This may be a justification for a pretty tricky approach in the preceding proof. The rest in the time expansion. The second and last term in (17) to be examined is 2 ˜ x, h) u(t + h, x) − u(t, x) . R(t, x, h) = E g u(t, We shall apply (12) to the random variables ξ := u(t, ˜ x, h) and Z := (u(t + h, x) − u(t, x))2 . For this, we have to make sure that the assumptions of Lemma 1.3 are satisfied. For Z := (u(t + h, x) − u(t, x))2 , and the two choices of ξ —u(t, x) and u(t + h, x)—this has been proved in [14]. Then it suffices to remark that the norm · HT as well as · −1 HT define convex functions and use the definition of u(t, ˜ x, h) to conclude. Consequently, 2 R(t, x, h) C u(t + h, x) − u(t, x)3,43 .


277

Owing to (7) and (16) we conclude that sup

(t,x)∈[0,T ]×D

R(t, x, h) Ch2α ,

(49)

1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. The estimates (18) and (49) show that

sup

E g u(t + h, x) − g u(t, x) Chα ,

(t,x)∈[0,T ]×D

1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. Therefore the mapping t ∈ ]0, T [ → pt,x (y) is Hölder con1+δ 3 tinuous of degree α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [, uniformly in y ∈ R varying on bounded sets.

Step 2 (Space increments). Fix t ∈ ]0, T ] and consider the Taylor expansion ¯ − u(t, x) E g u(t, x) ¯ − g u(t, x) = E g u(t, x) u(t, x) 2 + E g u(t, ˆ x, x) ¯ u(t, x) ¯ − u(t, x) ,

(50)

where x, x¯ ∈ D and u(t, ˆ x, x) ¯ denotes a random variable lying on the segment determined by u(t, x) ¯ and u(t, x). First order term. Our aim is to prove that ¯ α, ¯ − u(t, x) C|x − x| sup E g u(t, x) u(t, x)

t∈[0,T ]

(51)

1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. As we did for the time increments, we consider Eq. (6) and write 3 E g u(t, x) u(t, x) ¯ − u(t, x) = Si (t, x, x), ¯ i=1

with ¯ S1 (t, x, x) d d ¯ − G(t) ∗ v0 + G(t) ∗ v˜0 (x) G(t) ∗ v0 + G(t) ∗ v˜0 (x) , = E g u(t, x) dt dt S2 (t, x, x) ¯ t = E g u(t, x) ds G(t − s, dz) b u(s, x¯ − z) − b u(s, x − z) , 0

R3

278


S3 (t, x, x) ¯ t G(t − s, x¯ − z) − G(t − s, x − z) σ u(s, y) M(ds, dz) . = E g u(t, x) 0 R3

Let us consider S1 (t, x, x). ¯ As for the term T1,1 (t, x, h), we first apply the inequality (12) and notice that d d ¯ − G(t) ∗ v0 + G(t) ∗ v˜0 (x) G(t) ∗ v0 + G(t) ∗ v˜0 (x) Z(t; x, x) ¯ := dt dt is deterministic. Thus, it suffices to estimate the absolute value of the random variable Z(t; x, x) ¯ defined before. For this, we apply [5, Lemmas 4.2 and 4.4] which tell us that the fractional Sobolev norm of any integration degree p 2 and differential order ρ < γ1 ∧ γ2 is bounded. Hence, since p is arbitrary, by the Sobolev embedding theorem we have that ¯ ρ, ¯ C|x − x| ¯ sup C Z(t; x, x) (52) sup S1 (t, x, x) t∈[0,T ]

t∈[0,T ]

with ρ < γ1 ∧ γ2 . ¯ By virtue of (12), it suffices to We continue the proof with the study of the term S2 (t, x, x). find an upper bound of t ds G(t − s, dz) b u(s, x¯ − z) − b u(s, x − z) R3

0

2,42

in terms of a power of |x − x|. ¯ The measure on [0, t] × R3 defined by ds G(t − s, dz) is finite. Hence, we can apply Minkowski’s inequality and obtain for any p ∈ [1, ∞[ t ds G(t − s, dz) b u(s, x¯ − z) − b u(s, x − z) R3

0

t

ds 0

2,p

G(t − s, dz)b u(s, x¯ − z) − b u(s, x − z) 2,p

R3

C|x − x| ¯ α, 1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. The last inequality is obtained by using that b and its derivatives are Lipschitz continuous and bounded functions, and by applying (7) and (16). Hence,

¯ C|x − x| ¯ α, sup S2 (t, x, x)

t∈[0,T ]

for any α ∈ ]0, γ1 ∧ γ2 ∧

2−β 2

∧

1+δ 2 [.

(53)


279

To analyze S3 (t, x, x) ¯ we proceed in a similar manner as for T1,3 (t, x, h) by applying first [15, Proposition 3.9] and then (12). We obtain ¯ = E g u(t, x) D·,∗ u(t, x), σ u(·, ∗) S3 (t, x, x) × G(t − ·, x¯ − ∗) − G(t − ·, x − ∗) 1]0,t] (·) H T C D·,∗ u(t, x), σ u(·, ∗) G(t − ·, x¯ − ∗) − G(t − ·, x − ∗) 1]0,t] (·) H 3,43 . T

Notice that the last expression has a similar structure than the right-hand side of (25) where h (t, x) := G(t + h − ·, x − ∗) − G(t − ·, x − ∗) is replaced by G(t − ·, x¯ − ∗) − G(t − ·, x − ∗). B.∗ Hence, we can proceed as in the analysis of the time increments to see that it suffices to deduce an estimate for G(t − ·, x − ∗)σ u(·, ∗) , σ u(·, ∗) G(t − ·, x¯ − ∗) − G(t − ·, x − ∗)

HT 3,p ,

for any p ∈ [1, ∞[. To pursue the proof, we split the argument of the above expression into two terms

t S3,1 (t, x, x) ¯ =

dr

dη G(t − r, x − ξ )

dξ

0

R3

R3

t

× σ u(r, ξ ) f (ξ − η)σ u(r, η) G(t − r, x¯ − η),

S3,2 (t, x, x) ¯ =

dr 0

dη G(t − r, x − ξ )

dξ R3

R3

× σ u(r, ξ ) f (ξ − η)σ u(r, η) G(t − r, x − η),

and we apply the change of variables (ξ → x − ξ, η → x¯ − η), (ξ → x − ξ, η → x − η), respectively. We obtain S3,1 (t, x, x) ¯ − S3,2 (t, x, x) ¯

t =

dr

G(t − r, dξ )G(t − r, dη)f (ξ − η)σ u(r, x − ξ )

R3 R3

0

× σ u(r, x¯ − η) − σ u(r, x − η)

t +

dr 0

G(t − r, dξ )G(t − r, dη) f x − x¯ − (ξ − η) − f (ξ − η)

R3 R3

× σ u(r, x − ξ ) σ u(r, x¯ − η) .

(54)

By Minkowski’s inequality the · k,p -norm of the first term in the right-hand side of (54) is bounded by

280


t

G(t − r, dξ )G(t − r, dη)f (ξ − η)

dr R3

0

R3

× σ u(r, x − ξ ) σ u(r, x¯ − η) − σ u(r, x − η) k,p

C|x − x| ¯ α,

(55)

where the very last upper bound follows from (7), (16) and (14). For the second term of the right-hand side of (54) we apply [5, Lemma 6.1] which implies

t sup

t∈[0,T ]

dr 0

¯ α˜ , G(r, dξ )G(r, dη) f x − x¯ − (ξ − η) − f (ξ − η) C|x − x|

R3 R3

with α˜ ∈ ]0, (2 − β) ∧ 1[. From this and the properties (8), (14), we obtain the upper bound C|x − x| ¯ α˜ . Hence we conclude ¯ C|x − x| ¯ α, sup S3 (t, x, x)

t∈[0,T ]

for any α ∈ ]0, γ1 ∧ γ2 ∧

2−β 2

∧

1+δ 2 [.

(56)

With (52)–(56), we have proved (51).

The rest term in the space expansion. The contribution of the second order term in (50) comes from the estimate 2 E g u(t, ˆ x, x) ¯ u(t, x) ¯ − u(t, x) Cg ∞ h2α , which is a consequence of (7). Hence, we have proved that for any fixed y ∈ R3 the mapping x ∈ D → pt,x (y) is Hölder 1+δ continuous of degree α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [, uniformly in t ∈ ]0, T ]. The proof of the theorem is now complete.

2

Acknowledgments This paper has been written when the author was visiting the Institute Mittag-Leffler at Djursholm (Sweden) during a semester devoted to SPDEs. She would like to express her gratitude for the inspiring environment, the very kind hospitality and the financial support provided by this institution. References [1] V. Bally, E. Pardoux, Malliavin calculus for white noise driven parabolic SPDEs, Potential Anal. 9 (1998) 27–64. [2] R.C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous SPDE’s, Electron. J. Probab. 4 (1999). [3] R.C. Dalang, N.E. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab. 26 (1) (1998) 187–212.


281

[4] R.C. Dalang, C. Mueller, Some non-linear SPDE’s that are second order in time, Electron. J. Probab. 8 (1) (2003) 1–21. [5] R.C. Dalang, M. Sanz-Solé, Hölder–Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., in press. [6] S. Kusuoka, D. Stroock, Applications of Malliavin calculus, Part II, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 32 (1985) 1–76. [7] A. Millet, M. Sanz-Solé, A stochastic wave equation in two space dimensions: Smoothness of the law, Ann. Probab. 27 (1999) 803–844. [8] A. Millet, M. Sanz-Solé, Approximation and support theorem for a wave equation in two space dimensions, Bernoulli 6 (5) (2000) 887–915. [9] A. Millet, P.-L. Morien, On a stochastic wave equation in two space dimensions: Regularity of the solution and its density, Stochastic Process. Appl. 86 (1) (2000) 141–162. [10] P.-L. Morien, The Hölder and the Besov regularity of the density for the solution of a parabolic stochastic partial differential equation, Bernoulli 5 (2) (1999) 275–298. [11] D. Nualart, The Malliavin Calculus and Related Topics, second ed., Probab. Appl. (N. Y.), Springer-Verlag, Berlin, 2006. [12] D. Nualart, L. Quer-Sardanyons, Existence and smoothness of the density for spatially homogeneous SPDEs, Potential Anal. 27 (2007) 281–299. [13] L. Quer-Sardanyons, M. Sanz-Solé, Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation, J. Funct. Anal. 206 (1) (2004) 1–32. [14] L. Quer-Sardanyons, M. Sanz-Solé, A stochastic wave equation in dimension 3: Smoothness of the law, Bernoulli 10 (1) (2004) 165–186. [15] M. Sanz-Solé, Malliavin Calculus with Applications to Stochastic Partial Differential Equations, Fund. Sci. Math., EPFL Press, 2005, distributed by CCR Press. [16] L. Schwartz, Théorie des distributions, Hermann, Paris, 1966. [17] S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Inst. Fund. Res./SpringerVerlag, Bombay, 1984.


Gaussian bounds for degenerate parabolic equations D. Cruz-Uribe a,∗ , Cristian Rios b a Department of Mathematics, Trinity College, Hartford, CT 06106-3100, USA b University of Calgary, Calgary, AB T2N1N4, Canada

Received 5 June 2007; accepted 29 January 2008 Available online 15 May 2008 Communicated by Daniel W. Stroock

Abstract Let A be a real symmetric, degenerate elliptic matrix whose degeneracy is controlled by a weight w in the A2 or QC class. We show that there is a heat kernel Wt (x, y) associated to the parabolic equation wut = div A∇u, and Wt satisfies classic Gaussian bounds: 2 Wt (x, y) C1 exp −C2 |x − y| . t t n/2 We then use this bound to derive a number of other properties of the kernel. © 2008 Published by Elsevier Inc. Keywords: Kernel; Gaussian bounds; Degenerate elliptic; Degenerate parabolic

1. Introduction 1.1. Overview of the problem In this paper we study the degenerate parabolic equation div A∇u = w

∂u . ∂t


E-mail addresses: [email protected] (D. Cruz-Uribe), [email protected] (C. Rios). 0022-1236/$ – see front matter © 2008 Published by Elsevier Inc. doi:10.1016/j.jfa.2008.01.017

(1.1)

284

D. Cruz-Uribe, C. Rios / Journal of Functional Analysis 255 (2008) 283–312

where A = (Aij (x))ni,j =1 is a matrix of complex-valued, measurable functions satisfying the degenerate ellipticity condition ⎧ ⎪ ⎪ ⎨

λw(x)|ξ |2 ReAξ, ξ = Re

⎪ ⎪ ⎩

Aξ, η Λw(x)|ξ ||η|,

n

Aij (x)ξj ξ¯i ,

i,j =1

(1.2)

for some λ, Λ, 0 < λ Λ < ∞, and all ξ, η ∈ Cn . The weight w that controls the degeneracy is a non-negative, locally integrable function which we assume is either in the Muckenhoupt class A2 or in the class of QC weights, which arise in the study of quasi-conformal mappings. For ease of reference we make the following definition. Definition 1.1. Let En (w, λ, Λ) denote the class of n×n matrices of complex-valued, measurable functions satisfying the degenerate ellipticity condition (1.2). Remark 1.2. If A ∈ En (w, λ, Λ), then so is its adjoint, A∗ . Our goal is to show that if A is real symmetric, then there exists a heat kernel associated to this equation: an L∞ function Wt (x, y) such that given a function f ∈ Cc∞ ,

u(x, t) =

Wt (x, y)f (y) dy

(1.3)

Rn

is a solution of (1.1) satisfying the initial condition u(x, 0) = f (x). Further, we will show that the heat kernel satisfies Gaussian bounds: 2 Wt (x, y) C1 exp −C2 |x − y| . t t n/2

(1.4)

A central feature of our results is that the weight does not appear in (1.3) or in (1.4)—the kernel is defined with respect to Lebesgue measure. When w ≡ 1 (that is, A is uniformly elliptic), these results are well known. For the existence of the heat kernel, see Friedman [18]; Gaussian bounds are due to Aronson [1]. Solutions of Eq. (1.1) with A real symmetric were first treated by Chiarenza and Frasca [8]. Chiarenza and Serapioni [11], showed that if n 3 and w ∈ A2 , then solutions of (1.1) satisfied a Harnack inequality on standard parabolic cylinders. Chiarenza and Franciosi [7] proved the same result for n 2 and w a QC weight. (See Proposition 3.8 below.) More recently, Ishige [24] has proved a Harnack inequality and continuity of the solution for a more general parabolic equation with lower order terms, and for A any real matrix that satisfies (1.2) with w ∈ A2 . When A is a complex matrix, much less is known even when A is uniformly elliptic. Auscher [2] showed that Gaussian bounds hold for L∞ perturbations of real symmetric matrices. Auscher, McIntosh and Tchamitchian [5] showed that Gaussian bounds hold if A satisfies (1.2) and is Hölder continuous, and when n 2 smoothness is not necessary. Ouhabaz [27] has shown that if A is a Lipschitz perturbation of a real, uniformly elliptic matrix, then (1.4) holds. (Note that neither of the last two results assumes that A is symmetric.) On the other hand,


285

Auscher, Coulhon and Tchamitchian [4], building on an example in [26], showed that if n 5, then Gaussian bounds for the heat kernel need not hold in general. Given a degenerate elliptic matrix A, it might seem more natural to consider the usual parabolic equation ∂u . ∂t

div A∇u =

(1.5)

When w ∈ A2 this equation has been studied by several authors; their results indicate that, surprisingly, (1.1) is the “right” generalization to consider. Chiarenza and Serapioni [9,10] showed that the assumption w ∈ A2 did not imply local regularity of the solution. Local regularity required higher integrability of the weight w −1 ; e.g., the stronger hypothesis w ∈ A1+2/n , n 3. Further, they showed that in this case solutions to (1.5) satisfied a Harnack inequality on the weighted parabolic cylinders Qx,t (r) = B2r (x) × (t − hx (r), t), where

−n/2

hx (r) =

w(y)

2/n dy

.

Br (x)

Gutiérrez and Nelson [22] showed that with the same assumption the heat kernel associated with Eq. (1.5), W˜ t (x, y), satisfied a weighted Gaussian estimate: there exist C1 , C2 , α > 0 such that α W˜ t (x, y) C1 exp −C2 hx (|x − y|) . (1.6) n t h−1 x (t) A sharper but more complicated version of this inequality was later proved by Gutiérrez and Wheeden [23]. 1.2. Statement of the main results Hereafter, let w be either an A2 weight or a QC weight. If w ∈ QC, then we will also assume n 2; otherwise we can take n 1. Define the differential operator Lw = −w −1 div A∇, and let e−t Lw be the semigroup generated by Lw . Given any f in the domain of e−t Lw , u(x, t) = e−t Lw f (x) is a solution of the initial value problem ⎧ ⎨ ∂u = −Lw u, ∂t ⎩ u(x, 0) = f (x).

(1.7)

Our main result is the following. Theorem 1.3. If A ∈ En (w, λ, Λ) is real symmetric, then there exists a heat kernel Wt (x, y) associated to the operator e−t Lw such that for all f ∈ Cc∞ , e−t Lw f (x) =

Wt (x, y)f (y) dy. Rn

(1.8)

286


Furthermore, for all t > 0 and x, y ∈ Rn , the kernel Wt satisfies the Gaussian bounds 2 Wt (x, y) C1 exp −C2 |x − y| , t t n/2

(1.9)

and the Hölder continuity estimates μ |h| |x − y|2 Wt (x + h, y) − Wt (x, y) C1 , exp −C 2 t t n/2 t 1/2 + |x − y| μ |h| |x − y|2 Wt (x, y + h) − Wt (x, y) C1 , exp −C2 t t n/2 t 1/2 + |x − y|

(1.10) (1.11)

where h ∈ Rn is such that 2|h| t 1/2 + |x − y|. The constants C1 , C2 and μ depend only on n, w, λ, and Λ. Remark 1.4. Perhaps the most outstanding aspect of our results is that they appear to be independent of the weight. Thus the kernel Wt (x, y) is integrated with respect to Lebesgue measure and the upper bounds (1.9) on Wt are the classic Gaussian bounds for the heat kernel [1]. This is in sharp contrast to the weighted inequalities (1.6) obtained in [22,23] for a related class of degenerate parabolic equations. The classical nature of our estimates will be very useful for the weighted Kato square root problem, among other applications. Remark 1.5. We believe that Theorem 1.3 should hold for all real matrices (i.e., not necessarily symmetric) that satisfy (1.2). Towards proving this, we remark that the approach taken by Ouhabaz [27] for uniformly elliptic matrices can be adapted to the case of degenerate matrices. However, this technique relies on the Sobolev embedding theorem, and in the weighted case this result is only true on bounded domains (see [17]). Central to the proof of Theorem 1.3 is an estimate which is of independent interest. When w ≡ 1 this inequality appears in Auscher et al. [6], without proof; they note that in the special case of the Laplace–Beltrami operator the result is due to Gaffney [19] and that it can be proved by modifying an argument due to Davies [14]. Theorem 1.6. Given A ∈ En (w, λ, Λ), let E and F be two closed subsets of Rn and set d = dist(E, F ). Then there exist constants c, C > 0 such that given any function f ∈ L2 (w) with supp(f ) ⊂ E, −t L e w f

L2 (w,F )

Ce−cd

2 /t

f L2 (w,E) .

The constants depend only n, λ, and Λ. As a consequence of the Gaussian bounds we have the conservation property. Theorem 1.7. Given a matrix A ∈ En (w, λ, Λ), suppose that the heat kernel of the associated semigroup e−t Lw satisfies (1.4). Then e−t Lw 1 = 1, where 1 denotes the function on Rn which equals the constant 1.


287

As another consequence we get bounds for the derivative of the semigroup. Theorem 1.8. With the same hypotheses as Theorem 1.7, the semigroup Vt , t 0, defined by ∂ −t Lw f (x) = Lw e−t Lw f (x), e Vt f (x) = − ∂t

is given by a heat kernel Vt (x, y): for all f ∈ Cc∞ ,

Vt f (x) =

Vt (x, y)f (y) dy. Rn

Furthermore, for all t > 0 and x, y ∈ Rn , Vt satisfies the Gaussian bounds Vt (x, y)

C1 t n/2+1

|x − y|2 exp −C2 , t

and the Hölder continuity estimates

|x − y|2 , exp −C2 t t n/2+1 μ |h| |x − y|2 Vt (x, y + h) − Vt (x, y) C1 , exp −C2 t t n/2+1 t 1/2 + |x − y|

Vt (x + h, y) − Vt (x, y)

C1

|h| t 1/2 + |x − y|

μ

where h ∈ Rn is such that 2|h| t 1/2 + |x − y|. Finally, Vt has zero integral: for all x ∈ Rn ,

Vt 1 =

Vt (x, y) dy = 0. Rn

Remark 1.9. In both Theorems 1.7 and 1.8 we do not assume that A is real symmetric; we only assume that the associated heat kernel satisfies Gaussian bounds. 1.3. Organization The remainder of this paper is organized as follows. In Sections 2 and 3 we gather some basic result about weighted Sobolev spaces and semigroups. In Section 4 we prove the Gaffney-type estimate in Theorem 1.6. In Section 5 we prove Theorems 1.3 and 1.8. In Section 6 we prove Theorem 1.7. Throughout this paper, all notation will be standard or defined as needed. Λ and λ will always denote the ellipticity constants in (1.2). Unless otherwise specified, C, c, etc., will denote an arbitrary constant which may depend on the dimension n, λ and Λ, and the A2 or QC constant associated with a weight w. Sometimes for clarity we will specify the dependence of the constant by writing C(n), C(n, w), etc. Given an angle θ , define the sector Σ(θ ) = {z ∈ C: z = 0, |arg(z)| < θ }. Finally, given a bounded operator T on L2 (w, Ω), let

T B(L2 (w,Ω)) denote its operator norm.

288


2. Weighted Sobolev spaces The basic theory of weighted Sobolev spaces was developed by Fabes, Kenig and Serapioni [17] and we will follow their development. (See also Turesson [29].) One key difference, however, is that they only dealt with real-valued functions; complex-valued functions introduce a number of minor technical complications. 2.1. Weights By a weight we mean a non-negative, locally integrable function. Given a weight w and a measurable set E, we let

w(E) =

w(x) dx. E

We are concerned with two weight classes: the Muckenhoupt class A2 and the quasiconformal weights QC. The first class is straightforward to define; we give a slightly more general definition which we will need below. Given p, 1 < p < ∞, and a weight w, we say that w ∈ Ap if there exists a constant [w]Ap (referred to as the Ap constant of w) such that p−1

1 1 1−p sup w(x) dx w(x) dx = [w]Ap < ∞, |Q| |Q| Q Q

Q

where the supremum is taken over all cubes Q in Rn . Note that if w ∈ Ap , then it follows by a change of variables that for all a ∈ R, b ∈ Rn , wab (x) = w(ax + b) ∈ Ap , and [wab ]Ap = [w]Ap . Let A∞ denote the union of the Ap classes. The properties of these weights can be found in [16, 20,21]. To define the class QC, fix n 2 and let f : Rn → Rn be a bijection whose components fi have distributional derivatives in LnLoc . Let f (x) denote the Jacobian of f and let |f | denote its determinant. Then f is quasi-conformal if there exists a constant k such that

n ∂fi (x)2 ∂xj

1/2

1/n k f (x) .

i,j =1

Given such an f , the function w = |f |1−2/n is called a QC weight. We will denote the best constant k associated with f , and so with w, by [w]QC . As before we have that [wab ]QC = [w]QC . David and Semmes [12] showed that if w ∈ QC, then w ∈ A∞ , but QC weights have much more structure than an arbitrary A∞ weight. The classes QC and A2 are different. Thus, a QC weight need not be in A2 : for example, w(x) = |x|α(n−2) ∈ QC for all α > −1, but w is not in A2 if α n/(n − 2). Conversely, |x|β , −n < β −(n − 2), is in A2 but not in QC. (See [17, p. 106].)


289

2.2. Weighted Sobolev spaces Given an open set Ω in Rn , and a weight w in either A2 or QC, let Lp (w, Ω), 1 p < ∞, be the Banach function space of complex-valued functions with norm

f

Lp (w,Ω)

=

f (x)p w(x) dx

1/p .

Rn

L2 (w, Ω) is a Hilbert space with inner product

f, gw = f (x)g(x)w(x) dx. Ω

When p = ∞, L∞ (w, Ω) consists of functions essentially bounded with respect to the measure w(x) dx; since w ∈ A∞ , L∞ (w, Ω) = L∞ (Ω). Define the Sobolev space H01 (w, Ω) to be the closure of Cc∞ (Ω) with respect to the norm

f H 1 (w,Ω) = f L2 (w,Ω) + ∇f L2 (w,Ω) . 0

(If Ω = Rn , then we write simply L2 (w) and H01 (w).) That this closure is well defined is a consequence of the following lemma [17, pp. 90, 106]. Lemma 2.1. Given a weight w in A2 or QC, suppose {gk } ⊂ Cc∞ (Ω) is such that

gk L2 (w,Ω) → 0. If g is such that ∇gk − g L2 (w,Ω) → 0, then g = 0. Thus, if f ∈ H01 (w, Ω), there exists a vector-valued function in L2 (w, Ω), denoted ∇f , and there exists {gk } ⊂ Cc∞ (Ω) such that in L2 (w, Ω), {gk } converges to f and {∇gk } converges to ∇f . If w ∈ A2 , w −1 is locally integrable, so by Hölder’s inequality, f, ∇f ∈ L1Loc (Ω). Hence, we can integrate by parts: for all φ ∈ Cc∞ (Ω),

∇f (x) φ(x) dx = − f (x) ∇φ(x) dx. Ω

Ω

However, if w ∈ QC, then f and ∇f need not be locally integrable, so this formula does not make sense. This fact complicates the proof of some of the differentiation formulas given below. To establish additional details about the structure of H01 (w, Ω), note first that if f = u + iv ∈ 1 H0 (w, Ω), then by passing to the real and imaginary parts of an approximating sequence of Cc∞ functions, we have that u, v ∈ H01 (w, Ω). Lemma 2.2. Let θ : R2 → C have continuous partial derivatives and be such that |∇θ | M. If f = u + iv ∈ H01 (w, Ω), then the function θ (f ) = θ (u, v) ∈ H01 (w, Ω) and almost everywhere in Ω, ∂u ∂v ∂θ (f ) ∂θ ∂θ (u, v) (u, v) = + , ∂xi ∂s ∂xi ∂t ∂xi

1 i n.

(2.1)

290


Proof. Let {φk } ⊂ Cc∞ (Ω) be such that φk → f in H01 (w, Ω). By passing to a subsequence, we may assume that it converges pointwise almost everywhere as well. For each k 1, θ (φk ) is continuously differentiable and has compact support in Ω, and so is in H01 (w, Ω). Let uk = Re(φk ), vk = Im(φk ). By the chain rule, for 1 i n, ∂θ (φk ) ∂θ ∂uk ∂θ ∂vk = + . (uk , vk ) (uk , vk ) ∂xi ∂s ∂xi ∂t ∂xi By assumption, θ is Lipschitz, so |θ (φk )(x) − θ (f )(x)| M|φk (x) − f (x)|. Hence,

θ (φk )(x) − θ (f )(x)2 w(x) dx

Ω

1/2

M

φk (x) − f (x)2 w(x) dx

1/2 .

Ω

Since φk → f in L2 (w, Ω), we have that θ (φk ) → θ (f ) in L2 (w, Ω). Since u, v ∈ L2 (w, Ω) and ∇θ ∈ L∞ , the right-hand side of (2.1) is in L2 (w, Ω), 1 i n. Temporarily denote it by Fi . Then ∂θ (φk ) ∂θ ∂u ∂θ ∂u ∂uk ∂θ ∂uk ∂θ ∂x − Fi ∂s (uk , vk ) ∂x − ∂s (u, v) ∂x + ∂t (uk , vk ) ∂x − ∂t (u, v) ∂x i i i i i = Ak + Bk . To complete the proof it will suffice to show that Ak , Bk → 0 in L2 (w, Ω). For in that case, since H01 (w, Ω) is complete we have that θ (f ) ∈ H01 (w, Ω) and (2.1) holds. We will show Ak converges to 0; the proof for Bk is identical. We have that ∂θ ∂u ∂uk ∂θ ∂u ∂θ ∂u ∂θ A (uk , vk ) (uk , vk ) (uk , vk ) (u, v) + − − ∂s ∂xi ∂s ∂xi ∂s ∂xi ∂s ∂xi ∂u ∂uk ∂u ∂θ ∂θ . + (uk , vk ) − (u, v) M − ∂x ∂x ∂s ∂s ∂x i

i

i

Since u ∈ H01 (w, Ω), the first term tends to 0 in L2 (w, Ω). The second term is dominated point∂u |. Further, since φk → f pointwise and θ has continuous partial derivatives, the wise by 2M| ∂x i second term tends to 0 pointwise. Therefore, by the dominated convergence theorem it also tends to 0 in L2 (w, Ω), and we are done. 2 Lemma 2.3. Let f = u + iv ∈ H01 (w, Ω). Then |f | ∈ H01 (w, Ω) and for 1 i n and almost every x ∈ Ω, ∂u ∂v u(x) ∂x (x) + v(x) ∂x (x) ∂|f | i i χ{f (x)=0} . (x) = ∂xi |f (x)|

(2.2)

√ Proof. For each > 0, define the function θ (s, t) = s 2 + t 2 + 2 − . Then θ has continuous, bounded derivatives, and so by Lemma 2.2, θ (f ) ∈ H01 (w, Ω) and for 1 i n, ∂u ∂v ∂θ (f ) u(x) ∂xi (x) + v(x) ∂xi (x) = ∂xi u(x)2 + v(x)2 + 2


291

almost everywhere. Since θ (f )(x) |f (x)| and θ (f ) → |f | pointwise, by the dominated convergence theorem it converges in L2 (w, Ω). ∂u ∂u Denote the right-hand side of (2.2) by Fi . Then |Fi | | ∂x (x)| + | ∂x (x)| ∈ L2 (w, Ω). Furi i

(f ) (f ) ther, we have that | ∂θ∂x | |Fi | and | ∂θ∂x | converges pointwise to Fi . So again by dominated i i 2 convergence it converges in L (w, Ω). This yields the desired result. 2

Corollary 2.4. If f ∈ H01 (w, Ω) is real-valued, then f + = max(f, 0) and f − = min(f, 0) are both in H01 (w, Ω), and for 1 i n, ∂f + ∂f = χ{f >0} , ∂xi ∂xi

∂f − ∂f = χ{f 0: cos arg(z) − sin arg(z) / tan(τ ) = cos arg(z) 1 − tan arg(z) / tan(τ ) > cos arg(z) 1 − tan(τ )/ tan(τ ) = 0. Since A ∈ En (w, λ, Λ), for all ξ ∈ Cn , 2 2 2 ReAξ, ξ + ImAξ, ξ = Aξ, ξ Λ2 w(x)2 |ξ |4 , 2 ReAξ, ξ λ2 w(x)2 |ξ |4 . Together, these imply that ImAξ, ξ Λ2 − λ2 w(x)|ξ |2 , and so RezAξ, ξ = |z| cos(arg z) ReAξ, ξ − sin(arg z) ImAξ, ξ |z| cos(arg z)λw(x)|ξ |2 − sin(arg z) Λ2 − λ2 w(x)|ξ |2 Λ2 2 −1 = λ|z|w(x)|ξ | cos(arg z) − sin(arg z) λ2 λ|z|w(x)|ξ |2 cos arg(z) − sin arg(z) / tan(τ ) . On the other hand it is immediate that for all ξ , η, |zAξ, η| Λ|z|w(x)|ξ ||η|.

2

3.2. Resolvents and semigroups Though the operator Lw is not bounded, there are two closely related bounded operators associated to it: the resolvent (λI + Lw )−1 and the semigroup e−t Lw . The operator Lw is the infinitesimal generator of a strongly continuous, contraction semigroup e−t Lw . The range of e−t Lw is D(Lw ). Further, e−t Lw is a holomorphic semigroup in the sector Σ(π/2 − ω), where ω = arctan(Λ/λ). (See [27, Theorem 1.52].) Proposition 3.4. If the matrix A is real, then the semigroup e−t Lw is real and positive: if f ∈ L2 (w, Ω) is real-valued, then so is e−t Lw f ; if f is non-negative, then so is e−t Lw f .

294


Proof. By [27, Theorem 2.5], e−t Lw is real if given any f = u + iv, a(u, v) ∈ R. This is immediate from the definition of a given that A is real. By [27, Theorem 2.6], e−t Lw is positive if, in addition, u+ , u− ∈ H01 (w, Ω), and a(u+ , u− ) 0. The first condition is just Corollary 2.4. The second also follows from Corollary 2.4: ∇u+ and ∇u− have disjoint supports, so a(u+ , u− ) = 0. 2 For every t > 0, tI + Lw is an invertible operator from D(Lw ) into L2 (w, Ω). Its inverse, the resolvent of Lw , (tI + Lw )−1 , is a bounded operator on L2 (w, Ω) satisfying t (tI + Lw )−1 2 B(L (w,Ω)) 1.

(3.2)

An important consequence of the fact that e−t Lw is a holomorphic semigroup is the fact that the resolvent is bounded on a sector of the complex plane larger than the half-plane {Re z > 0}. (See [27, Theorem 1.45].) Proposition 3.5. Let ω = arctan(Λ/λ). Then for every θ ∈ (π/2, π − ω) there exists Mθ > 0 such that sup z(zI + Lw )−1 B(L2 (w,Ω)) Mθ .

(3.3)

z∈Σ(θ)

The next proposition shows that e−t Lw and (zI + Lw )−1 can be characterized in terms of one another. (See [25, pp. 482–489].) Proposition 3.6. Let ω = arctan(Λ/λ). Then for f ∈ L2 (w, Ω) and z ∈ Σ(π/2 − ω), e

−zLw

1 f= 2πi

ezζ (ζ I + Lw )−1 f dζ,

(3.4)

Γ

where the path Γ is the union of the rays γ ± = {z ∈ C: z = re±iθ , r R > 0} and the arc γ0 = {z ∈ C: z = Reiψ , |ψ| θ }, going around the origin counter-clockwise, and π − ω > θ > π/2 + |arg(z)|. Further, if Re(z) > 0, −1

(zI + Lw )

∞ =

e−zt e−t Lw dt.

(3.5)

0

More generally, if z ∈ Σ(π − ω), then this identity remains true if we instead integrate over the ray t = reiθ , r > 0, where θ is such that e−t Lw is bounded and Re(zt) > 0. Inequality (3.5) is referred to as the Laplace identity. Another consequence of the fact that e−t Lw is a holomorphic semigroup is that the operator

∂ −t Lw ∂ −t Lw e e f = −Lw e−t Lw f f= ∂t ∂t

(3.6)


295

is again a bounded operator on L2 (w, Ω). More precisely, there exists C > 0 such that for all t > 0, Lw e−t Lw

B(L2 (w,Ω))

C . t

(See Yosida [30, p. 239].) ∗ Finally, we note that the operators e−¯zLw and (¯zI + L∗w )−1 are the adjoints of e−zLw and (zI + Lw )−1 , respectively. 3.3. Solutions of the parabolic equation Here we define a weak solution to the parabolic equation (1.7) and show that the semigroup e−t Lw yields a solution. Our definition is the one given by Chiarenza and Frasca [8]. Fix T > 0; we say that u ∈ L2 ([0, T ], H01 (w, Ω)) if for any t ∈ [0, T ] the function u(·, t) ∈ H01 (w, Ω) and

T

u(·, t)2

H01 (w,Ω)

dt < ∞.

0

We define u ∈ L2 ([0, T ], L2 (w, Ω)) in the same way, but with the H01 norm replaced by the L2 norm. We say that a function v ∈ W0 ([0, T ]) if v ∈ L2 ([0, T ], H01 (w, Ω)), vt ∈ L2 ([0, T ], L2 (w, Ω)), and v(x, 0) = v(x, T ) = 0, x ∈ Ω. A function u is a weak solution of (1.1) if u ∈ L2 ([0, T ], H01 (w, Ω)) and for all v ∈ W0 ([0, T ]),

T

A∇u(x) · ∇v(x) − w(x)u(x)vt (x) dx dt = 0.

(3.7)

0 Ω

Proposition 3.7. Given f ∈ L2 (w, Ω), define the function u(x, t) = e−t Lw f (x), t > 0. Then u ∈ L2 ([0, T ], L2 (w, Ω)) and (3.7) holds. Since u(x, 0) = f (x), u is a solution of (1.7). Proof. We first show that u ∈ L2 ([0, T ], L2 (w, Ω)).

T

T

u(·, t)2

dt C H 1 (w,Ω) 0

u(x, t)2 w(x) dx dt + C

0 Ω

0

T

∇u(x, t)2 w(x) dx dt.

0 Ω

We estimate each integral separately. The first is straightforward: since u = e−t Lw f and the semigroup is a contraction on L2 (w, Ω),

T

0 Ω

u(x, t)2 w(x) dx dt

T

f (x)2 w(x) dx dt T f 2 2

L (w,Ω)

0 Ω

< ∞.

296


To estimate the second integral, note that u(·, t) = e−t Lw f (·) ∈ D(Lw ), and so a(u, u) = Lw u, uw . Therefore, by the ellipticity conditions (1.2), again using the fact that e−t Lw is a contraction,

T

∇u(x, t)2 w(x) dx dt 1 λ

0 Ω

T

Re a u(·, t), u(·, t) dt

0

=

1 λ

1 = λ

T

Lw u(x, t)u(x, t)w(x) dx dt

Re 0

Ω

T

Re

Lw e−t Lw f (x)e−t Lw f (x)w(x) dx dt

Ω

0

T

2 −1 ∂ −t Lw Re e f (x) w(x) dx dt 2λ ∂t

Ω 0 1 Re f (x)2 − e−T Lw f (x)2 w(x) dx = 2λ

=

Ω

C f 2L2 (w) < ∞. To show that u satisfies (3.7), first note that ut = −Lw e−t Lw f = −Lw u. Now fix v ∈ W0 ([0, T ]). Then, since v(x, 0) = v(x, T ) = 0, by Fubini’s theorem and integration by parts in t,

T

w(x)u(x, t)vt (x, t) dx dt = 0 Ω

T u(x, t)v(x, t)|T0

−

Ω

ut (x, t)v(x, t) dt w(x) dx 0

T =

Lw u(x, t)v(x, t) dt w(x) dx Ω 0

T

a u(·, t), v(·, t) dt

= 0

T

=

A∇u(x, t) · ∇v(x, t) dx dt. 0 Ω

Eq. (3.7) now follows immediately.

2


297

Finally, we give a precise statement of the Harnack inequality. Given a pair (x0 , t0 ), x0 ∈ Ω, t > 0, define the parabolic cylinders Qρ (x0 , t0 ) = (x, t): |t − t0 | < ρ 2 , |x − x0 | < 2ρ , 3 2 ρ + 2 Qρ (x0 , t0 ) = (x, t): ρ < t − t0 < ρ , |x − x0 | < , 4 2 3 2 1 2 ρ − . Qρ (x0 , t0 ) = (x, t): − ρ < t − t0 < − ρ , |x − x0 | < 4 4 2 Proposition 3.8. Let A ∈ En (w, λ, Λ) be real symmetric with w ∈ A2 (n 1) or w ∈ QC (n 2). Then there exists γ = γ (n, λ, Λ, w) > 0 such that if u(x, t) is a non-negative solution of (1.1) in Qρ (x0 , t0 ), ρ > 0, then sup

Q− ρ (x0 ,t0 )

u(x, t) γ

inf

Q+ ρ (x0 ,t0 )

u(x, t).

Proof. When w ∈ QC and n 2 this result is due to Chiarenza and Franciosi [7]. When w ∈ A2 and n 3, it is due to Chiarenza and Serapioni [11]; the cases n = 1 and 2 for w ∈ A2 follow from the higher-dimensional case via an argument shown to the second author by M. Safonov. Here we sketch the details for n = 2; the case n = 1 is treated in essentially the same way. Let u(x, y, t) be a non-negative solution of (1.1) in Qρ (x0 , y0 , t0 ), ρ > 0, i.e. div A∇u(x, y, t) = ∂t u(x, y, t), ˜ where A ∈ E2 (w, λ, Λ). For ρ z 3ρ define v(x, y, z, t) = zu(x, y, t) and let A(x, y, z) be the 3 × 3 matrix ˜ A(x, y, z) =

A(x, y) 0

0 w(x, y)

.

Then a short calculation shows that v is a solution of the three-dimensional parabolic equation ˜ is ˜ (x,y,z) v = vt in the set Qρ (x0 , y0 , 2ρ, t0 ) ⊂ Qρ (x0 , y0 , t0 ) × (ρ, 3ρ). The matrix A div(x,y,z) A∇ 3 ˜ ˜ λ, Λ), where w(x, ˜ y, z) = w(x, y) ∈ A2 (R ). Further, v is positive real symmetric and A ∈ E3 (w, in Qρ (x0 , y0 , 2ρ, t0 ). Therefore, by the Harnack inequality in dimension n = 3, sup

Q− ρ (x0 ,y0 ,2ρ,t0 )

v(x, y, z, t) γ

inf

Q+ ρ (x0 ,y0 ,2ρ,t0 )

v(x, y, z, t),

and so sup

Q− ρ (x0 ,y0 ,t0 )

u(x, y, t) 3γ

inf

Q+ ρ (x0 ,y0 ,t0 )

u(x, y, t).

2

298


4. Proof of Theorem 1.6 The proof of Theorem 1.6 is based on the following lemma. It is a generalization to degenerate elliptic operators and complex time of a result due to Auscher et al. [6, Lemma 2.1], and our proof is based on theirs. Throughout this section we assume that w ∈ A2 or QC and A ∈ En (w, λ, Λ). Lemma 4.1. Let E and F be two closed sets in Rn and let d = dist(E, F ). Let τ = arctan √

λ Λ2 −λ2

and fix ν, 0 < ν < π/2 + τ , and z ∈ Σ(ν). Then there exist positive constants C and c depending on n, Λ, λ, ν such that for all f ∈ L2 (w) with support in E,

√ z(zI + Lw )−1 f (x)2 w(x) dx Ce−cd |z| f (x)2 w(x) dx. (4.1) F

E

Proof. We consider first the case where ν < π/2. Without loss of generality we may also assume ν > π/4, so tan(ν) > 1. Since z(zI + Lw )−1 = (I + z−1 Lw )−1 and |z| > 0, if we make the change of variables z → z−1 , then to establish (4.1) it will suffice to prove

d (I + zLw )−1 f (x)2 w(x) dx Ce−c √|z| f (x)2 w(x) dx. (4.2) F

E

Define uz = (I + zLw )−1 f ; then f = uz + zLw uz . By the definition of Lw , if v ∈ H01 (w), then Lw uz , vw = a(uz , v), so

uz (x)v(x)w(x) dx + z A∇uz (x) · ∇v(x) dx = f (x)v(x)w(x) dx. Rn

Rn

Rn

Let v = η2 ut , where η ∈ C0∞ is a non-negative function with supp(η) ∈ Rn \ E that will be fixed below. Since f and η have disjoint supports, the right-hand side is zero. Hence, if we rearrange terms we get that

z 2 u (x) η(x)2 w(x) dx + z

Rn

= −2z

A∇uz (x) · ∇uz (x)η(x)2 dx

Rn

η(x)uz (x)A∇uz (x) · ∇η(x) dx.

(4.3)

Rn

Now take the absolute value of both sides of (4.3). Since A ∈ En (w, λ, Λ), we can apply (1.2) and Young’s inequality to get for any > 0 that

2z η(x)uz (x)A∇uz (x) · ∇η(x) dx Rn

2|z| Rn

η(x)uz (x)A∇uz (x) · ∇η dx


2|z|Λ

η(x)uz (x)∇uz (x)∇η(x)w(x) dx

Rn

|z|Λ

299

z 2 u (x) ∇η(x)2 w(x) dx + |z|Λ

2 η(x)2 ∇uz (x) w(x) dx.

(4.4)

Rn

To estimate the absolute value of the left-hand side of (4.3), let

R=

z 2 u (x) η(x)2 w(x) dx,

n s + it = z,

S + iT =

Rn

A∇uz (x) · ∇uz (x)η(x)2 dx. R

Clearly, R 0, and since z ∈ Σ(ν), s > 0 and |t| tan(ν)s. Again by (1.2),

η(x) Re A∇uz (x) · ∇uz (x) dx λ

S=

2

Rn

2 η(x)2 ∇uz (x) w(x) dx 0,

Rn

and

|T |

η(x)2 Im A∇uz (x) · ∇uz (x) dx Λ

Rn

2 Λ η(x)2 ∇uz (x) w(x) dx S. λ

Rn

Let γ = λ(tan(ν)Λ)−1 < 1. Then if we combine these inequalities we get that the absolute value of the left-hand side of (4.3) is equal to

1/2 (R + sS − tT )2 + (sT + tS)2 1/2 R 2 + s 2 S 2 + t 2 T 2 − 2R|tT | + 2RsS + t 2 S 2 1/2 R 2 + t 2 T 2 − 2(1 − γ )R|tT | + |z|2 S 2 1/2 γ R 2 + γ t 2 T 2 + |z|2 S 2 √ γ |z| R + S. 2 2

If we now combine this estimate with (4.4) we get that √

2 γ z 2 |z|λ 2 u (x) η(x) w(x) dx + η(x)2 ∇uz (x) w(x) dx 2 2 Rn

|z|Λ

Rn

z 2 u (x) ∇η(x)2 w(x) dx + |z|Λ

Rn

2 η(x)2 ∇uz (x) w(x) dx.

(4.5)

300


λ Let = 2Λ ; then the second term on each side of the inequality cancel. Now let η = eαθ − 1, where supp(θ ) ⊂ Rn \ E, 0 θ 1, θ = 1 on F , ∇θ ∞ Kd , and

1/2 √ λ γ d α= √ . 2 2 16Λ K |z| Then (4.5) yields √

√

2 2 γ 2αθ(x) γ e − 2eαθ(x) + 1 uz (x) w(x) dx e2αθ(x) uz (x) w(x) dx. 2 8 Rn

Rn

If we discard the integral of |uz |2 w on the left-hand side and rearrange terms, we get that

2 2 8 e2αθ(x) uz (x) w(x) dx eαθ(x) uz (x) w(x) dx. 3 Rn

Rn

Since θ 1 and θ = 1 on F , this yields

z 2 u (x) w(x) dx 8 e−α uz (x)2 w(x) dx. 3 Rn

F

Since uz = (I + zLw )−1 f and by Proposition 3.5 the resolvent is bounded on L2 (w) with a constant that depends on ν, inequality (4.2) follows immediately. Now suppose that ν ∈ (π/2, π/2 + τ ). In this case, we can find ν < π/2 and τ < τ such that z = z ζ , where |ζ | = 1, | arg(ζ )| < τ , and z ∈ Σ(ν ). Then we can rewrite the left-hand side of (4.2) as

(I + zLw )−1 f (x)2 w(x) dx = I + z L −1 f (x)2 w(x) dx, w F

F

where L w = ζ Lw is the densely defined operator associated to the sesquilinear form generated by the matrix ζ A. By Lemma 3.3, there exist 0 < λζ < Λζ such that ζ A ∈ En (w, λζ , Λζ ). Therefore, L w and its resolvent have all the properties of Lw that we used in the above argument, so we can repeat it to get the desired inequality. 2 Remark 4.2. Since A∗ ∈ En (w, λ, Λ), Lemma 4.1 holds for L∗w with the same constants. We will now prove Theorem 1.6. Fix t > 0; without loss of generality we may assume that d 2 t. By Proposition 3.6,

1 −t Lw f= etζ (ζ I + Lw )−1 f dζ, e 2πi Γ

where Γ is the union of the rays γ ± = {z ∈ C: z = re±iν , r R > 0} and the arc γ0 = {z ∈ C: z = Reiψ , |ψ| ν}, going around the origin counter-clockwise, and ν ∈ (π/2, π/2 + τ ),


301

where τ is defined as in Lemma 4.1. We will fix the value of R below. Then by Minkowski’s inequality and Lemma 4.1,

−t L e w f (x)2 w(x) dx

F

=

1 4π 2

2

etζ (ζ I + Lw )−1 f (x) dζ w(x) dx Γ

F

1/2 2 tζ

e 1 ζ (ζ I + Lw )−1 f (x)2 w(x) dx d|ζ | ζ 4π 2 Γ

C 4π 2

F

f (x)2 w(x) dx

tζ 2 e −c d √|ζ | e 2 d|ζ | . ζ

(4.6)

Γ

E

Let β = cos(ν) < 0 and parametrize the last integral using the definition of the path Γ to get

−tζ

∞ tβr

ν √ √ e −c d √|ζ | e −c d2 r −tR cos θ −c d2 R e 2 d|ζ | 2 dr + e e dθ e ζ r Γ

Now let

−ν

R

2 −c d √R e 2 R

∞

R √ d

2 −c e 2 tR|β|

d

etβr dr + 2πetR e−c 2

R tβR

e

d

+ 2πetR e−c 2

√

√

R

R

.

√ R = δd/t, where δ = c/4. Then, since d 2 t,

−tζ e −c d √|ζ | cδ d 2 cδ d 2 d2 2 2 d2 2 d2 ˜ −c˜ t . d|ζ | 2 eβδ t e− 2 t + 2πeδ t e− 2 t Ce ζ e 2 δ |β|

Γ

The desired inequality follows immediately if we substitute this estimate into (4.6). 5. Proof of Theorems 1.3 and 1.8 The bulk of this section is devoted to the proof of Theorem 1.3; at the end we discuss how to derive Theorem 1.8 from it. The heart of the proof of Theorem 1.3 is to show that perturbations of e−t Lw are bounded from L2 (w) to L∞ . More precisely, given any real-valued φ ∈ Cc∞ , there exist constants α and C such that for all f ∈ L2 (w), −φ −t L φ e e w e f

n

L∞

Ct − 4 eαtρ f L2 (w) , 2

(5.1)

where ρ = ∇φ L∞ . Given inequality (5.1), we get the existence of the heat kernel Wt (x, y) via functional analysis; then by an argument due to Davies (cf. [13]) we get the desired Gaussian

302


bounds. Hölder continuity estimates then follow from the Harnack inequality and a classical argument. 5.1. The L2 (w), L∞ estimates To prove (5.1) it will suffice to prove it for non-negative functions f and for x = 0 and t = 1: −φ −L φ e e w e f (0) Ceαρ 2 f

(5.2)

L2 (w) .

To see that it suffices to consider non-negative functions, we use the fact that e−t Lw is linear and that given f = u + iv, we can decompose f as u+ − u− + iv + − iv − , where g + = max(g, 0) and g − = − min(g, 0), g L2 (w) g − L2 (w) + g + L2 (w) 2 g L2 (w) . The second reduction follows from homogeneity and the fact that if w is in A2 or QC, then so is wab = w(a · + b) for all a ∈ R, b ∈ Rn . To show that we can take t = 1, suppose that (5.1) holds when√t = 1 for all operators Lw . Define the functions u(x, t) = e−φ e−t Lw eφ f (x) t t ts). Then a straightforward computation shows that v = e−φ e−s Lt eφ f t , and v(y, s) = u( ty, √ √ where f t (y) = f ( ty), φ t (y) = φ( ty), and Lt is the operator induced by the sesquilinear form

at (f, g) = At ∇f (x) · ∇g(x) dx, Rn

√ √ where At (y) = A( ty). The matrix At satisfies (1.2) with w replaced by w t (y) = w( ty). Since w ∈ A2 /QC, w t ∈ A2 /QC with the same constant. Therefore, if (5.1) holds for s = 1 for the operator Lt , then, with ρt = ∇φ t L∞ = t 1/2 ρ, u(·, t) ∞ = v(·, 1) ∞ Ceαρt2 v(·, 0) 2 t L L L (w ) = Ce

αtρ 2

√ 2 √ f ( ty) w( ty) dy

1 2

n

= Ct − 4 eαtρ f L2 (w) . 2

Rn

To show that it suffices to take x = 0, we can repeat the above argument, replacing f by f 0 (x) = f (x + x0 ), w by w 0 (x) = w(x + x0 ), etc., for some fixed x0 ∈ Rn . We will now prove (5.2). Fix f ∈ L2 (w). Let Q0 ⊂ Rn be the cube (with sides parallel to the coordinate axes) centered at the origin with (Q0 ) = 9, and let f 0 = f χQ0 . For each integer k 1, let Qk = 3k Q0 and define f k = f χQk \Qk−1 . Decompose Qk \Qk−1 into 3n − 1 disjoint cubes Qk,j , 1 j 3n − 1, of side length 3k+1 . For k 1, let f k,j = f χQk,j . Then we have that ∞ 3 −1 −L φ k,j −φ −L φ e w e f (0) + e−φ(0) e−Lw eφ f 0 (0). e e w e f (0) e−φ(0) n

(5.3)

k=1 j =1

We first estimate the sum. Let uk,j (x, t) = e−t Lw eφ f k,j (x); then by Proposition 3.4 uk,j is k,j k,j k k,j is a a non-negative solution of Lw u = ut in Rn+1 + . Define v (y, s) = u (3 y, s). Then v n+1 solution to Lk v = vs in R+ , where Lk is the operator induced by the sesquilinear form defined


303

as above with the matrix Ak (y) = A(3k y). The matrix Ak satisfies (1.2) with w replaced by w k (y) = w(3k y), which is again an A2 /QC weight. Therefore, by Proposition 3.8 applied to the parabolic cylinder Q1 (0, 13 8 ), sup

13 Q− 1 (0, 8 )

v k,j (y, s) γ

inf

13 Q+ 1 (0, 8 )

v k,j (y, s).

13 k,j k,j (y, s). Let B = {x: In particular, since (0, 1) ∈ Q− 13 v r 1 (0, 8 ), |v (0, 1)| γ infQ+ 1 (0, 8 ) |x| < r}. Then

k,j v (0, 1) C(n)γ w k (B 1 )− 12

218

k,j v (y, s)2 w k (y) dy ds

2

19 8

1 2

.

|y|< 12

By a change of variables this becomes k,j u (0, 1) C(n)γ w(B 3k )− 12

218

k,j u (x, t)2 w(x) dx dt

1 2

.

2 19 8

k

|x|< 32

Since eφ f k,j (x) is supported in Qk,j and dist(Qk,j , {|x| < orem 1.6 we have

3k k 2 }) dist(Qk , Qk−2 ) = 3 ,

by The-

e−φ(0) e−Lw eφ f k,j (0) = e−φ(0) uk,j (0, 1) − 12 −c32k

C(n)γ w(B 3k )

218

e

e

2 19 8

2(φ(x)−φ(0)) k,j

f

2 (x) w(x) dx dt

1 2

Rn

1 2k C(n)γ w(B 3k )− 2 e−c3 eφ(·)−φ(0) L∞ (Qk,j ) f k,j L2 (w) 2

1

C(n)γ w(B 3k )− 2 e−c3 e3 2

2k

k+1

√

n 2 ∇φ L∞

k,j f

L2 (w)

.

We estimate the last term on the right-hand side in (5.3) in a similar fashion, except that instead of Theorem 1.6 we use the fact that e−Lw is bounded on L2 (w) to obtain √ n e−φ(0) e−Lw eφ f 0 (0) Ce9 2 ∇φ L∞ f 0 L2 (w) .

Now substitute both these estimates in (5.3) and apply Hölder’s inequality twice to get

304


e−φ(0) e−Lw eφ f (0) ∞ 3 −1 n

C(n)γ

1

√

n 2 ∇φ L∞

w(B 3k )− 2 e−c3 e3 2k

k+1

k,j f

2

k=1 j =1

√

L2 (w)

+ Ce9

n 2 ∇φ L∞

0 f

L2 (w)

∞ √ 1/2 1 2k k+1 n ∇φ ∞ k L f 2 C(n)γ 3n − 1 w(B 3k )− 2 e−c3 e3 L2 (w) 2

k=1 √

+ Ce9

n 2 ∇φ L∞

C(n, γ )

∞

C(n, γ )

L2 (w)

1

w(B 3k )− 2 e−c3 e3

∞

2k

k+2

√

n 2 ∇φ L∞

k f

L2 (w)

2

k=0

0 f

1/2 −1

w(B 3k )

k=0

2

∞

2 √ exp −2c32k + 3k+2 n ∇φ L∞ f k L2 (w)

1/2 .

k=0

Since w ∈ A∞ , it satisfies a reverse doubling condition: there exists β > 1 such that βw(Br ) w(B3r ). Thus, ∞

w(B 3k )−1 2

k=0

∞

β −k w(B 1 )−1 < C(w) < ∞. 2

k=0

Therefore, we have shown that e−φ(0) e−Lw eφ f (0)

1/2 ∞ 81n 2 2 f k 2

∇φ 2L∞ C(n, γ , w) exp = C(n, γ , w)eαρ f L2 (w) , L (w) 8c k=0

where α =

81n 8c

and ρ = ∇φ L∞ . This proves (5.2) and so (5.1).

5.2. Gaussian bounds To find the heat kernel and show that it satisfies Gaussian bounds, first note that by duality, (5.1) implies that −φ −t L φ e e w e f

n

L2 (w)

Ct − 4 eαtρ f L1 . 2

Since e−φ e−t Lw eφ is a semigroup, we can combine this inequality with (5.1) to get −φ −t L φ e e w e f

Ct −n/2 eαtρ f L1 . 2

L∞

(5.4)


305

By a classical result of Dunford and Pettis [15] (see also [3, p. 42]), this implies that for each φ φ ∈ Cc∞ , there exists a kernel Wt (x, y) such that for all f ∈ L1 , e

−φ −t Lw φ

e

e f (x) =

φ

Wt (x, y)f (y) dy Rn

and φ Wt (x, y) Ct −n/2 eαtρ 2 . In particular, let φ ≡ 0 to get the kernel Wt (x, y) of e−t Lw ; then it is immediate that Wt (x, y) = φ eφ(x)−φ(y) Wt (x, y). Hence, Wt (x, y) Ceαtρ 2 eφ(x)−φ(y) t −n/2 .

(5.5)

Inequality (5.5) is true for every φ ∈ Cc∞ with ∇φ L∞ = ρ, ρ > 0. Therefore, by an approximation argument we may take φ to be a Lipschitz function that satisfies φ(x) − φ(y) = −ρ|x − y|. Then (5.5) becomes Wt (x, y) Ct −n/2 exp αtρ 2 − ρ|x − y| . If we optimize the value of ρ, we get ρ =

|x−y| 2αt ,

and thus

2 Wt (x, y) Ct −n/2 exp − |x − y| . 4αt This is the desired inequality. 5.3. Hölder continuity The proof of inequalities (1.10) and (1.11) follow by standard arguments from the classical theory of elliptic and parabolic operators. Therefore, here we will only briefly sketch the proof. First, since the heat kernel of L∗w is Wt (y, x), (1.11) follows from (1.10) by duality. Given that Wt (x, y) satisfies Gaussian bounds, it is well known (see [3, p. 30]) that to prove (1.10) it suffices to prove that there exist constants C and ν > 0 such that for all t > 0 and all x, y, h ∈ Rn , ν Wt (x + h, y) − Wt (x, y) C|h| . t n/2+ν/2

By a classical result (again see [3, p. 42]), this is equivalent to proving that e−t Lw maps L1 into C|h|ν the space of Hölder continuous functions C ν , with norm t n/2+ν/2 .

306


To prove that e−t Lw : L1 → C ν , we first use the Harnack inequality, Lemma 3.8, and an argument due to Trudinger [28] to show that if f ∈ L1 and u(x, t) √ = e−t Lw f (x), then there exist and 0 < ρ < ρ t/2, ν > 0 and C > 0 such that given (x, t) ∈ Rn+1 0 + osc u C

Qρ (x,t)

ρ ρ0

ν osc u,

Qρ0 (x,t)

where oscR u = supR u − infR u. Further, by (5.4) (with φ ≡ 0) we have that osc u 2 u L∞ (Qρ0 (x,t)) Ct −n/2 f L1 .

Qρ0 (x,t)

The desired norm inequality follows by combining these estimates. 5.4. Proof of Theorem 1.8 This result follows from the fact that since e−t Lw is an holomorphic semigroup the Gaussian bounds in Theorem 1.3 can be extended to complex time. More precisely, we have the following. Theorem 5.1. Let A ∈ En (w, λ, Λ) be real symmetric, and let ω = arctan(Λ/λ). Then for all ν, 0 < ν < π/2 − ω, if z ∈ Σ(ν), there exists a heat kernel Wz (x, y) associated to the operator e−zLw . Furthermore, for all x, y ∈ Rn , the kernel Wz satisfies 2 Wz (x, y) C1 exp −C2 |x − y| , |z| |z|n/2 and μ |h| |x − y|2 Wz (x + h, y) − Wz (x, y) C1 , exp −C2 |z| |z|n/2 |z|1/2 + |x − y| μ |h| |x − y|2 Wz (x, y + h) − Wz (x, y) C1 , exp −C2 |z| |z|n/2 |z|1/2 + |x − y| where h ∈ Rn is such that 2|h| |z|1/2 + |x − y|. The constants C1 , C2 and μ depend only on n, ν, λ, and Λ. The proofs of Theorem 1.8 and the Gaussian bounds in Theorem 5.1 are identical to the proofs in the unweighted case as given in Auscher and Tchamitchian [3, p. 48]. We refer the reader there for complete details. The proof that Vt 1 = 0 then follows at once from the conservation property, Theorem 1.7. 6. Proof of Theorem 1.7 Our proof is adapted from the one for uniformly elliptic operators given by Auscher, McIntosh and Tchamitchian [5, Lemma 5.8]. The weighted case differs in many technical details, so we present the complete proof. Hereafter, let ω = arctan(Λ/λ).


307

It is straightforward to show that since the kernel of e−zLw , z ∈ Σ(π/2−ω), satisfies Gaussian bounds, e−zLw : Lp → Lp , 1 p ∞, with a bound that depends only on arg(z). By the Laplace identity we get the same Lp estimates for (zI + Lw )−1 . The same proof shows that the weighted versions of these inequalities are true if 2 p < ∞; in the case 1 p < 2 we need to assume w ∈ Ap . This need not be the case if w ∈ A2 or QC. However, the following weaker result suffices for our purposes. Lemma 6.1. Let w ∈ A2 or QC, and let z ∈ Σ(π − ω). Then (zI + Lw )−1 is a densely defined operator from L1 (w) into itself with domain containing Cc∞ . In fact, if φ ∈ Cc∞ , then (zI + Lw )−1 φ(x) Ce−c|x| ,

(6.1)

where the constants depend on n, Λ, λ, arg(z) and φ. Proof. If inequality (6.1) holds, then it follows immediately that e−zLw φ ∈ L1 (w): since w ∈ A∞ , there exists D > 1 such that for every r > 0, w(B2r (0)) Dw(Br (0)) [21, p. 695]. Hence, ∞ k (zI + Lw )−1 φ 1 Cw B1 (0) + Ce−c2 w B2k+1 (0) \ B2k (0) L (w) k=0

Cw B1 (0)

∞

D k e−c2 < ∞. k

k=0

To prove (6.1), fix φ ∈ Cc∞ and suppose that supp(φ) ⊂ BR (0), R > 1. By Proposition 3.6, −1

(zI + Lw )

∞ φ(x) =

e−zνt e−νt Lw φ(x) dt,

0

where we take ν ∈ Σ(π/2 − ω) such that |ν| = 1 and Re(zν) > 0. Since e−νt Lw φ ∈ L∞ is uniformly bounded, assume without loss of generality that |x| > 2R, so if x ∈ supp(φ), |x − y| > |x|/2. Therefore, by Theorem 5.1, (zI + Lw )−1 φ(x)

∞ C

e

∞ e

− Re(zν)t

n R

0

∞ C φ ∞ R n 0

−n/2

|νt| Rn

0

C

− Re(zν)t

|x − y|2 φ(y) dy dt exp −c |νt|

c |x|2 φ(y) dy dt t −n/2 exp − 4 t

c |x|2 e− Re(zν)t t −n/2 exp − dt 4 t

308


C φ ∞ R

+1 |x|2 2j

c |x|2 dt e− Re(zν)t t −n/2 exp − 4 t

∞

n

j =−∞ ∞

C φ ∞ R n

2j |x|2

2j |x|2 e−2

j =−∞ ∞

C φ ∞ |x|

2

j =−∞

j |x|2 Re(zν)

j 2 −n/2 c 2 |x| exp − 2−j 8

n c −j j 2 −1 j . exp −2 |x| Re(zν) − 2 − log(2) 8 2

Since |x| > 2R 2, there exists J > 0 such that 2J |x| < 2J +1 . We estimate the sum in the last term by splitting it into three pieces depending on the size of j : ∞ j =−∞

c n −1 j exp −2j |x|2 Re(zν) − 2−j − log(2) 8 2

=

∞ j =0

∞

−1

+

j =−J

+

−J −1

j =−∞

c n −1 j exp −2j |x|2 Re(zν) − 2−j − log(2) 8 2

exp −2j |x|2 Re(zν)

j =0

−1 c −j n + exp −|x| Re(zν) −1 j exp − 2 − log(2) 8 2 j =−J

∞ c c n c −1 j exp 2J − 2j + log(2) + exp − |x| 8 4 8 2 j =J +1

Ce−c|x| . Combining these two estimates we get that (zI + Lw )−1 φ(x) C φ ∞ |x|2 e−c|x| C φ ∞ e−c|x| . This completes the proof.

2

Proof of Theorem 1.7. Since the semigroup and the resolvent are bounded on L∞ , by Proposition 3.6 we have that

1 e−t Lw 1 = e−tζ (ζ I + Lw )−1 1 dζ, (6.2) 2πi Γ

where in the definition of Γ we take R = 1/t. Suppose for the moment that we could prove for all ζ ∈ Γ that (ζ I + Lw )−1 1 = ζ −1 .

(6.3)


309

Then (6.2) becomes 1 2πi

e t Lw 1 =

ζ −1 e−tζ dζ = 1,

Γ

where the last equality follows from a standard contour integral argument. To prove (6.3) it suffices to show that for all φ ∈ Cc∞ ,

−1 −1 (ζ I + Lw ) 1 φ(x)w(x) dx = ζ φ(x)w(x) dx. Rn

By duality,

Rn

(ζ I + Lw )−1 1 φ(x)w(x) dx =

Rn

−1 ζ¯ I + L∗w φ(x)w(x) dx.

Rn

Let h(x) = (ζ¯ I + L∗w )−1 φ(x); then by Lemma 6.1 we have that h ∈ L1 (w). Furthermore, L∗w h = −ζ¯ h + φ, so L∗w h ∈ L1 (w). Therefore,

(ζ I + Lw )−1 1 φ(x)w(x) dx = ζ −1 φ(x)w(x) dx − ζ −1 L∗w h(x)w(x) dx. Rn

Rn

Rn

To complete the proof we will show that the last integral equals zero. Let supp(φ) ⊂ BR (0), and for all r > 2R, let χr ∈ Cc∞ be such that χr ≡ 1 on Br (0), supp(χr ) ⊂ B2r (0), and |∇χr | cr −1 . Then

n ∗ ∗ Lw h(x)w(x) dx = lim Lw h(x)χr (x)w(x) dx r→∞ Rn R

= lim A∗ ∇h(x) · ∇χr (x) dx r→∞

Rn

Λ lim

r→∞ Rn

lim Cr r→∞

∇h(x)∇χr (x)w(x) dx

−1

w B2r (0)

∇h(x)2 w(x) dx

1/2 .

supp(∇χr )

To bound the last integral, we first make a preliminary estimate. let ψ ∈ Cc∞ be such that supp(ψ) ∩ supp(φ) = ∅. Then for all δ > 0,

∇h(x)2 ψ(x)2 w(x) dx λ−1 Re A∗ ∇h(x) · ∇h(x)ψ(x)2 dx Rn

Rn

λ−1 A∗ ∇h(x) · ∇h(x)ψ(x)2 dx Rn

310


λ

∗ 2 Lw h(x) h(x)ψ(x) w(x) dx

−1

Rn

+ 2λ

∗ ψ(x)h(x)A ∇h(x) · ψ(x) dx

−1

Rn

λ−1 −ζ¯ h(x) + φ(x) h(x)ψ(x)2 w(x) dx Rn

+ δΛλ

−1

∇h(x)2 ψ(x)2 w(x) dx

Rn

+ δ −1 Λλ−1

h(x)2 ∇ψ(x)2 w(x) dx.

Rn

If we fix δ such that δΛλ−1 = 1/2, then we can rearrange terms to get

∇h(x)2 ψ(x)2 w(x) dx C|ζ |

Rn

h(x)2 ∇ψ(x)2 + ψ(x)2 w(x) dx.

(6.4)

Rn

Now for each r > 2R, fix ψr ∈ Cc∞ such that supp(ψ) ⊂ B3r (0) \ Br/2 (0), ψ ≡ 1 on supp(∇χr ) and |∇ψr | C. Then by (6.4) and Lemma 6.1 we have that

lim Cr

−1

r→∞

w B2r (0)

∇h(x)2 w(x) dx

1/2

supp(∇χr )

= lim Cr −1 w B2r (0)

∇h(x)2 ψr (x)2 w(x) dx

r→∞

1/2

supp(∇χr )

lim Cr

−1

r→∞

w B2r (0)

h(x)2 w(x) dx

1/2

B3r (0)\Br/2 (0)

3/2 lim Cr −1 w B3r (0) exp(−cr) r→∞

= 0. The last equality holds since w ∈ A∞ : as in the proof of Lemma 6.1, there exists D > 1 such that for all r > 1, if 3N < r 3N +1 , then 3/2 −cr 3/2 −c3N 3 e D 2 (N +1) w B1 (0) e C. w B3r (0) This completes the proof.

2


311

Acknowledgments The authors would like to thank Steven Hofmann for suggesting this project to us and for his help and guidance in completing it. Both authors gratefully acknowledge the support of the Stewart-Dorwart faculty development fund at Trinity College. References [1] D.G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967) 890–896. [2] Pascal Auscher, Regularity theorems and heat kernel for elliptic operators, J. London Math. Soc. (2) 54 (2) (1996) 284–296. [3] Pascal Auscher, Philippe Tchamitchian, Square root problem for divergence operators and related topics, Astérisque 249 (1998), viii + 172. [4] Pascal Auscher, Thierry Coulhon, Philippe Tchamitchian, Absence de principe du maximum pour certaines équations paraboliques complexes, Colloq. Math. 71 (1) (1996) 87–95. [5] Pascal Auscher, Alan McIntosh, Philippe Tchamitchian, Heat kernels of second order complex elliptic operators and applications, J. Funct. Anal. 152 (1) (1998) 22–73. [6] Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on Rn , Ann. of Math. (2) 156 (2) (2002) 633–654. [7] Filippo Chiarenza, Michelangelo Franciosi, Quasiconformal mappings and degenerate elliptic and parabolic equations, Matematiche (Catania) 42 (1–2) (1987) 163–170. [8] Filippo Chiarenza, Michele Frasca, Boundedness for the solutions of a degenerate parabolic equation, Appl. Anal. 17 (4) (1984) 243–261. [9] Filippo Chiarenza, Raul P. Serapioni, Degenerate parabolic equations and Harnack inequality, Ann. Mat. Pura Appl. (4) 137 (1984) 139–162. [10] Filippo M. Chiarenza, Raul P. Serapioni, A Harnack inequality for degenerate parabolic equations, Comm. Partial Differential Equations 9 (8) (1984) 719–749. [11] Filippo Chiarenza, Raul P. Serapioni, A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova 73 (1985) 179–190. [12] Guy David, Stephen Semmes, Strong A∞ weights, Sobolev inequalities and quasiconformal mappings, in: Analysis and Partial Differential Equations, in: Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 101–111. [13] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math., vol. 92, Cambridge Univ. Press, Cambridge, 1990. [14] E.B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. Anal. Math. 58 (1992) 99–119, Festschrift on the occasion of the 70th birthday of Shmuel Agmon. [15] Nelson Dunford, B.J. Pettis, Linear operations on summable functions, Trans. Amer. Math. Soc. 47 (1940) 323–392. [16] Javier Duoandikoetxea, Fourier Analysis, Grad. Stud. Math., vol. 29, Amer. Math. Soc., Providence, RI, 2001. [17] Eugene B. Fabes, Carlos E. Kenig, Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1) (1982) 77–116. [18] Avner Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs, NJ, 1964. [19] Matthew P. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959) 1–11. [20] José García-Cuerva, José L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud., vol. 116, North-Holland, Amsterdam, 1985. [21] Loukas Grafakos, Classical and Modern Fourier Analysis, Pearson/Prentice Hall, Upper Saddle River, NJ, 2004. [22] Cristian E. Gutiérrez, Gail S. Nelson, Bounds for the fundamental solution of degenerate parabolic equations, Comm. Partial Differential Equations 13 (5) (1988) 635–649. [23] Cristian E. Gutiérrez, Richard L. Wheeden, Bounds for the fundamental solution of degenerate parabolic equations, Comm. Partial Differential Equations 17 (7–8) (1992) 1287–1307. [24] Kazuhiro Ishige, On the behavior of the solutions of degenerate parabolic equations, Nagoya Math. J. 155 (1999) 1–26. [25] Tosio Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss., vol. 132, Springer-Verlag, New York, 1966.

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Stochastic scalar conservation laws Jin Feng ∗,1 , David Nualart 2 Department of Mathematics, University of Kansas, Lawrence, KS 66045, United States Received 11 February 2008; accepted 17 February 2008 Available online 9 April 2008 Communicated by Paul Malliavin

Abstract We introduce a notion of stochastic entropic solution à la Kruzkov, but with Ito’s calculus replacing deterministic calculus. This results in a rich family of stochastic inequalities defining what we mean by a solution. A uniqueness theory is then developed following a stochastic generalization of L1 contraction estimate. An existence theory is also developed by adapting compensated compactness arguments to stochastic setting. We use approximating models of vanishing viscosity solution type for the construction. While the uniqueness result applies to any spatial dimensions, the existence result, in the absence of special structural assumptions, is restricted to one spatial dimension only. Published by Elsevier Inc. Keywords: Stochastic analysis; Scalar conservation law; Stochastic compensated compactness

1. Introduction We are interested in the well-posedness (existence and uniqueness) for first order nonlinear stochastic partial differential equation (SPDE) of the following type ∂t u(t, x) + divx F u(t, x) =

z∈Z


E-mail address: [email protected] (J. Feng). 1 Supported in part by US ARO W911NF-08-1-0064. 2 Supported in part by US NSF DMS-0604207.

0022-1236/$ – see front matter Published by Elsevier Inc. doi:10.1016/j.jfa.2008.02.004

σ x, u(t, x); z ∂t W (t, dz).

(1)

314

J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373

In the above, x ∈ R d , t 0, u(t, x) is a random scalar-valued function, F = (F1 , . . . , Fd ) : R → R d is a vector field (the flux). Regarding the random term on the right-hand side of the equation, Z is a metric space, and W (t, dz) is a space–time Gaussian white noise martingale random measure with respect to a filtration {Ft } (e.g. Walsh [18], Kurtz and Protter [13]) with E W (t, A) ∩ W (t, B) = μ(A ∩ B)t (2) for measurable A, B ⊂ Z, where μ is a (deterministic) σ -finite Borel measure on the metric space Z. In addition, σ : R d × R × Z → R. In the case of σ = 0, (1) reduces to a deterministic partial differential equation known as the scalar conservation law ∂t u(t, x) + divx F u(t, x) = 0, (3) which has been extensively studied in nonlinear partial differential equation theory literature (e.g. Dafermos [3]). A well-known difficulty for (3) is that solutions cannot be interpreted in classical sense: non-differentiability in x for u(t, x) develops in finite time, even if u(0, x) is chosen to be smooth [3, Theorem 5.1.1]. On the other hand, because of the nonlinearity in F , (Schwartz) distributional weak solution will generally not be unique (e.g. Section 4.2 of [3]). Kruzkov [9,10] introduced a method for selecting a weak solution motivated by physical consideration (the entropic solution). Well-posedness of (3) in the entropic solution sense can be proved for u(t) ∈ L1 ∩ L∞ , t 0. There are also other methods of selecting weak solutions. Most of these different approaches can be shown to be equivalent, at least in one space dimension d = 1. It is worth mentioning that, from a physical point of view, vector-valued u version of (3) is ultimately more interesting. However, little is known about well-posedness in that case. A detailed exposition about deterministic conservation law, for scalar- as well as vector-valued u, is given by Dafermos [3]. See also Chen [2] for a survey. Chapter 11 of Evans [7] contains a brief but informative introduction to the scalar case. The goal of this article is to introduce a proper generalization of entropic solution to the stochastic case (1) (Definition 2.5). Such notion will enable us to prove uniqueness of solution under mild assumptions on F and σ (Theorem 3.5). We will also give existence result for slightly more restrictive situations in one space dimension in Section 4. The following example gives us a feel on the scope of application that model (1) covers. Example 1.1. Let Z = {1, 2, . . . , m} and μ be a counting measure on Z, (1) reduces to m σk x, u(t, x) ∂t Wk (t), ∂t u(t, x) + divx F u(t, x) =

(4)

k=1

where W1 , . . . , Wm are independent standard Brownian motions. In particular, taking d = 1 and F (u) = |u|2 /2, the equation reduces to the stochastic Burgers’ equation ∂t u(t, x) + u(t, x)ux (t, x) =

m

σk x, u(t, x) ∂t Wk (t).

k=1

The W (t, dz) term can be extended to general semi-martingale random measure, and the theory developed here is expected to hold as well. We do not pursue this direction in this article.


315

There is a well-known connection between Hamilton–Jacobi and conservation law equations. Such connection can be transferred to the stochastic case as well. Let scalar function φ = φ(t, x) : [0, ∞) × R d → R be a solution to (5) ∂t φ(t, x) + F ∇x φ(t, x) = σ x, ∇x φ(t, x); z ∂t W (t, dz). Z

Let vector-valued function u(t, x) = ∇x φ(t, x), then ∇x σ (x, u; z) + ∂u σ (x, u; z)(∇x · u) ∂t W (t, dz). ∂t u + ∇x F (u) = Z

The case of d = 1 and σ = σ (x; z) independent of u gives scalar conservation law as considered in (1). In a series of publications [14], Lions and Souganidis consider equations related to (5): m σk ∇φ(t, x) ◦ dWk (t), ∂t φ + F ∇φ, D 2 φ = k=1

where D 2 is the Hessian operator and ◦ stands for Stratonovich type integral. Stochastic generalizations of viscosity solution are used. 2. Stochastic entropic solution—definition and main result 2.1. Definitions Definition 2.1. (Φ, Ψ ) is called an entropy–entropy flux pair if Φ ∈ C 1 (R) and Ψ = (Ψ1 , . . . , Ψd ) : R d → R d is a vector field satisfying Ψk (r) = Φ (r)(Fk ) (r),

k = 1, . . . , d.

(6)

Remark 2.2. Ψk can be chosen as r Ψk (r) =

Φ (s)(Fk ) (s) ds,

for some fixed v ∈ R.

v

Note that, unlike the usual definition, we do not require Φ to be convex in this definition. A special class of entropy–entropy flux pairs will play a major role in later analysis. We define it next. For each ε > 0, let β = βε ∈ C ∞ (R) be convex, β(r) = 0,

r 0,

β(r) = Cε + r,

Cε > 0, r ε.

K = (Φ, Ψ ) is an entropy–entropy flux pair:

Φ(r) = Φ u (r) = β(u − r), or Φ(r) = Φv (r) = β(r − v), u, v ∈ R .

Throughout this article, we assume the following regularities.

(7)

316


Condition 2.3. (1) Fk ∈ C 2 (R), and Fk (s) have at most polynomial growth in s, for all k = 1, . . . , d; (2) For each compact subset K ⊂ R d × R d , there exists MK : Z → R and nonnegative, nondecreasing, continuous function ρK : R → R with ρK (0) = 0 such that

σ (y, v; z) − σ (x, u; z) |u − v|1/2 ρK |u − v| + |x − y| MK (z), ∀(x, y) ∈ K, z ∈ Z, where CK ≡

2 MK (z) μ(dz) < ∞.

z∈Z

Example 2.4. Let σk : R d × R → R be Lipschitz for each k = 1, . . . , m and consider (4), then the second part of the above conditions is satisfied. Definition 2.5 (Stochastic entropic solution). Let (Ω, {Ft : t 0} ⊂ F, P ) be a filtered probability space where W (t, ·) is adapted space–time Gaussian white noise martingale random measure satisfying (2). We call an L2 (R d )-valued {Ft } -adapted stochastic process u = u(t) = u(t, x) a stochastic entropic solution of (1), provided (1) for each T > 0, p = 2, 3, 4, . . . , p sup E u(t)p < ∞,

(8)

σ x, u(r, x); z 4 dx μ(dz) dr < ∞.

(9)

0tT

and for each N = 1, 2, . . . fixed,

T

E z∈Z |x|N

0

(2) For each 0 s t, each 0 ϕ ∈ Cc2 (R d ), and each (Φ, Ψ ) ∈ K, Φ u(t, ·) , ϕ − Φ u(s, ·) , ϕ t

Ψ u(r, ·) , ∇x ϕ dr +

s

+ (s,t]×Z

1 Φ u(r, ·) σ 2 ·, u(r, ·); z , ϕ μ(dz) dr 2

(s,t]×Z

Φ u(r, ·) · σ ·, u(r, ·); z , ϕ W (dr, dz).

(10)


317

Perhaps a more revealing way to re-state integral inequality (10) is to say that u is a (Schwartz distributional) weak solution to ∂t Φ u(t, x) + divx Ψ u(t, x) 1 Φ u(t, x) σ 2 x, u(t, x); z μ(dz) 2 Z

+

∂W (t, dz) . Φ u(t, x) σ x, u(t, x); z ∂t

(11)

Z

When σ = 0, the right-hand side of the above inequality drops to zero. (11) reduces to exactly the defining differential inequality in deterministic entropic solution initially introduced by Kruzkov [9]. Some explanation on the meaning of (10) is necessary: (Φ, Ψ ) ∈ K implies that Φ and Φ are bounded and Ψ has at most polynomial growth. Together with (8) and (9), each term in (10) is well defined. A significant special yet common case satisfying (9) is when σ is uniformly bounded supx,u,z |σ (x, u, z)| < +∞, and μ(Z) < ∞. By an interpolation argument, to verify that (8) holds for p = 2, 3, . . . , it is good enough to show for even positive integer valued cases of p = 2, 4, 6, . . . . Moreover, both imply that (8) holds for all p ∈ [2, ∞). Unlike deterministic scalar conservation law (i.e. the case σ = 0), to prove path-wise uniqueness, we also need to capture more explicitly “noise–noise interaction” between any two possibly different stochastic solutions. We strengthen the definition of solution as follows. Definition 2.6 (Stochastic strong entropic solution). We call an L2 (R d )-valued, {Ft }-adapted process v = v(t) = v(t, x) a stochastic strong entropic solution of (1) if the following holds: (1) it is a stochastic entropic solution (i.e. (8), (9) and (10) hold for u replaced by v); ˜ satisfying (2) for each L2 (R d )-valued, Ft -adapted process u(t) p ˜ p < ∞, T > 0, p = 2, 3, . . . , sup E u(t) 0tT

and for each β ∈ C ∞ (R) of the form (17), 0 ϕ ∈ Cc∞ (R d × R d ), and f (r, z; v, y) =

˜ x) − v σ x, u(r, ˜ x); z ϕ(x, y) dx, β u(r,

x∈R d

there exists a deterministic function {A(s, t): 0 s t} such that

E y (s,t]×Z

f r, z; v(t, y), y W (dr, dz) dy

E (s,t]×Z y

∂ f r, z; u(r, ˜ y), y σ y, v(r, y); z μ(dz) dy dr + A(s, t) ∂v

(12)

318


with the following property: for each T > 0, there exist partitions 0 = t1 < t2 < · · · < tm = T satisfying m

lim

maxi |ti+1 −ti |→0+

A(ti , ti+1 ) = 0,

t 0.

i=1

2.2. Main results We list another set of conditions stronger than those in Condition 2.3. Condition 2.7. (1) (2) (3)

d = 1, F ∈ C 2 (R) and the set {r ∈ R: F (r) = 0} is dense in R, ∞ d 2 d there 2exist f ∈ L (R ) ∩ L (R ), deterministic constant C > 0, and M : Z → R such that Z M (z) μ(dz) < ∞,

σ (x, u; z) f (x) 1 + |u| M(z),

(13)

σ (x, u; z) − σ (y, v; z) C |u − v| + |x − y| M(z) + σ (x, u; z) .

(14)

and

The main result of this article is the following. Theorem 2.8. Assume Condition 2.3 holds, and that u0 satisfies p p E u0 p + u0 2 < ∞,

p=1,2,... L

p (R d )-valued

random variable

p = 1, 2, . . . .

(Uniqueness) Suppose that u, v are two stochastic entropic solutions of (1) with the same initial condition u(0) = u0 = v(0), and that one of u, v is a strong stochastic entropic solution. Then almost surely u(t) = v(t) for t 0. (Existence) Assume furthermore that Condition 2.7 holds, then there exists a strong stochastic entropic solution (hence also entropic solution) for (1) with initial value u0 . 2.3. Notations Throughout, we denote the space of smooth, rapidly decreasing functions

S R d = f ∈ C ∞ : sup x m Dxn f (x) < ∞, m, n = 1, 2, . . . .

(15)

x

Let J ∈ Cc∞ (R d ) be the standard mollifier defined by J (x) =

C exp{ |x|21−1 } if |x| < 1, 0 if |x| 1,

(16)


319

where constant C > 0 is selected so that R d J (z) dz = 1. For each ε > 0, we set Jε (z) = ε −d J (ε −1 |z|). Jε ∈ Cc∞ (R d ) with supp(Jε ) ⊂ [−ε, ε]d . For each f ∈ Lloc (R d ), we define its mollification fε (x) = Jε ∗ f (x) = Jε (x − y)f (y) dy = Jε (y)f (x − y) dy. Rd

Rd

Let A ⊂ R d , then function χA (x) = 1 if x ∈ A and χA (x) = 0 if x ∈ / A. To simplify, with a slight abuse of notation, we denote χ(x) = χ[0,+∞) (x). For a ∈ R, we denote a+ = max{a, 0}. Then |a| = a+ + (−a)+ . We need smooth functions approximating β(r) = r+ ∈ C(R). We consider J in the special case of d = 1 and define r−ε ρε (r) = Jε (s) ds,

r βε (r) =

−∞

ρε (s) ds,

r ∈ R.

(17)

−∞

Then by direct verification, we have the following. Lemma 2.9. The above constructed ρε , βε ∈ C ∞ (R) have the following properties: βε = ρε , βε (r) = Jε (r − ε); ρε is a nondecreasing function and βε (r) = ρε (r) =

0 if r 0, 1 if r 2ε;

(18)

and βε is convex and βε (r) = where Cˆ =

0 if r 0, ε Cˆ + (r − 2ε) if r 2ε,

(19)

1 s −1 ( t=−1 J (t) dt) ds < 2. Furthermore, 0 βε (r) = Jε (r − ε) ε −1 C,

0 r 2ε,

implying 0 rβε (r) 2C,

for 0 r 2ε.

3. Uniqueness 3.1. A doubling lemma Let u be a stochastic entropic solutions and v be a stochastic strong entropic solution. We estimate the evolution of (u(t) − v(t))+ 1 = (u(t) − v(t))+ L1 . First, let β = βε (r) as constructed in Lemma 2.9. We approximate (u(t) − v(t))+ 1 by R d ×R d β(u(t, x)−v(t, y))ϕ(x, y) dx dy (by considering limits βε (r) → r+ and ϕ(x, y) dx dy → δx (dy) dx). Then we develop estimate for time evolution of such approximate. In the deterministic scalar conservation law setting (i.e. σ = 0), Kruzkov [9] appears to be the first who introduced

320


such an argument for a uniqueness proof. Our goal in this section is to generalize such arguments properly to a stochastic setting (see Lemma 3.2). Let 0 ϕ ∈ Cc∞ (R d × R d ). For t > s 0,

β u(t, x) − v(t, y) ϕ(x, y) dx dy −

R d ×R d

=

β u(s, x) − v(s, y) ϕ(x, y) dx dy

R d ×R d


β u(s, x) − v(t, y) ϕ(x, y) dx dy

β u(s, x) − v(t, y) ϕ(x, y) dx dy −

+

β u(s, x) − v(s, y) ϕ(x, y) dx dy

≡ I1 + I 2 . First, we estimate I2 . We introduce notation α(u, v) = α1 (u, v), . . . , αd (u, v) , where (noting β(r) = 0 for r < 0) ∞ αk (u, v) =

β

(u − w)Fk (w) dw

v

u =

β (u − w)Fk (w) dw.

(20)

v

Lemma 3.1. t

I2

α u(s, x), v(r, y) · ∇y ϕ(x, y) dx dy dr

s R d ×R d

1 + 2

t

β u(s, x) − v(r, y) σ 2 y, v(r, y); z ϕ(x, y) dx dy μ(dz) dr

s Z x,y

β u(s, x) − v(r, y) σ y, v(r, y); z ϕ(x, y) dx dy W (dr, dz).

−

(21)

(s,t]×Z x,y

Proof. Let u ∈ R be fixed. We take Φ(v) = β(u − v), Ψk (v) = apply (10) to v(t, y). Therefore, for each x ∈ R d ,

β u − v(t, y) ϕ(x, y) dy −

y

v

u (−β

)(u − w)F (w) dw, k

β u − v(s, y) ϕ(x, y) dy

y

t

v(r,y)

− s y∈R d

u

d k=1

∂ β (u − w)Fk (w) dw ϕ(x, y) dy dr ∂yk

and


1 + 2

321

β u − v(r, y) σ 2 y, v(r, y); z ϕ(x, y) dy μ(dz) dr

(s,t]×Z y

+

−β u − v(r, y) σ y, v(r, y); z ϕ(x, y) dy W (dr, dz).

(s,t]×Z y

Taking u = u(s, x) inside the above inequality and integrating x, and applying Fubini’s theorem, we arrive at (21). 2 We estimate I1 next. We introduce notation α(u, ˆ v) = αˆ 1 (u, v), . . . , αˆ d (u, v) , where (noting β (r) = 0 for r 0) u αˆ k (u, v) =

β

(w − v)Fk (w) dw

u =

−∞

β (w − v)Fk (w) dw.

(22)

v

For each v ∈ R d fixed, taking Φ(u) = β(u − v) and Ψ (u) = α(u, ˆ v), we apply (10) for u(t, x), then we take v = v(t, y),


R d ×R d

β u(s, x) − v(t, y) ϕ(x, y) dx dy

R d ×R d

t

αˆ u(r, x), v(t, y) · ∇x ϕ(x, y) dx dy dr

s R d ×R d

+

1 2

t

β u(r, x) − v(t, y) σ 2 x, u(r, x); z ϕ(x, y) dx dy μ(dz) dr

s Z R d ×R d

+

β u(r, x) − v(t, y) σ x, u(r, x); z ϕ(x, y) dx W (dr, dz) dy

y∈R d (s,t]×Z x∈R d

≡ I3 + I4 + I5 . To achieve simplicity in exposition, we have slightly abused notation for the term I5 as this should not be understood as an Ito’s integral (the integrand contains anticipative term v(t, y)). The rigorous meaning of it should be understood in the following sense. Let

322


f (r, z, v, y) =

β u(r, x) − v σ x, u(r, x); z ϕ(x, y) dx

Rd

and G5 (s, t; v, y) =

(23)

f (r, z, v, y) W (dr, dz).

(s,t]×Z

For each v, y fixed, the above is an Ito’s integral. Then, we define

G5 s, t; v(t, y), y dy.

I5 = I5 (s, t) = y∈R d

A key defining property (12) in stochastic strong entropic solution gives us E[I5 ] E − R d ×R d

β u(r, x) − v(r, y) σ y, v(r, y); z

(s,t]×Z

× σ x, u(r, x); z μ(dz) dr ϕ(x, y) dx dy + A(s, t). Together with the estimate on E[I3 ] and E[I4 ], we have E[I1 ] t

E

αˆ u(r, x), v(r, y) · ∇x ϕ(x, y) dx dr dy

s R d ×R d

1 + E 2

β u(r, x) − v(r, y) σ 2 x, u(r, x); z ϕ(x, y) dx dy μ(dz) dr

(s,t]×Z R d ×R d

−E

β u(¯r , x) − v(¯r , y) σ y, v(¯r , y); z

(s,t]×Z R d ×R d

× σ x, u(¯r , x); z ϕ(x, y) dx dy μ(dz) d r¯

+ A(s, t). Combine this with the estimate on I2 in (21), by arbitrariness of 0 s t, we can arrive at a stochastic version of the doubling of variable estimate first introduced by Kruzkov [9] in deterministic context (see for instance Evans [7, Theorem 3, p. 608]).


323

Lemma 3.2. For each t > 0, we have


E R d ×R d

β u(0, x) − v(0, y) ϕ(x, y) dx dy

R d ×R d

t

α u(r, x), v(r, y) · ∇y ϕ(x, y) + αˆ u(r, x), v(r, y) · ∇x ϕ(x, y) dx dy dr

E 0 R d ×R d

1 + E 2

β u(r, x) − v(r, y)

(0,t]×Z R d ×R d

2 × σ y, v(r, y); z − σ x, u(r, x), z ϕ(x, y) dx dy μ(dz) dr . Proof. We select the sequence of partitions of [0, t] appearing in the defining relation of strong entropic solution (Definition 2.6): 0 = t1 · · · tm = t < ∞. Using the above estimate on I1 and the estimate (21) on I2 (set the s = ti , t = ti+1 there), E

β u(ti+1 , x) − v(ti+1 , y) ϕ(x, y) dx dy − ti+1 ti R d ×R d

β u(ti , x) − v(ti , y) ϕ(x, y) dx dy


E 1 + E 2

β u(r, x) − v(r, y)

(ti ,ti+1 ]×Z R d ×R d

2 × σ y, v(r, y); z − σ x, u(r, x), z ϕ(x, y) dx dy μ(dz) dr

+ A(ti , ti+1 ). Summing over i,


E R d ×R d

t

β u(0, x) − v(0, y) ϕ(x, y) dx dy

R d ×R d

E


0 R d ×R d

1 + E 2

(0,t]×Z R d ×R d

β u(r, x) − v(r, y)

324


2 × σ y, v(r, y); z − σ x, u(r, x); z ϕ(x, y) dx dy μ(dz) dr +

m

A(ti , ti+1 ).

i=1

Taking limm→∞ , we arrive at the desired inequality.

2

3.2. Uniqueness Using Lemma 3.2 as the point of departure, we now let ϕ(x, y) dx dy → δx (dy) dx and β(r) → r+ to arrive at an L1 type estimate for u − v. First, we select test function ϕ in the following manner. Let J be a one-dimensional standard mollifier as defined by (16), and let 0 ψ ∈ Cc∞ (R d ). We choose

d − y x x +y k k −d ∈ Cc∞ R d × R d . ϕδ (x, y) = δ J ψ 2δ 2

(24)

k=1

Then 1 xk − yk x +y −d xj − yj J ∂xj ϕδ (x, y) = δ J ψ 2δ 2δ 2δ 2 k=j

d 1 −d xk − yk x+y , + J δ ∂j ψ 2 2δ 2

(25)

k=1

1 xk − yk x +y −d xj − yj ∂yj ϕδ (x, y) = − J δ J ψ 2δ 2δ 2δ 2 k=j

d 1 −d xk − yk x +y , + J δ ∂j ψ 2 2δ 2

(26)

k=1

and

d − y x x +y k k −d (∂xj + ∂yj )ϕδ (x, y) = δ ∈ Cc∞ R d × R d . J ∂j ψ 2δ 2

(27)

k=1

We let βε be defined according to (17), and take β = βε , ϕ = ϕδ in Lemma 3.2. We note that βε (r) → r+ uniformly in r ∈ R as ε → 0+. Recall the definition of αk , αˆ k in (20) and (22), using Lemma 2.9, for each k = 1, 2, . . . , each u, v ∈ R d fixed and α = (α1 , . . . , αd ),


∞ lim αk (u, v) = lim

ε→0+

ε→0+

325

βε (u − w)Fk (w) dw

v

∞ =

χ(u − w)Fk (w) dw = χ(u − v) Fk (u) − Fk (v)

v

= lim αˆ k (u, v). ε→0+

We recall that by earlier convention on notations, χ(r) = χ[0,+∞) (r). The right-hand side of the inequality in Lemma 3.2 consists of two terms. In view of the above limit for limε→0+ αk and limε→0+ αˆ k , the first term is easier to be controlled in the limδ→0+ limε→0+ limit. However, the second term is easier to be controlled in the limε→0+ limδ→0+ limit. Therefore, we need more careful estimates by considering ε → 0+, δ → 0+ at the same time with appropriate speeds. Lemma 3.3. Suppose that ε → 0+, δ → 0+ and εδ −1 → 0+ (e.g. let δ = ε 2/3 ), then

lim sup ε→0+, δ→0+, εδ −1 →0+

t

E

α u(r, x), v(r, y) · ∇y ϕ(x, y)

0 R d ×R d

+ αˆ u(r, x), v(r, y) · ∇x ϕ(x, y) dx dy

dr

t d

E χ u(r, x) − v(r, x) Fk u(r, x) − Fk v(r, x) ∂k ψ(x) dx dr .

0

Rd

k=1

Proof. We need to estimate the difference between αk (u, v) and χ(u − v)(Fk (u) − Fk (v)) more precisely. When u v, for w v, βε (u − w) = 0, therefore αk (u, v) = 0 = χ(u − v) Fk (u) − Fk (v) . When u > v, then by Lemma 2.9, u αk (u, v) =


v v∨(u−2ε)

Fk (w) dw +

= v

u


v∨(u−2ε)

= χ(u − v) Fk (u) − Fk (v) +

u

v∨(u−2ε)

βε (u − w) − 1 Fk (w) dw.

326


This, together with the at most polynomial growth assumption on F in Condition 2.3, implies

αk (u, v) − χ(u − v) Fk (u) − Fk (v) 2

u

F (w) dw εCp 1 + |u|p

u−ε

for some p 1, where Cp is independent of ε and u, v and can be chosen to be independent of k = 1, 2, . . . , d as well. Similar conclusion holds for αˆ k (u, v) = χ(u − v) Fk (u) − Fk (v) +

u∧(v+2ε)

β (w − v) − 1 Fk (w) dw.

v

Combine the above estimate with (25), (26) and (27), t

E

α u(r, x), v(r, y) · ∇y ϕ(x, y)

0 R d ×R d

+ αˆ u(r, x), v(r, y) · ∇x ϕ(x, y) dx dy

dr

t

E

d χ u(r, x) − v(r, y) Fk u(r, x) − Fk v(r, y)

R d ×R d

0

k=1

× ∂xk ϕδ (x, y) + ∂yk ϕδ (x, y) dx dy

dr ε + C E δ d

j =1

t

p

p

2 + u(r, x) + v(r, y)

0 R d ×R d

− y x x x + y − y j j k k

× δ −d

J J ψ dx dy dr

2δ 2δ 2 k=j

+ εC

d

t

j =1

× δ

−d

E

p

p

2 + u(r, x) + v(r, y)

0 R d ×R d

d xk − yk x +y dx dy dr . J ∂j ψ 2δ 2

(28)

k=1

This gives the conclusion of the lemma.

2

Next, we estimate the second term on the right-hand side of the inequality in Lemma 3.2.


327

Lemma 3.4. Suppose that ε → 0+, δ → 0+ and δ 2 ε −1 → 0+ (e.g. let δ = ε 2/3 ), then lim sup

βε u(r, x) − v(r, y)

E

ε→0+, δ→0+, δ 2 ε −1 →0+

(0,t]×Z R d ×R d

2 × σ y, v(r, y); z − σ x, u(r, x), z ϕδ (x, y) dx dy μ(dz) dr 0. Proof. Select compact K = Kψ ⊂ R d × R d to be such that supp(ϕδ ) ⊂ K for all 0 < δ < 1. Under Condition 2.3 on σ (x, u; z), and by the estimate 0 βε ε −1 C, 0 rβε (r) 2Cχ[0,2ε] (r) (e.g. Lemma 2.9) and that βε (r) = 0 for r 2ε or r 0, for (x, y) ∈ K,

2

β (u − v) σ (y, v; z) − σ (x, u, z)

2 2 |u − v| + 2β (u − v)|x − y|2 MK 2β (u − v)|u − v|ρK (z) 2 2 2 4CρK (2ε)MK (z) + 2Cε −1 |x − y|2 MK (z).

We have therefore E

2 β u(r, x) − v(r, y) σ y, v(r, y); z − σ x, u(r, x), z

(0,t]×Z R d ×R d

× ϕδ (x, y) dx dy μ(dz) dr

2 4CρK (2ε)

×

ϕδ (x, y) dx dy

K

2 MK (z) μ(dz)ψ∞ t.

2 MK (z) μ(dz) + 2(Cψ )ε −1 δ 2

Z

(29)

Z

The conclusion follows.

2

Theorem 3.5. Suppose u is a stochastic entropic solution of (1) and v is a stochastic strong entropic solution. Then (1) (L1 contraction) E u(t) − v(t) + 1 E u(0) − v(0) + 1 .

(30)

(2) (Comparison principle) Suppose that v(0, x) u(0, x) a.e. in x holds almost surely, and that E[(u(0, ·) − v(0, ·))+ 1 ] < ∞, then almost surely v(t, x) u(t, x)

a.e. in x.

328


Remark 3.6. The following proof can be simplified considerably if we assume sup E u(t)1 + v(t)1 < ∞. 0tT

However, we adapted a definition of entropic solution which does not assume the above. We have no effective way of establishing such estimates in the absence of additional structural assumptions on σ . Proof. We define ϕδ according to (24) where δ = ε 2/3 . Taking ε → 0+ limit to the inequality in Lemma 3.2, by Lemmas 3.3 and 3.4, E

u(t, x) − v(t, x) + ψ(x) dx −

Rd

u(0, x) − v(0, x) + ψ(x) dx

Rd

t d

χ u(r, x) − v(r, x) Fk u(r, x) − Fk v(r, x) ∂k ψ(x) dx dr . E

0

Rd

(31)

k=1

Let 0 ψN (x) = e−N

−1 |x|

∈ W 1,p (R d ), p = 1, 2, . . . , ∞. Then ∂k ψN (x) = −

1 xk ψN (x), N |x|

x = 0,

and ∂k ψN ∞ N −1 . Noting estimate (8) in the definition of entropic solution, by standard approximation by truncation and mollification arguments, (31) holds with ψ replaced by ψN . By the at most polynomial growth assumption on F in Condition 2.3, there exists p0 1, such that for any integer p > p0 1,

χ(u − v) Fk (u) − Fk (v) χ(u − v)

u

F (r) dr k

v

C1 (u − v)+ + |u|1+p + |v|1+p .

(32)

By (8) for u and v,

t

χ u(r, x) − v(r, x) Fk u(r, x) − Fk v(r, x) ∂k ψN (x) dx dr

E

0

x

C1 N

−1

t E 0

u(r, x) − v(r, x) + ψN (x) dx dr + θN (t),

Rd

where 1+p 1+p θN (t) = C1 N −1 sup E u(r)1+p + v(r)1+p . 0rt

(33)


329

Let wN (t) = E[ R d (u(t, x) − v(t, x))+ ψN (x) dx] and w(t) = E[ R d (u(t, x) − v(t, x))+ dx]. Then wN (t) wN (0) + θN (T ) + C1 N

−1

t wN (r) dr,

0 t T.

0

By (8) for u, v, sup0tT wN (t) < ∞. Therefore by the Gronwall’s inequality, −1 sup wN (t) wN (0) + θN (T ) eC1 N T . 0tT

Send N → ∞. By the monotone convergence theorem, wN (t) → w(t) for every t 0. We arrive at (30). From (30) it follows that, if v(0, x) u(0, x) a.e. in x almost surely, then v(t, x) u(t, x) a.e. in x almost surely. 2 4. A constructive existence theory 4.1. Heuristic outlines We refer to Dafermos [3] (in particular Chapter VI) for background discussions and references on physical motivation of deterministic conservation laws. The stochastic case can be considered similarly. We would like to view (1) as limit of some microscopic stochastic system behaving effectively like ∂t u(t, x) + divx F u(t, x) = σ x, u(t, x); z ∂t W (t, dz) + εxx u, u(0) = u0 , (34) z∈Z

with asymptotically vanishing ε. The εxx term (with ε > 0) has a smoothing effect on solution u. For now, let us pretend that u = uε is a solution of (34) which is sufficiently smooth so that spatial derivatives up to the second order exist in classical sense and are continuous (Lemma 4.10). Let Φ ∈ C 2 (R) and Ψ = (Ψ1 , . . . , Ψd ) be an entropy–entropy flux pair (Definition 2.1). Then by Ito’s formula, at least formally, ∂t Φ u(t, x) + divx Ψ u(t, x) Φ u(t, x) σ x, u(t, x); z ∂t W (t, dz) + εΦ u(t, x) xx u(t, x) = z∈Z

1 + Φ u(t, x) 2

σ 2 x, u(t, x); z μ(dz).

(35)

Z

It is tempting to send ε → 0 and arrive at a limit. This is not correct. With each ε > 0 fixed, we can establish second order derivative information about u by exploiting the smoothing effect of ε. However, we do not know the magnitude of fluctuation for the nonlinear term

330


εΦ (u(t, x))xx u(t, x) as ε vanishes (note that u = uε ). This is the exact same difficulty as in deterministic conservation law case (σ = 0) which is handled as follows. First, we observe

2

εΦ u(t, x) xx u(t, x) = εxx Φ u(t, x) − εΦ u(t, x) ∇x u(t, x) .

(36)

We can always view xx Φ(u(t, x)) in (Schwartz) distributional sense, as far as u is locally integrable. We do not have control over |∇x u(t, x)| uniformly over ε > 0, and it is not reasonable to hope so (this quantity can blow up in finite time, even if u(0) ∈ C ∞ ∩ Cb , in the case of σ = 0). However, since Φ is convex (i.e. Φ 0) for ϕ 0, we have a one-sided trivial bound

εΦ (u)|∇x u|2 , ϕ 0.

In summary, we have now for 0 ϕ ∈ Cc2 , Φ u(t, ·) , ϕ − Φ u(s, ·) , ϕ t

Ψ u(r, ·) , ∇x ϕ dr +

= s

1 Φ u(r, ·) σ 2 ·, u(r, ·); z , ϕ μ(dz) dr 2

(s,t]×Z

t +ε

2

Φ u(r, ·) , ϕ − Φ u(r, ·) ∇x u(r, ·) , ϕ dr

s

+

Φ u(r, ·) σ ·, u(r, ·); z , ϕ W (dr, dz)

(s,t]×Z

t

Ψ u(r, ·) , ∇x ϕ dr +

s

+ o(ε) +

1 Φ u(r, ·) σ 2 ·, u(r, ·); z , ϕ μ(dz) dr 2

(s,t]×Z

Φ u(r, ·) σ ·, u(r, ·); z , ϕ W (dr, dz).

(37)

(s,t]×Z

Both the left-hand and right-hand sides of the inequality are stable under ε → 0+ limit, provided p we have Lloc type stability of u = uε (which is a lot easier and possible to estimate than ∇uε or uε ). Sending ε → 0, (10) follows. As in the deterministic scalar conservation law case, we face two main issues in order to make the above rigorous. One, we need regularity estimates on the solution u for the approximate equation (34) so that the Ito’s formula can be applied to the transformation of u(t, x) = uε (t, x). Two, we also need to verify relative compactness on u = uε (in some appropriate topology) as ε goes to zero. The first issue is more or less known in various different contexts for slightly different models in stochastic analysis literature. We will adapt existing methods and discuss the issue more carefully (because of the generality here) in the first subsection below. The second issue, however, has never been considered in its current generality (i.e. with the stochastic term). Kim [8] modifies deterministic arguments to construct a very special SPDE which is essentially reformulated as a randomness in coefficients type of conservation law. To handle the general


331

situation here, we have to introduce non-trivial new technical ideas. In particular, we derive stochastic versions of the (originally deterministic) compensated compactness results. As in the deterministic case, the compensated compactness argument will eventually restrict our consideration to one space dimension (i.e. d = 1) only. However, most of our estimates are not restricted by dimensionality. In deterministic theory, one can also obtain compactness in C([0, ∞), Lp ) (e.g. p = 1 or p = ∞) by Ascoli–Arzela type argument by estimating corresponding modulus of continuities in time and in space variables (e.g. [3, Section 6.3]). In particular, the spatial modulus of continuity estimate is usually achieved by perturbing conservation law equation through a spatial translation of the solution and then prove a comparison result. Such procedure does not generalize well to the type of SPDEs as in (1). Because of the σ term, spatial translation of solution will generally not be another solution or even an approximate solution in some well controlled sense. Similar observation was also made by Kim [8] in a simpler model context. Theorem 4.1. Suppose that Condition 2.7 holds. Then there exists a stochastic strong entropic solution (hence also a stochastic entropic solution, see Definitions 2.6 and 2.5) u = u(t, x) for (1). We divide proof into several parts below. 4.2. Existence and regularity of approximate equation (34) Throughout this subsection, we assume that F = (F1 , . . . , Fd ) satisfies Fk ∈ C ∞ for each k, (m) and that the mth order derivative of Fk satisfies |Fk (r)| Cm < ∞, m = 0, 1, 2, . . . . We m m also assume that σ (x, u; z), Dx σ (x, u; z), ∂u σ (x, u; z) exist and are continuous and uniformly bounded and Dxm σ (·, u; z) ∈ S(R d ) (see (15)), for all m = 1, 2, . . . . Finally, Z supx,u |σ 2 (x, u; z)| μ(dz) < ∞. 4.2.1. Existence of solution when ε > 0 is held fixed Let the fundamental solution of the heat equation be denoted by G(t, x) = Gε (t, x) =

1 2 e−|x| /(4εt) , d/2 (4πεt)

t > 0.

Let E[u0 22 ] < ∞. First, we define successive approximates to (34): let u0 (t, x) = u0 (x), un (0, x) = u0 (x) and dun (t, x) + ∇ · F un−1 (t, x) dt = εun (t, x) dt + σ x, un−1 (t, x); z W (dt, dz). (38) Z

We consider the mild solution for the above equation given by

t

u (t, x) =

G(t, x − y)u0 (y) dy −

n

y

+ (0,t]×Z y

G(t − s, x − y) 0 y

d

∂yi Fi un−1 (s, y) dy ds

i=1

G(t − s, x − y)σ y, un−1 (s, y); z dy W (ds, dz).

(39)

332


The above right-hand side needs some explanation. The first term is always well defined because of E[u0 22 ] < ∞, the second term is well defined provided Dy un−1 ∈ L∞ ([0, T ]; L2 (R d )) for each T > 0; the third term is always well defined because of our earlier assumptions on σ . At this point, it is only clear that u1 is defined. We need regularity information on un−1 to conclude that un is defined for n 2. We claim that all un (t, x) are well defined and un (t, ·) ∈ S(R d ). To verify this claim, we need the following properties. Lemma 4.2. Let h = h(s) = h(s, x) be an adapted process in C([0, T ]; H p (R d )), p = 1, 2, . . . , T > 0 and h(s, ·) ∈ S(R d ). Let V (t, x) = G(t − s, x − y)h(s, y) dy ds. (0,t]×R d

Then V = V (t) = V (t, x) is an adapted process in C([0, T ]; H p (R d )), T > 0 and V (t, ·) ∈ S(R d ) and in particular, G(t − s, x − y)h(s, y) dy ds = − G(t − s, x − y)∂yk h(s, y) dy ds. ∂xk (0,t]×R d

(0,t]×R d

Similarly, if for each z ∈ Z fixed, f (·,·; z) = f (t, y; z) ∈ C([0, T ]; H p (R d )) is an Ft -adapted process in t for p = 1, 2, . . . , T > 0, and T E

m

2 2 f (s, y; z) + Dy f (s, y; z) dy μ(dz) ds < ∞,

0 Rd Z

where m = 1, 2, . . . , and f (s,·; z) ∈ S(R d ), then N(t, x) = G(t − s, x − y)f (s, y; z) dy W (ds, dz),

(40)

(0,t]×Z y

has the following property for each T > 0. Lemma 4.3. N (t) ∈ C([0, T ]; H p (R d )), p = 1, 2, . . . , and in particular, G(t − s, x − y)∂yk f (s, y; z) dy W (ds, dz), ∂xk N (t, x) = −

a.s.

(0,t]×Z y

and N(t, ·) ∈ C ∞ (R d ). Proof. Continuity of N as an L2 -valued process in t can be handled as in Proposition 7.3 of Da Prato and Zabczyk [4]. Regarding ∂xk N , all we need is to show that (Schwartz) distributional derivative of N agrees with the right-hand side. Then, since the right-hand side is continuous in x, the identify is established. For each ϕ ∈ Cc∞ (R d ),


E

333

G(t − s) ∗x f (s, x; z)∂xk ϕ(x)

[0,t]×Z x

2

− G(t − s) ∗x (−∂k f )(s, x; z)ϕ(x) dx W (ds, dz)

E

G(t − s) ∗x f (s, x; z)∂k ϕ(x)

= [0,t]×Z

2 − G(t − s) ∗x (−∂k f )(s, x; z)ϕ(x) dx

μ(dz) × ds = 0.

In the above, ∗x means convolution with respect to spatial variable x only. By such representation, it follows that ∂xk N has trajectory in C([0, T ]; L2 (R d )). Replace f by ∂xk f and repeat the above arguments, we conclude that N has trajectory in C([0, T ]; H p (R d )) and N (t, ·) ∈ C p (R d ) for p = 1, 2, . . . . 2 Lemma 4.4. N (t, ·) ∈ S(R d ) almost surely for each t fixed. Proof. From Lemma 4.3, we already know that N (t, ·) ∈ C ∞ (R d ). Therefore, we only need to show

sup |x|m N (t, x) < ∞,

a.s.

x∈R d

On the one hand, by a Sobolev (Morrey’s) inequality (e.g. [7, (23), p. 268]), there exists deterministic constant C > 0 when p d,

sup |x|m N (t, x) C |x|m N (t, ·)W 1,p (R d ) .

x∈R d

On the other hand, direct computation shows that for t > 0, there exist (constant coefficient) mth order polynomials of t, denoted by Ci (t), i = 0, 1, 2, . . . , m, (C0 = 0 is a constant), such that t m ∂xmk G(t, x − y) = C0 xkm + C1 (t)xkm−1 + · · · + Cm (t) G(t, x − y). Therefore, by induction, it is sufficient to prove that for all j = 0, 1, . . . , m,

|t

(0,t]×Z y

=

− s|j ∂xmk G(t

− s, x − y)f (s, y; z) dy W (ds, dz)

|t − s| G(t − s, x j

1,p

Wx

− y)∂ymk f (s, y; z) dy W (ds, dz)

(0,t]×Z y

The above holds because of another Sobolev embedding · W 1,p C · H 1+m ,

p 2, 2m > d,

1,p

Wx

< ∞,

a.s.

334


and that because for i = 0, 1, . . . ,

E

x

|t − s| G(t − s, x j

(0,t]×Z y

C

dx

2 n+i G(t − s) ∗x ∂yk f (s, x; z) μ(dz) ds dx

E x

2

− y)∂yn+i f (s, y; z) dy W (ds, dz)

k

(0,t]×Z

n+i ∂ f (s, ·; z)2 μ(dz) ds < ∞, yk 2

CE [0,t]×Z

where the first inequality follows from Burkholder–Davis–Gundy inequality, and the second follows from Young inequality for convolutions. 2 In view of Lemmas 4.2–4.4, by induction, we conclude the following. Lemma 4.5. For each n = 1, 2, . . . , un (t) ∈ C([0, T ]; H p (R d )), p = 1, 2, . . . , un (t, ·) ∈ S(R d ), t > 0. It is well known that, in the context of stochastic semi-linear equation, under moderate conditions, a mild solution is also a weak solution. The following is a statement of this kind in our present context. Its proof follows, for instance, from a straightforward adaptation of [4, Proposition 6.4]. Lemma 4.6. For each ϕ ∈ Cc∞ (R d ),

t

∇ϕ, F un−1 (s) ds

u (t), ϕ − u (0), ϕ = n

n

0

+

σ x, un−1 (r, x); z ϕ(x) dx W (dr, dz)

[0,t]×Z x

t +ε

ϕ, un−1 (r) dr.

0

We define energy functionals e2m : L2 (R d ) → [0, ∞]: 2 1 e2m (u) = m u2 , 2

m = 0, 1, 2, . . . .

Lemma 4.7. There exist finite constants Cε,m,T > 0 which is independent of n such that m n E e2k (u0 ) , t T . E e2m u (t) Cε,m,T 1 + k=0

(41)

(42)


335

Proof. Apply Ito’s formula to e2m , there exists finite constant C˜ ε,m,T > 0, t m n 2 n m n−1 m n n F u (r) , ∇u (r) − ε ∇u (r)2 dr E e2m u (t) = E e2m u (0) + 1 + E 2

0

m

σ x, un−1 (r, x); z 2 dx μ(dz) dr

x

[0,t]×Z x

E e2m (u0 ) + C˜ ε,m,T

t

m n−1 2 E F u (r) 2

0

+ 1 + E e0 un−1 (r) + · · · + e2m un−1 (r) dr. Denote

m n Mn (t) = 1 + E e2k u (t) ,

m M(0) = 1 + E e2k (u0 ) .

k=0

k=0

Then t Mn (t) cM(0) + c

Mn−1 (s) ds 0

where the constant c > 0 is independent of n. Choose K so large that c follows inductively (in n) that Mn (t) cM(0)eKt .

T 0

e−Kt dt < 1. Then it

2

We now show that un converges in appropriate sense to a limiting process. We adapt a wellknown fixed point argument which can be found in proof of part two of Theorem 7.4 in [4], for instance. Because of the term divx F (u(t, x)), the adaptation requires explanation. Lemma 4.8. There exists a Lp (R d )-valued (p 2), Ft -adapted process u satisfying p lim sup E u(t) − un (t)p = 0, n→∞ 0tT

(43)

and for m = 0, 1, 2, . . . , m E e2k (u0 ) , E e2m u(t) Cε,m,T 1 +

t T.

(44)

k=0

In addition, p sup E u(t)p < ∞.

0tT

(45)

336


Moreover, u is a mild solution to (34) in the sense that

t

u(t, x) =

G(t, x − y)u(0, y) dy − y

+

G(t − s, x − y)

d

∂yi Fi u(s, y) dy ds

i=1

0 y

G(t − s, x − y)σ y, u(s, y); z dy W (ds, dz).

(46)

(0,t]×Z y

Proof. First, by direct integration ∂x G(t, ·) = i 1

∂x G(t, x) dx = Ct −1/2 , i

t > 0.

x

We denote n L1 u (t, x) =

t

d

∂i G(t − s, x − y)Fi un (s, y) dy ds.

0 y∈R d i=1

For p 1, t 0, we define a deterministic measure on [0, t] by √ √ m(ds) = mt (ds) = ∂i G(t − s, ·)1 ds = 2Cd( t − t − s ). Then p E L1 un (t, ·) − L1 um (t, ·)p c1

d i=1

p t n E ∂i G(t − s) ∗ u (s) − um (s) (·) ds p

0

p t d

n

∂i G(t − s) ∗ u (s) − um (s) p ds

E

c2

i=1

0

p t

n

m u (s) − u (s) p m(ds)

c3 E

0

t c4 E 0

c5 t

1/2

t p n m u (s) − u (s) m(ds) = c4 E un (s) − um (s)p m(ds) p p

p sup E un (s) − um (s)p ,

0st

0

(47)


337

where the second inequality follows from the Minkowski’s inequality, the third one follows from the Young’s inequality for convolutions, the fourth one follows from Jensen’s inequality. Therefore, p p sup E L1 un (s) − L1 um (s)p ct 1/2 sup E un (s) − um (s)p .

0st

0st

Denote

n L2 u (t, x) =

G(t − s, x − y)σ y, un (s, y); z dy W (ds, dz).

(0,t]×Z y∈R d

By properties of stochastic integral for p 2, p E L2 un (t, ·) − L2 um (t, ·)p

G(t − s, x − y) σ y, un (s, y); z c6 E

x

(0,t]×Z y

p/2 2 m

− σ y, u (s, y); z dy μ(dz) ds dx t c7 E

G(t − s, ·) ∗ un (s) − um (s) (x) 2(p/2) dx ds

0 x

t c8 E

G(t − s, ·) un (s) − um (s) p ds 1 p

0

p c9 t sup E un (s) − um (s)p . 0st

Combine the above, and apply them to (39), there exist ρ ∈ (0, 1) and T0 > 0 which are independent of the initial conditions un (0), n = 1, 2, . . . , such that

n

u − um ≡ sup E un (t) − um (t)p 1/p ρ un−1 − um−1 . p 0tT0

By a fixed point argument and by “pasting” short time existence result to obtain global existence result, we have existence of u satisfying (43). The same type estimates can be used to show (45). Then (44) follows from (42) and Fatou’s lemma, and the mild solution property (46) follows from the fixed point argument applied to (39). 2 Taking limit n → ∞ to the equality in Lemma 4.6 we have

338


Lemma 4.9. For each ϕ ∈ Cc∞ (R d ),

u(t), ϕ − u(0), ϕ =

t

∇ϕ, F u(s) ds +

σ x, u(r, x); z ϕ(x) dx W (dr, dz)

[0,t]×Z x

0

t +ε

ϕ, u(r) dr.

0

4.2.2. Regularity of solution when ε > 0 is fixed As in the proof of Proposition 7.3 in [4], we can show that

U (t, x) =

G(t − s, x − y)σ y, u(s, y); z dy W (ds, dz)

(0,t]×Z y

has a continuous modification as L2 (R d )-valued process. Therefore, suppose that E[e2m (u0 )] < ∞, then not only do we have (44), it can also be shown a posterior that u ∈ C([0, ∞); L2 (R d )) for all d = 1, 2, . . . . In the rest of this subsection, we denote u = uε to emphasize its dependence on ε > 0. Lemma 4.10. Suppose that E[e2m (u0 )] < ∞ for 2m [d/2] + 3. Then there exists an Ft adapted process u = u(t) ∈ C([0, ∞); L2 (R d )) satisfying almost surely that (1) e2m (u(t)) < ∞, for all t > 0; (2) ∂ij u = ∂xi ,xj u(t, ·) ∈ C(R d ) for all i, j = 1, . . . , d. Therefore, (34) holds in the classical strong sense. That is, for each x fixed, (34) holds as a finite-dimensional stochastic differential equation. Proof. The conclusions then follow from (44) and from a Sobolev inequality—see Evans [7, Theorem 6, p. 270]. 2 Apply Ito’s formula to (34), we obtain the following. Lemma 4.11. Let Φ ∈ C 2 (R) and convex. If E[e2m (u0 )] < ∞ for 2m > [d/2] + 3, then there exists Ft -adapted solution with properties listed as in Lemma 4.10 such that (37) holds. 4.2.3. Uniform in ε estimate for solutions of (34) We now denote uε , Fε , σε to emphasize their dependence on ε. Throughout this subsection, we assume that those conditions on Fε , σε at the beginning of Section 4.2 still holds. Next, we derive some estimates which are uniform in ε. For such purpose, we require initial conditions satisfy, for some 2m > [d/2] + 3, E e2m uε (0) < ∞,

ε > 0,


339

and p p sup E uε (0)p + uε (0)2 < ∞,

p = 1, 2, . . . .

ε

(48)

Finally, we require that

sup σε (x, u; z) f (x) 1 + |u| M(z),

(49)

ε>0

where

ZM

2 (z) μ(dz) < ∞

and f ∈ L∞ (R d ) ∩ L2 (R d ).

Lemma 4.12. For even positive integers p = 2, 4, 6, . . . , p sup sup E uε (t, ·)p < ∞.

(50)

ε 0tT

p

Proof. By (45), we already know that sup0tT E[uε (t)p ] < ∞ for every p 2 and T 0. u Let Φ(u) = (p)−1 |u|p and Ψ (u) = (Ψ1 (u), . . . , Ψd (u)) be Ψk (u) = 0 Φ (r)(Fε )k (r) dr. Then for each x ∈ R d fixed, we apply (35), (36), Lemma 4.10 and integration with respect to x to arrive at p p E uε (t)p − E uε (0)p

t p(p − 1)

up−2 (s, x)σε2 ε

E

x, uε (s, x); z dx μ(dz) ds.

Z x

0

Gronwall inequality (noting (49)) then implies p p sup E uε (t)p CT sup E uε (0)p .

2

(51)

ε>0

0tT

In the case of p = 2, since by (37), uε (t)2 − uε (0)2 2

t

2

= 0 Z x

σε2 x, uε (s, x); z dx μ(dz) ds − 2ε

+

∇x uε (s, x) 2 dx ds

[0,t]×R d

uε (s, x)σ x, uε (s, x); z dx W (ds, dz).

(0,t]×Z x

This leads to the following estimates:

(52)

340


Lemma 4.13. For each p = 1, 2, . . . T p 2 sup E ε ∇uε (t)2 dt < ∞. ε>0

(53)

0 2p

Proof. Apply Ito’s formula to uε (t)2 using (52), by (50), (48), 2p 2p sup E uε (T )2 + uε (0)2 < ∞. ε>0

Note that, (49) and (50) give T σε2

sup E ε>0

0

x, uε (s, x); z dx μ(dz) ds

p 0

(0,T ]×Z x

c sup E

ε>0

p

uε (s, x)σ x, uε (s, x); z dx W (ds, dz)

u2ε (s, x)σ 2

p/2

< ∞. x, uε (s, x); z dx μ(dz) ds

[0,T ]×Z

In view of (52), the conclusion follows from the above estimates.

2

More generally, we have the following useful estimate. Lemma 4.14. Let Φ ∈ C 2 (R) with Φ, Φ , Φ having at most polynomial growth. Φ needs not be convex. Then

p T

2

sup E ε Φ uε (t, x) ∇x uε (t, x) dx dt < ∞,

ε>0 0

p = 1, 2, . . . , T > 0.

(54)

Rd

Proof. Let (Φ, Ψ ) be an entropy–entropy flux pair. The equality in (37) holds when u is replaced by uε , ϕ = 1, for general (possibly non-convex) Φ ∈ C 2 . Using (50), the rest of the proof follows that of the previous Lemma 4.13. 2 4.3. Convergence of {uε (t, x): ε > 0} as measure-valued processes We generalize L.C. Young’s relaxed measure approach to treat convergence of nonlinear PDEs in this stochastic setting. We identify uε (t, x) with a random measure-valued function νε (t, x, du) = δuε (t,x) (du).


341

With a slight abuse of notation, we also denote νε (t) = νε (t, dx, du) = νε (t, x, du) dx, and view it as a random measure-valued process in the following sense. Let M0 = M(R d × R) be the space of nonnegative Radon measures ν on R d × R with ν(dx, R) = dx. We endow M0 with a topology τ0 so that νn → ν ∈ M0 if and only if f, νn → f, ν for all f ∈ Cb (R d × R) satisfying f (x, u) = 0 when |x| > k for some k > 0. (M0 , τ0 ) is metrizable as follows. We denote Π ν1 ,ν2 = π ∈ M R d × R × R d × R : π dx, du; R d × R = ν1 (dx, du); π R d × R; dy, dv = ν2 (dy, dv) , ν1 , ν2 ∈ M0 ,

(55)

and introduce r(ν1 , ν2 ) =

∞ 1 qk (ν1 , ν2 ) , 2k 1 + qk (ν1 , ν2 ) k=1

where

qk2 (ν1 , ν2 ) = inf

2 2 ν1 ,ν2 . |x − y| + |u − v| ∧ 1 π(dx, du; dy, dv): π ∈ Π

|x|k, |y|k

Note that on each subspace Ak = {(x, u); |x| k, u ∈ R} ⊂ R d × R, ν(Ak ) = Ck is a finite constant which only depends on k; |x − y|2 + |u − v|2 ∧ 1 defines a metric on Ak which gives the same topology as the one induced by usual Euclidean distance. Consequently qk is just a 2-Wasserstein metric on space of measures on Ak with fixed total mass Ck . It induces the usual weak convergence topology on such a sub-space of finite measures. See Ambrosio, Gigli and Savaré [1, Chapter 7] for some properties of such metric. It follows that (M0 , r) is a complete separable metric space. It can be shown that each νε (t) has continuous trajectories in C([0, ∞), M0 ) ⊂ M([0, ∞); M0 ). Here and below, we write M([0, ∞); M0 ) to denote the space of Borel-measurable, M0 -valued processes on [0, ∞) topologized by a metric d ν1 (·), ν2 (·) =

∞

e−t 1 ∧ r ν1 (t), ν2 (t) dt.

(56)

0

(M([0, ∞); M0 ), d) is a complete separable metric space. For properties of such type of space, see Kurtz [12, Section 4]. We have trouble establishing convergence in probability (even along subsequences) of {νε (·): ε > 0} in C([0, ∞); M0 ). We will prove convergence in M([0, ∞); M0 ) instead. By existence of slicing measure (e.g. [6, Theorem 10, p. 14]), for each ν ∈ M0 , there exists a probability measure-valued function ν(x; ·) = ν(x; du) ∈ P(R) such that for each f ∈ Cb (R d × R),

342


(1) x → u f (x, u)ν(x; du) is Lebesgue measurable; (2) R d ×R f (x, u)ν(dx, du) = R d R f (x, u)ν(x; du) dx. Therefore, each process ν(t) ∈ M([0, ∞), M0 ) also admits a representation ν(t, dx, du) = ν(t, x; du) dx. From this point on, we assume Condition 2.7 holds for σ and for F , except the condition d = 1 (i.e. d may still be any positive integer). We also assume that p E u0 p < ∞,

p = 1, 2, . . . .

We take σε and Fε to be the following particular approximation of σ and F . Let J ∈ Cc∞ (R) be the one-dimensional mollifier in (16), and let φ ∈ Cc∞ (R) be such that 0 φ 1, φ(r) = 1 for |r| < 1 and φ(r) = 0 for |r| > 2 and |φ (r)| 2. Let Fε = (F1,ε , . . . , Fd,ε ) with Fk,ε (r) = φ ε|r|2 Fk (r) ∗ Jε (r), σε (x, u; z) =

d y v

Jε (xk − yk )Jε (u − v) φ ε |y|2 + |v|2 σ (y, v; z) dy dv,

k=1

where Jε (r) = ε −1 J (r/ε). Then Fε , σε satisfy the conditions required at the beginning of Section 4.2. Under (14),

σε (x, u; z) − σ (x, u, z)

d

Lφ,σ |y| + |v| Jε (yk )Jε (v) dyk dv M(z) + σ (x, u; z)

k=1

y v

εC M(z) + σ (x, u; z) ,

(57)

where Lφ,σ > 0 is a constant. Similarly, we can estimate the error for approximating F by Fε . By part one of Condition 2.3, there exist constants C > 0 and p0 ∈ {1, 2, . . .},

F (r + s) − F (r) |s|C 1 + |r|p0 , k

k

r ∈ R, |s| < 1.

Therefore

F (r) − F (r) εC1 1 + |r|p0 . k,ε k

(58)

We now construct a smooth approximation of u0 . Let uε (0, x) = y∈R d

Jε (x − y) u0 (y)φ ε|y|2 dy ∈ Cc∞ R d .

(59)


343

Then for each ε > 0 and m = 1, 2, . . . fixed, 2 1 E e2m uε (0) = E m Jε ∗ u0 φ ε|y|2 2 Cm,ε E u0 22 < ∞, 2 and p p sup E uε (0)p E u0 p < ∞,

p = 1, 2, . . . .

(60)

ε>0

Using the above error estimates, we can derive the following main result of this section. Lemma 4.15. There exists an Ft -adapted process ν0 (·) with trajectory in M([0, ∞); M0 ) such that lim E r νε (t), ν0 (t) = 0,

ε→0+

t 0.

This implies that {νε (·): ε > 0}, as metric space (M([0, ∞); M0 ), d)-valued random variables, converges in probability to ν0 (·). Therefore, for each 0 t1 · · · tm , lim νε (t1 ), . . . , νε (tm ) = ν0 (t1 ), . . . , ν0 (tm ) ,

ε→0+

in probability.

Proof. Let 0 ϕδ ∈ Cc∞ (R d × R d ) be of the form as in (24), and βε be of the form as in Lemma 2.9. We derive an approximate version of the inequality appearing in Lemma 3.2. Notice that xx βε uθ (t, x) − uκ (t, y)

2

= βε uθ (t, x) − uκ (t, y) xx uθ + βε uθ (t, x) − uκ (t, y) ∇x uθ (t, x) . By Ito’s formula,

βε uθ (t, x) − uκ (t, y) ϕδ (x, y) dx dy

R d ×R d

βε uθ (0, x) − uκ (0, y) ϕδ (x, y) dx dy + M(t) + A1 (t) + A2 (t) + A3 (t),

R d ×R d

with non-decreasing processes

1

A

(t) = A1ε,δ,θ,κ (t) =

t

0 R d ×R d

βε uθ (r, x) − uκ (r, y) − divx Fθ uθ (r, x)

+ divy Fκ uκ (r, y) ϕδ (x, y) dx dy

dr,

344


2

A

t

1 2

(t) = A2ε,δ,θ,κ (t) =

β uθ (r, x) − uκ (r, y)

ε

σθ x, uθ (r, x); z

z

0 R d ×R d

2 − σκ y, uκ (r, y); z μ(dz)ϕδ (x, y) dx dy dr, t 3

A

(t) = A3ε,δ,θ,κ (t) =

βε uθ (r, x) − uκ (r, y) (θ xx − κyy )ϕδ (x, y) dx dy dr,

0 R d ×R d

and martingale

βε uθ (r, x) − uκ (r, y)

M(t) = Mε,δ,θ,κ (t) = R d ×R d (0,t]×Z

× σθ x, uθ (r, x); z − σκ y, uκ (r, y); z W (dr, dz)ϕδ (x, y) dx dy.

We can invoke the a priori estimates in Lemma 4.12 to estimate the Ak s. First, since ϕδ has compact support in x, y, uniformly in δ > 0, and since 0 βε (r) r,

sup (θ xx − κyy )ϕδ (x, y) C(θ + κ)δ −2 , x,y

by (50), for each t > 0, lim

θ→0+, κ→0+, δ→0+, ε→0+, δ −2 (θ+κ)→0+

E A3 (t) = 0.

Noting 0 β (r) ε −1 C (Lemma 2.9), E A2 (t) I + II + III where 1 I= 2

t E 0

×

β uθ (r, x) − uκ (r, x)

ε

R d ×R d

σθ x, uθ (r, x); z − σκ y, uκ (r, y); z 2 μ(dz)ϕδ (x, y) dx dy dr,

z

II = ε −1 C

t

III = ε

t C

E 0

σθ x, uθ (r, x); z − σ x, uθ (r, x); z 2 μ(dz)ϕδ (x, y) dx dy dr,

R d ×R d z

0 −1

E

R d ×R d z

σ y, uκ (r, y); z − σκ y, uκ (r, y); z 2 μ(dz)ϕδ (x, y) dx dy dr.


345

By (57), (II + III) = 0.

lim

ε→0+, δ→0+, κ→0+, θ→0+; ε −1 (θ 2 +κ 2 )→0+

By identical arguments as in the proof of Lemma 3.4, we have (29) holds when u(r, x) and v(r, y) are replaced by uθ (r, x) and uκ (r, y), respectively. Therefore lim

ε→0+, δ→0+, θ→0+, κ→0+, δ 2 ε −1 →0+

I = 0.

In summary lim

θ→0+, κ→0+, δ→0+, ε→0+, ε −1 (θ 2 +κ 2 )→0+, ε −1 δ 2 →0+

E A2 (t) = 0.

We now estimate A1 . First, we approximate α = αε , αˆ = αˆ ε in (20) and (22) by αε,θ = (αε,θ;1 , . . .) and αˆ ε,θ = (αˆ ε,θ;1 , . . .) ∞ αε,θ;k (u, v) =

βε (u − w)Fθ,k (w) dw

u =

v

u αˆ ε,θ;k (u, v) =

βε (u − w)Fθ,k (w) dw,

v βε (w − v)Fθ,k (w) dw

−∞

u =

βε (w − v)Fθ,k (w) dw.

v

By (58), there exists p > 1 such that

αε,θ;k (u, v) − αε;k (u, v) Cθ 1 + |u|p + |v|p . Similar estimate holds for αˆ ε,θ;k − αˆ ε;k . Therefore t

A (t) =

E 1

0

αε,θ uθ (r, x), uκ (r, y) ∇y ϕδ (x, y)

R d ×R d

+ αˆ ε,κ uθ (r, x), uκ (r, y) ∇x ϕδ (x, y) dx dy

dr

t

E

0

αε uθ (r, x), uκ (r, y) ∇y ϕδ (x, y)

R d ×R d

+ αˆ ε uθ (r, x), uκ (r, y) ∇x ϕδ (x, y) dx dy

dr

t θ κ +C + E δ δ

s R d ×R d

p

p 1 + uθ (t, x) + vκ (t, y)

− y x x x + y − y j j k k

× δ −d

J J ψ dx dy dr .

2δ 2δ 2 k=j

346


We note that the estimates in (28) still hold with u, v replaced by uθ , vκ . We write lim = lim supε→0+,δ→0+,κ→0+,θ→0+, θ+κ →0+, ε →0+ . By (27) and (32), δ

δ

lim A1 (t) t

lim E

0

d χ uθ (r, x) − uκ (r, y) Fj uθ (r, x) − Fj uκ (r, y)

R d ×R d

j =1

x+y

dx dy dr × Jδ (x − y)∂j ψ

2

t

C lim E

x + y

uθ (r, x) − uκ (r, y) Jδ (x − y)∂j ψ dx dy dr , 2

0 R d ×R d

where C is a constant independent of the choice of ψ . By symmetry, we also have similar estimates when the roles of uθ (t, x) and uκ (t, y) are reversed. √ √ Now, we let θ, κ → 0+ and take ε = θ ∨ κ, δ = ε 2/3 . Then (θ + κ)δ −2 → 0+, (θ 2 + κ 2 )ε −1 → 0+, and δ 2 ε −1 → 0+, εδ −1 → 0+. From the construction of βε , there exists a constant C0 > 0 such that

βε (r) + βε (−r) − |r| εC0 . Denote

uθ (t, x) − uκ (t, y) Jδ (x − y)ψ x + y dx dy . 2

mψ (t) = lim E R d ×R d

It follows from the above estimates that

mψ (t) − mψ (s) C

t d

m∂j ψ (r) dr,

0 s t.

s j =1 −1

By simple approximation, the above still holds when ψ = ψN (x) = e−N |x| ∈ W 1,p (R d ), p = 1, 2, . . . , ∞. As in the proof of Theorem 3.5, it follows (by Gronwall inequality) then lim mψN (t) lim mψN (0).

N →∞

N →∞

We now recall the way uθ (0, x) is constructed in (59), by the integrability estimates in (60),


lim mψN (0) lim

N →∞

lim E

N →∞ θ→0+

uθ (0, x) − u0 (x) ψN (x) dx

x

uκ (0, y) − u0 (y) ψN (y) dy

+ lim E κ→0+

347

= 0.

y

We now estimate limN mψN (t) using qK (νθ (t), νκ (t)) for K = 1, 2, . . . . Let stochastic measure πt (dx, du; dy, dv) = δuκ (t,y) (dv) Jδ (x − y) dy δuθ (t,x) (du) dx. Then πt ∈ Π νθ (t),νκ (t) (see definition in (55)), for N K,

uθ (t, x) − uκ (t, y) Jδ (x − y)ψN x + y dx dy 2

R d ×R d

e−1

|u − v|πt (dx, du; dy, dv)

u,v∈R;|x|K,|y|K 2 e−1 qK

νθ (t), νκ (t) − e−1

|x − y|2 Jδ (x − y) dx dy.

|x| 0} is a sequence of M-valued random variables. Note that M([0, ∞), M0 ) can be continuously embedded into M. Let (Φ, Ψ ) be a given entropy–entropy flux pair with Φ, Φ , Φ having at most polynomial growth. We define Ito’s integral Mε (t, x) = Φ uε (r, x) σε x, uε (r, x); z W (dr, dz) [0,t]×Z

and let Φε (t, x) = Φ uε (t, x) ,

Ψε (t, x) = Ψ uε (t, x) ,

χε (t, x) = χε,1 + χε,2 ,

ψε (t, x) = ψε,1 + ψε,2 ,

and


349

where χε,1 (t, x) = ε∂x2 Φ uε (t, x) = ε∂x Φ uε (t, x) ∂x uε (t, x) , χε,2 (t, x) = ∂t Mε (t, x); and ψε,1 (t, x) =

1 2

Φ uε (t, x) σε2 x, uε (t, x); z μ(dz),

Z

2

ψε,2 (t, x) = −εΦ uε (t, x) ∂x uε (t, x) . The meaning of χε,2 is given as follows. Mε (t) = Mε (t, x, ω) is a continuous function in t for each x, ω ∈ Ω fixed. We can take Schwartz distributional derivative ∂t in t of Mε and such derivative is χε,2 . The equality in (37) (Lemma 4.11) is a statement that, ∂t Φε (t, x, ω) + ∂x Ψε (t, x, ω) = χε (t, x, ω) + ψε (t, x, ω)

(62)

holds ω-wise. Note again, ∂t , ∂x above should all be understood in Schwartz distributional sense. Let T > 0 be an arbitrarily given but fixed constant. We denote O = (0, T ) × R. 4.4.1. A priori estimates for several sequences of random fields The main result of this subsection is the following. −1 Lemma 4.18. {∂t Φε + ∂x Ψε : ε > 0} is a sequence of Hloc (O)-valued random variables. As such random variables, the sequence is tight.

Proof. We apply a stochastic generalization of the Murat lemma (see Lemma A.3) to show −1 that, as Hloc (O)-valued random variables, the left-hand side of (62) is tight. This only requires verifying conditions of Lemma A.3. By the integrability conditions on uε in Lemma 4.12, {Φε : ε > 0} and {Ψε : ε > 0} are both p stochastically bounded as Lloc (O)-valued random variables, 2 p < ∞. Therefore, the left−1,p hand side of (62) is a stochastically bounded sequence in Wloc (O). By the moment estimates (50) on uε in Lemma 4.13, {ψε,1 : ε > 0} is stochastically bounded in L2loc (O), hence it is stochastically bounded in as random variable in Mloc (O) (space of Radon signed-measures on each fixed bounded open subset of O) with total variation norm. By (54), {ψε,2 : ε > 0} is also stochastically bounded (in total variation norm), as Mloc (O)-valued random variables. −1 By (53), limε→0+ χε,1 = 0 in probability as sequence of Hloc (O)-valued random vari−1 ables, and is therefore tight. Finally, we claim that the set of Hloc (O)-valued random variables {χε,2 : ε > 0} is tight (which we will prove in Lemma 4.20 below). 2

350


We call a deterministic function 0 ϕ(t, x) 1 a cutoff function, if ϕ(t, x) ∈ Cc∞ ([0, ∞) × R d ). We use it to discuss local properties of functions such as local integrability and so on. We localize Mε by t Mεϕ (t, x) =

ϕ(r, x) Mε (dr, x). 0

ϕ

We set Mε (t, x) = 0 for t < 0. In order to estimate some fractional derivatives of Mε locally, it will be convenient for us to introduce the notion of Marchaud fractional derivative for α ∈ (0, 1) (Samko, Kilbas, Marichev [16, Section 5.4]). We define α D± φ (t) =

−α (1 − α)

∞

φ(t ∓ s) − φ(t) ds, s 1+α

0

for those φ where the integrand above is L1 integrable. At least for φ ∈ Cc∞ (R) [16, Section 5.7],

d2 − 2 dt

α/2

φ=

d dt

α α φ = D+ φ.

α f ∈ L (R) for some p > 1 and s −1 = p −1 − α, Provided f ∈ Lsloc (R), D− loc p

∞

∞ α f (t)D+ φ(t) dt

=−

−∞

α φ(t)D− f (t) dt,

φ ∈ Cc∞ (R).

−∞

See [16, Corollary 2 of Theorem 6.2]. Therefore, in such cases, Schwartz distributional derivative αf. ∂ α f = D− ϕ Recall that Mε (t, x) is a continuous (in time) local martingale with each x fixed. Hence it is almost surely Hölder continuous in t (almost everywhere) with exponent 0 < β < 1/2 when x ϕ is fixed (Revuz, Yor [15, Exercise 1.20, p. 187]). Consequently, for 0 < α < β < 1/2, ∂tα Mε = ϕ αM . D− ε Lemma 4.19. Assume that (49) holds. Let ϕ be a cutoff function. Then there exists an 0 < α < 1/2 ϕ −1+α such that, as Hloc (O)-valued random variables, {∂t Mε : ε > 0} are stochastically bounded. That is, for each δ > 0, there exists a constant Cδ > 0 with sup P ∂t Mεϕ −1+α > Cδ < δ. ε>0

Proof. First, the integrability estimates in (50) imply that 2 sup E Mεϕ 2 < ∞. ε>0

(63)


351

Therefore, we only need to prove that, for each δ > 0, there exists a constant Cδ > 0 satisfying sup P ∂tα Mεϕ 2 > Cδ < δ.

(64)

ε>0

We verify this next. Recall that we assume Φ(u) is of at most polynomial growth as u → ∞. Take γ > 6, then for 0 s < t T , there exists p0 > 2 such that γ E Mεϕ (t, ·) − Mεϕ (s, ·)γ C1 |x| 0.

(65)

By a normed space version of Garsia’s inequality (e.g. Stroock, Varadhan [17, Exercise 2.4.1, p. 60]), if T T 0 0

ϕ

ϕ

Mε (t) − Mε (s)Lγ |t − s|1/2−1/γ

γ ds dt λ

then ϕ M (t) − M ϕ (s) Cλ |t − s|−3/γ +1/2 , ε ε γ

352


for some deterministic constant Cλ . This implies that, for 0 < α < β ≡ 1/2 − 3/γ , α ϕ γ α ∂ M = D M ϕ γ C7 t ε γ −,t ε γ

|Mεϕ (t + s, x) − Mεϕ (t, x)| γ

dx dt ds

s (1+α) t

x

s

γ

∞ ϕ ϕ Mε (t + s) − Mε (t)γ

C8

ds dt C9 Cλ < ∞,

s (1+α) t

ε > 0,

(66)

s=0

α emphasizes that the Marchaud fractional derivative is taken with respect to t. where D−,t Combining of (65) and (66) gives (64). 2

Lemma 4.20. Assume that (49) holds. Then for each δ > 0 and T > 0, there exists a compact set −1 (O) such that K = K(δ, T ) Hloc inf P (χε,2 ∈ K) > 1 − δ.

ε>0

−1+α −1 Proof. The conclusion follows from the compact embedding of Hloc (O) to Hloc (O) and from the results in Lemma 4.19. 2

4.4.2. Identifying ν0 (t, x; du) as a function To simplify notation, for any function f = f (u), we denote f = f (t, x) =

f (u)ν0 (t, x, du)

u∈R

whenever the integral makes sense. In particular, u(t, x) =

u∈R uν0 (t, x; du).

We also write

D

X = Y for random variables X, Y having identical probability law/distribution. Let (Φi , Ψi ), i = 1, 2, be two choices of entropy–entropy flux pairs, where Φi has at most polynomial growth (therefore Ψi will have at most polynomial growth as well). Lemma 4.21. For every deterministic ϕ ∈ Cc∞ (O), D

ϕ, Ψ1 Φ2 − Φ1 Ψ2 = ϕ, Ψ1 · Φ2 − Φ1 · Ψ2 .

(67)

Proof. On the one hand, by Lemma 4.15 and the uniform in ε moment estimates in (50), for each ϕ ∈ Cc∞ ((0, T ) × R d ), the following convergence in probability result holds lim

ε→0+

ϕ(t, x) Ψ1 uε (t, x) Φ2 uε (t, x) − Φ1 uε (t, x) Ψ2 uε (t, x) dx dt

= lim

ε→0+ (t,x)∈[0,T ]×R

ϕ(t, x)

= ϕ, Ψ1 Φ2 − Φ1 Ψ2 .

u∈R

Ψ1 (u)Φ2 (u) − Φ1 (u)Ψ2 (u) νε (t, x, du) dt dx


353

On the other hand, we can apply Theorem A.2 by choosing Gε (t, x) = Φ1 uε (t, x) , Ψ1 uε (t, x) ,

Hε (t, x) = −Ψ2 uε (t, x) , Φ2 uε (t, x) .

In view of Lemmas 4.12 and 4.18, we have the following convergence in probability law result: lim

ε→0+

ϕ(t, x) Ψ1 uε (t, x) Φ2 uε (t, x) − Φ1 uε (t, x) Ψ2 uε (t, x) dt dx

D

= ϕ, Ψ1 · Φ2 − Φ1 · Ψ2 . We conclude the proof.

2

We now finish the proof of the main result of this section. ∞ Proof of Lemma 4.17. Let u deterministic 0 ϕ ∈ Cc (O). Take Φ1 (u) = u, Ψ1 (u) = F (u) and Φ2 (u) = F (u), Ψ2 (u) = 0 (F )2 (r) dr. Eq. (67) gives

D ϕ, F 2 − uΨ2 = ϕ, (F )2 − u · Ψ2 .

(68)

On the other hand, for t > 0, x ∈ R, ω ∈ Ω fixed, and u ∈ R, by Schwartz’s inequality, 2 F (u) − F u(t, x) =

u

2

F (v) dv

u − u(t, x) Ψ2 (u) − Ψ2 u(t, x) . (69)

u(t,x) ¯

Integrate the above as a function of u against ν0 (t, x, du), 2 F 2 + F (u) − 2F F (u) uΨ2 − u · Ψ2 .

(70)

Take expectation on both (68) and (70) and combine them, 2 ϕ(t, x)E F − F (u) dt dx 0. By the arbitrariness of ϕ, the following holds almost surely: F dt dx =

F (u)ν0 (t, x, du) dt dx = F u(t, x) dt dx = F (u) dt dx.

R

Therefore

ϕ(t, x)(dt dx) =

2 F (u) − F u(t, x, ω) ν0 (t, x, du; ω)P (dω)

ω∈Ω u∈R

2 ϕ(t, x)(dt dx)E F 2 − F (u) =

ϕ(t, x)(dt dx)E[uΨ2 − u · Ψ2 ]

(71)

354


=

ϕ(t, x)(dt dx)

u − u(t, x, ω) Ψ2 (u) − Ψ2 u(t, x; ω) ν0 (t, x, du; ω)P (dω).

ω u

In the above, the second equality follows from (71) and the third one follows from (68). We conclude then, almost surely (69) holds as an equality for all u on the support of random measure ν0 (t, x; du). On the other hand, from the property of Schwartz inequality, this cannot be true ¯ With the condition on F , the support of u has to be on single unless F is constant during u, u. point mass u¯ almost surely. 2 By (50) and Fatou’s lemma

|u| ν0 (t; x, du) dx < ∞,

sup E 0tT

p

p = 2, 4, . . . .

x u

If (61) holds, then

p

u(t, ¯ x) dx < ∞,

sup E 0tT

p = 2, 4, . . . .

(72)

x

4.5. Existence of stochastic entropic solution 4.5.1. Existence of measure-valued solution Let Fε (r) = (Fε,1 , . . . , Fε,d ) be as defined in last section. Let convex Φ ∈ C 2 (R) have at most polynomial growth. Define Ψε = (Ψε,1 , . . . , Ψε,d ) with r

Φ (s)(Fε,k ) (s) ds.

Ψε,k (r) = 0

Then (37) can be written as (Lemma 4.11)

Φϕ, νε (t) − Φϕ, νε (s)

(Ψε · ∇ϕ)νε (r, x, du) dx dr

(s,t]×R d ×R

+

1 2

2 Φ σε ϕ νε (r, x, du) dx dr μ(dz)

Z (s,t]×R d ×R

+ε

(Φϕ)νε (r, x, du) dx dr

(s,t]×R d ×R

+

Φ σε ϕ, νε (r) W (dr, dz),

(s,t]×Z

where shorthand notation Φϕ = Φ(u)ϕ(x) and so on are used. The main result of this section is

(73)


355

Lemma 4.22. Let u(0), F, σ satisfy conditions in Theorem 4.1. Then the Ft -adapted process ν0 (t) has trajectory in C([0, ∞), M0 ), and it satisfies (Ψ · ∇ϕ)ν0 (r, x, du) dx dr Φϕ, ν0 (t) − Φϕ, ν0 (s) (s,t]×R d ×R

+

1 2

2 Φ σ ϕ ν0 (r, x, du) dx dr μ(dz)

Z (s,t]×R d ×R

Φ σ ϕ, ν0 (r) W (dr, dz).

+

(74)

(s,t]×Z

We show the proof in two steps. First, we establish the following. Lemma 4.23. Assume that (61) holds, then lim E r ν0 (t), ν0 (s) = 0, t→s+

s 0.

In particular, this implies that ν0 (·) ∈ C([0, ∞); M0 ). Proof. Let y ∈ R d . Take Φ(u) = |u − uε (s, y)|2 and ϕ(x) = Jδ (x − y) and apply (37) (Lemma 4.11). We notice that u Ψε,k (u) = 2

Fε,k (r)

r − uε (s, y) dr = 2

0

r

Fε,k (r)r dr − 2uε (s, y) Fε,k (u) − Fε,k (0) ,

0

k = 1, 2, . . . , d, have at most polynomial growth, and that

∂x Jδ (x − y) c1 δ −1 ,

x Jδ (x − y) c2 δ −2 . k For ψ 0 and ψ ∈ Cc∞ (R d ), we denote O = int supp(ψ) ,

Oδ = x ∈ R d : dist(x, O) < δ .

Then

uε (t, x) − uε (s, y) 2 Jδ (x − y) dx ψ(y) dy

Fs

E y∈O x∈O δ

uε (s, x) − uε (s, y) 2 Jδ (x − y) dx ψ(y) dy

y∈O x∈O δ

c3 + δ

t

E s

y∈O x∈O δ

2

p1

1 + uε (s, y) + uε (r, x) dx ψ(y) dy Fs dr

356


t +

σ x, uε (r, x); z Jδ (x − y) dx ψ(y) dy μ(dz) Fs dr 2

E y∈O x∈O δ z

s

εc4 + 2 δ

t

E

p

1 + uε (r, x) 2 dx ψ(y) dy Fs dr

y∈O x∈O δ

s

for some p1 , p2 2. Let stochastic measure π(dx, du; dy, dv) = δuε (s,y) (dv) Jδ (x − y) dy δuε (t,x) (du) dx. Then direct verification shows that π ∈ Π νε (t),νε (s) (see definition in (55)). Furthermore, for each K > 0 fixed such that {x: |x| < K} ⊂ O, and for all δ > 0,

uε (t, x) − uε (s, y) 2 Jδ (x − y) dx dy

(x,y)∈O ×O δ

=

|u − v|2 π(dx, du; dy, dv)

(x,y)∈O ×O δ

qK νε (t), νε (s) −

|x − y|2 Jδ (x − y) dx dy

O ×O δ

qK νε (t), νε (s) − δ 2 c5 . In view of the convergence in probability result in Lemma 4.15, E qK ν0 (t), ν0 (s)

|u − v|2 Jδ (x − y)ν0 (s; dx, du)ν0 (s; dy, dv) c5 δ 2 + E y x

c6 + δ

t

sup E s

ε>0

+ c7

y∈O x∈O δ

t

2

p1

1 + uε (s, y) + uε (r, x) dx dy dr

sup E s

ε>0

σ x, uε (r, x); z Jδ (x − y) dx dy μ(dz) dr. 2

y∈O x∈O δ z

Noting (50) and (61), lim sup E qK ν0 (t), ν0 (s) c5 δ 2 + E t→s+

y∈O x∈O δ

2

u(s, ¯ x) − u(s, ¯ y) Jδ (x − y) dx dy .


357

By the arbitrariness of δ > 0 and integrability estimates (72) on u(s) ¯ we arrive at lim E qK ν0 (t), ν0 (s) = 0,

s 0, K = 1, 2, . . . .

t→s+

2

Hence conclude the lemma. Next, we show that Lemma 4.24.

lim E

ε→0+

Φ σε ϕ, νε (r) W (dr, dz) −

(s,t]×Z

2

Φ σ ϕ, ν0 (r) W (dr, dz)

= 0.

(s,t]×Z

Proof.

E

Φ σε ϕ, νε (r) W (dr, dz) −

(s,t]×Z

(s,t]

E

2

Φ σ ϕ, ν0 (r) W (dr, dz)

(s,t]×Z

2 Φ σε ϕ, νε (r) − Φ σ ϕ, ν0 (r) μ(dz) dr.

Z

Therefore, the result follows from Lemma 4.15 and from (50).

2

Assuming (61) holds, then (74) becomes (10). Combined with estimates (72) u(t, ¯ x) is a stochastic entropic solution. 4.6. Existence of stochastic strong entropic solution To be consistent with the notations in the definition of strong entropic solution (as well as the uniqueness proof), we write vε = uε where uε is constructed as in Lemma 4.10, and v = v(t, y) = u(t, y) = uν0 (t, y; dv). We assume all the conditions at the beginning of Section 4.2.3 regarding vε (0), Fε , and σε . We also assume that σ satisfies (13). We assume that (61), a conclusion of Lemma 4.17, holds and consider general dimensions d = 1, 2, . . . . (72) translates into p sup E v(t)p < ∞,

p = 2, 4, . . . .

(75)

0tT

Let u(t) ˜ = u(t, ˜ x) be an arbitrary Ft -adapted Lp (R d )-valued process with p ˜ p < ∞, sup E u(t)

∀T > 0, p = 2, 4, . . . .

0tT

Let β be of the form as in (17). We prove that (12) holds in the following lemma.

(76)

358


Lemma 4.25. For each T > 0, there exists a deterministic function {A(s, t): 0 s t < ∞} such that

E

˜ x) − v(t, y) σ x, u(r, ˜ x); z ϕ(x, y) dx dy W (dr, dz) β u(r,

(s,t]×Z y x

E −

R d ×R d

˜ x) − v(r, y) σ y, v(r, y); z β u(r,

Z×(s,t]

× σ x, u(r, ˜ x); z μ(dz) dr ϕ(x, y) dx dy + A(s, t) with the property that, for each sequence of partitions 0 t1 · · · tm T , lim

maxi |ti+1 −ti |→0+

A(ti , ti+1 ) = 0.

i

The proof consists of several involved estimates. We present them in further subsections. 4.6.1. A special martingale N , and its estimates For each α ∈ C 2 such that α, α , α ∈ Cb (R), and each ϕ ∈ Cc∞ (R d × R d ), we denote

N (α, ϕ)(s, t; y, v) =

α u(r, ˜ x) − v σ x, u(r, ˜ x); z ϕ(x, y) dx W (dr, dz), (77)

(s,t]×Z x

where 0 s t, (y, v) ∈ R d × R, and the integral is defined in Ito’s sense. For each s fixed, the above is a martingale in t s. Next, we derive some useful properties and a priori estimates regarding N . We note that N (α, ϕ)(s, t; y, v) = 0 whenever |y| > C for some large C = Cϕ depending only on the support of ϕ. Lemma 4.26. The following identities hold almost surely for each (y, v) ∈ R d × R fixed: ∂v N(α, ϕ)(s, t; y, v) = N (−α , ϕ)(s, t; y, v), ∂yi N(α, ϕ)(s, t; y, v) = N (α, ∂yi ϕ)(s, t; y, v). Proof. The proof of Theorem 7.6 of Kunita [11, p. 180] can be modified to show this.

2

Lemma 4.27. Suppose that α ∈ Cc (R). Then for each T > 0, p > 5, there exist a > 0, C2 > 0, for any δ > 0, E

sup

s,t∈[0,T ], |t−s| 2, there exist a constant C = C(β, ϕ, p) independent of ε, and a constant a > 0 such that p E Xε (ϕ)(s, t; ·, ·)∞ C|t − s|a . By a formal application of integration by parts (note that we are handling integral with anticipating integrand), we have (79). We justify this rigorously next. Lemma 4.31. The following representation holds almost surely: t Γ1,ε (s, t) = s

Xε (∂yk ϕ) s, r; y, vε (r, y) dy dr.

(79)

y

Proof. Let Jδ be a one-dimensional mollifier as defined before (smooth and have compact support). First, through integration by parts, N (β y

, ϕ)(s, r; y, v)Fε (v)Jδ

v − vε (r, y) dv ∂yk vε (r, y) dy

v

=

∂v Xε (ϕ)(s, r; y, v)Jδ v − vε (r, y) dv ∂yk v(r, y) dy

y v

=

Xε (ϕ)(s, r; y, v)Jδ v − vε (r, y) dv ∂yk v(r, y) dy

y v

=−

Xε (ϕ)(s, r; y, v)∂yk Jδ v − vε (r, y) dy dv

v y

= v y

∂yk Xε (ϕ)(s, r; y, v)Jδ v − vε (r, y) dy dv

362


Xε (∂yk ϕ)(s, r; y, v)Jδ v − vε (r, y) dv dy.

= y v

Sending δ → 0+ and noting the estimates in Lemmas 4.28, 4.30, we arrive at the conclusion.

2

Therefore, by Lemma 4.30,

E Γ1,ε (s, t)

t C1 |r − s|a dy dr C2 |t − s|1+a ≡ A1 (s, t) s |y|cϕ

satisfies that for all partitions of [0, T ], limmaxi |ti+1 −ti |→0+ i |A1 (ti , ti+1 )| = 0. Now we estimate Γ2,ε . First, we note that (36) holds even if Φ is not convex. In particular, β u(¯ ˜ r , x) − vε (t, y) yy vε (r, y)

2

˜ r , x) − vε (r, y) − β u(¯ ˜ r , x) − vε (r, y) ∇y vε (r, y) . = yy β u(¯ Similar to the last lemma, Lemma 4.32. t

N (β , ϕ) s, r; y, vε (r, y) εyy vε (r, y) dy dr

Γ2,ε (s, t) = s |y|0

a Cδm ,

for some a > 0, p −1 + q −1 = 1 with p > 8. In the above, we invoked (80) and Lemma 4.28 for the last inequality. The conclusion follows. 2 We conclude that Lemma 4.29 holds. 4.6.3. Proof of Lemma 4.25 With the above estimates, we are ready to prove the main result of this section. Proof. Let J, Jδ be mollifiers defined as in (16) in the special case of one dimension. Recall notation (77), we first let

Zε,δ (t) =

N (β , ϕ) s, t; y, v − vε (t, y) Jδ (v) dv dy

|y| 0}, we mean the family of probability distributions (on S) of the random variables is tight. ¯ be a sequence of H p (O; R m ) × H q (O; R m )Lemma A.1. Let {(Fε , Gε ): ε > 0} and (F¯ , G) valued random variables, where p = −q ∈ {0, ±1, ±2, . . .}. Suppose the following conditions hold. (1) {Fε : ε > 0} is stochastically bounded in H p (O; R m ). That is, for each δ > 0, there exists a deterministic constant Cδ ∈ (0, ∞) such that sup P Fε H p > Cδ < δ. ε>0


367

(2) Let (·,·)H p denote the inner product in H p (O; R m ). For each finite collection of deterministic φ1 , . . . , φk ∈ H p (O; R m ), k = 1, 2, . . . , sequence of R k × H q (O; R m )-valued random variables converges: D ¯ . lim (φ1 , Fε )H p , . . . , (φk , Fε )H p , Gε = (φ1 , F¯ )H p , . . . , (φk , F¯ )H p , G

ε→0+

(A.1)

Let ·,· be the continuous bilinear pairing between H p (O; R m ) and H q (O; R m ), p = −q. Then D ¯ lim Fε , Gε = F¯ , G,

ε→0+

where the above is joint convergence with that in (A.1). That is, D ¯ F¯ , G ¯ . lim (φ1 , Fε )H p , . . . , (φk , Fε )H p ; Gε ; Fε , Gε = (φ1 , F¯ )H p , . . . , (φk , F¯ )H p ; G;

ε→0+

Proof. By Skorokhod representation theorem (e.g. [5, Theorem 3.18]), without loss of generality, we assume that all random variables are defined on the same probability space and all convergence in probability distribution is convergence almost surely. D ¯ First, for each h, δ > 0, by condition (1) and the assumption that limε→0+ Gε = G,

¯ > h lim P Fε H p Gε − G ¯ Hq > h lim P Fε , Gε − G

ε→0+

ε→0+

¯ H q > hC −1 + P Fε H p > Cδ < δ. lim P Gε − G δ ε→0+

D ¯ ¯ ¯ = F , G. Therefore, to conclude the lemma, it is sufficient to prove that limε→0+ Fε , G Let {f1 , . . . , fk , . . .} and {g1 , . . . , gk , . . .} be a dual system of (deterministic) complete orthonormal bases for H p (O; R m ) and H q (O; R m ), respectively. That is,

f¯, g ¯ =

∞ (f¯, fk )H p (g, ¯ gk )H q ,

∀f¯ ∈ H p , g¯ ∈ H q .

k=1

For every h, δ > 0, by condition (1),

2 ∞

¯ lim sup P

(Fε , fk )H p (G, gk )H q > h

N →∞ ε>0 k=N +1 ∞ 2 2 ¯ gk )H q > h lim sup P Fε H p (G, N →∞ ε>0

lim

N →∞

P

k=N +1 ∞

¯ gk )2H q > hC −2 (G, δ

k=N +1

+ sup P Fε H p > Cδ δ. ε>0

By the above uniform in ε estimate, and by condition (2),

368


¯ = lim lim lim Fε , G

ε→∞ N →∞

ε→0+

D

= lim

lim

N →∞ ε→0+

D

= lim

N →∞

N

∞

¯ gk )H q − (Fε , fk )H p (G,

∞

¯ gk )H q (Fε , fk )H p (G,

k=N +1

k=1 N ¯ gk )H q (Fε , fk )H p (G, k=1

¯ gk )H q = F¯ , G. ¯ (F¯ , fk )H p (G,

2

k=1

¯ H¯ ) be a sequence of L2 (O; R m ) × Theorem A.2 (Div–curl). Let (Gε , Hε ), ε > 0, (G, L2 (O; R m )-valued random variables. Suppose the following holds: (1) {Gε : ε > 0} and {Hε : ε > 0} are both stochastically bounded as L2 (O; R m )-valued random variables. That is, for each δ > 0, there exists a deterministic constant Cδ ∈ (0, ∞) such that sup P Gε L2 + Hε L2 > Cδ < δ. ε>0

(2) For each finite collection of deterministic φ1 , . . . , φk ∈ L2 (O; R m ), lim φ1 , Gε , . . . , φk , Gε ; φ1 , Hε , . . . , φk , Hε

ε→0+

D ¯ . . . , φk , G; ¯ φ1 , H¯ , . . . , φk , H¯ . = φ1 , G, (3) Both {∇ · Gε : ε > 0} and {∇ × Hε : ε > 0} are tight as sequences of H −1 (O)-valued random variables. Then for each finite collection of deterministic ϕ1 , . . . , ϕk ∈ Cc∞ (O), D ¯ · H¯ , . . . , ϕk , G ¯ · H¯ , lim ϕ1 , Gε · Hε , . . . , ϕk , Gε · Hε = ϕ1 , G

ε→0+

where the convergence is joint convergence in probability law/distribution with that in condition (2). Proof. Let H 2 (O, R m )-valued random variables hε be defined as (weak) solution to −hε = Hε ,

x ∈ O,

hε = 0,

x ∈ ∂O.

(A.2)

Condition (1) of the theorem implies that {hε : ε > 0} is stochastically bounded as H 2 (O; R m )valued random variables. Since any bounded set in H 2 (O; R m ) is a compact set in L2 (O; R m ), {hε : ε > 0} is a tight sequence as L2 (O; R m )-valued random variable. Selecting subsequence if necessary, there exists a L2 (O; R m )-valued random variable h0 such that D

lim hε = h0 .

ε→0+


369

Indeed, h0 has to be H 2 (O; R m )-valued. To see this, we invoke the Skorohod representation and assume limε→0+ hε − h0 L2 = 0 almost surely. Then by lower semicontinuity of · H 2 , h0 (·, ω)

H2

lim infhε (·, ω)H 2 . ε→0+

Consequently for each δ > 0, there exists Cδ > 0, P h0 H 2 > Cδ P lim inf hε H 2 > Cδ = P ε→0+

! " hε H 2 > Cδ κ>0 01

∀r ∈ R.

−∞

For an arbitrarily given ψ ∈ L1 ∩ L2 (R), we define on (K , B(K ), Λ(b,β) ) +∞

uψ(t) N (dt, du) ≡ lim

n→∞ |t|n |u|>1

−∞ |u|>1

uψ(t) N (dt, du)

(3.7)

in probability, provided that such a limit exists. Since the right-hand integrals in (3.7) are in +∞ finitely divisible, so is −∞ |u|>1 uψ(t) N(dt, du). By [18, Lemma 7.7], "

+∞

log E(b,β) exp i r −∞ |u|>1

# uψ(t) N (dt, du)

= lim

n→∞ |t|n

hβ rψ(t) dt

(3.8)

exists. Thus ψ ∈ Θβ , and vice versa. Summing up the above argument we have the following result.

668

H.-H. Shih / Journal of Functional Analysis 255 (2008) 657–680

+∞ Proposition 3.5. For ψ ∈ L1 ∩ L2 (R), −∞ |u|>1 uψ(t) N (dt, du) exists on the probability space (K , B(K ), Λ(b,β) ) if and only if ψ ∈ Θβ . Moreover, Λ(b,β) -a.e. on K , +∞ +∞ ·, ψ (b,β) = b ψ(t) dt + Ib,β;1 (ψ · 1R×[−1,1] ) + −∞

(3.9)

uψ(t) N (dt, du).

−∞ |u|>1

Proof. The formula (3.9) immediately follows from Theorem 2.3, (3.6), and (3.8).

2

For η ∈ K and x ∈ Ω(b,β) , it follows from Remark 3.2 that ˙ x), η = (x, η) = X(·;

+∞ η(t) X(dt; x)

−∞

+∞ +∞ +∞ =b η(t) dt + η(t) B(b,β) (dt; x) + η(t) Yβ (dt; x), −∞

−∞

(3.10)

−∞

where all of the above integrals in (3.10) are understood pathwise in the sense of Riemann– Stieltjes integrals; and for c < d it follows from Theorem 3.1 that Λ(b,β) -a.e. on Ω(b,β) , Yβ (d) − Yβ (c) =

β (dt, du) + u1[c,d]×{u;01

R2∗

The above argument leads to the following definition concerning Lévy stochastic integrals. Definition 3.6. For any ψ ∈ Θβ , we define Lévy stochastic integrals: +∞ −∞ ψ(t) X(dt) on (K , B(K ), Λ(b,β) ) by +∞ +∞ ψ(t) Yβ (dt) = uψ(t) · 1{u;01

R2∗

and +∞ +∞ +∞ +∞ ψ(t) X(dt) = b ψ(t) dt + ψ(t) B(b,β) (dt) + ψ(t) Yβ (dt). −∞

−∞

−∞

−∞

4. The Segal–Bargmann transform To begin with, the following two notations are fixed in this section: 1. Dn ≡ {u ∈ R \ {0}; 1/n |u| n};


669

2. For any β ∈ M+ (R), β(n) (E) ≡ β(E ∩ ({0} ∪ Dn )), ∀E ∈ B(R), n ∈ N. For (b, β) ∈ R × M+ (R), let ϕ ∈ L2 (K , Λ(b,β) ) with ϕ ∼ (φm )b,β . Then the Segal– Bargmann (or the S-) transform S(b,β) ϕ of ϕ is a complex-valued analytic functional on L2c (R2 , λβ ) defined by S(b,β) ϕ(g) =

ϕ(x)E(b,β) (g)(x) Λ(b,β) (dx),

g ∈ L2c R2 , λβ ,

K

1 ⊗m ), called the coherent state functionals associated with g where E(b,β) (g) = ∞ m=0 m! Ib,β;m (g corresponding to (b, β). In fact, S(b,β) ϕ(g) =

∞ m=0

...

R2

φm (s1 , . . . , sm )g(s1 ) · · · g(sm ) λβ (ds1 ) . . . λβ (dsm ).

(4.1)

R2

In this section, we will be devoted to derive the closed form of the Segal–Bargmann transform of square-integrable Lévy white noise functionals. Equivalently, we are looking for an explicit form of the coherent state functionals. Let H be a complex Hilbert space and F 1 (H ) be the class of all analytic functions on H with norm ·F 1 (H ) given by f 2F 1 (H ) =

∞ 2 1 m D f (0) L(m) (H ) , (2) m!

m=0

(m)

where D is the Fréchet derivative of f and L(2) (H ) is the space of all m-linear Hilbert–Schmidt operators on H . Then F 1 (H ) is called the Bargmann–Segal–Dwyer space. It is well known that the S(b,β) -transform is a unitary operator from L2 (K , Λ(b,β) ) onto F 1 (L2c (R2 , λβ )) (see [9]). Proposition 4.1. Let H be a complex separable Hilbert space and K the closed subspace of H . on H by For any F ∈ F 1 (K), we define a functional F (h) = F (hK ), F

h ∈ H,

⊥ ⊥ where h = hK + h⊥ K , hK ∈ K and hK ∈ K , the orthogonal complement of K on H . Then 1 F ∈ F (H ) and F F 1 (H ) = F F 1 (K) .

is an analytic function on K. Let {ei }, {fj } be orthonormal Proof. First, it is obvious that F bases of K and K ⊥ , respectively. We note that for any xi ∈ H , i = 1, 2, . . . , m, if there is k ∈ {1, 2, . . . , m} such that xk ∈ K ⊥ , then (0)(x1 , . . . , xm ) = D m F (0) (x1 )K , . . . , (xk−1 )K , 0, (xk+1 )K , . . . , (xm )K = 0. DmF It implies that

670


m D F (0)2 (m)

= L(2) (H ) =

∞

m D F (0)(ei1 , . . . , eim ) 2

i1 ,...,im =0 ∞

m D F (0)(ei , . . . , ei ) 2 m 1

i1 ,...,im =0

2 = D m F (0)L(m) (K) , (2)

F 1 (H ) = F F 1 (K) . whence F

∀m ∈ N ∪ {0},

2

By Proposition 4.1, F 1 (K) is unitarily isometric to a closed subspace of F 1 (H ). Since, for any n ∈ N, L2c (R2 , λβ(n) ) can be regarded as a closed subspace of L2c (R2 , λβ ) by the isometry: h ∈ L2c (R2 , λβ(n) ) → h · 1R×({0}∪Dn ) ∈ L2c (R2 , λβ ), we apply Proposition 4.1 to define a map Φb,β(n) ,β from L2 (K , Λ(b,β(n) ) ) into L2 (K , Λ(b,β) ) by −1 (S(b,β Φb,β(n) ,β (ϕ) = S(b,β) (n) ) ϕ),

ϕ ∈ L2 (K , Λ(b,β(n) ) ).

Therefore Φb,β

(n) ,β

(ϕ)L2 (K ,Λ

(b,β) )

= S(b,β = ϕL2 (K ,Λ(b,β 2 (n) ) ϕF 1 (L2 c (R ,λβ ))

(n) )

).

On the other hand, let Φb,β,β(n) be the map from L2 (K , Λ(b,β) ) into L2 (K , Λ(b,β(n) ) ) by −1 (S(b,β) ϕ|L2c (R2 ,λβ Φb,β,β(n) (ϕ) = S(b,β (n) )

(n)

) ),

ϕ ∈ L2 (K , Λ(b,β) ).

Proposition 4.2. Let (b, β) ∈ R × M+ (R) and n ∈ N. (i) If ϕ ∈ L2 (K , Λ(b,β(n) ) ) with the chaos decomposition ϕ ∼ (φm )b,β(n) , then Φb,β(n) ,β (ϕ) ∼ φm · (1R×({0}∪Dn ) )⊗m b,β . (ii) If ψ ∈ L2 (K , Λ(b,β) ) with the chaos decomposition ψ ∼ (ψm )b,β , then Φb,β,β(n) (ψ) ∼ (ψm )b,β(n) . Proof. It is straightforward by comparing the S-transform of both-side functionals. For any n ∈ N, we define a mapping Jb,β,n from K into K by Jb,β,n (x) =

b + B˙ (b,β) (·; x) + Y˙β,n (·; x) 0

if x ∈ Ω(b,β) , if x ∈ / Ω(b,β) .

2


671

For each η ∈ K and x ∈ K ,

+∞ +∞ +∞ η(t) dt + η(t) B(b,β) (dt; x) + η(t) Yβ,n (dt; x) · 1Ω(b,β) (x), Jb,β,n (x), η = b

−∞

−∞

−∞

hence the function x ∈ K → (Jb,β,n (x), η) is obviously B(K )-measurable, and so J−1 b,β,n ({x ∈ K ; (x, η) a}) lies in B(K ) for any a ∈ R. Then Jb,β,n is B(K )-measurable by applying the fact that B(K ) is a σ -field generated by all cylinder sets.

Proposition 4.3. Let (b, β) ∈ R × M+ (R) and n ∈ N. (i) For any t ∈ R and x ∈ Ω(b,β) , jX t; Jb,β,n (x) = Yβ,n (t; x) − Yβ,n (t−; x) = jX (t; x) · 1Dn jX (t; x) ; Yβ(n) t; Jb,β,n (x) = Yβ,n (t; x); X t; Jb,β,n (x) = bt + B(b,β) (t; x) + Yβ,n (t; x). (ii) For any B ∈ B(K ), Λ(b,β(n) ) (B) = Λ(b,β) (J−1 b,β,n (B)). (iii) For any complex-valued bounded continuous function ϕ on K ,

ϕ(x) Λ(b,β(n) ) (dx) =

lim

n→∞

K

ϕ(x) Λ(b,β) (dx). K

(iv) For any t ∈ R, B(b,β(n) ) (t; Jb,β,n (x)) = B(b,β) (t; x), Λ(b,β) -a.e. x ∈ Ω(b,β) . Proof. Since Jb,β,n (x) = b + T(B(b,β) (·; x) + Yβ,n (·; x)) for any x ∈ Ω(b,β) , the assertion in (i) immediately follows from the definitions of Jb,β,n , Yβ(n) , Yβ,n , and Theorem 2.7, where T is the map defined several lines before Theorem 2.7. Observe that for any η ∈ K, +∞ C(b,β(n) ) (η) = exp f(b,β(n) ) η(t) dt −∞

=

+∞ +∞ exp i b η(t) dt + i η(t) B(b,β) (dt; x) −∞

K

−∞

+∞ +i η(t) Yβ,n (dt; x) Λ(b,β) (dx)

=

−∞

ei (Jb,β,n (x),η) Λ(b,β) (dx) K

672


=

ei (x,η) Λ(b,β) J−1 b,β,n (dx) .

K

This implies that Λ(b,β(n) ) and Λ(b,β) ◦ J−1 b,β,n coincide on the σ -field generated by all random variables (·, η), η ∈ K, hence on B(K ) and we obtain the assertion in (ii). The assertion in (iii) is a simple consequence of (ii) and Remark 3.2. Finally, we observe that 1=

exp i X(t; x) − bt − B(b,β(n) ) (t; x) − Yβ(n) (t; x) Λ(b,β(n) ) (dx)

K

(by Theorem 3.1) = exp i X t; Jb,β,n (x) − bt − B(b,β(n) ) t; Jb,β,n (x) − Yβ(n) t; Jb,β,n (x) Λ(b,β) (dx) K

by (ii) = exp i X t; Jb,β,n (x) − bt − B(b,β(n) ) t; Jb,β,n (x) − Yβ,n (t; x) Λ(b,β) (dx). K

by (i) .

Then X t; Jb,β,n (x) = bt + B(b,β(n) ) t; Jb,β,n (x) + Yβ,n (t; x), Combining this with (i), the assertion in (iv) immediately follows.

Λ(b,β) -a.e. x ∈ Ω(b,β) . 2

For any (b, β) ∈ R × M+ (R), by Proposition 4.3, there is Ξ(b,β) ∈ Ω(b,β) ∩ B(K ) with Λ(b,β) (Ξ(b,β) ) = 1 such that for any x ∈ Ξ(b,β) , Jb,β,n (x) ∈ Ω(b,β(n) ) and the formula in Proposition 4.3(iv) holds for any n ∈ N and t ∈ R. Proposition 4.4. Let (b, β) ∈ R × M+ (R) and ϕ ∈ L2 (K , Λ(b,β(n) ) ) for some n ∈ N. Then Φb,β(n) ,β (ϕ) = ϕ ◦ Jb,β,n ,

Λ(b,β) -a.e. on Ξ(b,β) .

2 ((R2 )m , λ⊗m ) of the form Proof. Let Sm , m ∈ N, be the class of all functions gm ∈ L c β(n) gm =

i=1

1E i ×···×E i , ai 1

m

ai ’s ∈ C,

where Eji ∈ B(R2 ) with λβ(n) (Eji ) < +∞, Eji < Eji +1 , and Eji ∩ (R × {0}) is a union of disjoint intervals of the form (p, q] for any i, j . By Proposition 4.3, we see that


673

M(b,β(n) ) Eji ; Jb,β,n (x) +∞ +∞ = 1E i (t, 0) B(b,β(n) ) dt; Jb,β,n (x) + j

−∞

β(n) dt, du; Jb,β,n (x) u1E i (t, u) N j

−∞ |u|>0

+∞ +∞ β (dt, du; x) = 1E i (t, 0) B(b,β) (dt; x) + u1E i (t, u) N j

j

−∞

−∞ Dn

= M(b,β) Eji ∩ R × {0} ∪ Dn ; x ,

∀x ∈ Ξ(b,β) ,

(4.2)

and therefore, for gm ∈ Sm as above, m ai M(b,β(n) ) Eji ; Jb,β,n (x) Ib,β(n) ;m (gm ) Jb,β,n (x) = j =1

i=1

=

i=1

ai

m

by (4.2) M(b,β) Eji ∩ R × {0} ∪ Dn ; x

j =1

= Ib,β;m gm · (1R×({0}∪Dn ) )⊗m (x).

(4.3)

2 ((R2 )m , λ⊗m ) (see [4, Theorem Now, assume that ϕ ∼ (φm )b,β(n) . Since Sm , m ∈ N, is dense in L c β(n) 2.1]), given an > 0, we can choose ψm ∈ Sm such that |ψm − φm |2L2 ((R2 )m ,λ⊗m ) c

β(n)

, m! · 2m+4

for each m ∈ N ∪ {0}.

By the Cauchy–Schwarz inequality, (4.3), Propositions 4.3(ii), and 4.2(i), ϕ ◦ Jb,β,n − Φb,β

(n) ,β

2 (ϕ)L2 (K ,Λ

(b,β) )

2 N ⊗m (x) Λ(b,β) (dx) 2 Ib,β;m ψm · (1R×({0}∪Dn ) ) ϕ Jb,β,n (x) − m=0

Ξ(b,β)

N 2 + 2 Ib,β;m ψm · (1R×({0}∪Dn ) )⊗m − Φb,β(n) ,β (ϕ) 2

L (K ,Λ(b,β) )

m=0

2 N =2 Ib,β(n) ;m (ψm ) Jb,β,n (x) Λ(b,β) (dx) ϕ Jb,β,n (x) − m=0

Ξ(b,β)

N 2 + 2 Ib,β;m ψm · (1R×({0}∪Dn ) )⊗m − Φb,β(n) ,β (ϕ) 2 m=0

L (K ,Λ(b,β) )

674


2 N = 4ϕ − Ib,β(n) ;m (ψm ) 2

L (K ,Λ(b,β(n) ) )

m=0

=4

N m=0

m!|φm − ψm |2L2 ((R2 )m ,λ⊗m ) c β (n)

∞

+

m=N +1

∞ +4 m!|φm |2L2 ((R2 )m ,λ⊗m ) → c β(n) 2 2

m!|φm |2L2 ((R2 )m ,λ⊗m ) c β (n)

as N → ∞.

m=N +1

Since > 0 is arbitrarily given, the proposition is proved.

2

Corollary 4.5. Let (b, β) ∈ R × M+ (R) and g ∈ L2c (R2 , λβ(n) ), n ∈ N. Then E(b,β(n) ) (g) ◦ Jb,β,n = E(b,β) (g · 1R×({0}∪Dn ) ),

Λ(b,β) -a.e. on K .

In Lee, Shih [9] the closed form of the Segal–Bargmann transform was derived for those +∞ β ∈ M+ (R) satisfying the moment condition: −∞ |u|n β(du) < +∞ for any n ∈ N. In the rest of this paper, we will show that this result is actually fulfilled by all β ∈ M+ (R), even without the moment condition. In fact, we will show that the limit “limn→∞ E(b,β(n) ) (Jb,β,n (x))” exists for Λ(b,β) -a.e. x ∈ K , and find its explicit form. First, for (b, β) ∈ R × M+ (R), let Dβ = h ∈ L2c (R2 , λβ ); h∗ ∈ L1c (R2∗ , νβ ) ,

h∗ (t, u) = uh(t, u), (t, u) ∈ R2∗ .

Then Dβ is a complex Banach space with the |·|β -norm defined by |h|β =

1/2 h(t, u) 2 λβ (dt, du) uh(t, u) νβ (dt, du), +

R2

h ∈ Dβ ,

R2∗

and Dβ ⊂ Dβ(n) , n ∈ N. For each h ∈ Dβ , by [9, Proposition 2.1] we can take a Δ(b,β) (h) ∈ B(K ) with Λ(b,β) (Δ(b,β) (h)) = 1 such that ∗ h∗ t, jX (t; x) = h (t, u) N (dt, du; x) < +∞, t∈R

x ∈ Δ(b,β) (h).

R2∗

∗ We note that for x ∈ Δ(b,β) (h), $ there are ∗only a finite number of 1 + h (t, jX (t; x)) which are zero; and the infinite product t∈R (1 + h (t, jX (t; x))) is absolutely convergent. Thus, for each h ∈ Dβ , there is associated a functional on K defined by

$ Υ(b,β) (h; x) =

0,

∗ t∈R (1 + h (t, jX (t; x)))

if x ∈ Δ(b,β) (h), otherwise.

(4.4)

In addition, define two complex functions G(b,β) and P(b,β) on Dβ × (K , B(K ), Λ(b,β) ) by


β({0}) G(b,β) (h; ·) = exp − 2 P(b,β) (h; x) =

exp(−

675

+∞ +∞ h(t, 0)2 dt + h(t, 0) B(b,β) (dt) −∞

−∞

∗ R2∗ h (t, u) νβ (dt, du))Υ(b,β) (h; x)

0,

if x ∈ Δ(b,β) (h), otherwise.

(4.5)

Proposition 4.6. Let (b, β) ∈ R × M+ (R). Then, for any h ∈ Dβ and n ∈ N, (i) G(b,β(n) ) (h; Jb,β,n (x)) = G(b,β) (h; x), Λ(b,β) -a.e. x ∈ Ξ(b,β) ; (ii) P(b,β(n) ) (h; Jb,β,n (x)) = P(b,β) (h · 1R×Dn ; x), Λ(b,β) -a.e. x ∈ Ξ(b,β) ∩ Δ(b,β) (h). Proof. First, we observe that

β({0}) G(b,β) (h; ·) = exp − 2

+∞ 2 h(t, 0) dt + Ib,β;1 (h · 1R×{0} ) . −∞

Let ϕ = Ib,β(n) ;1 (h · 1R×{0} ). By Proposition 4.2, Φb,β(n) ,β (ϕ) = Ib,β;1 (h · 1R×{0} ). The assertion (i) immediately follows from Proposition 4.4. Next, by Proposition 4.3(ii), we see that for Λ(b,β) a.e. x ∈ Ξ(b,β) ∩ Δ(b,β) (h), Jb,β,n (x) ∈ Δ(b,β(n) ) (h). Then P(b,β(n) ) h; Jb,β,n (x)

∗ h (t, u) νβ(n) (dt, du) Υ(b,β(n) ) h; Jb,β,n (x) = exp − R2∗

= exp − (h · 1R×Dn )∗ (t, u) νβ (dt, du) Υ(b,β) (h · 1R×Dn ; x) R2∗

= P(b,β) (h · 1R×Dn ; x). We obtain the assertion (ii) and the proof is complete. Lemma 4.7. (See [9, Theorem 5.8].) Let (b, β) ∈ R × M+ (R). Assume that β satisfies the moment condition. Then, for any ϕ ∈ L2 (K , Λ(b,β) ) and h ∈ Dβ , S(b,β) ϕ(h) =

ϕ(x)G(b,β) (h; x)P(b,β) (h; x) Λ(b,β) (dx). K

Now, we are ready to prove the main theorem. Theorem 4.8. Let (b, β) ∈ R × M+ (R). Then, for any ϕ ∈ L2 (K , Λ(b,β) ) and h ∈ Dβ , S(b,β) ϕ(h) =

ϕ(x)G(b,β) (h; x)P(b,β) (h; x) Λ(b,β) (dx). K

676


Proof. Assume that ϕ is a bounded continuous function on K . Then ϕ ∈ L2 (K , Λ(b,β(n) ) ) for any n ∈ N. Observe that for h ∈ Dβ , S(b,β) ϕ(h) = lim S(b,β) Φb,β(n) ,β (ϕ) (h · 1R×({0}∪Dn ) ) n→∞ ϕ Jb,β,n (x) E(b,β(n) ) (h) Jb,β,n (x) Λ(b,β) (dx), = lim n→∞

(4.6)

K

where the first equality is obtained by (4.1) and Proposition 4.2, and the second equality by Proposition 4.4 and Corollary 4.5. Since β(n) satisfies the moment condition, by Proposition 4.3(ii) and Lemma 4.7 we see that

ϕ Jb,β,n (x) E(b,β(n) ) (h) Jb,β,n (x) Λ(b,β) (dx)

K

=

ϕ(x)E(b,β(n) ) (h)(x) Λ(b,β(n) ) (dx) K

=

ϕ(x)G(b,β(n) ) (h; x)P(b,β(n) ) (h; x) Λ(b,β(n) ) (dx) K

=

ϕ Jb,β,n (x) G(b,β(n) ) h; Jb,β,n (x) P(b,β(n) ) h; Jb,β,n (x) Λ(b,β) (dx)

K

=

ϕ Jb,β,n (x) G(b,β) (h; x)P(b,β) (h · 1R×Dn ; x) Λ(b,β) (dx),

(4.7)

K

where the last equality is obtained by Proposition 4.6. We note that ϕ(Jb,β,n (x)) → ϕ(x) as n → ∞ since Jb,β,n (x) → x in the weak topology of K , x ∈ Ω(b,β) . Now, for x ∈ Δ(b,β) (h) ∩

∞ %

Δ(b,β) (h · 1R×Dn ),

n=1

the Λ(b,β) -measure of which is 1, we have P(b,β) (h; x) − P(b,β) (h · 1R×D ; x) n |h∗ |L1 (R2 ,ν ) Υ(b,β) (h; x) − Υ(b,β) (h · 1R×D ; x) c ∗ β e n + exp

R2∗ \(R×Dn )

!

∗ h (t, u) νβ (dt, du) − 1 Υ(b,β) (h; x) .

(4.8)


677

Let Rn (x) denote the term in the brace of (4.8). Then Rn (x) → 0 as n → ∞, and

∗ |h∗ | +1 Rn (x) 2 + e L1c (R2∗ ,νβ ) exp ln 1 + h (t, u) N (dt, du; x) , R2∗

where the right-hand term is in L1 (K , Λ(b,β) ). Thus, by applying the Lebesgue dominated argument to (4.7) we conclude that this theorem holds for any bounded continuous function ϕ in L2 (K , Λ(b,β) ). Since the class of all bounded continuous functions on K is dense in L2 (K , Λ(b,β) ), the theorem follows. 2 The following proposition gives the analyticity of G(b,β) and P(b,β) for any (b, β) ∈ R × M+ (R). Proposition 4.9. Let (b, β) ∈ R × M+ (R). Then, the following two functions: h → G(b,β) (h; ·)

and h → P(b,β) (h; ·)

are entire from the complex Banach space Dβ into L2 (K , Λ(b,β) ). Proof. From (4.5) it is easy to see that for any x ∈ K ,

∗ Υ(b,β) (h; x) exp ln 1 + h (t, u) N (dt, du; x) , R2∗

which implies that P(b,β) (h; ·) 2 L (K ,Λ

(b,β) )

exp 2|h∗ |L1c (R2∗ ,νβ ) + (1/2)|h|2L2 (R2 ,λ ) . c

β

(4.9)

On the other hand, G(b,β) (h; ·)

β({0}) = exp L2 (K ,Λ(b,β) ) 2

+∞ 2 h(t, 0) dt . −∞

Thus, the maps h → G(b,β) (h; ·) and h → P(b,β) (h; ·), h ∈ Dβ , are locally bounded. In fact, it is obvious that h → G(b,β) (h; ·) is analytic. To see the analyticity of the map: h → 2 P(b,β) (h; ·), it is sufficient to show that for any g, h ∈ Dβ and ϕ ∈ L (K , Λ(b,β) ), the function K ϕ(x)P(b,β) (zg + h; x) Λ(b,β) (dx), z ∈ C, is entire in C. Take a sequence {zn } ⊂ C which converges to z. Let Ξ = Δ(b,β) (g) ∩ Δ(b,β) (h) ∩ Δ(b,β) (zg + h) ∩

∞ %

Δ(b,β) (zn g + h) .

n=1

Then Λ(b,β) (Ξ ) = 1. Fix an x ∈ Ξ . Let {t1x , t2x , . . .} be the enumerable set of all t ∈ R with jX (t; x) = 0. Since, for any n, k ∈ N,

678

H.-H. Shih / Journal of Functional Analysis 255 (2008) 657–680 ∞ (zn g ∗ + h∗ ) t x , jX (t x ; x) j

j

j =k ∞ ∗ 1 + sup |zn |; n ∈ N |g | + |h∗ | tjx , jX tjx ; x ,

(4.10)

j =k

where the last sum tends to zero as k → ∞, there is k0 ∈ N such that (zn g ∗ + h∗ ) t x , jX t x ; x < 1 j j 2 for all n ∈ N and j k0 + 1. So, for each j k0 + 1, lim log 1 + (zn g ∗ + h∗ ) tjx , jX tjx ; x = lim log 1 + (zg ∗ + h∗ ) tjx , jX tjx ; x ,

n→∞

n→∞

(4.11) where log stands for the principle branch of the logarithm on C \ {w ∈ R; w 0}. By (4.11), using the inequality: log(1 + w) (3/2)|w| for |w| < 1/2, (4.12) and applying the dominated convergence argument, ∞

lim

n→∞

log 1 + (zn g ∗ + h∗ ) tjx , jX tjx ; x

j =k0 +1

=

log 1 + (zn g ∗ + h∗ )(t, u) Cx (dt, du)

{(tjx ,jX (tjx ;x));j k0 +1}

=

∞

log 1 + (zg ∗ + h∗ ) tjx , jX tjx ; x ,

(4.13)

j =k0 +1

where Cx is the counting measure on {(tjx , jX (tjx ; x)); j k0 + 1}. By (4.11) and (4.13), lim Υ(b,β) (zn g ∗ + h∗ ; x)

n→∞

=

k 1 + (zg ∗ + h∗ ) tjx , jX tjx ; x j =1

× exp

∞

log 1 + (zg ∗ + h∗ ) tjx , jX tjx ; x ,

k k0 + 1.

j =k+1

Since, by (4.10) and (4.12), the above sum tends to zero as k → ∞, we obtain that lim Υ(b,β) (zn g ∗ + h∗ ; x) = Υ(b,β) (zg ∗ + h∗ ; x),

n→∞

x ∈ Ξ.


679

Therefore, ϕ(x)P(b,β) (zn g + h; x) Λ(b,β) (dx)

lim

n→∞

K

=

ϕ(x) lim P(b,β) (zn g + h; x) Λ(b,β) (dx) n→∞

by (4.9)

Ξ

(zg ∗ + h∗ )(t, u) νβ (dt, du) = exp − × =

R2∗

ϕ(x) lim Υ(b,β) (zn g ∗ + h∗ ; x) Λ(b,β) (dx) n→∞

Ξ

ϕ(x)P(b,β) (zg + h; x) Λ(b,β) (dx), K

by which we conclude that the map: z ∈ C → K ϕ(x)P(b,β) (zg + h; x) Λ(b,β) (dx) is continuous. For every triangular path T in the complex plane, by (4.9) and applying the dominated convergence argument we have ϕ(x)P(b,β) (zg + h; x) Λ(b,β) (dx) dz T K

=

∗ ∗ ϕ(x) lim exp − (zg + h )(t, u) νβ (dt, du) r→∞

Ξ

×

T

R2∗

! r 1 + (zg ∗ + h∗ ) tjx , jX tjx ; x dz Λ(b,β) (dx) = 0. j =1

By the Morera’s theorem, the function z ∈ C → P(b,β) (zg + h; ·) is entire. The proof is complete. 2 Acknowledgments The author would like to thank the referee for pointing out several misprints and making useful comments to improve this paper. He would like to express his sincere gratitude to Professor YuhJia Lee for his constant encouragement and many valuable suggestions during the preparation of this paper. References [1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Univ. Press, Cambridge, 2004. [2] T. Hida, Analysis of Brownian Functionals, second ed., Carleton Math. Lecture Notes, vol. 13, Carleton Univ. Press, Carleton, 1978.

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Inviscid limit for the energy-critical complex Ginzburg–Landau equation Chunyan Huang, Baoxiang Wang ∗ LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China Received 22 October 2007; accepted 25 April 2008 Available online 29 May 2008 Communicated by C. Kenig

Abstract In this paper, we consider the limit behavior for the solution of the Cauchy problem of the energy-critical complex Ginzburg–Landau equation in Rn , n 3. In lower dimension case (3 n 6), we show that its solution converges to that of the energy-critical nonlinear Schrödinger equation in C(0, T , H˙ s (Rn )), T > 0, s = 0, 1, as a by-product, we get the regularity of solutions in H 3 for the nonlinear Schrödinger equation. In higher dimension case (n > 6), we get the similar convergent behavior in C(0, T , L2 (Rn )). In both cases we obtain the optimal convergent rate. © 2008 Elsevier Inc. All rights reserved. Keywords: Complex Ginzburg–Landau equation; Nonlinear Schrödinger equation; Inviscid limit; Energy-critical power

1. Introduction We are interested in the inviscid limit behavior between the energy-critical complex Ginzburg–Landau equation (CGL) ut − (μ + i)u + (a + i)|u|α u = 0,

u(0, x) = u0 (x),

(1.1)

and the nonlinear Schrödinger equation (NLS) vt − iv + i|v|α v = 0,

v(0, x) = v0 (x),


E-mail addresses: [email protected] (C. Huang), [email protected] (B. Wang). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.017

(1.2)

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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725

+ n + where √ u(t, x) and v(t, x) are complex-valued functions of (t, x) ∈ R × R , R = [0, ∞), i = −1, μ > 0, a > 0, ut = ∂u/∂t, vt = ∂v/∂t, = ∂ 2 /∂ 2 x1 + · · · + ∂ 2 /∂ 2 xn , u0 and v0 are known complex-valued functions of x ∈ Rn . In Eqs. (1.1) and (1.2), 0 < α < 4/(n − 2) (0 < α < ∞, if n = 1, 2) is said to be an energy-subcritical power, α = 4/(n − 2) is said to be an energy-critical power. Eq. (1.1) was first discovered by Ginzburg and Landau for a phase transition in superconductivity [8], which was also subsequently derived in many other areas, such as instability waves in hydrodynamics and pattern formation; cf. [11]. When α is an energysubcritical or an energy-critical power, the global well-posedness of (1.1) has been established, see for instance, Ginibre and Velo [7] and references therein. Eq. (1.2) is a basic model equation for diverse physical phenomena, including Bose–Einstein condensates, description of the envelop dynamics of a general dispersive wave in a weakly nonlinear medium (cf. [10,16]). The global well-posedness of solutions of (1.2) has been studied by many authors (cf. [3–5,13]). When α is an energy-subcritical power, the global well-posedness of (1.2) can be found in Kato [12]. When α is an energy-critical power, Bourgain [3] and Grillakis [9] showed the global well-posedness and the existence of scattering operators for Eq. (1.2) with radial and finite energy data in three spatial dimension, and the radial condition was removed by Colliander, Keel, Staffilani, Takaoka and Tao [5]. Tao [17] showed the global well-posedness of Eq. (1.2) with radial and finite energy data in higher spatial dimensions. Recently, the radial condition in higher spatial dimensions was removed by Ryckman and Visan (cf. [15,19]). In this paper, we consider the following question: does the solution of (1.1) tend to the solution of (1.2) as the parameters μ and a tend to zero? This problem has been studied by several authors; cf. Wu [23], Bechouche and Jüngel [1], Wang [20] and Machihara and Nakamura [14]. But as far as the authors can see, in the energy-critical case α = 4/(n − 2), except that Bechouche and Jüngel [1] obtained the convergent behavior in the weak-star topology, the inviscid limit behavior in the strong topology between (1.1) and (1.2) remains unsolved; cf. [20]. When α = 4/(n − 2), (1.1) and (1.2) can be rewritten as 4

ut − (μ + i)u + (a + i)|u| n−2 u = 0, vt − iv + i|v|

4 n−2

v = 0,

u(0, x) = u0 (x),

v(0, x) = v0 (x).

(1.3) (1.4)

We will show that when the spatial dimension n 3, the solution of (1.3) strongly converges to that of (1.4) in C(0, T ; L2 ) for any T > 0. Moreover, when 3 n 6, we can prove the stronger convergent behavior in C(0, T ; H˙ 1 ) for any T > 0. Let Lr (Rn ) be the Lebesgue space, f r := f Lr (Rn ) , f ˙ k := ∇ k f r , H˙ k := H˙ k , H k := L2 ∩ H˙ k . Put Hr

E u(t) =

2

Rn

2 n − 2 2n 1 u(t, x) n−2 dx, ∇u(t, x) + 2 2n

(1.5)

which is said to be the energy for the CGL (1.3) and the NLS (1.4). E(u) is invariant under the scaling u(t, x) → uλ (t, x) = λ Our main results are the following.

n−2 2

u λ2 t, λx ,

λ > 0.

(1.6)


683

Theorem 1.1. Let 3 n 6. Let u0 ∈ H 1 , v0 ∈ H 3 , 0 < δ 1. Assume that T > 1 is arbitrary and u0 − v0 H 1 δ,

0 < μT , a δ.

(1.7)

Then the solutions of (1.3) and (1.4) satisfy the following approximate behavior: k ∇ (u − v)

2 n L∞ t Lx ([0,T ]×R )

C ∇ k (u0 − v0 )2 + μT + a ,

k = 0, 1,

(1.8)

where C := C(u0 H 1 , v0 H 3 ) denotes a constant that depends only on u0 H 1 and v0 H 3 . Theorem 1.2. Let n 5. Let u0 ∈ H 1 , v0 ∈ H 1 , 0 < δ 1. Assume that T > 1 is arbitrary and u0 − v0 H 1 δ,

0 < μT , a δ.

(1.9)

Then the solutions of (1.3) and (1.4) satisfy the following approximate behavior: 1/2 +a , u − vL∞ 2 n C u0 − v0 2 + (μT ) t Lx ([0,T ]×R )

(1.10)

where C := C(n, u0 H 1 , v0 H 1 ) denotes a constant that depends only on n, u0 H 1 and v0 H 1 . Theorem 1.3. Let n > 6. u0 ∈ H 1 , 0 < a, μ 1. Then the solutions u of (1.3) have the following upper bound estimate: u

2(n+2) Lt,xn−2 (R+ ×Rn )

C E(u0 ) ,

(1.11)

where C(E(u0 )) denotes a constant that depends only on E(u0 ) and is independent of μ, a. We now first briefly sketch the proofs of Theorems 1.1 and 1.2. On the basis of our earlier work [20], if we get the upper bounds of the solutions u and v of Eqs. (1.3) and (1.4) in the spaces L2(n+2)/(n−2) (R+ × Rn ), then we can prove our Theorems 1.1 and 1.2 by iteration method. Using Bourgain’s, Colliander, Keel, Staffilani, Takaoka and Tao’s, and Ryckman and Visan’s results (cf. [3,5,15,19]), we see that v in the space L2(n+2)/(n−2) (R+ × Rn ) has an upper bound that depends only on v0 H˙ 1 . A natural idea seems to apply a similar way as in [5,15,19] obtaining a uniform upper bound of u in L2(n+2)/(n−2) (R+ × Rn ) for all 0 < μ, a 1. However, in [5,15,19], the proofs of the global well-posedness of (1.4) rely upon the symmetric property of the Schrödinger semi-group eit (i.e., eit has the same dispersive properties in the cases t < 0 and t > 0) and this symmetric property is invalid for the Ginzburg–Landau semi-group e(μ+i)t . Moreover, the proof will become very complicated if we imitate the procedures as in [5,15,19]. When 3 n 6, we will give a simple proof of Theorem 1.1 by using the perturbation analysis method, i.e., treating the NLS as a perturbative equation of the CGL: wt − (μ + i)w + g(v, w) = 0,

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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725 4

4

4

where w = u − v, g(v, w) = −μv + (a + i)(|w + v| n−2 (w + v) − |v| n−2 v) + a|v| n−2 v. Hence, it suffices to show that w is uniformly bounded in L2(n+2)/(n−2) (R+ × Rn ) for all 0 < μ, a 1 as the initial data w0 is small enough. Following the idea as in [20], we treat μv as a part of the nonlinear terms. We consider the following integral equation: t w=e

(μ+i)t

w0 −

e(μ+i)(t−τ ) g(v, w)(τ ) dτ. 0

In view of the Strichartz type estimates for the Ginzburg–Landau semi-group (which are similar to those for the Schrödinger semi-group), in order to control w in the space L2(n+2)/(n−2) (R+ × Rn ), we need to control v in H 3 . For n = 3, 4, the upper bound of v in H 3 can be easily obtained, since the nonlinearity |v|4/(n−2) v is a C ∞ function (cf. [3,20]). For n = 5, 6, |v|4/(n−2) v is only two-times differentiable, we need to use the “formal time-differentiation method” to overcome the regularity loss of the nonlinearity. After showing the upper bound of v in H 3 , we can perform the perturbation analysis and finally obtain our result, as desired. For higher spatial dimensions n > 6, since |v|4/(n−2) v is not two-times differentiable, the “formal time-differentiation method” seems invalid to get the upper bound of v in H 3 . This is why we assume that n 6 in Theorem 1.1. When n > 6, we shall bound u in L2(n+2)/(n−2) (R+ × Rn ) by applying the ideas as in Bourgain [3], Colliander, Keel, Staffilani, Takaoka and Tao [5], and Visan [19], i.e., the induction method on the energy E. In detail, for any energy E 0 and T > 0, define MT (E) := sup u

2(n+2)

,

Lt,xn−2 ([0,T ]×Rn )

where u ranges over all of the S˙ 1 solutions of (1.3) on [0, T ] × Rn with E(u) E and the parameters 0 μ, a 1. Put M(E) := sup MT (E): T > 0 . Noticing that both E(u) and MT (E) are invariant under the scaling (1.6), we immediately have Proposition 1.4. For any E, T > 0, we have M(E) = MT (E) ≡ M1 (E).

(1.12)

For E < 0, define M(E) = 0. Our goal is to show M(E) < ∞ for any energy E. To be more comprehensive, we write Q = E: M(E) < ∞ . The perturbation analysis implies that if E ∈ Q then E + ε ∈ Q for some 0 < ε 1, and so, Q is open, see Lemma 2.9 below for details. Moreover, Q contains zero and is connected. We use the induction-on-energy argument as in [3,5,19]. Assume for contradiction that M(E) can be infinite, consider the minimal energy Ec = inf E: M(E) = ∞ < ∞,


685

then naturally M(E) < ∞ for all E < Ec . Since Q is open and contains zero, then Ec > 0. Using this critical energy Ec , we can construct a near blowup solution whose energy is near to Ec with 2(n+2)

an enormous Lt,xn−2 norm. We will prove that such near blowup solution does not exist, then eventually we get a contradiction. In detail, by the critical property of Ec and Lemma 2.9, we have Lemma 1.5 (Induction hypothesis). Let 0 < μ, a 1 be variable parameters. Let t0 ∈ R+ , suppose u(t0 ) is an H˙ 1 function with E(u(t0 )) Ec − η for some η > 0, then there exists a global solution u to (1.3) on [t0 , +∞) × Rn with initial data u(t0 ) at time t0 satisfying u

2(n+2)

Lt,xn−2 ([t0 ,+∞)×Rn )

M(Ec − η).

(1.13)

We need eight small parameters 1 η0 η 1 η 2 η 3 η 4 η 5 η 6 η 7 > 0 where each ηj is chosen sufficiently small depending on previous η0 , . . . , ηj −1 and the critical energy Ec . Since M(Ec ) is infinite, we can find an S˙ 1 solution u of (1.3) with Ec /2 E(u) Ec such that for some T > 0, μ, a ∈ (0, 1], u

2(n+2) Lt,xn−2 ([0,T ]×Rn )

=

1 , η7

(1.14)

which leads to the following definition. Definition 1.6. A minimal energy blowup solution (MEBS for short) to (1.3) is an S˙ 1 Schwartz solution which satisfies (1.14) and with energy 1 Ec E(u) Ec . 2

(1.15)

Using the energy estimate for the CGL, we claim that for the MEBS, both μ and a must be very small. In fact, we have Proposition 1.7. If u is a MEBS, then we have max(μ, a) η6 . Proof. By the energy estimate for the CGL (see Theorem 2.7 below), we have T μ

u(τ, x)2 dx dτ + a

0 Rn

T

2(n+2) u(τ, x) n−2 dx dτ E(u0 ).

0 Rn

If a η6 , we see that u

2(n+2) Lt,xn−2 ([0,T ]×Rn )

E(u0 ) η6

(n−2)/2(n+2) ,

686

C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725 2(n+2)

which contradicts (1.14). So, it suffices to consider the case μ η6 . Interpolating Lt,xn−2 be2n

2n

n−2 tween L∞ and L2t Lxn−4 , we have t Lx

u

2(n+2) Lt,xn−2 ([0,T ]×Rn )

u

4/(n+2) 2n n−2 L∞ t Lx

u

(n−2)/(n+2) 2n

L2t Lxn−4

E(u0 )4/(n+2) u E(u0 )

4/(n+2)

which also contradicts (1.14).

(n−2)/(n+2) L2t,x

E(u0 ) η6

(n−2)/2(n+2) ,

2

In the following we always assume that 0 μ, a η6 . By choosing a subinterval [0, T ∗ ] ⊂ [0, T ], such that u

2(n+2) Lt,xn−2 ([0,T ∗ ]×Rn )

=

1 . η5

(1.16)

We divide [0, T ∗ ] = I0 ∪ I1

(1.17)

2(n+2)

with each subinterval containing half of the Lt,xn−2 ([0, T ∗ ] × Rn ) norm. We want to show that u

2(n+2) Lt,xn−2 (I0 ×Rn )

C η1 , . . . , η4 , E(u0 ) .

(1.18)

Following [5,19], the proof of (1.18) proceeds in the following four steps. (1) Using the same way as in [5,19], we can show that the MEBS is localized in physical and frequency space for any time t ∈ I0 . (2) We generalize the frequency localized Morawetz estimate (FLME) of the NLS to the CGL and get a uniform version of the FLME which is independent of μ, a. Our 4 idea is to apply the known Morawetz estimate for the NLS [5,19] and treat μu and a|u| n−2 u as the perturbative terms, which can be controlled by the energy estimates for the CGL. (3) We show the frequency-localized L2 mass conservation law to prevent energy evacuation to the high frequency. Comparing with the NLS, the new difficulty lies in the fact that the L2 norm of the solutions of the CGL has some dissipation (see (2.13) for details) and we need to show that this dissipation vanishes when μ, a 0. Another difficulty is that the Ginzburg–Landau semi-group e(μ+i)t is not symmetric with respect to time t, one needs to carefully deal with the backward case t < 0. (4) Following [5] and using the perturbation analysis, we can finish the proof of (1.18).


687

2. Notations and known results In this section, we recall some notations, basic facts and the Strichartz estimates for the NLS and the CGL. 2.1. Notations q

Let I ⊂ R be an interval. We use Lt Lrx to denote the space–time norm: uLq Lr (I ×Rn ) := t

x

I

u(t, x)r dx

q/r

1/q (2.1)

dt

Rn

with the usual modification when q or r is equal to infinity. We denote uLq (I,H˙ k ) := r q q ∇ k uLqt Lr (I ×Rn ) . If there is no confusion, we will write Lt∈I Lrx := Lt Lrx (I ×Rn ) and Lrt∈I,x := x Lrt Lrx (I × Rn ). We introduce the Littlewood–Paley projection operators in following way. Let ϕ(ξ ) be a radially symmetric bump function adapted to the ball {ξ ∈ Rn : |ξ | 2} which equals 1 on the unit ball. Let ψ(ξ ) := ϕ(ξ ) − ϕ(2ξ ). For each dyadic number N , we define the Fourier multipliers: PN f := F −1 ϕ(ξ/N )Ff, P>N f := F −1 1 − ϕ(ξ/N ) Ff, PN f := F −1 ψ(ξ/N )Ff. Recall that for any dyadic number N , there hold the telescoping identities PN f =

PM f ;

P>N f =

MN

PM f ;

f=

M>N

PM f,

M

for all Schwartz f , where M ranges over dyadic numbers. Define PM 0, 2

Sμ (t)ϕ C S(t)ϕ Ct −n(1/2−1/p) ϕp , p p

(2.8)

where C is independent of μ and t, ϕ ∈ Lp , 1/p + 1/p = 1. Combining (2.8) with the standard Strichartz estimates for the Schrödinger equation, one can prove the following time–space Lp –Lp estimates. We omit the details here, one can see Wang [20] for nonendpoint case and [21] for endpoint case. Lemma 2.2. Let n 3, k ∈ N ∪ {0}. Let (q, r) be any admissible pair and (q1 , r1 ) be any dual admissible pair. Then we have Sμ (t)ϕ

Lq (0,∞;H˙ rk )

ϕH˙ k ,

Aμ f Lq (0,T ;H˙ k ) f r

(2.9)

q

L 1 (0,T ;H˙ k )

(2.10)

,

r1

for all ϕ ∈ H˙ k , f ∈ Lq1 (0, T ; H˙ rk ), 0 < T ∞. When μ = 0, it reduces to the standard 1 Strichartz estimates for the Schrödinger equation (see [13]).

We emphasize that the estimates (2.9) and (2.10) are independent of μ > 0. Using Lemma 2.2 together with Duhamel’s formula (2.5) and the triangle inequality, we can verify the following multi-linear form of the Strichartz estimates. Corollary 2.3. Let I be a time interval, k = 0, 1, and let u : I × Rn → C be a solution to ut − (μ + i)u =

M

Fm

m=1

for some functions F1 , . . . , FM , where μ 0, then for any admissible pair (q, r),

M k u(t0 ) ˙ k n + ∇ Fm q r C n H (R ) L L (I ×R )

k ∇ u

t

x

m=1

q

r

Lt m Lxm (I ×Rn )

,

(2.11)

, r ) are dual admissible pairs. where t0 = min I , (q1 , r1 ), . . . , (qm m

We need the following improved Strichartz estimate for the nonhomogeneous term, which can be proved by following the method in [6] and the Strichartz estimates for the CGL.

690


Proposition 2.4. Let I ⊂ [0, +∞) be a compact time interval. Let (q, r) and (q, ˜ r˜ ) be two Schrödinger-acceptable pairs satisfying the scaling condition q1 + q1˜ = n2 (1 − 1r − 1r˜ ) and either n−2 r n < < , n r˜ n − 2

1 1 + = 1, q q˜

1 1 , r q

and

1 1 r˜ q˜

or n−2 r n . n r˜ n − 2

1 1 + < 1, q q˜ Then e(μ+i)(t−s) F (s) ds s 6. Let I ⊂ [0, +∞) be a compact time interval and let u˜ be an approximate solution to (1.3) on I × Rn in the sense that 4

˜ n−2 u˜ + e, iu˜ t + (1 − μi)u˜ = (1 − ai)|u| for some function e. Assume that u ˜

2(n+2)

Lt,xn−2 (I ×Rn )

M,

u ˜ L∞ n E ˙1 t H (I ×R ) x

˜ 0 ) in the sense that for some constants M, E > 0. Let t0 = min I and let u(t0 ) close to u(t u(t0 ) − u(t ˜ 0 )H˙ 1 (Rn ) E (2.14) for some E > 0. Assume also the smallness conditions

2 ∇PN e(μ+i)(t−t0 ) u(t0 ) − u(t ˜ 0 ) 2(n+2)

2n(n+2) 2 Lt n−2 Lx n +4 (I ×Rn )

N

∇e

2n

L2t Lxn+2 (I ×Rn )

1 2

ε,

ε,

(2.15) (2.16)

for some 0 < ε < ε1 , where ε1 = ε1 (E, E , M) > 0 is a small constant. Then there exists a solution u to (1.3) on I × Rn with initial data u(t0 ) satisfying ∇(u − u) ˜

2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (I ×Rn )

7 C(E, E , M) ε + ε (n−2)2 ,

7 u − u ˜ S˙ 1 (I ×Rn ) C(E, E , M) E + ε + ε (n−2)2 ,

uS˙ 1 (I ×Rn ) C(E, E , M).

(2.17) (2.18) (2.19)

692


Here C(E, E , M) > 0 is a non-decreasing function of E, E , M, and the dimension n. Remark 2.10. Recall that the Ginzburg–Landau semi-group Sμ (t) satisfies the same Strichartz estimates as the corresponding Schrödinger semi-group S(t). Noticing that t0 = min I , we only need to consider the CGL forward in Lemma 2.9, then the proof is parallel to that of the NLS (see Tao and Visan [18]) and we omit the details of the proof. 3. Proof of Theorem 1.1 In this section, we prove the main Theorem 1.1. We will use the perturbation analysis to give a short proof of Theorem 1.1 when 3 n 6. 3.1. Nonlinear estimates The purpose of this subsection is to derive some nonlinear estimates, which will be applied in 4 the next subsection. Write F (z) = |z| n−2 z, let Fz , Fz¯ be the complex derivatives

∂F ∂F 1 ∂F 1 ∂F Fz := −i , Fz¯ := +i , 2 ∂x ∂y 2 ∂x ∂y then we see that Fz (z) =

4 n |z| n−2 , n−2

Fz¯ (z) =

4 z2 2 |z| n−2 2 , n−2 |z|

(3.1)

4

which implies that Fz (z) and Fz¯ (z) are both O(|z| n−2 ). Note that the difference of two nonlinear terms satisfies the integral identity: 1 F (u) − F (v) =

Fz v + θ (u − v) (u − v) + Fz¯ v + θ (u − v) (u − v) dθ.

(3.2)

0

Hence, 4 4 F (u) − F (v) |u − v| |u| n−2 + |v| n−2 .

(3.3)

Observe that F (u) obeys the fractional chain rule: ∇F u(x) = Fz u(x) ∇u(x) + Fz¯ u(x) ∇u(x).

(3.4)

By (3.1), (3.4) and Hölder’s inequality, we can get the following estimate. Proposition 3.1. For k = 0, 1, we have. k ∇ F (u)

4

2n L2t Lxn+2 (I ×Rn )

u n−2 2(n+2) Lt,xn−2 (I ×Rn )

k ∇ u

2(n+2) n−2

Lt

2n(n+2) 2 +4

Lx n

. (I ×Rn )

(3.5)


693

Similarly, we can invoke Hölder’s inequality and (3.1), (3.3), (3.4) to get: Proposition 3.2. For k = 0, 1, we have k ∇ F (u) F (u) − F (v)

(2∗ ) Lt,x (I ×Rn )

k ∇ u

4

(2∗ ) Lt,x (I ×Rn )

u n−2 2(n+2)

4 u n−2 2(n+2)

∗

L2t,x (I ×Rn )

Lt,xn−2 (I ×Rn ) 4

+ v n−2 2(n+2)

Lt,xn−2 (I ×Rn )

Lt,xn−2 (I ×Rn )

(3.6)

,

u − vL2∗ (I ×Rn ) .

(3.7)

t,x

Now we estimate the difference with one-order derivative. Write w = u − v. When n 6, in view of the chain rule (3.4), we calculate ∇ F (u) − F (v) = Fz (u)∇u + Fz¯ (u)∇ u¯ − Fz (v)∇v − Fz¯ (v)∇ v¯ = Fz (u) − Fz (v) ∇v + Fz¯ (u) − Fz¯ (v) ∇ v¯ + Fz (u)∇(u − v) + Fz¯ (u)∇(u − v) Fz (u) − Fz (v) + Fz¯ (u) − Fz¯ (v) |∇v| + Fz (u) + Fz¯ (u)∇(u − v) 6−n 6−n 4 |∇v||u − v| |u| n−2 + |v| n−2 + ∇(u − v)|u| n−2 6−n 6−n 4 |∇v||w| |v| n−2 + |w| n−2 + |∇w||w + v| n−2 . (3.8) Therefore by Hölder’s inequality and (3.8), we get Proposition 3.3. When n 6, there hold ∇ F (u) − F (v)

(2∗ ) Lt,x

6−n 6−n n−2 ∇vL2∗ v n−2 + w 2(n+2) 2(n+2) w t,x

Lt,xn−2

Lt,xn−2

2(n+2)

Lt,xn−2

4 4 n−2 + ∇wL2∗ v n−2 2(n+2) + w 2(n+2) , t,x

∇ F (u) − F (v)

2n L2t Lxn+2

∇v

Lt,xn−2

2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4

+ ∇w

6−n 6−n n−2 v n−2 2(n+2) + w 2(n+2) w

2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4

(3.9)

Lt,xn−2

Lt,xn−2

Lt,xn−2

4 4 n−2 v n−2 2(n+2) + w 2(n+2) .

Lt,xn−2

2(n+2)

Lt,xn−2

(3.10)

Lt,xn−2

We need the following estimates. Proposition 3.4. We have F (u) ∇F (u)

L2

Cu

4

L2

6−n n−2 H˙ 1

CuHn−2 ˙ 1 uH˙ 2 ,

u2H˙ 2 Cu

4 n−2 H˙ 1

uH˙ 3 ,

(3.11) n 6.

(3.12)

694


Proof. By Hölder’s inequality and Sobolev’s embedding, F (u)

L2

n+2 4 n−2 n+2 n+2 n+2 n−2 Cu n−2 . 2n+4 C ∇uL2 uL2

L

n−2

Similarly when n 6, ∇F (u)

L2

4 4 C |u| n−2 ∇uL2 C |u| n−2 Ln ∇u

C u

6−n 4 H˙ 1

u

n−2 4 H˙ 2

4 n−2

4

2n L n−2

Cu n−24n uL2 L n−2

uH˙ 2 .

(3.13)

When n 6, we need to compute F (u). Noticing that Fzz (z) = 6−n 2n z |z| n−2 |z| , (n−2)2

Fz¯ z¯ =

6−n 3 8−2n z |z| n−2 |z| 3, (n−2)2

6−n 2n |z| n−2 |z| z , Fz¯z (n−2)2

= Fz¯ z =

we find that F (u(x)) obeys the chain rule:

2 2 2 F u(x) = Fzz u(x) ∇u(x) + 2Fz¯z u(x) ∇u(x) + Fz¯ z¯ u(x) ∇u(x) + Fz u(x) u(x) + Fz¯ u(x) u(x). (3.14) Therefore, it satisfies the estimate: 4 F u(x) C |u| 6−n n−2 |∇u|2 + |u| n−2 |u| ,

n 6.

2

(3.15)

3.2. Proof of Theorem 1.1 Suppose that u and v are solutions of (1.3) and (1.4), respectively. Defining w = u − v, then we see that w satisfies:

4 4 4 wt − (μ + i)w − μv + (a + i) |w + v| n−2 (w + v) − |v| n−2 v + a|v| n−2 v = 0, w(0) = w0 .

w can be rewritten as 4 4 4 w = Sμ (t)w0 − Aμ −μv + (a + i) |w + v| n−2 (w + v) − |v| n−2 v + a|v| n−2 v .

(3.16)

If the initial data w0 is small enough, then we can prove the following estimate for the solution of (3.16), whose proof will be given in the end of this section. Proposition 3.5. Let 3 n 6. Suppose that w is a solution to (3.16), T > 0 is arbitrary, w0 H 1 δ 1, u0 ∈ H 1 , v0 ∈ H 3 , 0 < μT , a δ. Then we have w

2(n+2) Lt,xn−2 ([0,T ]×Rn )

C u0 H 1 , v0 H 3 .


695

Proposition 3.6. Let 3 n 6. Suppose v is a solution to (1.4) and v0 ∈ H 3 , then for k = 0, 1, 2, 3, v(t)

n ˙k + L∞ t Hx (R ×R )

C.

(3.17)

Proof. If n = 3, 4, (3.17) is shown by Bourgain [3]. If n = 5, 6, since |v|4/(n−2) v is not threetimes differentiable, we need to use the “formal time-differentiation method” to overcome the regularity loss of the nonlinearity. This method has been used by Kato [12], Cazenave and Weissler [4]. The case k = 0, 1 follows immediately from the mass and energy preserving fact u(t)2 = u0 2 and E(v(t)) = E(v0 ) k ∇ v

2 + n L∞ t Lx (R ×R )

C ∇ k v0 2 ,

k = 0, 1.

(3.18)

When k = 2, v can be written as the following integral form,1 t v = S(t)v0 − i

S(t − τ )F (v) dτ. 0

In view of Theorem 2.8, we see that v

2(n+2) Lt,xn−2 (R+ ×Rn )

+ ∇v

2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (R+ ×Rn )

C v0 H˙ 1 .

(3.19)

So, for a given small parameter 0 < η 1, we can divide R+ into J disjoint intervals Ij such that R+ = Jj=1 Ij and v

2(n+2)

Lt,xn−2 (Ij ×Rn )

+ ∇v

2(n+2) n−2

Lt

η,

2n(n+2) 2 +4

Lx n

j = 1, . . . , J.

(3.20)

(Ij ×Rn )

By Corollary 2.3 and (3.15), 4 vS˙ 0 (Ij ×Rn ) v(min Ij )2 + |v| n−2 v 6−n + |v| n−2 |∇v|2

(2∗ )

Lt,x (Ij ×Rn )

2n

L2t Lxn+2 (Ij ×Rn )

.

(3.21)

Using Hölder’s inequality and Sobolev’s embedding, one has that 6−n |v| n−2 |∇v|2

2n


1 Strictly speaking, in order to guarantee the following the a priori estimates hold, we should first show the local wellposedness of solutions in H 3 by using the standard contraction mapping argument, namely, one needs to construct a metric space in which every norm of the solution appeared in the following estimates is finite, see Remark 3.7 below.

696

C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725 6−n

v n−2 2(n+2)

∇v

Lt,xn−2 (Ij ×Rn ) 6−n

v n−2 2(n+2)

∇v


2(n+2)


∇v

2(n+2) n−2

Lt

2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )

2n(n+2) 2 +4

Lx n

2 ∇ v

(Ij ×Rn )

2(n+2) n−2

Lt

2n(n+2) 2 +4

Lx n

(Ij ×Rn )

4

η n−2 vS˙ 0 (Ij ×Rn ) ,

(3.22)

and 4 |v| n−2 v

(2∗ ) Lt,x (Ij ×Rn )

4

v n−2 2(n+2) Lt,xn−2 (Ij ×Rn )

vL2∗ (Ij ×Rn ) t,x

4

η n−2 vS˙ 0 (Ij ×Rn ) .

(3.23)

Hence, in view of (3.20)–(3.23), we have 4 vS˙ 0 (Ij ×Rn ) v(min Ij )2 + η n−2 vS˙ 0 (Ij ×Rn ) .

(3.24)

Taking j = 1 and noticing that v(min I1 ) = v0 , we obtain that vS˙ 0 (I1 ×Rn ) 2v0 H˙ 2 ,

vL∞ 2 n 2v0 H˙ 2 , t Lx (I1 ×R )

(3.25)

which implies that v(min I2 )2 2v0 H˙ 2 . Repeating the procedure above, we have vS˙ 0 (I2 ×Rn ) 22 v0 H˙ 2 ,

2 vL∞ 2 n 2 v0 H˙ 2 . t Lx (I2 ×R )

By iteration, for any j = 1, . . . , J , we see that vS˙ 0 (Ij ×Rn ) 2j v0 H˙ 2 ,

j vL∞ 2 n 2 v0 H˙ 2 . t Lx (Ij ×R )

Hence, vL∞ 2 (R+ ×Rn ) C v0 H˙ 1 v0 H˙ 2 , L t x

(3.26)

which concludes the conclusion in the case k = 2. Finally, we consider the case k = 3 and use the equation 4 ∇vt − i∇v = −i∇ |v| n−2 v = −i∇F (v)

(3.27)

to estimate vH˙ 3 . It suffices to calculate ∇vt 2 and ∇(F (v))2 . It is easy to see that ∇vt satisfies the following integral equation: t ∇vt = iS(t)∇v0 − iS(t)∇F (v0 ) − i 0

S(t − τ )∇∂τ F v(τ ) dτ.

(3.28)


697

Let Ij (j = 1, . . . , J ) be the decomposition of R+ as in (3.20). We have 4 ∇vt S˙ 0 (Ij ×Rn ) v(minIj )H˙ 3 + ∇F v(min Ij ) 2 + |v| n−2 ∇vt

(2∗ )

Lt,x (Ij ×Rn )

6−n + |v| n−2 ∇vvt

2n


(3.29)

.

Using the same way as in Proposition 3.2, we get 4 |v| n−2 ∇vt

4

(2∗ ) Lt,x (Ij ×Rn )

v n−2 2(n+2) Lt,xn−2 (Ij ×Rn )

∇vt L2∗ (Ij ×Rn ) t,x

4

η n−2 ∇vt S˙ 0 (Ij ×Rn ) .

(3.30)

In view of Hölder’s inequality and Sobolev’s embedding, we have 6−n |v| n−2 ∇vvt

2n


6−n

v n−2 2(n+2)

∇v

Lt,xn−2 (Ij ×Rn ) 6−n

v n−2 2(n+2)

∇v


η

4 n−2

2(n+2) n−2

Lt

2(n+2) n−2

Lt

2n(n+2) 2 +4

Lx n

2n(n+2) 2 +4

Lx n

vt (Ij ×Rn )

2(n+2)


∇vt (Ij ×Rn )

2(n+2) n−2

Lt

2n(n+2) 2 +4

Lx n

(Ij ×Rn )

∇vt S˙ 0 (Ij ×Rn ) .

(3.31)

Inserting the estimates above into (3.29) and using Proposition 3.4, we conclude that ∇vt S˙ 0 (Ij ×Rn ) v(min Ij )H˙ 3 + ∇F v(min Ij ) 2 4/(n−2) . v(min Ij )H˙ 3 1 + v(min Ij )H˙ 1

(3.32)

Taking notice of v(min I1 ) = v0 , we have 4 n−2 1 + v . ∇vt L∞ v 2 n 3 ˙ 0 0 H t Lx (I1 ×R ) H˙ 1

(3.33)

On the other hand, from Proposition 3.4, (3.18) and (3.26), it follows that for any t > 0, ∇F v(t)

L2

6−n 2 v(t)Hn−2 ˙ 1 v(t) H˙ 2 C v0 H˙ 1 v0 H˙ 3 .

(3.34)

Hence, by Eq. (3.27), ∇vL∞ 2 n ∇vt L∞ L2 (I ×Rn ) + ∇F v(t) ∞ 2 Lt Lx (I1 ×Rn ) t Lx (I1 ×R ) t x 1 C v0 H˙ 1 v0 H˙ 3 .

(3.35)

698


(3.35) implies that v(min I2 )

H˙ 3

C v0 H˙ 1 v0 H˙ 3 .

(3.36)

Combining (3.32) with (3.34) and (3.36), we have ∇vt S˙ 0 (I2 ×Rn ) C v0 H˙ 1 v0 H˙ 3 . Again, using the same way as in (3.35), we obtain that ∇vL∞ 2 n C v0 H˙ 1 v0 H˙ 3 . t Lx (I2 ×R ) Using standard iteration method, we get for any j = 1, . . . , J , ∇vL∞ 2 (I ×Rn ) C v0 H˙ 1 v0 H˙ 3 , L j t x which implies that ∇vL∞ 2 + n C v0 H˙ 1 v0 H˙ 3 . t Lx (R ×R ) This completes the conclusion.

(3.37)

2

Remark 3.7. In order to guarantee Proposition 3.6 holds, we should first prove the local well posedness in H 3 , which can be shown by using a standard contraction mapping argument (see [4], for instance) and combining the proof of Proposition 3.6. Define t

S(t − τ )F v(τ ) dτ.

T : v → S(t)v0 − i 0

Now let η, T be sufficiently small numbers, M = 4Cv0 H 3 . Define D = v: ∇ j v S˙ 0 ([0,T ]×Rn ) M, j = 0, 1, 2, ∂t vS˙ 1 ([0,T ]×Rn ) M, ∇v

2(n+2) 2n(n+2) 2 ∗ L2t,x ∩Lt n−2 Lx n +4 ([0,T ]×Rn )

η, ∇vL∞ 2 n M t Lx ([0,T ]×R )

equipped with the metric d(u, v) = u − vL∞ L2 ∩L2∗ ([0,T ]×Rn ) . t

x

t,x

Then (D, d) is a complete metric space. We claim that T : D → D . Indeed, for any v ∈ D , by the Strichartz estimate and ∇v

∗

2(n+2)

2n(n+2) 2 +4

n−2 n L2t,x ([0,T ]×Rn )∩Lt∈[0,T ] Lx

Following the proof of Proposition 3.6, one has that

η 1.


j ∇ T v ˙ 0

S ([0,T ]×Rn )

∇T v

M,

0 j 2,

2(n+2)

∗

699

2n(n+2) 2 +4

η.

n−2 n L2t,x ([0,T ]×Rn )∩Lt∈[0,T ] Lx

Taking formal time derivative and spatial derivative with respect to T v, we have i∂t ∇T v + ∇T v = ∇F (v). Using the same way as in (3.29)–(3.31), we estimate 4

∂t T vS˙ 1 ([0,T ]×Rn ) Cv0 H˙ 3 + Cη n−2 M. 4

Since Cη n−2 1, we get ∂t T vS˙ 1 ([0,T ]×Rn ) 3M/4. In (3.34), we have verified that ∇F (v)L∞ 2 n M/4, therefore ∇T vL∞ L2 ([0,T ]×Rn ) M. So, T : D → D . For t Lx ([0,T ]×R ) t x any u, v ∈ D , 4 d(T u, T v) u n−2 2(n+2)

4

+ v n−2 2(n+2)

Lt,xn−2 ([0,T ]×Rn )

Lt,xn−2 ([0,T ]×Rn )

u − vL2∗ ([0,T ]×Rn ) . t,x

So T : D → D is strictly contractive, then there exists a unique fixed point v ∈ D . We have shown that there exists a solution v ∈ C([0, T ]; H 3 ) to (1.4). We can repeat the above procedures step by step, then find a T ∗ > 0 such that v ∈ C([0, T ∗ ); H 3 ) is a solution of (1.4). Now we can prove our main result. Proof of Theorem 1.1. First, we consider the convergence in L2 space. In view of Corollary 2.3 and Proposition 3.2, we have 4 wS˙ 0 (I ×Rn ) C w(min I )2 + μCvL1 L2 (I ×Rn ) + aC |v| n−2 v t

4 4 + C |w + v| n−2 (w + v) − |v| n−2 v

(2∗ )

Lt,x (I ×Rn )

x

(2∗ )

Lt,x (I ×Rn )

4 n−2 C w(min I )2 + μC|I |vL∞ 2 n + aCv 2(n+2) t Lx (I ×R )

Lt,xn−2 (I ×Rn )

4 + C w + v n−2 2(n+2)

4

+ v n−2 2(n+2)

Lt,xn−2 (I ×Rn )

Lt,xn−2 (I ×Rn )

wL2∗ (I ×Rn ) . t,x

vL2∗ (I ×Rn ) t,x

(3.38)

By Theorem 2.8 and Proposition 3.5, we can divide [0, T ] into N = N (u0 H 1 , v0 H 3 ) disjoint time intervals {Ik }N k=1 such that 4 C w + v n−2 2n+4

n−2 Lt,x (Ik ×Rn )

4

+ v n−2 2n+4

n−2 Lt,x (Ik ×Rn )

Replace I by Ik in (3.38), and with (3.39), then

1 , 2

1 k N.

(3.39)

700


wS˙ 0 (Ik ×Rn ) CwL∞ 2 n + μC|Ik |vL∞ L2 (I ×Rn ) t Lx (Ik−1 ×R ) t x k a 1 + vL2∗ (Ik ×Rn ) + wL2∗ (Ik ×Rn ) . t,x t,x 2 2

(3.40)

Using iteration, we get wS˙ 0 (Ik ×Rn ) 2CwL∞ 2 n + 2Cμ|Ik |vL∞ L2 (I ×Rn ) + av 2∗ Lt,x (Ik ×Rn ) t Lx (Ik−1 ×R ) t x k 2C 2CwL∞ 2 n + 2Cμ|Ik−1 |vL∞ L2 (I n + av 2∗ L (Ik−1 ×Rn ) t Lx (Ik−2 ×R ) t x k−1 ×R ) t,x

+ 2Cμ|Ik |vL∞ 2 n + av 2∗ L (Ik ×Rn ) t Lx (Ik ×R ) t,x

2 · · · (2C) w0 2 + μ(2C)|Ik |vL∞ 2 n + μ(2C) |Ik−1 |vL∞ L2 (I n t Lx (Ik ×R ) t x k−1 ×R ) k

+ · · · + μ(2C)k |I1 |vL∞ 2 n + av 2∗ L (Ik ×Rn ) t Lx (I1 ×R ) t,x

+ 2CavL2∗ (Ik−1 ×Rn ) + · · · + (2C) avL2∗ (I1 ×Rn ) t,x t,x

k (2C) w0 2 + μ |Ij |vL∞ 2 n t Lx ( j k Ij ×R ) k

j k

+ a sup vL2∗ (Ij ×Rn ) 1 + 2C + · · · + (2C)k j k

t,x

∗ |Ij |vL∞ + a sup v (2C)k+1 w0 2 + μ 2 n 2 n L (Ij ×R ) . t Lx ( j k Ij ×R ) j k

j k

t,x

(3.41)

In view of Corollary 2.3 and inequality (3.39), one can easily check that sup vL2∗ (Ij ×Rn ) 2C E(v0 ) v0 2 .

j k

t,x

(3.42)

Summarizing the above discussion, we obtain that N +2 wL∞ w0 2 + μT vL∞ 2 n (2C) 2 n + av0 2 , t Lx ([0,T ]×R ) t Lx ([0,T ]×R )

(3.43)

where N = N(u0 H 1 , v0 H 3 ) is a large constant that depends only onu0 H 1 and v0 H 3 , and is independent of μ and a. Combining (3.43) with (3.26), we get for any T > 0 N +2 w0 2 + μT v0 H 2 + av0 2 wL∞ 2 n (2C) t Lx ([0,T ]×R )

(3.44)

which implies the desired results. The proof of (1.8) for k = 1 proceeds in a similar way as that for k = 0, which is essentially implied by the proof of Proposition 3.5, see below and we omit the details. 2 Proof of Proposition 3.5. Let T > 0 and 0 < η 1 be a sufficiently small parameter. Following the proof of Proposition 3.6, we can divide [0, T ] into J (depending on v0 H 3 ) disjoint intervals {Ij }Jj=1 such that on each interval Ij there holds


v

2(n+2)


+ ∇v

2(n+2) n−2

Lt

2n(n+2) 2 +4

Lx n

701

η.

(3.45)

(Ij ×Rn )

Choose 0 < δ 1 small enough satisfying (4C)J δ η. Let w0 H 1 δ/2, 0 < μT , a δ/4. By Sobolev’s embedding, Corollary 2.3, Propositions 3.1–3.3 and 3.6, w

+ ∇wL∞ 2 n t Lx (Ij ×R )

2(n+2)


∇w

2(n+2) n−2

Lt

2n(n+2) 2 +4

Lx n

(Ij ×Rn )

+ ∇wL∞ 2 n t Lx (Ij ×R )

C w(min Ij )H˙ 1 + μC∇vL1 L2 (Ij ×Rn ) + Ca ∇F (v) t

+ C ∇ F (u) − F (v)

x

2n


2n


C w(min Ij )H˙ 1 + μC|Ij |∇vL∞ 2 n t Lx (Ij ×R ) 4

+ Cav n−2 2(n+2)

∇v


+ C∇v

2(n+2) 2n(n+2) 2 Lt∈In−2 Lx n +4 j

+ C∇w

2(n+2) n−2

Lt

Lt∈In−2,x

Lt∈In−2,x

j

H˙ 1

v

2(n+2)

Lt∈In−2,x j

j

4 n−2 2(n+2) Lt,xn−2 (Ij ×Rn )

4

+ w n−2 2(n+2) Lt,xn−2 (Ij ×Rn )

+ Cδ

6−n 6−n + Cη η n−2 + ∇w n−2 2(n+2)

2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )

+ C∇w

(Ij ×Rn )

6−n 6−n n−2 v n−2 2(n+2) + w 2(n+2) w

2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )

C w(min Ij )

2n(n+2) 2 +4

Lx n

2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )

∇w

2(n+2) n−2

Lt

2n(n+2) 2 +4

Lx n

(Ij ×Rn )

4

4

η n−2 + C∇w n−2 2(n+2)

4 C w(min Ij )H˙ 1 + Cδ + Cη n−2 ∇w

2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )

2(n+2) n−2

Lt 4

2n(n+2) 2 +4

Lx n

(Ij ×Rn )

n+2

+ Cη∇w n−2 2(n+2)

2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )

+ C∇w n−2 2(n+2) Lt

n−2

2n(n+2) 2 Lx n +4 (Ij ×Rn )

.

(3.46)

Taking j = 1 and noticing that w(min I1 ) = w0 , by the standard continuity method, we get ∇w

2(n+2) n−2

Lt

2n(n+2) 2 +4

Lx n

(I1 ×Rn )

+ ∇wL∞ 2 n 2Cδ, t Lx (I1 ×R )

(3.47)

where the constant C depends only on u0 H 1 and v0 H 3 . By (3.47), ∇w(min I2 )2 2Cδ. Again, it follows from (3.46) that

702


∇w

2(n+2) n−2

Lt

2n(n+2) 2 +4

Lx n

(I2 ×Rn )

+ ∇wL∞ 2 n t Lx (I2 ×R ) 4

4

4C 2 δ + Cη n−2 ∇w + C∇w

2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (I2 ×Rn )

n+2 n−2 2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (I2 ×Rn )

+ Cη∇w n−2 2(n+2) Lt

n−2

2n(n+2) 2 +4

Lx n

(I2 ×Rn )

(3.48)

.

The continuity method yields that ∇w

2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (I2 ×Rn )

2 + ∇wL∞ 2 n (4C) δ. t Lx (I2 ×R )

Noticing that (4C)J δ η, we can repeat the procedure above step by step and finally get the result, as desired. 2 4. Proof of Theorem 1.3 4.1. Concentration for the MEBS Using the induction hypothesis (Lemma 1.5) and bilinear estimate, we construct a near solution to (1.3), then applying the long-time perturbation lemma, one can prove the following proposition which means that the MEBS cannot be included in this class, that is to say, the MEBS can only concentrate in one particular frequency in the phase space. Since the proof is similar to NLS (1.4) as in [5,19], we omit the details. Proposition 4.1 (Frequency localization of energy at each time). Let u be a minimal energy blowup solution to (1.3), then for any t ∈ I0 , there exists a dyadic frequency N (t) ∈ 2Z such that for every η4 η η0 , we have Pc(η)N (t) u(t)

PC(η)N (t) u(t)

H˙ x1

η,

(4.1)

H˙ x1

η,

(4.2)

and Pc(η)N (t) t0 , and a point x1 ∈ Rn such that (μ+i)(t −t ) 1 0 e Pme u(t0 )(x1 ) c(η0 ).

(4.4)

Choosing time t2 which is symmetric to t1 with respect to t0 , i.e., t0 − t2 = t1 − t0 . Define f (t2 ) := Pme δx1 , where δx1 is the Dirac mass at x1 . Define f (t) := e(μ+i)(t−t2 ) f (t2 ) for any t > t2 . Then n n − we can easily check that for any t > t2 , 1 p ∞, there holds f (t)Lpx C(η0 )t − t2 p 2 . Hence c(η0 ) e(μ+i)(t0 −t2 ) Pme u(t0 ), δx1 = u(t0 ), f (t0 ) f (t0 ) 2n u(t0 ) 2n η1 C(η0 )t1 − t0 , Lxn+2

(4.5)

Lxn−2

which means that both t1 and t2 are far from t0 . Hence f

2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 ([t0 ,∞)×Rn )

n−2 C(η0 )· − t2 − n+2

2(n+2) n−2

Lt n−2

([t0 ,∞)) n−2

C(η0 )|t2 − t0 |− 2(n+2) C(η0 )η12(n+2) .

(4.6)

704


Then, following the same step as in Visan [19], we can show that u

2(n+2)

Lt,xn−2 ([t0 ,+∞)×Rn )

C(η0 , η1 ).

Since [t0 , +∞) contains the interval I1 , this contradicts (1.16).

2

We now state an interpolation lemma which is useful in proving the spatial concentration of energy. Lemma 4.3. (See [22, Corollary 4.2].) Suppose that 1 p0 < p < ∞, −∞ < s1 < s < s0 < ∞, 0 < θ < 1 and 1 θ 1−θ = , + p p0 ∞

s = θ s0 + (1 − θ )s1 ,

then we have uH˙ s (Rn ) Cuθ˙ s0

Bp0 ,p0 (Rn )

p

u1−θ ˙ s1

B∞,∞ (Rn )

.

Using the ideas in Bourgain [3] and Lemma 4.3, we can prove the following. Proposition 4.4 (Spatial concentration of energy at each time). For any minimal energy blowup solution u to (1.3) (or only suppose u 2n η1 , uH˙ 1 1), and for any t ∈ I0 , there exist a dyadic number M(t) and a position x0 n/2

η1 M(t)

n−2 2

x

Lxn−2

(t) ∈ Rn

such that

n−2 PM(t) u t, x0 (t) M(t) 2 .

Furthermore, there holds the inverse Sobolev inequality: M(t)

n−2 2 p−n+kp

p PM(t) (−)k/2 u(t, x)Lp (|x−x

0 (t)|C(η1 )/M(t))

np/2

η1

M(t)

n−2 2 p−n+kp

,

k = 0, 1,

(4.7)

for all 1 < p < ∞. In particular, we have

∇PM(t) u(t, x)2 dx ηn . 1

(4.8)

|x−x0 (t)|C(η1 )/M(t)

Proof. We only prove the second inequality in (4.7), by Proposition 4.2 and Lemma 4.3, we have for any t ∈ I0 , η1 u

n−2

2n Lxn−2

2

2

2 B˙ ∞,∞

2 B˙ ∞,∞

uH˙n1 u n − n−2 u n − n−2 . x

(4.9)


705

It follows that there exist a dyadic number M(t) and x0 (t) ∈ Rn such that n PM(t) u t, x0 (t) M(t) n−2 2 η2.

1

Recall that PN = F −1 ψ(ξ/N )F . Define φ(ξ ) = ψ(ξ/2) + ψ(ξ ) + ψ(2ξ ), then φ is a smooth function with compact support in [1/4, 4]. Thus n

n−2

M(t) 2 η12 PM(t) u(t, x0 ) = F −1 φ ξ/M(t) |ξ |−k ∗ PM(t) (−)k/2 u(t, x0 ) −1 −k n−k k/2 F = M(t) |ξ | φ M(t)(x0 − x) PM(t) (−) u(t, x) dx

M(t)

−1 −k k/2 F |ξ | φ M(t)(x0 − x) PM(t) (−) u(t, x) dx

n−k

|x−x0 |C(η1 )/M(t)

n−k

+ M(t)

−1 −k k/2 F |ξ | φ M(t)(x0 − x) PM(t) (−) u(t, x) dx

|x−x0 |>C(η1 )/M(t)

−1 −k PM(t) (−)k/2 u p F M(t) |ξ | φ Lp L (|x−x0 |C(η1 )/M(t)) + PM(t) (−)k/2 uL2 F −1 |ξ |−k φ L2 (|x−x |>C(η )/M(t)) . n p −k

0

1

(4.10)

Since φ is a rapidly decreasing function, we can choose C(η1 ) sufficiently large such that the second term in the last inequality is smaller than M(t) PM(t) (−)k/2 u p L (|x−x

n−2 n 2 − p +k

0 |C(η1 )/M(t))

thus the conclusion follows.

n

η12 /2, which implies that

M(t)

n−2 n 2 − p +k

n

η12 /2,

2

Remark 4.5. M(t) in Proposition 4.4 is equivalent to N (t) in Proposition 4.1. More precisely, M(t) ∈ c(η1 )N (t), C(η1 )N (t) . Proof. By Proposition 4.4, for any t ∈ I0 , there exists a dyadic number M(t) such that PM(t) u(t, x)

n/2

2n n−2

η1 .

(4.11)

While Proposition 4.1 tells us that there exists dyadic number N (t) satisfying Pc(η )N (t) u(t) 2n + PC(η )N (t) u(t) 2n η100n . 1 1 1 n−2

(4.12)

n−2

Then M(t) ∈ c(η1 )N (t), C(η1 )N (t) .

(4.13)

706


Hence, for the MEBS, we can regard M(t) in Proposition 4.4 as to be identical with N (t) in Proposition 4.1. 2 Corollary 4.6. Let I0 be defined as in (1.17), then −2 |I0 | η5−1 Nmin .

(4.14)

Proof. By Proposition 4.4, we know that 2 Nmin

N (t) C(η1 )

|PN (t) u|

2

2(n+2) n−2

dx.

|x−x0 (t)|C(η1 )/N (t)

Integrating it on I0 , and together with (1.16), we get 2 |I0 |Nmin

C(η1 )

|PN (t) u|

2(n+2) n−2

dx C(η1 )η5−1 ,

I0 |x−x0 (t)|C(η1 )/N (t)

2

which is the result, as desired.

Remark 4.7. If we normalize Nmin = 1, this corollary tells us that the length of I0 on which the MEBS is defined is not too long, in fact it can be controlled by Cη5−1 . 4.2. N(t) takes finitely many values Lemma 4.8. Let u be a MEBS. Let I = [t1 , t2 ] be any subinterval of I0 with n/2

|I |N (t1 )2 η1 /4C. Then we have c(η1 )N (t1 ) N (t) C(η1 )N (t1 ),

∀t ∈ I.

Proof. By Proposition 4.4 we see that for any t = t1 , there exists N (t1 ) such that N (t1 )−

n−2 2

PN (t ) u(t1 ) ∞ ηn/2 . 1 1 L x

(4.15)

On the other hand, for any t2 t1 , we use Bernstein’s and Hölder’s inequalities to estimate N(t1 )−

n−2 2

PN (t ) u(t1 ) − PN (t ) u(t2 ) ∞ 1 1 L

x

PN (t1 ) u(t1 ) − u(t2 ) 2

n − n−2 2 +2

N(t1 )

t2 N(t1 ) t1

PN (t ) ∂τ u(τ ) dτ 1 2


t2 N (t1 )

707

4 PN (t ) u(τ ) + PN (t ) |u| n−2 u2 dτ 1 1 2

t1

t2 N(t1 ) |t1 − t2 |∇uL∞ ([t1 ,t2 ],L2 ) + N (t1 ) 2

4 N (t1 )|u| n−2 u

2n n+2

dτ

t1

CN (t1 )2 |t1 − t2 | + CN (t1 )2 |t1 − t2 |u

n+2 n−2

.

2n

n−2 L∞ ([t1 ,t2 ]×Rn ) t Lx

Since the energy is conserved, we obtain N(t1 )−

n−2 2

PN (t ) u(t1 ) − PN (t ) u(t2 ) ∞ ηn/2 /2, 1 1 1 L x

t1 , t2 ∈ I.

(4.16)

(4.15) and (4.16) yield N (t1 )−

n−2 2

PN (t ) u(t2 ) ∞ ηn/2 /2. 1 1 L x

Following the same step as in Proposition 4.4, we can show that for p =

2n n−2 ,

PN (t ) u(t2 , x)p dx ηnp/2 N (t1 ) n−2 2 p−n , 1 1

|x−x0 (t2 )|C(η1 )/N (t1 )

which gives that PN (t ) u(t2 ) 1

n/2

2n n−2

η1 .

(4.17)

On the other hand, Proposition 4.1 tells us that Pc(η

1 )N (t2 )

u(t2 )

2n n−2

+ PC(η1 )N (t2 ) u(t2 )

2n n−2

η1100n .

(4.18)

By (4.17) and (4.18), we get N (t1 ) ∈ c(η1 )N (t2 ), C(η1 )N (t2 ) , which is the result, as desired.

2

Corollary 4.9. Let {Ij }Jj=1 be a pairwise disjoint decomposition of I0 with Ij = [tj , tj +1 ] and n/2

|Ij |N (tj )2 η1 /4C. Then we have N(t) ≡ N (tj ) for any t ∈ Ij , i.e., N (t) is a step function.

708


Proof. By Lemma 4.8, we can regard N (t) as N (t) ≡ N (tj ), So, the result follows.

t ∈ [tj , tj +1 ].

2

Corollary 4.10. We have Nmin := min N (t) > 0 t∈I0

and N(t)−1 is a Lebesgue integrable function on I0 . Proof. Since N(t) takes finitely many values N (tj ) (1 j J ) and each N (tj ) is a dyadic number, we have the result. 2 4.3. Energy is almost conserved as μ, a → 0 By Theorem 2.7, we see that the energy has a dissipation for the solutions of the CGL. However, for the MEBS u, we show that the energy E(u(t)) tends to Ec as μ, a → 0. More precisely, we have Lemma 4.11. Let u be a minimal energy blowup solution, then μ


I 0 Rn

2(n+2) u(τ, x) n−2 dx dτ η4 .

I 0 Rn

Proof. Let I0 = [0, t0 ]. Since u is a solution of Eq. (1.3), we see from Theorem 2.7 that E u(t0 ) + μ

t0


0 Rn

t0

2(n+2) u(τ, x) n−2 dx dτ Ec .

0 Rn

If the conclusion of Lemma 4.11 does not hold, then E u(t0 ) Ec − η4 . Therefore the induction hypothesis Lemma 1.5 tells us that u

2(n+2)

Lt,xn−2 (I1 ×Rn )

M(Ec − η4 ),

which contradicts the definition of minimal energy blowup solution on interval I1 .

2


709

4.4. Morawetz estimate Proposition 4.12. Suppose n > 6. Let u be a minimal energy blowup solution to (1.3) and N∗ < c(η2 )Nmin , Nmin := inft∈I0 N (t), suppose furthermore that 0 < a, μ < η6 , then I 0 Rn Rn

|PN∗ u(t, y)|2 |PN∗ u(t, x)|2 dx dy dt |x − y|3

+ I 0 Rn Rn

2n

|PN∗ u(t, y)|2 |PN∗ u(t, x)| n−2 dx dy dt η1 N∗−3 . |x − y|

(4.19)

To prove this proposition, we first recall a known statement for general Schrödinger equation: iφt + φ = N .

(4.20)

Proposition 4.13. (Interaction Morawetz inequality for general NLS [19].) Let φ be a solution to (4.20), then φ satisfies (n − 1)(n − 3) I 0 Rn Rn

+2

I 0 Rn Rn

|φ(t, y)|2 |φ(t, x)|2 dx dy dt |x − y|3

φ(t, y)2 x − y {N , φ}p (t, x) dx dy dt |x − y|

4φ3L∞ L2 φL∞ ˙1 t Hx t x

+4 I0

Rn

{N , φ}m (t, y)∇φ(t, x)φ(t, x) dx dy dt,

(4.21)

Rn

where we denote the mass bracket {f, g}m := Im(f g) ¯ and the momentum bracket {f, g}p := Re(f ∇ g¯ − g∇ f¯). It is easy to see that the CGL equation (1.3) can be rewritten as iut + u = μiu + 4 4 (1 − ai)|u| n−2 u := |u| n−2 u + N . We can regard N as an additional nonlinear term for the NLS, then we need to compute the new terms {N , u}p and {N , u}m . On the other hand, the first term φL∞ 2 in the right-hand side of (4.21) is not scaling invariant and it can become t Lx very large after scaling, so we should throw away the low frequency part of φ when we apply Proposition 4.13. Let us now take up the main business of this section. Proof of Proposition 4.12. Writing the left-hand side of (4.19) as L(N∗ , u), we claim that we only need to consider the case N∗ = 1, i.e., L(1, u) η1 for any minimal energy blowup solution u to (1.3) on I0 . In fact, if u is a minimal energy blowup solution to (1.3) on I0 , then by the scaling 2(n+2)

n−2

invariance of Lt,xn−2 norm and energy norm, uλ = λ− 2 u( λt2 , xλ ) is a minimal energy blowup solution to (1.3) on λ2 I0 . So there holds L(1, uN∗ ) η1 , that is

710


L 1, uN∗ =

N∗2 I0 Rn Rn

|P1 uN∗ (t, y)|2 |P1 uN∗ (t, x)|2 dx dy dt |x − y|3

+ N∗2 I0 Rn Rn

2n

|P1 uN∗ (t, y)|2 |P1 uN∗ (t, x)| n−2 dx dy dt |x − y|

η1 ,

(4.22)

in which

N∗

P1 u

1− n (t, x) = N∗ 2 (PN∗ u)

t x . , N∗2 N∗

(4.23)

By (4.22), we obtain the result, as desired. So our main task is to show the following L(1, u) = I 0 Rn Rn

|P1 u(t, y)|2 |P1 u(t, x)|2 dx dy dt |x − y|3

+ I 0 Rn Rn

2n

|P1 u(t, y)|2 |P1 u(t, x)| n−2 dx dy dt |x − y|

η1 .

(4.24)

To do this, we define uhi = P1 u, ulo = Pη−1 uLt L2x (I0 ×Rn ) t Lx (I0 ×R ) 2

2

P1·η−1 uL∞ n + η2 P>η−1 uL∞ H˙ 1 (I ×Rn ) ˙1 0 t H (I0 ×R ) t 2

x

2

η2 .

x

(4.27)

Our goal is to prove (4.24), which implies particularly that I0

Rn

Rn

|uhi (t, y)|2 |uhi (t, x)|2 dx dy dt η1 , |x − y|3

(4.28)


711

which can be rewritten as |uhi |2

1

− n−3 L2t H˙ x 2 (I0 ×Rn )

η12 .

(4.29)

By the standard continuity argument, it suffices to prove (4.29) under the bootstrap hypothesis: |uhi |2

1

− n−3 L2t H˙ x 2 (I0 ×Rn )

(C0 η1 ) 2 ,

(4.30)

where C0 is a large constant depending on energy but not on any ηi . We claim that (4.30) implies − n−3 |∇| 4 uhi

1

L4t,x (I0 ×Rn )

(C0 η1 ) 4 ,

(4.31)

as can be seen by taking f = uhi in the following lemma. Lemma 4.14. (See [19, Lemma 5.6].) − n−3 |∇| 4 f

L4x

1 n−3 |∇|− 2 |f |2 22 .

To prove Proposition 4.12, we first prove a weaken form. Proposition 4.15. Under the same assumption as Proposition 4.12, we have I 0 Rn Rn

|uhi (t, y)|2 |uhi (t, x)|2 dx dy dt + |x − y|3

η2 + η 2

I 0 Rn Rn

2n

|uhi (t, y)|2 |uhi (t, x)| n−2 dx dy dt |x − y|

4 4 4 uhi (t, x)Phi |u| n−2 u − |uhi | n−2 uhi − |ulo | n−2 ulo (t, x) dx dt

(4.32)

I 0 Rn

+ η2 I0

4 uhi (t, x)Plo |uhi | n−2 uhi (t, x) dx dt

(4.33)

4 uhi (t, x)Phi |ulo | n−2 ulo (t, x) dx dt

(4.34)

4 ∇ulo (t, x)ulo (t, x) n−2 uhi (t, x) dx dt

(4.35)

n+2 ∇ulo (t, x)uhi (t, x) n−2 dx dt

(4.36)

4 ∇Plo |u| n−2 u (t, x)uhi (t, x) dx dt

(4.37)

Rn

+ η2 I 0 Rn

+ η22 I0

Rn

+ η22 I 0 Rn

+ η22 I0

Rn

712


n+2

|uhi (t, y)|2 |ulo (t, x)| n−2 |uhi (t, x)| dx dy dt |x − y|

+ I0

Rn

Rn

n+2

|uhi (t, y)|2 |ulo (t, x)||uhi (t, x)| n−2 dx dy dt |x − y|

+ I0

Rn

Rn

I0

(4.39)

4

|uhi (t, y)|2 |Plo (|uhi | n−2 uhi )(t, x)||uhi (t, x)| dx dy dt. |x − y|

+ Rn

(4.38)

Rn

4

(4.40)

4

Proof. We apply Proposition 4.13 with φ = uhi , N = Phi (|u| n−2 u)+μiuhi −aiPhi (|u| n−2 u) := N1 + N2 + N3 , then we obtain (n − 1)(n − 3) I 0 Rn Rn

+2

I 0 Rn Rn

|uhi (t, y)|2 |uhi (t, x)|2 dx dy dt |x − y|3

uhi (t, y)2 x − y {N1 + N2 + N3 , uhi }p dx dy dt |x − y|

4uhi 3L∞ L2 uhi L∞ ˙1 t Hx t x {N1 + N2 + N3 , uhi }m (t, y)∇uhi (t, x)uhi (t, x) dx dy dt. +4 I0

Rn

(4.41)

Rn

By energy conservation and (4.27), 3 4uhi 3L∞ L2 uhi L∞ ˙ 1 η2 . t H t

x

x

Visan [19] has shown that

{N1 , uhi }m (t, y)∇uhi (t, x)uhi (t, x) dx dy dt (4.32) + (4.33) + (4.34),

I 0 Rn Rn

I 0 Rn Rn

uhi (t, y)2 x − y {N1 , uhi }p (t, x) dx dy dt |x − y|

I 0 Rn Rn

2n

|uhi (t, y)|2 |uhi (t, x)| n−2 dx dy dt + (4.35) + · · · + (4.40). |x − y|

So, it suffices to consider the corresponding estimates for N2 and N3 . We first begin with the momentum bracket term:


{N2 , uhi }p = μ Re(iuhi ∇uhi − uhi ∇iuhi ) = −μ Im ∇(uhi uhi ) − 2∇uhi uhi ,

713

(4.42)

and 4 4 {N3 , uhi }p = a −i|u| n−2 u, uhi p − a −iPlo |u| n−2 u , uhi p 4 4 = a −i|u| n−2 u, u p − a −i|ulo | n−2 ulo , ulo p 4 4 4 − a −i |u| n−2 u − |ulo | n−2 ulo , ulo p − a −iPlo |u| n−2 u , uhi p 4 4 4 = −a −i |u| n−2 u − |ulo | n−2 ulo , ulo p − a −iPlo |u| n−2 u , uhi p := I1 + I2 ,

(4.43) 4

where in the last equality, we use the fact that {−i|u| n−2 u, u}p = 0. This can be seen in the following: 4 4 4 −i|u| n−2 u, u p = Re −i|u| n−2 u∇ u¯ − u∇ −i|u| n−2 u 4 4 = Im |u| n−2 u∇ u¯ − u∇ |u| n−2 u¯ = 0. Now we handle the mass bracket term, {N2 , uhi }m = μ Im(iuhi uhi ) = μ Re(uhi uhi ),

(4.44)

and 4 {N3 , uhi }m = −a Re Phi |u| n−2 u uhi 4 4 2n = −a Re Phi |u| n−2 u uhi − |uhi | n−2 uhi uhi − a|uhi | n−2 4 4 4 = −a Re Phi |u| n−2 u − |uhi | n−2 uhi − |ulo | n−2 ulo uhi 4 4 2n + a Re Plo |uhi | n−2 uhi uhi − a Re Phi |ulo | n−2 ulo uhi − a|uhi | n−2 . Then (4.42) corresponds to uhi (t, y)2 x − y {N2 , uhi }p (t, x) dx dy dt |x − y| I 0 Rn Rn

= (n − 1)μ I 0 Rn Rn

+ 2μ

I 0 Rn Rn

= II1 + II2 .

uhi (t, y)2

1 Im(uhi uhi ) dx dy dt |x − y|

uhi (t, y)2 x − y Im ∇uhi uhi (t, x) dx dy dt |x − y|

(4.45)

714


We first estimate II1 , using the Hardy–Littlewood–Sobolev inequality and Hölder’s inequality, II1 = (n − 1)μ I 0 Rn

1 ∗ |uhi |2 (x) Im(uhi uhi ) dx dt |·|

1 2 μ ∗ |uhi | |·| I0

μ

p

uhi uhi

Lx

|uhi |2 q uhi L

2p 2−p

x

uhi 2 2q uhi Lx

I0

uhi L2x dt

2p 2−p Lx

uhi L2x dt, 1 q

where p is chosen large enough (at least p > n), and 1 uhi L2q uhi θL12 ∇uhi 1−θ , L2 x

dt

Lx

I0

μ

p

Lx

x

uhi

x

=

1 p

2p 2−p Lx

+

(4.46)

n−1 n .

Using interpolation,

2 uhi θL22 ∇uhi 1−θ , L2 x

x

3n , θ1 = 56 , θ2 = 13 . where 0 < θi < 1, i = 1, 2. For convenience, we choose p = 32 n, then q = 3n−1 Bernstein’s inequality tells us that uhi 2 ∇uhi 2 uhi 2 . Replace the above estimates into (4.46), and applying the energy estimate in Theorem 2.7, μu2 2 E(u0 ), we get Lt,x

II1 μ

5

1

1

2

uhi L3 2 ∇uhi L3 2 uhi L3 2 ∇uhi L3 2 uhi L2x dt x

I0 5 3

x

η2 μ

x

x

5

uhi 22 dt η23 .

(4.47)

I0

The second term can be bounded by II2 μ

∇uhi L2x uhi L2x uhi 2L2 dt y

I0

μη22

uhi 22 dt η22 . I0

In the sequel, we deal with the contribution of the momentum bracket (4.43). We denote by = ˙ ˙ ∇(f g) + f ∇g, then that we omit the conjugate symbols in the equality. Writing {f, g}p = 4 4 I1 = −a −i |u| n−2 u − |ulo | n−2 ulo , ulo p 4 4 4 4 = ˙ a∇ i |u| n−2 u − |ulo | n−2 ulo ulo + a i |u| n−2 u − |ulo | n−2 ulo ∇ulo .

(4.48)


715

Integrating by parts, we find that the first term in (4.48) corresponds to a scalar multiple of

4

I 0 Rn Rn

4

|uhi (t, y)|2 ||u| n−2 u − |ulo | n−2 ulo |(t, x)|ulo (t, x)| dx dy dt (4.38) + (4.39). |x − y|

The second term corresponds to

4 4 uhi (t, y)2 |u| n−2 u − |ulo | n−2 ulo (t, x)∇ulo (t, x) dx dy dt (4.35) + (4.36).

I 0 Rn Rn 4

Now we estimate I2 in (4.43), when the derivative is taken on Plo (|u| n−2 u), we estimate it by 4 a I0 Rn Rn |uhi (t, y)|2 |∇Plo (|u| n−2 u)||uhi (t, x)| dx dy dt (4.37), while when the derivative falls on uhi , it is a bad term, so we integrate by parts, then we obtain a

4 uhi (t, y)2 ∇Plo |u| n−2 u (t, x)uhi (t, x)

I 0 Rn Rn 4

+

|uhi (t, y)|2 |Plo (|u| n−2 u)(t, x)||uhi (t, x)| dx dy dt |x − y|

(4.37) + · · · + (4.40). Now let us deal with the mass bracket as in (4.41). (4.44) corresponds to Cμ

Re uhi uhi (t, y)∇uhi (t, x)uhi (t, x) dx dy dt

I 0 Rn Rn

uhi L∞ 2 uhi L∞ L2 μ t Lx t y

∇uhi L2x uhi L2y dt η22 .

(4.49)

I0

(4.45) corresponds to

{N3 , uhi }m (t, y)∇uhi (t, x)uhi (t, x) dx dy dt

I 0 Rn Rn

(4.32) + (4.33) + (4.34) + a

2n |uhi | n−2 (t, y)∇uhi (t, x)uhi (t, x) dx dy dt

I 0 Rn Rn

(4.32) + (4.33) + (4.34) + a

2n

∇uhi L2x uhi L2x uhi n−22n dt I0

Lyn−2

(4.32) + (4.33) + (4.34) + aη2

2n

uhi n−22n dt, I0

Lyn−2

716


in which

2n

uhi n−22n dt auhi L2

a I0

n uhi x,t (I0 ×R )

Lyn−2

n+2 n−2 2(n+2) Lt,xn−2 (I0 ×Rn )

n−2 a 1/2 |I0 |1/2 η2 a 2(n+2) uhi 1/2 −1/2

η6 η5

1/2

η2 η4

n+2 2(n+2) Lt,xn−2 (I0 ×Rn )

n−2

η2 ,

(4.50)

where in the last second inequality we apply Corollary 4.6 and Lemma 4.11. Up to now, we finish the proof of Proposition 4.15. 2 Now let us develop estimates on the low and high frequency parts to u, which we will use to bound the error terms in Proposition 4.15. Proposition 4.16. The low frequency part of u satisfies 4 2

ulo S˙ 1 (I0 ×Rn ) C1 η2(n−2) .

(4.51)

The high frequency part of u can be split into a good and a bad term, i.e., uhi = g + b, they satisfy the estimates 2

gS˙ 0 (I0 ×Rn ) C1 η2n−2 ,

(4.52)

gS˙ 1 (I0 ×Rn ) C1 , − 2 |∇| n−2 b

2n(n−2) 2 −3n−2

L2t Lxn

(4.53) 1 4

(I0 ×Rn )

C1 η 1 .

(4.54)

Proof. We define g and b to be the unique solutions to the following initial value problems respectively i∂t + (1 − μi) g = G + Phi F (ulo ) + Phi F (ulo + g) − F (g) − F (ulo ) , g(t0 ) = uhi (t0 ) and i∂t + (1 − μi) b = B + Phi F (u) − F (ulo + g) , b(t0 ) = 0 4

where F (z) = (1 − ai)|z| n−2 z, G + B = Phi (F (g)). The aim of defining b(t0 ) here is to make use of the improved Strichartz estimate (Proposition 2.4). Then following the same way as in [19], using bootstrap argument, we can prove our Proposition 4.16. 2 With Proposition 4.16 at hand, we can bound the error terms in the right-hand side of Proposition 4.15, we note here that ulo is a good term, we apply the estimation (4.51) directly, while


717

uhi is a bad term, we split it into g and b, making use of (4.52)–(4.54), we can also treat it, since the argument is the same as in [19], we omit it. So, we get that (4.32)–(4.40) can all be bounded by η1 , then Proposition 4.12 is proven. 2 By Proposition 4.16 and scaling, there holds Corollary 4.17. Suppose n > 6, u is a minimal energy blowup solution to (1.3), N∗ < c(η2 )Nmin . Then we can split PN∗ u = g + b satisfying: 2

gS˙ 0 (I0 ×Rn ) η2n−2 N∗−1 ,

(4.55)

gS˙ 1 (I0 ×Rn ) 1,

(4.56)

− 2 |∇| n−2 b

1

−3

η14 N∗ 2 .

(4.57)

1 − 3 n−2 η14 N∗ 2 n ,

(4.58)

2n(n−2) 2 L2t Lxn −3n−2 (I0 ×Rn )

Upon scaling, we can prove b b

2n2 2n (n+1)(n−2) Ltn−2 Lx (I0 ×Rn )

2n(n+2) 2(n+2) (n−2)(n+3) Lt n−2 Lx (I0 ×Rn )

PN∗ u

1 − 3 n−2 η14 N∗ 2 n+2 ,

(4.59)

1

6n L3t Lx3n−4 (I0 ×Rn )

η16 N∗−1 .

(4.60)

4.5. Contradiction argument Proposition 4.18. Let u be a minimal energy blowup solution to (1.3), 0 < a, μ η6 . Then for any t ∈ I0 , we have N (t) C(η4 )Nmin . Proof. N(t) is a step function, therefore there exists tmin ∈ I0 such that N (tmin ) = Nmin . By Bernstein inequality and energy conservation, we have −1 u(tmin )L2 c(η0 )Nmin , x P>C(η )N u(tmin ) 2 c(η0 )N −1 . 0 min min L

Pc(η

0 )Nmin 0 there exists T2 () > 0 s.t. lim sup 3 R→∞ R

−T 2 ()

|u|2 dx dt . −∞ BR

ω

744

L. Vega, N. Visciglia / Journal of Functional Analysis 255 (2008) 726–754

In particular, if we choose T () = max{T1 (), T2 ()}, then we get: ∀ > 0 there exists T () > 0 s.t. 1 |u|2 dx dt . lim sup 3 R R→∞

(4.12)

R\(−T ();T ()) BR

Hence the proof of Proposition 4.3 (in the case n = 3) will follow from the following fact: 1 ∀T > 0 we have lim sup 3 R R→∞

T |u|2 dx dt = 0.

(4.13)

−T BR

Notice that by using the Hölder inequality we get

u(t)2 dx R 2 u(t)2 6 , Lx

BR

and this implies 1 R3

T

C |u| dx dt R

T

2

−T BR

−T

u(t)2 6 dt 2CT u 2 ∞ 6 . Lx Lt Lx R

By combining this fact with (1.1) and with the Sobolev embedding H˙ x1 ⊂ L6x , we finally get (4.13). 2 Proof of Theorem 0.2. First of all let us recall the following identity 2 ∇x uD ¯ x2 ψ∇x u = ∂|x| ψ|∂|x| u|2 +

∂|x| ψ |∇τ u|2 , |x|

(4.14)

where ψ is a radially symmetric function. By using this identity and by choosing in the identity (0.8) the function ψ ≡ x , then it is easy to deduce that |x|>1

|∇τ u|2 dx dt < ∞. |x|

In particular we deduce lim

R→∞

and

|x|>R

|∇τ u|2 dx = 0 |x|

(4.15)


1 lim R→∞ R

745

|∇τ u|2 dx dt = 0.

(4.16)

BR

By combining (4.2) with (4.16) we get (0.10). Next we shall prove (0.9). For any k ∈ N we fix a function hk (r) ∈ C0∞ (R; [0, 1]) such that: hk (r) = 1

∀r ∈ R s.t. |r| < 1,

hk (r) = 0

∀r ∈ R s.t. |r| >

hk (r) = hk (−r)

∀r ∈ R.

k+1 , k (4.17)

Let us introduce the functions ψk (r), Hk (r) ∈ C ∞ (R): r ψk (r) =

r (r − s)hk (s) ds

and Hk (r) =

0

(4.18)

hk (s) ds. 0

Notice that

ψk

(r) = hk (r),

ψk (r) = Hk (r)

∞ ∀r ∈ R and

lim ∂r ψk (r) =

hk (s) ds.

r→∞

(4.19)

0

Moreover an elementary computation shows that x ψk

C x

∀x ∈ Rn , n 3,

and 2x ψk (x) =

C |x|3

∀x ∈ Rn s.t. |x| 2 and n 4,

2x ψk (x) = 0 ∀x ∈ R3 s.t. |x| 2,

(4.20) (4.21)

where 2x is the biLaplacian operator. Thus the functions φ ≡ ψk satisfy the assumptions of Proposition 4.2. In the sequel we shall need the rescaled functions x ∀x ∈ Rn , k ∈ N and R > 0, ψk,R (x) ≡ Rψk R

(4.22)

where ψk is defined in (4.18). Notice that by combining the general identity (4.14) with (0.8), where we choose ψ = ψk,R defined in (4.22), and recalling (4.19) we get:

746


∂|x| ψk,R 1 λ ∗ |∇τ u|2 − |u|2 2x ψk,R + |u|2 x ψk,R dx dt |x| 4 n ∞ 2λ ∗ |∇x f |2 + ∗ |f |2 + |g|2 dx ∀k ∈ N, R > 0. = (4.23) hk (s) ds 2 2 ∂|x| ψk,R |∂|x| u|2 +

0

Notice also that due to (4.21) we get: 2 ψk,R |u|2 dx dt C |u|2 dx dt x R3 BR

R3

provided that n = 3, and in particular by using (4.5) we get 2 ψk,R |u|2 dx dt = 0. lim x

(4.24)

R→∞

R3

In the case n 4 we use (4.20) in order to deduce:

2 ψk,R |u|2 dx dt C x

Rn

1 R3

|u|2 dx dt +

Rn \BR

BR

|u|2 dx dt . |x|3

(4.25)

On the other hand, an explicit computation shows that if we choose in (0.8) ψ ≡ x , when n 4, then we get: Rn

|u|2 dx dt < ∞ for n 4, |x|3

(4.26)

that in conjunction with the Lebesgue dominated convergence theorem, (4.5) and (4.25) implies 2 ψk,R |u|2 dx dt = 0 for n 4. (4.27) lim x R→∞

Rn

By using (4.24), (4.27), (4.3) and (4.15) we get: lim

∂|x| ψk,R

R→∞

|∇τ u|2 1 2 λ ∗ − x ψk,R |u|2 + x ψk,R |u|2 dx dt = 0 |x| 4 n

(4.28)

for every k ∈ N and for every dimension n 3. We can combine this fact with (4.23) in order to deduce: 2 lim ψk,R |∂|x| u|2 dx dt ∂|x| R→∞

∞

=

hk (s) ds 0

|∇x f |2 +

2λ 2∗ 2 |f | + |g| dx 2∗

∀k ∈ N.

(4.29)


747

On the other hand, due to the properties of hk (see (4.17)), we get 1 R

2 ∂|x| ψk,R |∂|x| u|2 dt dx

|∂|x| u| dx dt 2

BR

1 R

1 = R

x |∂|x| u|2 dt dx hk R

|∂|x| u|2 dx dt |x|< k+1 k R

that due to (4.29) implies: 1 lim sup R→∞ R

∞

|∂|x| u| dx dt 2

BR

0

1 k+1 lim inf k R→∞ R

2λ 2∗ 2 |f | + |g| dx 2∗

|∇x f |2 +

hk (s) ds

|∂|x| u|2 dx dt

∀k ∈ N.

(4.30)

BR

Since k ∈ N is arbitrary and since the following identity is trivially satisfied: ∞ hk (s) ds = 1,

lim

k→∞ 0

we can deduce (0.9) by using (4.30). The proof is complete. 2 5. Proof of Theorem 0.3 First step: proof of (0.11). Following [15] we multiply Eq. (0.1) by ϕu and integrating the corresponding identity on the strip (−T , T ) we get: T −T

Rn

1 ∗ |∂t u|2 − |∇x u|2 − λ|u|2 ϕ + |u|2 x ϕ dx dt = ± 2 ±

∂t u(±T )u(±T )ϕ dx. (5.1)

By taking the limit as T → ∞ and by using Proposition 3.2 we get (0.11). Second step: proof of (0.12). For any k ∈ N we fix a function ϕk (r) ∈ C0∞ (R; [0, 1]) such that: ϕk (r) = 1

∀r ∈ R s.t. |r| < 1,

ϕk (r) = 0

∀r ∈ R s.t. |r| >

ϕk (r) = ϕk (−r)

∀r ∈ R.

We also introduce the rescaled functions

k+1 , k (5.2)

748


ϕk,R ≡

x 1 ϕk . R R

Notice by combining the cut-off property of the functions ϕk with (4.2) and (4.5), we get: ∗ lim |u|2 ϕk,R dx dt = lim |u|2 x ϕk,R dx dt = 0 ∀k ∈ N, R→∞

R→∞

in any dimension n 3. By using this fact in conjunction with (0.11), where we choose ϕ ≡ ϕk,R , we get: lim (5.3) |∂t u|2 − |∇x u|2 ϕk,R dx dt = 0 ∀k ∈ N. R→∞

Notice that by combining (5.3) with the cut–off properties of ϕk we get:

1 R

BR

∀k ∈ N there exists R(k) > 0 s.t. 1 k+1 1 2 |∂t u| dx dt |∇x u|2 dx dt + k+1 k R( k ) k

∀R > R(k).

BR( k+1 ) k

By combining (5.4) with (0.9) and (0.10) , we get: lim sup R→∞

1 R

|∂t u|2 dx dt

k+1 k

2λ 1 ∗ |∇x f |2 + ∗ |f |2 + |g|2 dx + 2 k

∀k ∈ N,

BR

and in particular 1 lim sup R R→∞

2λ 2∗ 2 2 |∇x f | + ∗ |f | + |g| dx. |∂t u| dx dt 2 2

BR

Similarly one can show that 1 lim inf R→∞ R

|∂t u| dx dt 2

BR

2λ 2∗ 2 |∇x f | + ∗ |f | + |g| dx, 2 2

Rn

and finally we get 1 R→∞ R

|∂t u|2 dx dt =

lim

2λ ∗ |∇x f |2 + ∗ |f |2 + |g|2 dx 2

BR

1 R→∞ R

= lim

|∇x u|2 dx dt BR

where at the last step we have combined (0.9) with (0.10). The proof of (0.12) is complete. Finally notice that by combining (0.9), (0.10) and (0.12), we get (0.13). 2

(5.4)


749

Acknowledgment The authors should like to thank the anonymous referee for interesting suggestions that improved the original version of this paper. Appendix A ∗

The aim of this appendix is to show that the L2 -norm of the solution to the following Cauchy problem: 2u = 0, u(0) = f ∈ H˙ x1 ,

∂t u(0) = g ∈ L2x ,

(A.1)

goes to zero as t → ±∞. Notice that this fact represents a slight improvement compared with the usual Strichartz estimate u(t, x)

∗

2 L∞ t Lx

C f H˙ 1 + g L2x . x

On the other hand, in Proposition A.2 we shall show that in general no better result can be expected. In fact we shall show that there cannot exist a priori any rate on the decay of the ∗ L2 -norm of the solution to (A.1). Along this section, when it is not better specified, we shall denote by T (t)(f, g) the solution to the Cauchy problem (A.1) with initial data (f, g) computed at time t, i.e.: T (t) : H˙ x1 × L2x (f, g) → u(t) ∈ H˙ x1 , where u(t, x) solves (A.1). Proposition A.1. Let u(t, x) ∈ Ct (H˙ x1 ) ∩ Ct1 (L2x ) be the unique solution to (A.1), then lim u(t)L2∗ = 0.

t→±∞

x

Proof. We treat for simplicity the case t → ∞ (the case t → −∞ can be treated in a similar way). Notice that due to the Sobolev embedding and the conservation of the energy we have: u(t)2 2∗ S ∇x u(t)2 2 + ∂t u(t)2 2 Lx Lx Lx = S ∇x f 2L2 + g 2L2 ∀(f, g) ∈ H˙ x1 × L2x . x

x

(A.2)

In particular the operators T (t) introduced above, are uniformly bounded for every t > 0 in the ∗ space L(H˙ x1 × L2x , L2x ). On the other hand, we have the following dispersive estimate (see [17]): u(t, x)

L∞ x

t

C

f

n−1 2

m−1 B˙ 1,12

+ g

m+1 B˙ 1,12

(A.3)

750


s denotes the standard Besov spaces). Notice also that the Fourier representation of the (here B˙ p,q solution to (A.1) implies

u(t)

L2x

C f L2x + g H˙ −1 . x

(A.4)

In particular by combining (A.3) with (A.4) we deduce u(t)

∗

L2x

n−2 2 C u(t)Ln ∞ u Ln2 n−1 x x t n

∀(f, g) ∈ C0∞ Rn × C0∞ Rn ,

where C ≡ C(f, g) > 0. As a consequence we get lim u(t)L2∗ = 0 ∀(f, g) ∈ C0∞ Rn × C0∞ Rn .

t→∞

(A.5)

It is now easy to remove in (A.5) the regularity assumption (f, g) ∈ C0∞ (Rn ) × C0∞ (Rn ) by a classical density argument. 2 Notice that the previous result represents a slight improvement compared with the usual Strichartz estimate: u(t, x)

∗

2 L∞ t Lx

C f H˙ 1 + g L2x . x

On the other hand, next proposition shows that in general no better result can be expected, ∗ since there cannot exist a priori any rate on the decay of the L2 -norm of the solution to (A.1). Proposition A.2. Let γ ∈ C([0, ∞); R) be any function such that lim γ (t) = ∞.

t→∞

Then there exists g ∈ L2x such that u(tn )

∗

L2x

>

1 , γ (tn )

where {tn } is a suitable sequence that goes to +∞ and 2u = 0, ∂t u(0) = g ∈ L2x .

u(0) = 0,

(A.6)

Proof. We claim the following fact: S(t)

L(L2x ,L2x∗ )

∗

0 > 0,

(A.7)

where S(t) : L2x g → u(t) ∈ L2x is the solution operator associated to the Cauchy problem (A.6).


751

Notice that due to (A.7) we get lim γ (t)S(t)L(L2 ,L2∗ ) = ∞,

t→∞

x

x

and in particular due to the Banach–Steinhaus theorem the operators γ (t)S(t) cannot be pointwisely bounded, or in an equivalent way there exists at least one g ∈ L2x such that sup γ (t)S(t)g L2∗ = ∞.

(A.8)

x

[0,∞)

On the other hand, the function γ (t) S(t)g L2∗ is bounded on bounded sets of [0, ∞), and hence x (A.8) implies that lim sup γ (t)S(t)g L2∗ = ∞ x

t→∞

and it completes the proof. Next we shall prove (A.7). Let us fix h ∈ L2x such that:

h L2x = 1 and S(1)hL2∗ = u(1, x)L2∗ = η0 > 0 x

x

where u(t, x) denotes the unique solution to (A.6) with g = h. n A rescaling argument implies that u (t, x) ≡ 2 −1 u(t, x) solves (A.6) with initial data g ≡ n h ≡ 2 h(x). In particular this implies that: n 1 h = 2 −1 u(1, x) S

and h L2x = 1,

and hence: 1 1 n −1 S u(1, x)L2∗ = u(1)L2∗ 2 2∗ S h 2∗ = 2 x x L(Lx ,Lx ) Lx = S(1)hL2∗ = η0 > 0 ∀ > 0. x

2

The proof of (A.7) is complete. Appendix B

This section is devoted to the proof of Proposition 2.1. Let us underline that its content is well known in the literature, in particular it contains the equipartition of the energy principle first proved in [4] by using Fourier analysis. The aim of this section is to present a proof that involves the conformal energy. Proof of Proposition 2.1. First of all notice that (2.1) implies: lim

t→∞

∂t u(t)2 = lim

t→∞

∂|x| u(t)2 dx.

752


By combining this fact with (2.2) and with the following trivial identity: |∇x u|2 = |∂|x| u|2 + |∇τ u|2 , we can deduce (2.4). Hence it is enough to prove (2.1) and (2.2) in order to deduce (2.4). Next notice that by a density argument it is sufficient to prove (2.2), (2.5) and (2.6) under the stronger assumption (f, g) ∈ C0∞ (Rn ) × C0∞ (Rn ) in order to deduce (2.1)–(2.3) under the weaker assumption (f, g) ∈ H˙ x1 × L2x . Since now on we shall assume that (f, g) ∈ C0∞ (Rn ) × C0∞ (Rn ). Following [5] we introduce the conformal energy factor 2 t + |x|2 ∂t u + 2ntxj ∂j u . Since 2u = 0 we get for every T > 0 the following identity: 0=

n T 2 t + |x|2 ∂t u + 2txj ∂j u 2u dx dt j =1 0

2 2 n 2 T + |x|2 ∂t u(T ) + ∇x u(T ) + 2nT r∂|x| u(T )∂t u(T ) dx 2

=

−

n 2 |x| |∇x f |2 + |g|2 dx + n(n − 1) 2

T

t |∂t u|2 + |∇x u|2 dx dt,

0

where we have used the Stokes formula. Notice that this identity implies the following inequality: 2 2 T + |x|2 ∂t u(T ) + |∇x u|2 + 4T |x|∂|x| u(T )∂t u(T ) dx |x|2 |∇x f |2 + |g|2 dx.

(B.1)

On the other hand, we have the trivial pointwise inequality |∂|x| u|2 |∇x u|2 that can be combined with (B.1) in order to give: 2 2 T + |x|2 ∂t u(T ) + |∂|x| u|2 + 4T |x|∂|x| u(T )∂t u(T ) dx |x|2 |∇x f |2 + |g|2 dx, (B.2) and

2 2 2 T + |x|2 ∂t u(T ) + ∇x u(T ) − 4T |x|∇x u(T )|∂t u(T )| dx |x|2 |∇x f |2 + |g|2 dx.

(B.3)


753

Next recall the basic inequality (a + b)2 (c + d)2 + (a − b)2 (c − d)2 4 a 2 + b2 c2 + d 2 + 16abcd,

∀a, b, c, d ∈ R,

whose proof is completely elementary. By combining this inequality with (B.2) and (B.3) we get respectively:

2 2 2 2 T + |x| ∂t u(T ) + ∂|x| u(T ) + T − |x| ∂t u(T ) − ∂|x| u(T ) dx 4 |x|2 |∇x f |2 + |g|2 dx,

and

2 2 2 2 T + |x| ∂t u(T ) − ∇x u(T ) + T − |x| ∂t u(T ) + ∇x u(T ) dx 4 |x|2 |∇x f |2 + |g|2 dx.

This in turn implies:

∂t u(T ) + ∂|x| u(T )2 dx 4 |x|2 |∇x f |2 + |g|2 dx, 2 T 2 ∂t u(T ) − ∇x u(T ) dx 4 |x|2 |∇x f |2 + |g|2 dx, 2 T

(B.4) (B.5)

and

∂t u(T ) − ∇x u(T )2 + ∂t u(T ) + ∇x u(T )2 dx

2|x| 0, τ > 0,

(1.1)

where θ is the Laplace–Beltrami operator on the unit sphere S = {x ∈ Rd : |x| = 1} and ζ is the characteristic function of the complement to the unit ball {x ∈ Rd : |x| 1}. Before describing the results we need to give a definition of the operator θ . Let us introduce the polar coordinates E-mail address: [email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.019

756

O. Safronov / Journal of Functional Analysis 255 (2008) 755–767

in Rd and denote them by r = |x| and θ = operator is hidden in the relation =

1 r d−1

x |x| .

One of the definitions of the Laplace–Beltrami

1 ∂ d−1 ∂ r + 2 θ . ∂r ∂r r

The standard argument with separation of variables allows one to define the operator (1.1) as the orthogonal sum of one-dimensional Schrödinger operators, which implies that H0 is essentially self-adjoint on C0∞ (Rd ). The spectrum of this operator has an absolutely continuous component, which coincides with the positive half-line [0, ∞) as a set. We perturb now the operator H0 by a real-valued potential V = Vω that depends on the random parameter ω = {ωn }n∈Zd : ωn χ(x − n). Vω = n∈Zd

The function χ in this formula is the characteristic function of the cube [0, 1)d , and ωn are independent random variables taking their values in the interval [−1, 1]. We will assume that E[ωn ] = 0 for all n. This condition guarantees the presence of oscillations of V . Set H± = H0 ± Vω . Theorem 1.1. Let 0 < ε < 2/(d + 1). Then, for any τ > 0, the operator H± has an absolutely continuous spectrum, whose essential support covers the positive half-line [0, ∞). In other words, the spectral projection EH± (δ) corresponding to a set δ ⊂ R+ of positive Lebesgue measure, is different from zero. Remark. The condition d 5 is related to the necessity to use the estimate (1.13) for the mean value V¯ of the potential over the sphere of radius |x|. Our method requires that V¯ ∈ L(d+1)/2 , while V¯ behaves as |x|−(d−1)/2 at the infinity. Proof of Theorem 1.1. We shall consider only the case when τ = 1, because the only property of τ that matters is that it is positive. The proof of the theorem is based on two sufficiently deep observations. 1. The entropy of the spectral measure of the operator H+ (as well as H− ) can be estimated by the negative eigenvalues of the operators H+ and H− . Let us clarify this statement. Let μ be the spectral measure of H+ , constructed for the element f , which means that (H+ − z)−1 f, f =

∞

−∞

dμ(t) . t −z

Let λj (∓V ) be the negative eigenvalues of the operator H± . Then one can find such an element f of the space L2 (Rd ), that the measure constructed for this element will satisfy the condition b a

λj (V )1/2 + λj (−V )1/2 , log μ (λ) dλ −C 1 + j

j

(1.2)


757

where 0 < a < b < ∞ and the constant C depends only on a and b. The proof of this statement b can be found in [5]. Due to Jensen’s inequality, the integral a log μ (λ) dλ can diverge only to −∞. But, if it converges, then μ (λ) > 0 almost everywhere on [a, b], which leads to a certain conclusion about the absolutely continuous spectrum of H+ . Let us draw attention of the reader to the main difficulty of application of (1.2): it is derived only for compactly supported perturbations and one has to make sure that it survives in the limit, when V is approximated by compactly supported functions. 2. Within the conditions of the theorem,

λj (±V )1/2 < ∞, E

(1.3)

j

which implies λj (±V )1/2 < ∞,

almost surely.

j

Actually, it is much better to take the expectation in both sides of (1.2) and then talk about approximations of V by compactly supported functions, instead of doing it directly. Let us introduce the notation V¯ for the mean value of Vω over the sphere of radius |x| 1 V¯ (x) = |S|

Vω |x|θ dθ.

S

In order to establish (1.3) we will show that Vω = V¯ + div Q where Q is a vector potential having no radial component, i.e. x, Q(x) = 0, ∀x. Besides this, we will show that Q can be chosen in such a way that

E |x|=R

|Q|p dS CE

|Vω |p dS ,

R > 2,

(1.4)

|x|=R

where C depends only on the dimension d and the parameter p 2. Our arguments will be based on the fact that the operator H0 + ζ (x) ±2V ∓ 2V¯ + 4|x|−2+ε Q2 0

(1.5)

is positive, and therefore it does not have negative eigenvalues. The reason why the relation (1.5) holds is that the operator in its left-hand side is representable in the form ∗ − + ζ (x) |x|−ε/2 ∇θ ∓ 2|x|−1+ε/2 Q |x|−ε/2 ∇θ ∓ 2|x|−1+ε/2 Q , ∂ + 1r ∇θ (here er = x/|x|). We will keep the relawhere ∇θ is defined by the relation ∇ = er ∂r tion (1.5) in mind and leave it for the moment. In order to apply (1.5) we have to understand how

758


the eigenvalue sums j |λj (±V )|1/2 behave with respect to operation of addition of two potentials. Unfortunately the eigenvalue sums are not additive, but there is something that is similar to additivity. We will show that λj (V1 + V2 )γ λj ε −1 V1 γ + λj (1 − ε0 )−1 V2 γ 0 j

j

(1.6)

j

for any ε0 ∈ (0, 1). In order to do that we need to recall the Birman–Schwinger principle that reduces the study of eigenvalues λj (V ) to the study of the spectrum of a certain compact operator. For any self-adjoint operator T and s > 0 we define n+ (s, T ) = rank ET (s, +∞), where ET (·) denotes the spectral measure of T . Recall the following relation (see [4]) n+ (s + t, T + S) n+ (s, T ) + n+ (t, S).

(1.7)

The next statement is known as the Birman–Schwinger principle. Lemma 1.1. Let V be a real-valued function defined on the space Rd . Let N (λ, V ) be the number of eigenvalues of H0 − V below λ < 0. Then N (λ, V ) = n+ 1, (H0 − λ)−1/2 V (H0 − λ)−1/2 . Combining this lemma with (1.7) we obtain Corollary 1.1. For any ε0 ∈ (0, 1), N (λ, V1 + V2 ) N λ, ε0−1 V1 + N λ, (1 − ε0 )−1 V2 .

(1.8)

∞ Now, since j |λj (V )|γ = 0 γ s γ −1 N (−s, V ) ds, the inequality (1.6) holds for the Lieb– Thirring sums. Without loss of generality we can assume that V and Q are equal to zero for |x| < 1. This can be achieved by the methods of the scattering theory: one can pass from V to ζ V without changing the absolutely continuous spectrum of the differential operator. Due to the representation V = (V − V¯ − 2|x|−2+ε Q2 ) + V¯ + 2|x|−2+ε Q2 , we obtain from (1.6), that λj (V )γ λj 2V − 2V¯ − 4|x|−2+ε Q2 γ + λj 2V¯ + 4|x|−2+ε Q2 γ . (1.9) Since the operator (1.5) is positive, the first sum in the right-hand side of (1.9) equals zero. Thus λj (Vω )γ λj 2V¯ + 4|x|−2+ε Q2 γ .

(1.10)


759

Now formula (1.10) and the classical Lieb–Thirring estimate (see [6] and [7]) lead to the following important intermediate result: −1+ε/2 d+2γ d/2+γ λj (Vω )γ C |x| ¯ Q dx + |V | dx . (1.11) j

Indeed, Theorem 1.2 (Lieb–Thirring). If d 3. Then the negative eigenvalues νj of − − V satisfy the estimate γ +d/2 |νj |γ C V (x) dx, γ 0. j

The Lieb–Thirring estimate is valid for the eigenvalues of a Schrödinger operator. The operator whose eigenvalues we have to estimate has an additional term −|x|− θ . The quadratic form of this operator is positive. Therefore (1.11) follows by monotonicity. What is left to prove at this moment? If we take the expectation in both sides of (1.11), then we shall reduce the problem to the proof of the two relations:

−1+ε/2 d+1 |x| Q dx < ∞ (1.12) E |x|>1

and

(d+1)/2 ¯ E dx < ∞. |V |

(1.13)

The relation (1.12) for ε < 2/(d + 1) immediately follows from (1.4). We shall establish (1.13) in the next section. 2. Proof of (1.4) and (1.13) Let us now prove the necessary estimates (1.4) and (1.13). The proof of these relations is based on the observation that if one has independent random variables τn with zero expectations, then

2

= E τn E (τn )2 . n

n

We shall begin with the following statement. Lemma 2.1. The relation (1.4) holds for even integers p = 2q with q 1. Proof. The mapping V → Q is given by the formula ¯ Q = |x|∇θ −1 θ (V − V ).

(2.1)

760


The kernel k(x, y) of this mapping has a singularity of order |x − y|−(d−2) on the diagonal, but this singularity is integrable. Thus, k0 (x, ˆ y) ˆ , |x − y|d−2

k(x, y) =

xˆ =

y x , yˆ = , |x| |y|

where k0 ∈ L∞ , Q(x) =

k(x, y)Vω (y) dS(y).

{y: |y|=|x|}

All the statements about the kernel k will be proved in Appendix A (see Proposition A.1). We represent Q in the form of a sum Q = Q1 + Q2 , where

k0 χ0 (x − y)V (y) dS(y) |x − y|d−2

Q1 = {y: |y|=|x|}

and χ0 is the characteristic function of the unit ball {x: |x| < 1}. We will establish the estimates

E |Qj |2q dS(x) CR d−1 ,

j = 1, 2,

(2.2)

|x|=R

separately. The estimate (2.2) for Q1 is obvious. Indeed,

Q1 (x) C

{y: |y|=|x|}

χ0 (x − y) dS(y). |x − y|d−2

For large values of the radius R = |x| the piece of the sphere intersecting the ball {y: |y − x| < 1} becomes flat. Therefore the latter integral approximately equals the integral over the unit ball in Rd−1 : Q1 (x) C˜

{y∈Rd−1 : |y| 0 is sufficiently small, then one of these pieces of the sphere will be sufficiently flat, so one can substitute this piece by the plane of dimension d − 1: {y: |y|=|x|, |x−y| 1/2,

in the mathematical literature. One can study either the discrete or the continuous model of this operator. For the discrete model, the presence of the absolutely continuous spectrum in dimension d = 2 was proved by Bourgain [2,3]. In dimension d = 3, the corresponding result was obtained by Denissov [1]. In spite the fact that the main result of this paper pertains to the theory of random operators, we would like now to formulate a result of the deterministic type. This result will be related to the operator − − εΓ + V ,

(3.1)

where Γ u(x) =

1 |S|

u |x|θ dθ.

S

The potential V in this model is not random: on the contrary, it is a fixed potential. Theorem 3.1. Let d 2 and ε > 0. Assume that V ∈ L∞ , V (rθ ) dθ = 0,

∀r > 1,

(3.2)

S

and

V 2 (x) dx < ∞. (1 + |x|)d−1

(3.3)

Then the absolutely continuous spectrum of the operator (3.1) is essentially supported by [−ε, ∞). Proof. The proof of this theorem relies on the fact that the negative eigenvalues βj (∓V ) of the operator − − εΓ + εI ± V satisfy the condition βj (∓V )1/2 < ∞.

(3.4)

j

The proof of (3.4) is based on the circumstance that the behaviour of the eigenvalues near zero depends only on the structure of the edge of the spectrum of the unperturbed operator. But in the suggested model, this edge has the same structure as the one of the one-dimensional Schrödinger

764


operator. Let us introduce the notation P1 for the spectral projection of the operator A = − − εΓ + εI corresponding to the interval [0, ε). Set also P2 = I − P1 . Then V = 2 Re P1 V P2 + P2 V P2 , due to the condition (3.2) on the mean values of V . It was noticed before that βj (V )1/2 = n+ 1, (A + λ)−1/2 V (A + λ)−1/2 dλ √ . 2 λ j ∞

0

Besides the distribution function n+ we shall need distribution functions of singular values of non-selfadjoint operators n(s, T ) = n+ s 2 , T ∗ T , s > 0 (here T is a compact operator). Two of the important properties of this function are s ,T , n(s, T S) n S and n(s1 + s2 , T1 + T2 ) n(s1 , T1 ) + n(s2 , T2 ). Using these properties, we obtain dλ βj (V )1/2 n c1 , (A + λ)−1/2 P1 V √ λ j ∞

0

∞ + 0

dλ n+ c2 , (A + λ)−1/2 P2 V P2 (A + λ)−1/2 √ . 2 λ

Let us remark that the second term equals zero if the norm V L∞ is sufficiently small. In the general case, this term can be well estimated by the integral (3.3). It remains to consider the first term ∞ 0

dλ n c1 , (A + λ)−1/2 P1 V √ λ ∞ = 0

∞ 0

dλ n+ c12 , (A + λ)−1/2 P1 V 2 P1 (A + λ)−1/2 √ λ dλ n+ c12 , (A + λ)−1/2 Γ V 2 Γ (A + λ)−1/2 √ = 2 |Λj |1/2 , λ j


765

where Λj are the eigenvalues of the operator −Γ − c1−2 Γ V 2 Γ, which, as a matter of fact, is a one-dimensional Schrödinger operator with the potential αd /r 2 − V 2 . Therefore, according to the Lieb–Thirring bound for a one-dimensional operator (see [6,7]), the sum j |Λj |1/2 can be estimated by the integral (3.3). 2 At the end of this section the author would like to mention one idea, which might be useful for the reader in the study of Anderson’s model. Let us look at the result from a different point of view. Consider an operator which is in a certain sense close to the operator ωn χ(x − n). (3.5) − + Vω , where Vω = n

First, we introduce the class S of perturbations B for which the wave operators exist: B ∈ S if and only if ∃ s-lim exp −it (− + B) exp it (−) . t→±∞

Note, that this class is very rich (meaning “large”) and it can include even differential operators whose coefficients do not decay at infinity. For example, the operator −ζ (x)|x|−s θ ,

s > 1,

belongs to the class S (here ζ is the characteristic function of the exterior of the unit ball), but the coefficients of this operator behave at infinity as |x|2−s . Theorem 3.2. Let ε > 0 and let d 3. Assume that ωn are bounded independent random variables with the property E[ωn ] = 0, for all n. Then for almost every ω, there exists a perturbation B ∈ S such that the operator − + B + (1 + |x|)−ε Vω has a.c. spectrum all over the positive half-line [0, ∞). Proof. Indeed, let Q and V¯ be the same as in the proof of Theorem 1.1. In particular, it means that x, Q(x) = 0, ∀x, div Q = Vω , and V¯ = |S|−1 Vω |x|θ dθ. S

Define −ε −1−ε 2 Q . B = −ζ (x)|x|−(1+ε) θ − ζ (x) 1 + |x| V¯ + 1 + |x| It is easy to check that B ∈ S. On the other hand the operator − + B + ζ (x)(1 + |x|)−ε Vω is positive. So, there is no necessity to estimate eigenvalues of this operator and the trace formula obtained in [5] gives the relation b E

−1−ε 2 1−d log μ (λ) dλ −C 1 + 1 + |x| E Q |x| dx

a

for the spectral measure μ of the operator − + B + ζ (x)(1 + |x|)−ε Vω .

2

766


Appendix A 1. Finally consider a technical and less interesting question about the kernel of the operator 2 ∇θ −1 θ on the unit sphere. Let Γ be the orthogonal projection in L (S) onto the subspace of constant functions. 2 Proposition A.1. The kernel of the operator ∇θ −1 θ (I − Γ ) acting in L (S) is a function of the form

k(x, y) =

k0 (x, y) , |x − y|d−2

x, y ∈ S,

where k0 is bounded. Proof. Let us fix the point y. Clearly, the kernel k(x, y) possesses the symmetry with respect to the axis connecting y and −y. Let s be the distance between x and y along the geodesic curve. Then k(x, y) = ρ(s)e(s), where ρ(s) is a certain scalar function and e(x) is the unit vector, tangent to the mentioned geodesic curve at the point x. That means s = 2 arcsin(|x − y|/2). Since div k = δ(x − y) − 1/|S|, we obtain that ρ is a solution of an equation of the form ρ + q(s)ρ = −1/|S|,

s ∈ (0, π),

where the function q(s) = div(e) has two singularities: at the point s = 0 corresponding to y and at the point s = π corresponding to −y. Moreover the character of the singularities at both points y and −y is the same, the only difference is the sign of the leading term: q(s) ∼

d −2 , s

s → 0,

q(s) ∼

d −2 , s−π

s → π.

x−y Indeed, if x ∈ S is close to y, then e is close to the vector |x−y| . Therefore div(e) ∼ function ρ(s) must be smooth at the point s = π , therefore

1 ρ= |S|f (s)

π f (s) ds,

d−2 |x−y| ,

The

where f (s) = exp q(s) ds .

s

Let us clarify the situation with the point s = 0. Since div ρe = −1/|S| everywhere except the point y, we conclude automatically that the function div ρe has to have a singularity at y. The c as s → 0, with some constant c. In other only possible singularity is the one of the type ρ ∼ f (s) words, k(x, y) ∼ as x → y.

2

c |x − y|d−2


767

2. We see that an important role in the proof of Theorem 1.1 was played by one of the results of the paper [5]. It would be nice to understand which exactly result of the paper [5] is refered to. Suppose that we have an operator A+ = −

d2 + T (r), dr 2

where T (r) is a matrix-valued function. Certainly this operator has to act in the space of vectorvalued functions, for example in L2 (R+ , Cm ). Assume that T has a finite support. Then there exists a function f ∈ L2 (R+ , Cm ) such that the spectral measure corresponding to f satisfies b

∞ 1/2 λj (A+ ) + T (r)e, e dr , log μ (λ) dλ −C 1 +

a

0

where λj (A+ ) are negative eigenvalues of A+ and e is the vector with coordinates (1, 0, . . . , 0). Moreover, b

λj (A+ )1/2 + λj (A− )1/2 , log μ (λ) dλ −C 1 +

a 2

d where λj (A− ) are negative eigenvalues of A− = − dr 2 − T (r). Here a > 0 and b < ∞.

References [1] S.A. Denisov, Absolutely continuous spectrum of multidimensional Schrödinger operator, Int. Math. Res. Not. 74 (2004) 3963–3982. [2] J. Bourgain, On random Schrödinger operators on Z2 , Discrete Contin. Dyn. Syst. 8 (1) (2002) 1–15. [3] J. Bourgain, Random lattice Schrödinger operators with decaying potential: Some multidimensional phenomena, in: V.D. Milman, G. Schechtman (Eds.), Geometric Aspects of Functional Analysis, Israel Seminar 2001–2002, in: Lecture Notes in Math., vol. 1807, Springer, Berlin, 2003, pp. 70–98. [4] K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. 37 (1951) 760–766. [5] A. Laptev, S. Naboko, O. Safronov, Absolutely continuous spectrum of Schrödinger operators with slowly decaying and oscillating potentials, Comm. Math. Phys. 253 (3) (2005) 611–631. [6] E.H. Lieb, Bounds on the eigenvalues of the Laplace and Schrödinger operators, Bull. Amer. Math. Soc. 82 (1976) 751–753. See also: The number of bound states of one body Schrödinger operators and the Weyl problem, in: Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, RI, 1980, pp. 241–252. [7] E.H. Lieb, W. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, in: Studies in Math. Phys., Essays in Honor of Valentine Bargmann, Princeton, NJ, 1976, pp. 269–303.


The fixed point property via dual space properties P.N. Dowling ∗ , B. Randrianantoanina, B. Turett 1 Department of Mathematics and Statistics, Miami University, Oxford, OH 45056, USA Received 3 April 2008; accepted 24 April 2008 Available online 6 June 2008 Communicated by N. Kalton

Abstract A Banach space has the weak fixed point property if its dual space has a weak∗ sequentially compact unit ball and the dual space satisfies the weak∗ uniform Kadec–Klee property; and it has the fixed point property if there exists ε > 0 such that, for every infinite subset A of the unit sphere of the dual space, A ∪ (−A) fails to be (2 − ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property. © 2008 Elsevier Inc. All rights reserved. Keywords: Fixed point property; E-convexity

Determining conditions on a Banach space X so that every nonexpansive mapping from a nonempty, closed, bounded, convex subset of X into itself has a fixed point has been of considerable interest for many years. A Banach space has the fixed point property if, for each nonempty, closed, bounded, convex subset C of X, every nonexpansive mapping of C into itself has a fixed point. A Banach space is said to have the weak fixed point property if the class of sets C above is restricted to the set of weakly compact convex sets; and a Banach space is said to have the weak∗ fixed point property if X is a dual space and the class of sets C is restricted to the set of weak∗ compact convex subsets of X. A well-known open problem in Banach spaces is whether every reflexive Banach space has the fixed point property for nonexpansive mappings. The question of whether more restrictive * Corresponding author.

E-mail addresses: [email protected] (P.N. Dowling), [email protected] (B. Randrianantoanina), [email protected] (B. Turett). 1 Current address: Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.021

P.N. Dowling et al. / Journal of Functional Analysis 255 (2008) 768–775

769

classes of reflexive spaces, such as the class of superreflexive Banach spaces or Banach spaces isomorphic to the Hilbert space 2 , have the fixed point property has also long been open and has been investigated by many authors [8,14,15,17]. Recently, J. García-Falset, E. Llorens-Fuster, and E.M. Mazcuñan-Navarro [7] proved that uniformly nonsquare Banach spaces, a sub-class of the superreflexive spaces, have the fixed point property. In this article, it is shown that the larger class of E-convex Banach spaces have the fixed point property. The E-convex Banach spaces, introduced by S.V.R. Naidu and K.P.R. Sastry [18], are a class of Banach spaces lying strictly between the uniformly nonsquare Banach spaces and the superreflexive spaces (see also [1]). The second geometric property of Banach spaces that is considered in this article is the weak∗ uniform Kadec–Klee property in a dual Banach space. A dual space X ∗ has the weak∗ uniform Kadec–Klee property if, for every ε > 0, there exists δ > 0 such that, if (xn∗ ) is a sequence in def

the unit ball of X ∗ converging weak∗ to x ∗ and the separation constant sep(xn∗ ) = inf{xn∗ − ∗ : m = n} > ε, then x ∗ < 1 − δ. It is well known [6] that, if X ∗ has weak∗ uniform Kadec– xm Klee property, then X ∗ has the weak∗ fixed point property. If, in addition, the unit ball of X ∗ is weak∗ sequentially compact, more is true: Theorem 3 notes that, if X ∗ has weak∗ uniform Kadec–Klee property and the unit ball of X ∗ is weak∗ sequentially compact, then X has the weak fixed point property. As a consequence of Theorem 3, it is noted that several nonreflexive Banach spaces such as quotients of c0 and C(T )/A0 , the predual of H 1 , have the weak fixed point property. Since the proofs of the main theorems in this paper will require elements of the proof that uniformly nonsquare Banach spaces have the fixed point property, a complete proof of this known result is presented. The proof presented here is a distillation of the original proof and combines elements from [5, Theorem 2.2] and [7, Theorem 3.3]. Recall that a Banach space X is uniformly nonsquare [10] if there exists δ > 0 such that, if x and y are in the unit ball of X, then either (x + y)/2 < 1 − δ or (x − y)/2 < 1 − δ. The general set-up in proving that a Banach space has the weak fixed point property has, by now, become standard fare. If a Banach space X fails to have the weak fixed point property, there exists a nonempty, weakly compact, convex set C in X and a nonexpansive mapping T : C → C without a fixed point. Since C is weakly compact, it is possible by Zorn’s lemma to find a minimal T -invariant, weakly compact, convex subset K of C such that T has no fixed point in K. Since the diameter of K is positive (otherwise K would be a singleton and T would have a fixed point in K), it can be assumed that the diameter of K is 1. It is well known that there exists an approximate fixed point sequence (xn ) for T in K and, without loss of generality, we may assume that (xn ) converges weakly to 0. For details on this general set-up, see [8, Chapter 3]. Theorem 1. (See García-Falset, Llorens-Fuster, and E.M. Mazcuñan-Navarro [7].) Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. Proof. Assume that a Banach space X fails to have the fixed point property. Since uniformly nonsquare spaces are reflexive [10], the fixed point property and the weak fixed point property coincide for X. Therefore there exists a nonexpansive map T : K → K without a fixed point where K is a minimal T -invariant set in X with diameter 1. Let (xn ) be an approximate fixed point sequence for T in K and assume that (xn ) converges weakly to 0. Consider the space ∞ (X)/c0 (X) endowed with the quotient norm given by [wn ] = lim supn wn where [wn ] denotes the equivalence class of (wn ) ∈ ∞ (X). For a bounded set

770


C in X, the set [C] in ∞ (X)/c0 (X) is defined by [C] = {[wn ]: wn ∈ C for all n ∈ N}. Using the notation in [5, p. 840], let [W ] = [zn ] ∈ [K]: [zn ] − [xn ] 1/2 and lim sup lim supzm − zn 1/2 . n

m

It is easy to check that [W ] is a closed, bounded, convex, nonempty (since [ 12 xn ] is in [W ]) subset of [K], and is [T ]-invariant where [T ] : [K] → [K] is defined by [T ][zn ] = [T (zn )]. So, by a result of Lin [13], sup[zn ]∈[W ] [zn ] − x = 1 for each x ∈ K. In particular, with x = 0, sup[zn ]∈[W ] [zn ] = 1. Let ε > 0 and choose [zn ] ∈ [W ] with [zn ] > 1 − ε. Let (yj ) = (znj ) be a subsequence of (zn ) such that limyn = [zn ] and (yn ) converges weakly to an element y in K. There is no loss in generality in assuming that yn > 1 − ε for all n ∈ N and in choosing yn∗ ∈ X ∗ so that yn∗ = 1, yn∗ (yn ) = yn , and (yn∗ ) converges weak∗ to y ∗ . (This is possible because the fixed point property is separably-determined [8, p. 35]; so there is no loss in generality in assuming that BX∗ is weak∗ -sequentially compact.) From the definition of [W ] and the weak lower semicontinuity of the norm, it follows that, if n is large enough, yn − y lim infyn − ym < m

Therefore, with un =

2 1+ε (yn

1+ε 2

− y) and u =

1 and y lim infyj − xnj . j 2

2 1+ε y,

2 2 = 2 yn > 2 1 − ε > 2(1 − 2ε) (y y − y) + un + u = 1 + ε n 1+ε 1+ε 1+ε if n ∈ N is large enough. Applying the weak lower semicontinuity of the norm again, it follows that lim inf(un − um ) + u un + u > 2(1 − 2ε) m

if n ∈ N is large enough. So, by taking another subsequence if necessary, we can assume that un + u > 2(1 − 2ε) and (un − um ) + u > 2(1 − 2ε) for all n and all m > n. w∗ ∗ − Furthermore, since ym −→ y ∗ , lim inf(un − um ) − u = lim inf(um + u) − un m m ∗ lim inf ym (um + u) − un m ∗ (un ) = lim inf um + u − ym m

2(1 − 2ε) − y ∗ (un ). w → 0, it follows that lim infm (un − um ) − u > 2(1 − 3ε) if n is large enough. Then, since un − Therefore, for n large enough and m > n also large enough, both (un − um ) + u > 2(1 − 3ε) and (un − um ) − u > 2(1 − 3ε)


771

hold. Since ε > 0 is arbitrary, un − um < 1, and u < 1, the above inequalities imply that X fails to be uniformly nonsquare, a contradiction which finishes the proof. 2 We want to refine the sequences (xnj ), (yj ), and (yj∗ ) that appear in the proof of Theorem 1. Recall the result of Goebel and Karlovitz [8, p. 124]: If K is a minimal T -invariant, weakly compact, convex set for the nonexpansive map T and (xn ) is an approximate fixed point sequence for T in K, then the sequence (xn − x) converges to the diameter of K for every x in K. Fix ε > 0 and set x˜1 = xn1 , y˜1 = y1 , and y˜1∗ = y1∗ . Then, by the Goebel–Karlovitz Lemma, there exists j1 > 1 such that min{x˜1 − xnj1 , y˜1 − xnj1 } > 1 − ε. Set x˜2 = xnj1 , y˜2 = yj1 , and y˜2∗ = yj∗1 . Another application of the Goebel–Karlovitz Lemma yields j2 > j1 such that mini=1,2 {x˜i − xnj2 , y˜i − xnj2 } > 1 − ε. Set x˜3 = xnj2 , y˜3 = yj2 , and y˜3∗ = yj∗2 . Continuing in this manner, we obtain sequences (xñ ) and (yñ ) in K (and BX ) and a sequence (yñ∗ ) in SX∗ satisfying mini 1 − ε for all n ∈ N and yñ∗ (yñ ) = yñ > 1 − ε for all n ∈ N. In the following, these sequences are renamed by omitting the tildes. The following result is a summary of several easy computations. Lemma 2. Let X be a Banach space whose dual unit ball is weak∗ sequentially compact and assume that X fails the weak fixed point property. Given ε > 0, there exist sequences (yn ) in BX and (yn∗ ) in SX∗ and elements y ∈ BX and y ∗ ∈ BX∗ satisfying: (1) (2) (3) (4) (5) (6) (7)

∗

w w → y and yn∗ − −→ y ∗ ; yn − for every n ∈ N, 1 − ε < yn = yn∗ (yn ) 1; 1+ε ∗ for every n ∈ N, 1−3ε 2 < yn (y) y < 2 ; 1−3ε 1+ε 2 < yn − y < 2 ; 1+ε if n = m, then 1−3ε 2 < yn − ym < 2 ; 1−3ε 1+2ε ∗ if n = m, then 2 < yn (ym ) < 2 ; 1−3ε 1+ε ∗ 2 y (y) 2 .

Proof. Claims (1) and (2), the third inequality in (3), and the second inequalities in (4) and (5) are immediate from the proof of Theorem 1. Then y yn∗ (y) = yn∗ (yn ) − yn∗ (yn − y) > (1 − ε) −

1 + ε 1 − 3ε = 2 2

proving (3). Also yn − y yn∗ (yn − y) = yn − yn∗ (y) > (1 − ε) −

1 + ε 1 − 3ε = 2 2

which finishes the proof of (4). From our refinement of (xn ) and (yn ) done just prior to the lemma and the definition of [W ] in the proof of Theorem 1, if n > m, yn − ym = (yn − xn ) + (xn − ym ) xn − ym − yn − xn

772


> (1 − ε) − =

1+ε 2

1 − 3ε 2

showing that (5) holds. The lower inequality in (6) follows from (2) and (5): If n = m, 1 − ε < yn∗ (yn ) = yn∗ (yn − ym ) + yn∗ (ym ) yn − ym + yn∗ (ym )
m, first combine (3) with the weak∗ convergence of (yn∗ ) to y ∗ to obtain (7). Then, since (yn ) converges weakly to y, there is no loss in generality 1+2ε ∗ ∗ in assuming that y ∗ (yn ) < 1+2ε 2 for all n ∈ N. In particular, y (y1 ) < 2 . Therefore, since (yn ) converges weak∗ to y ∗ , there exists n1 ∈ N such that, if n n1 , yn∗ (y1 ) < 1+2ε 2 . Setting y˜1 = y1 , ∗ (y˜ ) < 1+2ε and (y ∗ ) y˜1∗ = y1∗ , y˜2 = yn1 , and y˜2∗ = yn∗1 gives y˜2∗ (y˜1 ) < 1+2ε . Then, since y 2 n 2 2 . Set converges weak∗ to y ∗ , there exists n2 ∈ N such that n2 > n1 , and, if n n2 , yn∗ (y˜2 ) < 1+2ε 2 y˜3 = yn2 and y˜3∗ = yn∗2 . Continuing in this manner generates sequences (yñ ) and (yñ∗ ) satisfying all of the conditions of the lemma. 2

As a consequence of these computations, we have the following Theorem 3. Let X be a Banach space such that BX∗ is weak∗ sequentially compact. If X ∗ has the weak∗ uniform Kadec–Klee property, then X has the weak fixed point property for nonexpansive mappings. Proof. If X fails to have the weak fixed point property, consider the sequences (yn ) and (yn∗ ) determined in Lemma 2. In particular, note that yn∗ = 1 for all n ∈ N and that (yn∗ ) converges weak∗ to y ∗ . Note also that, if n = m, ∗ ∗ ∗ ∗ yn − ym (yn − ym ) = yn∗ (yn ) − yn∗ (ym ) − ym (yn ) + ym (ym ) > (1 − ε) − = 1 − 4ε.

1 + 2ε 1 + 2ε − + (1 − ε) 2 2


773

It follows that ∗ yn − ym 1 − 4ε 1 − 4ε ∗ ∗ yn − ym > =2 > 2 − 10ε. 2 yn∗ − ym yn − ym (1 + ε)/2 1+ε 1 Thus, if ε < 10 , (yn∗ ) is a sequence in the unit sphere of X ∗ , (yn∗ ) converges weak∗ to y ∗ , and ∗ sep{yn } > 1. Therefore, by the weak∗ uniform Kadec–Klee property of X ∗ , there exists δ > 0 such that

y ∗ < 1 − δ.

(∗)

But, by (3) and (7), 1+ε ∗ 1 − 3ε y ∗ (y) yy ∗ < y . 2 2 1 δ Therefore, if ε < min{ 10 , 4}

y ∗ >

1 − 3ε > 1 − 4ε > 1 − δ, 1+ε

a contradiction to (∗). Therefore X has the weak fixed point property for nonexpansive mappings. 2 Of course the theorem implies that c0 with its usual norm has the weak fixed point property which was a result first proven by Maurey [16]. Since H 1 has the weak∗ uniform Kadec–Klee property [2], its predual C(T )/A0 has the weak fixed point property by this theorem. In the same manner, since C1 (H ), the ideal of trace class operators on a Hilbert space H , has the weak∗ uniform Kadec–Klee property [12], its predual C∞ (H ), the ideal of compact operators in B(H ), has the weak fixed point property. Since quotients of Banach spaces with weak∗ sequentially compact dual unit balls have weak∗ sequentially compact dual unit balls [4, p. 227], it is easy to check that the following holds. Corollary 4. Let X be a Banach space such that BX∗ is weak∗ sequentially compact. If X ∗ has the weak∗ uniform Kadec–Klee property and Y is a closed subspace of X, then X/Y has the weak fixed point property for nonexpansive mappings. We note that the corollary implies that the quotients of c0 have the weak fixed point property. This is implicit in the work of Borwein and Sims [3]. The authors had hoped that the sequences identified in Lemma 2 would be useful in establishing connections between superreflexive Banach spaces and the fixed point property for nonexpansive mappings. Consider the sequences generated in Lemma 2 for each ε = k1 , k ∈ N. ∗ ) denote the sequences (y ) and (y ∗ ) constructed in That is, for each k ∈ N, let (yk,n ) and (yk,n n n ∗ Lemma 2 with ε = k1 ; let yk,∞ denote the weak limit of the sequence (yk,n ); and let yk,∞ denote ∗ ∗ the weak limit of the sequence (yk,n ). For a non-trivial ultrafilter U on N, let XU denote the ultrapower of X with respect to U . (For information on ultraproducts in Banach space theory, see [9] or [20].) Define sequences (yn ) in XU , (y∗n ) in (X ∗ )U , and the point y in XU by

774


yn = (y1,n , y2,n , y3,n , . . .)U , ∗ ∗ ∗ , y2,n , y3,n , . . . U , and y∗n = y1,n y = (y1,∞ , y2,∞ , y3,∞ , . . .)U . elements The pair of sequences (2(yn − y)) and (y∗n ) forms a biorthogonal system of norm-one ∗ ) and, for each sequence (α ) of nonnegative real numbers, ∞ α y∗ = and (X in X n U n=1 n n ∞ U ∗ ∗ n=1 αn . Moreover, as is clear from the proof of Theorem 3, ym − yn = 2 for all m = n. Initially, the authors felt that, if X was a renorming of 2 , this “positive 1 -type behavior” should not occur in (X ∗ )U since (X ∗ )U would also be a renorming of a Hilbert space. However, as first pointed out to us by Professor V.D. Milman, there do exist renormings of 2 with this behavior. A second example resulted from a discussion with Professors A. Pełczyński and M. Wojciechowski. In fact, every infinite-dimensional Banach space can be renormed to exhibit this 1 -type behavior for nonnegative linear combinations. To see this, let (xi , xi∗ ) in X × X ∗ be a biorthogonal system with xi = 1 and xi∗ 2. (Such a biorthogonal system exists by applying a theorem of Ovsepian and Pełczyński [4, p. 56] to a separable subspace of X and then extending to functionals on all of X via the Hahn–Banach theorem.) Then |||x||| = max{|x1∗ (x)|, 12 x, supi=j ; i,j 2 (|xi∗ (x)| + |xj∗ (x)|)} defines an equivalent norm on X ∞ with |||x1 + xn ||| = 1 and ||| ∞ n=1 αn (x1 + xn )||| = n=1 αn if αn 0. (For related examples, see Example 3.13 in [18] and Theorem 7 in [11].) Despite the above disappointment, the sequence (y∗n ) in (XU )∗ or the sequences (yn∗ ) in X ∗ for a given ε in Lemma 2 can be used to generalize Theorem 1. A subset A of X is symmetrically ε-separated if the distance between any two distinct points of A ∪ (−A) is at least ε and a Banach space X is O-convex if the unit ball BX contains no symmetrically (2 − ε)-separated subset of cardinality n for some ε > 0 and some n ∈ N [18]. O-convex spaces are superreflexive. Therefore the proof of Theorem 3 combines with property (3) in Lemma 2 to show that, if X fails to have the fixed point property, then, for every ε > 0, there exists a countably infinite set A = {y1∗ , y2∗ , . . .} in the unit sphere of X ∗ such that A ∪ (−A) is (2 − ε)-separated. In particular, this implies: Theorem 5. If X ∗ is O-convex, then the Banach space X has the fixed point property for nonexpansive mappings. Since uniformly nonsquare Banach spaces are O-convex, Theorem 5 is a generalization of Theorem 1. Naidu and Sastry [18] also characterized the dual property to O-convexity. For ε > 0, a convex subset A of BX is an ε-flat if A ∩ (1 − ε)BX = ∅. Note that the convex hulls of the sets {y1∗ , y2∗ , . . .} from Lemma 2 are 3ε-flats. A collection D of ε-flats is jointly complemented if, for each distinct ε-flats A and B in D, the sets A ∩ B and A ∩ (−B) are nonempty. Define E(n, X) = inf{ε: BX contains a jointly complemented collection of ε-flats of cardinality n}. A Banach space X is E-convex if E(n, X) > 0 for some n ∈ N. In [19], S. Saejung noted that E-convex spaces may fail to have normal structure and asked if E convex spaces have the fixed point property. Since a Banach space is E-convex if and only if its dual space is O-convex, Theorem 5 can be restated to give a positive answer to Saejung’s question. Theorem 6. E-convex spaces have the fixed point property for nonexpansive mappings.


775

For a detailed analysis of O-convex, E-convex, and related properties in the hierarchy between Hilbert spaces and reflexive spaces, see [1,18,19]. Note added in proof The authors thank Jesús García-Falset for pointing out that X ∗ having the weak∗ uniform Kadec–Klee property implies that R(X) < 2 which, by Theorem 3 in [J. García-Falset, The fixed point property in Banach spaces with the NUS-property, J. Math. Anal. Appl. 215 (2) (1997) 532–542], shows that X has the weak fixed point property. Thus Theorem 3 in this article is a special case of Theorem 3 in the article of García-Falset. Acknowledgments The authors thank V.D. Milman, A. Pełczyński, N. Randrianantoanina, and M. Wojciechowski for helpful discussions during the preparation of this article. We also thank the referee for several helpful comments. References [1] D. Amir, C. Franchetti, The radius ratio and convexity properties in normed linear spaces, Trans. Amer. Math. Soc. 282 (1984) 275–291. [2] M. Besbes, S.J. Dilworth, P.N. Dowling, C.J. Lennard, New convexity and fixed point properties in Hardy and Lebesgue–Bochner spaces, J. Funct. Anal. 119 (1994) 340–357. [3] Jon M. Borwein, Brailey Sims, Nonexpansive mappings on Banach lattices and related topics, Houston J. Math. 10 (1984) 339–356. [4] J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York, 1984. [5] T. Domínguez-Benavides, A geometrical coefficient implying the fixed point property and stability results, Houston J. Math. 22 (1996) 835–849. [6] D. van Dulst, B. Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), in: Lecture Notes in Math., vol. 991, 1983, pp. 35–43. [7] J. García-Falset, E. Llorens-Fuster, E.M. Mazcuñan-Navarro, Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings, J. Funct. Anal. 233 (2006) 494–514. [8] Kazimierz Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990. [9] S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1981) 221–234. [10] R.C. James, Uniformly nonsquare Banach spaces, Ann. of Math. 80 (1964) 542–550. [11] Clifford A. Kottman, Subsets of the unit ball that are separated by more than one, Studia Math. 53 (1975) 15–27. [12] Chris Lennard, C1 is uniformly Kadec–Klee, Proc. Amer. Math. Soc. 109 (1990) 71–77. [13] P.-K. Lin, Unconditional bases and fixed points of nonexpansive mappings, Pacific J. Math. 116 (1985) 69–76. [14] P.-K. Lin, Stability of the fixed point property of Hilbert spaces, Proc. Amer. Math. Soc. 127 (1999) 3573–3581. [15] E. Llorens-Fuster, The fixed point property for renormings of 2 , in: Seminar of Mathematical Analysis, Univ. Sevilla Secr. Publ., Seville, 2006, pp. 121–159. [16] B. Maurey, Points fixes des contractions de certains faiblement compacts de L1 , Seminar on Functional Analysis, 1980–1981, École Polytech., Palaiseau, Exp. No. VIII, 1981, 19 pp. [17] E.M. Mazcuñan-Navarro, Stability of the fixed point property in Hilbert spaces, Proc. Amer. Math. Soc. 134 (2005) 129–138. [18] S.V.R. Naidu, K.P.R. Sastry, Convexity conditions in normed linear spaces, J. Reine Angew. Math. 297 (1976) 35–53. [19] Satit Saejung, Convexity conditions and normal structure of Banach spaces, J. Math. Anal. Appl., in press. [20] Brailey Sims, “Ultra”-Techniques in Banach Spaces, Queen’s Papers in Pure and Appl. Math., vol. 60, Queen’s Univ., Kingston, ON, 1982.


Reduced Weyl asymptotics for pseudodifferential operators on bounded domains I. The finite group case Pablo Ramacher 1 Georg-August-Universität Göttingen, Institut für Mathematik, Bunsenstr. 3-5, 37073 Göttingen, Germany Received 4 October 2007; accepted 25 February 2008 Available online 9 April 2008 Communicated by Paul Malliavin

Abstract Let G ⊂ O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn , and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A0 in L2 (Rn ) with G-invariant Weyl symbol, and assume that it is semi-bounded from below. We show that the spectrum of the Friedrichs extension A of the operator res ◦ 2 A0 ◦ ext : C∞ c (X) → L (X) is discrete, and derive asymptotics for the number Nχ (λ) of eigenvalues of A less or equal λ and with eigenfunctions in the χ -isotypic component of L2 (X) as λ → ∞, giving also an estimate for the remainder term in case that G is a finite group. In particular, we show that the multiplicity of each unitary irreducible representation in L2 (X) is asymptotically proportional to its dimension. © 2008 Elsevier Inc. All rights reserved. Keywords: Pseudodifferential operators; Asymptotic distribution of eigenvalues; Multiplicities of representations of finite groups; Peter–Weyl decomposition

Contents 1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Weyl calculus for pseudodifferential operators in Rn . . . . . . . . . . . . . . . . Reduced spectral asymptotics and the approximate spectral projection operators Estimates from below for the reduced spectral counting function . . . . . . . . . . .

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E-mail address: [email protected]. 1 The author was supported by the grant RA 1370/1-1 of the German Research Foundation (DFG) during the

preparation of this work. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.02.012

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5. 6. 7.

P. Ramacher / Journal of Functional Analysis 255 (2008) 777–818

Estimates from above for the reduced spectral counting function . . . . . . . . . . . . . . . . . . . . . . 802 The finite group case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814

1. Introduction Let G ⊂ O(n) be a compact group of isometries acting on Euclidean space Rn , and let X be a bounded domain in Rn which is transformed into itself under the action of G. Consider the regular representation of G T(g)φ(x) = φ g −1 x in the Hilbert spaces L2 (Rn ) and L2 (X) of square integrable functions by unitary operators. As a consequence of the Peter–Weyl theorem, the representation T decomposes into isotypic components according to Hχ , L2 Rn =

L2 (X) =

ˆ χ∈G

res Hχ ,

ˆ χ∈G

ˆ denotes the set of irreducible characters of G, and res : L2 (Rn ) → L2 (X) is the natural where G 2 n restriction operator. Similarly, ext : C∞ c (X) → L (R ) will denote the natural extension operator. Let A0 be a symmetric, classical pseudodifferential operator in L2 (Rn ) of order 2m with Ginvariant Weyl symbol a and principal symbol a2m , and assume that (A0 u, u) cu2m for some s n c > 0 and all u ∈ C∞ c (X), where · s is a norm in the Sobolev space H (R ). Consider further the Friedrichs extension of the lower semi-bounded operator 2 res ◦A0 ◦ ext : C∞ c (X) −→ L (X),

which is a self-adjoint operator in L2 (X), and denote it by A. Finally, let ∂X be the boundary of X, which is not assumed to be smooth, and assume that for some sufficiently small ρ > 0, vol(∂X)ρ Cρ, where (∂X)ρ = {x ∈ Rn : dist(x, ∂X) < ρ}. Since A commutes with the action of G due to the invariance of a, the eigenspaces of A are unitary G-modules that decompose into irreducible subspaces. In 1972, Arnol’d [1] conjectured that by studying the asymptotic behavior of the spectral counting function Nχ (λ) = dχ

μχ (t),

tλ

where μχ (λ) is the multiplicity of any irreducible representation of dimension dχ corresponding to the character χ in the eigenspace of A with eigenvalue λ, one should be able to show that the multiplicity of each unitary irreducible representation in the above decomposition of L2 (X) is asymptotically proportional to its dimension as λ → +∞. The aim of this work is to show that this is indeed the case, giving a precise description of the leading term of Nχ (λ), together with an estimate for the remainder.


779

The asymptotic distribution of eigenvalues was first studied by Weyl [18] for certain second order differential operators in Rn using variational techniques. Another approach, which also gives an asymptotic description for the eigenfunctions, was introduced by Carleman [4]. His idea was to study the kernel of the resolvent, combined with a Tauberian argument. Minakshisundaram and Pleijel [15] showed that one can study the Laplace transform of the spectral function as well, and extended the results of Weyl to closed manifolds, and Gårding [7] generalized Carleman’s approach to higher order elliptic operators on bounded sets in Rn . Hörmander [11] then extended these results to elliptic differential operators on closed manifolds using the theory of Fourier integral operators. Further developments in this direction were given by Duistermaat and Guillemin, Helffer and Robert, and Ivrii. The first ones to study Weyl asymptotics for elliptic operators on closed Riemannian manifolds in the presence of a compact group of isometries in a systematic way were Donnelly [5] together with Brüning and Heintze [3], giving first-order Weyl asymptotics for the spectral distribution function for each of the isotypic components, together with an estimate for the remainder in some special cases. Later, Guillemin and Uribe [8] described the relation between the spectrum of the considered operators, and the reduction of the corresponding bicharacteristic flow, and Helffer and Robert [9,10] studied the situation in Rn . Our approach is based on the Weyl calculus of pseudodifferential operators developed by Hörmander [12], and the method of approximate spectral projections, first introduced by Tulovsky and Shubin [17]. This method is somehow more closely related to the original work of Weyl, and starts from the observation that the asymptotic distribution function N (λ) for the eigenvalues of an elliptic, self-adjoint operator is given by the trace of the orthogonal projection on the space spanned by the eigenvectors corresponding to eigenvalues λ. By introducing suitable approximations to these spectral projections in terms of pseudodifferential operators, one can then derive asymptotics for N (λ), and also obtain estimates for the remainder term. Nevertheless, due to the presence of the boundary, the original method of Shubin and Tulovsky cannot be applied to our situation, and one is forced to use more elaborate techniques, which were subsequently developed by Feigin [6] and Levendorskii [14]. Recently, Bronstein and Ivrii have obtained even sharp estimates for the remainder term in the case of differential operators on manifolds with boundaries satisfying the conditions specified above [2,13]. This work is structured as follows. Part I provides the foundations of the calculus of approximate spectral projection operators, and addresses the case where G is a finite group of isometries. The case of a compact group of isometries will be the subject of Part II. The main result of Part I is Theorem 8. It states that if G is a finite group, the spectrum of A is discrete, and that, as λ → +∞, the spectral counting function Nχ (λ) is given by

Nχ (λ) = dχ

μχ (t) =

tλ

−1 vol(a2m ((−∞, 1])) n/2m λ + O λ(n−)/2m (2π)n |G|

dχ2

for arbitrary ∈ (0, 12 ), where |G| denotes the cardinality of G, and dχ the dimension of any irreducible representation of G corresponding to the character χ . Consequently, the multiplicity in L2 (X) of any irreducible representation corresponding to the character χ is given asymptotically by

−1 ((−∞,1])) n/2m dχ vol(a2m λ . (2π)n |G|

780


2. The Weyl calculus for pseudodifferential operators in Rn We begin by reviewing some basic facts in the Weyl calculus for pseudodifferential operators that will be needed in the sequel. We study then the pullback of symbols, and the composition of pseudodifferential operators with linear transformations. In order to introduce the relevant symbol classes, let g be a slowly varying Riemannian metric in Rl , regarded as a positive definite quadratic form, and assume that m is a positive, g-continuous function on Rl as defined in [12, Definitions 2.1 and 2.2]. Definition 1. The class of symbols S(g, m) is defined as the set of all functions u ∈ C∞ (Rl ) such that, for every integer k 0, k

(k) 1/2 gx (tj ) m(x) < ∞. νk (g, m; u) = sup sup u (x; t1 , . . . , tk ) x∈Rl tj ∈Rl

j =1

Here u(k) stands for the kth differential of u. Note that with the topology defined by the above semi-norms, S(g, m) becomes a Fréchet space. Consider now Rl = Rn ⊕ Rn , regarded as a symplectic space with the symplectic form σ (x, ξ ; y, η) = ξ, y − x, η , where ·,· denotes the usual Euclidean product of two vectors. Thus, σ = dξj ∧ dxj . Assume that g is σ -temperate, and that m is σ, g-temperate (see [12, Definition 4.1]). If a ∈ S(g, m) is interpreted as a Weyl symbol, the corresponding pseudodifferential operator is given by

x +y , ξ u(y) dy dξ, ei(x−y)ξ a Opw (a)u(x) = ¯ 2 where dξ ¯ = (2π)−n dξ . Here and it what follows, it will be understood that each integral is to be performed over Rn , unless otherwise specified. According to [12, Theorem 5.2], Opw (a) defines a continuous linear map from S(Rn ) to S(Rn ), and from S (Rn ) to S (Rn ), and the corresponding class of operators will be denoted by L(g, m). Moreover, one has the following result concerning the L2 -continuity of pseudodifferential operators. Theorem 1. Let g be a σ -temperate metric in Rn ⊕ Rn , g σ the dual metric to g with respect to σ , and g g σ . Let a ∈ S(g, m), and assume that m is σ , g-temperate. Then Opw (a) : L2 (Rn ) → L2 (Rn ) is a continuous operator if, and only if, m is bounded. Proof. See [12, Theorem 5.3].

2

The composition of pseudodifferential operators is described by the main theorem of Weyl calculus. Theorem 2. Let g be a σ -temperate metric in Rn ⊕ Rn , and g g σ . Assume that a1 ∈ S(g, m1 ), a2 ∈ S(g, m2 ), where m1 , m2 are σ, g-temperate functions. Then the composition of Opw (a1 ) with Opw (a2 ) in each of the spaces S(Rn ) or S (Rn ) is a pseudodifferential operator with Weyl symbol σ w (Opw (a1 ) Opw (a2 )) in the class S(g, m1 m2 ). Moreover,


781

j 1 w w iσ (Dx , Dξ ; Dy , Dη ) a1 (x, ξ )a2 (y, η)|(y,η)=(x,ξ ) /j ! σ Op (a1 ) Op (a2 ) (x, ξ ) − 2 w

j 1. In this case, one writes a ∼

aj .

We will further write S −∞ (g, m) =

∞ S g, hN σm , N =1

and denote the corresponding operator class by L−∞ (g, m). We introduce now certain hypoelliptic symbols which will be needed in the sequel. They were introduced by Levendorskii in [14]. Definition 2. The class of symbols SI(g, m) consists of all a ∈ S(g, m) that can be represented in the form a = a1 + a2 , where cm < |a1 | and a2 ∈ S(g, hσ m) for some constants c, > 0. The corresponding class of operators is denoted by LI(g, m). If instead of cm < |a1 | one has cm < a1 , one writes a ∈ SI + (g, m) and LI + (g, m), respectively. For a proof of the following lemmas, we refer the reader to [14, Lemmas 5.5, 8.1, and 8.2]. Lemma 1. Let a ∈ SI(g, m). Then there exists a symbol b ∈ SI(g, m−1 ) such that Opw (a) Opw (b) − 1 ∈ L−∞ (g, 1),

Opw (b) Opw (a) − 1 ∈ L−∞ (g, 1).

The operator Opw (b) is called a parametrix for Opw (a).

782


Lemma 2. If a ∈ SI + (g, m), then there exists a symbol b ∈ S(g, m1/2 ) such that Opw (a) − Opw (b)∗ Opw (b) ∈ L−∞ (g, m), where Opw (b)∗ is the adjoint of Opw (b). Lemma 3. Let > 0, and at ∈ S(g, hσ ), t ∈ R, be a family of symbols depending on a parameter. Furthermore, assume that the corresponding seminorms νk (g, hσ ; at ) are bounded by some constants independent of t, and let c > 0 be arbitrary. Then there exists a subspace L ⊂ L2 (Rn ) of finite codimension such that w Op (at )u

L2

cuL2

for all u ∈ L and all t ∈ R.

Remark 1. Lemma 3 is a consequence of the fact that, for a ∈ S(g, 1), one has the uniform bound w Op (a)

L2

C max νk (g, 1; a), kN

where C > 0 and N ∈ N depend only on the constants characterizing g, but not on a (see the proof of the sufficiency in [12, Theorem 5.3], and [14, Theorem 4.2]). In general, the pullback of symbols under C∞ mappings is described by the following lemma.

Lemma 4. Let g1 , g2 be slowly varying metrics on Rl , respectively Rl , and χ ∈ C∞ (Rl , Rl ). Then χ ∗ S(g2 , 1) ⊂ S(g1 , 1) if, and only if, for every k > 0, k g2χ(x) χ (k) (x; t1 , . . . , tk ) Ck g1x (tj ),

x, t1 , . . . , tk ∈ Rl .

j =1

In particular, if m is g2 -continuous, then χ ∗ m is g1 -continuous and χ ∗ S(g2 , m) ⊂ S(g1 , χ ∗ m). Proof. See [12, Lemma 8.1].

2

In all our applications, we will be dealing mainly with metrics g on R2n of the form δ −ρ 2 gx,ξ (y, η) = 1 + |x|2 + |ξ |2 |y|2 + 1 + |x|2 + |ξ |2 |η| ,

(3)

where 1 ρ > δ 0. The conditions of Theorem 2 are satisfied then, and h2σ (x, ξ ) = (1 + |x|2 + |ξ |2 )δ−ρ by (2). For the rest of this section, assume that g is of the form (3), and put h(x, ξ ) = (1 + |x|2 + |ξ |2 )−1/2 . In this case, the space of symbols S(g, m) can also be characterized as follows.


783

Definition 3. Let g be of the form (3), and m a g-continuous function. The class Γρ,δ (m, R2n ), 0 δ < ρ 1, consists of all functions u ∈ C∞ (R2n ) which for all multiindices α, β satisfy the estimates α β ∂ ∂ u(x, ξ ) Cαβ m(x, ξ ) 1 + |x|2 + |ξ |2 (−ρ|α|+δ|β|)/2 . ξ x

l (R2n ) for Γ (h−l , R2n ), where l ∈ R. In particular, we will write Γρ,δ ρ,δ

An easy computation then shows that S(g, m) = Γρ,δ (m, R2n ). For future reference, note β that u ∈ S(g, m) implies ∂ξα ∂x u ∈ S(g, mhρ|α|−δ|β| ). The pullback of symbols for metrics of the form (3) can now be described as follows. Lemma 5. Let δ +ρ 1, and g be a metric of the form (3). Assume that χ(x, ξ ) = (y(x), η(x, ξ )) is a diffeomorphism in R2n such that η is linear in ξ , and the derivatives of y and η are bounded for |ξ | < 1. Furthermore, let 1 gx,ξ (t) gχ(x,ξ ) (t) Cgx,ξ (t), C

1 m(x, ξ ) χ ∗ m(x, ξ ) Cm(x, ξ ), C

where m is a g-continuous function, and C > 0 is a suitable constant. Then χ ∗ S(g, m) ⊂ S(g, χ ∗ m). Proof. Instead of verifying the necessary and sufficient condition in Lemma 4, we will prove the statement directly. Let b ∈ S(g, m), and let s, t, . . . be k vectors in R2n . The kth differential (b ◦ χ)(k) (x, ξ ; s, t, . . .) = t, D s, D . . . (b ◦ χ)(x, ξ ) is given by a sum of terms of the form si tj . . . ∂ α (b ◦ χ)(x, ξ ), where we can assume that all the coefficients si , tj , . . . are different from zero; in particular, (b ◦ χ)(1) (x, ξ ) = b(1) (χ(x, ξ ))χ (1) (x, ξ ), where

χ

(1)

(x, ξ ) =

y (1) (x) A(x, ξ )

0 , B(x)

A being linear in ξ . The derivatives ∂ α (b ◦ χ)(x, ξ ) are sums of expressions of the form β ∂ b χ(x, ξ ) ∂ γ1 χi1 (x, ξ ) . . . ∂ γl χil (x, ξ ), where γ1 + · · · + γl = α and l = |β|. Since additional powers of ξ only appear in companion with additional derivatives of b with respect to η that originate from derivatives of b ◦ χ with respect to x, each of the terms of (b ◦ χ)(k) (x, ξ ; s, t, . . .) can be estimated from above by some constant times an expression of the form si tj . . . ∂ β ∂ β b χ(x, ξ ) P d (x, ξ ), y

η

784


where P d (x, ξ ) is a homogeneous polynomial in ξ of degree d which is bounded for |ξ | < 1, and d = |β | − N = |β | − k + N ; here N = |α | and N = |α | denote the number of x- and ξ -components in the product si tj , . . . , respectively. Indeed, if we differentiate in ∂ α (b ◦ χ)(x, ξ ) first with respect to ξ we get n

∂ξα (b ◦ χ)(x, ξ ) =

ηj1 ,...,ηj

where ∂ξα = ∂ξi1 . . . ∂ξi that

N

|β |.

|α |

|α |

=1

(∂ηj1 . . . ∂ηj

|α |

∂ηj|α | ∂ηj1 b) χ(x, ξ ) (x) . . . (x), ∂ξi1 ∂ξi|α |

, and differentiating now with respect to x yields the assertion. Note

In order to prove the assertion of the lemma, we have to show that β β

sup sup x,ξ s,t,...

|si tj . . . (∂y ∂η b)(χ(x, ξ ))P d (x, ξ )| 1/2

1/2

gx,ξ (s)gx,ξ (t) . . . m(x, ξ )

< ∞,

(4)

where it suffices to consider only those s, t, . . . whose only non-zero components are si , tj , . . . . Since N d, there are d vectors p, q, . . . among the vectors s, t, . . . contributing with xcomponents to the product si tj pk ql . . . . Furthermore, let w, z, . . . be d vectors such that wn+k = pk , zn+l = ql , . . . , their other components being zero. We then obtain the estimate β β

1/2

|si tj pk ql . . . (∂x ∂ξ b)(χ(x, ξ ))| 1/2

1/2

1/2

1/2

m(χ(x, ξ ))gχ(x,ξ ) (s)gχ(x,ξ ) (t) . . . gχ(x,ξ ) (w)gχ(x,ξ ) (z) . . . d(1−δ−ρ)/2 C 1 + |x|2 + |ξ |2

·

1/2

|P d (ξ )gx,ξ (w)gx,ξ (z) . . . | 1/2

1/2

gx,ξ (p)gx,ξ (q) . . .

for all x, ξ, s, t, . . . . Indeed, −ρ/2 1/2 gx,ξ (w) = |pk | 1 + |x|2 + |ξ |2 ,

δ/2 1/2 gx,ξ (p) = |pk | 1 + |x|2 + |ξ |2 ,

....

On the other hand, besides the d vectors p, q, . . . there are still N − d = k − |β | |β | vectors among the remaining vectors s, t, . . . contributing with x-components to the product si tj . . . . 1/2 Since the corresponding quotients |rl |/gχ(x,ξ ) (r) can be estimated from above by some constant, we can assume that there are precisely |β | of them. Also note that there are exactly d + N = |β | vectors among the vectors s, t, . . . , w, z, . . . contributing with ξ -components to si tj . . . . We can therefore assume that the components of s, t, . . . , w, z, . . . are prescribed by the multiindex β = (β , β ) in such a way that β si tj pk ql . . . ∂xβ ∂ξ b χ(x, ξ ) = b(|β|) χ(x, ξ ); s, t, . . . , w, z, . . . . The desired estimate (4) now follows by using the assumptions that b ∈ S(g, m) and δ + ρ 1. 2


785

If a ∈ S(g, m) is regarded as a right, respectively left, symbol, the corresponding pseudodifferential operators are given by Op (a)u(x) = l

ei(x−y)ξ a(x, ξ )u(y) dy dξ, ¯

Op (a)u(x) = r

ei(x−y)ξ a(y, ξ )u(y) dy dξ, ¯

where g is assumed to be of the form (3). By [12, Theorem 4.5], the three sets of operators Opw (a), Opl (a), and Opr (a) coincide. Theorem 2 can then also be formulated in terms of left and right symbols. In what follows, we would like to treat left, right, and Weyl symbols on the same grounding by introducing the notion of the τ -symbol. To do so, we introduce yet another l (R2n ), compare [16, Chapter 4]. class of amplitudes which is closely related to the space Γρ,δ l (R3n ) consists of all functions u ∈ C∞ (R3n ) which for a suitable Definition 4. The class Πρ,δ l ∈ R satisfy the estimates

α β γ ∂ ∂ ∂y u(x, y, ξ ) Cαβγ 1 + |x|2 + |y|2 + |ξ |2 (l−ρ|α|+δ|β+γ |)/2 ξ x (l +ρ|α|+δ|β+γ |)/2 × 1 + |x − y|2 . l (R3n ) and Γ l (R2n ) is described by the following The relationship between the spaces Πρ,δ ρ,δ lemma.

Lemma 6. Let 0 δ < ρ 1, and p : R2n → Rn be a linear map such that (x, y) → l (R2n ), and define (p(x, y), x − y) is an isomorphism. Let a(w, η) ∈ Γρ,δ b(x, y, ξ ) = a p(x, y), ψ(x, y)ξ , where ψ : Ξ → GL(n, R) is a C∞ mapping on some open subset Ξ ⊂ R2n , having bounded l (Ξ × Rn ). derivatives. If δ + ρ 1, then b ∈ Πρ,δ β γ

Proof. We will proof the assertion by induction on |α + β + γ |. First note that ∂ξα ∂x ∂y b(x, y, ξ ) is given by a sum of terms of the form α β ∂η ∂w a p(x, y), ψ(x, y)ξ P d (x, y, ξ ),

(5)

where P d (x, y, ξ ) is a polynomial in ξ of degree d. Each of these summands can be estimated from above by 2 (l−ρ|α |+δ|β |)/2 d 2 P (ξ ), C 1 + p(x, y) + ψ(x, y)ξ where P d (ξ ) is a polynomial in ξ of degree d with constant coefficients, and C > 0 is a constant. We assert that the inequality −ρ|α | + δ|β | + d −ρ|α| + δ|β + γ |

(6)

786


holds for all |α + β + γ | = N , and all occurring combinations of α , β , and d. It is not difficult to verify the assertion for N = 1. Let us now assume that (6) holds for |α + β + γ | = N . Differentiating (5) with respect to ξj yields n ∂ηi ∂ηα ∂wβ a p(x, y), ψ(x, y)ξ ψ(x, y)ij P d (x, y, ξ ) i=1

+ ∂ηα ∂wβ a p(x, y), ψ(x, y)ξ ∂ξj P d (x, y, ξ ),

and we get the inequalities −ρ |α | + 1 + δ|β | + d −ρ(|α| + 1) + δ|β + γ |, −ρ|α | + δ|β | + d − 1 −ρ(|α| + 1) + δ|β + γ |. Similarly, differentiation with respect to, say xj , gives n ∂wi ∂ηα ∂wβ a p(x, y), ψ(x, y)ξ (∂xj pi )(x, y)P d (x, y, ξ ) i=1

+

n ∂ηi ∂ηα ∂wβ a p(x, y), ψ(x, y)ξ ∂xj ψ(x, y)ξ i P d (x, y, ξ ) i=1

+ ∂ηα ∂wβ a p(x, y), ψ(x, y)ξ ∂xj P d (x, y, ξ ), and we arrive at the inequalities −ρ|α | + δ |β | + 1 + d −ρ|α| + δ |β + γ | + 1 ,

−ρ |α | + 1 + δ|β | + d + 1 −ρ|α| + δ|β + γ | − ρ + 1 −ρ|α| + δ |β + γ | + 1 , −ρ|α | + δ|β | + d −ρ|α| + δ |β + γ | + 1 , where, in particular, we made use of the assumption δ + ρ 1. This proves (6) for |α + β + γ | = N + 1. Summing up, we get the estimate α β γ ∂ ∂ ∂y b(x, y, ξ ) ξ x l−ρ|α|+δ|β+γ | C1 1 + p(x, y) + |ξ | l−ρ|α|+δ|β+γ | |l|+ρ|α|+δ|β+γ | 1 + |x − y| C2 1 + p(x, y) + |x − y| + |ξ | , where the latter inequality follows by using the easily verified inequality |s| (1 + |p(x, y)| + |ξ |)s C 1 + |x − y| , (1 + |p(x, y)| + |x − y| + |ξ |)s

s ∈ R,

compare the proof of Proposition 23.3 in [16]. Since |x| + |y| and |p(x, y)| + |x − y| define equivalent metrics, the assertion of the lemma follows. 2


787

l (R3n ), where 1 ρ > δ 0. Then the oscillatory integral Proposition 2. Let a(x, y, ξ ) ∈ Πρ,δ

Au(x) =

ei(x−y)ξ a(x, y, ξ )u(y) dy dξ ¯

(7)

defines a continuous linear operator from S(Rn ) to S(Rn ), and from S (Rn ) to S (Rn ). 3n ∞ n Proof. Consider first the case a ∈ C∞ c (R ), and assume that u ∈ C (R ) has bounded derivatives. Then the integration in (7) is carried out over a compact set, and partial integration gives Au(x) = ei(x−y)ξ x − y −M Dξ M Dy N ξ −N a(x, y, ξ )u(y) dy dξ, ¯

where M, N are even non-negative integers, and x stands for (1 + x12 + · · · + xn2 )1/2 . Let now l (R3n ), and assume that M, N are such that l − N (1 − δ) < −n, l + l + 2δN − M < −n. a ∈ Πρ,δ The latter integral then becomes absolutely convergent, defining a continuous function of x, and represents the regularization of the oscillatory integral (7). Increasing M and N we will obtain integrals which are convergent also after differentiation with respect to x. In view of the inequality x k y k x − y k , where k > 0, one finally sees that A defines a continuous map from S(Rn ) to S(Rn ), which, by duality, can be extended to a continuous map from S (Rn ) to S (Rn ). 2 We can now introduce the notion of the τ -symbol. In what follows, m will be a g-continuous function. Corollary 1. Let a ∈ S(g, m) = Γρ,δ (m, R2n ), 0 1 − ρ δ < ρ 1, and τ ∈ R. Then Au(x) =

ei(x−y)ξ a (1 − τ )x + τy, ξ u(y) dy dξ ¯

defines a continuous operator in S(Rn ), respectively S (Rn ). In this case, a is called the τ symbol of A, and the operator A is denoted by Opτ (a). Proof. For simplicity, we restrict ourselves to the case m = h−l . By Lemma 6 we then have l (R3n ), and the assertion follows with the previous propob(x, y, ξ ) = a((1 − τ )x + τy, ξ ) ∈ Πρ,δ sition. The case of a general m is proved in a similar way. 2 Our next aim is to prove the following. Theorem 3. Let 0 1 − ρ δ < ρ 1, τ, τ ∈ R be arbitrary, a(x, ξ ) ∈ S(g, m) = Γρ,δ (m, R2n ), and assume that κ : Rn → Rn is an invertible linear map. Furthermore, assume that m is invariant under κ in the sense that m(κ −1 (x), t κ(ξ )) = m(x, ξ ), and set A = Opτ (a). Then A1 u = A(u ◦ κ) ◦ κ −1 ,

u ∈ S Rn ,

defines a pseudodifferential operator with a uniquely defined τ -symbol σ τ (A1 ) ∈ S(g, m).

788


Proof. Let us consider first the case m = h−l . Putting κ1 = κ −1 , one sees that A1 is a Fourier integral operator given by ei(κ1 (x)−y)·ξ a (1 − τ )κ1 (x) + τ y, ξ u κ(y) dy dξ A1 u(x) = ¯ = ei(κ1 (x)−κ1 (y))·ξ a (1 − τ )κ1 (x) + τ κ1 (y), ξ det κ1 (y)u(y) dy dξ, ¯ and performing the change of variables ξ → t κ (ξ ), we get A1 u(x) = ei(x−y)·ξ a1 (x, y, ξ ) u(y)dy dξ, ¯ where we put a1 (x, y, ξ ) = a((1 − τ )κ1 (x) + τ κ1 (y),t κ (ξ ))|det κ1 ||det t κ|. Applying Lemma 6 l (R3n ) for arbitrary a ∈ with p(x, y) = (1 − τ )κ1 (x) + τ κ1 (y), one obtains a1 (x, y, ξ ) ∈ Πρ,δ l (R2n ) = S(g, h−l ). Next, let us introduce the coordinates v = (1 − τ )x + τy, w = x − y, and Γρ,δ expand a1 (x, y, ξ ) = a1 (v + τ w, v − (1 − τ )w, ξ ) into a Taylor series at w = 0, compare [16, pp. 180–182]. This yields

a1 (x, y, ξ ) =

|β+γ |N −1

(−1)|γ | |β| γ τ (1 − τ )|γ | (x − y)β+γ ∂xβ ∂y a1 (v, v, ξ ) + rN (x, y, ξ ), β!γ !

where rN (x, y, ξ ) =

1

cβγ (x − y)

β+γ

|β+γ |=N

γ (1 − t)N −1 ∂xβ ∂y a1 v + tτ w, v − t (1 − τ )w, ξ dt,

0 β γ

cβγ being constants. Since the operator with amplitude (x − y)β+γ (∂x ∂y a1 )(v, v, ξ ) coincides β+γ β γ with the one with amplitude (−1)|β+γ | (∂ξ Dx Dy a1 )(v, v, ξ ), we can write A1 also as A1 = BN + RN , where BN is the operator with τ -symbol bN (x, ξ ) =

|β+γ |N −1

1 |β| β+γ γ τ (1 − τ )|γ | ∂ξ (−Dx )β Dy a1 (x, y, ξ )|y=x , β!γ !

and RN has amplitude rN (x, y, ξ ). Similarly, we can assume that RN is given by a sum of terms having amplitudes of the form 1

β+γ β γ ∂ξ ∂x ∂y a1 v + tτ w, v − t (1 − τ )w, ξ (1 − t)N −1 dt,

0

where |β + γ | = N . In view of the estimate β+γ β γ ∂ ∂ ∂y a1 v + tτ w, v − t (1 − τ )w, ξ ξ

x

l−N (ρ−δ) l +N (ρ+δ) C 1 + |v| + |wt| + |ξ | 1 + |tw| ,


789

for some l and |β + γ | = N , one can then show that rN (x, y, ξ ) ∈ Πρ,δ (R3n ), where, by assumption, ρ − δ > 0. Define now A1 as the pseudodifferential operator with τ -symbol l−N (ρ−δ)

a1 (x, ξ ) ∼

∞ bN (x, ξ ) − bN −1 (x, ξ ) .

(8)

N =0

Then A1 − A 1 has kernel and τ -symbol belonging to S(R2n ). Since bN (x, ξ ) ∈ S(g, h−l ) for all N , the assertion of the theorem follows in view of the uniqueness of the τ -symbol, and l (R2n ). Let us consider now the case of a general m. By examining the proof of σ τ (A1 ) ∈ Γρ,δ Lemma 6, we see that a1 (x, y, ξ ) must satisfy an estimate of the form α β γ ∂ ∂ ∂y a1 (x, y, ξ ) C1 m p(x, y), t κ(ξ ) 1 + p(x, y) + |ξ | −ρ|α|+δ|β+γ | ξ x

−ρ|α|+δ|β+γ | C2 m p(x, y), t κ(ξ ) 1 + |x| + |y| + |ξ | ρ|α|+δ|β+γ | × 1 + |x − y| .

Consequently, β+γ β γ ∂ ∂ ∂y a1 v + tτ w, v − t (1 − τ )w, ξ ξ

x

−N (ρ−δ) Cm p v + tτ w, v − t (1 − τ )w , t κ(ξ ) 1 + |v| + |wt| + |ξ | l +N (ρ+δ) × 1 + |tw| , where |β + γ | = N , and we can again define A 1 = Opτ (a1 ) by the asymptotic expansion (8), such that A1 − A 1 has kernel and τ -symbol belonging to S(R2n ). The assertion of the theorem now follows by noting that bN (x, ξ ) ∈ S(g, m) = Γρ,δ (m, R2n ) for all N , due to the invariance of m. In particular, one has the asymptotic expansion r

σ τ (A1 )(x, ξ ) − ∈ S g, hN σm

|β+γ |N −1

1 |β| β+γ γ τ (1 − τ )|γ | ∂ξ (−Dx )β Dy a1 (x, y, ξ )|y=x β!γ ! (9)

for arbitrary integers N , where the first summand is given by a1 (x, x, ξ ) = a(κ −1 (x), t κ(ξ )).

2

Theorem 3 allows us, in particular, to express the τ -symbol of an operator in terms of its τ -symbol. More generally, one has the following. Corollary 2. In the setting of Theorem 3 assume that, in addition, a(κ −1 (x), t κ(ξ )) = a(x, ξ ), and det κ = ±1. Then A1 = A, and the τ -symbol of A = Opτ (a) is given by σ τ (A)(x, ξ ) ∼

1 |γ | β+γ β+γ τ |β| (1 − τ )|γ | (τ − 1)|β| τ ∂ξ Dx a(x, ξ ). β!γ ! β,γ

(10)

790


Proof. With a1 (x, y, ξ ) defined as in the proof of Theorem 3, we have a1 (x, y, ξ ) = a((1−τ )x + τ y, ξ ), so that A1 = Opτ (a) = A. The corollary then follows with the asymptotic expansion (9). 2 3. Reduced spectral asymptotics and the approximate spectral projection operators In this section, we shall introduce the method of approximate spectral projection operators, and apply it to the problem considered in the introduction. Thus, let G ⊂ O(n) be a compact group of isometries acting on Euclidean space Rn , and X a bounded domain in Rn which is invariant under G. Consider the regular representation T in the Hilbert spaces L2 (Rn ) and L2 (X) of square integrable functions by unitary operators. Let A0 be a symmetric, classical pseudodifferential operator of order 2m with principal symbol a2m as defined in [16], and regard it as an n operator in L2 (Rn ) with domain C∞ c (R ). Furthermore, assume that A0 is G-invariant, i.e. that it commutes with the operators T (g) for all g ∈ G, and that (A0 u, u) cu2m ,

u ∈ C∞ c (X),

(11)

for some c > 0, where (·,·) denotes the scalar product in L2 (Rn ), and · s is a norm in the Sobolev space H s (Rn ). Consider next the decomposition of L2 (Rn ) and L2 (X) into isotypic components, Hχ , L2 (X) = res Hχ , L2 Rn = ˆ χ∈G

ˆ χ∈G

ˆ is the set of all irreducible characters of G, and res denotes the restriction of functions where G 2 defined on Rn to X. Similarly, ext : C∞ c (X) → L (X) will denote the natural extension operator. The Hχ are closed subspaces, and the corresponding projection operators are given by Pχ = dχ χ(k)T (k) dk, G

where dχ is the dimension of any irreducible representation corresponding to the character χ , and dk denotes Haar measure on G. If G is just finite, dk becomes the counting measure, and one simply has Pχ =

dχ χ(k)T (k). |G| k∈G

Since T (k) is unitary, one computes for u, v ∈ L2 (Rn ) (u, Pχ v) = dχ χ(k) u, T (k)v dk = dχ χ k −1 T k −1 u, v dk = (Pχ u, v), G

G

where we made use of χ(g) = χ(g −1 ). Hence Pχ is self-adjoint. Let now A be the Friedrichs extension of the lower semi-bounded operator 2 res ◦A0 ◦ ext : C∞ c (X) −→ L (X).


791

A is a self-adjoint operator in L2 (X), and is itself lower semi-bounded. Its spectrum is real, and consists of the point spectrum and the continuous spectrum. Recall that, in general, a symmetric operator S in a separable Hilbert space is called lower semi-bounded, if there exists a real number c such that for all u ∈ D(S),

(Su, u) cu2

where D(S) denotes the domain of S. Now, if V is a subspace contained in D(S), the quantity N (S, V ) = sup dim L: (S u, u) < 0 ∀0 = u ∈ L , L⊂V

can be used to give a qualitative description of the spectrum of S. More precisely, one has the following classical variational result of Glazman. Lemma 7. Let S be a self-adjoint, lower semi-bounded operator in a separable Hilbert space, and define N (λ, S) to be equal to the number of eigenvalues of S, counting multiplicities, less or equal λ, if (−∞, λ) contains no points of the essential spectrum, and equal to ∞, otherwise. Then N (λ, S) = N S − λ1, D(S) . Proof. See [14, Lemma A.1].

2

In particular, the lemma above allows one to determine whether S has essential spectrum or not, where the latter is given by the continuous spectrum and the eigenvalues of infinite multiplicity. Let us now return to the situation above. Since A commutes with the action of G on L2 (X), the eigenspaces of A are unitary G-modules that decompose into irreducible subspaces. The purpose of this paper is to investigate the spectral counting function Nχ (λ) = dχ

μχ (t),

tλ

where μχ (λ) is the multiplicity of any irreducible representation of dimension dχ corresponding to the character χ in the eigenspace of A with eigenvalue λ. Nχ (λ) is equal to the number of eigenvalues of A, counting multiplicities, less or equal λ and with eigenfunctions in res Hχ , if (−∞, λ) contains no points of the essential spectrum, and equal to ∞, otherwise. One has then the following. Lemma 8. Nχ (λ) = N (A0 − λ1, Hχ ∩ C∞ c (X)). Proof. Let Aχ be the Friedrichs extension of res ◦A0 ◦ ext : C∞ c (X) ∩ Hχ → res Hχ . Then Nχ (λ) = N(λ, Aχ ), and the assertion follows with [14, Lemma A.2]. 2 In order to estimate N (A0 − λ1, Hχ ∩ C∞ c (X)), we will apply the method of approximate spectral projection operators. It was first introduced by Tulovsky and Shubin, and later developed

792


and generalized by Feigin and Levendorskii, and we will mainly follow [14] in our construction. Thus, let us consider on R2n the metric gx,ξ (y, η) = |y|2 + h(x, ξ )2 |η|2 ,

−1/2 h(x, ξ ) = 1 + |x|2 + |ξ |2

(12)

which is clearly of the form (3). Our symbol classes will be mainly of the form S(h−2δ g, p) = Γ1−δ,δ (p, R2n ) where p is a σ, h−2δ g-temperate function, and 0 δ < 1/2. In this case, 2δ−1 h2σ (x, ξ ) = 1 + |x|2 + |ξ |2 , by Eq. (2), which amounts to hσ = h1−2δ . Also note that u ∈ S(h−2δ g, p) implies ∂ξα ∂x u ∈ l S(h−2δ g, h(1−δ)|α|−δ|β| p). In particular, S(h−2δ g, h−l ) = Γ1−δ,δ (R2n ), where l ∈ R. The symbols and functions used will also depend on the spectral parameter λ. Nevertheless, their membership to specific symbol classes will be uniform in λ, which means that the values of their seminorms in the corresponding symbol classes will be bounded by some constant independent of λ. Now, if a denotes the left symbol of the classical pseudodifferential operator A0 , clearly a ∈ S(g, h−2m , K × Rn ) for any compact set K ⊂ Rn , so that σ l (A0 − λ1) ∈ S(g, q˜λ2 , K × Rn ) uniformly in λ 1, where β

q˜λ2 (x, ξ ) = h−2m (x, ξ ) + λ

(13)

is a σ, g-temperate function. But for u ∈ C∞ c (X), the quadratic form ((A0 − λ1)u, u) entering in l n the definition of N (A0 − λ1, Hχ ∩ C∞ c (X)) depends only on values of σ (A0 − λ1) on X × R . 2 n l By changing the latter symbol outside X × R we can achieve that σ (A0 − λ1) ∈ S(g, q˜λ ) uniformly in λ 1. In view of Corollary 2 we can therefore assume that A0 − λ1 can be represented as a pseudodifferential operator with Weyl symbol a˜ λ = σ w (A0 − λ1) ∈ S(g, q˜λ2 ). In particular, we may take σ w (A0 ) ∈ S(g, h−2m ). But by Eq. (11) and [14, Lemma 13.1] we even have a2m (x, ξ ) c

for all (x, ξ ) ∈ X × S n−1 and some constant c > 0.

Since a − a2m ∈ S(g, h−2m+1 , K × Rn ), we can therefore assume that A0 ∈ LI + (g, h−2m ), obtaining the following. Lemma 9. Let A0 be a classical pseudodifferential operator satisfying (11). Then A0 and A0 −λ1 can be represented as pseudodifferential operators with Weyl symbols σ w (A0 ) ∈ SI + (g, h−2m ) and a˜ λ ∈ SI + (g, q˜λ2 ), respectively. Note that if σ w (A0 ), and consequently also a˜ λ , are G-invariant in the sense that σ w (A0 ) σg (x, ξ ) = σ w (A0 )(x, ξ ),

a˜ λ σg (x, ξ ) = a˜ λ (x, ξ ),

where σg is the symplectic transformation given by σg (x, ξ ) = (κg (x), t κg (x)−1 (ξ )) = (κg (x), κg (ξ )), and κg (x) = gx denotes the action of g, the operators A0 and A0 − λ1 will commute with the action of G by Corollary 2. We can therefore formulate the assumption about the G-invariance of A0 also in terms of its Weyl symbol, and shall henceforth assume that the Weyl symbol and the principal symbol a2m of A0 are invariant under σg for all g ∈ G. In order


793

to define the approximate spectral projection operators, we will introduce now the relevant symbols. Having in mind Lemma 5, let aλ ∈ S(g, 1), and d ∈ S(g, d) be G-invariant symbols which, on Xρ × {ξ : |ξ | > 1}, Xρ = {x: dist(x, X) < ρ}, are given by

λ 1 1− , aλ (x, ξ ) = a2m (x, ξ ) 1 + λ|ξ |−2m d(x, ξ ) = |ξ |−1 , where ρ > 0 is some fixed constant, and in addition assume that d is positive and that d(x, ξ ) → 0 as |x| → ∞. We need to define smooth approximations to the Heaviside function, and to certain characteristic functions on X. Thus, let χ˜ be a smooth function on the real line satisfying 0 χ˜ 1, and 1 for s < 0, χ(s) ˜ = 0 for s > 1. Let C0 > 0 and δ ∈ (1/4, 1/2) be constants, and put ω = 1/2 − δ. We then define the G-invariant function (14) χλ = χ˜ ◦ aλ + 4hδ−ω + 8C0 d h−δ , where 0 < δ − ω < 1/2. 0 (R2n ) uniformly in λ. Lemma 10. χλ ∈ S(h−2δ g, 1) = Γ1−δ,δ

Proof. We first note that h−δ ∈ S g, h−δ ,

aλ + 4hδ−ω + 8C0 d ∈ S(g, 1),

since d ∈ S(g, d) ⊂ S(g, 1), and hδ−ω ∈ S(g, hδ−ω ) ⊂ S(g, 1). Now, each of the derivatives of χλ with respect to x and ξ can be estimated by a sum of derivatives of (aλ + 4hδ−ω + 8C0 d)h−δ . β β But because of ∂ξα ∂x (aλ + 4hδ−ω + 8C0 d) ∈ S(g, h|α| ), ∂ξα ∂x h−δ ∈ S(g, h−δ+|α| ), we obtain α β ∂ ∂ χλ (x, ξ ) Cα,β h(1−δ)|α| = Cα,β 1 + |x|2 + |ξ |2 −(1−δ)|α|/2 , ξ x

0 0 where Cα,β is independent of λ. We therefore obtain χλ ∈ Γ1−δ,0 (R2n ) ⊂ Γ1−δ,δ (R2n ) uniformly in λ, and the assertion follows. 2

Next, let U be a subset in R2n , c > 0, and put U (c, g) = (x, ξ ) ∈ R2n : ∃(y, η) ∈ U : g(x,ξ ) (x − y, ξ − η) < c ; according to Levendorskii [14, Corollary 1.2], there exists a smoothened characteristic function ψc ∈ S(g, 1) belonging to the set U and the parameter c, such that supp ψc ⊂ U (2c, g), and ψc |U (c,g) = 1. Let now Mλ = (x, ξ ) ∈ R2n : aλ < 4hδ−ω + 8C0 d .

(15)

794


Both Mλ and ∂X × Rn are invariant under σk for all k ∈ G, as well as (∂X × Rn )(c, h−2δ g), and Mλ (c, h−2δ g), due to the invariance of a2m (x, ξ ), and the considered metrics and symbols. Now, let η˜ c , ψλ,c ∈ S(h−2δ g, 1) be smoothened characteristic functions corresponding to the parameter c, and the sets ∂X × Rn and Mλ , respectively. According to Lemma 5, we can assume that they are invariant under σk for all k ∈ G; otherwise consider G η˜ c ◦ σk dk, G ψλ,c ◦ σk dk, respectively. We then define the functions ηλ,−c (x, ξ ) = ηc (x, ξ ) =

0, (1 − η˜ c (x, ξ ))ψλ,1/c (x, ξ ), η˜ c (x, ξ ), 1,

x∈ / X, x ∈ X,

x∈ / X, x ∈ X.

(16) (17)

Only the support of ψλ,c depends on λ, but not its growth properties, so that ηc , ηλ,−c ∈ S(h−2δ g, 1) uniformly in λ. Furthermore, since η˜ 2c = 1 on supp η˜ c , and ψλ,1/c = 1 on supp ψλ,1/2c , one has ηλ,−c = 1 on supp ηλ,−2c , which implies ηλ,−2c ηλ,−c = ηλ,−2c . Similarly, one verifies ηc η2c = ηc . We are now ready to define the approximate spectral projection operators. Definition 5. The approximate spectral projection operators of the first kind are defined by E˜λ = Opw (ηλ,−2 ) Opw (χλ ) Opw (ηλ,−2 ), while the approximate spectral projection operators of the second kind are Eλ = E˜λ2 (3 − 2E˜λ ). Remark 2. E˜λ is a smooth approximation to the spectral projection operator Eλ of A using Weyl calculus, while Eλ is an approximation to Eλ2 (3 − 2Eλ ) = Eλ . Note that, since ηλ,−2 and χλ are G-invariant, Corollary 2 implies that the operators Opw (ηλ,−2 ), Opw (χλ ), and consequently also E˜λ and Eλ , commute with the action T (g) of G. The choice of Eλ was originally due to the fact that its trace class norm can be estimated from above by the operator norm of 3 − 2E˜λ , and the Hilbert–Schmidt norm of E˜λ , which are easier to handle. This construction was first used by Feigin [6]. Both E˜λ and Eλ are integral operators with kernels in S(R2n ). Indeed, the asymptotic expansion (1), together with Proposition 1, imply that the Weyl symbols of E˜λ and Eλ can be written in the form a + r, where a has compact support, and r ∈ S −∞ (h−2δ g, 1), because χλ has compact support in ξ , and ηλ,−2 has x-support in X. Thus, σ w (E˜λ ) and σ w (Eλ ) are rapidly decreasing Schwartz functions, and the same holds for the corresponding τ -symbols. By [12, Lemma 7.2], this also implies that E˜λ and Eλ are of trace class and, in particular, compact operators in L2 (Rn ). In addition, by Theorem 2, and the asymptotic expansion (10), one has σ τ (E˜λ ), σ τ (Eλ ) ∈ S(h−2δ g, 1) uniformly in λ. On the other hand, the functions ηλ,−2 and χλ are realvalued, which by general Weyl calculus implies that Opw (ηλ,−2 ), Opw (χλ ), and consequently also E˜λ , and Eλ , are self-adjoint operators in L2 (Rn ). By construction, Eλ commutes with the projection Pχ , so that Pχ Eλ = Eλ Pχ is a self-adjoint operator of trace class as well. Although the decay properties of σ τ (Eλ ) are independent of λ, its support does depend on λ, which will


795

result in estimates for the trace of Pχ Eλ in terms of λ that will be used in order to prove Theorem 8. In particular, the estimate for the remainder term in Theorem 8 is determined by the particular choice of the range (1/4, /1/2) for the parameter δ, which guarantees that 1 − δ > δ. By the general theory of compact, self-adjoint operators, zero is the only accumulation point of the point spectra of E˜λ and Eλ , as well as the only point that could possibly belong to the continuous spectrum. The following proposition and its corollary give uniform bounds for the number of eigenvalues away from zero. They are based on certain L2 -estimates for pseudodifferential operators. Proposition 3. The number of eigenvalues of E˜λ lying outside the interval [− 14 , 54 ] is bounded by some constant independent of λ. Proof. Since χλ , ηλ,−c ∈ S(h−2δ g, 1), Theorem 2 yields σ w (E˜λ ) ∈ S(h−2δ g, 1) uniformly in λ. Furthermore, taking into account the asymptotic expansion (1), we have 2 σ w (E˜λ ) = ηλ,−2 χλ + rλ , 2 where rλ ∈ S(h−2δ g, h1−2δ ). Now, since 0 χλ , ηλ,−2 1, for each > 0 there exists a constant 2 2 c > 0 such that + ηλ,−2 χλ c and (1 + ) − ηλ,−2 χλ c. Consequently, the symbols of 1 + E˜λ and (1+)1− E˜λ admit a representation of the form a1 +a2 , where a1 c, a2 ∈ S(h−2δ g, h1−2δ ); thus

1 + E˜λ ∈ LI + h−2δ g, 1 ,

(1 + )1 − E˜λ ∈ LI + h−2δ g, 1

uniformly in λ. According to Lemma 2, this implies that for each λ there exist two operators T1 , T2 such that 1 + E˜λ T1 and (1 + )1 − E˜λ T2 , and Ti ∈ L−∞ (g, 1) uniformly in λ. Therefore, by Lemma 3, there exist two subspaces Li ⊂ L2 (Rn ) of finite codimension such that Ti uL2 uL2 for u ∈ Li and all λ, which implies, via Cauchy–Schwarz, that −u2L2 (Ti u, u) u2L2 on Li . Putting everything together we arrive at the L2 -estimates (E˜λ u, u) (T1 − 1)u, u −2u2L2 , (E˜λ u, u) (1 + )1 − T2 u, u (1 + 2)u2L2 , where u ∈ L1 ∩ L2 , and taking =

1 8

yields the desired result, since codim L1 ∩ L2 < ∞.

2

Corollary 3. The number of eigenvalues of Eλ lying outside the interval [0, 1] is bounded by some constant independent of λ. Proof. If ν˜ i denote the eigenvalues of E˜λ , then the eigenvalues of Eλ are given by νi = ν˜ i2 (3 − 2˜νi ). 2 Let now NχEλ denote the number of eigenvalues of Eλ which are 1/2, and whose eigenfunctions are contained in the χ -isotypic component Hχ of L2 (Rn ). Since zero is the only accumulation point of the point spectrum of Eλ , NχEλ is clearly finite. The next lemma will show that it

796


can be estimated by the trace of the operator Pχ Eλ , and its square, so that it is natural to expect that it will provide a good approximation for Nχ (λ) = tr Pχ Eλ = N (A0 − λ1, Hχ ∩ C∞ c (X)). Lemma 11. There exist constants c1 , c2 > 0 independent of λ such that 2 tr(Pχ Eλ )2 − tr Pχ Eλ − c1 NχEλ 3 tr Pχ Eλ − 2 tr(Pχ Eλ )2 + c2 .

(18)

Proof. Since Eλ ∈ L(h−2δ g, 1), Theorem 1 implies that Eλ is L2 -continuous. Moreover, by Remark 1, there is a constant C independent of λ such that Eλ L2 C; hence all eigenvalues of the operators Eλ are bounded by C. Let now νi,χ denote the eigenvalues of Eλ with eigenfunctions in Hχ . Taking into account Corollary 3 and the previous remark, we obtain the estimate NχEλ

νi,χ 1/2

νi,χ +

(1 − νi,χ ) + c1

1/2νi,χ 1

νi,χ + 2

νi,χ 1/2

νi,χ (1 − νi,χ ) + c1 ,

1/2νi,χ 1

where c1 > 0, like all other constants ci > 0 occurring in this proof, can be chosen independent of λ. Consequently, the right-hand side can be estimated from above by 3 tr Pχ Eλ − 2 tr Pχ Eλ · Pχ Eλ + c2 . In the same way one computes NχEλ =

νi,χ +

νi,χ 1/2

i

νi,χ − 2

(1 − νi,χ )

νi,χ 1/2

i

νi,χ (1 − νi,χ ) − c3

0νi,χ 1/2

νi,χ −

νi,χ − c3

0νi,χ 1/2

i

νi,χ − 2

νi,χ (1 − νi,χ ) − c4 ,

i

where the right-hand side can be estimated from below by 2 tr Pχ Eλ · Pχ Eλ − tr Pχ Eλ − c4 . This completes the proof of (18). 2 As the next section will show, NχEλ will provide us with a lower bound for the spectral counting function Nχ (λ). Nevertheless, in order to obtain an upper bound as well, it will be necessary to introduce new approximations to the spectral projection operators. Namely, let χλ+ = χ˜ aλ+ h−δ ,

aλ+ = aλ − 4hδ−ω − 8C0 d,

where χ˜ is defined as in (14). As in Lemma 10, one verifies that χλ+ ∈ S(h−2δ g, 1) uniformly in λ. Definition 6. The approximate spectral projection operators of the third kind are F˜ λ = Opw η22 χλ+ , while the approximate spectral projection operators of the fourth kind are Fλ = F˜ λ2 (3 − 2F˜ λ ).


797

Like the projection operators of the first and second kind, F˜ λ and Fλ are self-adjoint operators in L2 (Rn ) with kernels in S(R2n ), and therefore of trace class. Since Fλ commutes with T (k), Pχ Fλ is a self-adjoint operator of trace class, too. Let MχFλ denote the number of eigenvalues of Fλ which are 1/2, and whose eigenfunctions are contained in the χ -isotypic component Hχ . Since Proposition 3 and Corollary 3 hold for F˜ λ and Fλ as well, we obtain Lemma 12. There exist constants c1 , c2 > 0 independent of λ such that 2 tr(Pχ Fλ )2 − tr Pχ Fλ − c1 MχFλ 3 tr Pχ Fλ − 2 tr(Pχ Fλ )2 + c2 . Proof. The proof is a verbatim repetition of the proof of Lemma 11 with Eλ replaced by Fλ .

(19) 2

4. Estimates from below for the reduced spectral counting function In this section, we shall estimate the spectral counting function Nχ (λ) = N (A0 − λ1, Hχ ∩ C∞ c (X)) from below by adapting techniques developed in [14] to our situation. The main result will be Theorem 4. Let NχEλ be the number of eigenvalues of Eλ which are 1/2, and whose eigenfunctions are contained in the χ -isotypic component Hχ . Then there exists a constant C > 0 independent of λ such that Eλ N A0 − λ1, Hχ ∩ C∞ c (X) Nχ − C.

(20)

As a first step towards the proof, let q˜λ be defined as in (13), and qλ ∈ SI(g, q˜λ−1 ) be a Ginvariant symbol which, on X × {ξ : |ξ | > 1} is given by −1/2 , qλ (x, ξ ) = a2m (x, ξ ) 1 + |ξ |−2m λ and consider the G-invariant function π = (hδ−ω + C0 d)−1/2 ∈ SI(g, π), together with the operators Π = Opw (π),

Qλ = Opw (qλ ).

Since π q˜λ−1 is bounded, ΠQλ is a continuous operator in L2 (Rn ). The parametrices of Π and Qλ , which exist according to Lemma 1, will be denoted by RΠ and RQλ . Furthermore, an examination of the proof of Lemma 1 shows that if a ∈ SI(g, m) is G-invariant, then the Weyl symbol b of the parametrix of Opw (a) can be assumed to be G-invariant. Consequently, the parametrices RΠ and RQλ commute with the operators T (k). Lemma 13. Let LEχλ = Span{u ∈ S(Rn ) ∩ Hχ : Eλ u = νu, ν 12 } and L˜ Eχλ = Opl (ηλ,−1 ) · Qλ ΠLEχλ . Then dim L˜ Eχλ dim LEχλ − C = NχEλ − C for some constant C > 0 independent of λ.

(21)

798


Proof. Let us first note that since ηλ,−1 has support in X × Rn , and Opl (ηλ,−1 ) Qλ Π commutes with Pχ , we have L˜ Eχλ ⊂ C∞ c (X) ∩ Hχ . Next, we will prove that RΠ RQλ Opl (ηλ,−1 )Qλ ΠEλ = Eλ + T ,

(22)

where T ∈ L−∞ (g, 1). Indeed, the Weyl symbol of Opl (ηλ,−1 )Qλ ΠEλ is given by a linear combination of products of derivatives of the Weyl symbols of Qλ , Π , Eλ , and Opl (ηλ,−1 ). By the asymptotic expansion (10), 1 −1 |β| β l σ Op (ηλ,−1 ) (x, ξ ) ∼ ∂ξ Dxβ ηλ,−1 (x, ξ ). β! 2 w

β

Now, Eq. (51) implies that, up to terms of order −∞, the support of σ w (Eλ ) is contained in supp ηλ,−2 , and we shall express this by writing supp∞ σ w (Eλ ) ⊂ supp ηλ,−2 . For the same reason, we must have supp∞ σ w (Opl (ηλ,−1 )Qλ ΠEλ ) ⊂ supp ηλ,−2 . But ηλ,−1 = 1 on supp ηλ,−2 implies that all terms in the expansion of σ w (Opl (ηλ,−1 )) vanish on supp ηλ,−2 , except for the zero order terms. Proposition 1 then yields σ w Opl (ηλ,−1 ) (x, ξ ) = ηλ,−1 (x, ξ ) on supp ηλ,−2 , up to a term of order −∞. On this set, the Weyl symbol of Opl (ηλ,−1 )Qλ ΠEλ therefore reduces to ηλ,−1 = 1 times a linear combination of products of derivatives of the Weyl symbols of Qλ , Π and Eλ supported in supp ηλ,−2 , which corresponds to the Weyl symbol of Qλ ΠEλ , plus an additional term of order −∞. Thus, Opl (ηλ,−1 )Qλ ΠEλ = Qλ ΠEλ + T˜ ,

T˜ ∈ L−∞ g, π q˜λ−1 ,

(23)

and (22) follows by taking into account the definition of the parametrix. Now, Eλ : LEχλ → LEχλ is clearly surjective, and 1 Eλ u u, 2

u ∈ LEχλ ,

implies that Eλ is injective on LEχλ as well. Eq. (22) therefore means that on LEχλ RΠ RQλ Opl (ηλ,−1 )Qλ Π = 1LEλ + T Eλ−1 . χ

(24)

According to Lemma 3, there exists a subspace of finite codimension M such that T u u/8 for all u ∈ M and all λ. This gives −1 T E u 2T u 1 u for all u ∈ LEλ ∩ M. χ λ 4


799

Let now v, w ∈ LEχλ ∩ M, and assume that Opl (ηλ,−1 )Qλ Πv = Opl (ηλ,−1 )Qλ Πw. By (24) we deduce w + T Eλ−1 w = v + T Eλ−1 v and consequently (1 + T Eλ−1 )(v − w) = 0. But for u ∈ M ∩ LEχλ one computes

1 + T E −1 u u − T E −1 u 1 − 1 u; λ λ 4 hence 1 + T Eλ−1 is injective, and v = w. Thus we have shown that Opl (ηλ,−1 )Qλ Π : LEχλ ∩ M −→ L˜ Eχλ is injective, and the assertion of the lemma follows with C = codim M < ∞.

(25) 2

Since L˜ Eχλ ⊂ C∞ c (X) ∩ Hχ , the next proposition will provide us with a suitable reference subspace in order to prove Theorem 4. Its dimension will be estimated from below with the help of the preceding lemma. Note that the parametrices of Π and Qλ were needed to show the injectivity of (25). Proposition 4. There exists a subspace L ⊂ L˜ Eχλ such that dim L dim LEχλ − C for some constant C > 0 independent of λ, and (A0 − λ1)u, u L2 < 0 for all 0 = u ∈ L. Eλ ∞ Note that, by construction, L˜ Eχλ ⊂ C∞ c (X) ∩ Hχ , while Lχ ⊂ Cc (X). It is this proposition n that accomplishes the transition from R to X, which, according to (23), is achieved by a perturbation of order −∞.

Proof. Let v ∈ LEχλ and w = Opl (ηλ,−1 )Qλ ΠEλ v ∈ L˜ Eχλ . Eq. (23) implies that w = Qλ ΠEλ v + T˜ v,

T˜ ∈ L−∞ g, π q˜λ−1 .

Consequently, one computes ∗ (A0 − λ1)w, w = Π ∗ Q∗λ A0 − λ1 + 4RQ Opw hδ−ω + C0 d RQλ Qλ ΠEλ v, Eλ v λ ∗ − 4 Π ∗ Q∗λ RQ Opw hδ−ω + C0 d RQλ Qλ ΠEλ v, Eλ v + (T v, v) λ =: (D1 Eλ v, Eλ v) − 4(D2 Eλ v, Eλ v) + (T v, v),

(26)

where T is of order −∞. Now, since Qλ RQλ − 1 ∈ L−∞ (g, 1), we have D2 = 1 + K2 ,

K2 ∈ L(g, h);

indeed, by definition, the Weyl symbol of Π is equal to π = (hδ−ω + C0 d)−1/2 ∈ SI(g, π). Now, according to Lemma 9, A0 − λ1 = Opw (a˜ λ ), where a˜ λ ∈ SI + (g, q˜λ2 ). Thus, D1 = Π ∗ Q∗λ Opw (a˜ λ )Qλ + 4 Opw (hδ−ω + C0 d) Π + K1 ,

(27)

800


where K1 ∈ L−∞ (g, 1). Furthermore, we can assume that qλ ∈ S(g, q˜λ−1 ) is such that aλ = (a2m − λ)qλ2 ∈ S(g, 1), and using Theorem 2 one computes aλ − σ w Q∗λ Opw (a˜ λ )Qλ = aλ − qλ2 a˜ λ + r = qλ2 (a2m − λ − a˜ λ ) + r ∈ S(g, d),

(28)

where r ∈ S(g, h). But this implies aλ − σ w (Q∗λ Opw (a˜ λ )Qλ ) + 4C0 d cd for some sufficiently large C0 and some c > 0; hence aλ − σ w Q∗λ Opw (a˜ λ )Qλ + 4C0 d ∈ SI + (g, d).

(29)

Using Lemma 2, we conclude from (29) that there exists a T4 ∈ L−∞ (g, d) such that Q∗λ Opw (a˜ λ )Qλ Opw (aλ ) + 4C0 Opw (d) + T4 .

(30)

Together with Eλ v2 14 v2 , Eqs. (26)–(30) therefore yield the estimate (A0 − λ1)w, w = (T v, v) − 4(K2 Eλ v, Eλ v) − 4(Eλ v, Eλ v) + (K1 Eλ v, Eλ v) + Π ∗ Q∗λ Opw (a˜ λ )Qλ + 4C0 Opw (d) + 4 Opw hδ−ω ΠEλ v, Eλ v Π ∗ Opw (aλ ) + 8C0 Opw (d) + 4 Opw hδ−ω ΠEλ v, Eλ v − v2 + (K3 v, v), where K3 ∈ S(h−2δ g, h). We therefore set aλ− := aλ + 8C0 d + 4hδ−ω ∈ S(g, 1), and obtain the estimate (A0 − λ1)w, w Π ∗ Opw aλ− ΠEλ v, Eλ v − v2 + (K3 v, v).

(31)

Thus, it remains to show that Eλ∗ Π ∗ Opw (aλ− )ΠEλ − 1 + K3 is negative definite on some subspace of finite codimension. In order to do so, we will show that Eλ∗ Π ∗ Opw (aλ− )ΠEλ − 1 + K3 −1 + K4 , where K4 ∈ L(h−2δ g, hω ) and ω > 0. As it shall become apparent in the following discussion, the key to this is contained in the fact that, although aλ− ∈ S(g, 1), there exists a K5 ∈ L(h−2δ g, hδ ) such that Opw (χλ aλ− χλ ) K5 ! Now, ΠEλ = Π E˜λ Dλ = Π Opw (ηλ,−2 ) Opw (χλ ) Opw (ηλ,−2 )Dλ = Π Opw (ηλ,−2 ), Opw (χλ ) Opw (ηλ,−2 )Dλ + Opw (χλ )Π Opw (ηλ,−2 ) Opw (ηλ,−2 ))Dλ =: W1 + W2 , where we put Dλ = E˜λ (3 − 2E˜λ ). Since Π and E˜λ are self-adjoint, we obtain Eλ Π Opw aλ− ΠEλ = W2∗ Opw aλ− W2 + R,

(32)


801

where R = W1∗ Opw (aλ− )W2 + W2∗ Opw (aλ− )W1 + W1∗ Opw (aλ− )W1 is given by a sum of terms which contain either [Π Opw (ηλ,−2 ), Opw (χλ )], or its adjoint [Opw (χλ ), Opw (ηλ,−2 )Π], as factors. Now, the crucial remark is that supp∞ σ w Π Opw (ηλ,−2 ), Opw (χλ ) ⊂ suppdiff χλ ⊂ (x, ξ ): aλ− (x, ξ ) hδ (x, ξ ) , (33) (k)

where suppdiff χλ = {(x, ξ ): ∃k > 0: χλ (x, ξ ) = 0}. To see this, first note that by Theorem 2 and Proposition 1, we have the trivial inclusion supp∞ σ w ([Π Opw (ηλ,−2 ), Opw (χλ )]) ⊂ supp χλ . But since the terms in the asymptotic expansion of the Weyl symbol of [Π Opw (ηλ,−2 ), Opw (χλ )] are of order 1, they vanish unless (x, ξ ) ∈ suppdiff χλ , and one obtains the first inclusion. The second inclusion follows by noting the implications (k)

χλ = 0 ∀k > 0

⇐

χλ = 0

or χλ = 1

⇐

aλ− h−δ 1 or aλ− h−δ 0.

While computing the Weyl symbol of R, we can therefore replace aλ− with bλ−

= aλ− θλ ,

1 − −δ θλ = θ , a h 2 λ

(34)

where θ ∈ C∞ c (R) is a real-valued function taking values between 0 and 1, which is equal 1 on [−1, 1], and which vanishes outside [−2, 2], so that θλ = 1 on {(x, ξ ): |aλ− (x, ξ )| hδ (x, ξ )}. Indeed, this replacement adds at most a term of order −∞ to the Weyl symbol of R. Now, the advantage of performing this replacement resides in the fact that, on supp θλ , one has |aλ− | 4hδ , which together with α β − −|α| −δ−(1−δ)|α|+δ|β| 2 ∂ ∂ a (x, ξ ) C 1 + |x|2 + |ξ |2 2 C 1 + |x|2 + |ξ |2 , ξ x λ

|α| 1,

i.e. νk (h−2δ g, hδ ; aλ− ) < ∞, k 1, yields aλ− ∈ S(h−2δ g, hδ , supp θλ ), in contraposition to aλ− ∈ S(g, 1). Consequently, bλ− ∈ S(h−2δ g, hδ ), and we obtain R ∈ L h−2δ g, hδ π 2 ⊂ L h−2δ g, hω ,

(35)

since W1 , W2 ∈ L(h−2δ g, π), Dλ ∈ L(h−2δ g, 1), and hδ π 2 = hδ (hδ−ω + C0 d)−1 = (h−ω + C0 h−δ d)−1 ∼ hω . Eqs. (31), (32), and (35) therefore yield the estimate (A0 − λ1)w, w W2∗ Opw aλ− W2 v, v − v2 + (K4 v, v),

(36)

where K4 = K3 + R ∈ L(h−2δ g, hω ). To examine W2∗ Opw (aλ− )W2 more closely, let us consider the operator S = Opw (χλ ) Opw aλ− Opw (χλ ) − Opw χλ aλ− χλ . By the usual argument, the asymptotic expansion (1) and Proposition 1 yield supp∞ σ w (S) ⊂ suppdiff χλ . In the computation of the Weyl symbol of S we can therefore again replace aλ−

802


with bλ− , getting at most an additional term of order −∞. Since Opw (χλ ) ∈ L(h−2δ g, 1) by Lemma 10, we obtain S ∈ L h−2δ g, hδ .

(37)

Now, by construction, aλ− χλ hδ , since 0 χλ 1 and χλ = 0 for aλ− h−δ > 1, so that one infers α β ∂ ∂ χλ a − χλ (x, ξ ) C 1 + |x|2 + |ξ |2 (−δ−(1−δ)|α|+δ|β|)/2 ξ x

λ

for some constant C > 0. But this implies Opw (χλ aλ− χλ ) ∈ L(h−2δ g, hδ ). Using (36) and (37), we therefore get (A0 − λ1)w, w W3∗ Opw χλ aλ− χλ W3 v, v − v2 + (K5 v, v) = −v2 + (K6 v, v), with W3 = Π Opw (ηλ,−2 ) Opw (ηλ,−2 )Dλ ∈ L(h−2δ g, π), K5 = K4 + W3∗ Opw (χλ ) Opw aλ− Opw (χλ ) − Opw χλ aλ− χλ W3 ∈ L h−2δ g, hω , K6 = K5 + W3∗ Opw χλ aλ− χλ W3 ∈ L h−2δ g, hω . Since hσ = h1−2δ , Lemma 3 implies that the operator −1 + K6 is negative definite on a subspace U ⊂ L2 (Rn ) of finite codimension which does not depend on λ. Putting L := Opl (ηλ,−1 )Qλ ΠEλ (U ∩ LEχλ ∩ M) ⊂ L˜ Eχλ with M as in (25), we finally get (A0 − λ1)w, w < 0 ∀0 = w ∈ L,

(38)

where dim U ∩ LEχλ ∩ M − codimM dim M ∩ Eλ (U ∩ LEχλ ∩ M) dim L, since Eλ is bijective on LEχλ , and dim LEχλ dim U ∩ LEχλ ∩ M + codimU ∩ M. The assertion of the proposition now follows. 2 We can now prove Theorem 4. Proof of Theorem 4. Let L ⊂ L˜ Eχλ ⊂ C∞ c (X) ∩ Hχ be as in the previous proposition. Then (38) ∞ holds, and N (A0 − λ1, Hχ ∩ Cc (X)) dim L. Furthermore, dim L dim LEχλ − C = NχEλ − C, and the assertion of the theorem follows. 2 5. Estimates from above for the reduced spectral counting function In what follows, we shall prove an estimate from above for Nχ (λ) = N (A0 − λ1, Hχ ∩ in terms of the number MχFλ of eigenvalues of Fλ which are 1/2, and whose eigenfunctions are contained in the χ -isotypic component Hχ . In order to do so, we first prove the following. C∞ c (X))


803

Proposition 5. There exists a constant C > 0 independent of λ such that n l w + r ∞ + C. N A0 − λ1, Hχ ∩ C∞ c (X) N Op (η1 )Π Op aλ Π Op (η1 ) + 1, Hχ ∩ Cc R Note that this proposition accomplishes the transition from variational quantities related to Rn to quantities related to the bounded subdomain X. Now, the proof of Proposition 5 relies on the following. ∞ Lemma 14. There exists a subspace L ⊂ C∞ c (X) of finite codimension in Cc (X) such that

(A0 − λ1)u, u Opl (η1 )Π Opw aλ+ Π Opr (η1 ) + 1 RΠ RQλ u, RΠ RQλ u for all 0 = u ∈ L, and all λ. Proof of Proposition 5. Let us assume Lemma 14 for a moment, and introduce the notation Aλ [u] = (A0 − λ1)u, u ,

Bλ [u] =

l Op (η1 )Π Opw aλ+ Π Opr (η1 ) + 1 u, u .

According to that lemma, there exists a subspace L in C∞ c (X) of finite codimension such that Aλ [u] Bλ [RΠ RQλ u],

0 = u ∈ L,

for all λ. Let now m be a positive σ, g-tempered function such that 1/m is bounded. Following [14], we introduce the weight spaces of Sobolev type H(g, m) = span T w: w ∈ L2 Rn , T ∈ L(g, 1/m) ⊂ L2 Rn , and endow them with the strongest topology in which each of the operators T : L2 (Rn ) → H(g, m), T ∈ L(g, 1/m), is continuous. It can then be shown that there exists an operator Λm ∈ L(g, m) such that Λm : H(g, m) → L2 (Rn ) is a topological isomorphism. In particular, H(g, m) becomes a Hilbert space with the norm um = Λm uL2 . Furthermore, we have the continuous embedding S(Rn ) ⊂ H(g, m), and if m1 is a bounded, σ, g-tempered function, and A ∈ L(g, mm1 ), then A : H(g, m) → H(g, m−1 1 ) defines a continuous map. Now, by Theorem 3 and the asymptotic expansion (9), RΠ RQλ ∈ LI(g, π −1 q˜λ ), so that by Lemma 1 the opera−1 tor Λπ RΠ RQλ Λ−1 q˜λ ∈ LI(g, 1) has a parametrix Z ∈ LI(g, 1) satisfying ZΛπ RΠ RQλ Λq˜λ = −∞ 1 + K, where K ∈ L (g, 1). Since by Lemma 3 the kernel of 1 + K must be finite-dimensional, Ker Λπ RΠ RQλ Λ−1 q˜λ < ∞; consequently r = dim Ker RΠ RQλ : H(g, q˜λ ) → H (g, π) < ∞.

(39)

Next, let U ⊂ C∞ c (X) ∩ Hχ be a subspace such that Aλ [u] < 0,

∀0 = u ∈ U.

Then, for all 0 = u ∈ V := U ∩ L ∩ H(g,q˜λ ) (Ker RΠ RQλ : H(g, q˜λ ) → H(g, π)), 0 > Bλ [RΠ RQλ u].

(40)

804


Because RΠ RQλ is injective on V , (39) yields the inequality dim U dim V + C dim RΠ RQλ V + C for some constant C > 0 independent of λ. Since RΠ RQλ commutes with the operators T (k) of the representation of G, RΠ RQλ V ⊂ Hχ ∩ H(g, π), and we obtain the estimate dim U sup dim W : Bλ [w] < 0 ∀0 = w ∈ W + C. W ∈H(g,π)∩Hχ

n But C∞ c (R ) ∩ Hχ is dense in H(g, π) ∩ Hχ , and the assertion of the proposition follows.

2

Let us now prove Lemma 14. Proof of Lemma 14. Let u ∈ C∞ c (X). Then r ei(x−y)ξ ηc (y, ξ )u(y) dy dξ Op (ηc )u(x) = ¯ = u(x), since ηc is equal to oneon X × Rn . Now, for general B ∈ L(g, p), σ r (Opr (ηc )B) is given by an asymptotic expansion j aj , where the first term is equal to ηc σ r (B). Consequently, σ r Opr (ηc )B = ηc σ r (B) + a − ηc σ r (B) + r, with a as in Proposition 1, and r ∈ S −∞ (h−2δ g, p). But a − ηc σ r (B) = 0 on X × Rn , and we obtain Opr (ηc )Bu = Bu + T u, T ∈ L−∞ h−2δ g, p . (41) Using Lemma 9, and setting u˜ = RΠ RQλ u, one computes (A0 − λ1)u, u = Opw (a˜ λ )Qλ Π u, ˜ Qλ Π u˜ + (T1 u, u) ˜ u˜ = Π ∗ Q∗λ Opw (a˜ λ )Qλ − 4 Opw hδ−ω + C0 d Π u, ˜ u˜ + (T1 u, u) + 4 Π ∗ Opw hδ−ω + C0 d Π u, =: Π ∗ D1 Π Opr (η1 )u, ˜ Opr (η1 )u˜ + 4(D2 u, ˜ u) ˜ + (T2 u, u), where we took (41) into account together with RΠ RQλ − RΠ RQλ ∈ L−∞ (g, q˜λ π −1 ), and Ti ∈ L−∞ . The reason for including Opr (η1 ) will become apparent in the proof of the next theorem. Now, by (28), aλ − σ w (Q∗λ Opw (a˜ λ )Qλ ) ∈ S(g, d), which implies that for sufficiently large C0 D1 − Opw aλ+ = Q∗λ Opw (a˜ λ )Qλ + 4C0 Opw (d) − Opw (aλ ) ∈ LI + (g, d), so that according to Lemma 2, there exists a T3 ∈ L−∞ (g, d) such that D1 − Opw (aλ+ ) T3 . On the other hand, since π 2 = (hδ−ω + C0 d)−1 , D2 − 1 ∈ L(g, h), and we obtain (A0 − λ1)u, u Opl (η1 )Π ∗ Opw aλ+ Π Opr (η1 )u, ˜ u˜ + 2u ˜ 2 + (T4 u, u), (42)


805

where T4 ∈ L(g, π −2 q˜λ2 h); hereby we used the fact that Opl (η1 ) is the adjoint of Opr (η1 ), compare [16, p. 26]. Furthermore, since by (9) the Weyl symbol of RΠ RQλ is equal to π −1 q˜λ modulo terms of lower order, (RΠ RQλ )∗ RΠ RQλ + T4 ∈ LI + g, π −2 q˜λ2 . Lemmata 1–3 now allow us to deduce the existence of a subspace L ⊂ C∞ c (X) of finite codimension in L2 (X) such that u ˜ 2 + (T4 u, u) =

(RΠ RQλ )∗ RΠ RQλ + T4 u, u > 0

(43)

for all 0 = u ∈ L, and all λ. Indeed, according to Lemma 2, Λπ 2 [(RΠ RQλ )∗ RΠ RQλ + T4 ]Λ−1 2 ∈ q˜λ

LI + (g, 1) can be written in the form B ∗ B + T5 , where B ∈ LI(g, 1) and T ∈ L−∞ (g, 1). By a reasoning similar to the one that led to (39), one can infer from Lemma 1 that the kernel of B must be finite-dimensional, and together with Lemma 3 conclude that there exists a subspace L˜ ⊂ L2 (Rn ) of finite codimension such that BuL2 cuL2 ,

T5 uL2
0. Thus, we obtain (43), and together with (42) we get (A0 − λ1)u, u Opl (η1 )Π ∗ Opw aλ+ Π Opr (η1 ) + 1 u, ˜ u˜ for all 0 = u ∈ L. This concludes the proof of the lemma.

2

We are now in position to prove an estimate from above for N (A0 − λ1, Hχ ∩ C∞ c (X)). Theorem 5. Let MχFλ be the number of eigenvalues of Fλ which are 1/2, and whose eigenfunctions are contained in the χ -isotypic component Hχ . Then there exists a constant C > 0 independent of λ such that Fλ N A0 − λ1, Hχ ∩ C∞ c (X) Mχ + C. Proof. We shall continue with the notation introduced in the proof of Proposition 5. According to that proposition, it suffices to prove a similar estimate for n N Opl (η1 )Π Opw aλ+ Π Opr (η1 ) + 1, Hχ ∩ C∞ c R from above. For this sake, we will show that there exists a subspace L ⊂ UχFλ = Span u ∈ S(Rn ) ∩ Hχ : Fλ u = νu, ν < 1/2 , whose codimension in UχFλ is finite and uniformly bounded in λ, such that Bλ [u] 0 for all u ∈ L.

806


Indeed, let us assume this statement for a moment. Since Fλ is a compact self-adjoint operan tor in L2 (Rn ), there exists an orthonormal basis of eigenfunctions {uj }∞ j =1 in S(R ). But Fλ commutes with the action T (g) of G, so that each of the eigenspaces of Fλ is an invariant subspace, and must therefore decompose into a sum of irreducible G-modules. Consequently, Hχ has an orthonormal basis of eigenfunctions lying in S(Rn ) ∩ Hχ . Hence, Hχ = UχFλ ⊕ VχFλ , where VχFλ = Span{u ∈ S(Rn ) ∩ Hχ : Fλ u = νu, ν 1/2}. Now, if W ⊂ S(Rn ) ∩ Hχ is a subspace with Bλ [u] < 0 for all 0 = u ∈ W, then L ∩ W = {0}, and therefore W ⊂ VχFλ ⊕ U , where U is a finite-dimensional subspace of UχFλ whose dimension is bounded by some constant C > 0 independent of λ. Consequently, dim W dim VχFλ + C. But this implies n N Opl (η1 )Π Opw aλ+ Π Opr (η1 ) + 1 , Hχ ∩ C∞ c R sup dim W : Opl (η1 )Π Opw aλ+ Π Opr (η1 ) + 1 u, u < 0 ∀0 = u ∈ W W ⊂S (R n )∩Hχ

dim VχFλ + C = MχFλ + C, and the assertion of the theorem follows with the previous proposition. Let us now show the existence of the subspace L. Take v ∈ UχFλ ⊂ L2 (Rn ), and put v˜ = (1 − Fλ )v. We then expect ˜ 0. Now, one computes that Bλ [v] 2 1 − Fλ Opl (η1 )Π Opw aλ+ Π Opr (η1 ) 1 − Fλ v, v + (1 − Fλ )v + (K1 v, v) 2 (Dv, v) + v − Fλ v + (K1 v, v)

˜ = Bλ [v]

1 (Dv, v) + v2 + (K1 v, v), 4

(44)

where we put Fλ = Opw (χλ+ )2 (3 − 2 Opw (χλ+ )), D = 1 − Fλ Opl (η1 )Π Opw aλ+ Π Opr (η1 ) 1 − Fλ , and K1 ∈ L−∞ . Indeed, one has Fλ v 12 v, and Opr (η1 )Fλ −Opr (η1 )Fλ ∈ L−∞ (h−2δ g, 1), since the terms in the asymptotic expansions of the Weyl symbols of Opr (η1 )Fλ and Opr (η1 )Fλ coincide because of η2 = 1 on supp η1 . Next we note that, similarly to (33), (45) supp∞ σ w Fλ , Opl (η1 )Π ⊂ suppdiff χλ+ ⊂ (x, ξ ): aλ+ (x, ξ ) hδ (x, ξ ) , and we set bλ+ = aλ+ θλ ,

θλ = θ

1 + −δ aλ h , 2


807

with θ as in (34). An argument similar to that concerning bλ− shows that bλ+ ∈ S(h−2δ g, hδ ). Now, because of bλ+ = aλ+ on supp∞ σ w ([Fλ , Opl (η1 )Π]), we have (Dv, v) =

(1 − Fλ ), Opl (η1 )Π Opw bλ+ Π Opr (η1 ) 1 − Fλ v, v + Opl (η1 )Π 1 − Fλ Opw aλ+ Π Opr (η1 ) 1 − Fλ v, v + (K2 v, v),

where K2 is of order −∞. Since [(1 − Fλ ), Opl (η1 )Π] Opw (bλ+ )Π Opr (η1 )(1 − Fλ ) ∈ L(h−2δ g, hδ π 2 ) ⊂ L(h−2δ g, hω ), we therefore obtain (Dv, v) = Opl (η1 )Π 1 − Fλ Opw aλ+ Π Opr (η1 ) 1 − Fλ v, v + (K3 v, v), where K3 ∈ L(h−2δ g, hω ). Using a similar argument to commute Π Opr (η1 ) with 1 − Fλ , we finally get (46) (Dv, v) = Opl (η1 )Π 1 − Fλ Opw aλ+ (1 − Fλ )Π Opr (η1 )v, v + (K3 v, v), where K3 ∈ L(h−2δ g, hω ). Now, the asymptotic expansion of the Weyl symbol of the operator (1 − Fλ ) Opw (aλ+ )(1 − Fλ ) gives 2 2 σ w 1 − Fλ Opw aλ+ 1 − Fλ = 1 − χλ+ 3 − 2χλ+ aλ+ + r

(47)

with supp∞ r ⊂ suppdiff χλ+ . While computing r, we can therefore replace aλ+ by bλ+ , so that r ∈ S(h−2δ g, hδ ). As a consequence, (46) and (47) yield 2 (Dv, v) = Opl (η1 )Π Opw 1 − (χλ+ )2 3 − 2χλ+ aλ+ Π Opr (η1 )v, v + (K4 v, v), where K4 ∈ L(h−2δ g, hω ). Hereby we used again the fact that π 2 hδ ∼ hω . Next, one verifies that [1 − (χλ+ )2 (3 − 2χλ+ )]2 aλ+ + C1 hδ ∈ SI + (h−2δ g, [1 − (χλ+ )2 (3 − 2χλ+ )]2 aλ+ + C1 hδ ) for some C1 > 0, since χλ+ = 1 for aλ+ < 0, so that [1 − (χλ+ )2 (3 − 2χλ+ )]2 aλ+ 0. According to Lemma 2, we therefore have 2 2 Opw 1 − χλ+ 3 − 2χλ+ aλ+ K5 ∈ L h−2δ g, hδ , and we arrive at the estimate K6 ∈ L h−2δ g, hω .

(Dv, v) (K6 v, v), Together with (44) we finally obtain the estimate

1 ˜ (v, v) + (K7 v, v), Bλ [v] 4

K7 ∈ L h−2δ g, hω .

Using the already familiar argument of Lemma 3, one infers the existence of a subspace M ⊂ L2 (Rn ) of finite codimension on which 1/4 + K7 is positive definite. Putting L := (1 − Fλ )(UχFλ ∩ M) ⊂ UχFλ we therefore get Bλ [w] 0,

for all w ∈ L.

808


Furthermore, since 1 − Fλ is injective on UχFλ , codimU Fλ L = codimU Fλ (M ∩ UχFλ ) codimM, χ

as desired. This completes the proof of the theorem.

χ

2

Remark 3. The leading idea in the proof of the last theorem was that each v ∈ UχFλ has to be, approximately, an eigenvector of the corresponding spectral projection operator of A with eigenvalue zero. For this reason, such a v cannot satisfy (Av, v) < λv2 , nor be an element of W . 6. The finite group case For the rest of Part I, we shall concentrate on the case where G is a finite group, while the compact group case will be treated in Part II. The two preceding sections showed that, in view of Lemmata 11 and 12, the spectral counting function Nχ (λ) = N (A0 − λ1, Hχ ∩ C∞ c (X)) can be estimated from below and from above in terms of the traces of Pχ Eλ and Pχ Fλ , and their squares. We will therefore now proceed to estimate these traces in terms of the reduced Weyl volume. For this sake, we introduce first certain sets associated to the support of the symbols of the approximate spectral projection operators; their significance will become apparent later. Thus, let Wλ = (x, ξ ) ∈ X × Rn : aλ < 0 , Ac,λ = (x, ξ ) ∈ X × Rn : aλ < c hδ−ω + d , Bc,λ = X × Rn − Ac,λ , Dc = ∂X × Rn c, h−2δ g , Fλ = (x, ξ ) ∈ X × Rn : χλ = 0 or ηλ,−2 = 0 or χλ = ηλ,−2 = 1 , RV c,λ = (x, ξ ) ∈ X × Rn : |aλ | < c hδ−ω + d ∪ (x, ξ ) ∈ Dc : x ∈ X, aλ < c hδ−ω + d . Note that Dc = {(x, ξ ) ∈ R2n : dist(x, ∂X) < −2δ

h

√

c(1 + |x|2 + |ξ |2 )−δ/2 }, since for

δ (x, ξ )g(x,ξ ) (x − y, ξ − η) = 1 + |x|2 + |ξ |2

|ξ − η|2 2 + |x − y| < c 1 + |x|2 + |ξ |2

to hold for some (y, η) ∈ ∂X × Rn , it is necessary and sufficient that |x − y|2 (1 + |x|2 + |ξ |2 )δ < c is satisfied for some y ∈ ∂X. Now, recall that |G| = χ∈Gˆ dχ2 . We then have the following. Proposition 6. For sufficiently large c > 0 we have tr Pχ Eλ − Vχ X × Rn , aλ c vol RV c,λ ,

(48)

where dχ2 Vχ X × Rn , aλ = |G|

X×Rn

dχ2 vol Wλ 1(−∞,0] aλ (x, ξ ) dx dξ ¯ = (2π)n |G|

(49)


809

is the expected approximation given in terms of the reduced Weyl volume, and 1(−∞,0] denotes the characteristic function of the interval (−∞, 0]. Furthermore, a similar estimate holds for tr Pχ Eλ · Pχ Eλ , too. Proof. The proof will require several steps. Let σ r (Eλ )(x, ξ ) denote the right symbol of Eλ . n Then, for u ∈ C∞ c (R ), dχ −1 χ(h) ei(h x−y)ξ σ r (Eλ )(y, ξ )u(y) dy dξ. Pχ Eλ u(x) = ¯ |G| h∈G

The kernel of Pχ Eλ , which is a rapidly decreasing function, is given by dχ −1 χ(h) ei(h x−y)ξ σ r (Eλ )(y, ξ ) dξ. KPχ Eλ (x, y) = ¯ |G| h∈G

The trace of Pχ Eλ can therefore be computed by tr Pχ Eλ =

KPχ Eλ (x, x) dx

dχ = χ(h) tr Eλ + |G| |G| dχ2

−1 x−x)ξ

ei(h

σ r (Eλ )(x, ξ ) dx dξ, ¯

h=e

where we made use of the relation χ(e) = dχ , and the fact that tr Eλ = σ r (Eλ )(x, ξ ) dx dξ ¯ . As a next step, we will prove that, for all e = h ∈ G, there exists a sufficiently large constant c > 0 such that i(h−1 x−x)ξ r e σ (Eλ )(x, ξ ) dx dξ (50) ¯ c vol(RV c,λ ). As already noticed, the decay properties of σ τ (Eλ )(x, ξ ) ∈ S(h−2δ g, 1) are independent of λ for arbitrary τ ∈ R, while its support does depend on λ. Indeed, by Theorem 2 and Corollary 2, together with the asymptotic expansions (1) and (10) and Proposition 1, 2 2 2 χλ 3 − 2ηλ,−2 χλ + fλ + rλ , σ τ (Eλ ) = ηλ,−2

(51)

where rλ ∈ S −∞ (h−2δ g, 1), and fλ ∈ S(h−2δ g, h1−2δ ), everything uniformly in λ; in addition, fλ (x, ξ ) = 0 if (x, ξ ) ∈ Fλ . To see this, note that σ τ (Eλ )(x, ξ ) is given asymptotically as a linear combination of products of derivatives of σ w (Eλ ) at (x, ξ ), which in turn is given asymptotically by a linear combination of terms involving derivatives of ηλ,−2 , χλ . The τ -symbol of Eλ is therefore asymptotically given by aj ∈ S h−2δ g, h(1−2δ)N , aj ∈ S h−2δ g, h(1−2δ)j , σ τ (Eλ ) − 0j 0 such that d(gx, x) κd(x, Σg ) for all x ∈ Rn , and arbitrary e = g ∈ G. Proof. Let x ∈ Rn − Σg be an arbitrary point, and p the closest point to x belonging to Σg . Write x = expp t0 X, where expp denotes the exponential mapping of Rn , and (p, X) ∈ Tp Rn ,

|X| = 1.

Then t0 = (x, Σg ). Consider next the direct sum decomposition Tp (Rn ) = U ⊕ V , where U = (p, Y ) ∈ Tp Rn : dgp (Y ) = Y , and V = U ⊥ . Since p is a fixed point of g, we also have the identity g expp Y = expp dgp (Y ), which implies expp tY ∈ Σg

if, and only if,

(p, Y ) ∈ U,

where t ∈ R.

Consequently, U = Tp (Σg ). Now, with expp tY = p + tY , and x, p as above, one computes


811

2 2 |gx − x|2 = expp t0 dgp (X) − expp t0 X = p + t0 dgp (X) − p − t0 X 2 = d 2 (x, Σg )dgp (X) − X . Because of (x − p) ⊥ Σg , we must have (p, X) ∈ Tp (Σg )⊥ = V , and therefore dgp (X) − X 2 = 0. The latter expression depends continuously on (p, X) ∈ {(p, Y ) ∈ Tp (Σg )⊥ : |Y | = 1}, and is actually independent of p, so that it can be estimated from below by some positive constant uniformly for all x. The assertion of the proposition now follows. 2 Returning now to our previous computations, we split the integral in (52) in an integral over −δ/2 , D = (x, ξ ) ∈ X × Rn : dist(x, Σ) 1 + |ξ |2 and a second integral over the complement of D in X × Rn . Since supp χλ ⊂ {(x, ξ ): aλ + 4hδ−ω + 8C0 hδ }, the integral over Ω×X D can be estimated by a constant times the volume of the set {(x, ξ ) ∈ Ω×X D: aλ + 4hδ−ω + 8C0 d hδ }, which is contained in the set {(x, ξ ) ∈ X × Rn : dist(x, Σ) < (1 + |ξ |2 )−δ/2 , aλ c(hδ−ω + d)} for some sufficiently large c > 0. By examining the proof of Lemma 18, one sees that the volume of the latter can be estimated from above by vol(Σc2 |ξ |−δ ∩ X) dξ + c3 K|ξ |c1 λ1/2m

for some suitable constants K, ci > 0, and consequently has the same asymptotic behaviour in λ as the volume of RV c,λ . In studying the asymptotic behaviour of the integral (52), we can therefore restrict the domain of integration to D. By the previous lemma, there exists a constant κ > 0 such that −1 h x − x κ 1 + |ξ |2 −δ/2 for all (x, ξ ) ∈ D and e = h ∈ G. 2 2 χλ )2 (3 − 2ηλ,−2 χλ ) + fλ has compact support in ξ , this implies that Since (ηλ,−2 −1

2 ei(h x−x)ξ α 2 2 ∂ξ ηλ,−2 χλ 3 − 2ηλ,−2 χλ + fλ (x, ξ ) −1 2 |h x − x| is integrable on D, as well as rapidly decreasing in ξ . Integrating by parts with respect to ξ we therefore get for (52) the expression D

−1

2 2 ei(h x−x)ξ 2 2 −∂ξ1 − · · · − ∂ξ2n ηλ,−2 χλ 3 − 2ηλ,−2 χλ + fλ (x, ξ ) dx dξ ¯ ; (53) −1 2 |h x − x|

in particular notice that, by Fubini’s theorem, the boundary contributions vanish. Now, if (x, ξ ) ∈ 2 2 χλ )2 (3 − 2ηλ,−2 χλ ) + fλ is constant, so its derivatives with respect to ξ Fλ , the function (ηλ,−2

812


are zeros, and we can restrict the integration in (53) to the set X×Rn Fλ ∩ D, where X×Rn Fλ denotes the complement of Fλ in X × Rn . Lemma 16. For sufficiently large c > 0, the set X×Rn Fλ is contained in RV c,λ . Proof. This assertion is already stated in [14, p. 55]. For the sake of completeness, we give a proof here. Thus, consider / D4 , aλ < −4hδ−ω − 8C0 d , Eλ = (x, ξ ) ∈ X × Rn : (x, ξ ) ∈ and let Mλ be defined as in (15). Since supp η˜ 2 ⊂ D4 , and ψλ,1/2 = 1 on Mλ (1/2, h−2δ g), it is clear that (54) Eλ ⊂ (x, ξ ) ∈ X × Rn : χλ = ηλ,−2 = 1 ⊂ Fλ , and consequently X×Rn Fλ ⊂ X×Rn Eλ . Next, we are going to prove that, for sufficiently large c, / Mλ (1, h−2δ g). Thus, assume (x, ξ ) ∈ Bc,λ ; on X × {ξ : |ξ | > 1} (x, ξ ) ∈ Bc,λ implies (x, ξ ) ∈ we have

|ξ |2m 1 λ 1 2m + . 1− c |ξ | (1 + |x|2 + |ξ |2 )(δ−ω)/2 a2m (x, ξ ) |ξ | + λ Therefore, as c becomes large, |ξ | must become large, too. On the other hand, if (y, η) ∈ Mλ , |η| must be bounded. For large c we therefore have |ξ − η|2 ∼ |ξ |2 , which means that h−2δ (x, ξ )g(x,ξ ) (x − y, ξ − η) ∼ (1 + |x|2 + |ξ |2 )δ → ∞ as c → ∞. Hence, for sufficiently large c, (x, ξ ) ∈ / Mλ (1, h−2δ g). Since supp ψλ,1/2 ⊂ Mλ (1, h−2δ g), we arrive in this case at the inclusions (55) Bc,λ ⊂ (x, ξ ) ∈ X × Rn : ηλ,−2 (x, ξ ) = 0 ⊂ Fλ , and combining (54) and (55) we get X×Rn Fλ ⊂ Ac,λ ∩ X×Rn Eλ ⊂ RV c,λ , as desired.

(56)

2

As a consequence of the foregoing lemma, the integral in (53) is bounded from above by the volume of RV c,λ , times a constant independent of λ, since the integrand is uniformly bounded with respect to λ. Thus, we have shown (50). The assertion of the proposition now follows by observing that tr Eλ − vol Wλ c vol RV c,λ . (57) (2π)n −1 Indeed, similarly to our previous discussion of the integral ei(h x−x)ξ σ r (Eλ )(x, ξ ) dx dξ ¯ , the integral 2 2 2 σ r (Eλ )(x, ξ ) dx dξ ηλ,−2 χλ 3 − 2ηλ,−2 χλ + fλ + rλ (x, ξ ) dx dξ tr Eλ = ¯ ¯ =


813

can be split into three parts; the contribution coming from rλ (x, ξ ) is bounded by some constant independent of λ, while the contribution coming from fλ can be estimated in terms of the volume of RV c,λ , since supp fλ ⊂ X×Rn Fλ ⊂ RV c,λ , by the previous lemma. Now, 2 2 χλ )2 (3 − 2ηλ,−2 χλ ) must be equal 1 on Wλ ∩ X×Rn RV c,λ , since according to (56) we (ηλ,−2 have X×Rn RV c,λ ⊂ Bc,λ ∪ Eλ , and hence Wλ ∩ X×Rn RV c,λ ⊂ Eλ ⊂ {(x, ξ ) ∈ X × Rn : χλ = 2 2 χλ )2 (3 − 2ηλ,−2 χλ ) vanishes ηλ,−2 = 1}, due to the fact that Wλ ∩ Bc,λ = ∅. Furthermore, (ηλ,−2 −2δ / Mλ (1, h g), by the proof of the preon Bc,λ , since for large c, (x, ξ ) ∈ Bc,λ implies (x, ξ ) ∈ vious lemma. Taking into account that Wλ and RV c,λ are subsets of Ac,λ , we therefore obtain for sufficiently large c 2 2 2 χλ (x, ξ ) dx dξ ηλ,−2 χλ 3 − 2ηλ,−2 ¯ vol(Wλ ∩ Ac,λ RV c,λ ) = + (2π)n

2 2 2 ηλ,−2 χλ 3 − 2ηλ,−2 χλ (x, ξ ) dx dξ. ¯

Ac,λ −(Wλ ∩Ac,λ RV c,λ )

Now, since Ac,λ RV c,λ ⊂ Wλ , one has Ac,λ − Wλ ∩ Ac,λ RV c,λ = RV c,λ . The estimate (57) now follows, and together with (50) we obtain (48). Finally, if in the previous computations Eλ is replaced by Eλ2 , we obtain a similar estimate for the trace of Pχ Eλ · Pχ Eλ = Pχ Eλ2 . This concludes the proof of Proposition 6. 2 As a consequence, we get the following. Theorem 6. Let NχEλ be the number of eigenvalues of Eλ which are 1/2 and whose eigenfunctions are contained in the χ -isotypic component Hχ of L2 (Rn ). Then E N λ − Vχ X × Rn , aλ c vol RV c,λ χ

(58)

for some sufficiently large c > 0. Proof. From the preceding proposition, and the estimate (18), one deduces that for some sufficiently large c > 0 NχEλ 3 tr Pχ Eλ − 2 tr Pχ Eλ · Pχ Eλ + c2 Vχ X × Rn , aλ + c vol RV c,λ , NχEλ 2 tr Pχ Eλ · Pχ Eλ − tr Pχ Eλ − c1 Vχ X × Rn , aλ − c vol RV c,λ , which completes the proof of (58).

2

In analogy to the previous considerations, one proves the following. Theorem 7. For sufficiently large c > 0 one has the estimate F M λ − Vχ X × Rn , aλ c vol RV c,λ , χ where MχFλ is the number of eigenvalues of Fλ , counting multiplicities, greater or equal 1/2, and whose eigenfunctions are contained in the χ -isotypic component Hχ of L2 (Rn ).

814


Proof. The proof is similar to the one of Theorem 6, and uses Lemma 12. In particular, as in Eq. (51), one has 2 σ τ (Fλ ) = η22 χλ+ 3 − 2η22 χλ+ + fλ + rλ ,

(59)

where rλ ∈ S −∞ (h−2δ g, 1), and fλ ∈ S(h−2δ g, h1−2δ ), everything uniformly in λ. The asymptotic analysis for tr Pχ Fλ and tr(Pχ Fλ )2 now follows the lines of the proof of Proposition 6. 2 7. Proof of the main result We can now give a description of the asymptotic behavior of the spectral counting function Nχ (λ) as λ → +∞ in the finite group case. Collecting all the results obtained so far one deduces Proposition 7. There exist constants C1 , C2 > 0 which do not depend on λ, such that for all λ N A0 − λ1, Hχ ∩ C∞ (X) − Vχ X × Rn , aλ C1 vol RV C ,λ + C2 . c 1 Proof. By Theorems 4 and 5, there exist constants Ci > 0 independent of λ such that NχEλ −C1 Fλ N (A0 − λ1, Hχ ∩ C∞ c (X)) Mχ + C2 . Theorems 6 and 7 then yield the estimate n −c vol RV c,λ − C1 N A0 − λ1, Hχ ∩ C∞ c (X) − Vχ X × R , aλ c vol RV c,λ + C2 for some sufficiently large c > 0.

2

In order to formulate the main result, we need two last lemmata. Lemma 17. For λ > 0 one has −1 dχ2 vol(a2m ((−∞, 1])) n/2m λ + C, Vχ X × Rn , aλ = n (2π) |G|

where C > 0 is some constant independent of λ. Proof. Let us define bλ (x, ξ ) = aλ (x, λ1/2m ξ ). Then for x ∈ X and |ξ | > λ−1/2m one has

1 1 bλ (x, ξ ) = 1 − . a2m (x, ξ ) 1 + |ξ |−2m Furthermore, by [14, Lemma 13.1], condition (11) implies that a2m (x, ξ ) ι > 0 for all (x, ξ ) ∈ X × S n−1 , so that for x ∈ X and |ξ | > λ−1/2m the condition bλ (x, ξ ) < 0 is equivalent to a2m (x, ξ ) < 1. Now, departing from (49) one computes dχ2 Vχ X × Rn , aλ = |G| =

X Rn

dχ2 (2π)n |G|

1(−∞,0] bλ x, λ−1/2m ξ dx dξ ¯ X Rn

1(−∞,0] bλ (x, ξ ) dx dξ · λn/2m


=

815

−1 vol(a2m ((−∞, 1])) n/2m + O(1), λ (2π)n |G|

dχ2

and the assertion of the lemma follows.

2

Lemma 18. Assume that for some sufficiently small ρ > 0 there exists a constant C > 0 such that vol(∂X)ρ Cρ. Then, as λ → +∞, vol RV c,λ = O(λ(n−)/2m ), where ∈ (0, 12 ). Proof. According to the definition of RV c,λ at the beginning of Section 6, we have vol RV c,λ vol (x, ξ ) ∈ X × Rn : |aλ | − c hδ−ω + d < 0 + vol (x, ξ ) ∈ Dc : x ∈ X, aλ < c hδ−ω + d , √ where Dc = {(x, ξ ): dist(x, ∂X) < c(1 + |x|2 + |ξ |2 )−δ/2 }, and 0 < δ − ω < 1/2. In what follows, let us assume that λ 1. It is not difficult to see that, for |ξ | > 1, there exists a constant c1 > 0 independent of λ such that aλ (x, ξ ) − c hδ−ω + d (x, ξ ) < 0

⇒

|ξ | < c1 λ1/2m .

(60)

Indeed, let c1 be such that c12m max(2, 2/ι),

1 c hδ−ω + d (x, ξ ) , 3 x∈X, |ξ |>c1 sup

where ι > 0 is a lower bound for a2m (x, ξ ) on X × S n−1 . Since 1−

λ 1 a2m (x, ξ ) 2

⇐⇒

|ξ |2m

2λ , a2m (x, ξ/|ξ |)

one computes for |ξ | c1 λ1/2m that aλ (x, ξ )

1 1 1 1 1 , 2 1 + λ|ξ |−2m 2 1 + c1−2m 3

while, on the other hand, c(hδ−ω + d)(x, ξ ) 13 , so that aλ (x, ξ ) − c(hδ−ω + d)(x, ξ ) 0. This proves (60). As a consequence, we obtain the estimate vol (x, ξ ) ∈ Dc : x ∈ X, aλ < c hδ−ω + d vol (x, ξ ) ∈ X × Rn : |ξ | K, dist(x, ∂X) < c2 |ξ |−δ , aλ < c hδ−ω + d + vol (x, ξ ) ∈ X × Rn : |ξ | K vol (∂X)c2 |ξ |−δ ∩ X dξ + c3 , K|ξ |c1 λ1/2m

816


where δ ∈ ( 14 , 12 ), and K 1 is some sufficiently large constant; here and in all what follows, ci > 0 denote suitable, positive constants independent of λ. Now, since vol(∂X)ρ Cρ, for some ρ > 0, vol (x, ξ ) ∈ Dc : x ∈ X, aλ < c hδ−ω + d c2 c4

r n−1−δ dr dS n−1 (η) + c3

S n−1 Krc1 λ1/2m

= c5 λ(n−δ)/2m − K n−δ + c3 . Next, let |ξ | K, and assume that the inequality |aλ (x, ξ )| c(hδ−ω + d)(x, ξ ) is fulfilled. As before, we have |ξ |2m < c12m λ, as well as λ c 1 + λ|ξ |−2m d + hδ−ω (x, ξ ) c6 1 + λ|ξ |−2m |ξ |−(δ−ω) . 1 − a2m (x, ξ )

(61)

Combining (60) and (61), one deduces for sufficiently large K that |ξ |2m −c6 |ξ |2m + λ |ξ |−(δ−ω) +

λ c7 λ. a2m (x, ξ/|ξ |)

Let us now introduce the variable R(x, ξ ) = λ/a2m (x, ξ ) = λ|ξ |−2m /a2m (x, ξ/|ξ |). Performing the corresponding change of variables one computes vol (x, ξ ) ∈ X × Rn : |aλ | − c hδ−ω + d < 0 vol (x, ξ ) ∈ X × Rn : c1 λ1/2m > |ξ | K, 1 − R(x, ξ ) c6 1 + λ|ξ |−2m |ξ |−(δ−ω) + c8 r n−1 dr dS n−1 (η) dx + c8 X S n−1 {rK: |1−R|c9 λ−(δ−ω)/2m }

c10

R X S n−1 {R: |1−R|c9 λ−(δ−ω)/2m }

c11 λ

n 2m

−1

λ Ra2m (x, η)

n 2m

dR dS n−1 (η) dx + c8

n R − 2m −1 dR + c8 = O λ(n−(δ−ω))/2m + c8 .

{R: |1−R|c9 λ−(δ−ω)/2m }

Hereby we made use of the fact that (1 + z)β − (1 − z)β = O(|z|) for arbitrary z ∈ C, |z| < 1, and β ∈ R. 2 We are now in position to prove the main result of Part I, which generalizes [14, Theorem 13.1] to bounded domains with symmetries in the finite group case. Theorem 8. Let G be a finite group of isometries in Euclidean space Rn , and X ⊂ Rn a bounded domain which is invariant under G such that, for some sufficiently small ρ > 0, vol(∂X)ρ Cρ. Let further A0 be a symmetric, classical pseudodifferential operator in L2 (Rn ) of order 2m with G-invariant Weyl symbol σ w (A0 ) ∈ S(g, h−2m ) and principal symbol a2m , and assume that


817

(A0 u, u) cu2m for some c > 0 and all u ∈ C∞ c (X). Consider further the Friedrichs extension of the operator 2 res ◦A0 ◦ ext : C∞ c (X) −→ L (X),

and denote it by A. Finally, let Nχ (λ) be the number of eigenvalues of A less or equal λ and with eigenfunctions in the χ -isotypic component res Hχ of L2 (X), if (−∞, λ) contains no points of the essential spectrum, and equal to ∞, otherwise. Then, for all ∈ (0, 12 ), Nχ (λ) =

−1 −1 vol(a2m ((−∞, 1])) vol(a2m ((−∞, 1])) n/2m λ + O λ(n−)/2m n (2π) |G| |G|

dχ2

as λ → +∞, where dχ denotes the dimension of any irreducible representation of G corresponding to the character χ . In particular, A has discrete spectrum. Proof. By Lemma 8 and Proposition 7 we have Nχ (λ) − Vχ X × Rn , aλ C1 vol RV C

1 ,λ

+ C2

for some suitable constants C1 , C2 > 0 independent of λ. Lemmata 17 and 18 then imply −1 dχ2 vol(a2m ((−∞, 1])) n/2m λ −O λ(n−)/2m Nχ (λ) − O λ(n−)/2m n (2π) |G|

with arbitrary ∈ (0, 1/2). In particular, Nχ (λ) remains finite for λ < ∞, so that the essential spectrum of A must be empty. The assertion of the theorem now follows. 2 Remark 4. An alternative description of the expected approximation Vχ (X × Rn , aλ ) defined 1 −n/2m dx dS n−1 (η). Then for in (49) can be given as follows. Let γ = n|G| X S n−1 (a2m (x, η)) λ > 0 one has dχ2 γ · λn/2m + O(1). Vχ X × Rn , aλ = (2π)n Indeed, as already noticed, a2m (x, ξ ) ι > 0 for all (x, ξ ) ∈ X × S n−1 , so that aλ is strictly negative on Xρ × {ξ : |ξ | > 1} if, and only if, a2m (x, ξ ) − λ < 0, which in turn is equivalent to −1 1/2m |ξ | < λa2m x, ξ/|ξ | , due to the homogeneity of the principal symbol. For λ > 0 one therefore concludes from this Vχ X × Rn , aλ =

vol (x, ξ ) ∈ X × Rn : |ξ | 1 (2π)n |G| −1 + vol (x, ξ ) ∈ X × Rn : |ξ |2m < λa2m x, ξ/|ξ | dχ2

818


= O(1) +

= O(1) +

dχ2 (2π)n |G| dχ2 (2π)n |G|

(λ/a2m (x,η))1/2m

r n−1 dr dS n−1 (η) dx X S n−1

0

1 (λ/a2m (x, η))n/2m dS n−1 (η) dx. n

X S n−1

Acknowledgment The author wishes to thank Professor Mikhail Shubin for introducing him to this subject, and for many helpful discussions and useful remarks. References [1] V. Arnol’d, Frequent representations, Moscow Math. J. 3 (4) (2003) 1209–1221. [2] M. Bronstein, V. Ivrii, Sharp spectral asymptotics for operators with irregular coefficients I. Pushing the limits, Comm. Partial Differential Equations 28 (2003) 83–102. [3] J. Brüning, E. Heintze, Representations of compact Lie groups and elliptic operators, Invent. Math. 50 (1979) 169– 203. [4] T. Carleman, Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes, in: C.R. Séme Cong. Math. Scand., Stockholm, 1934, Lund, 1935, pp. 34–44. [5] H. Donnelly, G-spaces, the asymptotic splitting of L2 (M) into irreducibles, Math. Ann. 237 (1978) 23–40. [6] V.I. Feigin, Asymptotic distribution of eigenvalues for hypoelliptic systems in Rn , Math. USSR Sb. 28 (4) (1976) 533–552. [7] L. Gårding, On the asymptotic distribution of eigenvalues and eigenfunctions of elliptic differential operators, Math. Scand. 1 (1953) 237–255. [8] V. Guillemin, A. Uribe, Reduction and the trace formula, J. Differential Geom. 32 (2) (1990) 315–347. [9] B. Helffer, D. Robert, Etude du spectre pour un opératour globalement elliptique dont le symbole de Weyl présente des symétries I, Amer. J. Math. 106 (1984) 1199–1236. [10] B. Helffer, D. Robert, Etude du spectre pour un opératour globalement elliptique dont le symbole de Weyl présente des symétries II, Amer. J. Math. 108 (1986) 973–1000. [11] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968) 193–218. [12] L. Hörmander, The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32 (1979) 359–443. [13] V. Ivrii, Sharp spectral asymptotics for operators with irregular coefficients II. Domains with boundaries and degenerations, Comm. Partial Differential Equations 28 (2003) 103–128. [14] S.Z. Levendorskii, Asymptotic Distribution of Eigenvalues, Kluwer Academic, Dordrecht, Boston, MA, 1990. [15] S. Minakshisundaram, Å. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math. 1 (1949) 242–256. [16] M.A. Shubin, Pseudodifferential Operators and Spectral Theory, 2nd ed., Springer-Verlag, Berlin, 2001. [17] V.N. Tulovsky, M.A. Shubin, On the asymptotic distribution of eigenvalues of pseudodifferential operators in Rn , Math. Trans. 92 (4) (1973) 571–588. [18] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912) 441–479.


Atomic representations of rank 2 graph algebras Kenneth R. Davidson a,∗,1 , Stephen C. Power b,2 , Dilian Yang a,3 a Pure Mathematics Department, University of Waterloo, Waterloo, ON N2L-3G1, Canada b Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK

Received 30 May 2007; accepted 16 May 2008 Available online 19 June 2008 Communicated by D. Voiculescu

Abstract We provide a detailed analysis of atomic ∗-representations of rank 2 graphs on a single vertex. They are completely classified up to unitary equivalence, and decomposed into a direct sum or direct integral of irreducible atomic representations. The building blocks are described as the minimal ∗-dilations of defect free representations modelled on finite groups of rank 2. © 2008 Elsevier Inc. All rights reserved. Keywords: Higher rank graph; Atomic ∗-representation; Dilation

1. Introduction Kumjian and Pask [10] introduced higher rank graphs and their associated C*-algebras as a generalization of graph C*-algebras that are related to the generalized Cuntz–Kreiger algebras of Robertson and Steger [16]. The C*-algebras of higher rank graphs have been studied in a variety of papers [7,11,14,15,17]. See also Raeburn’s survey [13]. In [9], Kribs and Power examined the nonself-adjoint operator algebras which are associated with these higher rank graphs. More recently, Power [12] presented a detailed analysis of single * Corresponding author.

E-mail addresses: [email protected] (K.R. Davidson), [email protected] (S.C. Power), [email protected] (D. Yang). 1 Partially supported by an NSERC grant. 2 Partially supported by EPSRC grant EP/E002625/1. 3 Partially supported by the Fields Institute. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.008

820

K.R. Davidson et al. / Journal of Functional Analysis 255 (2008) 819–853

vertex rank 2 case. As this case already contains many new and intriguing algebras, we were motivated to study them more closely. The rank 2 graphs on one vertex form an intriguing family of semigroups with a rich representation theory. There are already many interesting and non-trivial issues. Our purpose in this paper is to completely classify the atomic ∗-representations of these semigroups. These representations combine analysis with some interesting combinatorial considerations. They provide a rich class of representations of the associated C*-algebra which have proven effective in understanding the underlying structure. The algebras are given concretely in terms of a finite set of generators and relations of a special type. Given a permutation θ of m × n, form a unital semigroup F+ θ with generators e1 , . . . , em and f1 , . . . , fn which is free in the ei ’s and free in the fj ’s, and has the commutation relations ei fj = fj ei where θ (i, j ) = (i , j ) for 1 i m and 1 j n. This is a cancellative semigroup with unique factorization [10]. + A ∗-representation of the semigroup F+ θ is a representation π of Fθ as isometries with the property that m

π(ei )π(ei )∗ = I =

i=1

n

π(fj )π(fj )∗ .

j =1

An atomic ∗-representation acts on a Hilbert space with an orthonormal basis which is permuted, up to unimodular scalars, by each of the generators. The C*-algebra C∗ (F+ θ ) is the universal C*algebra generated by these ∗-representations. The motivation for studying these representations comes from the case of the free semigroup ∗ + F+ m generated by e1 , . . . , em with no relations. The C*-algebra C (Fm ) is just the Cuntz algebra Om [1] (see also [2]). Davidson and Pitts [3] classified the atomic ∗-representations of F+ m and showed that the irreducibles fall into two types, known as ring representations and infinite tail representations. They provide an interesting class of C*-algebra representations of Om which are amenable to analysis. Furthermore the atomic ∗-representations feature significantly in the dilation theory of row contractions [5]. The 2-graph situation turns out to be considerably more complicated than the case of the free semigroup. Whereas the irreducible atomic ∗-representations of F+ n are of two types, the irreducible atomic ∗-representations of F+ fall into six types. Nevertheless, we are able to put θ them all into a common framework modelled on abelian groups of rank 2. 2. Background Rank 2 graphs. The semigroup F+ θ is generated by e1 , . . . , em and f1 , . . . , fn . The identity is denoted as ∅. There are no relations among the e’s, so they generate a copy of the free semigroup + on m letters, F+ m ; and there are no relations on the f ’s, so they generate a copy of Fn . There are commutation relations between the e’s and f ’s given by a permutation θ in Sm×n of m × n: ei f j = f j e i

where θ (i, j ) = (i , j ).

A word w ∈ F+ θ has a fixed number of e’s and f ’s regardless of the factorization; and the degree of w is d(w) := (k, l) if there are k e’s and l f ’s. The degree map is a homomorphism of + 2 F+ θ into N0 . The length of w is |w| = k + l. The commutation relations allow any word w ∈ Fθ


821

to be written with all e’s first, or with all f ’s first, say w = eu fv = fv eu . Indeed, one can factor w with any prescribed pattern of e’s and f ’s as long as the degree is (k, l). It is straightforward to see that the factorization is uniquely determined by the pattern and that F+ θ has the unique factorization property and cancellation. See also [9,10,12]. Example 2.1. With n = m = 2 we note that the relations e1 f 1 = f 2 e1 ,

e1 f 2 = f 1 e2 ,

e2 f 1 = f 1 e1 ,

e2 f 2 = f 2 e2 ,

arise from the 3-cycle permutation θ = ((1, 1), (1, 2), (2, 1)) in S2×2 . We will refer to F+ θ as the forward 3-cycle semigroup. The reverse 3-cycle semigroup is the one arising from the 3-cycle ((1, 1), (2, 1), (1, 2)). It was shown by Power in [12] that the 24 permutations of S2×2 give rise to 9 isomorphism classes of semigroups F+ θ , where we allow isomorphisms to exchange the ei ’s for fj ’s. In particular, the forward and reverse 3-cycles give non-isomorphic semigroups. Example 2.2. With n = m = 2 the relations e1 f 1 = f 1 e 1 ,

e1 f 2 = f 1 e2 ,

e2 f 1 = f 2 e1 ,

e2 f 2 = f 2 e2 ,

are those arising from the 2-cycle permutation ((1, 2), (2, 1)) and we refer to F+ θ in this case as the flip semigroup because of the commutation rule: ei fj = fi ej for 1 i, j 2. This is an example which has periodicity, a concept which will be explained in more detail later. Representations. We now define two families of representations which will be considered here: the ∗-representations and defect free (partially isometric) representations. Definition 2.3. A partially isometric representation of F+ θ is a semigroup homomorphism σ : → B(H) whose range consists of partial isometries on a Hilbert space H. The representation F+ θ σ is isometric if the range consists of isometries. A representation is atomic if it is partially isometric and there is an orthonormal basis which is permuted, up to scalars, by each partial isometry. That is, σ is atomic if there is a basis {ξk : k 1} so that for each w ∈ F+ θ , σ (w)ξk = αξl for some l and some α ∈ T ∪ {0}. We say that σ is defect free if m i=1

σ (ei )σ (ei )∗ = I =

n

σ (fj )σ (fj )∗ .

j =1

An isometric defect free representation is called a ∗-representation of F+ θ . For a (partially) isometric representation, the defect free condition is equivalent to saying that the ranges of the σ (ei )’s are pairwise orthogonal and sum to the identity, and that the same holds for the ranges of the σ (fj )’s. Equivalently, this says that [σ (e1 ) . . . σ (em )] is a (partial)

822


isometry from the direct sum of m copies of H onto H, and the likewise for [σ (f1 ) . . . σ (fn )]. A representation which satisfies the property that these row operators are isometries into H is called row isometric. The left regular representation λ of F+ θ is an example of a representation which is row isometric, but is not defect free. Definition 2.4. The C*-algebra C∗ (F+ θ ) is the universal C*-algebra for ∗-representations. This is the C*-algebra generated by isometries E1 , . . . , Em and F1 , . . . , Fn which are defect free: m n ∗=I = ∗ , and satisfies the commutation relations of F+ , with the universal E E F F i j i=1 j =1 θ i j ∗ + property that every ∗-representation σ of F+ θ extends to a ∗-homomorphism of C (Fθ ) onto C∗ (σ (F+ θ )). In the case of the free semigroup F+ n , Davidson and Pitts [3] classified the atomic row isometric representations. They showed that in the irreducible case there are three possibilities, namely (i) the left regular representation, (ii) a ring representation, determined by a primitive word u in F+ n and a unimodular scalar λ, and (iii) a tail representation, constructed from an aperiodic infinite word in the generators of F+ n . The left regular representation is the only one which is not defect free. The ring and tail representations provide the irreducible atomic ∗-representations. The universal C*-algebra of F+ n is the Cuntz algebra On . A ring representation is determined by a set of k basis vectors which are cyclically permuted, modulo λ, according to the k letters of u. The primitivity of u means that u is not a proper power of a smaller word. On the other hand, a tail representation σ is determined by an infinite word z = z0 z−1 z−2 . . . in the generators of F+ n . There is a subset of basis elements ξ0 , ξ−1 , ξ−2 , . . . for which σ (zk )ξk−1 = ξk for k 0. In both cases the subspace M spanned by the basis vector subset is coinvariant for σ (i.e. σ (w)∗ M ⊂ M for all w ∈ F+ ) and cyclic for σ (i.e. σ (w)M = n w∈F+ n ⊥ H). On the complementary invariant subspace M , σ decomposes as a direct sum of copies of the left regular representation. We shall meet these representations, as restrictions, in the classification of irreducible atomic representations of F+ θ . + Dilations. If σ is a representation of F+ θ on a Hilbert space H, say that a representation π of Fθ on a Hilbert space K ⊃ H is a dilation of σ if

σ (w) = PH π(w)|H

for all w ∈ F+ θ .

The dilation π of σ is a minimal isometric dilation if π is isometric and the smallest invariant subspace containing H is all of K, which means that K = w∈F+ π(w)H. This minimal isometθ ric dilation is called unique if for any two minimal isometric dilations πi on Ki , there is a unitary operator U from K1 to K2 such that U |H is the identity map and π2 = Ad U π1 . It is straightforward to see that an isometric dilation of a defect free representation is still defect free, and hence is a ∗-representation. An important result from our paper [6] is that every defect free representation has a unique minimal ∗-dilation. Both existence and uniqueness are of critical importance. Moreover, if the original representation is atomic, then so is the ∗-dilation. Theorem 2.5. (See [6, Theorems 5.1, 5.5].) Let σ be a defect free (atomic) representation. Then σ has a unique minimal dilation to a (atomic) ∗-representation.


823

Conversely, if π is a ∗-representation, one can obtain a defect free representation σ on a subspace H if H is co-invariant, i.e. H⊥ is invariant for π(F+ θ ), by setting σ (w) = PH π(w)|H . By Dilation Theorem 2.5, σ has a unique minimal ∗-dilation. But such a ∗-dilation is evidently the restriction of π to the reducing subspace K = w∈F+ π(w)H. If the subspace H is cyclic, θ i.e. K = K, then this minimal ∗-dilation is π itself. In this case, π is uniquely determined by σ . The significance of this for us is that every atomic ∗-representation has a particularly nice coinvariant subspace on which the compression σ has a very tractable form. It is by determining these smaller defect free representations that we are able to classify the atomic ∗-representations. 3. Examples of atomic representations We begin with a few examples of atomic representations. Example 3.1. Given a permutation θ of m × n, select a cycle of θ , say (i1 , j1 ), (i2 , j2 ), . . . , (ik , jk ) . Form a Hilbert space of dimension k with basis ξs for 1 s k. Define σ (ei ) =

∗ ξs ξs−1

and σ (fj ) =

is =i

ξs−1 ξs∗ .

js =j

That is, σ (fjs ) maps ξs to ξs−1 and σ (eis ) maps ξs−1 back to ξs . Likewise σ (eis+1 ) maps ξs to ξs+1 and σ (fjs+1 ) maps ξs+1 back to ξs . This corresponds to the commutation relation eis fjs = fjs+1 eis+1 indicated by the cycle of θ . i1 i3

i2

ξ2

ξ1 j2

ik

...... j3

ξk jk

j1

So it is not difficult to verify that this defines a defect free atomic representation of F+ θ . By . Dilation theorem 2.5, this can be dilated to a unique ∗-representation of F+ θ One can further adjust this example by introducing scalars. For example, if α, β ∈ T, define σα,β (ei ) = ασ (ei ) and σα,β (fj ) = βσ (fj ). Two such representations will be shown to be unitarily equivalent if and only if α1k = α2k and β1k = β2k . Example 3.2 (Inductive representations). An important family of ∗-representations were introduced in [6]. Recall that the left regular representation λ acts on 2 (F+ θ ), which has orthonormal }, by left multiplication: λ(w)ξ = ξ for all w, v ∈ F+ basis {ξv : v ∈ F+ v wv θ θ . Start with an arbitrary infinite word or tail τ = ei0 fj0 ei1 fj1 . . . . Let Fs = F := F+ θ , for s = act as injective maps by right 0, 1, 2, . . . , viewed as a discrete set on which the generators of F+ θ multiplication, namely,

824


ρ(w)g = gw

for all g ∈ F.

Consider ρs = ρ(eis fjs ) as a map from Fs into Fs+1 . Define Fτ to be the injective limit set Fτ = lim(Fs , ρs ); →

and let ιs denote the injections of Fs into Fτ . Thus Fτ may be viewed as the union of F0 , F1 , . . . with respect to these inclusions. The left regular action λ of F+ θ on itself induces corresponding maps on Fs by λs (w)g = wg. Observe that ρs λs = λs+1 ρs . The injective limit of these actions is an action λτ of F+ θ on Fτ . Let + 2 λτ also denote the corresponding representation of Fθ on (Fτ ). The standard basis of 2 (Fτ ) is {ξg : g ∈ Fτ }. A moment’s reflection shows that this provides a defect free, isometric (atomic) representation of F+ θ ; i.e. it is a ∗-representation. We now describe a coinvariant cyclic subspace that contains all of the essential information ∗ about this representation. Let H = λτ (F+ θ ) ξι0 (∅) . This is coinvariant by construction. As it contains ξιs (∅) for all s 1, it is easily seen to be cyclic. Let στ be the compression of λτ to H. ...

i−2,0

•

i−1,0

•

j−2,0

...

i−2,−1

•

i−1,−1

i−2,−2

•

i−1,−2

•

.. .

i0,−1

j0,0

•

j−1,−1

•

j−2,−2

.. .. .. ...

•

j−1,0

j−2,−1

...

i0,0

i0,−2

j0,−1

•

j−1,−2

.. .

j0,−2

.. .

Since λτ is a ∗-representation, for each (s, t) ∈ (−N0 )2 , there is a unique word eu fv of degree (|s|, |t|) such that ξι0 (∅) is in the range of λτ (eu fv ). Set ξs,t = λτ (eu fv )∗ ξι0 (∅) . It is not hard to see that this forms an orthonormal basis for H. For each (s, t) ∈ (−N0 )2 , there are unique integers is,t ∈ {1, . . . , m} and js,t ∈ {1, . . . , n} so that στ (eis,t )ξs−1,t = ξs,t

for s 0 and t 0,

στ (fjs,t )ξs,t−1 = ξs,t

for s 0 and t 0,

στ (ei )ξs,t = 0

if i = is+1,t or s = 0,

στ (fj )ξs,t = 0

if j = js,t+1 or t = 0.

Note that we label the edges leading into each vertex, rather than leading out. This choice reflects the fact that the basis vectors for a general atomic partial isometry representation are each in the range of at most one of the partial isometries π(ei ) and at most one of the π(fj ).


825

Consider how the tail τ = ei0 fj0 ei1 fj1 . . . determines these integers. It defines the path down the diagonal; that is, is,s := i|s|

and js−1,s := j|s|

for s 0.

This determines the whole representation uniquely. Indeed, for any vertex ξs,t with s, t 0, take T |s|, |t|, and select a path from (−T , −T ) to (0, 0) that passes through (s, t). The word τT = ei0 fj0 . . . eiT −1 fjT −1 satisfies στ (τT )ξ−T ,−T = ξ0,0 . Factor it as τT = w1 w2 with d(w1 ) = (T − |s|, T − |t|) and d(w2 ) = (|s|, |t|), so that στ (w2 )ξ−T ,−T = ξs,t and στ (w1 )ξs,t = ξ0,0 . Then w1 = eis,t w = fjs,t w . It is evident that each στ (ei ) and στ (fj ) is a partial isometry. Moreover, each basis vector is in the range of a unique στ (ei ) and στ (fj ). So this is a defect free, atomic representation with minimal ∗-dilation λτ . The symmetry group. An important part of the analysis of these atomic representations is the recognition of symmetry. Definition 3.3. The tail τ determines the integer data Σ(τ ) = (is,t , js,t ): s, t 0 . (k)

(k)

Two tails τ1 and τ2 with data Σ(τk ) = {(is,t , js,t ): s, t 0} are said to be tail equivalent if the two sets of integer data eventually coincide; i.e. there is an integer T so that (1) (1) (2) (2) for all s, t T . is,t , js,t = is,t , js,t Say that τ1 and τ2 are (p, q)-shift tail equivalent for some (p, q) ∈ Z2 if there is an integer T so that (2) (2) (1) (1) for all s, t T . is+p,t+q , js+p,t+q = is,t , js,t Then τ1 and τ2 are shift tail equivalent if they are (p, q)-shift tail equivalent for some (p, q) ∈ Z2 . The symmetry group of τ is the subgroup Hτ = (p, q) ∈ Z2 : Σ(τ ) is (p, q)-shift tail equivalent to itself . A sequence τ is called aperiodic if Hτ = {(0, 0)}. The semigroup F+ θ is said to satisfy the aperiodicity condition if there is an aperiodic infinite word. Otherwise we say that F+ θ is periodic. In our classification of atomic ∗-representations, an important step is to define a symmetry group for the more general representations which occur. The ∗-representation will be irreducible precisely when the symmetry group is trivial. This will yield a method to decompose atomic ∗-representations as direct integrals of irreducible atomic ∗-representations. The graph of an atomic representation. We need to develop a bit more notation. Let σ be an atomic representation. Let the corresponding basis be {ξk : k ∈ S}. Write ξ˙k to denote the

826


subspace Cξk . Form a graph Gσ with vertices ξ˙k . If σ (ei )ξ˙k = ξ˙l , draw a directed blue edge from ξ˙k to ξ˙l labelled i; and if σ (fj )ξ˙k = ξ˙l , draw a directed red edge from ξ˙k to ξ˙l labelled j . This is the graph of the representation, and it contains all of the information about σ except for the scalars in T. In our analysis of atomic representations, one can easily split a representation into a direct sum of atomic representations which have a connected graph. So we will generally work with representations with connected graph. Lemma 3.4. Let σ be a defect free atomic representation with connected graph Gτ . Let ξ˙1 and ˙ ξ˙2 be two vertices in Gσ . Then there is a vertex η˙ and words w1 , w2 ∈ F+ θ so that ξi = σ (wi )η˙ for i = 1, 2. Proof. The connectedness of the graph means that there is a path from ξ˙1 to ξ˙2 . We will modify this path to first pull back along a path leading into ξ˙1 , and then move forward to ξ˙2 . The original path can be written formally as ak ak−1 . . . a1 where each al has the form ei or fj if it is moving forward or ei∗ or fj∗ if pulling back. After deleting redundancies, we may assume that there are no adjacent terms fj∗ fj or ei∗ ei . At each vertex, there is a unique blue (red) edge leading in; so moving forward along a blue (red) edge and pulling back along the same colour is just one of these redundancies. Thus if there are adjacent terms of the form a ∗ b which do not cancel, then one of a, b is an e and the other an f . For definiteness, suppose that this section of the path moves from η˙ 1 to η˙ 2 along the path fj∗ ei . That means that σ (ei )η˙ 1 = η˙ = σ (fj )η˙ 2 . As σ is defect free, there is a basis vector η˙ 0 and an fj so that σ (fj )η˙ 0 = η˙ 1 . Factor ei fj in the other order as fj ei . Then σ (fj )η˙ 2 = η˙ = σ (ei fj )η˙ 0 = σ (fj ei )η˙ 0 = σ (fj ) σ (ei )η˙ 0 . However there is a unique red edge into η, and thus fj = fj

and σ (ei )η˙ 0 = η˙ 2 .

It is now clear that there is a red–blue diamond in the graph with apex η˙ 0 , and red and blue edges leading to η˙ 1 and η˙ 2 , respectively, and from there, red and blue edges, respectively, leading into η. ˙ So the path fj∗ ei may be replaced by ei fj∗ . Similarly, the path ei∗ fj from η˙ 2 to η˙ 1 may be replaced by fj ei∗ . Repeated use of this procedure replaces any path from ξ˙1 to ξ˙2 by a path of the form w2 w1∗ ; and hence η = σ (w1 )∗ ξ1 is the intermediary vector. 2 4. Classifying atomic representations Consider an atomic ∗-representation π of F+ θ with connected graph Gπ . Observe that if we restrict π to the subalgebra generated by the ei ’s, we obtain an atomic, defect free representation of the free semigroup F+ m . The graph splits into the union of its blue components. By [3], this decomposes the restriction into a direct sum of ring representations and infinite tail representations. Our first result shows that the connections provided by the red edges force a parallel structure in the blue components.


827

Lemma 4.1. Let π be an atomic ∗-representation of F+ θ with connected graph Gπ . Let H1 and H2 be two blue components of Gπ and assume that there is a red edge leading from a vertex in H1 to a vertex in H2 . Then every red edge into H2 comes from H1 . The two components are either both of infinite tail type or both are of ring type; and in the ring case the length of the (unique) ring for H2 is an integer multiple t of the ring length in H1 , where 1 t n. Proof. This is an exercise in using the commutation relations. Suppose first that H1 is an infinite tail graph. Fix a vertex ξ˙0 ∈ H1 and a red edge fj0 so that π(fj0 )ξ˙0 = ζ˙0 is a vertex in H2 . There is a unique infinite sequence of vertices ξk for k < 0 and integers ik so that π(eik )ξ˙k−1 = ξ˙k

for k 0.

Now there are unique integers ik and jk so that ...i f jk fj0 ei0 i−1 ...ik+1 = ei0 i−1 k+1

for k < 0.

Let ζ˙k := π(fjk )ξ˙k for k < 0. Then it is evident that π(eik )ζ˙k−1 = ζ˙k

for k 0.

The images of each vertex under the various red edges are all distinct. So in particular, the ζ˙k are all distinct, and so H2 is also an infinite tail graph. Now one similarly can follow each vertex ζ˙k forward under a blue path eu to reach any vertex ζ˙ in H2 . Then ζ˙ = π(eu )ζ˙k = π(eu )π(fjk )ξ˙k = π(fj )π(eu )ξ˙k . Thus the vertex π(eu )ξ˙k in H1 is mapped to ζ˙ by π(fj ). Hence the red edge leading into ζ˙ comes from H1 . The case of a ring representation is similar. Starting with any vertex in H1 which is connected to H2 by a red edge, one can pull back along the blue edges until one is in the ring. So we may suppose that ξ˙0 lies in the ring of H1 and that π(fj0 )ξ˙0 = ζ˙0 in H2 . Let u be the unique minimal word such that π(eu )ξ˙0 = ξ˙0 , and let p = |u|. As in the first paragraph, we continue to pull back from ζ˙0 and from ξ˙0 along the blue edges. Call these edges ζ˙k and ξ˙k , respectively, for k < 0. After p steps, we return to ξ˙0 = ξ˙−p in H1 , and we have obtained p distinct vertices in H2 and reach ζ˙−p . So there is a word u of length p so that π(eu )ζ˙−p = ζ˙0 . The commutation relations yield π(eu )ζ˙−p = ζ˙0 = π(fj0 )π(eu )ξ˙0 = π(eu )π(fj )ξ˙0 . There is a unique blue path of length p leading into ζ˙0 . Therefore u = u and π(fj )ξ˙0 = ζ˙−p . Notice that if j = j0 , then ζ˙−p = ζ˙0 . However if j = j0 , then ζ˙−p is a different vertex in H2 . Repeat the process, pulling back another p blue steps, to reach a vertex ζ˙−2p . Along the way, we obtain vertices which are distinct from the previous ones, and each is the image of some vertex in the ring of H1 . As before, ζ˙−2p is the image of ξ˙0 under some red edge. Eventually this

828


process must repeat, because there are only n red edges out of ξ˙0 . That is, there are integers s and t with 1 t n so that ζ˙−(s+t)p = ζ˙−sp . This is a ring in H2 of length tp. The argument that each edge in H2 is in the range of a red edge coming from H1 is identical to the infinite tail case. 2 One might hope that the red edges from H1 to H2 provide a nice bijection, or a t-to-1 map that preserves the graph structure. Even though these edges are determined algebraically by the commutation relations, such a nice pairing does not occur as the following examples demonstrate. Example 4.2. Consider m = n = 3 with θ given by ((1, 2), (2, 1)), or equivalently by the relations e1 f 2 = f 1 e 2 ,

e2 f 1 = f 2 e1 ,

and ei fj = fj ei ,

otherwise.

There is a 1-dimensional defect free representation ρ(e3 ) = ρ(f3 ) = 1 and ρ(ei ) = ρ(fi ) = 0 for i = 1, 2. This has a dilation to a ∗-representation π . Let the initial vector be called ξ0 . Define ζj 0 = π(fj )ξ0 for j = 1, 2; and let ξi = π(ei )ξ0 and ζj i = π(ei )ζj 0 for i = 1, 2 and j = 1, 2. The blue component H0 containing ξ0 has a ring of length one. 3

3

ξ0

1

2 1

3

3

ξ1

2

ζ10 2

1

3

ζ20 2

1

2

ξ2

3

1 2

1

ζ11

ζ12

ζ21

ζ22

It is easy to check that π(e3 )ζj 0 = ζj 0 for j = 1, 2. Hence each is a ring in a separate blue component Hj . But the commutation relations also show that π(fj )ξi = ζij . This means that the vertex ξ˙1 has two red edges leading to the component H1 ; and ξ˙2 has two red edges leading to H2 . Example 4.3. With the same algebra, consider the 1-dimensional representation ρ(e1 ) = ρ(f3 ) = 1 and ρ(ei ) = ρ(fj ) = 0 otherwise. Again this has a dilation to an isometric defect free representation π . Let the initial vector be called ξ . Define ζj = π(fj )ξ for j = 1, 2. A computation with the relations shows that π(e1 )ζ1 = ζ1 . So this is a ring of length one in a component H1 . But π(e2 )ζ1 = ζ2 . So ζ2 is also in H1 ; but even though it is in the range of a red edge from the ring of H0 , it does not lie on the ring of H1 .


1

3

ξ

1

1

829

2 2

ζ1

ζ2

Example 4.4. Consider m = n = 2 and the reverse 3-cycle semigroup of Example 2.1 given by the 3-cycle ((1, 1), (2, 1), (1, 2)). There is a 1-dimensional defect free representation ρ(e2 ) = ρ(f2 ) = 1 and ρ(e1 ) = ρ(f1 ) = 0. This has a dilation to a ∗-representation π . Let the initial vector be called ξ . Define η = π(f1 )ξ and ζj = π(fj )η for j = 1, 2. Then an exercise with the relations shows that π(e1 )η = η and π(e1 )ζ2 = ζ1 and π(e2 )ζ1 = ζ2 . 2

ξ

2

1 1

η

1

2 2

ζ1

ζ2 1

Thus the initial component H0 has a ring of length one at ξ , as does the component H1 connected to it (at η), but it connects to a component H2 in which ζi are the vertices of a ring of length 2. One can show that there are components with rings of length 2k for all k 0. Splitting into cases. Evidently the reasoning of Lemma 4.1 also applies when we decompose the graph into its red components. It is now possible to split the analysis into several cases. 1. (Ring by ring type.) Both blue and red components are ring representations. In this case, the set of vertices which are in both a red and a blue ring determines a finite-dimensional coinvariant subspace on which the representation is defect free, and there is exactly one red and blue edge beginning at each vertex. Moreover this is a cyclic subspace for the representation because starting at any basis vector, pulling back along the blue and red edges eventually ends in the ring by ring portion. So this finite-dimensional piece determines the full representation. Since each ring is obtained by pulling back from any of the others, it follows from Lemma 4.1 that all of the blue rings have the same length, say k; and likewise all of the red rings have the same length, say l. 2. (Mixing type.) For one colour, the components are ring representations while the for the other colour, the components are infinite tail representations. There are two subcases.

830


2a. (Ring by tail type.) The blue components are ring representations, and the red components are infinite tail type. If one begins at any blue ring and pulls back along the red edges, one obtains an infinite sequence of blue ring components. By Lemma 4.1, the size of the ring is decreasing as one pulls back. Hence it is eventually constant. From this point on back, there is a unique red edge from each ring to the corresponding point on the next. Thus one obtains a semi-infinite cylinder of fixed circumference k which is coinvariant and cyclic, and determines the full representation. 2b. (Tail by ring type.) The red components are ring representations, and the blue components are infinite tail type. 3. (Tail by tail type.) Both the red and the blue components are of infinite tail type. This actually has several subtypes. Start with a basis vector ξ˙ in a blue component H0 . Pull back along the red edges to get an infinite sequence of blue components Ht for t 0. The union of these components for t 0 forms a coinvariant cyclic subspace that determines the full representation. This sequence may be eventually periodic, or they may all be distinct. If they are eventually periodic, we may assume that we begin with a component H0 in the periodic sequence. 3a. (Inductive type.) The sequence of components are all distinct. In this case, starting at any basis vector ξ˙0,0 , one may pull back along both blue and red edges to obtain basis vectors ξs,t for (s, t) ∈ (−N0 )2 . The restriction of the representation to this coinvariant subspace is defect free, and determines the whole representation as in Example 3.2. So this is an inductive representation. 3b. The sequence of components repeats after l steps. Thus by Lemma 4.1, there are blue components H0 , . . . , Hl−1 so that every red edge into each Hi comes from Hi−1 (mod l) . This is further refined by comparing the point of return to the starting point ξ˙ . It is not apparent at this point that the only possibilities are the following. 3bi. (Return below.) The return is eventually below the start. In this case, there is a vertex ξ˙0 in H0 and a word u0 so that the vertex ζ˙0 obtained by pulling back l red steps using the word v0 from ξ˙0 satisfies π(eu0 )ξ˙0 = ζ˙0

and π(fv0 )ζ˙0 = ξ˙0 .

This same type of relationship persists when pulling back along both blue and red edges from ξ˙0 . 3bii. (Return above.) The return is eventually above the start. Here there is a vertex ξ˙0 in H0 and a word u0 so that the vertex ζ˙0 obtained by pulling back l red steps using the word v0 from ξ˙0 satisfies π(eu0 )ζ˙0 = ξ˙0 = π(fv0 )ζ˙0 . This same type of relationship persists when pulling back along both blue and red edges from ξ˙0 .


831

The ring by ring representations are finitely correlated, meaning that there is a finitedimensional coinvariant, cyclic subspace. Equivalently, this means that the ∗-representation is the minimal ∗-dilation of a finite-dimensional defect free representation. Conversely, every finitely correlated atomic ∗-representation with connected graph is of ring by ring type, as the other cases clearly do not have a finite-dimensional non-zero coinvariant and cyclic subspace. In the following sections, each case will be considered in more detail. Eventually a common structure emerges. This will be codified by the general construction given in the next section. Symmetry. In each case, we associate a symmetry group to the picture. In the ring by ring case, it is easiest to describe because there is no equivalence relation. If the minimal coinvariant subspace consists of blue cycles of length k and red cycles of length l, then we associate the representation to a quotient group G of Ck × Cl ; and the symmetry group is a subgroup of G. We show that there is a finite-dimensional representation on Ckl which reflects the full symmetry, and that certain quotients yield a decomposition into irreducible summands. In type 3a, the inductive case, we have already seen how to define a symmetry subgroup of Z2 . Again, an inductive representation will be irreducible precisely when this symmetry group is trivial. In the other cases, the symmetry group is a subgroup of Ck × Z in the type 2a case or of Z2 /Z(a, b) in the 3b cases. In types 2 and 3, the decomposition into irreducibles may require a direct integral rather than a direct sum. 5. A group construction In this section, we will describe a general class of examples, and explain how to decompose them into irreducible representations. Start with an abelian group G with two designated generators g1 and g2 . We consider a defect 2 free atomic representation of F+ θ on (G) which is given by the following data: i : G → {1, . . . , m},

i(g) =: ig ,

j : G → {1, . . . , n},

j (g) =: jg ,

α : G → T,

α(g) =: αg ,

β : G → T,

β(g) =: βg .

We wish to define a representation σ of F+ θ by σ (ei )ξg = δi ig αg ξg+g1 , σ (fj )ξg = δj jg βg ξg+g2 . In order for this to actually be a representation, we require that eig+g2 fjg = fjg+g1 eig

for all g ∈ G

832


and αg+g2 βg = βg+g1 αg

for all g ∈ G.

Let Gσ be the graph of this representation, which has vertices ξ˙g for g ∈ G and blue edges labelled ig from ξ˙g to ξ˙g+g1 , and red edges labelled jg from ξ˙g to ξ˙g+g2 . These will be called group construction representations of F+ θ . It is evident that there is a unique blue and red edge leading into each vertex ξ˙g , and so this is a defect free atomic representation. Thus it has a unique minimal ∗-dilation. Decomposing this representation into irreducible summands or a direct integral must simultaneously decompose the ∗-dilation into a direct sum or direct integral of the minimal ∗-dilations of the summands. The symmetry group of σ is defined as H = {h ∈ G: ig+h = ig and jg+h = jg for all g ∈ G}. This will play a central role in this decomposition. We will address the non-trivial issue of how to actually define the functions i and j later when considering the various cases. There are obstructions, and the way by which this difficulty is overcome is not immediately apparent. Example 3.1 is an example of this type in which G is the cyclic group Ck and g1 = −g2 = 1. The case of G = Z2 can be seen in Example 3.2. Here we describe a coinvariant subspace which is identified with (−N0 )2 , but it can be extended to all of Z2 . (We apologize to the reader that the notation is not consistent between Example 3.2 and this section.) The issue of the scalar functions α and β is more elementary. We shall see that it suffices to consider the case in which α and β are constant, and will determine when two are unitarily equivalent. For the moment, we assume that such a representation is given and consider how to analyze it. The group G, being abelian with two generators, is a quotient of Z2 . The subgroups of Z2 are {0}, singly generated Z(a, b), or doubly generated. In the doubly generated case, it is an easy exercise to see that the quotient is finite. In this case, if gi has order pi for i = 1, 2, then G is in fact a quotient of Cp1 × Cp2 . We shall show that the different groups correspond to the different representation types as follows: Type 1. G finite. Type 2. (a) G = Ck × Z; (b) G = Z × Cl . Type 3. (a) G = Z2 ; (bi) G = Z2 /Z(k, l), kl > 0; (bii) G = Z2 /Z(k, l), kl < 0. Scalars. Let us dispense with the scalars first. This is actually quite straightforward in spite of the notation. For each group G, there is a canonical homomorphism κ of Z2 onto G sending the standard generators (1, 0) and (0, 1) to g1 and g2 . Let K be the kernel of κ, so that G Z2 /K.


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Theorem 5.1. Let G = Z2 /K as above. A group construction representation σ on 2 (G) is unitarily equivalent to another of the same type with the same functions i, j for which the scalar functions are constants α0 and β0 . The constants determine a unique character ψ of K. They are unique up to a choice of an extension ϕ of ψ to a character on Z2 ; and they are given by α0 = ϕ(1, 0) and β0 = ϕ(0, 1). The choice of ϕ is unique up to a character χ of G, which changes α0 and β0 to α0 χ(g1 ) and β0 χ(g2 ). Proof. For each (s, t) ∈ Z2 , there is a unique path from ξ˙0 to ξ˙κ(s,t) in Gσ with s blue edges and t red edges, where a negative number indicates traversing the arrow in the backward direction. Corresponding to this, there is a unique partial isometry W in C∗ (σ (F+ θ )) of the form σ (eu )σ (fv ), σ (fv )∗ σ (eu ), σ (eu )∗ σ (fv ) or σ (eu )∗ σ (fv )∗ with |u| = |s| and |v| = |t|, depending on the signs of s and t, so that W ξ˙0 = ξ˙κ(s,t) . The subgroup K corresponds to those paths which return ξ˙0 to itself. Thus for (s, t) ∈ K, W ξ0 = ψ(s, t)ξ0 for a unique scalar ψ(s, t) ∈ T. In any unitarily equivalent representation given in the same form on 2 (G), the vector come from sent to ξ0 must be a vector with exactly the same functions i, j ; and thus must span{ξh : h ∈ H }. Moreover there are constraints on the scalars, namely if U ∗ ξ0 = h∈A ah ξh where A = {h: ah = 0}, then α(h + g) = α(h + g) and β(h + g) = β(h + g) for all h, h ∈ A and all g ∈ G. (For otherwise, the representation would not correspond to a graph.) Hence exactly the same words return each ξh to itself, for h ∈ A. We claim that the function ψ on K is independent of the unitary equivalence. Let ψ be the function obtained in this equivalent representation. The argument of the previous paragraph shows that instead of computing the function ψ using U ∗ ξ0 , we can use ξh for any h ∈ A and obtain the same result. Let W be the partial isometry found in the first paragraph which carries ξ˙0 to ξ˙h . Let (s, t) ∈ K, and let W0 and Wh be the partial isometries of degree (s, t) which take ξ˙0 and ξ˙h to themselves, respectively. Now W ∗ W W0 and W ∗ Wh W are partial isometries of the same combined and absolute degrees which map ξ˙0 to itself. The uniqueness of factorization means that these two words are equal! Therefore

ψ(s, t) = W ∗ W W0 ξ0 , ξ0 = W ∗ Wh W ξ0 , ξ0 = ψ (s, t). So we see that the original choice of scalars forces certain values, namely the function ψ, to be fixed independent of unitary equivalence. Next we show that ψ is a character. Given two words (s1 , t1 ) and (s2 , t2 ) in K, let W1 and W2 be the corresponding partial isometries. Then the partial isometry W corresponding to the sum (s1 + s2 , t1 + t2 ) need not equal W1 W2 . However the unique factorization means that W1 W2 = V ∗ V W for some partial isometry V containing ξ0 in its domain. Thus one computes

ψ(s1 + s2 , t1 + t2 ) = V ∗ V W ξ0 , ξ0 = W1 W2 ξ0 , ξ0 = ψ(s1 , t1 )ψ(s2 , t2 ). So ψ is multiplicative. It is routine to extend ψ to a character ϕ of Z2 . In our context, one can easily do this ‘bare hands.’ But it is a general fact for characters on any subgroup of any abelian group [8, Corollary 24.12]. If ϕ1 and ϕ2 are two characters extending ψ , then ϕ2 ϕ1 is a character which takes the constant value 1 on all of K. Thus it induces a character χ of G. We see that ϕ2 (s, t) = ϕ1 (s, t)χ(sg1 + tg2 ). Conversely, any choice of χ yields an extension.

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The unitary equivalence between σ and the representation σ on the same graph but with constants α0 = ϕ(1, 0) and β0 = ϕ(0, 1) can be accomplished by a diagonal unitary U = diag(γg ). Define γg by selecting a partial isometry W as in the first paragraph so that W ξ˙0 = ξ˙g . Let (a, b) be the degree of the word W . Define γg = ϕ(a, b) W ξ0 , ξg . While the choice of W is not unique, if W is another such partial isometry, W ∗ W ξ˙0 = ξ˙0 corresponds to a word (s, t) in K. Then W has degree (a + s, b + t). Thus

ϕ(a + s, b + t) W ξ0 , ξg = ϕ(a, b)ϕ(s, t) W W ∗ W ξ0 , ξg = ϕ(a, b)ϕ(s, t)ϕ(s, t) W ξ0 , ξg = ϕ(a, b) W ξ0 , ξg . So U is well defined. Now if γg is computed using W , calculate

γg+g1 = ϕ(a + 1, b) σ (eig )W ξ0 , ξg+g1 = ϕ(a + 1, b)αg W ξ0 , ξg . Then we see that U σ (eig )U ∗ ξg = U σ (eig )ϕ(a, b) W ξ0 , ξg ξg = U αg ϕ(a, b) W ξ0 , ξg ξg+g1 = ϕ(a + 1, b)αg W ξ0 , ξg αg ϕ(a, b) W ξ0 , ξg ξg+g1 = ϕ(1, 0)ξg+g1 = α0 ξg+g1 . One deals with σ (fjg )ξg in the same manner. Thus σ is unitarily equivalent to the representation σ with scalars α0 and β0 . It was irrelevant which extension ϕ of ψ was used; so all are unitarily equivalent to each other. 2 Symmetry. The key to the decomposition is to look for symmetry in the graph of σ . Recall that H = {h ∈ G: ig+h = ig and jg+h = jg for all g ∈ G}. It is clear that H is a subgroup of G. Note that we ignore the scalars for this purpose which is justified by Theorem 5.1. Indeed, we shall suppose that the scalars are constants α0 and β0 . For each coset [g] = g + H of G/H , let W[g] = span{ξg+h : h ∈ H }. Pick a representative gk for each coset of G/H , selecting 0 ∈ [0]. Each of the subspaces W[gk ] can be identified with 2 (H ). This identification depends on the choice of representative. Define J[gk ] : 2 (H ) → W[gk ] by J[gk ] ξh = ξgk +h .


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Since ig = ig+h and jg = jg+h , these integers are dependent only on the coset. So we will write i[g] and j[g] . Observe that σ (ei[g] ) maps W[g] to W[g+g1 ] . To understand this map, note that for each gk , there is a unique element hk1 ∈ H so that the representative for [gk + g1 ] is gk + g1 − hk1 . Then it is easy to see that ∗ σ (ei[gk ] )|W[gk ] = α0 J[gk +g1 ] Lhk1 J[g , k]

where Lh is the (left) regular translation by h on 2 (H ). Likewise, there is an hk2 ∈ H so that gk + g2 − hk2 is the chosen representative for [gk + g2 ]. Then ∗ σ (fj[gk ] )|W[gk ] = β0 J[gk +g2 ] Lhk2 J[g . k]

It is now a routine matter to diagonalize σ . The unitary operators Lh for h ∈ H all commute, and so can be simultaneously diagonalized by the Fourier transform which identifies 2 (H ) with L2 (Hˆ ), and carries Lh to the multiplication operator Mh given by Mh f (χ) = χ(h)f (χ). We explain in more detail how this works in the finite case. Here Lh are unitary matrices, and σ decomposes as a finite direct sum of irreducible representations. For each χ ∈ Hˆ , let

ζ0 = |H |−1/2 χ

χ(h)ξh .

h∈H χ

χ

Then a routine calculation shows that Lh ζ0 = χ(h)ζ0 . Consider the subspaces χ χ Mχ = span ζ[g] := J[g] ζ0 : [g] ∈ G/H . The choice of representative for each coset only affects the scalar multiple of the vectors, and the subspace Mχ is independent of this choice. It is easy to see that these are reducing subspaces for 2 σ (F+ θ ). This decomposes σ into a direct sum of representations σχ acting on (G/H ). Indeed, if [gk + g1 ] = [gl ], then we calculate σχ (ei[gk ] )ζ[gk ] = σ (ei[gk ] )J[gk ] |H |−1/2 χ

−1/2

= σ (ei[gk ] )|H | = α0 |H |−1/2

χ(h)ξh

h∈H

χ(h)ξgk +h

h∈H

χ(h)ξgk +g1 +h

h∈H

= α0 |H |−1/2

χ(h)ξgl +hk1 +h

h∈H

= α0 J[gl ] χ(hk1 )|H |−1/2

χ(h + hk1 )ξh+hk1

h∈H χ

χ

= α0 χ(hk1 )ζ[gl ] = α0 χ(hk1 )ζ[gk +g1 ] .

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Similarly, χ

χ

σχ (fj[gk ] )ζ[gk ] = β0 χ(hk2 )ζ[gk +g2 ] . So the representations σχ all act on 2 (G/H ) with the same functions i, j , but with different constants. Since H is finite, so is G; and we may write G = Z2 /K where K is a subgroup of finite index. Then G/H = Z2 /H K =: Z2 /L. We wish to calculate the character ψχ on L which distinguishes σχ . Lemma 5.2. Let σ be a group construction representation on 2 (Z2 /K) with symmetry group ˆ Then the summands σχ are determined L/K, and scalars determined by the character ψ ∈ K. by the set of characters ψχ ∈ Lˆ satisfying ψχ |K = ψ. The enumeration is given by elements which are related by ψχ (l) = ψ0 (l)χ(l + K); and this enumerates all possible of Hˆ = L/K extensions of ψ from K to L. ˙χ ˙χ Proof. For each l ∈ L, there is a unique word wl ∈ F+ θ of degree l so that σχ (wl )ξ0 = ξ0 ; and then χ

χ

σχ (wl )ξ0 = ψχ (l)ξ0 . When k ∈ K, one has σ (wk )ξh = ψ(k)ξh

for all h ∈ H.

Therefore it follows that χ

χ

σχ (wk )ξ0 = ψ(k)ξ0

for all h ∈ H.

So ψχ |K = ψ . In general, we fix an extension ψ0 of ψ to Z2 and use this to calculate ψχ as in Theorem 5.1. For l ∈ L, let hl := l + K ∈ L/K = H . 1 σ (wl ) ψχ (l) = χ(h1 )ξh1 , χ(h2 )ξh2 |H | h1 ∈H

=

1 |H |

h2 ∈H

χ(h1 )χ(h2 )ψ0 (l) ξh1 +hl , ξh2

h1 ∈H h2 ∈H

= χ(hl )ψ0 (l). Hence we obtain the desired relationship between ψ0 and ψχ . Note that the definition of the subspaces Mχ depends on the choice of ϕ, but that the decomposition is unique. All extensions of ψ occur in this manner, so we have enumerated all possibilities. 2 Lastly we explain why these summands are irreducible. Clearly, if H = {0}, the representation is reducible because we have exhibited a non-trivial collection of reducing subspaces. The


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restriction to each of these subspaces yields a representation of G/H on 2 (G/H ). By construction, the symmetry group of each of these representations is H /H = {0}. So irreducibility follows from the following lemma. 2 Lemma 5.3. If a group construction representation of F+ θ on (G) has symmetry group {0}, then it is irreducible.

Proof. The complete lack of symmetry means that for any element g ∈ G \ {0}, there is a word w ∈ F+ θ so that σ (w)ξ0 = 0 and σ (w)ξg = 0. Indeed, if we write G = {g0 = 0, g1 , g2 , . . .}, there are words wk so that σ (wk )ξ0 = 0 and σ (wk )ξgi = 0

for 1 i k.

Thus Pk = σ (wk )∗ σ (wk ) are projections such that Pk ξ0 = ξ0 for all k 1 and Pk ξgi = 0 when k i. Therefore WOT-lim Pk = ξ0 ξ0∗ . ∗ For any gi ∈ G, there are words wi , xi ∈ F+ θ so that σ (wi ) σ (xi )ξ0 = ξgi . Set Vi := ∗ ∗ ∗ ∗ σ (wi ) σ (xi ). Then Vi ξ0 ξ0 Vj = ξgi ξgj . This is a complete set of matrix units for the compact operators in the von Neumann algebra generated by σ (F+ θ ). Therefore σ is irreducible. 2 A similar analysis works in the case of infinite groups. Of course, there are no subspaces corresponding to the representations σχ ; but the procedure is just a measure theoretic version of the same. Putting all of this together, we obtain the following decomposition. 2 Theorem 5.4. Let σ be a group construction representation of F+ θ on G = Z /K. If the symmetry group H is finite, then σ decomposes as a direct sum of irreducible atomic representations σχ on 2 (G/H ), one for each χ ∈ Hˆ . If H is infinite, then σ decomposes as a direct integral over the dual group Hˆ of irreducible atomic representations σχ on 2 (G/H ). The representations σχ all have the same graph, and the scalars are given by all possible extensions of the character ψ on K to L = H K.

Further considerations. Because of Dilation theorem 2.5, we know that an atomic ∗-dilation is uniquely determined by its restriction to any cyclic coinvariant subspace. In the cases examined in this section, one can often select a smaller subspace which will suffice. In the case of a finite group, the subspace has no proper subspace which is cyclic and coinvariant. But when the group is infinite, there are many such subspaces—and none are minimal. For example, in Example 3.2 we saw that the restriction of a representation on Z2 to (−N0 )2 is such a subspace. Indeed, if the subspace is spanned by standard basis vectors, then whenever if contains ξs0 ,t0 , it must contain all ξs,t for s s0 and t t0 . Likewise, if G = Z2 /Z(p, q) with pq 0, then span{ξ[s,t] : s s0 , t t0 } is a proper cyclic coinvariant subspace. In the case pq > 0, it is easy to see that there is no proper cyclic invariant subspace. However we shall also see that Z(p, q) with pq > 0 can never be the full symmetry group of any representation of F+ θ . This discussion suggests that we need to put an equivalence relation on these representations. It is evident that two equivalent representations must have unitarily equivalent ∗-dilations. 2 Definition 5.5. Two group construction representations σ and σ of F+ θ on G = Z /K, with data {α0 , β0 , ig , jg : g ∈ G} and {α0 , β0 , ig , jg : g ∈ G}, respectively, are said to be equivalent if there ˆ so that is an integer T and a character χ ∈ G

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α0 = χ(g1 )α0 , ig = ig

β0 = χ(g2 )β0 ,

and jg = jg

for all g = [s, t], s, t T .

Conversely, a group-like construction which is defined only on a coinvariant cyclic atomic subspace span{ξg : g = [s, t], s, t T } may be extended to the full group, usually in many ways. To see this, take the minimal ∗-dilation π . There are two cases, G = Z2 and G = Z2 /Z(p, q) with pq 0. In the first case, take the basis vector ξT ,T and any infinite word . . . fjk eik . . . fjT eiT . Define ξk,k = π(fjk−1 eik−1 . . . fjT eiT )ξT ,T . The minimal coinvariant subspace Mk generated by ξk,k can be identified with an atomic basis {ξg : g = [s, t], s, t k}. These subspaces are nested, and their union yields a defect free atomic representation on 2 (G) as desired. In the case of G = Z2 /Z(p, q) with pq 0, we may suppose that q = 0. Then using an infinite word . . . eik . . . eiT will work in exactly the same way. The other issue to discuss here is how the decomposition of the ∗-dilation corresponds to the decomposition of the restriction to the coinvariant subspace. This is a direct consequence of the uniqueness of minimal dilations. Indeed in the case of a representation which decomposes into a direct sum of irreducible representations, the direct sum of the ∗-dilations of the summands is clearly a minimal ∗-dilation. Hence it is the unique ∗-dilation. That is, the ∗-dilation of a direct sum is the direct sum of the ∗-dilations of the summands. 2 Theorem 5.6. Let σ be a group construction representation of F+ θ on (G) with symmetry group H . Then the minimal ∗-dilation decomposes as a direct integral of the ∗-dilations of the irreducible integrands in the direct integral decomposition of σ .

Proof. Let π denote the minimal ∗-dilation of σ on K. We first show that there is a spectral measure on K over measurable subsets of Hˆ which is absolutely continuous with respect to Haar measure and extends the spectral measure on 2 (G). Write σ as a direct integral over Hˆ . For each measurable subset A ⊂ Hˆ , let E(A) be the spectral projection onto span{Mχ : χ ∈ A}. The restriction σA of σ to this subspace has a minimal ∗-dilation πA . By uniqueness, π πA ⊕ πAc . This decomposition splits K F (A)K ⊕ F (Ac )K. It is routine to check that F is countably additive, that E(A) = PH F (A) = F (A)PH , and that F is absolutely continuous. The rest follows from standard arguments. Since we do not actually need any explicit formulae for the direct integral decomposition for any of our analyses, we will not subject the reader to the technicalities. 2 The main theorem. The central result of this paper is the following: Theorem 5.7. Every atomic ∗-representation of F+ θ with connected graph is the minimal ∗dilation of a group construction representation. It is irreducible if and only if its symmetry group is trivial. In general, it decomposes as a direct sum or direct integral of irreducible group construction representations. The proof evolves from a case by case analysis of the various types.


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6. Finitely correlated atomic representations In this section, we restrict our attention to atomic ∗-representations of F+ θ which are finitely correlated. Such representations are particularly tractable. As in the case of the free semigroup [5], the whole class of finitely correlated ∗-representations may turn out to be classifiable. This general problem is not considered here. We assume that the graph is connected. By the discussion in Section 4, we obtain a coinvariant cyclic subspace spanned by the ring by ring portion of the graph. We wish to show that it always arises from a group construction. To this end, we need a criterion for when this construction is possible. ˙ Lemma 6.1. Let ρ be an atomic representation of F+ θ . If ξ is a vertex of Gρ and u, v are words ˙ ˙ ˙ such that ρ(eu )ξ = ξ = ρ(fv )ξ , then eu fv = fv eu . Proof. There are words u and v so that eu fv = fv eu . So ρ(fv eu )ξ˙ = ξ˙ = ρ(fv )ξ˙ . Since |v | = |v|, ρ(fv ) would have range orthogonal to ρ(fv ) unless v = v. Therefore they must be equal. Similarly u = u. 2 Lemma 6.2. Suppose that eu0 fv0 = fv0 eu0 where |u0 | = k and |v0 | = l. Then there is an atomic defect free representation σ on Ck × Cl with σ (eu0 )ξ0 = ξ0 = σ (fv0 )ξ0 . Arbitrary constants α, β ∈ T yield a representation σα,β given by σα,β (ei ) = ασ (ei ) and σα,β (fj ) = βσ (fj ). Then σα,β σα ,β if and only if α k = α k and β l = β l . Proof. Write u0 = ik−1,0 . . . i0,0 and v0 = j0,l−1 . . . j0,0 . The commutation relations show that there are unique words ut for 0 t l, so that fv0 eu0 = fj0,l−1 . . . fj0,t eut fj0,t−1 . . . fj0,0 . Write ut = ik−1,t . . . is,t . . . i0,t for 0 t l and note that ul = u0 . Similarly, there are unique words vs so that fv0 eu0 = eik−1,0 . . . eis,0 fvs eis−1,0 . . . ei0,0 . Write vs = js,l−1 . . . js,0 for 0 s k; and again one has vk = v0 . It follows from unique factorization that fv0 eu0 = eik−1,0 . . . eis+1,0 fjs+1,l−1 . . . fjs+1,t eis,t fjs,t−1 . . . fjs,0 eis−1,0 . . . ei0,0 = eik−1,0 . . . eis+1,0 fjs+1,l−1 . . . fjs+1,t+1 eis,t+1 fjs,t . . . fjs,0 eis−1,0 . . . ei0,0 . Now cancellation shows that fjs+1,t eis,t = eis,t+1 fjs,t

for all s ∈ Ck and t ∈ Cl .

These are the relations needed to allow the construction of a homomorphism on 2 (Ck × Cl ) as in Section 5. Namely,

840


σα,β (eis,t )ξs,t = αξs+1,t

and σα,β (fjs,t )ξs,t = βξs,t+1 .

One calculates that σα,β (eu0 )ξ0,0 = α k ξ0,0 and σα,β (fv0 )ξ0,0 = β l ξ0,0 . Clearly σα,β is also a defect free atomic representation for any α, β ∈ T. Indeed, scalars can be assigned to each edge arbitrarily; but by Theorem 5.1, there is no loss in making the constants all the same. The characters of Ck × Cl have the form χ(s, t) = ω1s ω2t where ω1k = 1 = ω2l . Thus Theorem 5.1 also shows that σα,β σα ,β if and only if α = χ(g1 )α = ω1 α and β = χ(g2 )β = ω2 β; and this is equivalent to α k = α k and β l = β l . 2 Corollary 6.3. Let H be the symmetry subgroup for the representation σα,β constructed in Lemma 6.2. For any subgroup K H , there is a group construction representation on 2 (G/K) such that σ (eu0 )ξ0 = α k ξ0

and σ (fv0 )ξ0 = β l ξ0 .

This representation decomposes as a direct sum of irreducible atomic defect free representations on 2 (G/H ) indexed by H /K. Proof. It is evident that the induced representation on 2 (G/K) fits the conditions of Section 5. The symmetry group is clearly H /K. Since H /K is finite, Decomposition theorem 5.4 yields a finite direct sum of irreducible representations on 2 (G/H ). 2 We are now ready to establish the following result. Theorem 6.4. Any defect free atomic representation of F+ θ on a finite-dimensional space with connected graph is isometrically isomorphic to a dilation of a group construction representation for a finite group G. Proof. Let σ be a given finitely correlated defect free representation on a finite-dimensional Hilbert space H, and with a connected graph. Pulling back along the blue (respectively red) edges eventually reaches a periodic state in the ring by ring portion of the representation. Fix a standard basis vector ξ0 in the ring by ring. One can find the unique minimal words u0 and v0 so that eu0 ξ˙0 = ξ˙0 and fv0 ξ˙0 = ξ˙0 . Then eu0 fv0 = fv0 eu0 by Lemma 6.1. For each (s, t) ∈ (N0 )2 , there is a unique word ws,t ∈ F+ θ of degree (s, t) such that σ (ws,t )ξ˙0 =: ξ˙s,t = 0. In particular, wk,0 = eu0 and w0,l = fv0 . From the uniqueness of this path, ws+ak,t+bl = ws,t eua0 fvb0 . So ξ˙s+ak,t+bl = ξ˙s,t ; thus the set of vectors ξs,t is periodic, and so may be indexed by an element g of Ck × Cl , say ξg . Let K denote the set K = {g ∈ Ck × Cl : ξ˙g = ξ˙0 }. Clearly K is closed under addition, and hence is a subgroup of Ck × Cl . The cosets each determine a distinct basis vector. So the subspace H0 spanned by all the basis vectors ξs,t in the ring by ring portion of the graph is naturally identified with 2 (G), where G = Ck × Cl /K. Observe that H0 is a cyclic coinvariant subspace for σ . Thus there is a setup exactly as in Section 5 using the group G, with the functions i, j , α and β determined by the action of σ on the basis {ξg : g ∈ G} for H. The compression σ of σ to H0 is therefore unitarily equivalent to this group construction. The representation σ has a unique


841

minimal ∗-dilation π , and this evidently coincides with the minimal ∗-dilation of σ . Hence there is a subspace of the ∗-dilation of σ which corresponds to H, exhibiting σ as a dilation of σ . 2 By Theorem 6.4 one can now easily obtain the following. It is sufficient to decompose the graph into connected components. Corollary 6.5. Any finitely correlated atomic ∗-representation of F+ θ is unitarily equivalent to the direct sum of irreducible atomic ∗-representations which dilate group construction representations on finite abelian groups. Example 6.6. Let us take another look at Example 3.1. Let θ be a permutation in Smn , and fix a cycle of θ : (i0 , j0 ), (i1 , j1 ), . . . , (ik−1 , jk−1 ) . Let G = Ck with g1 = 1 and g2 = −1. Set i(g) = ig and j (g) = jg−1 for g ∈ G. Use this to define a representation ρα,β . Then at each vertex ξg , ρα,β (fjg eig )ξg = αρα,β (fjg )ξg+1 = αβξg = βρα,β (eig−1 )ξg−1 = ρα,β (eig−1 fjg−1 )ξg . Thus the commutation relations show that this is a representation. It is not difficult to see that there are no symmetries. So it is irreducible. Let u0 = ik−1 . . . i1 i0 and v0 = j0 j1 . . . jk−1 . These elements satisfy the identities ρα,β (eu0 )ξ0 = α k ξ0 and ρα,β (fv0 )ξ0 = β k ξ0 . It is easy to check that eu0 fv0 = fv0 eu0 . Therefore we may consider the atomic representation σ on Ck × Ck given by Lemma 6.2. It is easy to obtain, for 0 s < k, that us = ik−1−s . . . i1 i0 ik−1 . . . ik−s

and vs = js js+1 . . . jk−1 j0 . . . js−1 .

It follows that the subgroup of symmetries includes H = (0, 0), (1, 1), . . . , (k − 1, k − 1) . However, as this comes from a cycle of θ , a little thought shows that there are no other symmetries. Compute Ck × Ck /H Ck . The characters of Ck are given by χ(1) = ω where ωk = 1. Therefore by Theorem 5.4, we can decompose σ ωk =1 ρω,ω . Long commuting words. We now show that there are many infinitely many finitely correlated representations for any F+ θ by exhibiting arbitrarily long primitive commuting words. Proposition 6.7. For any given F+ θ , there are commuting pairs eu and fv which determine irreducible atomic representations of arbitrarily large dimension.

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Proof. Consider the two sets of words U := {eu : |u| = N !} and V := {fv : |v| = N !}. Since |U| = mN ! and |V| = nN ! , for simplified notation, we write U = {u1 , . . . , umN! } and V = {v1 , . . . , vnN! }. Given ui ∈ U and vj ∈ V, the relation θ uniquely determines ui ∈ U and vj ∈ V such that eui fvj = fvj eui . We obtain a permutation θ of mN ! × nN ! so that θ (ui , vj ) = (ui , vj ). If eu fv = fv eu for all u ∈ U and v ∈ V, then pick any primitive words u ∈ U and v ∈ V. Then eu fv = fv eu leads to an atomic representation on a quotient G of CN ! × CN ! by the symmetry subgroup H . Since u has no symmetries, (s, 0) ∈ / H for any 1 s < |u|. So |G| N !. Otherwise, θ has a cycle, say C, of length t 2. Without loss of generality, we can assume that C = ((u0 , v0 ), . . . , (ut−1 , vt−1 )). As in Example 6.6, we obtain a representation ρ on 2 (CN !t ). Therefore N! does not belong to the symmetry subgroup H . But H = p where p = 0 or p divides N!t. The t pairs (ui , vi ) are all distinct, and so at least one of the words U or V is not N !-periodic. So p does not divide N !. Therefore p N + 1 implying |CN !t /H | N + 1. Consequently there is an irreducible representation on a space of finite dimension greater than N . 2 Example 6.8. Consider the forward 3-cycle algebra of Example 2.1 given by the permutation ((1, 1), (1, 2), (2, 1)) in S2×2 . The relations have the succinct form fj ei = ei+j fi where addition is modulo 2. Observe that fik ei1 ,i2 ,...,ik = eik +i1 ,i1 +i2 ,...,ik−1 +ik fik , fik−1 +ik eik +i1 ,i1 +i2 ,...,ik−1 +ik = eik−1 +2ik +i1 ,ik +2i1 +i2 ,...,ik−2 +2ik−1 +ik fik−1 +ik , .. . f (n−1)i p

k−p

e (n−1)i p

1−p ,...,

= e (n )i1−p ,..., (n )ik−p f (n−1)i . (n−1 k−p p p p p )ik−p

n The binomial sums are over p 0. If we take k = l = 2n − 1, then 2 p−1 ≡ 1 for all p. So in order that the last e term equal the original, it suffices that ls=1 is ≡ 0. The f word is then uniquely determined so that fv eu = eu fv . For example, f1222212 e1121212 = e1121212 f1222212 . It follows that there are primitive commuting pairs of arbitrarily great length. 7. The ring by tail case Now consider the case 2a, the ring by tail type in which the blue components are ring representations and the red components are infinite tail representations. To simplify the presentation, we introduce some notation. For a word u = ik−1 . . . i0 of length k 1, define u(s, 0] := is−1 . . . i0

and u(s) := is−1 . . . i0 ik−1 . . . is

for 0 s < k,

and observe that u(k) = u. Also if v = v−1 v−2 v−3 . . . is an infinite word, for 0 t < t write v(−t, −t ] = v−t−1 . . . v−t

and v(−t, −∞) = v−t−1 v−t−2 v−t−3 . . . .


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Start with a basis vector, ξ0,0 say, in one of the blue rings of minimal size k. One can find a unique word u0 = ik−1,0 . . . i0,0 so that ρ(eu0 )ξ˙0,0 = ξ˙0,0 . Choose a scalar α so that ρ(eu0 )ξ0,0 = α k ξ0,0 . For 1 s k − 1, renormalize the basis vectors so that ξs,0 = α s ρ(eu0 (s,0] )ξ0,0 . Then ρ(eis,0 )ξs,0 = αξs+1,0

for s ∈ Ck .

Pull back along the red edges from ξ0,0 to obtain an infinite red tail v0 := j0,−1 j0,−2 . . . and basis vectors ξ˙0,t for t < 0 satisfying ρ(fj0,t )ξ˙0,t = ξ˙0,t+1

for t < 0.

Normalize the basis vectors so that ρ(fj0,−t )ξ0,−t = ξ0,1−t for t 1. There is a unique word ut = ik−1,t . . . i0,t of length k for each t < 0 so that eu0 fv0 (0,t] = fv0 (0,t] eut . It follows that ρ(eut )ξ˙0,t = ξ˙0,t ; and hence one can deduce that ρ(eut )ξ0,t = α k ξ0,t for t < 0. Define ξs,t = α s ρ(ut (s, 0])ξ0,t . Then ρ(eis,t )ξs,t = αξs+1,t

for s ∈ Ck and t 0.

Similarly one can pull back along the red edges from ξs,0 to obtain an infinite word vs := js,−1 js,−2 . . . for 1 s k − 1. Again using the commutation relations, one obtains that ρ(fjs,t )ξs,t = ξs,t+1 (s)

It is also easy to verify that the cycles ut

satisfy

ρ(eu(s) )ξs,t = α k ξs,t t

for t < 0 and s ∈ Ck .

for t 0 and s ∈ Ck .

It is evident that the vectors {ξs,t : 0 s < k and t 0} span a coinvariant subspace. By the connectedness of the graph, it is also a cyclic subspace. Thus this subspace determines the representation by the uniqueness of the isometric dilation in Theorem 2.5. There are only mk words of length k in m letters. So by the pigeonhole principle, there is some word u which is repeated infinitely often in the sequence {ut : t 0}. Without loss of generality, we may assume that there is a sequence t0 = 0 > t1 > t2 > · · · such that utk = u0 for k 1. Then eu0 fv0 (0,tk ] = fv0 (0,tk ] eu0

for k 1.

Conversely, given a word u0 , α ∈ T, an infinite tail v0 = j0,−1 j0,−2 . . . and a sequence 0 < t1 < t2 < · · · such that eu0 fv0 (0,−tk ] = fv0 (0,−tk ] eu0 for k 1, one can build a representation of type case 2a. We will do this by our group construction by extending the definition of this subspace indexed by Ck × −N0 to the group G = Ck × Z. This may be accomplished in many ways, but a simple way is to make it periodic on Ck × N0 by repeating the segment on Ck × [0, t1 ) using the fact that ut1 = u0 and eu0 fv0 (0,t1 ] = fv0 (0,t1 ] eu0 . The finitely correlated representation of Lemma 6.2 can be unfolded to obtain a representation on Ck × Z. Just cut off the left half

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of the cylinder and glue it to the one we have. Any choice of extension is tail equivalent to any other. Note that this analysis is necessary, and it is not the case that an arbitrary tail eu0 and infinite word fv0 determines a representation. For example, this may force a red edge and blue edge into the same vertex which is not possible from the commutation relations. We have obtained: Theorem 7.1. A ring by tail representation is determined by a word u0 of length k, a scalar α ∈ T, an infinite word v0 := j0,−1 j0,−2 . . . and a sequence 0 > t1 > t2 > · · · such that eu0 fv0 (0,tk ] = fv0 (0,tk ] eu0 for k 1. It corresponds to a group construction on the group Ck × Z. Since a ring by tail representation is given by the group construction on G = Ck × Z, we can use the decomposition results of Section 5. Theorem 7.2. Let π be a (connected) ring by tail representation with symmetry subgroup H Ck × Z. If H Ck × {0}, then π decomposes as a finite direct sum of irreducible ring by tail representations. Otherwise, it decomposes as a direct integral of irreducible ring by ring atomic representations. Proof. In the first case, H = (d, 0) is a subgroup of Ck × {0}. Since H is finite, Theorem 5.4 splits π as a direct sum of finitely many irreducible representations for the quotient group G/H Ck/d × Z. So these are irreducible ring by tail representations. In the second case, H contains an element (a, b) with b = 0. Therefore H contains the element (0, kb). Thus G = Ck × Z/H is a quotient of Ck × Ckb ; and in particular, G is finite. Since H is infinite, Theorem 5.4 yields a direct integral of irreducible representations on 2 (G). Evidently, these are ring by ring type. 2 The case 2b is handled by exchanging the role of the red and blue edges in the case 2a. 8. The tail by tail case In the case 3, both the red and the blue components are infinite tail representations. The most basic is the case 3a, in which the red tail never intersects the original blue component again. Case 3a. As we saw in Example 3.2, the 3a case is an inductive limit of copies of the left regular representation. So in a certain sense, they are the easiest. Given a representation of type 3a, start with any basis vector ξ0,0 . Pull back on both blue and red edges to determine integers is,t and js,t and basis vectors ξ˙s,t for s, t 0 so that π(eis,t )ξ˙s−1,t = ξ˙s,t

and

π(fjs,t )ξ˙s,t−1 = ξ˙s,t

for s, t 0.

The assumption of the case 3a ensures that the ξ˙s,t are all distinct. Thus we have found a cyclic coinvariant subspace M = span{ξs,t : s, t 0} on which we have a representation on (−N0 )2 . The representation is determined by the tail


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τ = e0,0 f−1,0 e−1,−1 f−2,−1 . . . since the other is,t and js,t for s, t 0 are determined by the commutation relations. One could use Theorem 5.1 to make the scalars all equal to 1; but in fact that is not necessary. Instead one selects the appropriate unit vector ξs,t in ξ˙s,t recursively so that π(eis,t )ξs−1,t = ξs,t

and

π(fjs,t )ξs,t−1 = ξs,t

for s, t 0.

Example 3.2 explains how to dilate this to a ∗-representation which is an inductive limit of copies of the left regular representation. By Theorem 2.5, this is the unique minimal ∗-dilation; and hence it is π . While the infinite tail is sufficient to describe the representation, as is done in Example 3.2, it is not unique, and the equivalence relation on tails is not at all transparent. A much more useful collection of data associated to τ is the set Σ(π, ξ0,0 ) = Σ(τ ) = (is,t , js,t ): s, t 0 . In Definition 3.3, we put an equivalence relation of shift tail equivalence on these sets. Definition 8.1. For each inductive ∗-representation π , define Σ(π) to be the equivalence class of Σ(π, ξ ) modulo shift tail equivalence for any standard basis vector ξ ∈ Hπ . That this definition makes sense is part of the following result. Theorem 8.2. If π is an inductive (type 3a) atomic ∗-representation, then Σ(π) is independent of the choice of initial vector. Two inductive ∗-representations π1 and π2 are unitarily equivalent if and only if Σ(π1 ) = Σ(π2 ). Proof. Start with two standard basis vectors, ξ0,0 and ζ0,0 . The connectedness of the graph means that there is a path from ξ0,0 to ζ0,0 . By Lemma 3.4, there is a path from ξ0,0 to ζ0,0 of the form uv ∗ . Let d(v) = (s1 , t1 ) and d(u) = (s2 , t2 ). Then ξ˙−s1 ,−t1 = ζ˙−s2 ,−t2 . Therefore the data agrees on all basis vectors obtained by pulling back from that common vector. So the two data sets are (s1 − s2 , t1 − t2 )-shift tail equivalent. Clearly, if two inductive ∗-representations σ1 and σ2 have shift tail equivalent data Σ(πi , ξi ), then they are unitarily equivalent. Consider the converse. Suppose that π1 and π2 are unitarily equivalent, say via a unitary U in B(Hπ1 , Hπ2 ). Fix the basis vector ξ0,0 for π1 , and corresponding basis ξs,t for s, t 0 and data Σ(π1 , ξ0,0 ). From the unitary equivalence, ξ0,0 is identified with a vector η0,0 = U ξ ∈ Hπ2 . Write η in the standard basis for π2 , say η = ai ζi ; and choose a standard basis vector ζ = ζi0 for which ai0 = 0. For any s, t 0, there is a word ws,t ∈ F+ θ with d(ws,t ) = (|s|, |t|) so that π1 (ws,t )ξs,t = ξ0,0 . Therefore η0,0 = π2 (ws,t )U ξs,t is in the range of the partial isometry π2 (ws,t ). Since π2 is atomic, the range of π2 (ws,t ) is spanned by standard basis vectors. Consequently ζ is in the range of π2 (ws,t ) for every s, t 0. Restating this another way, it says that Σ(π2 , ζ ) = Σ(π1 , ξ0,0 ). Hence Σ(π1 ) = Σ(π2 ). 2

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Example 8.3. It is tempting to think that an inductive ∗-representation can be determined by the blue and red components containing a standard basis vector. That is, start at a basis vector ξ0,0 . Pull back along the blue edges to obtain the infinite word u = i−1 i−2 . . . determining the blue infinite tail component containing ξ0,0 . Likewise pull back along the red edges to get a red infinite tail v = j−1 j−2 . . . determining the red component containing ξ0,0 . The infinite words u and v may not determine the representation uniquely! Consider the permutation ((1, 1), (1, 3))((1, 2), (2, 1)) in S3×3 with two 2-cycles; and the two infinite words u = e1 e3 e3 e3 . . . and v = f1 f3 f3 f3 . . . . Two inequivalent ∗-representations π1 and π2 will be constructed to produce a coinvariant subspace spanned by vectors ξ−s,−t for s, t 0 with πi (e1 )ξ−1,0 = ξ0,0 ,

πi (e3 )ξ−s−1,0 = ξ−s,0

πi (f1 )ξ0,−1 = ξ0,0 ,

πi (f3 )ξ0,−t−1 = ξ−0,−t

for s 1, for t 1.

So they have the same infinite blue and red components. We define in addition π1 (e3 )ξ−s−1,−t = ξ−s,−t π1 (e2 )ξ−1,−t = ξ0,−t π1 (f3 )ξ−s,−t−1 = ξ−s,−t π1 (f2 )ξ−s,−1 = ξ−s,0

for s 1 and t 0, for t 1, for s 0 and t 1, for s 1,

and π2 (e3 )ξ−s−1,−t = ξ−s,−t π2 (e1 )ξ−1,−t = ξ0,−t π2 (f1 )ξ−s,−t−1 = ξ−s,−t π2 (f3 )ξ−s,−1 = ξ−s,0

for s 1 and t 0, for t 1, for s 0 and t 1, for s 1.

These two ∗-representations are evidently not shift tail equivalent. Thus they are not unitarily equivalent. This means that these representations are not some form of twisted product of an infinite tail representation for Am with an infinite tail representation for An . Case 3b. Now suppose that we have a ∗-representation π of type 3b. Pulling back along the red edges yields a periodic sequence of blue components of infinite tail type, say H0 , . . . , Hl−1 , so that every red edge into each Hi comes from Hi−1 (mod l) . Start with a basis vertex ξ˙ . Let v be the unique word of length l so that fv maps onto ξ˙ . Then there is a vertex ζ˙ ∈ H0 so that π(fv )ζ˙ = ξ˙ . Our first goal is to explain why these two vertices are comparable in H0 if ξ˙ is sufficiently far up the tail.


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Lemma 8.4. Suppose that π is a type 3b atomic ∗-representation in which blue components H0 , . . . , Hl−1 are periodic of period l. There is a vertex ξ˙0 in H0 and a word u0 so that the vertex ζ˙0 on H0 obtained by pulling back l red steps from ξ˙0 via a word v0 satisfies either π(fv0 )ζ˙0 = ξ˙0

and π(eu0 )ξ˙0 = ζ˙0

(3bi)

or π(fv0 )ζ˙0 = ξ˙0 = π(eu0 )ζ˙0 .

(3bii)

This same type of relationship persists for each vertex obtained from pulling back along every path leading into ξ˙0 . Proof. Since H0 is an infinite tail representation of Am , there is a vertex ζ˙0 in H0 and words u1 and u2 so that π(eu1 )ζ˙0 = ξ˙

and π(eu2 )ζ˙0 = ζ˙ ;

and thus π(fv eu2 )ζ˙0 = ξ˙ . Use the commutation relations to write fv eu2 = eu2 fv0 . Set ξ˙0 = π(fv0 )ζ˙0 . Then π(eu1 )ζ˙0 = ξ˙ = π(fv eu2 )ζ˙0 = π(eu )π(fv0 )ζ˙0 = π(eu )ξ˙0 . 2

2

It follows that the vertices ζ˙0 and ξ˙0 are obtained from ξ˙ by pulling back along the blue edges |u1 | and |u2 | = |u2 | steps, respectively. So they are comparable, and the relationship depends on whether |u2 | is less than, equal to or greater than |u1 |. If |u2 | < |u1 |, then uniqueness of the pull back along blue edges means that u1 = u2 u0 . So π(u0 )ζ˙0 = ξ˙0 . Similarly, if |u2 | > |u1 |, then u2 = u1 u0 and π(eu0 )ζ˙0 = ξ˙0 . In the case |u2 | = |u1 |, we have ξ˙0 = ζ˙0 and so ξ˙0 = π(fv0 )ξ˙0 . This is the tail by ring case, which has been excluded. In the first case, suppose that η˙ is any vertex in these l components which is obtained by ˙ pulling back from ξ0 . That is, there is a word w ∈ F+ θ so that π(w)η˙ = ξ0 . There is a basis vector ζ˙ in the same component and a word v with |v | = l so that π(fv )ζ˙ = η. ˙ From the commutation relations, there are words v of length l and w of degree d(w) so that wfv = fv w . Therefore, π(fv )π(w )ζ˙ = ξ˙0 = π(fv0 )ζ˙0 . Uniqueness implies that v = v0 and π(w )ζ˙ = ζ˙0 . Now factor eu0 w = w eu with |u | = |u0 | and d(w ) = d(w ). Then ξ˙0 = π(eu0 w )ζ˙ = π(w )π(eu )ζ˙ . Since w is a word with the same degree, say (s, t), as w and w, we deduce that π(eu )ζ˙ is the unique vertex obtained by pulling back from ξ˙0 by s blue and t red edges, namely η. ˙ That is, π(eu )ζ˙ = η; ˙ and ζ˙ lies above η˙ in its blue component. The other case is handled in a similar manner. 2

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We now proceed as in the case 3a. Start with the basis vertex ξ˙0,0 in H0 provided by Lemma 8.4. Pull back on both blue and red edges to determine integers is,t and js,t for s, t 0 and vertices ξ˙s,t so that π(eis,t )ξ˙s−1,t = ξ˙s,t

and

π(fjs,t )ξ˙s,t−1 = ξ˙s,t

for s, t 0.

The difference in the case 3b is that there is periodicity. In the case 3bi, there are unique words u0 of length k and v0 of length l so that π(fv0 eu0 )ξ˙0 = ˙ξ0 . By Lemma 8.4, for each vertex ξs,t , pulling back k blue edges and l red edges will return to the same vertex. That is, ξ˙s−k,t−l = ξ˙s,t for all s, t 0. So there is Z(k, l) periodicity. This allows us to extend the definitions to all of Z2 using the periodicity; and to then collapse this to a representation on Z2 /Z(k, l). Write ξ[s,t] or ξg for the vector associated to an element g = [s, t] coming from the equivalence class of (s, t). It is now easy to see that the vectors ξg are distinct because one can always choose the representative (s, t) with 0 t < l, determining the component Ht , and within this component, s determines the position on the infinite tail. Indeed, the (k, l)-periodicity allows us to select a distinguished spine because pulling back l steps along the red edges moves us forward k (specific) blue edges. We shall see soon that in this case, there is always additional symmetry. In the case 3bii, there are unique words u0 of length k and v0 of length l so that π(fv0 )ζ˙0 = π(eu0 )ζ˙0 = ξ˙0,0 . Hence ζ˙0 = ξ˙0,−l = ξ˙−k,0 . By Lemma 8.4, for each vertex ξs,t , pulling back k blue edges or pulling l red edges will result in the same vertex. That is, ξ˙s−k,t = ξ˙s,t−l for all s, t 0. So there is Z(k, −l) periodicity. In this case, there is no canonical way to carry forward. However, as in the previous case, we obtain a parameterization of the basis vectors as a semi-infinite subset of Z2 /Z(k, −l); and we will write ξ[s,t] or ξg for the vector associated to an element g = [s, t] coming from the equivalence class of (s, t) when there is a representative with s, t 0. In both cases, the subspace M = span{ξ[s,t] : s, t 0} is a coinvariant cyclic subspace. So it is sufficient to determine shift tail equivalence. As usual, we define the symmetry subgroup Hπ from the shift tail symmetry of the data Σ(π, ξ0,0 ) = {(i[s,t] , j[s,t] ): s, t 0}, but consider it as a subgroup of Z2 /Z(k, ±l) by modding out by the known symmetry. Let Σ(π) denote the shift tail equivalence class of Σ(π, ξ0,0 ). The following result follows in an identical manner to the case 3a, so it will be stated without proof. The converse to the second statement does not follow because there are issues with the scalars that will be dealt with soon. Theorem 8.5. If π is a type 3b atomic ∗-representation, then Σ(π) is independent of the choice of initial vector. If two type 3b ∗-representations π1 and π2 are unitarily equivalent, then Σ(π1 ) = Σ(π2 ). We now consider these two cases in more detail. Case 3bi. Here we have symmetry Z(k, l) with k, l > 0. In this case, there are words u0 = i[−1,0] . . . i[−k,0] and v0 = j[0,l−1] . . . j[0,0] so that π(eu0 fv0 )ξ˙[0,0] = ξ˙[0,0] .


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The basis ξ˙[s,t] satisfies ξ˙[ks,0] = ξ˙[0,−ls] . Since we can pull back along either blue or red edges, we find that these vectors are defined for all s ∈ Z. Indeed, all ξ˙[s,t] are defined for (s, t) ∈ Z2 . This leads to the important observation about the case 3bi that this representation is determined by u0 and v0 . Lemma 8.6. An atomic ∗-representation of type 3bi is determined by words u0 and v0 and a constant β ∈ T satisfying π(eu0 fv0 )ξ0,0 = βξ0,0 . The symmetry subgroup Hπ is always nonzero. Indeed, there is full symmetry (without tail equivalence), namely there is an integer p > 0 so that i[s−p,t] = i[s,t]

and j[s−p,t] = j[s,t]

for all s, t ∈ Z.

Proof. We can find words u1 , v1 , u−1 and v−1 to factor e u 0 f v0 = f v1 e u 1

and fv0 eu0 = eu−1 fv−1 .

Continuing recursively, we obtain words ur in {1, . . . , m}k and vr in {1, . . . , n}l for r ∈ Z so that eur fvr = fvr+1 eur+1

for all r ∈ Z.

This determines the doubly infinite paths τe = . . . u1 u0 u−1 . . .

and τf = . . . v1 v0 v−1 . . . .

The e’s and f ’s move in opposite directions, and the two spines intersect every k steps forward along the blue path for every l steps backward along the red path. The commutation relations allow us to compute the l infinite blue paths and k infinite red paths, completing the picture of this coinvariant subspace. The commutation of words eu of length k with words fv of length l is given by a permutation θ in Smk ×nl determined by θ so that eu fv = fv eu , where θ (u, v) = (u , v ). The pairs (ur , vr ) therefore satisfy θ (ur , vr ) = (ur+1 , vr+1 ). It follows that the pairs (ur , vr ) move repeatedly through a cycle of the permutation θ . Consequently, the sequence (ur , vr ) is periodic of length p, where p is the length of the cycle. It follows that i[s,t] = i[s+pk,t]

and j[s,t] = j[s,t−pl] = j[s+pk,t] .

Therefore Hπ contains [pk, 0] = [0, −pl]; and thus is a non-trivial symmetry subgroup. Moreover the symmetry is global, not just for s, t T . The scalars are determined by Theorem 5.1. The character ψ of Z(k, l) is given by ψ(k, l) = β where π(eu0 fv0 )ξ0,0 = βξ0,0 . One can extend this to a character on Z2 ; and we will do this by setting ϕ(1, 0) = 1 and ϕ(0, 1) = β0 where β0 is an lth root of β. 2 As an immediate consequence of the previous two results, we obtain: Corollary 8.7. Two type 3bi ∗-representations π1 and π2 are unitarily equivalent if and only if Σ(π1 ) = Σ(π2 ) and β1 = β2 , where β1 and β2 are the scalars of Lemma 8.6.

850


Conversely, we obtain Theorem 8.8. Given words u0 in {1, . . . , m}k and v0 in {1, . . . , n}l and scalar β ∈ T, there is an atomic ∗-representation of F+ θ of type 3bi determined by this data. Proof. The proof of Lemma 8.6 explains how this is done. The pair (u0 , v0 ) lies in a cycle of θ (u0 , v0 ), (u1 , v1 ), . . . , (up−1 , vp−1 ) . So eui fvi = fvi+1 eui+1 for i ∈ Cp . As in the construction of Examples 3.1 and 6.6, we obtain that eup−1 ...u1 u0 commutes with fv0 v1 ...vp−1 . Indeed, one has the useful relations eui fvi vi+1 ...vp−1 v0 ...vi−1 = fvi+1 ...vp−1 v0 ...vi eui . This is the algebraic form of the (k, l)-periodicity. By Lemma 6.2, there is a finitely correlated defect free representation on Cpk × Cpl determined by this commuting pair. The relations above ensure (k, l) periodicity; so there is a corresponding construction on Cpk × Cpl / (k, l) . The idea is to ‘unfold’ this to obtain a 3bi ∗-representation. It is probably easier to envisage unfolding the representation on Cpk × Cpl to a representation on Z2 with pkZ × plZ symmetry, and then observing that the (k, l)-periodicity of the original picture becomes Z(k, l) periodicity of the type 3a representation. So one now goes to the quotient Z2 /Z(k, l) to obtain the desired representation of type 3bi. One can deal with scalars as before. 2 Thus we obtain: Corollary 8.9. Every ∗-representation of type 3bi comes from a group construction for a group of the form Z2 /Z(k, l) with kl > 0. It always has a non-trivial symmetry group, and so is never irreducible. It decomposes as a direct integral of irreducible ∗-representations of type 1. Case 3bii. This is the trickiest case. Here we have Z(k, −l) symmetry, where k, l > 0. That is, our basis is ξ[s,t] for s, t 0 where the equivalence class consists of cosets of Z(k, −l). We have a word u0 of length k and v0 of length l so that π(eu0 )ξ˙[−k,0] = π(fv0 )ξ˙[−k,0] = ξ˙[0,0] . Thus there is a scalar β ∈ T so that π(fv0 )ξ[−k,0] = βπ(eu0 )ξ[−k,0] . Pulling back from ξ˙[0,0] along the blue edges yields the infinite tail τe = i[0,0] i[−1,0] i[−2,0] . . . = u0 u1 u2 . . . , where ud = i[dk,0] i[dk−1,0] . . . i[(d−1)k+1,0] for d 0 are the consecutive words of length k. Similarly, pulling back from ξ˙[0,0] along the red edges yields the infinite tail τf = j[0,0] j[0,−1] j[0,−2] . . . = v0 v−1 v−2 . . .


851

where vd = i[0,dl] i[0,dl−1] . . . i[0,(d−1)l+1] for d 0 are the consecutive words of length l. Let β0 be chosen so that β0l = β. Then as before, we may normalize the basis vectors so that π(ei[s,t] )ξ[s−1,t] = ξ[s,t]

and π(fj[s,t] )ξ[s,t−1] = β0 ξ[s,t]

for sl + tk 0. Since [dk, 0] = [0, dl], we have π(fvd+1 eud )ξ˙[dk,0] = ξ˙[(d+2)k,0] = π(eud+1 fvd )ξ˙[dk,0] ; and therefore we obtain the commutation relations fvd+1 eud = eud+1 fvd

for d < 0.

This is a rather strong compatibility condition, and suggests why not all pairings are possible. For each d 0, there are at most mk nl possible pairs (ud , vd ). By the Pigeonhole principle, one of these pairs is repeated infinitely often. Without loss of generality, we may suppose that this sequence begins at 0; so that there are integers 0 = d0 > d1 > d2 > · · · so that udr = u0

and vdr = v0

for r 1.

A computation now shows that eu0 u1 ...udr +1 and fv0 v−1 ...vdr +1 commute. Indeed, one readily computes that eui ...uj −1 fvj = fvi eui+1 ...uj

and eui fvi+1 ...vj = fvi ...vj −1 euj .

This encodes the Z(k, −l) symmetry. Repeated application of this yields eu0 u1 ...udr +1 fv0 v−1 ...vdr +1 = eu0 u1 ...udr +2 eudr +1 fvdr fv−1 v−2 ...vdr +1 = eu0 u1 ...udr +2 fvdr +1 eudr fv−1 v−2 ...vdr +1 = fv0 eu1 ...udr +1 eu0 fv−1 v−2 ...vdr +1 = fv0 eu1 ...udr +1 fv0 ...vdr +2 eudr +1 .. . = fv0 v−1 ...vdr +1 eu0 u1 ...udr +1 . Now we can construct a sequence of ring by ring representations on the finite groups Gr = C|dr |k × C|dr |l / (k, −l) by building a ring by ring representation on Gr using these commuting words and the fact that they have the Z(k, −l) symmetry. That is, consider a finitely correlated representation ρr on a basis ζg for g = [s, t] in Gr for dr k s < 0 and 0 t < l by ρr (ei )ζ[s−1,t] = δi is,t ζ[s,t]

and ρr (fj )ζ[s,t−1] = δj js,t β0 ζ[s,t] .

852


These observations also provide a way (indeed many ways) to extend the definition of the coinvariant subspace to the full group Z2 /Z(k, l) by making it (d1 k, −d1 l) periodic moving forward. That is, one may define i[s,t] = i[s ,t ]

and j[s,t] = i[s ,t ]

for sk + tl > 0,

where (s , t ) is chosen in (d1 k, 0] × (d1 l, 0] so that s ≡ s (mod d1 k) and t ≡ t (mod d1 l). Lemma 8.10. An atomic ∗-representation of type 3bii is determined by a scalar β ∈ T and two infinite tails τe = u0 u1 u2 . . . and τf = v0 v−1 v−2 . . . , where |ud | = k and |vd | = l, satisfying fvd+1 eud = eud+1 fvd

for d < 0.

Proof. The discussion above shows that β and two infinite words with the desired properties are associated to each ∗-representation of type 3bii. Conversely, suppose that such data is given. Then by Section 6, there are finitely correlated representations ρr as defined above for r 1. The data {(is,t , js,t ): s, t 0} has Z(k, −l) symmetry, and may be extended in a compatible way to all of Z2 /Z(k, −l). Therefore we may construct a representation on Z2 /Z(k, −l) using the group construction of Section 5. 2 Again we consider Σ(π, ξ ) to be defined on the group Z2 /Z(k, −l); and define Σ(π) to be its equivalence class modulo shift tail equivalence. The scalars are determined by Theorem 5.1 by the character ψ on Z(k, −l) given by π(eu0 )π(fv0 )∗ ξ0,0 = ψ(k, −l)ξ0,0 . Combining the previous results yields: Corollary 8.11. Two type 3bii ∗-representations π1 and π2 are unitarily equivalent if and only if Σ(π1 ) = Σ(π2 ) and ψ1 = ψ2 , where ψi is the character on Z(k, −l) determined by πi . Decomposition. It now follows that a type 3 ∗-representation has the form of one of the group constructions. In particular, by Lemma 5.3, a ∗-representation π is irreducible if and only if the symmetry group Hπ is trivial. In general, we obtain a direct integral decomposition into irreducibles. The case 3bi is never irreducible, so these representations decompose as a direct integral of finitely correlated ∗-representations. Theorem 8.12. A (connected atomic) tail by tail ∗-representation π with symmetry group Hπ Gπ decomposes as a direct integral of irreducible atomic ∗-representations dilating a family of representations on 2 (Gπ /Hπ ). Consider the possibilities for the case 3a, the inductive type. The subgroup Hπ = (k, l) for kl > 0 cannot occur, because this would require an irreducible ∗-representation of type 3bi. Corollary 8.13. If π is a tail by tail ∗-representation of inductive type, then the symmetry group is one of the following: (1) Hπ = {(0, 0)} when π is irreducible. (2) Hπ = (k, l) where kl < 0, and π is a direct integral of irreducible 3bii ∗-representations. (3) Hπ = (k, l) where kl = 0, and π is a direct integral of irreducible type 2 ∗-representations.


853

(4) Hπ has rank 2, and π is a direct integral of irreducible finitely correlated ∗-representations. It is natural to ask whether F+ θ has irreducible ∗-representations of inductive type. This is equivalent to the existence of an infinite tail without any periodicity. This is exactly the aperiodicity condition introduced by Kumjian and Pask [10]. This property is generic, but there are periodic examples such as the flip semigroup of Example 2.2. In [4], this property is explored in detail. The semigroup F+ θ is either aperiodic, or it has Z(k, −l) periodicity for some kl > 0. This leads to structural differences in C∗ (F+ θ ). It would be interesting to determine whether all aperiodic F+ θ have irreducible ∗-representations of type 3bii. References [1] J. Cuntz, Simple C*-algebras generated by isometries, Comm. Math. Phys. 57 (1977) 173–185. [2] K.R. Davidson, C*-Algebras by Example, Fields Institute Monogr. Ser., vol. 6, Amer. Math. Soc., Providence, RI, 1996. [3] K.R. Davidson, D.R. Pitts, Invariant subspaces and hyper-reflexivity for free semi-group algebras, Proc. London Math. Soc. 78 (1999) 401–430. [4] K.R. Davidson, D. Yang, Periodicity in rank 2 graph algebras, Canad. J. Math., in press. [5] K.R. Davidson, D.W. Kribs, M.E. Shpigel, Isometric dilations of non-commuting finite rank n-tuples, Canad. J. Math. 53 (2001) 506–545. [6] K.R. Davidson, S.C. Power, D. Yang, Dilation theory for rank 2 graph algebras, J. Operator Theory, in press. [7] C. Farthing, P.S. Muhly, T. Yeend, Higher-rank graph C*-algebras: An inverse semigroup and groupoid approach, Semigroup Forum 71 (2005) 159–187. [8] E. Hewitt, K. Ross, Abstract Harmonic Analysis, vol. I, Grundlehren Math. Wiss., vol. 115, Springer, New York, 1963. [9] D.W. Kribs, S.C. Power, The analytic algebras of higher rank graphs, Math. Proc. R. Ir. Acad. 106 (2006) 199–218. [10] A. Kumjian, D. Pask, Higher rank graph C*-algebras, New York J. Math. 6 (2000) 1–20. [11] D. Pask, I. Raeburn, M. Rordam, A. Sims, Rank-two graphs whose C*-algebras are direct limits of circle algebras, J. Funct. Anal. 239 (2006) 137–178. [12] S.C. Power, Classifying higher rank analytic Toeplitz algebras, New York J. Math. 13 (2007) 271–298. [13] I. Raeburn, Graph Algebras, CBMS Reg. Conf. Ser. Math., vol. 103, Amer. Math. Soc., Providence, RI, 2005. [14] I. Raeburn, A. Sims, T. Yeend, Higher-rank graphs and their C*-algebras, Proc. Edinb. Math. Soc. (2) 46 (2003) 99–115. [15] I. Raeburn, A. Sims, T. Yeend, The C*-algebras of finitely aligned higher-rank graphs, J. Funct. Anal. 213 (2004) 206–240. [16] G. Robertson, T. Steger, Affine buildings, tiling systems and higher rank Cuntz–Krieger algebras, J. Reine Angew. Math. 513 (1999) 115–144. [17] A. Sims, Gauge-invariant ideals in the C*-algebras of finitely aligned higher-rank graphs, Canad. J. Math. 58 (2006) 1268–1290.


Norm controlled inversions and a corona theorem for H ∞-quotient algebras Pamela Gorkin a , Raymond Mortini b,∗ , Nikolai Nikolski c,1 a Department of Mathematics, Bucknell University, Lewisburg, 17837 PA, USA b Département de Mathématiques, LMAM, UMR 7122, Université Paul Verlaine, Ile du Saulcy, F-57045 Metz, France c IMB, Université Bordeaux 1, 351, cours de la Libération, F-33405 Talence, France

Received 8 November 2007; accepted 19 May 2008 Available online 13 June 2008 Communicated by N. Kalton

Abstract Let Θ be an inner function on the unit disc D. We give a description of those Θ for which the quotient algebra H ∞ /ΘH ∞ has no corona with respect to the visible part of its spectrum, that is for which ∞ M(H ∞ /ΘH ∞ ) = {z ∈ D: Θ(z) = 0}M(H ) . It happens that this property is equivalent to the norm con∞ ∞ trolled inversion property for H /ΘH , as well as to a kind of weakened Carleson type embedding theorem. The quotient algebra A(D)/ΘH ∞ is also considered. An interpretation of our main results in terms of model operators is given, too. © 2008 Elsevier Inc. All rights reserved. Keywords: H ∞ -quotient algebras; Corona theorems; Inner functions; Weak embedding property; Norm estimates of corona-solutions; Model operators; H ∞ -calculus

1. Introduction The problem we are interested in here is that of efficient inversion, as it is treated in [1,11], and subsequent papers. This problem is described in detail below.


E-mail addresses: [email protected] (P. Gorkin), [email protected] (R. Mortini), [email protected] (N. Nikolski). 1 Partially supported by the Marie Curie Actions European Grant 030042. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.011

P. Gorkin et al. / Journal of Functional Analysis 255 (2008) 854–876

855

Let A be a commutative unital Banach algebra with unit e and M(A) its maximal ideal space. For a ∈ A, let aˆ denote the Gelfand transform of a. We let ˆ . δ(a) = min a(t) t∈M(A)

Note that δ(a) a ˆ ∞ a. When a = (a1 , . . . , an ) ∈ An we define ˆ , δn (a) = min a(t) t∈M(A)

where |a(t)| ˆ = ( nj=1 |aˆ j (t)|2 )1/2 for t ∈ M(A) and we let a =

n

1/2 aj

2

.

j =1

Let δ be a real number satisfying 0 < δ 1. We are interested in finding, or bounding, the functions c1 (δ) = sup a −1 : a 1, δ(a) δ and

cn (δ) = sup inf b:

n

aj bj = e , a 1, δn (a) δ .

j =1

In fact, often only a part of M(A) is available, say Λ ⊂ M(A). We call this a visible part of M(A). (We will present examples below.) In this case, we would like to bound a −1 in terms of the visible part of the spectral data. Thus, we define the counterpart of δ(a) as ˆ . δ(a, Λ) = inf a(t) t∈Λ

In a similar way, we consider δn (a, Λ). This, in turn, yields modified functions cn (Λ, δ) for n 1 (if it is not readily apparent which algebra we mean, we write cn (A, Λ, δ)). If a is not invertible, we define a −1 = ∞. If δn (a, Λ) δ and a 1, we let b = ∞, if the set of all b = (b1 , . . . , bn ) for which nj=1 aj bj = e is empty. It should be clear that 1 cn (Λ, δ) cn+1 (Λ, δ) and, if 0 < δ δ 1, then cn (Λ, δ) cn (Λ, δ ). This implies the existence of a critical constant, denoted here by δn (A, Λ) (or simply δn (A) if Λ = M(A)) such that cn (Λ, δ) = ∞

for 0 < δ < δn (A, Λ)

and cn (Λ, δ) < ∞ for δn (A, Λ) < δ 1.

If A is a uniform algebra, then δ1 (A) = 0. For elements a with 0 < δn (a, Λ) < δn (A, Λ) we can say that the inversion problem is ill posed in the sense that there is no control of inverses in terms of visible spectral data. For elements with δn (A, Λ) < δn (a, Λ) 1 the problem is said to be well posed; that is, there is such an estimate.

856


Let H ∞ be the algebra of bounded holomorphic functions on the unit disc D = {z ∈ C: |z| < 1} endowed with the supremum norm f ∞ = sup|z| 0 there exists η > 0 such that

z ∈ D: Θ(z) < η ⊂ z ∈ D: inf bλ (z) < . λ∈Λ


857

We call this an embedding property because the latter condition is related to the famous Carleson embedding property (CEP) appearing in the Carleson interpolation theorem. Namely, the Carleson embedding means that (1 − |λj |2 )(1 − |z|2 ) j 1

|1 − λj z|2

C

for every z ∈ D (see [10, p. 151]), whereas for a Blaschke product Θ with zero sequence (λj ) the WEP is equivalent to (1 − |λj |2 )(1 − |z|2 ) j 1

|1 − λj z|2

C

for every z ∈ D \ λ∈Λ {ζ : |bλ (ζ )| < }. It is known that CEP is equivalent to the set Λ being a finite union of Carleson interpolating sequences. Therefore, the condition “Θ = B is a finite product of interpolating Blaschke products,” is sufficient for H ∞ /BH ∞ to have no corona. A surprising example (Example 3.7, invented in a collaboration with S. Treil and V. Vasyunin) shows that there exist Blaschke products satisfying WEP but not CEP, and so there exist algebras H ∞ /BH ∞ without corona (or, equivalently, having the norm controlled inversion property) such that the generating Blaschke product B is not a finite product of interpolating Blaschke products. We mention that, according to Theorems 3.3 and 3.4, the norm controlled inversion property (meaning δ1 (H ∞ /ΘH ∞ ) = 0 and/or the existence of the joint majorant for inverses (solutions of Bezout equations) cn (Λ, δ) < ∞ for all 0 < δ < 1) is equivalent to the simple inverse stability of the restriction algebra H ∞ |Λ: f ∈ H ∞ , δ(f, Λ) = inf f (λ) > 0 λ∈Λ

⇒

f + ΘH ∞ is invertible in H ∞ /ΘH ∞ .

The latter equivalence fails for the quotient disc algebras A(D)/ΘH ∞ . Indeed, Lemma 4.1 shows that every function f ∈ A(D) satisfying δ(f, σ (Θ)) > 0 is invertible in A(D)/ΘH ∞ if and only if m(σ (Θ) ∩ T) = 0, whereas the norm controlled inversions (i.e. δ1 (A(D)/ΘH ∞ ) = 0) hold true if and only if m(σ (Θ) ∩ T) = 0 and the WEP are satisfied (see Theorem 4.2 below). Moreover, in the case where m(σ (Θ) ∩ T) = 0, the maximal ideal space M(A(D)/ΘH ∞ ) can be identified with the spectrum σ (Θ). Therefore, as for some other Banach algebras (see examples in [11,12]), the lack of the norm controlled inversion property for quotient algebras is not necessarily related to the existence of a corona (meaning the existence of a large “invisible spectrum” or difference between the entire maximal ideal space and the “visible part”), but rather a subtle discrepancy between the Gelfand transform norm and the original Banach algebra norm. The geometric meaning of WEP sequences is still, unfortunately, unclear. Nothing similar to the Carleson density characterization for CEP sequences (see [10, p. 153]) is known. It is worth mentioning a couple of known related properties (for details see the end of Section 3 below): a separated WEP sequence is interpolating and a WEP sequence that is a finite union of separated sequences is a CEP sequence. Recall that a sequence, (λj ), is separated if infj =k |bλj (λk )| > 0. Finally, we interpret our results in terms of the spectral properties of the model operators. In fact, the interest in studying just this class of operators was the primary motivation for this paper. Now, let Θ be an inner function and

858


KΘ = H 2 ΘH 2 be the model space; that is, the orthogonal complement of the shift-invariant subspace ΘH 2 of the Hardy space 2 k 2 ˆ ˆ H = f= f (k)z : f = f (k) < ∞ 2

k0

k0

on the disc D. The model operator (having Θ as its characteristic function, see [10,13]) is MΘ : KΘ → KΘ , MΘ f = PΘ (zf ),

f ∈ KΘ ,

where PΘ denotes the orthogonal projection on KΘ . It is known that every C00 Hilbert space contraction T having unit defect indices, rank(I − T ∗ T ) = rank(I − T T ∗ ) = 1, is unitarily equivalent to a model operator MΘ . Now we can provide an operator theoretic interpretation of our function theoretic results (see [10,13,18]). (i) The spectrum of MΘ coincides with σ (Θ), and the point spectrum (the set of eigenvalues) is Λ = σ (Θ) ∩ D. (ii) The commutant of MΘ , {MΘ } = {A : KΘ → KΘ , AMΘ = MΘ A} coincides with the set of H ∞ functions of MΘ , and is isometrically isomorphic to the quotient algebra H ∞ /ΘH ∞ . (iii) For f ∈ H ∞ , f (λ) are eigenvalues of A = f (MΘ ), where λ ∈ Λ. (iv) The operator f (MΘ ) is invertible if and only if the class f + ΘH ∞ is invertible in the algebra H ∞ /ΘH ∞ . (v) The inner function Θ is a Blaschke product if and only if the eigen- and associated-vectors of MΘ are complete in KΘ . We are interested in the following questions. Is it possible to guarantee the invertibility of f (MΘ ) knowing that the eigenvalues f (λ), for λ ∈ Λ, are bounded away from zero? Is it possible to bound the norm f (MΘ )−1 in terms of the minimum modulus of the eigenvalues δ(f, Λ) = infλ∈Λ |f (λ)| > 0? Clearly, these questions are equivalent to those answered in Theorems 3.3, 3.4, and 4.2. The paper is organized as follows. Section 2 contains the main notation and preliminaries. Section 3 is devoted to the equivalence of the norm controlled inversions and the WEP, as well as to a discussion of the latter property. Section 4 treats the case of the algebra A(D)/ΘH ∞ . 2. Preliminaries This section contains the notation and definitions we will need. The main contribution of this section is the newly-developed definition of WEP, or the Weak Embedding Property. λ λ−z Let bλ = |λ| be a single Blaschke factor and let N = {0, 1, 2, . . .} denote the natu1−λz ral numbers. Letting κ : λ → kλ be a map from D to N satisfying the Blaschke condition k (1 − |λ|) < ∞ (that is the Blaschke mass of κ is finite) we may define the Blaschke λ λ∈D product B = B(κ, ·) by


B(κ, z) =

859

bλ (z)kλ .

kλ >0

Note that kλ is just the multiplicity of the zero λ. A singular inner function S is defined by

ξ +z dμ(ξ ) for z ∈ D, Sμ (z) = exp − ξ −z T

where μ is a nonnegative, finite Borel measure on T that is singular with respect to Lebesgue measure m. Every inner function Θ on D can be factored as Θ = eiθ BSμ , where B is a Blaschke product and S is a singular inner function. We let Λ denote the sequence of zeros of Θ inside D. Recall that a Blaschke product B with (simple) zeros (zn ) is said to be an interpolating Blaschke product if H ∞ |Λ = ∞ (Λ). The classical Carleson theorem says that B is an interpolating Blaschke product if and only if B δC (B) := inf (zn ) = inf 1 − |zn |2 B (zn ) > 0. n bzn n The quantity δC (B) is called the Carleson separation constant for B (or for the corresponding zero-sequence Λ). If we write supp(κ) = {λ ∈ D: kλ > 0} for the zeros of the Blaschke factor in D, the spectrum of an inner function Θ is (by definition) σ (Θ) := supp(κ) ∪ supp(μ) (see, for example, [10, p. 62]). If Θ = B is a Blaschke product with simple zeros (that is, kλ 1 on D), then H ∞ /ΘH ∞ is the space of traces of H ∞ on the set Λ; that is, H ∞ /BH ∞ = H ∞ |Λ endowed with the trace norm given by f = inf{g∞ : g ∈ H ∞ , g|Λ = f |Λ}. For Blaschke products with higher multiplicities, the algebra H ∞ /BH ∞ can be similarly interpreted as a space of germs of height kλ on Λ. Let ρ(z, w) = |(z − w)/(1 − zw)| denote the pseudohyperbolic distance between two points z, w ∈ D. For λ ∈ D and > 0, let Dρ (λ, ) denote the pseudohyperbolic disc with center λ and radius . For a function ϕ : D → C and η > 0 we let Ωη (ϕ) = z ∈ D: ϕ(z) < η denote the η-level set of ϕ. The following concept will be the major feature in what follows.

860


Definition. Let Θ = BSμ be an inner function and let Λ be the zero sequence of Θ in D. We say that Θ satisfies the Weak Embedding Property (WEP) [for short: Θ ∈ WEP], if for every > 0 there exists η > 0 such that Ωη (Θ) ⊆

Dρ (λ, ).

λ∈Λ

For a WEP inner function we define Dρ (λ, ) , ηΘ () = sup η > 0: Ωη (Θ) ⊆ λ∈Λ

and we call ηΘ the WEP characteristic of Θ. For a Blaschke product B = BΛ associated with a Blaschke sequence Λ, we also write ηB = ηΛ . We note that

Dρ (λ, ) = Ω inf |bλ | . λ∈Λ

λ∈Λ

We also note that every WEP inner function has a nontrivial Blaschke factor and the union in our definition above does not depend on the multiplicities of the zeros of the Blaschke factor. The motivation as to why this property refers to a “weak embedding” is given in the introduction. The WEP inner functions are studied in Sections 3, 4 of this paper. However, we mention here four basic properties of WEP inner functions and WEP characteristics. (P1) If infλ∈Λ |bλ (z)| > 0, then |Θ(z)| ηΘ (). (P2) ηΘ () for every > 0. Proof. Indeed, since bλ is a factor of Θ for every λ ∈ Λ we see that |Θ(z)| |bλ (z)| for every z ∈ D and λ ∈ Λ. Therefore, if ηΘ () > we would be able to find an η and a z such that ηΘ () > |Θ(z)| = η > . Applying the definition of ηΘ () we see that for some λ we would have < Θ(z) bλ (z) < , a contradiction.

2

(P3) Let B be a Blaschke product that is the product of N interpolating Blaschke products Bj , 1 j N . Then B is a WEP Blaschke product and ηB () c · N for all with 0 < < 1 and a convenient constant c > 0. Proof. To see this, we note that it is well known (see, for example [10, p. 218]) that an interpolating Blaschke product Bj associated with the zero set Λj is characterized by the lower estimate


861

|Bj (z)| cj infλ∈Λj |bλ (z)| for every z ∈ D, where cj > 0 is a convenient constant. Multiplying these inequalities we obtain a constant c such that B(z) c · inf bλ (z)N λ∈Λ

for every z ∈ D. The result follows.

2

(P4) Let Λ be a sequence in D satisfying CEP (the definition of which appears in the introduction). Then Λ satisfies WEP with a lower bound for ηΛ , as indicated in (P3) above. Proof. Indeed, it is well known (see [2,10], or [13]) that a CEP sequence is a finite union of interpolating sequences. The result follows from (P3) above. 2 It will be convenient to have two notations for the zeros of an inner function when dealing with purely topological properties (see (6) below). Thus, we write Z(Θ) for the zero set {m ∈ M(H ∞ ): m(Θ) = 0} of Θ in the maximal ideal space M(H ∞ ) and ZD (Θ) for the zero set of Θ in the disc. 3. Norm controlled inversions in H ∞ /ΘH ∞ and the WEP The main results of this section are that the norm controlled inversion property for the quotient algebra H ∞ /ΘH ∞ is equivalent to the WEP (Theorems 3.3 and 3.4) and that the WEP is not equivalent to the CEP (Examples 3.7 and 3.8). We also give an upper estimate of the function cn (H ∞ /ΘH ∞ , Λ, δ) in terms of the WEP characteristic ηΘ , where Λ is the zero set of Θ in D. We begin by recalling two known lemmas. The first one is a version of a classical result of Kerr-Lawson [9] and Hoffman [6]. For a proof, see [2, Lemma 1.4, p. 404]. Lemma 3.1 (Hoffman’s lemma). Suppose b is an interpolating Blaschke product with zeros {zn : n √∈ N} and let δ(b) be its Carleson separation constant. If 0 < δ < δ(b) and 0 < < (1 − 1 − δ 2 )/δ, then {z ∈ D: |b(z)| < 2 } is the union of pairwise disjoint domains Vn with zn ∈ Vn and Vn ⊂ {z: ρ(z, zn ) < }. The second lemma is also well known (see [10, p. 218]). Lemma 3.2. Suppose that for z, λ ∈ D we have ρ(z, λ) < δ/3. Then |λ − z| δ/2 min 1 − |λ|2 , 1 − |z|2 . In this section we prove the following theorem. Theorem 3.3. Let Θ be a inner function on D. The following are equivalent. (1) The quotient algebra H ∞ /ΘH ∞ has no corona; that is, Λ = σ (Θ) ∩ D is dense in M(H ∞ /ΘH ∞ ). (2) If f ∈ H ∞ and δ1 (f, Λ) > 0, then f + ΘH ∞ is invertible in H ∞ /ΘH ∞ . (3) δn (H ∞ /ΘH ∞ , Λ) = 0 for every n 1. (4) δ1 (H ∞ /ΘH ∞ , Λ) = 0.

862


(5) Θ satisfies the WEP. (6) Θ = BSμ where ZD (B) = Z(B) = Z(Θ). Moreover, if (1)–(6) hold, then 1 ) log( ηΘ (δ/3) √ , cn H ∞ /ΘH ∞ , Λ, δ 18 n + 1 2 [ηΘ (δ/3)]

where ηΘ () is the WEP-characteristic of Θ. There is an alternative way of looking at the theorem above. For instance, statement (2) says that f + ΘH ∞ ∈ H ∞ /ΘH ∞ is invertible if and only if infλ∈Λ |f (λ)| > 0. Statement (3) means that for every f = (f1 , . . . , fn ) ∈ (H ∞ )n satisfying 0 < δn (f, Λ) f 1 there exist solutions h ∈ H ∞ and g = (g1 , . . . , gn ) ∈ (H ∞ )n to the Bezout equation n

fj gj + Θh = 1

j =1

with a norm control in terms of δn (f, Λ) only: g =

n

1/2 gj 2A

cn Λ, δn (f, Λ) .

j =1

The proof of Theorem 3.3 makes use of Carleson’s corona theorem. The aforementioned estimate will follow from the known estimates in the corona theorem (the best known one is given by S. Treil and B. √ Wick in [19]). The extra factor of n + 1 appears because, in place of the norm used in [10,19] or [2], which is n+1 1/2 2 Fj (z) F H ∞ = sup F (z) , where F (z) = , n+1

z∈D

j =1

we work (following the general setting) with the norm F =

n+1 j =1

1/2 Fj 2∞

that arises from the corresponding expression in H ∞ /ΘH ∞ satisfying √ ∞ . F n + 1 F Hn+1 This choice is quite natural, because H ∞ /ΘH ∞ is not, in general, a uniform algebra and therefore · Hn∞ is not defined. Moreover, for other Banach algebras in which Bezout equations are studied (see [12,14]) the use of norms like · is common. We will prove the following equivalent form of Theorem 3.3.


863

Theorem 3.4. Let Θ be an inner function on D and let Λ be its zero set in D. The following are equivalent. (1) The quotient algebra H ∞ /ΘH ∞ has no corona; that is, Λ is dense in M(H ∞ /ΘH ∞ ). (2) Given f ∈ H ∞ there exist g, h ∈ H ∞ such that f g + Θh = 1 if and only if δ(f, Λ) := infλ∈Λ |f (λ)| > 0. (3) Given f = (f1 , . . . , fn ) ∈ (H ∞ )n with f :=

n j =1

1/2 fj 2∞

1

and δn (f, Λ) = inf

λ∈Λ

n fj (λ)2

1/2 > 0,

j =1

the Bezout equation n

fj gj + Θh = 1

j =1

has a solution (g, h) := (g1 , . . . , gn , h) ∈ (H ∞ )n+1 such that g :=

n

1/2 gj

2

cn H ∞ /ΘH ∞ , Λ, δn (f, Λ) .

j =1

(4) For f ∈ H ∞ with 0 < δ(f, Λ) f 1, there is a solution to the Bezout equation f g + Θh = 1 with g c1 (δ(f, Λ)). (5) Θ satisfies the WEP. (6) Θ = BSμ where ZD (B) = Z(B) = Z(Θ); in particular Z(Sμ ) ⊆ Z(B). Moreover, if (1)–(6) hold, then, for any δ ∈ ]0, 1[, 1

log( ηΘ (δ/3) ) √ , cn H ∞ /ΘH ∞ , Λ, δ 18 n + 1 [ηΘ (δ/3)]2 where ηΘ () is the WEP-characteristic of Θ. Proof. Since M(H ∞ /ΘH ∞ ) can be identified with Z(Θ) (which is well known), it follows that (1) ⇔ (6). Moreover, the implications (3) ⇒ (4) ⇒ (2) as well as (3) ⇒ (1) ⇒ (2) are obvious. So it remains to prove (2) ⇒ (5) and (5) ⇒ (3). . . . , fn ) ∈ (H ∞ )n and suppose 0 < δ = δn (f, Λ) = First we show that (5) ⇒ (3). Let f = (f1 , infλ∈Λ |f (λ)| f 1, where |f (λ)| = ( nj=1 |fj (λ)|2 )1/2 and f = ( nj=1 fj 2∞ )1/2 . Let z ∈ D satisfy infλ∈Λ |bλ (z)| < δ/3 and let λ be such that |bλ (z)| δ/3. As in [10, p. 218]

864


we note that for h ∈ H ∞ if we define the conformal map τ (z) = (w − z)(1 − wz)−1 and write g = h ◦ τ −1 , then h (w) = g (0)τ (w) g∞ τ (w) = h∞ 1 − |w|2 −1 , where the inequality follows from Cauchy’s formula. Therefore, for λ ∈ Λ, f (z) f (λ) − f (λ) − f (z)

2 1/2 δ − |λ − z| · max fj (t) j

δ − |λ − z| ·

j

= δ − |λ − z| ·

t∈[λ,z]

−1 max fj ∞ 1 − t 2

t∈[λ,z]

fj / min 1 − |λ|2 , 1 − |z|2

j

δ − |λ − z|/ min 1 − |λ|2 , 1 − |z|2 . If |bλ (z)| δ/3, by Lemma 3.2 |λ − z| δ/2 · min 1 − |λ|2 , 1 − |z|2 . Consequently, for z satisfying infλ∈Λ |bλ (z)| < δ/3 we have f (z) δ/2. On the other hand, if z satisfies infλ∈Λ |bλ (z)| δ/3, then (see property (P1), Section 2), |Θ(z)| ηΘ (δ/3). Thus by property (P2), Section 2, ηΘ (δ/3) δ/3 < 1/e, we obtain f (z)2 + Θ(z)2 min (δ/2)2 , ηΘ (δ/3)2 = ηΘ (δ/3)2 for every z ∈ D. By Carleson’s corona theorem and the estimates of solutions of corona equations from [19], there exists a solution (g, h) := (g1 , . . . , gn , h) of the Bezout equation n

fj gj + Θh = 1

j =1

such that g

√

n gj (z)2 + h(z)2 n + 1 sup z

√

1/2

j =1

17 1 1 + n+1 log ηΘ (δ/3) ηΘ (δ/3)2 ηΘ (δ/3)

√ 18 1 log n+1 . ηΘ (δ/3) ηΘ (δ/3)2


865

This completes the proof of (5) ⇒ (3) and the estimate of inverses claimed in Theorems 3.3 and 3.4. We turn to the final implication showing that (2) ⇒ (5). k First assume that Θ is a Blaschke product; that is, Θ = B = j bλjj , where (λj ) denotes the zero sequence of Θ. Let be a positive number so that < 1. Consider the sets z ∈ D: bλj (z) < , Ω() := Ω inf |bλj | = j

j

and Ω (B) = z: B(z) < . Clearly, Ω() ⊆ Ω (B). Note also that for every r ∈ ]0, 1[ and every > 0, there exists an η > 0 such that Ωη (B) ∩ z ∈ D: |z| r ⊆ Ω(); N kj for example, we may take η = M min|z|r ∞ j =N +1 |bλj (z)|, where M = j =1 kj and N is chosen so large that |λj | > r for every j N . We want to prove that for every > 0 there exists η > 0 such that Ωη (B) ⊆ Ω()

(3.1)

(in other words, {|B| < η} ⊆ j Dρ (λj , )). To this end, suppose to the contrary that there exists > 0 such that for all η > 0 we have Ωη (B) \ Ω() = ∅. Thus, if z1 ∈ Ω1/2 (B) \ Ω(), then |bλj (z1 )| > for every j and |B(z1 )| 1/2. Now let r = 1 − (1 − |z1 |)/2 and choose n2 > 2 so that Ω1/n2 (B) ∩ z: |z| r ⊆ Ω(). Then Ω1/n2 (B) \ Ω() ⊆ z ∈ D: 1 − |z| < 1 − |z1 | /2 . Taking z2 ∈ Ω1/n2 (B)\Ω() we get 1−|z2 | < (1−|z1 |)/2, |bλj (z2 )| > for all j , and |B(z2 )| 1/n2 . Continuing in this way, we obtain sequences (nk ) and (zk ) such that (a) |bλj (zk )| = |bzk (λj )| > for all j and all k; (b) |B(zk )| 1/nk where nk > nk−1 ; (c) 1 − |zk | < (1 − |zk−1 |)/2. Using [10, p. 159] and property (c), we see that (zk ) is an interpolating sequence. Let B1 = ∞ k=1 bzk be the corresponding interpolating Blaschke product. Then [10, p. 218] implies that there exists a constant γ such that B1 (z) γ infbz (z). (3.2) j j

866


Now property (a) implies that |B1 (λj )| γ for every j . Property (b) implies that 2 2 2 inf B1 (z) + B(z) : z ∈ D infB(zk ) = 0. k

Therefore, we cannot find H ∞ functions g and h such that B1 g + Bh = 1. This contradicts our hypothesis (2) of Theorem 3.4. Hence our assumption that Ωη (B) \ Ω() = ∅ was wrong. Thus we have established (3.1) and so we proved statement (5) of Theorem 3.4 in the case when Θ is a Blaschke product. Now suppose that Θ = BSμ is an arbitrary inner function satisfying (2). Note that since ΘH ∞ = BSμ H ∞ ⊆ BH ∞ , it follows that for all f if f + ΘH ∞ is invertible in H ∞ /ΘH ∞ , then f + BH ∞ is invertible in H ∞ /BH ∞ . Therefore, the Blaschke factor of Θ also satisfies property (2). We turn now to the singular factor of Θ. First we show that for every α > 0 there exists β > 0 such that {|Sμ | < β} ⊆ {|B| < α}. Suppose this is not the case. Then there exists α0 > 0 such that for every β = 1/m, where m = 1, 2, . . . , there is a point zm ∈ D such that |S(zm )| < 1/m, but |B(zm )| α0 . Without loss of generality, we may assume that (zm ) is an interpolating sequence. Let b be the associated interpolating Blaschke product. Recall that (λk ) denotes the zero sequence of B. Since |bλk (zm )| |B(zm )| α0 for every m and k, we obtain (by Lemma 3.1 or formula (3.2)), that |b(λk )| is bounded away from zero. Thus assertion (2) implies that b + ΘH ∞ is invertible in H ∞ /ΘH ∞ ; that is there exist f and g in H ∞ such that f b + gΘ = 1. But 2 2 2 inf b(z) + Θ(z) : z ∈ D infSμ (zm ) = 0. m

This is a contradiction. Thus, for every α > 0 there exists β > 0 such that Ωβ (Sμ ) ⊆ Ωα (B). Without loss of generality, we may assume that β α. We have already shown that B ∈ WEP, so for > 0 we may choose α > 0 so that Ωα (B) ⊆ Ω (infλ |bλ |). Let β α be as above. We claim that by setting η = β 2 , we obtain Ωη (Θ) ⊆ Ω (infλ |bλ |); that is, we claim that assertion (5) of the theorem holds. In fact, if |Θ(z)| = |B(z)Sμ (z)| < β 2 , then either |B(z)| < β α and hence z ∈ Ω (infλ |bλ |), or |Sμ (z)| < β and then |B(z)| < α, and again z ∈ Ω (infλ |bλ |). 2 We derive two immediate corollaries. First we formalize the “splitting property” of WEP inner functions that appeared at the end of the previous proof. Corollary 3.5. Let Θ = BSμ be an inner function. The following assertions are equivalent: (a) Θ ∈ WEP; (b) B ∈ WEP and for every α > 0 there exists β > 0 such that Ωβ (Sμ ) ⊆ Ωα (B). Proof. By the theorem above, (a) implies assertion (2) of Theorems 3.3, 3.4. Then the second to the last paragraph of the previous proof shows that (b) holds. The last paragraph of the same proof above shows that (b) implies (a). 2


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Corollary 3.6. Let N be an integer and B a product of N interpolating Blaschke products. Then assertions (1)–(6) in Theorem 3.4 hold true for B. Moreover, for every δ ∈ ]0, 1[ and n 1 there exists a constant a > 0 (depending on B, and in particular on N ), such that 1

log( δ/3 ) √ cn H ∞ /BH ∞ , Λ, δ a n + 1 . δ 2N Proof. This follows from Theorem 3.4 and property (3) of Section 2.

2

In Section 2 we discussed Kerr-Lawson’s lemma, which shows that if B is a finite product of interpolating Blaschke products with zero sequence (zn ), then for every ∈ ]0, 1[ there exists η > 0 such that z ∈ D: B(z) < η ⊆ Dρ (zn , ). (3.3) n

Thus, as we noted in Section 2 (property (P3)), every such Blaschke product satisfies the weak embedding property. Kerr-Lawson [9] also proved that if u is an inner function and for some η > 0 and ∈ ]0, 1[ the set {z ∈ D: |u(z)| < η} is contained in the disjoint union n Dρ (zn , ) of pseudohyperbolic discs of fixed radius and centers zn , then u is a finite product of interpolating Blaschke products. In the last paragraph of his paper he asserted that this implies, in particular, that every Blaschke product that is not a finite product of interpolating Blaschke products is arbitrarily small arbitrarily far away from its zeros inside D. The following example shows that this is not the case; indeed the class of WEP Blaschke products is strictly larger than the class of finite products of interpolating Blaschke products. As indicated in the introduction, the main idea for these examples is due to S. Treil and was realized in a correspondence between V. Vasyunin and the third author of this paper. We are indebted to Professors Treil and Vasyunin for permitting us to include the example in this paper. Clearly, WEP, as defined in Section 2, is conformally invariant. In particular, we can replace the disc D by the upper-half plane C+ = {z ∈ C: Im z > 0} without changing the definitions. Since the pseudohyperbolic geometry of C+ is more transparent than that of D, we give our principal example in C+ (Example 3.7 below). Example 3.7. There exists a WEP Blaschke product B that does not satisfy the CEP (that is, B satisfies the WEP, but B is not a finite product of interpolating Blaschke products). Proof. The construction is done using techniques from [10]. Let a > 0 and let Θ(z) = eiaz be an inner function on C+ = {z: Im z > 0}. For t > 0, consider the Frostman shift Ba,t of Θ given by Ba,t (z) =

Θ(z) − e−t . 1 − e−t Θ(z)

Then Ba,t is a Blaschke product with zeros zk,a,t = 2πk/a + it/a, k ∈ Z. It satisfies the Carleson interpolation condition and the Blaschke mass of its zeros is k

t2

at ∼ a. + 4π 2 k 2

868


Now take a = 1/n2 and t = n for n = 1, 2, . . . . Define the Blaschke product B by B=

B1/n2 ,n .

n

This will be the Blaschke product we seek. Note that B has the zeros zk,n = 2πkn2 + in3 , for k ∈ Z and n 1. Moreover, we have the following: (1) For every > 0 the level set Ω =

z ∈ C+ : |z − zk,n | < n3 k,n

3 }. The zeros of B outside contains a half-plane; for example Π = {z ∈ C+ : Im z > (100/) (1) Π form a Carleson interpolating sequence, hence B := n100/ B1/n2 ,n is an interpolating Blaschke product. (2) A direct look at

eiz/n2 − e−n 1 − e−n eiz/n2 shows that the subproduct B (2) := n>100/ B1/n2 ,n is bounded away from zero on the strip S = {z ∈ C+ : 0 < Im z (100/)3 }. (3) Since (1) and (2) are valid for every > 0 and B = B (1) B (2) , for sufficiently small η > 0, the part of the level set S ∩ {z ∈ C+ : |B(z)| < η} is included in the union

z ∈ C+ : |z − zk,n | < n3 ,

k∈Z, n 0 there exists η > 0 such that {z ∈ C+ : |B(z)| < η} ⊆ Ω , but B is not a finite product of interpolating Blaschke products (because, for example, the Carleson embedding theorem is not satisfied: max x+iy∈Q y, where the summation runs over all zeros x + iy of B in the square Q = {0 x, y a}, is at least of the order a 4/3 as a → ∞, and not of the order of a as required for Carleson measures). 2 It is of interest to know whether there are inner functions satisfying the WEP that are not Blaschke products. The next example shows that such inner functions exist. Example 3.8. Let B be the Blaschke product of Example 3.7 and let S(z) = eiaz , where a > 0. Then the function Θ = SB is a WEP inner function. Proof. Indeed, keeping the notation of the previous example, we will prove that for every > 0, there exists η > 0 such that Ωη (Θ) ⊂ Ω . Let Π be the half-plane defined in Example 3.7 above. For z ∈ C+ \ Π , we have Θ(z) e−α B(z),


869

where α = a(100/)3 . Hence, by Example 3.7, Ω(Θ) ∩ (C+ \ Π ) ⊆ Ωηeα (B) ⊂ Ω for η > 0 sufficiently small. As we saw in Example 3.7 above, Π ⊂ Ω , so the inclusion Ωη (Θ) ⊂ Ω follows. 2 At this point, we mention a few additional properties of WEP Blaschke products and WEP sequences. Properties (P1)–(P4) can be found in Section 2. (P5) A separated WEP sequence is interpolating. Proof. Let Λ = (λj ) be a separated WEP-sequence and let B be the associated Blaschke product. Then, for every ∈ ]0, 1[ there exists η = η() > 0 such that |B| < η ⊆ Dρ (λ, ). λ∈Λ

Since Λ is separated, there exists > 0 such that the discs D = Dρ (λ, ), λ ∈ Λ, are pairwise disjoint. On the boundary, ∂D, of such a disc we have B(z) η( ) = η( ) bλ (z).

By the minimum modulus principle, the function ϕ : z → B(z)/bλ (z), which is holomorphic and |ϕ(z)| η( )/ on this disc. Taking z = λ, we get zero free on D, satisfies the same estimate the Carleson interpolation condition j =k |bλj (λk )| η( )/ > 0 for every λk ∈ Λ. 2 (P6) Let Λ be a finite union of separated sequences and suppose that Λ ∈ WEP. Then Λ ∈ CEP. Proof. Assume that Λ is a union of N separated sequences and let B = λ∈Λ bλ .1 Consider the open set Ω (infλ∈Λ |bλ |) = λ∈Λ Dρ (λ, ) and its open connected components Ω , Ω 2 , . . . . The triangle inequality implies that for > 0 small enough every Ω j contains no more than N points of Λ. Fixing such an > 0, we apply WEP, which gives Ωj Ωη (B) ⊆ Ω inf |bλ | = λ∈Λ

j

for some η > 0. Then every connected component of Ωη (B) is contained in one of Ω j , hence contains no more than N points of Λ. But it is known that a Blaschke product is the product of at most N interpolating Blaschke products if and only if there exists η > 0 such that every connected component of the level set {z: |B(z)| < η} contains at most N zeros of B. This fact is implicitly contained in [11, pp. 229, 230]. It can also be found in [13,15] and [14, vol. 2, p. 189]. Thus Λ ∈ WEP. 2 (P7) Let ζ ∈ ∂D and let 0 < θ < π/2. Let Λ be a sequence in a Stolz angle S(ζ, θ ) (that is the convex hull of the disk |z| sin θ and the point ζ ). Then Λ ∈ WEP if and only Λ ∈ CEP.

870


Proof. By definition, we see that a function satisfying CEP also satisfies WEP. To prove the converse, since a separated sequence in a Stolz angle is interpolating, it suffices to show that Λ is a finite union of separated sequences. It is also clear that for the latter property it is sufficient to prove that there exist β and N , 0 < β < 1, such that every disc Dρ (λ, β), λ ∈ Λ, contains no more than N points of Λ (the standard dyadic “white-and-black-boxes” reasoning gives the proof, see for instance [2], or [10, p. 159]). In order to ensure the last property, observe that if θ < θ < π/2, then α := inf bz (λ): z ∈ ∂S(ζ, θ ), λ ∈ Λ > 0. On the other hand, there exists ∈ ]0, 1[, close to 1, such that Dρ (w, ) ∩ ∂S(ζ, θ ) = ∅ for every w ∈ S(ζ, θ ). Hence, for every λ ∈ Λ, there exists z = z(λ) ∈ ∂S(ζ, θ ) such that |bz (λ)| < . Therefore, if β > 0 and γ := + β < 1, then we have Dρ (λ, β) ⊆ Dρ (z(λ), γ ). Now, denoting N(λ) = card Dρ (λ, β) ∩ Λ, we get B z(λ) bμ z(λ) γ N (λ) , μ

where the product is taken over all μ ∈ Dρ (λ, β) ∩ Λ. Since the WEP implies that |B(z(λ))| ηB (α), where B is the Blaschke product associated with Λ (see (P1), Section 2), we get supλ∈Λ N (λ) < ∞. 2 Alternatively, by using maximal ideal space techniques, we can prove the statement in (P7) as follows. Let Θ satisfy the WEP and assume that the zero set Λ of Θ is contained in a Stolz-angle. By Theorem 3.3, (1) ⇔ (6), we have that ZD (Θ) = Z(Θ). By [6, Theorem 6.4] every point m ∈ M(H ∞ ) in the closure of a Stolz-angle belongs to the closure of an interpolating sequence. Thus, by [4], Θ is a finite product of interpolating Blaschke products and so Λ ∈ CEP. 2 (P8) A finite product of WEP Blaschke products has the WEP property. Proof. Let Bj be the corresponding Blaschke products with zero sequences Λj , j = 1, . . . , n. By the WEP for Bj , there exists for every > 0 some δj such that Ωδj (Bj ) ⊆ Ω Now, if B =

n

j =1 Bj ,

Λ=

Ωδ (B) ⊆

n

n

j =1 Λj ,

and δ =

Ωδj (Bj ) ⊆

j =1

Thus B is a WEP-inner function.

n j =1

inf |bλ | .

λ∈Λj

n

j =1 δj ,

Ω

then

inf |bλ | ⊆ Ω inf |bλ | .

λ∈Λj

λ∈Λ

2

Since the definition of a WEP sequence does not involve the multiplicities of the zeros of the associated Blaschke product B, one can ask whether there exist WEP sequences whose multiplicities are unbounded.


871

(P9) There exist WEP sequences whose point multiplicities are not bounded. Proof. In fact, using√ the notation of the Example for a slowly growing sequence pn → ∞ 3.7, pn (for example, pn = [ n]), the product B = n1 B1/n 2 ,n satisfies the WEP property. This can be checked in the same manner as Example 3.7. 2 (P10) Let Λ = (λj ) be a WEP sequence with associated Blaschke product B and for > 0 let Λ() its subsequence of -isolated points; that is, Λ() = λj ∈ Λ: inf bλj (λk ) . λk =λj

Then, the multiplicities Nj of points λj in Λ() are uniformly bounded, namely, Nj

log ηΛ (/2) . log(/2)

Proof. To see this, let λj ∈ Λ() and z ∈ D such that |bλj (z)| = /2. Then |bλ (z)| /2 for every λ ∈ Λ. By the WEP-property (property (P1) of Section 2), |B(z)| ηB (/2). On the other hand, (/2)Nj = |bλj (z)|Nj |B(z)|. The result now follows. 2 Finally we remark that by [3], if Θ is WEP-inner function, then for every a ∈ D \ {0} with |a| sufficiently small, the Frostman transform (Θ − a)/(1 − aΘ) is in CEP. In particular, Θ can be uniformly approximated by interpolating Blaschke products. D)/ΘH ∞ 4. The quotient algebra A(D We note that it is still possible to make sense of Theorem 3.3 when we replace Λ = σ (Θ) ∩ D with Λ = σ (Θ). Rather than looking at H ∞ functions, we consider functions from the disc algebra, denoted here by A(D). Recall that A(D)/ΘH ∞ is the canonical image of A(D) in the quotient algebra H ∞ /ΘH ∞ . As mentioned in the introduction, the algebra A(D)/ΘH ∞ is closed in H ∞ /ΘH ∞ if and only if either m(σ (Θ) ∩ T) = 0 or m(σ (Θ) ∩ T) = 1. As we will see later on (Theorem 4.2), the latter case is not of interest for the efficient inversion problem. The former one, to the contrary, is very interesting. In this case, for the problem of norm controlled inversions, the algebra A(D)/ΘH ∞ is even more significant than H ∞ /ΘH ∞ . The reason is that for H ∞ /ΘH ∞ , the norm controlled inversion property (incidentally) coincides with the corona property, and hence the metric problem on the critical constants δn is, in a sense, hidden behind the topological fact that the visible spectrum Λ is dense in M(H ∞ /ΘH ∞ ). For A(D)/ΘH ∞ , these two properties are distinct: the algebra A(D)/ΘH ∞ never has a corona with respect to σ (Θ), but it may or may not have the norm-controlled inversion property. This phenomenon, which does appear in different situations (see [2,14,16]), is a specific internal property of a Banach algebra measuring the discrepancy between the Gelfand transform norm and the original algebra norm. Lemma 4.1. Let Θ be an inner function. The natural restriction embedding f + ΘH ∞ → f |σ (Θ) is a (contractive) homomorphism from the quotient algebra A(D)/ΘH ∞ into C(σ (Θ)). If m(σ (Θ) ∩ T) = 0, the maximal ideal space M(A(D)/ΘH ∞ ) coincides with σ (Θ) with respect to this mapping. Consequently, given fj ∈ A(D) the Bezout equation nj=1 gj fj = 1 is

872


solvable in A(D)/ΘH ∞ if and only if nj=1 |fj (λ)|2 > 0 for every λ ∈ σ (Θ) (or if and only if n infλ∈Λ ( j =1 |fj (λ)|2 ) > 0, in the case in which Θ is a Blaschke product B and Λ = B −1 ({0})). Proof. It is known that σ (Θ) = {λ ∈ D: lim infζ →λ |Θ(ζ )| = 0} (see [13, p. 63]). Hence, for every λ ∈ σ (Θ), f ∈ A(D), and g ∈ H ∞ , we have f (λ) = lim f (ζ ) lim supf (ζ ) + Θ(ζ )g(ζ ) f + Θg∞ , ζ →λ

ζ →λ

and therefore, f |σ (Θ)C(σ (Θ)) f H ∞ /ΘH ∞ . Now, assume that m(σ (Θ) ∩ T) = 0. If ϕ is a complex continuous homomorphism on A(D)/ΘH ∞ and λ = ϕ(z + ΘH ∞ ) (where z(ζ ) ≡ ζ ), then λ ∈ σ (Θ). Indeed, λ ∈ D since ϕ 1, and hence ϕ extends naturally to a homomorphism of A(D), f → f (λ). Moreover, there exists an outer function g in A(D) such that the zero set of g is σ (Θ) ∩ T, which gives 0 = ϕ(Θg) = (Θg)(λ); that is, λ ∈ σ (Θ). Since the polynomials are dense in A(D), we get ϕ(f + ΘH ∞ ) = f (λ) for every f ∈ A(D). Therefore, M(A(D)/ΘH ∞ ) ⊆ σ (Θ). Note that A(D) distinguishes the points of σ (Θ). The reverse inclusion is obvious. 2 Theorem 4.2. Let Θ be an inner function on D. The following are equivalent. (1) δn (A(D)/ΘH ∞ , σ (Θ)) = 0 for every n 1. (2) δ1 (A(D)/ΘH ∞ , σ (Θ)) = 0. (3) m(σ (Θ) ∩ T) = 0 and Θ ∈ WEP. Moreover, if (1)–(3) hold, then for every δ ∈ ]0, 1[, 1

log( ηΘ (δ/3) ) √ cn A(D)/ΘH ∞ , Λ, δ 18 n + 1 , [ηΘ (δ/3)]2 where, as usual, Λ = σ (Θ) ∩ D is the zero set of Θ and ηΘ () is the WEP-characteristic of Θ. Before proceeding to the proof, we note that the essential part of Theorem 4.2 will follow from estimates of solutions of the Bezout equation in the disc algebra A(D) as well as a generalization of the Rudin–Carleson theorem on free A(D)-interpolation. By the latter, we mean the following: let E ⊂ D be a closed set. Then A(D)|E = C(E) if and only if m(E ∩ D) = 0 and E ∩ D is an interpolating sequence (see [5]). For our proof we will need the following lemma suggested by Sergei Treil. This lemma can also be found in [16]. Lemma 4.3. Let 0 < δ 1. Then cn (H ∞ , D, δ) = cn (A(D), D, δ). Proof. The inequality cn (H ∞ , D, δ) cn (A(D), D, δ) is well known (see, for example, [10, Appendix 3]). For the reverse inequality, let n 1 and let f = (f1 , . . . , fn ) ∈ (A(D))n satisfy δn (f, D) = ∞ n ∞ solution of the equation inf nλ∈D |f (λ)| > 0. Let φ = (φ1 , . . . , φn ) ∈ (H ) be an H j =1 φj fj = 1.


873

Now, given > 0 there exists r with 0 < r < 1 such that n n φj (rz)fj (z) − 1 = φj (rz) fj (z) − fj (rz) < , j =1

j =1

for every z ∈ D. Therefore, the functions φj (rz) , j =1 φj (rz)fj (z)

gj (z) = n

z ∈ D,

give a solution in A(D) satisfying nj=1 gj fj = 1 with norm arbitrarly close to that of the solution given by φj . The inequality follows. 2 Proof of Theorem 4.2. It is clear that (1) ⇒ (2). We show that (2) ⇒ (3). Note first that if F ∈ A(D), then F is continuous at every point of the unit circle. If the inner factor of F is discontinuous at a point λ, its cluster set at the point λ is the closed unit disc [2, p. 80]. Therefore, F must vanish at the point λ. Now suppose f, g ∈ A(D) and f g + Θh = 1. Letting Θh play the role of F above we conclude that f g = 1 on σ (Θ). If m(σ (Θ) ∩ T) > 0, then f g ≡ 1. Since this must hold for every f that is bounded away from zero on σ (Θ) (and, in particular, for those f having a zero outside σ (Θ)) we see that the condition m(σ (Θ) ∩ T) = 0 is necessary for (2). Now let Λ = σ (Θ) ∩ D and let f ∈ H ∞ be such that 0 < δ = δ(f, Λ) = inf f (λ) f ∞ < 1. λ∈Λ

By R. Nevanlinna’s theorem (see [10, p. 204] or [13, vol. 1, p. 234]) there exists an inner function ϕ such that ϕ|Λ = f |Λ. Moreover, the same is true for any Blaschke set Λ ⊃ Λ. By using this fact for (1 + n1 )f instead of f and for Λn = Λ ∪ { 12 , 13 , . . . , n1 } instead of Λ, and uniformly approximating the corresponding inner function ϕ by Blaschke products, we obtain a sequence of Blaschke products Bn such that 1 f (z) − Bn (z) δ/n. sup 1 + n z∈Λn In particular |Bn | (1 + n1 )|f | − |Bn − (1 + n1 )f | δ on Λ. Choosing convenient finite Blaschke subproducts Cn of Bn , we get a sequence (Cn ) of functions in A(D) such that |Cn | δ on Λ and such that (Cn ) converges locally uniformly on D to f . Now Cj ∈ A(D) and 0 < δ = δ(f, Λ) |Cj (λ)| Cj ∞ = 1 for every λ ∈ σ (Θ). By (2), there exist gn ∈ A(D) and hn ∈ H ∞ satisfying Cn gn + Θhn = 1 and gn ∞ c1 A(D)/ΘH ∞ , δ . Using Montel’s theorem we obtain functions g and h in H ∞ with g c1 A(D)/ΘH ∞ , δ and f g + Θh = 1.

874


In particular, statement (2) of Theorem 3.4 holds. Thus, it follows from Theorem 3.4 (statement (5)) that Θ has the desired property. The last assertion of the theorem follows from the corresponding statement in Theorem 3.3 and Lemma 4.3. Next we show (3) ⇒ (1). Let Λ = σ (Θ) ∩ D and f = (f1 , . . . , fn ) ∈ (A(D))n be such that 0 < δ = δn (f, Λ) = minf (λ) f 1. λ∈Λ

The proof now proceeds in the same manner as that of Theorem 3.4. This yields the following estimation: f (z)2 + Θ(z)2 ηΘ (δ/3)2 for every z ∈ D. Since f is continuous on D and σ (Θ) ⊆ Λ, we see that |f | ηΘ (δ/3) on σ (Θ). Also, since m(σ (Θ) ∩ D) = 0, there exists a peak function p ∈ A(D) such that p = 1 on σ (Θ) ∩ T and |p(z)| < 1 for z ∈ D \ (σ (Θ) ∩ T) [7, p. 80]. Now given > 0, we may choose n sufficiently large, so that the function φ = 1 − p n satisfies f (z)2 + Θ(z)φ(z)2 (1 − )ηΘ (δ/3)2 for every z ∈ D. Now all the data, fj and Θφ, are in A(D), so we may use Lemma 4.3 to conclude that there exists g1 , . . . , gn+1 ∈ A(D) such that n

fj gj + Θφgn+1 = 1

j =1

and the norm g of g = (g1 , . . . , gn+1 ) is arbitrarily close to the norm of the best possible H ∞ solution. This shows that the lower bound for A(D)-solutions is the same as for the best H ∞ -solutions. 2 5. Open questions We conclude this paper with several open questions. (1) Find a geometric description of WEP sequences, introducing a “weak Carleson density” in place of the classical one that gives a description of the CEP sequences: μ(Qh ) ch, where μ = κ(λ)>0 (1 − |λ|2 )δλ and where Qh is a Carleson square with side h (see [2] or [10] for information on Carleson measures). (2) Describe possible singular factors Sμ of WEP inner functions Θ = Sμ B. We remark that it follows from Example 3.8 that a singular inner function Sμ with finite support, supp(μ), is admissible. (3) Is it true that finite products of interpolating Blaschke products are characterized by property (P3) of Section 2? It is sufficient to prove that if Λ is not in WEP, then c1 (Λ, δ) grows faster than any power of 1/δ as δ → 0.


875

(4) Let Θ be a non-WEP inner function, Θ = Sμ BΛ . Is it true that δ1 H ∞ /ΘH ∞ , Λ = 1 or

δ1 A(D)/ΘH ∞ , σ (Θ) = 1?

The latter statement makes sense even when Λ = ∅ (in which case the question is about an estimate of f −1 A(D)/ΘH ∞ for functions f ∈ A(D) satisfying 0 < δ f (z) f ∞ 1 for all z ∈ σ (Θ)); in this case the answer is known to be “yes,” see [11]). (5) Can every inner function be multiplied into the class WEP by a WEP-Blaschke product? This would solve, in particular, the long-standing open problem whether every point outside the Shilov boundary and having a trivial Gleason part lies in the closure of a Blaschke sequence. Acknowledgments The authors are grateful to Sergei Treil and his wife Marina who improvised an informal 48-hour seminar with the third author on the WEP property at their home in Providence (Fall 2006), and to Vassily Vasyunin who made a 1000-mile trip to Providence in order to add his mathematical expertise to these efforts. After several late night attempts to prove or to disprove the conjecture “WEP = CEP,” Sergei arrived on the morning of the second day with glasses of juice and the inspired idea of an example of a sequence that is WEP but not CEP (presented in Example 3.7). References [1] O. El-Fallah, N.K. Nikolski, M. Zarrabi, Estimates for resolvents in Beurling–Sobolev algebras, Algebra i Analiz 10 (6) (1998) 1–92 (in Russian, Russian summary); translated in: St. Petersburg Math. J. 10 (6) (1999) 901–964. [2] J.B. Garnett, Bounded Analytic Functions, Pure Appl. Math., vol. 96, Academic Press, New York, 1981, xvi+467. [3] P. Gorkin, R. Mortini, Two new characterizations of Carleson–Newman Blaschke products, preprint, 2007. [4] C. Guillory, K. Izuchi, D. Sarason, Interpolating Blaschke products and division in Douglas algebras, Proc. R. Ir. Acad. 84 (1984) 1–7. [5] E. Heard, J. Wells, An interpolation problem for subalgebras of H ∞ , Pacific J. Math. 28 (1969) 543–553. [6] K. Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967) 74–111. [7] K. Hoffman, Banach Spaces of Analytic Functions, reprint of the 1962 original. Dover, New York, 1988. [8] K. Izuchi, Y. Izuchi, Algebras of bounded analytic functions containing the disk algebra, Canad. J. Math. 38 (1986) 87–108. [9] A. Kerr-Lawson, Some lemmas on interpolating Blaschke products and a correction, Canad. J. Math. 21 (1969) 531–534. [10] N.K. Nikolski, Treatise on the Shift Operator. Spectral Function Theory, with an appendix by S.V. Hrušˇcev [S.V. Khrushchev] and V.V. Peller; translated from the Russian by J. Peetre, Grundlehren Math. Wiss., vol. 273, Springer-Verlag, Berlin, 1986. [11] N.K. Nikolski, In search of the invisible spectrum, Ann. Inst. Fourier (Grenoble) 49 (6) (1999) 1925–1998. [12] N.K. Nikolski, The problem of efficient inversions and Bezout equations, in: Twentieth Century Harmonic Analysis—A Celebration, Il Ciocco, 2000, in: NATO Sci. Ser. II Math. Phys. Chem., vol. 33, Kluwer Acad. Publ., Dordrecht, 2001, pp. 235–269. [13] N.K. Nikolski, Operators, Functions, and Systems: An Easy Reading, vols. 1, 2. Model Operators and Systems, Math. Surveys Monogr., vol. 93, Amer. Math. Soc., Providence, RI, 2002. [14] N.K. Nikolski, Condition numbers of large matrices, and analytic capacities, Algebra i Analiz 17 (4) (2005) 125–180 (in English, English summary).

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[15] N.K. Nikolski, S.V. Khrushchëv, A functional model and some problems of the spectral theory of functions, Tr. Mat. Inst. Steklov. 176 (1987) 97–210, 327 (in Russian); translated in: Proc. Steklov Inst. Math. 3 (1988) 101–214. [16] S. Sidney, Sunwook Hwang, Sequence spaces of continuous functions, Rocky Mountain J. Math. 31 (2001) 641– 659. [17] D. Stegenga, Sums of invariant subspaces, Pacific J. Math. 70 (1977) 567–584. [18] B. Szökefalvi-Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, New York, 1970. [19] S. Treil, B. Wick, The matrix-valued H p corona problem in the disk and polydisk, J. Funct. Anal. 226 (2005) 138–172.


Ornstein–Uhlenbeck processes on Lie groups ✩ Fabrice Baudoin a , Martin Hairer b , Josef Teichmann c,∗ a Université Paul Sabatier, Institut des Mathématiques, 118, Rue de Narbonne, Toulouse, Cedex 31062, France b The University of Warwick, Mathematics Department, CV4 7AL Coventry, United Kingdom c Department of Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstrasse 8–10,

A-1040 Wien, Austria Received 15 November 2007; accepted 12 May 2008 Available online 20 June 2008 Communicated by Daniel W. Stroock

Abstract We consider Ornstein–Uhlenbeck processes (OU-processes) associated to hypo-elliptic diffusion processes on finite-dimensional Lie groups: let L be a hypo-elliptic, left-invariant “sum of the squares”-operator on a Lie group G with associated Markov process X, then we construct OU-processes by adding negative horizontal gradient drifts of functions U . In the natural case U (x) = − log p(1, x), where p(1, x) is the density of the law of X starting at identity e at time t = 1 with respect to the right-invariant Haar measure on G, we show the Poincaré inequality by applying the Driver–Melcher inequality for “sum of the squares” operators on Lie groups. The resulting Markov process is called the natural OU-process associated to the hypo-elliptic diffusion on G. We prove the global strong existence of these OU-type processes on G under an integrability assumption on U . The Poincaré inequality for a large class of potentials U is then shown by a perturbation technique. These results are applied to obtain a hypo-elliptic equivalent of standard results on cooling schedules for simulated annealing on compact homogeneous spaces M. © 2008 Elsevier Inc. All rights reserved. Keywords: Lie group; Hypo-elliptic diffusion; Spectral gap; Simulated annealing

✩ The first and third authors gratefully acknowledge the support from the FWF-grant Y 328 (START prize from the Austrian Science Fund). The second author gratefully acknowledges the support from the EPSRC fellowship EP/D071593/1. All authors are grateful for the warm hospitality at the Mittag-Leffler Institute in Stockholm and at the Hausdorff Institute in Bonn. * Corresponding author. E-mail addresses: [email protected] (F. Baudoin), [email protected] (M. Hairer), [email protected] (J. Teichmann).


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F. Baudoin et al. / Journal of Functional Analysis 255 (2008) 877–890

1. Introduction We consider a left-invariant hypo-elliptic diffusion process X, dXtx =

d

Vi Xtx ◦ dBti ,

X0x = x,

i=1

on a connected Lie group G together with a right-invariant Haar measure μ, which is then also invariant for the diffusion process X. The vector fields V1 , . . . , Vd are assumed to be left-invariant vector fields, and their brackets generate the Lie algebra. In the spirit of sub-Riemannian geometry the hypo-elliptic diffusion process can be used to define a metric (the Carnot–Carathéodory metric d), a geodesic structure and the notion of a gradient (the horizontal gradient gradhor ) on G. The density of the law of Xte with respect to μ is denoted by p(t, .) and smooth by Hörmander’s theorem. Fix t > 0. The natural Ornstein–Uhlenbeck process on G associated to X will be 1 gradhor p(τ, Yt ) dt + Vi (Yt ) ◦ dBti 2 d

dYt =

i=1

out of two reasons: first, the law of Xte is an invariant measure for Y and, second, there is a spectral gap for Y if X satisfies a Driver–Melcher inequality (see Section 4 for precise details). We can easily extend all results to compact homogeneous spaces M. The possible exponential convergence rate is then applied for new simulated annealing algorithms. The interest in those new algorithms lies in the fact that there are less Brownian motions than space dimensions of the optimization problem involved, and that the stochastic differential equations might have a smaller complexity and is therefore easier to evaluate. For this purpose we consider cases where the size of the spectral gap for the previously intro1 for some constant K > 0 (see Section 5 for precise details). This holds duced process Y is 2Kτ true for instance on SU(2) or on the Heisenberg tori. Let U be a smooth potential on M. We regard an equation of the type 1 dZt = − gradhor U (Zt ) dt + ε Vi (Zt ) ◦ dBti 2 d

i=1

as a compact perturbation of the natural OU process in order to get an estimate for its spectral gap, which can be expressed by U (x) + ε 2 log p ε 2 , x0 , x D for all x ∈ M and some constant D. This amounts to a comparison of U with the square of the Carnot–Carathéodory metric d(x0 , x) due to short-time asymptotics of the heat kernel on the compact manifold M. A concatenation of trajectories of such equations under a cooling schedule t → ε(t) = √ c leads then to the desired simulated annealing algorithms. The analytical arguments log(R+t)

are strongly inspired by the seminal works of R. Holley, S. Kusuoka and D. Stroock [7] and [8].


879

2. Preparations from functional analysis We consider a finite-dimensional, connected Lie group G with Lie algebra g, its right-invariant Haar measure μ and a family of left-invariant vector fields V1 , . . . , Vd ∈ g. We assume that Hörmander’s condition holds, i.e. the sub-algebra generated by V1 , . . . , Vd coincides with g. We consider furthermore a stochastic basis (Ω, F, P) with a d-dimensional Brownian motion B and the Lie group-valued process dXtx =

d

Vi Xtx ◦ dBti ,

X0x = x ∈ G.

(2.1)

i=1

The generator of this process is denoted by L, we have 1 2 Vi , 2 d

L=

i=1

where we interpret the vector fields as first-order differential operators on C ∞ (G, R). Furthermore, we define a semigroup Pt acting on bounded measurable functions f : G → R by Pt f (x) = E f Xtx . This semigroup can be extended to a strongly continuous semigroup on L2 (G, μ), which we will denote by the same letter Pt . The carré du champ operator Γ is defined for functions f , where it makes sense, by Γ (f, g) = L(f g) − f Lg − gLf.

(2.2)

In our particular case, we obtain immediately Γ (f, f ) =

d

(Vi f )2 .

i=1

Notice that the carré du champ operator does not change if we add a drift to the generator L. Due to the right-invariance of the Haar measure μ and the left-invariance of the vector fields Vi , the operator L is symmetric (reversible) with respect to μ and therefore μ is an infinitesimal invariant measure in the sense that Lf (x) μ(dx) = 0 for all smooth compactly supported test functions f . Furthermore, due to the symmetry of L we have from (2.2) the relation 2 f Lg μ = − Γ (f, g) μ (2.3) for all f ∈ C0∞ (G) and g ∈ C ∞ (G). Let now U : G → R be an arbitrary smooth function such that Z := exp −U (x) μ(dx) < ∞. G

880


We consider the modified generator 1 LU := L − Γ (U, ·). 2 Notice that μU = exp(−U )μ is an infinitesimal invariant (finite) measure for LU , since, by (2.3),

1 L f μ = (Lf ) exp(−U ) μ − Γ (U, f ) exp(−U ) μ 2 1 1 Γ f, exp(−U ) μ − Γ (f, U ) exp(−U ) μ = 0, =− 2 2 U

U

and notice that LU is symmetric on L2 (μU ) in the sense that 1 f LU g μU = gLU f μU = − Γ (f, g) μU 2 for all smooth compactly supported test functions f, g : G → R. The last equality is often referred to as integration by parts. By definition and by integration by parts the operator LU has a spectral gap at 0 of size a > 0 in L2 (μU ) if and only if Γ (f, f )(x) μU (dx) 2a f (x)2 μU (dx), G

G

for all compactly supported smooth functions f on G satisfying f (x) μU (dx) = 0. G

If we want to write an inequality for all test functions f , it reads like

Γ (f, f )(x) μ (dx) 2a G

μ (dx) −

U

U

f (x) μ (dx) G

2

U

G

2 (2.4) f (x) μ (dx) U

G

for all test functions f ∈ C0∞ (G). 3. Strong existence of OU-processes with values in Lie groups Let G be a finite-dimensional, connected Lie group. We consider now the special case of the ‘potential’ Wt (x) = − log p(t, x), t > 0, where p(t, x) is the density of the law of Xte with respect to μ. By Hörmander’s theorem [9,10], the function (t, x) → p(t, x) is a positive and smooth function, hence the potential Wt is as in the previous section. We write for short Lt instead of LWt and we call the associated Markov process the natural OU-process on G associated to the diffusion X. We show that we have in fact global strong solutions for the corresponding Stratonovich SDE with values in G. The next proposition is slightly more general.


881

Proposition 3.1. Consider a smooth potential U : G → R such that exp −U (x) μ(dx) < ∞. Consider the following Stratonovich SDE with values in G: d y y y Vi Yt ◦ dBti , dYt = V0 Yt dt +

y

Y0 = y ∈ G,

(3.1)

i=1

where V0 f = − 12 Γ (U, f ) for smooth test functions f . Then there is a global strong solution to (3.1) for all initial values y ∈ G. Proof. Since the coefficients defining (3.1) are smooth by assumption, there exists a unique strong solution up to the explosion time

ζy = inf t: lim Yτy = ∞ . τ →t

We then define a semigroup Pt on L2 (μU ) by y (Pt f )(y) = E f Yt 1ζy >t .

(3.2)

It can be shown in exactly the same way as in [4,14] that Pt is a strongly continuous contraction semigroup and that its generator A coincides with LU on the set C0∞ (G) of compactly supported smooth functions. On the other hand, setting D(LU ) = C0∞ (G), one can show as in [4,14] that LU is essentially self-adjoint, so that one must have A = LU = (LU )∗ . In particular, since the constant function 1 belongs to L2 (G, μU ) by the integrability of exp(−U ) and since (LU ψ)(x) μU (dx) = 0 for any test function ψ ∈ C0∞ (G), 1 belongs to the domain of (LU )∗ and therefore also to the domain of A. This then implies that Pt 1 = 1 by the same argument as in [14]. In particular, coming back to the definition (3.2) of Pt , we see that P(ζy = ∞) = 1 for every y, which is precisely the non-explosion result that we were looking for. 2 Remark 3.2. While this argument shows that, given a fixed initial condition y, there exists a y unique global strong solution Yt to (3.1), it does not prevent more subtle kinds of explosions, see for example [6]. By Proposition 3.1 and since p(t, x) is smooth and integrable, it follows immediately that the OU-process exists globally in a strong sense. Corollary 3.3. For any given τ > 0, the process d y y y dYt = V0 Yt dt + Vi Yt ◦ dBti , i=1

with V0 f = − 12 Γ (Wτ , f ) has a global strong solution.

y

Y0 = y ∈ G,

882


Remark 3.4. More traditional Lyapunov-function based techniques seem to be highly non-trivial to apply for this situation, due to the lack of information on the behaviour of U (y) at large y. In view of [1,11,12], it is tempting to conjecture that one has the asymptotic lim τ 2 ∂τ log p(τ, y) = d 2 (e, y),

τ →0

(3.3)

and that the limit holds uniformly over compact sets K that do not contain the origin e. (Note that it follows from [1] that this is true provided that K does not intersect the cut-locus.) If it were the case that (3.3) holds, possible space–time scaling properties of p(τ, x) could imply that, for / K. On the every τ > 0, there exists a compact set K such that Lp(τ, x) = ∂τ p(τ, x) > 0 for x ∈ other hand, one has 1 Lp(τ, .) , Lτ Wτ = − Γ (Wτ , Wτ ) + LWτ = − 2 p(τ, .) so that this would imply that Wτ is a Lyapunov function for the corresponding OU-process leading to another proof of the previous corollary. 4. Spectral gaps for natural OU-processes Next we consider the question if Lt admits a spectral gap on L2 (pt μ) for t > 0, which turns out to be a consequence of the Driver–Melcher inequality (see [15]). Theorem 4.1. The following assertions are equivalent: • The operator Lt has a spectral gap of size at > 0 on L2 (pt μ) for all t > 0, and a positive function a : R>0 → R>0 . • The local estimate 2 Pt Γ (f, f ) (g) 2at Pt f 2 (g) − (Pt f )(g) holds true for all test functions f : G → R, for all t > 0 and a positive function a : R>0 → R>0 at one (and therefore all) point g ∈ G. Furthermore, if we know that Γ (Pt f, Pt f )(e) φ(t)Pt Γ (f, f ) (e) holds true for all test functions f ∈ C0∞ (G), all t 0, and a strictly positive locally integrable function φ : R0 → R>0 , then we can choose at by t at

1 φ(t − s) ds = , 2

0

for t > 0 and the two equivalent assertions hold true.


883

Proof. Since μWt is equal to the law of Xte , one has f μWt = Pt f (e) for every f ∈ C0∞ (G). The equivalence of the first two statements then follows from (2.4) and the fact that the translation invariance of (2.1) implies that if the bound holds at some g, it must hold for all g ∈ G. We fix a test function f : G → R as well as t > 0 and consider H (s) = Ps (Pt−s f )2 for 0 s t. The derivative of this function equals H (s) = Ps Γ (Pt−s f, Pt−s f ) and therefore—assuming the third statement—we obtain H (s) φ(t − s)Pt Γ (f, f ) . Whence we can conclude t H (t) − H (0)

φ(t − s) dsPt Γ (f, f ) ,

0

which is the second of the two equivalent assertions for an appropriately chosen a.

2

Remark 4.2. We can replace the Lie group G by a general manifold M, on which we are given a hypo-elliptic, reversible diffusion X with “sum of the squares” generator L. Then the analogous statement holds, in particular local Poincaré inequalities on M for L lead to a spectral gap for the OU-type process Lt with t > 0. Corollary 4.3. Let G be a free, nilpotent Lie group with d generators e1 , . . . , ed of step m 1, and consider 1 2 ei , 2 d

L=

i=1

then the operator Lt has a spectral gap of size at =

1 2Kt

on L2 (pt μ) for some constant K > 0.

Proof. Due to the results of the very interesting PhD thesis [15] (see also [5]), there is a constant K such that the bound Γ (Pt f, Pt f )(e) KPt Γ (f, f ) (e) holds true for all test functions f ∈ C0∞ (G) and for all times t 0. This shows that at Kt = 12 , due to the assertions of Theorem 4.1. 2

884


Corollary 4.4. Let G = SU(2) be the Lie group of unitary matrices on C2 with Lie algebra g = su(2) = i, j, kR with the usual commutation relations, i.e. [i, j ] = 2k and its cyclic variants. Consider L=

1 2 i + j2 , 2

where we understand the elements i, j as left-invariant vector fields on G, then Lt has a spectral 1 gap of size at = 2Kt on L2 (pt μ) for some constant K > 0. Proof. The result follows from [3, Proposition 4.20].

2

4.1. Generalisation to homogeneous spaces Consider now M a compact homogeneous space with respect to the Lie group G, i.e. we have a (right) transitive action πˆ : G × M → M of G on M. We assume that there exists a measure μM on M which is invariant with respect to this action. We also assume that we are given a family V1 , . . . , Vd of left-invariant vector fields on G that generate its entire Lie algebra g as before. These vector fields induce fundamental vector fields ViM on M by means of the action πˆ . By choosing an ‘origin’ o ∈ M, we obtain a surjection π : G → M by π(g) = π(g, ˆ o). The vector fields V1 , . . . , Vd and V1M , . . . , VdM are consequently π -related. Due to the invariance of μM with respect to the action πˆ , the vector fields ViM are anti-symmetric operators on L2 (μM ) and the generator 1 M 2 Vi 2 d

LM =

i=1

is consequently symmetric on L2 (μM ). In particular we have M Vi f ◦ π = Vi (f ◦ π) for i = 1, . . . , d. The local Driver–Melcher inequality on G translates to the same inequality on M by means of PtM (f ) ◦ π = Pt (f ◦ π) for test functions f : M → R, hence we obtain the corresponding Driver–Melcher inequality on M with the same constants, too. 5. A simple result on simulated annealing By comparison with natural OU-processes on the homogeneous space M we can obtain spectral gap results for quite general classes of potentials. For later purposes, namely for applications to simulated annealing algorithms, we shall state a parametrized version of a simple perturbation result.


885

Theorem 5.1. Let M be a homogeneous space. Let Vε : M → R be a family of potentials Vε with Vε + log p(ε, ·) Dε for 0 < ε 1, and constants Dε > 0, where p(t, ·) denotes the density of the invariant measure of the process Xt starting at x0 . Assume furthermore that a Poincaré inequality holds for Lε , i.e. aε Pε f 2 (e) Pε Γ (f, f ) (e)

(5.1)

for test functions f ∈ C0∞ (M) with Pε f (e) = 0 and some constant aε > 0 and 0 < ε 1. Then one has exp(−Vε ) ∈ L1 (μ) and the Poincaré inequality

f 2 (x) exp −Vε (x) μ(dx) Cε

Γ (f, f )(x) exp −Vε (x) μ(dx)

(5.2)

holds for all test functions f ∈ C0∞ (M) with f (x) exp(−Vε (x)) μ(dx) = 0 and some constant ε) Cε = exp(2D > 0. In particular, this leads to a spectral gap of size at least aε 1 aε = Cε exp(2Dε ) for LVε . Proof. It follows immediately from the inequality p(ε, x) = exp −Vε (x) exp Vε (x) p(ε, x) exp(−Dε ) exp −Vε (x) that exp(−Vε ) ∈ L1 (μ). Furthermore, exp −Vε (x) =

1 p(ε, x) exp(−Dε )p(ε, x) p(ε, x) exp(Vε (x))

for all x ∈ M by assumption. Hence we deduce (5.2) with Cε =

exp(2Dε ) aε

from (5.1).

2

Remark 5.2. See [2] for results on unbounded perturbations. Throughout the remainder of this section we assume that M is a compact homogeneous space with respect to a connected Lie group G. We consider the same structures as in Section 4.1 on M, but we omit the index M on vector fields and measures in order to improve readability. We shall furthermore impose the following assumption on the spectral gap. Assumption 5.3. There is a constant K > 0 such that Γ (Pt f, Pt f )(e) KPt Γ (f, f ) (e) holds true for all test functions f ∈ C0∞ (M) and 0 t 1.

886


We prepare now a quantitative simulated annealing result under the previous Assumption 5.3 on M and follow closely the lines of [7]. Let U : M → R be a smooth potential. The idea is to search for minima of U by sampling the measure 1 U exp − 2 μ. Zε ε Recall that we do always assume that Zε = M exp(− εU2 )μ < ∞. Sampling this measure is performed by looking at the invariant measure of U 1 Lε = L − Γ 2 , · . 2 ε Notice that the previous operator satisfies 1 ε 2 Lε = ε 2 L − Γ (U, .), 2 and a spectral gap for ε 2 Lε is a spectral gap for the diffusion process d y y y dYt = V0 Yt dt + εVi Yt ◦ dBti ,

y

Y0 = y ∈ G,

i=1 y

with V0 f = − 12 Γ (U, f ). Consequently we know—given strong existence—that the law of Yt converges to Z1ε exp(− εU2 )μ and concentrates therefore around the minima of U . In this consideration ε is considered to be fixed. Next we try to obtain a time-dependent version of the previous considerations, leading to a process concentrating precisely at the minima of U . In the following theorem we quantify the speed of convergence towards the invariant measure. We denote by με the probability measure invariant for Lε and we use the notation 2 varε (f ) = f − f ε ε with f ε = M f (g) με (dg) for the variance with respect to this measure. First we estimate the spectral gap along a cooling schedule t → ε(t), then we prove that the measure concentrates around the minima of U even in a time-dependent setting. Theorem 5.4. Let U : M → R be a smooth functions, D a constant, x0 ∈ M a point such that U (x) + ε 2 log p ε 2 , x0 , x D, for all x ∈ M. Then there exist constants R, c > 0 such that for ε(t) = √

c log(R+t)

1 varε(t) (f ) K(R + t) Γ (f, f ) , 2 ε(t) for all test functions f ∈ C0∞ (M) and t 0.


887

Proof. We can start to collect results. Combining Assumption 5.3 with Theorem 5.1 applied for Vε = εU2 , we obtain that spectral gap for the operator Lε has size at least 1 2D exp − 2 Kε 2 ε for 0 < ε 1, so that ε 2 Lε has a spectral gap of size at least 2D 1 exp − 2 . K ε We choose c2 = 2D and R sufficiently large so that ε(t) 1 for t 0, and we conclude that 2D K(R + t) K exp ε(t)2 for all t 0, which yields the desired result.

2

We denote by Z the process with cooling schedule t → ε(t) as in the previous theorem, d dZtz = V0 Ztz dt + ε(t)Vi Ztz ◦ dBti , i=1

where the drift vector field is given through V0 f = − 12 Γ (U, f ). Then the previous conclusion leads to the following proposition. Proposition 5.5. Let ft denote the Radon–Nikodym derivative of the law of Ztz with respect to με(t) and let u(t) := ft − 1 2L2 (μ

ε(t) )

denote its distance in L2 (με(t) ) to 1 (which corresponds to varε(t) (ft )), then u (t) −

N 2N 1 u(t) + 2 u(t) + 2 u(t) K(R + t) c (R + t) c (R + t)

for the constants R, c and K from Theorem 5.4, and N = max U − min U . Remark 5.6. We find c2 > 3N K, such that supt0 u(t) is bounded from above by a constant depending on f0 , c, R, N and K. Proof. The proof follows closely the lines of [7]. By assumption we know that varε(t) (ft ) = u(t) = ft 2L2 (μ ) − 1 and hence with the notation β(t) = 1 2 , ε(t)

ε(t)

888


u (t) = − Γ (ft , ft ) ε(t) − β (t)

U − U ε(t) ft2 μt

= − Γ (ft , ft ) ε(t) − β (t) U − U ε(t) (ft − 1)2 μt − 2β (t) (U − U ε(t) )(ft − 1) μt

−

2N 1 N u(t) + 2 u(t). u(t) + 2 K(R + t) c (R + t) c (R + t)

Here, we used the Cauchy–Schwarz inequality and the conclusions of the previous Theorem 5.4 in the last line. 2 Assumption 5.7. Let U : M → R be a potential on M such that U (x) − d(x, x0 )2 D1 for some positive constant D1 , a point x0 ∈ M and for all x ∈ M. Here we denote by d(x, x0 ) the Carnot–Carathéodory metric on M, see for instance [11,12] and [16]. Remark 5.8. For non-compact manifolds M the limit lim t log p(t, x0 , x) = −d(x0 , x)2

t→0

is uniform on compact subsets of M, but usually not on the whole of M. An abelian, noncompact example where the limit is globally uniform is M = Rd . On the simplest non-compact and non-abelian example, the Heisenberg group G2d , the limit is not uniform, see recent work of H.-Q. Li [13]. Therefore we consider in our perturbation argument only compact manifolds M. On compact manifolds M we know due to R. Léandre’s beautiful results [11,12] that there is D2 such that d(x0 , x)2 + ε 2 log p ε 2 , x0 , x D2 for x ∈ M. Hence we can conclude by the triangle inequality that the potential U satisfies the assumptions of Theorem 5.4. Theorem 5.9. Assume Assumptions 5.3 and 5.7, and assume that sup ft L2 (με(t) ) < ∞. t0

Define U0 = infx∈M U (x) and, for every δ > 0, denote by Aδ the set Aδ = {x ∈ M | U (x) U0 + δ}. Then we can conclude that P Ztz ∈ Aδ M με(t) (Aδ ) for every t > 0 and every δ 0.


889

Proof. It follows from the Cauchy–Schwarz inequality that P Ztz ∈ Aδ = ft με(t) M με(t) (Aδ ), Aδ

as required.

2

Remark 5.10. Since limε→0 με (Aδ ) = 0 for every δ > 0, we obtain that for all continuous bounded test functions f , we have E f Ztz → f (xmin ), provided that there is only one element xmin ∈ M such that U (xmin ) = U0 . Remark 5.11. We can improve the previous result from L2 -estimates to Lq -estimates for ft − 1 Lq (μt ) , for q > 2: this follows from [8, Theorem 2.2] and the fact that the proof of [8, Theorem 2.7] applies due to a valid Sobolev inequality, i.e. for every q > 2 there is a constant C0 > 0 such that 1 2 2 f Lq (μ) C0 Γ (f, f )μ + f L2 (μ) 2 M

holds for all test functions f : M → R. Such a Sobolev inequality can be found for instance in [9, Section 3] or in [16]. Remark 5.12. We can apply hypo-elliptic simulated annealing to potentials on compact nilmanifolds with the respective sub-Riemannian structure due to Corollary 4.3, or we can apply it to potentials on SU(2) due to Corollary 4.4. The implementation of those new algorithms can yield some advantages with respect to elliptic simulated annealing algorithms on Riemannian manifolds as described in [8]. On the one hand the number of Brownian motions involved is smaller, such as the complexity of the SDE d dZtz = V0 Ztz dt + ε(t)Vi Ztz ◦ dBti i=1

as a whole, since less vector fields have to evaluate and the gradient V0 is less complex being a horizontal gradient. The price to pay is a larger constant c > 0 in the rate of convergence. In cases where hypo-elliptic simulated annealing can be directly compared with elliptic simulated annealing on flat space (for instance on 3-torus T3 , where we have the flat Euclidean structure and the sub-Riemmanian structure of the Heisenberg torus with two generators) the elliptic algorithm is superior. This is due to the fact the one can choose the vector fields in the elliptic algorithm constant (on flat T3 ) which reduces the complexity considerably, whereas one has to apply more sophisticated vector fields in the case of the Heisenberg group. For optimization on SU(2), where our theory applies due to Corollary 4.4, we have a visible advantage over the elliptic simulated annealing, since we need one more Brownian motion and one more vector field for the elliptic algorithm, and we cannot simplify the vector fields in the elliptic algorithm.

890


Acknowledgment We would like to thank an anonymous referee for helpful comments on a first version of our paper. References [1] G. Ben Arous, Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus, Ann. Sci. École Norm. Sup. (4) 21 (3) (1988) 307–331. [2] D. Bakry, M. Ledoux, F.-Y. Wang, Perturbations of functional inequalities using growth conditions, J. Math. Pures Appl. (9) 87 (4) (2007) 394–407. [3] F. Baudoin, M. Bonnefont, The subelliptic heat-kernel on SU(2): Representations, asymptotics and gradient bounds, preprint, arXiv: 0802.3320. [4] P.R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Funct. Anal. 12 (1973) 401–414. [5] B.K. Driver, T. Melcher, Hypoelliptic heat kernel inequalities on Lie groups, Stoch. Process. Appl., in press. [6] K.D. Elworthy, Stochastic dynamical systems and their flows, in: Stochastic Analysis, Proc. Internat. Conf., Northwestern Univ., Evanston, IL, 1978, Academic Press, New York, 1978, pp. 79–95. [7] R. Holley, D. Stroock, Simulated annealing via Sobolev inequalities, Comm. Math. Phys. 115 (4) (1988) 553–569. [8] R.A. Holley, S. Kusuoka, D.W. Stroock, Asymptotics of the spectral gap with applications to the theory of simulated annealing, J. Funct. Anal. 83 (2) (1989) 333–347. [9] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967) 147–171. [10] L. Hörmander, The Analysis of Linear Partial Differential Operators III. Pseudo-Differential Operators, Classics in Math., Springer, Berlin, 2007. Reprint of the 1994 edition. [11] R. Léandre, Majoration en temps petit de la densité d’une diffusion dégénérée, Probab. Theory Related Fields 74 (2) (1987) 289–294. [12] R. Léandre, Minoration en temps petit de la densité d’une diffusion dégénérée, J. Funct. Anal. 74 (2) (1987) 399– 414. [13] H.-Q. Li, Estimations asymptotiques du noyau de la chaleur sur les groupes de Heisenberg, C. R. Math. Acad. Sci. Paris. [14] X.-M. Li, Stochastic flows on noncompact manifolds, PhD thesis, 1992. [15] T. Melcher, Hypoelliptic heat kernel inequalities on Lie groups, PhD thesis, University of California at San Diego. [16] A. Sánchez-Calle, Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math. 78 (1) (1984) 143–160.


Free pluriharmonic majorants and commutant lifting ✩ Gelu Popescu Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA Received 13 December 2007; accepted 30 April 2008 Available online 3 June 2008 Communicated by D. Voiculescu

Abstract In this paper we initiate the study of sub-pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball 1/2

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