Asymptotic geometric analysis, harmonic analysis and related topics, Proc. CMA-AMSI Res. Symp

Preface This volume contains the proceedings of the CMA/AMSI Research Symposium on “Asymptotic Geometric Analysis, Harmonic Analysis and Related Topics”, organized by Andrew Hassell, Alan McIntosh, Shahar Mendelson, Pierre Portal, and Fyodor Sukochev at Murramarang (NSW) in February 2006. The meeting was sponsored by the Centre for Mathematics and its Applications (Australian National University) and the Australian Mathematical Sciences Institute whose support is gratefully acknowledged. The Symposium covered a variety of topics in functional, geometric, and harmonic analysis, and brought together experts, early career researchers, and doctoral students from Australia, Canada, Finland, France, Germany, Israel, and the USA. It is our hope that this volume reflects the lively research atmosphere of this conference, and we are glad to open it with a result of Ian Doust, Florence Lancien, and Gilles Lancien, which was essentially discovered during the symposium. We also wish to express our appreciation to the participants, the authors who contributed to this volume, and the CMA support staff (Chris Wetherell and Annette Hugues) who made this symposium what it was. Each article in this volume was peer refereed. Alan McIntosh and Pierre Portal (Editors)

i

Spectral theory for linear operators on L1 or C(K) spaces Ian Doust, Florence Lancien, and Gilles Lancien Abstract It is known that on a Hilbert space, the sum of a real scalar-type operator and a commuting well-bounded operator is well-bounded. The corresponding property has been shown to be fail on Lp spaces, for 1 < p != 2 < ∞. We show that it does hold however on every Banach space X such that X or X ∗ is a Grothendieck space. This class notably includes L1 and C(K) spaces. MSC (2000): 47B40, 46B03, 46B20 (primary) 46B10, 46B22, 46B26 (secondary). Received: 13 June 2006 / Accepted: 19 December 2006.

1

Introduction

This paper is concerned with scalar-type spectral and well-bounded operators on a Banach space X. The theory of scalar-type spectral operators was initiated by N. Dunford (see his survey [6]) in order to generalize the theory of self-adjoint operators to operators on general Banach spaces. These operators are those which admit an integral representation with respect to a countably additive spectral measure (the precise definition is given in section 2) and therefore a functional calculus for bounded measurable functions on their spectrum. In particular, the spectral expansion of such an operator converges unconditionally. An old result of J. Wermer ( [19]) asserts that finitely many commuting scalar-type spectral operators on a Hilbert space can be simultaneously transformed into normal operators, by passing to an equivalent inner product. An application of this is that the sum and product of two commuting scalar-type spectral operators on a Hilbert space are also scalar-type spectral. This result has been extended by many authors, beginning with C.A. McCarthy ( [14], [15]) in the 1960s who considered operators acting on Lp spaces for 1 < p < ∞. 1

More recently T.A. Gillespie ( [8]) showed that, on a Banach lattice, the Boolean algebra generated by two commuting bounded Boolean algebras is itself bounded. As a consequence he obtains that the sum and product of two commuting scalar-type spectral operators on a weakly complete Banach lattice (and also on a wide class of subspaces of Banach lattices) are scalartype spectral. It has been long known however that counterexamples to this result exist even within the class of super-reflexive Banach spaces. Indeed, a counterexample can be given on the von Neumann-Schattten classes Cp , for 1 < p != 2 < ∞. One may wonder under what conditions the sum of two commuting scalartype spectral operators would have the weaker property of being well-bounded. Well-bounded operators, introduced by D.R. Smart ( [18]), are defined as having a functional calculus for the absolutely continuous functions on some compact interval. They coincide with the operators having, in some weaker sense, a spectral decomposition that converges only conditionally. It is shown in [3] that on spaces with property (∆), the sum of two commuting scalar-type spectral operators is always well-bounded. (The class of spaces with property (∆), which was introduced by N. Kalton and L. Weis in [10], includes all UMD spaces.) On the other hand, T.A Gillespie proved in [7] that the sum of two commuting well-bounded operators is not always well-bounded, even on a Hilbert space. We address now the question of the well-boundedness of the sum of a real scalar-type spectral operator and a commuting well-bounded operator. It follows from the same work of T.A. Gillespie [7] that the answer is positive for Hilbert spaces. However, it is shown in [4], how one may construct a counterexample in any reflexive non-Hilbertian Banach lattice. The aim of this note is to prove that the answer is positive however on an abstract class of Banach spaces which includes L1 and C(K) spaces. Before proceeding, we would like to point out that much of this theory bears a close resemblance with that which arises from questions concerning whether the sum of two commuting unbounded operators either has an H ∞ functional calculus or is sectorial. We refer the reader to [1], [9], [10], [11] and [16] for the relevant definitions and, among other things, theorems analogous to the above mentioned results. One common ground for these two theories is certainly the classical work on unconditional bases as is expounded in [13].

2

2

Notation

Throughout this paper, X will denote a complex Banach space, BX its closed unit ball and B(X) the algebra of all bounded linear operators on X. Let Σ be the family of all Borel subsets of C. An operator T ∈ B(X) is said to be scalar-type spectral if there exists a spectral measure F defined on Σ, whose values are projections in B(X) and satisfying the following properties: (i) $F$ = sup{$F(A)$, A ∈ Σ} < +∞. (ii) T F(A) = F(A)T , for all A ∈ Σ. (iii) σ(T |F(A)X) ⊂ A, for all A ∈ Σ. (iv) F is !countably additive in the strong operator topology. (v) T = λ F(dλ). If in addition σ(T ) ⊂ R, then T is said to be real scalar-type spectral. Every scalar-type spectral operator T admits a functional calculus defined on the space B∞ (σ(T )) of all bounded Borel measurable functions on σ(T ) by the formula " f (T ) = f (λ) F(dλ) σ(T )

and satisfying the standard estimate

$f (T )$ ≤ 4$F$ sup |f (λ)|. λ∈σ(T )

Details can be found in [5] (the constant 4 can be replaced by 2 if f is real valued). An operator S ∈ B(X) is said to be well-bounded if there exist a constant K and a compact interval J = [a, b] such that for all complex polynomials p, " b $p(S)$ ≤ K$p$AC(J) , where $p$AC(J) = sup |p(t)| + |p$ (t)| dt. t∈J

a

Equivalently, there is a Banach algebra homomorphism f '→ f (T ), from the algebra AC(J) of all absolutely continuous functions on J into B(X), extending the natural definition for polynomials and satisfying: ∀f ∈ AC(J), $f (T )$ ≤ K$f $AC(J) .

On a general Banach space X, an operator S is well-bounded if and only if it admits a so-called decomposition of the identity. This a family (H(t))t∈R ⊂ 3

B(X ∗ ) of projections enjoying a few more properties and providing the following integral representation for the functional calculus: for any f in AC(J), x ∈ X and x∗ ∈ X ∗ , " b ∗ ∗ *f (T )x, x + = f (b)*x, x + − *x, H(t)x∗ +f $ (t) dt. a

Since we shall not need to use this decomposition of the identity, we refer the reader to [5] for the complete definition. We now recall that a Banach space X is a Grothendieck space (in short GT-space) if there is a constant C such that for every bounded linear operator T from X to #2 and all x1 , .., xn ∈ X: n # k=1

$T xk $#2 ≤ C $T $ sup x∗ ∈B

X∗

n # k=1

|x∗ (xk )|.

Such an operator is called absolutely summing (see [17] for a complete study of this notion).

3

A stronger functional calculus for scalar-type spectral operators on certain Banach spaces

The key step is to show that the unconditionality of the spectral decomposition of a scalar-type spectral operator is automatically strengthened in a GT-space. This is clearly inspired by the last section of [10] and by the fundamental work of J. Lindenstrauss and A. Pełczyński [12] on the uniqueness of unconditional bases in #1 . In fact, our next proposition is just a variation of Corollary 8 of Theorem 6.1 in [12]. Proposition 3.1. Suppose that X is a GT-space and F is a bounded finitely additive spectral measure defined on some algebra of subsets of C. Then there is a constant C such that for any x ∈ X and any A1 , .., An ∈ Σ which are pairwise disjoint: n # $F(Aj )x$ ≤ C$x$. j=1

In particular, if F is the spectral measure of a scalar-type spectral operator T on X, then for any x ∈ X, the X-valued vector measure µx , defined by 4

µx (A) = F(A)x, is a measure of bounded variation, whose total variation is dominated by C$x$. Proof. Let A1 , .., An ∈ Σ be pairwise disjoint. For every u ∈ X and 1 ≤ k ≤ n, we pick u∗k = u∗k (u) ∈ BX ∗ such that *F(Ak )u, u∗k + = $F(Ak )u$. Then, for a = (ak )nk=1 ∈ Cn , we define Tu,a : X → #n2 by $ %n ∀y ∈ X, Tu,a y = ak *F(Ak )y, u∗k + k=1 .

We clearly have

∀y ∈ X,

$Tu,a y$#n2 ≤ $a$#n2 $F$ $y$.

Since X is a GT-space, there exists C1 > 0 such that for all a ∈ #n2 , u ∈ X and u1 , .., un ∈ X, n # j=1

$Tu,a uj $#n2 ≤ C1 $F$ $a$#n2

sup

x∗ ∈BX ∗

n # j=1

|x∗ (uj )|.

(3.1)

We now apply the above inequality for uj = F(Aj )u and aj = $F(Aj )u$. We have that $ %n $ %n Tu,a uj = ak *F(Ak )F(Aj )u, u∗k + k=1 = δk,j $F(Aj )u$2 k=1 . Cancelling $a$#n2 from both sides of (3.1) gives ∀u ∈ X,

n n # # ( $F(Aj )u$2 )1/2 ≤ C1 $F$ sup |x∗ (uj )|. x∗ ∈BX ∗

j=1

j=1

∗ Note now that for any ∈ BX ∗ there&exist complex numbers of modulus &x n one, α1 , .., αn , so that j=1 |x∗ (uj )| = | nj=1 *αj F(Aj )u, x∗ +|. Then, it follows from the disjointness of the Aj ’s and the functional calculus bounds for T that n # ∀u ∈ X, sup |x∗ (uj )| ≤ 4$F$$u$ x∗ ∈BX ∗

and so

∀u ∈ X,

j=1

n # ( $F(Aj )u$2 )1/2 ≤ 4 C1 $F$2 $u$.

(3.2)

j=1

5

Suppose now that x ∈ X. Denote by Tx the operator Tx,a where a = (1, . . . , 1). For y ∈ X, inequality (3.2) implies n # $Tx y$ ≤ ( $F(Aj )y$2 )1/2 ≤ 4 C1 $F$2 $y$. j=1

Note that if xj = F(Aj )x then $Tx xj $#n2 = $F(Aj )x$. Using again the fact that X is a GT-space we get that n # j=1

$F(Aj )x$ =

n # j=1

$Tx xj $#n2 ≤ C1 $Tx $ sup

x∗ ∈BX ∗

n # j=1

|x∗ (xj )| ≤ 16 C12 $F$3 $x$.

We will also need the following dual statement. Proposition 3.2. Suppose X ∗ is a GT-space. Let T be a scalar-type spectral operator on X and let F be its spectral measure. Then there is a constant C such that for any x∗ ∈ X ∗ and any A1 , .., An ∈ Σ which are pairwise disjoint: n # j=1

$F(Aj )∗ x∗ $ ≤ C$x∗ $.

In other words, for any x∗ ∈ X ∗ , the total variation of the X ∗ -valued finitely additive vector measure νx∗ , defined by νx∗ (A) = F(A)∗ x∗ , is dominated by C$x∗ $. Proof. We just apply Proposition 3.1 to F∗ .

We now need to introduce more notation. For T ∈ B(X), we denote by {T }$ the commutant of T (namely, the closed subalgebra of B(X) consisting of all operators commuting with T ). Let ST denote the algebra of all {T }$ valued Borel simple functions defined on σ(T ), and let B∞ (σ(T ), {T }$ ) be the uniform closure of ST . We can now state the main result of this section. Theorem 3.1. Suppose that X or X ∗ is a GT-space (for instance X is an L1 -space, or X is a C(K)-space). Let T be a scalar-type spectral operator on X and let F be its spectral measure. For any finite families (Ai )ni=1 of & pairwise disjoint&Borel subsets of σ(T ) and (Si )ni=1 in {T }$ , we define n Φ( ni=1 Si Ai ) = i=1 Si F(Ai ). Then Φ can be extended into a bounded algebra homomorphism from B∞ (σ(T ), {T }$ ) into B(X). 6

Proof. Since T is a spectral operator, each operator that commutes with T also commutes with its spectral measure (see [6] or Theorem 6.6 in [5]). It is therefore simple to check that Φ is an algebra homomorphism on ST . The conclusion will therefore follow immediately once we can show that Φ is &n bounded on ST . Suppose then that f = i=1 Si Ai ∈ S. If X is a GT-space then it follows from Proposition 3.1 that for all x ∈ X, $Φ(f )x$ ≤

n # i=1

$Si F(Ai )x$ ≤ C sup $Si $ $x$ = C$f $∞ $x$. 1≤i≤n

If X ∗ is a GT-space, we apply Proposition 3.2 and the fact that Si and F(Ai ) commute to obtain that ∀x ∈ X ∀x∗ ∈ X ∗ |*Φ(f )x, x∗ +| ≤ C$f $∞ $x$ $x∗ $. In each case, our estimate clearly yields the conclusion.

4

Application to the well-boundedness of sums of operators

Our result is the following Theorem 4.1. Suppose that X or X ∗ is a GT-space. Let T be a real scalartype operator on X and let S be a well-bounded operator on X which commutes with T . Then S + T is a well-bounded operator on X. Proof. Let Φ be the functional calculus map from B∞ (σ(T ), {T }$ ) into B(X), associated with T in Theorem 3.1. Note first that if f (λ) = g(λ)IdX , where IdX is the identity operator on X and g is bounded, Borel measurable and scalar valued, then Φ(f ) = g(T ). On the other hand, if U ∈ {T }$ and f (λ) = U for all λ ∈ σ(T ), then Φ(f ) = U . For p a complex polynomial, define the map fp : σ(T ) → {T }$ by fp (λ) = p(S + λ). Combining the above remarks, with the fact that Φ is an algebra homomorphism, we get that Φ(fp ) = p(S + T ). It follows that there is a constant C1 such that for every polynomial p: $p(S + T )$ ≤ C1 sup $fp (λ)$B(X) .

(4.1)

λ∈σ(T )

7

Since S is well-bounded, there exist a compact interval J and a constant C2 such that for any complex polynomial p, $p(S)$ ≤ C2 $p$AC(J) . Let K be a compact interval containing σ(T ) + J. It is a standard fact [5, Lemma 18.7] that for all λ ∈ σ(T ) and all complex polynomials p, $fp (λ)$B(X) = $p(S + λ)$ ≤ C2 $p$AC(K) . Combining this with (4.1) shows that there is a constant C such that for every polynomial p, $p(S + T )$ ≤ C$p$AC(K) , and therefore that S + T is a well-bounded operator. Remark 4.1. Under the same assumptions, one can show with a similar proof that q(S, T ) is well-bounded for every real polynomial q. Remark 4.2. We conclude this note by showing that if T is a scalar-type spectral operator on a Hilbert space H, then it also admits a functional calculus defined on B∞ (σ(T ), {T }$ ). This gives an alternative quick proof of Gillespie’s result ( [7]). n n $ So, let i )i=1 be pairwise disjoint &(A &n Borel subsets of σ(T ), (Si )i=1 in {T } . n For f = i=1 Si Ai , set Φ(f ) = i=1 Si F(Ai ). Writing ∀t ∈ [0, 1] ∀x ∈ H Φ(f )x = &n

n $# i=1

n %$ # % ri (t)F(Ai ) ri (t)F(Ai )Si x , i=1

noting that $ i=1 ri (t)F(Ai )$ ≤ 2$F$ and using the parallelogram law, we obtain that for any x ∈ H: " 1 # n n # 2 1/2 $Φ(f )x$ ≤ 2$F$( $ ri (t)F(Ai )Si x$ dt) = 2$F$( $F(Ai )Si x$2 )1/2 . 0

i=1

i=1

But n # i=1

2

$F(Ai )Si x$ ≤

n #

$f $2∞ (

Thus

This finishes our proof.

i=1

2

$F(Ai )x$ ) =

"

$f $2∞ (

0

1

$

n #

ri (t)F(Ai )x$2 dt).

i=1

$Φ(f )x$ ≤ 4$F$2 $f $∞ $x$.

Acknowledgements. We wish to thank the referee for his helpful comments. 8

References [1] M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded H∞ functional calculus, Journal of Aust. Math. Society, Ser. A 60 1996, 51-89. [2] J. Diestel and J.J. Uhl, Vector measures, Math. Surveys, 15, Amer. Math. Soc., 1977. [3] I. Doust and T.A. Gillespie, Well-boundedness of sums and products of operators, J. London Math. Soc., (2)68 2003, 183-192. [4] I. Doust and G. Lancien, The spectral type of sums of operators on non-hilbertian Banach lattices, preprint. [5] H.R. Dowson, Spectral theory of linear operators, London Mathematical Society Monographs 12, Academic Press, London, 1978. [6] N. Dunford, A survey of the theory of spectral operators, Proc. Amer. Math. Soc., 64 1958, 217-274. [7] T.A. Gillespie, Commuting well-bounded operators on Hilbert spaces, Proc. Edimburgh Math. Soc., (2)20 1976, 167-172. [8] T.A. Gillespie, Boundedness criteria for Boolean algebras of projections, J. Funct. Anal., 148 1997, 70-85. [9] N.J. Kalton and G. Lancien, A solution to the problem of Lp -maximal regularity, Math. Zeit., 235 2000, 559-568. [10] N.J. Kalton and L. Weis, The H ∞ -calculus and sums of operators, Math. Ann., 321 2001, 319-345. [11] F. Lancien, G. Lancien and C. Le Merdy, A joint functional calculus for sectorial operators with commuting resolvents, Proc. London Math. Soc., 77 1998, 387-414. [12] J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in Lp -spaces and their applications, Studia Math., 29 1968, 275-326. [13] J. Lindenstrauss and L. Tzafriri, Classical Banch spaces I, Springer Berlin 1977. 9

[14] C.A. McCarthy, Commuting Boolean algebras of projections, Pacific J. Math., 11 1961, 295-307. [15] C.A. McCarthy, Commuting Boolean algebras of projections. II. Boundedness in Lp , Proc. Amer. Math. Soc., 15 1964, 781-787. [16] A. McIntosh, Operators which have an H∞ functional calculus, Miniconference on Operator Theory and Partial Differential Equations, Proc. Cent. Math. Anal. A.N.U. Canberra 14 1986, 210-231. [17] G. Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conference Series in Mathematics 60, Amer. Math. Soc., Providence, R.I., 1986. [18] D.R. Smart, Conditinally convergent spectral expansions, J. Austral. Math. Soc., Ser. A 1 1960, 319-333. [19] J. Wermer, Commuting spectral operators in Hilbert spaces, Pacific J. Math., 4 1954, 355-361. Ian Doust, School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia. [email protected] Florence Lancien, Laboratoire de Mathématiques, UMR 6623, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France. [email protected] Gilles Lancien, Laboratoire de Mathématiques, UMR 6623, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France. [email protected]

10

Vector-valued singular integrals, and the border between the one-parameter and the multi-parameter theories Tuomas P. Hytönen

∗

Abstract We survey the vector-valued theory of Fourier multipliers and singular integrals, especially concentrating on identifying the border between the one-parameter theory valid in UMD spaces and the multiparameter theory valid in UMD spaces with property (α). Some new results are also proved which clarify this question. MSC (2000): 42B15 (Primary); 42B20, 46E40 (Secondary). Keywords: UMD space, property (α), Calderón–Zygmund operator, R-boundedness. Received 10 May 2006 / Accepted 1 December 2006.

1

Introduction

After its introduction in the early 1980’s by D. L. Burkholder [10], the probabilistic UMD property (unconditionality of martingale differences; see Def. 2.1) has become the central notion in Harmonic Analysis of functions with values in infinite-dimensional spaces. Indeed, several results from the classical Littlewood–Paley and Calderón–Zygmund theories, including their more recent extensions, remain valid in the context of X-valued functions if and only if the Banach space X has UMD. Out of the vast amount of examples, we record here the continuity in Lp (R, X) of (the tensor extension of) the Hilbert transform [6, 11], as well as the extensions to Lp (Rn , X) of the Marcinkiewicz–Mihlin multiplier theorem [8, 45] and the David–Journé T (1) theorem [24]. While the mentioned results were all obtained by the end The author was partially supported by the Finnish Academy of Science and Letters (Vilho, Yrjö and Kalle Väisälä Foundation). ∗

11

of the 1980’s, there has been a revival of interest in the related questions since the turn of the millennium. This has been boosted by the realization of the connection of another probabilistic notion, the R-boundedness (Def. 2.3), to the continuity of more general vector-valued Calderón–Zygmund transformations with an operator-valued kernel [14, 57], as well as the successful applications of these ideas to Partial Differential Equations. The UMD spaces have retained their central position also for these recent developments. However, another probabilistic Banach space condition, property (α) (see Def. 2.2), has also frequently appeared in the assumptions of various results on vector-valued Harmonic Analysis, and it is now known that in many cases it cannot be avoided. Already in the late 1980’s it was shown that the more general (compared to Mihlin’s) Marcinkiewicz–Lizorkin multiplier theorem is not valid in certain UMD spaces [61], and more recently it was realized [43] that a characterization of the spaces with this multiplier theorem is UMD combined with (α). Since this observation, several further results have considerably clarified the interplay of the UMD and (α) properties, and the need for the latter has been clearly related to the “multi-parameter” nature of certain results [34, 36, 37]. The present article has a two-fold purpose: • to survey the state-of-art of the theory of vector-valued Fourier multipliers and singular integrals, with emphasis on the rôle of property (α) (and heavily biased towards the author’s own interests), and • to supplement some new results, which establish a fairly sharp border between the parts of the theory requiring or not requiring (α). The paper will concentrate on the vector-valued Calderón–Zygmund theory per se. For the applications to Partial Differential Equations we refer to the recent monograph [19] and the lecture notes [42]. The latter also contains a detailed presentation of the H ∞ -calculus of sectorial operators in UMD spaces (not dealt with here), for which there is also the recent survey [58].

2

Probabilistic preliminaries

We first recall the two fundamental Banach space properties which determine the behaviour of singular integrals of vector-valued functions:

12

Definition 2.1 ( [10]). A Banach space X is UMD if for some (and then all) p ∈ ]1, ∞[ there is a C < ∞ such that N N "p $1/p "p $1/p ! "# ! "# " " " " dk " "k dk " ≤ C E" E" k=1

X

k=1

X

N N p N whenever N ∈ Z+ , ("k )N is k=1 ∈ {−1, +1} , and (dk )k=1 ∈ L (Ω, F , P; X) a martingale difference sequence on some probability space (Ω, F , P) with % expectation E := Ω dP; i.e., there are sub-σ-algebras F0 ⊆ F1 ⊆ · · · ⊆ FN ⊆ F so that dk ∈ Lp (Ω, Fk , P; X) and E(1A dk ) = 0 for all A ∈ Fk−1 , for all k = 1, · · · , N .

Examples of UMD spaces include the reflexive Lebesgue Lp , Lorentz Lp,q and Schatten–von Neumann C q spaces, 1 < p, q < ∞. If X is any UMD space, so are its dual X # , the Bôchner spaces Lp (µ, X) for 1 < p < ∞, and the closed subspaces and quotients of X. UMD spaces are (super)reflexive, and they have non-trivial type (and then also Fourier-type) as well as cotype. There are a number of useful surveys of UMD spaces [12, 13, 51]. (1) In the following definition, and always thereafter, the εi , ε˜j , εk , · · · are i.i.d. (independent identically distributed) random signs with distribution P(εi = +1) = P(εi = −1) = 1/2. They are called the Rademacher variables. Definition 2.2 ( [47]). A Banach space X has property (α) if there is a C < ∞ such that N N "# " "# " " " " " E" εi ε˜j αij xij " ≤ CE" εi ε˜j xij " i,j=1

X

i,j=1

X

N ×N N ×N whenever N ∈ Z+ , (xij )N and (αij )N . i,j=1 ∈ X i,j=1 ∈ {−1, +1}

This property holds for the commutative Lp spaces for all 1 ≤ p < ∞, and is also inherited from X by Lp (µ, X) for p in the same range; on the other hand, the (infinite-dimensional) non-commutative C q spaces have (α) only when q = 2. Unlike UMD, property (α) is not self-dual (e.g., %1 has (α) while %∞ does not); however, it is important that the joint property “UMD and (α)” is inherited by the dual space. Every Banach space with a local unconditional structure (l.u.st.), in particular every Banach lattice, has property (α) if and only if it has finite cotype. A good reference is Pisier’s original paper [47], where property (α) was introduced. We also recall the main property that one typically needs to impose on the range of operator-valued singular integral kernels: 13

Definition 2.3 ( [1]). An operator collection T ⊂ L (X) is R-bounded if there is a C < ∞ such that N N "# " "# " " " " " εk Tk xk " ≤ CE" εk x k " E" X

k=1

k=1

X

N N whenever N ∈ Z+ , (xk )N and (Tk )N k=1 ∈ X k=1 ∈ T . The least number C is called the R-bound and denoted by R(T ).

Even earlier, this notion made anonymous appearances in [8, 61], and Bourgain [8] proved the useful fact that R(abcoT ) ≤ 2R(T ), where the bar designates the strong operator closure, and abco stands for the absolute (or balanced) convex hull. A scalar-valued version of this inequality was implicitly used already in Marcinkiewicz’ original proof of his multiplier theorem [44], which may explain why R-boundedness has become such a central concept in the operator-valued extensions of this classical result. This notion is studied in detail in [14, 57]. Another frequently used result in connection with the randomized norms is the inequality of Kahane which provides the second, non-trivial comparison in the following chain, where Kp < ∞ is constant only depending on p: N N N "# " "p $1/p "# " ! "# " " " " " " E" εk x k " ≤ E " εk x k " ≤ Kp E " εk x k " , k=1

X

k=1

X

X

k=1

1 < p < ∞.

This implies that one can replace the L1 norms in Definitions 2.2 and 2.3 by other Lp norms for 1 < p < ∞. (The p invariance of Def. 2.1 is due to a different reason.) Another useful inequality of Kahane is the contraction principle, for which the notion of R-boundedness gives the compact formulation R(Λ · idX ) ≤ 2 sup{|λ| : λ ∈ Λ}, whenever Λ ⊂ C. It is sometimes handy to transform R-boundedness on X into usual boundedness on a certain larger space, Rad(X). The use of this space (wellknown in the Banach space theory) in the context of R-boundedness originates from Girardi and Weis [27]. Definition 2.4. The Rademacher space Rad(X) is the completion of all Z finitely non-zero sequences (xj )∞ j=−∞ ∈ X in any of the following equivalent norms, where p ∈ [1, ∞[: ∞ "p $1/p ! " # & & " " &(xj )∞ & ε x . := E " " p j j −∞ Rad (X) j=−∞

X

14

Unless otherwise said, we use the L1 norm on Rad(X). Z Let us identify a finitely non-zero sequence (Tj )∞ −∞ ∈ L (X) with the ∞ operator T˜(xj )∞ −∞ := (Tj xj )−∞ on Rad(X). For T ⊂ L (X), let us denote Z T˜ := {(Tj )∞ finitely non-zero}. −∞ ∈ T

Then clearly R(T ) = sup{(T˜(L (Rad(X)) : T˜ ∈ T˜ } ≤ R(T˜ ). One of the main implications, and in fact a characterization, of property (α) of X is the converse estimate R(T˜ ) ≤ CR(T ) for T ⊂ L (X), where C depends on the (α) property constant of X only [14].

3

Littlewood–Paley decompositions

The first step in Bourgain’s [8] approach to the estimation of singular integrals in UMD spaces is transforming the defining unconditionality property of martingale differences into another unconditionality estimate of more analytic flavour. This is an analogue of the classical inequality of Littlewood and Paley concerning the dyadic spectral decomposition of a function. For −1 an interval I ⊂ R, we denote ∆[I] ' := F 1I F , where F is the Fourier ' k k+1 : η ∈ {−1, +1}, k ∈ Z} be the collection transform. Let I := {η 2 , 2 of dyadic intervals on R. The vector-valued Littlewood–Paley inequality is the following: Theorem 3.1 ( [8, 45]). Let X be a UMD space and 1 < p < ∞. Then there are constant 0 < c ≤ C < ∞ such that &# & & & f ∈ Lp (R, X). c (f (p ≤ E& εI ∆[I]f & ≤ C (f (p , I∈I

p

Conversely, this estimate implies that X is UMD and 1 < p < ∞.

The proof of Bourgain [8] consists of writing the UMD inequality for the −k translated dyadic filtrations (Dk − u)∞ [j, j + 1[ : k=0 of R, where Dk := {2 j ∈ Z} and 0 ≤ u < 1, and averaging over the values of the translation parameter u. This yields an inequality similar to that of Theorem 3.1 but k −2 with smooth and overlapping cut-offs (decaying like (2k ξ)2 resp. ξ) ' k (2 ' as k+1 ξ → 0 resp. ξ → ∞) in place of the indicators of the intervals 2 , 2 . The desired sharp cut-offs are then reached by a perturbation argument, which 15

uses the R-boundedness of the family of spectral projections ∆[J], where J ranges over all intervals of R. This R-boundedness is a consequence of the boundedness of the single projection ∆[R+ ] = (id +iH)/2, where H is the Hilbert transform, the identity ∆[a + J] = ei2πax ∆[J]e−i2πax , and the basic properties of R-bounded sets. A different approach to the vector-valued Littlewood–Paley decomposition is due to McConnell [45], who proved a Mihlin-type multiplier theorem directly from the UMD inequality by means of heavy stochastic machinery, and derived Theorem 3.1 as a corollary of his multiplier estimate. In Bourgain’s approach, on the other hand, Theorem 3.1 is used to obtain the multiplier theorem (see next section), which turns out to be sharper than that proved by McConnell. Theorem 3.1 shows that on the one-dimensional Euclidean domain R, the classical spectral decomposition extends to the UMD-valued situation, and in fact reflects the one-parameter decomposition postulated in the definition of UMD. When we want to move to Rn with n > 1, however, a one-parameter decomposition cannot adequately capture the full n-dimensional structure of the product domain Rn = R × · · · × R, and this is where property (α) enters the scene. Of course, by simply iterating Theorem 3.1, we obtain & # & & & (1) (n) c (f (p ≤ E& εI1 · · · εIn ∆[I]f & ≤ C (f (p , f ∈ Lp (Rn , X), (3.1) p

I∈I n

(1)

(n)

where I n := {I = I1 × · · · × In : I1 , · · · , In ∈ I }, and εI1 , · · · , εIn are i.i.d. (1) (n) sequences of Rademacher variables. The products εI1 · · · εIn , however, are not quite the same as one independent sequence εI = εI1 ×···×In indexed by the product intervals. Property (α), on the other hand, is precisely the condition under which the two random sums are equivalent, and Zimmermann proved the following: Theorem 3.2 ( [61]). Let n > 1. There are constant 0 < c ≤ C < ∞ such that the following estimates hold, if and only if X is a UMD space with property (α) and 1 < p < ∞: & # & & & c (f (p ≤ E& εI ∆[I]f & ≤ C (f (p , f ∈ Lp (Rn , X). I∈I n

p

All hope is not lost in general UMD spaces, either, but we have to content ourselves with a coarser decomposition, the so-called blocking by squares of 16

"

"

!

!

Figure 1: The dyadic product decomposition (left) and its blocking by squares (right). the product decomposition I n . This consists of the intervals r−1 ( i=1

n ( ' ) ki +1 ' ' kr kr +1 ' ) ηi 0, 2 × ηr 2 , 2 × ηi 0, 2ki

(3.2)

i=r+1

with η ∈ {−1, +1}n , k ∈ Zn . Denoting the set of these intervals by In , the result reads: Theorem 3.3 ( [61]). Let X be a UMD space, 1 < p < ∞ and n ≥ 1. Then there are constants 0 < c ≤ C < ∞ such that & &# & & c (f (p ≤ E& εI ∆[I]f & ≤ C (f (p , f ∈ Lp (Rn , X). I∈In

p

The two decompositions for n = 2 are illustrated in Fig. 1. Unconditional decompositions and their products and blockings are studied in detail in Witvliet’s thesis [59].

4

Vector-valued Fourier multipliers

While various classes of Calderón–Zygmund operators and their generalizations have been treated in UMD spaces by now, many of the typical vector17

valued phenomena are most easily illustrated in the context of Fourier multipliers, whose boundedness properties are very closely related to the unconditionality of Littlewood–Paley decompositions. In fact, the two ingredients used in Bourgain’s approach to the vector-valued Mihlin multiplier theorem are Theorem 3.1 and an R-boundedness estimate for the class of multipliers of bounded variation. The point is then to apply this R-boundedness result to the “dyadic pieces” m1I of a Mihlin multiplier to obtain the boundedness of the full multiplier m. Essentially the same idea goes through for operator-valued multipliers; we only need to strengthen bounded variation to R-bounded variation: Definition 4.1. We say that a set M of functions m : Rn → L (X) has uniformly R-bounded variation if there exists a fixed R-bounded set T ⊂ L (X), and for each m ∈ M a probability measure µ on [−∞, ¯ ∞[ ¯ and a strongly measurable τ : [−∞, ¯ ∞[ ¯ → T , such that for all x ∈ X * m(ξ)x = τ (y)x µ(dy). [−∞,ξ] ¯

We denoted ∞ ¯ := (∞, · · · , ∞) and [−∞, ¯ ξ] := [−∞, ξ1 ] × · · · × [−∞, ξn ]. Typical examples of R-bounded variation arise from multipliers m supported on intervals J = J1 × · · · × Jn and having derivatives Dα m(ξ), α ∈ {0, 1}n , ξ ∈ J, such that ((Dα m(ξ)(T (L1 (J α ) ≤ C < ∞, where (·(T is the Minkowski + + functional of an R-bounded set T , and J α := i:αi =1 Ji × i:αi =0 {inf Ji }.

Theorem 4.1 ( [8, 42, 61]). Let X be a UMD space and 1 < p < ∞. If M is a uniformly R-bounded collection of functions on Rn , then the set of Fourier multiplier operators Tm := F −1 mF , m ∈ M , is R-bounded on L (Lp (Rn , X)). This result is actually equivalent to the Lp (R, X) boundedness of the Hilbert transform, since H itself has a scalar multiplier of bounded variation, whereas every multiplier Tm with m ∈ M belongs to abco(T · S ), where T is the R-bounded set appearing in Def. 4.1, and S := {∆[ξ + Rn+ ] : ξ ∈ Rn } whose R-boundedness follows from the boundedness of H. We can now formulate a generic multiplier theorem: Theorem 4.2. Let X be a UMD space and 1 < p < ∞. Let J be a collection of intervals ⊂ Rn such that {∆[J] : J ∈ J } satisfies, for some 18

0 < c ≤ C < ∞, the unconditionality estimate &# & & & c (f (p ≤ E& εJ ∆[J]f & ≤ C (f (p , J∈J

p

f ∈ Lp (Rn , X).

If a multiplier m : Rn → L (X) has the property that the m1J , J ∈ J , are of uniformly R-bounded variation, then Tm ∈ L (Lp (Rn , X)). In fact, K := R(Tm1J : J ∈ J ) < ∞ by Theorem 4.1, and then & & 1 & 1 & &# & &# & εJ ∆[J]Tm f & = E& εJ Tm1J ∆[J]f & (Tm f (p ≤ E& c J∈J c J∈J p p &# & 1 C & & ≤ KE& εJ ∆[J]f & ≤ K (f (p . c c p J∈J

The following two results, which provide vector-valued extensions of the classical Mihlin and Marcinkiewicz–Lizorkin multiplier theorems, are now consequences of the generic Theorem 4.2 and the Littlewood–Paley decompositions of the previous section: Theorem 4.3 ( [8, 54, 57, 61]). Let n ≥ 1. If (and only if ) X is a UMD space and 1 < p < ∞, then every multiplier m : Rn \ {0} → L (X) such that R(|ξ||α| Dα m(ξ) : α ∈ {0, 1}n , ξ ∈ Rn \ {0}) < ∞ satisfies Tm ∈ L (Lp (Rn , X)). Theorem 4.4 ( [54, 61]). Let n > 1. If (and only if ) X is a UMD space with property (α) and 1 < p < ∞, then every multiplier m : Rn \ {0} → L (X) such that R(|ξ α | Dα m(ξ) : α ∈ {0, 1}n , ξ ∈ (R \ {0})n ) < ∞ satisfies Tm ∈ L (Lp (Rn , X)). The case of Theorem 4.3 when n = 1 and m is scalar-valued was proved by Bourgain [8] and extended to n > 1 by Zimmermann [61]; a slightly weaker statement had already been obtained by McConnell [45] using different methods. The operator-valued statement was first achieved by Weis [57] for n = 1 and extended to n > 1 by Štrkalj and Weis [54]; other proofs of 19

this result are given in [19, 29, 42]. Zimmermann’s original formulation of the scalar-multiplier case of Theorem 4.4 was in the slightly smaller class of UMD spaces with l.u.st.; a similar but yet more restricted statement in Banach lattices was proved in [21], based on [45]. The class of UMD spaces with (α) used by Štrkalj and Weis is the largest possible, as observed by Lancien [43]. Finally, it should be pointed out that the R-boundedness assumptions are not only a matter of technical convenience, but a necessity to obtain multiplier theorems, as shown by Clément and Prüss [15]: Theorem 4.5 ( [15]). Let n ≥ 1 and 1 < p < ∞. There is a constant C < ∞ such that if X is an arbitrary Banach space and m ∈ L1loc (Rn , L (X)), then R(m(ξ) : ξ ∈ Rn a Lebesgue point of m) ≤ C (Tm (L (Lp (Rn ,X)) .

5

Bootstrapping and induction

In the context of vector-valued estimates, various possibilities of self-improvement are looming around. A basic observation is the isomorphic (thanks to Kahane’s inequality) identification of spaces Rad(Lp (Rn , X)) ! Lp (Rn , Rad(X)), which gives rise to an identification of operators (Tmj )∞ , where −∞ ! T(mj )∞ −∞ on the right we have the Fourier multiplier with the sequence-valued kernel ξ ,→ M (ξ) := (mj (ξ))∞ −∞ ∈ L (Rad(X)). Thus, proving the R-boundedness of a family of Fourier multiplier operators amounts to checking the boundedness of the single L (Rad(X))-valued multiplier M (ξ). In the presence of property (α), it is possible to conclude that M inherits the required Rboundedness estimates from the original multipliers mj . These observations lead to the following: Theorem 5.1 ( [9, 27, 41, 56]). Let n ≥ 1. If (and only if ) 1 < p < ∞ and X is a UMD-space with property (α), then every family M of L (X)-valued multipliers on Rn such that R(|ξ α | Dα m(ξ) : α ∈ {0, 1}n , ξ ∈ (R \ {0})n , m ∈ M ) < ∞ induces an R-bounded family of operators {Tm : m ∈ M } ⊂ L (Lp (Rn , X)). 20

This was first proved for scalar-valued multipliers by Venni [56]; the operator-valued extension was found by Bu [9] and by Girardi and Weis [27] (apparently independently). The necessity of (α) is shown in [41]. Another basic identification is the Fubini isometry Lp (Rn+1 , X) ! Lp (R, Lp (Rn , X)), where Lp (Rn , X) has UMD and/or (α) if X has, and 1 < p < ∞. This again gives an identification of Fourier multipliers, Tξ∈Rn+1 &→m(ξ)∈L (X) ! Tξ1 ∈R&→Tm(ξ1 ,·) ∈L (Lp (Rn ,X)) . As it turns out, this kind of identification can be used, as done in [31], to reprove Theorems 4.4 and 5.1 for general n ≥ 1 using only Theorem 5.1 for n = 1 and a simple induction on the dimension. While the inductive method is not necessary for getting these results, as we saw, it may offer at least some conceptual simplification, which has been exploited in proving new estimates for singular integrals by Portal and the author [38]. It seems that the inductive method is most useful in the presence of (α), for otherwise it is difficult to ensure the required R-bounds of the sequence-valued multipliers, but it may also be useful in some special situations without (α), as we see in the proof of Proposition 6.1 below.

6

Scope of the two multiplier theorems

The Mihlin Multiplier Theorem 4.3 is a result typical of the classical Calderón– Zygmund theory, which deals with classes of operators invariant under the natural one-parameter family of dilations ξ ,→ δξ, δ > 0, of Rn . The Marcinkiewicz–Lizorkin Multiplier Theorem 4.4, on the other hand, is typical of the multi-parameter or product theory, allowing independent dilations ξ ,→ (δ1 ξ1 , · · · , δn ξn ), δ¯ = (δ1 , · · · , δn ) > ¯0, in the different coordinate directions. Since |ξ α | = |ξ1α1 · · · ξnαn | ≤ |ξ||α| (typically with strict inequality), Theorem 4.4 imposes less stringent conditions on the multiplier m, at the cost of restricting the admissible Banach spaces. Since the main interest in results like Theorems 4.3 and 4.4 comes from their applications, it is natural to enquire about the difference of the two conditions in practice: What are the typical multipliers for which one needs to use the more general Marcinkiewicz–Lizorkin theorem? In his classic book 21

Singular integrals, Stein [52] gives two examples of multipliers failing the Mihlin condition but falling under the scope of the Marcinkiewicz–Lizorkin theorem. The first one is the multiplier m(ξ) =

ξ1 +

i(ξ22

ξ1 + ξ32 + · · · + ξn2 )

related to parabolic equations, whereas the other example n ! ( |ξi | $αi |ξ α | |ξ1 |α1 |ξ2 |α2 · · · |ξn |αn = = , m(ξ) = 2 (ξ1 + ξ22 + · · · + ξn2 )|α|/2 |ξ| |ξ||α| i=1

(6.1)

(6.2)

where α = (α1 , α2 , · · · , αn ) ≥ ¯0, “is not untypical of a class arising in connection with the study of spaces of fractional potentials”. We will study the multipliers of the parabolic type more systematically in the next section and now take a closer look at the second example (6.2). This multiplier has the curious feature of formally belonging to the one-parameter class, having invariance under the standard dilations ξ ,→ δξ, but failing the Mihlin conditions for the derivatives. Nevertheless, there holds:

Proposition 6.1. For the multiplier m in (6.2), we have Tm ∈ L (Lp (Rn , X)) for all 1 < p < ∞ and all UMD spaces X.

Proof. Let us first observe that it suffices to treat m(ξ) = m$ (ξ) := (|ξ1 | / |ξ|)$ , " > 0: this is the special case with α = (", 0, · · · , 0), but if we know the boundedness of these multipliers, the general case of (6.2) also follows, since it is just the product of our special multiplier and its rotated versions. We make use of the identification Tm ! Tξ1 &→Tm(ξ1 ,·) , and verify the operator-valued Mihlin conditions for the one-dimensional multiplier M (ξ1 ) := Tm(ξ1 ,·) . Let us show that {M (ξ1 ) : ξ1 ∈ R\{0}} is R-bounded in L (Lp (Rn−1 , X)), for which it suffices by Theorem 4.1 that the scalar multipliers m(ξ1 , ·) have uniformly bounded variation. For α ∈ {0} × {0, 1}n−1 , it is easy to compute Dα m$ (ξ) = C$,α m$ (ξ)

ξα |ξ||α|

= C$,α

where ξ # := (ξ2 , · · · , ξn ), and then * ( * α |D m(ξ)| dξα ≤ C$,α Rα

= C$,α

i:αi =1 ∞

!*

−∞

(

|ξ1 |$/|α| ξi , 2 # |2 )$/2|α|+1 (ξ + |ξ 1 i:αi =1

|ξ1 |$/|α| |ξi | dξi 2 2 $/2|α|+1 −∞ (ξ1 + ξi ) $|α| |x| dx , (1 + x2 )1+$/2|α| ∞

22

which is a finite constant. We still need the R-boundedness of ξ1 M # (ξ1 ), but this is immediate from the- work already done, once we observe that , ξ1 D1 m$ (ξ) = " m$ (ξ) − m$+2 (ξ) .

Thus we observe that even if the full Marcinkiewicz–Lizorkin Multiplier Theorem 4.4 does not apply in a general UMD space, we can still go somewhat beyond the Mihlin Multiplier Theorem 4.3. The next section elaborates on this theme even more.

7

Parabolic theory of multipliers

The multiplier (6.1) is not well-behaved under the radial dilations, but it has the property of being invariant under the anisotropic one-parameter dilations ξ ,→ (δ 2 ξ1 , δξ2 , · · · , δξn ), δ > 0. It turns out that this is essentially as good as the radial dilations, since there is also an anisotropic version of the Littlewood–Paley inequality, which can be proved almost by a repetition of the construction of the blocking-by-squares decomposition of Theorem 3.3. The key observations to this end are the facts that • the one-dimensional Littlewood–Paley Theorem 3.1 remains valid if ' ' we' replace 'the dyadic intervals ± 2k , 2k+1 by any lacunary intervals ± λk , λk+1 , λ > 1, (this follows easily from the case λ = 2) and

• in constructing the blocking by squares of the product decomposition, we may take different one-dimensional decompositions in the different coordinate directions. Using the decomposition with λ = 2θi in the ith coordinate, this construction results in the decomposition In (θ), whose intervals are r−1 ( i=1

n ( ' ) θi (ki +1) ' ' θr kr θr (kr +1) ' ) ηi 0, 2 × ηr 2 , 2 × ηi 0, 2θi ki , i=r+1

where η ∈ {−1, +1} and k ∈ Z . This decomposition for n = 2 is illustrated in Fig. 2. We obtain the following theorems: Theorem 7.1 ( [34]). Let n ≥ 1, θ = (θ1 , · · · , θn ) > ¯0, X be a UMD space and 1 < p < ∞. Then there are 0 < c ≤ C < ∞ such that & # & & & c (f (p ≤ E& εI ∆[I]f & ≤ C (f (p , f ∈ Lp (Rn , X). n

I∈In (θ)

n

p

23

"

!

Figure 2: The anisotropic blocking decomposition with (2θ1 , 2θ2 ) = ( 52 , 2). Theorem 7.2 ( [34]). Let n ≥ 1, θ = (θ1 , · · · , θn ) > ¯0, X be a UMD space and 1 < p < ∞. Then every multiplier m : Rn \ {0} → L (X) such that R(,θ (ξ)θ·α Dα m(ξ) : α ∈ {0, 1}n , ξ ∈ Rn \ {0}) < ∞, . where ,θ (ξ) is the unique positive solution of ni=1 ξi2 ,θ (ξ)−2θi = 1, satisfies Tm ∈ L (Lp (Rn , X)). Theorem 7.2 follows from 7.1 in the same way as Theorem 4.3 follows from 3.3. It contains Theorem 4.3 as the special case θ = (1, · · · , 1), but also applies to more general multipliers of the parabolic type, like (6.1).

8

Multipliers for Sobolev-type inequalities

The following question of generalized Sobolev-type inequalities leads to an interesting class of Fourier multipliers: Find the conditions under which the following estimate for partial derivatives holds for all test functions, say u ∈ D(Rn , X): # & β & &D u& ≤ C (Dα u(p . (8.1) p α∈A

The classical problem for X = C has the simple answer, for p ∈ ]1, ∞[, that we must have β ∈ conv A , the convex hull of A (see e.g. [5]). The necessity 24

follows by writing (8.1) for appropriate sums of translates and dilates of u. A sketch for the sufficiency is as follows: By the boundedness of ∆[Rη1 × · · · × Rηn ], η ∈ {−, +}n , which commute with Dα , and by symmetry we may assume that F u is supported in Rn+ . By induction on the size of A , it suffices to treat #A = 2. The task is reduced to proving that m(ξ) =

1Rn+ (ξ) ξ tα0 +(1−t)α1 n 1 (ξ) = R+ ξ α0 + ξ α1 ξ (1−t)(α0 −α1 ) + ξ t(α1 −α0 )

(8.2)

is a Fourier multiplier of Lp (Rn , X). It is readily checked to be a Marcinkiewicz–Lizorkin multiplier, so we have (8.1) in UMD spaces with (α). But does it hold in general UMD spaces? For n = 1, clearly yes. For n > 1, we may observe that m(λθ ξ) = m(ξ) for all λ > 0, provided that we choose θ so that θ · (α0 − α1 ) = 0. This is always possible with some θ -= ¯0, but may be not with θ > ¯0. The homogeneity of m is not in general of the parabolic type covered by Theorem 7.2, which leads to the question: Does there exist a “hyperbolic” version of Theorem 7.2, allowing θ ∈ (R \ {0})n , and still valid in all UMD spaces? The answer to this general question turns out to be negative, as demonstrated in the following section, but for more specific reasons there nevertheless holds: Theorem 8.1 ( [36]). Let X be a UMD space, 1 < p < ∞, and β ∈ conv A . Then there is C < ∞ such that (8.1) holds for all u ∈ D(Rn , X). The proof exploits the Littlewood–Paley type estimate (3.1), the fact that the multipliers ξ α are products ξ1α1 · · · ξnαn of one-dimensional multipliers, and finally a complex interpolation argument.

9

No hyperbolic theory of multipliers

We now prove the impossibility of the hyperbolic theory of multipliers that one could have hoped for by the considerations in the previous section. Let us first look at some consequences that such a theory would " have. If" m ∈ C 2 (R\{0}) satisfies the second-order Mihlin-type condition "ξ k Dk m(ξ)" ≤ C for k = 0, 1, 2, then (ξ1 , ξ2 ) ∈ R2 ,→ m(ξ1 ξ2 ) is a hyperbolic multiplier with dilation invariance under ξ ,→ (λξ1 , λ−1 ξ2 ). If it was to be bounded on Lp (R2 , X) ! Lp (R, Lp (R, X)), then by Theorem 4.5, the set of operators 25

Tm(ξ1 ·) induced by the dilations of m would be R-bounded in L (Lp (R, X)). That such a result can only hold under the property (α) is the main content of the following: ∞ Proposition " k k 9.1. " There exists a Mihlin multiplier m ∈ C (R \ {0}) satisfying "ξ D m(ξ)" ≤ Ck for all k ∈ N, such that the following hold for every 1 < p < ∞ and every Banach space X:

• Tm ∈ L (Lp (R, X)) if and only if X is UMD, and

• {Tm(2j ·) : j ∈ Z} is R-bounded on Lp (R, X) if and only if X is UMD with (α). Proof. The building blocks of our. construction are a function ψˆ ∈ D(R) ˆ j supported in [1/2, 2] and satisfying ∞ j=−∞ ψ(2 ξ) = 1]0,∞[ (ξ), and a sequence Z (αj )∞ j=−∞ ∈ {0, 1} chosen so as to contain as a subsequence every one of the (countably many) finite bit sequences. Our multiplier is then defined as # ˆ −j ξ). m(ξ) := αj ψ(2 j∈Z

This clearly satisfies the Mihlin conditions of any order, so the asserted boundedness follows Theorem 4.3 and the R-boundedness from Theorem 5.1. Concerning the necessity of UMD, we observe that it suffices to prove (F −1 (1]0,∞[ fˆ)(p ≤ C (f (p for all f ∈ S (R, X) with supp fˆ a compact subset of R \ {0}. Given such an f , let us denote ρ := sup{|ξ| / |η| : ξ, η ∈ supp fˆ} < ∞. By definition, (αj )∞ j=−∞ contains a subsequence of 1’s of length N + 1 where N > log2 ρ, say αj ≡ 1 for j ∈ [j0 , j0 + N ] ∩ Z. An appropriate dilation gˆ := fˆ(λ·), λ > 0, has the positive half of its support on [2j0 , ρ2j0 ] ⊂ [2j0 , 2j0 +N ], where m(ξ) ≡ 1. Thus (F −1 (1]0,∞[ gˆ)(p = (Tm g(p ≤ C (g(p , and by dilation invariance we have the same with f in place of g. We then come to the necessity of (α) for the R-boundedness of the dilated multipliers. Recall that this is equivalent to the boundedness of the operatorvalued Fourier multiplier TM on Lp (R, Rad(X)), where j ∞ M (ξ)(xj )∞ −∞ := (m(2 ξ)xj )−∞

= (αi+j xj )∞ j=−∞ ,

if ξ = 2i . 26

For this, it is necessary by Theorem 4.5 that the essential range of M is R-bounded on Rad(X). This requires in particular that " "## " "## " " ˜ "" ˜ "" (9.1) εi ε˜j xi,j " εi ε˜j αi+j xi,j " ≤ CEE EE i∈F j∈G

X

i∈F j∈G

X

for all finite F, G ⊂ Z and xi,j ∈ X. But we may choose F = {n0 + N k : k = 0, · · · , N − 1}, G = {% : % = 0, · · · , N − 1}, so that {i + j : i ∈ F, j ∈ G} = [n0 , n0 + N 2 − 1] ∩ Z will be any desired sequence of N 2 consecutive integers, and the corresponding αi+j = αn0 +N k+( may be chosen as an arbitrary N × N array of bits. Thus (9.1) becomes precisely the defining condition of property (α).

We saw earlier that the UMD condition alone suffices for much of the oneparameter theory of multipliers, which goes somewhat beyond the standard Calderón–Zygmund theory related to the radial dilations. In contrast to this, the previous Proposition shows that we cannot go much further: there is no “hyperbolic one-parameter theory” of multipliers on its own right; it only exists as a special case of the multi-parameter theory, which requires the additional property (α). Incidentally, the fact that hyperbolic dilations, although having only one independent variable, should already be regarded as part of the multiparameter theory, has been observed in other contexts, too. As pointed out in [22] in connection to near-L1 estimates for maximal functions * 1 |f (y)| dy MR f (x) := sup x∈R∈R |R| R

related to different families of rectangles R, “the two-dimensional collection of rectangles of the form s × 1/s is already a two-parameter family.” We have now found a fairly sharp border between the multiplier theory valid in all UMD spaces and only in UMD spaces with (α), but some questions still remain. E.g., it would be interesting to know if the multiplier in Prop. 9.1 could be replaced by some more naturally occurring one. Observe that the m constructed in the proof in some sense contains all information in the universe, and one may wonder if a little less would be sufficient. Furthermore, we have mainly excluded the possibility of certain general statements in all UMD spaces, but the continuity or its failure of a particular operator arising from a specific application is not implied by these 27

results, leaving the possibility for various interesting special theorems, like the inequality (8.1) discussed in the previous section.

10

Other Banach space properties

In the remaining three sections, we briefly survey other developments in vector-valued Harmonic Analysis, which are independent of the above discussion of the border between one-parameter and multi-parameter theories. We first consider the effect on the multiplier theory of some further Banach space properties (besides UMD and (α)), which have been studied in a number of papers.

Fourier-type In the scalar-valued context, it is a classical result of Hörmander that Mihlin’s multiplier theorem remains valid when changing the set of derivatives for which the estimate |ξ||α| |Dα m(ξ)| ≤ C is required from α ∈ {0, 1}n to |α| ≤ 0n/21 + 1. In the vector-valued case, a similar reduction of the needed total order of differentiation is caused by the Fourier-type of the Banach space X. Recall that X has Fourier-type t ∈ [1, 2] if the Hausdorff–Young inequality (F f (t# ≤ C (f (t holds for f ∈ Lt (Rn , X) for one (and then all) n ∈ Z+ . Theorem 10.1 ( [26, 30, 33]). Let n ≥ 1, 1 < p < ∞, and let X be a UMD space (resp. UMD space with (α)) with Fourier-type t ∈ ]1, 2]. If ,

R(|ξ||α| Dα m(ξ) : ξ ∈ Rn \ {0}) < ∞,

resp. R(|ξ α | Dα m(ξ) : ξ ∈ (R \ {0})n ) < ∞ ,

for all α ∈ {0, 1}n with |α| ≤ 0n/t1 + 1, then Tm ∈ L (Lp (Rn , X)). The original realization of the condition |α| ≤ 0n/t1 + 1 under Fouriertype t is due to Girardi and Weis [26]. The possibility of intersecting this with Mihlin’s resp. Marcinkiewicz–Lizorkin’s condition was observed in [30] resp. [33]. The assumptions may be further weakened somewhat by considering appropriate fractional order smoothness, which allows to approach the critical index n/t.

28

Lattices Rubio de Francia [51] wrote about twenty years ago: “Our present knowledge of the properties and structure of UMD lattices is deeper than for general UMD spaces.” The state of affairs is still very much the same today. The additional tools available in UMD lattices, especially maximal functions and techniques using Muckenhoupt’s Ap weights, have made it possible to prove results like the following, which remain interesting open problems for general UMD spaces: Theorem 10.2 ( [51]). Every UMD lattice X is a complex interpolation space [H, Y ]θ , 0 < θ < 1, between a Hilbert space H and another UMD lattice Y . Theorem 10.3 ( [50, 51]). Let X be a UMD lattice and 1 < p < ∞. If f ∈ Lp (T, X), then the Fourier series of f converges to f almost everywhere. Theorem 10.3 was first proved by Rubio de Francia [50] for UMD spaces with an unconditional basis and extended by the same author [51] to the generality stated above. Other results in UMD lattices or UMD spaces with an unconditional basis have been proved in [7, 28, 55].

Interpolation spaces Theorem 10.2 motivated the following definition by Berkson and Gillespie: Definition 10.1 ( [2]). The class I consists of those UMD spaces X which are isomorphic to a closed subspace of a complex interpolation space [H, Y ]θ , 0 < θ < 1, between a Hilbert space H and another UMD space Y . By Theorem 10.2, every UMD lattice belongs to I, but I also contains other UMD spaces like the Schatten–von Neumann ideals C p = [C 2 , C q ]θ , 1/p = (1 − θ)/2 + θ/q. More generally, the interpolation properties of noncommutative spaces coincide with those of their commutative analogues under fairly broad conditions [20], which implies the membership in I for many further operator spaces. In some cases [2, 35, 38], improved results (compared to general UMD spaces) have been proved in the spaces of class I by interpolating with the estimates available in arbitrary UMD spaces and the stronger ones that one can get in a Hilbert space. Thus it would be interesting to know if I actually contains all UMD spaces. 29

Littlewood–Paley–Rubio property Berkson, Gillespie and Torrea have recently introduced the following notion: Definition 10.2 ( [3, 25]). Let 2 ≤ p < ∞. A Banach space X has the Littlewood–Paley–Rubio property LP Rp if there is C < ∞ so that for every collection J of disjoint intervals J ⊂ R there holds: &# & & & E& εJ ∆[J]f & ≤ C (f (p , f ∈ Lp (R, X). p

J∈J

The scalar field X = C, and then by Fubini also Lp (µ), satisfies LP Rp by an inequality of Rubio de Francia [49]. This inequality was used by Coifman, Rubio de Francia and Semmes [16] to improve the Marcinkiewicz multiplier theorem. Similarly, the LP Rp property of a Banach space X implies an improvement of the vector-valued multiplier theorem, which was obtained by Potapov and the present author: Theorem 10.4 ( [38]). Let 1 ≤ s < 2 ≤ p < ∞ and X be a Banach space with LP Rp . Let ' m : R' → L (X) be a function such that for all dyadic intervals I = ± 2k , 2k+1 we have sup

!

(f (ξ0 )(sT

+

N # j=1

(f (ξj−1 ) −

f (ξj )(sT

$1/s

≤ c < ∞,

where T is an R-bounded set, and the supremum is over all partitions inf I = ξ0 < ξ1 < · · · < ξN = sup I. Then Tm ∈ L (Lp (R, X)). The case s = 1 above is the (vector-valued) Marcinkiewicz multiplier theorem valid in every UMD space and 1 < p < ∞ (although we have only given the somewhat weaker formulation in Theorem 4.3 above). The assumption becomes weaker with increasing s. If we also assume that X ∈ I, then we may take s = 2 in Theorem 10.4.

11

Singular convolution operators

In the vector-valued treatment of Calderón–Zygmund operators beyond those represented by Mihlin or Marcinkiewicz–Lizorkin type multipliers, the following “restricted R-boundedness” estimate for the translations τh f := f (· − h) has become an indispensable companion to the Littlewood–Paley inequalities discussed earlier: 30

Theorem 11.1 ( [8]). Let X be a UMD space and 1 < p < ∞. Then there is a constant C < ∞ such that, whenever the functions fj ∈ Lp (Rn , X) and the colinear points hj ∈ Rn , j ∈ Z, satisfy supp F fj ⊆ {ξ : |ξ| ≤ 2j },

|hj | ≤ K2−j

for some K ≥ 2, there holds &# & &# & & & & & E& εj τhj fj & ≤ C log K · E& εj fj & . j

p

j

p

This result of Bourgain’s [8] had apparently no scalar-valued predecessor, but the scalar version of the theorem was independently discovered around the same time by Yamazaki [60]. The original statement in [8] concerns n = 1, but the transference to n > 1 is standard and may be found from [26]. Like the proof of Theorem 3.1, also that of Theorem 11.1 starts from the defining inequality of UMD spaces, but now applied to a less obvious choice of the filtration. Every second σ-algebra will be generated by simple intervals N −1 [k, k + ,1[, k ∈ Z, but the atoms of the - intermediate σ-algebras will be unions N −1 [k, k + 1[ ∪ [n + k, n + k + 1[ , k ≡ 0, 1, · · · , n − 1 (mod 2n), of two separated intervals, where the separation reflects the translations that we are aiming at. Like in the case of Theorem 3.1, the estimate obtained from the martingale inequality introduces extra smoothing in addition to the desired translations. Moreover, we can only reach translations which satisfy certain algebraic restrictions. In order to remove these deficiencies, a perturbation argument based on Theorem 4.1 is required. To make this argument, we have to split the functions fj into approximately log K subsets for separate treatment, which gives rise to the logarithmic factor in the final estimate. The scalar-valued version in [60] is somewhat easier, since the trivial case p = 2 can be extrapolated to the whole range of 1 < p < ∞ by standard Calderón–Zygmund methods. The first application of Theorem 11.1 is to the boundedness of singular integrals of convolution type: Theorem 11.2 ( [8, 39]). Let X be a UMD space and 1 < p < ∞. Let K ∈ C(Rn \ {0}, L (X)) satisfy the size and cancellation conditions

R(|x|n K(x), |x|n+δ |y|−δ [K(x + y) − K(x)] : |x| > 2 |y| > 0) < ∞ % for some δ > 0, and R( r 0) < ∞, as well as the % existence of the limit lim$↓0 $ 0) < ∞.

Let T : S (Rn ) → L (S (Rn ), L (X)) be a linear operator such that ** 4φ1 , T φ0 5 = φ1 (x)K(x, y)φ0 (y) dx dy

(12.1) 33

for all disjointly supported φ0 , φ1 ∈ D(Rn ), and such that R(rn 4φ1 (r · +h), T (φ0 (r · +h))5 : r > 0, h ∈ Rn ) ≤ C < ∞

(12.2)

for all “bump functions” φi ∈ D(B(0, 1)) with (Dα φi (∞ ≤ 1 for all |α| ≤ N (some large number ). If T (1) = 0, T # (1) = 0 (in the sense of distributions modulo constants), then T extends to an operator in L (Lp (Rn , X)) for 1 < p < ∞. This result was first obtained by Figiel [24] for scalar-valued kernels using a clever decomposition of the operator and martingale estimates. For scalar kernels, one can even replace the conditions T (1) = 0, T # (1) = 0 by T (1), T # (1) ∈ BMO(Rn ), and this condition is both necessary and sufficient for the conclusion T ∈ L (Lp (Rn , X)) under the other stated assumptions, so that a full analogue of the David–Journé theorem is valid. The operatorvalued extension, and a new proof building on the Fourier-analytic techniques discussed in the previous sections, was recently found by Weis and the present author [40]. It is possible to state sufficient BMO-type conditions for T (1) and T # (1) even in the operator setting, but they are probably far stronger than necessary. The problem may be reduced to the question of boundedness of operator-valued paraproducts, but the precise condition for this is unknown already in infinite-dimensional Hilbert spaces [4, 46]. While Figiel’s martingale approach to the T (1) theorem differs quite a lot from the techniques discussed in this paper, there still exist parallel ingredients. In particular, a key rôle is played by estimates for translations of the Haar functions from [23], which are analogous to Bourgain’s Translation Theorem 11.1. With appropriate modifications of Figiel’s ideas, it is also possible to get an operator-valued extension of the T (b) theorem of David, Journé and Semmes [18]: Theorem 12.2 ( [32]). Let X be a UMD space and let K and T be as in Theorem 12.1, except that in (12.1) we have bi φi in place of φi , and in (12.2) bi (·)φi (r · +h) in place of φi (r · +h),1 where b0 , b1 ∈ L∞ (Rn ) satisfy the accretivity condition Re bi ≥ c > 0. If T (b0 ) = 0 and T # (b1 ) = 0, then T ∈ L (Lp (Rn , X)) for 1 < p < ∞. 1

The analogue of (12.2) is assumed in a stronger form in [32], where also the φi are required to vary inside the R-bound; however, it is easy to see that only the uniform boundedness over the φi is actually required in the proof.

34

Again, for a scalar-valued kernel it is sufficient (and necessary) for the conclusion that T (b0 ), T # (b1 ) ∈ BMO(Rn ), and there is an analogous sufficient condition in the operator-valued case. The accretivity assumption may be replaced by more general para-accretivity; see [32]. The scalar-kernel case of Theorem 12.2 can be easily deduced from Theorem 12.1 and the original T (b) theorem from [18]. Various results have also been proved for vector-valued pseudo-differential operators, of which the following is representative: Theorem 12.3 ( [37, 48, 53]). Let X be a UMD space and 1 < p < ∞. Let a ∈ L∞ (Rn × Rn , L (X)) satisfy ,

k

R (1 + |ξ|)

Dξki a(x, ξ), (1

k

+ |ξ|)

Dξki a(x, ξ) − Dξki a(y, ξ) |x − y|δ

: ξ ∈ Rn

-

≤C 0 such that ζ + Sn (ε) ⊂ U \ N (ζ) and ! f (ζ) = Gz (ζ)ω(dz)f (z). (4.12) ζ+Sn (ε)

52

If Ω is any nonempty open connected subset of Rn+1 and f : κ(Ω) −→ C"(Cn ) is a complex left monogenic function, then for each ζ ∈ κ(Ω) \ Ω, following [16] we determine a simple real n-cycle M (ζ) ⊂ κ(Ω) \ N (ζ) close to γC (ζ) ⊂ Rn+1 in an n-dimensional complex affine subspace of Cn+1 and homologous to the n-sphere ζ + Sn (ε) in the image of κ(Ω) \ N (ζ) in complex projective space CPn . An application of Stokes’ theorem ensures that the representation ! f (ζ) = Gz (ζ)ω(dz)f (z) (4.13) M (ζ)

is valid. The point of difference with the representation (4.12) is that ζ+Sn (ε) lies in the (n + 1)-dimensional affine subspace ζ + Rn+1 of Cn+1 . We look at the cases of n odd and even separately.

4.1

n odd

Let ζ ∈ Cn+1 \ Rn+1 . By equation (4.5), the monogenic spectrum γC (ζ) = N (ζ) ∩ Rn+1 of ζ is an (n − 1)-dimensional sphere in Rn+1 in the hyperplane orthogonal to η = Im ζ (= 0 and passing through ξ = Re ζ. For each x ∈ γC (ζ), let Cx (ζ) denote the complex line {z ∈ Cn+1 : z = ξ + λ(x − ξ), λ ∈ C }

(4.14)

passing through x and its antipodal point in γC (ζ). For each 0 < ε < |η|, let 0 Σ(n+1) (ζ) = {z ∈ Cx (ζ) : |z − x| < ε } (4.15) ε x∈γC (ζ)

∂Σ(n+1) (ζ) = ε

0

x∈γC (ζ)

{z ∈ Cx (ζ) : |z − x| = ε }.

(4.16)

(n+1)

Then ∂Σε (ζ) is a real n-dimensional submanifold of Cn+1 —an S 1 -fibration (n+1) of γC (ζ) ⊂ Rn+1 mentioned in [16, p. 416]. The set ∂Σε (ζ) is the (n+1) boundary of the open set Σε (ζ) containing γC (ζ) and contained in an ε# -neighbourhood of γC (ζ) in Cn+1 for all ε# > ε. It is oriented from the standard orientations of Rn+1 and the unit circles in Cx (ζ) for each x ∈ γC (ζ), so (n+1) that ∂Σε (ζ) has the orientation induced by the (n+1)-form dζ0 ∧· · ·∧dζn . Proposition 4.3. Suppose that n ∈ N is odd and Ω is an open subset of Rn+1 . Let ζ ∈ κ(Ω) \ Ω and suppose that the convex hull of γC (ζ) in Rn+1 is 53

contained in Ω. If f : κ(Ω) → C"(Cn ) is a complex left monogenic function, (n+1) (ζ) ⊂ κ(Ω) \ N (ζ) and then there exists ε > 0 such that ∂Σε ! f (ζ) = Gz (ζ)ω(dz)f (z). (4.17) (n+1)

∂Σε

(ζ)

Proof. First, we show that the half-open line segment [w, ζ) = {λw+(1−λ)ζ : (n+1) 0 < λ ≤ 1 } joining w ∈ ∂Σε (ζ) and ζ lies in the complement of the null cone N (ζ) of ζ. Let ζ = ξ + iη with ξ, η ∈ Rn+1 . Representing w as w = ξ + u(1 + z/|u|) for u ∈ {η}⊥ in Rn+1 with |u| = |η| and z ∈ C with |z| = ε, we have |λw + (1 − λ)ζ − ζ|2C = = = =

λ2 |w − ζ|2C λ2 | − iη + u(1 + z/|u|)|2C λ2 (−|η|2 + |u|2 (1 + z/|u|)2 ) λ2 (2|η| + z)z, 0 < λ ≤ 1.

Because |2|η| + z| ≥ 2|η| − ε > |η| > 0, it follows that [w, ζ) ⊂ Cn+1 \ N (ζ). Next we see that [w, ζ) ⊂ κ(Ω). Let uˆ = u/|u|. For each 0 < λ ≤ 1, the set γC (λw + (1 − λ)ζ) = γC (ζ + λ(−iη + u(1 + z/|u|))) is the (n − 1)-dimensional sphere centred at ξ + λ(u + Re(z)ˆ u), with radius |(1 − λ)η + λ Im(z)ˆ u| and contained in the hyperplane {x ∈ Rn+1 : %x − (ξ + λ(u + Re(z)ˆ u)), (1 − λ)η + λ Im(z)ˆ u& = 0 }. Some open neighbourhood of the convex hull of γC (ζ) is contained in Ω, so for ε sufficiently small, γC (λw+(1−λ)ζ) is contained in Ω for each 0 ≤ λ ≤ 1, (n+1) proving that that [w, ζ] ⊂ κ(Ω) for every w ∈ ∂Σε (ζ). Consequently, we (n+1) can translate ∂Σε (ζ) along the line segments [w, ζ) ⊂ κ(Ω) \ N (ζ) for (n+1) w ∈ ∂Σε (ζ). Now suppose that ε# > 0 is so small that ζ +Bε! (0) ⊂ κ(Ω), where Bε! (0) is the closed ball of radius ε# > 0 centred at zero in Cn+1 . Choose δ > 0 so that (n+1) δ(2|η| + ε) < ε# . For each w ∈ ∂Σε (ζ) there exists u ∈ Rn+1 as above such that w − ζ = −iη + u(1 + z/|u|). Then δ|w − ζ| ≤ δ(|η| + |u(1 + z/|u|)|) < ε# (n+1) and ∂Σε (ζ) is homologous to the cycle {ζ + δ(w − ζ) : w ∈ ∂Σ(n+1) (ζ) } ⊂ Bε! (0) ε 54

in κ(Ω) \ N (ζ). For each u ∈ {η}⊥ with |u| = |η|, let uˆ = u/|u| and ηˆ = η/|η| be unit vectors in the u and η directions and let V denote the real three dimensional subspace spanned by the vectors uˆ, iˆ u and iˆ η . According to formula (4.4), we have V ∩ N (0) = {ζ ∈ V : |ζ|2C = 0 } = {z uˆ + aiˆ η : z ∈ C, a ∈ R, z2 − a2 = 0 } = {a(ˆ u ± iˆ η) : a ∈ R } The cycle defined by −iη ± u + z uˆ, z ∈ C with |z| = ε and the positive orientation can be deformed in V \ N (0) into the circle of radius |η| + ε (n+1) centred at −iη and then into z uˆ, z ∈ C with |z| = ε# , so that ∂Σε (ζ) is homologous to the cycle ζ + {z uˆ : u ∈ Rn+1 , u ∈ {η}⊥ \ {0}, z ∈ C, |z| = ε# }

(4.18)

zC ω(dz) is homogeneous on Cn+1 |z|n+1 C and so defines a closed n-form on complex projective space CPn . Because the (n+1) images of the cycle (4.18) and ζ+Sn (ε) in CPn are homologous and ∂Σε (ζ) and (4.18) are homologous, the equality (4.17) is now a consequence of Stokes’ theorem and Proposition 4.2. in κ(Ω) \ N (ζ). Moreover, the n-form

4.2

n even

Let ζ ∈ Cn+1 \ Rn+1 . By equation (4.3), the monogenic spectrum γC (ζ) = N (ζ) ∩ Rn+1 of ζ is an n-dimensional ball in Rn+1 in the hyperplane orthogonal to η = Im ζ (= 0 and passing through ξ = Re ζ. In particular, γC (ζ) is a convex subset of Rn+1 . For each x ∈ Rn+1 \ {ξ}, let Cx (ζ) denote the complex line (4.14). In polar coordinates, we have 2 1 n+1 iθ x − ξ , r ≥ 0, 0 ≤ θ < 2π . Cx (ζ) = z ∈ C : z = ξ + re |x − ξ|

55

For each ε > 0, let 1 2 1 sin2 θ cos2 θ = z ∈ Cx (ζ) : 2 > + , 0 ≤ θ < (4.19) 2π r ε2 (1 + ε2 )|η|2 x∈∂γC (ζ) 2 0 1 sin2 θ cos2 θ 1 (n+1) ∂Σε + , 0 ≤ θ < (4.20) 2π (ζ) = z ∈ Cx (ζ) : 2 = r ε2 (1 + ε2 )|η|2 Σ(n+1) (ζ) ε

0

x∈∂γC (ζ) (n+1)

(n+1)

The ellipsoid ∂Σε (ζ) is an n-cycle in Cn+1 . The set ∂Σε (ζ) is the (n+1) boundary of the open set Σε (ζ) containing γC (ζ) and contained in an (n+1) # n+1 ε -neighbourhood of γC (ζ) in C for ε# > ε; ∂Σε (ζ) has the orientation induced by the (n + 1)-form dζ0 ∧ · · · ∧ dζn . If n is odd and the convex hull of (n+1) γC (ζ) in Rn+1 is contained in Ω, then the cycles ∂Σε (ζ) defined by (4.16) and (4.20) are homologous in κC (Ω) \ N (ζ). A proof similar to the case for n odd gives the following result. Proposition 4.4. Suppose that n ∈ N is even and Ω is a nonempty connected open subset of Rn+1 . Let ζ ∈ κ(Ω) \ Ω. If f : κ(Ω) → C"(Cn ) is a complex left monogenic function, then there exists ε > 0 such that (n+1) ∂Σε (ζ) ⊂ κ(Ω) \ N (ζ) and ! f (ζ) = Gz (ζ)ω(dz)f (z). (4.21) (n+1)

∂Σε

(ζ)

(n+1)

In the case of n odd or even, for ζ = ξ+iη ∈ κ(Ω)\Ω, the set Σε (ζ) lies in a small neighbourhood of γC (ζ) contained in the intersection of κ(Ω)\N (ζ) with the n-dimensional complex hyperplane ξ + C{η}⊥ in Cn+1 . Because the representations (4.17) and (4.21) depend only on the values of f in an ndimensional complex hyperplane in Cn+1 , they may be modified to obtain the complex monogenic extension to Cn+1 of a holomorphic function defined on an open subset of Cn . We show how this is done for the case of sectors in the next section.

56

5

Holomorphic and Monogenic Functions on Sectors

Let Sν (Rn+1 ) = {x ∈ Rn+1 : x = x0 e0 + x, |x0 | ≤ tan ν|x| }, Sν◦ (Rn+1 ) = {x ∈ Rn+1 : x = x0 e0 + x, |x0 | < tan ν|x| }. It is clear that if ζ = ξ + iη lies in a sector in Cn , say, |η| ≤ |ξ| tan ν, then the monogenic spectrum γ(ζ) lies in a corresponding sector in Rn+1 . More precisely, we have Proposition 5.1 ( [4, Proposition 6.10]). Let ζ ∈ Cn \{0} and 0 < ω < π/2. Then γ(ζ) ⊂ Sω (Rn+1 ) if and only if |ζ|2C (= (−∞, 0] and | Im(ζ)| ≤ Re(|ζ|C ) tan ω.

(5.1)

Let Sω (Cn ) denote the set of complex vectors ζ ∈ Cn satisfying the inequality (5.1), that is, the complex sector Sω (Cn ) = {ζ ∈ Cn : γ(ζ) ⊂ Sω (Rn+1 )}

(5.2)

in Cn . The interior of Sω (Cn ) is written as Sω◦ (Cn ). For n = 1, we have Sω (C) = {z ∈ C : | Im z| ≤ tan ω| Re z| }, Sω◦ (C) = {z ∈ C : | Im z| < tan ω| Re z| }. Theorem 5.1. Let n be a nonnegative integer and 0 < ω < π/2. If f˜ is a C"(Cn )-valued holomorphic function defined on Sω◦ (Cn ), then there exists a unique left monogenic function f defined on Sω◦ (Rn+1 ) such that f˜(x) = f (x1 e1 + · · · + xn en ) for all x ∈ Rn , x (= 0. The linear map f˜ "−→ f is continuous for the compact-open topology. Sketch of the Proof. We describe the case that n is an odd integer. The case of n even is similar but a little more complicated. A calculation like that in (4.3) shows that for each ζ ∈ Cn+1 , z ∈ Cn and θ ∈ R, the intersection N (ζ) ∩ (z + eiθ Rn ) is either empty, a point or an (n − 2) sphere contained in a real (n − 1) dimensional hyperplane Hz,ζ,θ in Cn . Suppose that inf diam(N (ζ) ∩ (z + eiθ Rn )) > 2ε > 0, θ

57

so that the radius of the (n − 2) sphere N (ζ) ∩ (z + eiθ Rn ) is bounded below by ε as θ ∈ R varies. For each u ∈ N (ζ) ∩ (z + eiθ Rn ) with u = z + eiθ x, let Cu (z, ζ, θ) denote the 2 dimensional plane in Cn passing through u and the ⊥ centre of N (ζ) ∩ (z + eiθ Rn ) and parallel to Hz,ζ,θ in z + eiθ Rn . Then 0 0 Γ(n+1) (ζ, z) = {w ∈ Cu (z, ζ, θ) : |w − u| < ε (5.3) }, ε θ∈(−π,π] u∈N (ζ)∩(z+eiθ .Rn )

∂Γ(n+1) (ζ, z) = ε

0

0

θ∈(−π,π] u∈N (ζ)∩(z+eiθ .Rn )

{w ∈ Cu (z, ζ, θ) : |w − u| = ε (5.4) }.

(n+1)

The subset Γε (ζ, z) of Cn is a real (n + 1)-dimensional manifold embed(n+1) ded in Cn with an n-dimensional boundary ∂Γε (ζ, z). Furthermore, the formula ! # f (ζ ) = Gw (ζ # )ω(dw)f˜(w) (5.5) (n+1)

∂Γε

(ζ,z)

defines a complex left monogenic function for every ζ # ∈ Cn+1 such that (n+1) (n+1) N (ζ # ) intersects Γε (ζ, z) but is disjoint from ∂Γε (ζ, z) [16]. Because f˜ has a unique complex left monogenic extension from an open subset U of its domain in Cn to some neighbourhood of U in Cn+1 [16, p. 422], by Stokes’ theorem the same representation holds for all orientable n-dimensional mani(n+1) folds homologous in Cn \ N (ζ # ) to ∂Γε (ζ, z). As ζ ∈ Cn+1 and z ∈ Sω◦ (Cn ) vary, we obtain a well-defined complex left monogenic function f [16, p. 418]. Let Uω (Cn+1 ) denote the set of all ζ ∈ Cn+1 for which there exists z ∈ Sω◦ (Cn ) such that inf diam(N (ζ) ∩ (z + eiθ Rn )) > 0 and N(ζ) ∩ (z + eiθ Rn ) ⊂ S◦ω (Cn ), ∀θ ∈ R. θ

(5.6) By virtue of the formula (5.5), we obtain a unique complex left monogenic extension of f˜ to Uω (Cn+1 ) for which the linear map f˜ "−→ f is continuous for the compact-open topology for holomorphic functions defined on Sω◦ (Cn ), to the space of complex left monogenic functions defined on Uω (Cn+1 ). To complete the proof, it suffices to show that Sω◦ (Rn+1 ) \ Rn ⊂ Uω (Cn+1 ) ∩ (Rn+1 \ Rn ). Let x = x0 e0 + x ∈ Sω◦ (Rn+1 ) with x0 (= 0. Then x (= 0. Let z = x(1 + i tan β) for tan−1 (|x0 |/|x|) < β < ω. The centre of the (n − 2)-sphere N (x) ∩ (z + eiθ Rn ) is $ # x20 iθ z − e sin θ Im z 1 − | Im z|2 58

. and it has radius (sin2 θx20 + cos2 θ| Im z|2 )(1 − x20 /| Im z|2 ) > 0, so that for any ε > 0, there exists tan−1 (|x0 |/|x|) < β < ω such that N (x) ∩ (z + eiθ Rn ) is contained in a ball of radius ε about z in Cn , for every θ ∈ R. Because |z|C = |x|(1 + i tan β), an application of Proposition 5.1 guarantees that z ∈ Sω◦ (Cn ). With a sufficiently small choice of β > tan−1 (|x0 |/|x|), we can ensure that 0 γ(N (x) ∩ (z + eiθ Rn )) ⊂ S◦ω (Rn+1 ). θ∈R

This shows that

Sω◦ (Rn+1 )

\ Rn ⊂ Uω (Cn+1 ) ∩ (Rn+1 \ Rn ). Corollary 5.1. The linear map f "−→ f˜ defined by formula (1.2) maps the space of all left monogenic functions defined on Sω◦ (Rn+1 ) bijectively onto the space of all C"(Cn )-valued holomorphic functions defined on Sω◦ (Cn ). It is continuous between the compact-open topologies on each space.

A C"(Cn )-valued real analytic function defined on an open subset U of Rn necessarily has a left monogenic extension to an open subset of Rn+1 containing U by taking an expansion about each point of U in monogenic polynomials [2, Theorem 14.8]. In particular, the product φ.ψ : V ∩ Rn → C"(Cn ) of two left monogenic functions defined in an open subset V of Rn+1 intersecting Rn ≡ {0} × Rn has a unique left monogenic extension φ ·) ψ to a neighbourhood of V ∩ Rn called the (left) Cauchy-Kowalewski product of φ and ψ. The following corollary shows that in the case V = Sω◦ (Rn+1 ), the product is actually defined on all of V . Corollary 5.2. Let φ, ψ be left monogenic functions defined on Sω◦ (Rn+1 ). Then there exists a unique left monogenic function φ·) ψ defined on Sω◦ (Rn+1 ) such that φ ·) ψ(x) = φ(x).ψ(x) for every x ∈ Rn \ {0}. Proof. Let φ˜ and ψ˜ be the holomorphic counterparts of the left monogenic ˜ ψ˜ functions φ and ψ defined by formula (1.2). Then the product function φ. is certainly a C"(Cn )-valued holomorphic function defined on the open sector Sω◦ (Cn ). According to Corollary 5.1, there exists a unique left monogenic function defined on Sω◦ (Rn+1 ), which we denote by φ ·) ψ, such that (φ ·) ψ)˜= ˜ ψ˜ as holomorphic functions defined on S ◦ (Cn ). An appeal to the Cauchy φ. ω integral formula (2.2) of Clifford analysis ensures that φ ·) ψ(x) = (φ ·) ψ)˜(x) = φ(x).ψ(x)

for every x ∈ Rn \ {0}.

59

6

The analytic functional calculus for systems of operators of type ω

As mentioned in the Introduction, one the difficulties in forming a function f (A) of the n-tuple A = (A1 , . . . , An ) of commuting linear operators acting in a Banach space X for a holomorphic function f of n complex variables, is the absence of a suitably general integral representation formula for functions of several complex variables. Now we formulate an analytic functional calculus f˜ "−→ f˜(A) for holomorphic functions f˜ defined on a sector Sω◦ (Cn ) by using the representation (1.1) with the left monogenic counterpart f : Sω◦ (Rn+1 ) → C"(Cn )) of f˜ obtained from Theorem 5.1 above. Suppose that T : D(T ) −→ H is a single closed densely defined linear operator acting in the Hilbert space H. The spectrum of T is denoted by σ(T ). If 0 ≤ ω < π/2, then T is said to be of type ω, if σ(T ) ⊂ Sω (C) and for each ν > ω, there exists Cν > 0 such that 2(zI − T )−1 2 ≤ Cν |z|−1 ,

z∈ / Sν (C).

(6.1)

Then the bounded linear operator f (T ) is defined by the Riesz-Dunford formula ! 1 f (T ) = (λI − T )−1 f (λ) dλ. (6.2) 2πi C for any function f satisfying the bounds |f (z)| ≤ Kν

|z|s , 1 + |z|2s

z ∈ Sν◦ (C).

The contour C can be taken to be {z ∈ C : | Im(z)| = tan θ| Re(z)| }, with ω < θ < ν. The operator T of type ω is said to have a bounded H ∞ -functional calculus if for each ω < ν < π/2, there exists an algebra homomorphism f "−→ f (T ) from H ∞ (Sν◦ (C)) to L(H) agreeing with (6.2) and a positive number Cν such that 2f (T )2 ≤ Cν 2f 2∞ for all f ∈ H ∞ (Sν◦ (C)). The following result is from [10, Theorem 6.2.2] Theorem 6.1. Suppose that T is a one-to-one operator of type ω in H. Then T has a bounded H ∞ -functional calculus if and only if for every ω < ν < π/2, there exists cν > 0 such that T and its adjoint T ∗ satisfy the square function 60

estimates

!

∞

dt ≤ cν 2u22 , t ! 0∞ dt 2ψt (T ∗ )u22 ≤ cν 2u22 , t 0 2ψt (T )u22

u ∈ H,

(6.3)

u ∈ H,

(6.4)

for some function (every function) ψ ∈ H ∞ (Sν◦ (C)), which satisfies ! ∞ ! ∞ dt dt 3 = ψ 3 (−t) = 1, and ψ (t) t t 0 0 s |z| |ψ(z)| ≤ Cν , z ∈ Sν◦ (C), 1 + |z|2s

(6.5) (6.6)

for some s > 0. Here ψt (z) = ψ(tz) for z ∈ Sν◦ (C). We now use formula (1.1) to generalise this result to n-tuples of commuting operators acting in a Hilbert space H. The (n − 1)-sphere in Rn is denoted by Sn−1 . The set of s ∈ Sn−1 with nonzero coordinates sj for every j = 1, . . . , n is denoted by Sn−1 . Then Sn−1 0 0 is a dense open subset of Sn−1 with full surface measure. Definition 6.1. Let X be a Banach space and let A = (A1 , . . . , An ) be an n-tuple of densely defined linear operators Aj : D(Aj ) −→ X acting in X such that ∩nj=1 D(Aj ) is dense in X and let 0 ≤ ω < π2 . Then A is said to be uniformly of type ω if for all s ∈ Sn−1 , the densely defined operator 0 "n %A, s& := j=1 Aj sj is closed, σ(%A, s&) ⊂ Sω (C) and for each ν > ω, there exists Cν > 0 such that 2(zI − %A, s&)−1 2 ≤ Cν |z|−1 ,

z∈ / Sν (C), s ∈ Sn−1 . 0

(6.7)

It follows that s "−→ %A, s& is continuous on Sn−1 in the sense of strong 0 resolvent convergence [7, Theorem VIII.1.5]. If A is uniformly of type ω, it turns out that we can define the Cauchy kernel Gx (A) for the n-tuple A by the plane wave formula # $n ! (n − 1)! i n−1 sgn(x0 ) (e0 +is) (%x, s&I − %A, s& − x0 sI)−n ds Gx (A) = 2 2π n−1 S (6.8) n for all x = x0 e0 + x with x0 a nonzero real number, x ∈ R and x ∈ / n+1 n−1 n Sω (R ). Here S is the unit (n − 1)-sphere in R , ds is surface measure 61

and the inverse power (%xI − A, s& − x0 sI)−n is taken in the Clifford module L(X) ⊗ C"(Cn ), which is identified with the set L(n) (X(n) ) of all left module homomorphisms of X(n) = X ⊗ C"(Cn ), see [4, Equation (6.14)]. Suppose that 0 < ω < ν < π/2 and f is a left monogenic function defined on Sν◦ (Rn+1 ) such that for every ω < θ < ν there exists Cθ > 0 and s > 0 such that |f (x)| ≤ Cθ

|x|s , 1 + |x|2s

x ∈ Sθ◦ (Rn+1 ) \ Sω (Rn+1 ).

(6.9)

Now if ω < θ < ν and Hθ = {x ∈ Rn+1 : x = x0 e0 + x, |x0 |/|x| = tan θ} ⊂ S◦ν (Rn+1 ),

(6.10)

then it is easy to check that x "−→ Gx (A)n(x)f (x) is integrable with respect to n-dimensional surface measure µ on Hθ . Therefore, we define ! f (A) = Gx (A)n(x)f (x) dµ(x). (6.11) Hθ

The hypersurface Hθ can be varied in the region where x "−→ Gx (A) is two-sided monogenic in the Clifford module L(X) ⊗ C"(Cn ) and f is left monogenic in Sν◦ (Rn+1 ). Theorem 6.2. Let A = (A1 , . . . , An ) be an n-tuple of densely defined commuting linear operators Aj : D(Aj ) −→ H acting in a Hilbert space H such that ∩nj=1 D(Aj ) is dense in H. Suppose that 0 ≤ ω < π2 and A is uniformly of type ω. Suppose in addition, that T = i(A1 e1 +· · ·+An en ) is a one-to-one operator of type ω acting in H(n) = C"(Cn ) ⊗ H and T has an H ∞ -functional calculus. Then the n-tuple A has a bounded H ∞ -functional calculus on Sν◦ (Cn ) for any ω < ν < π/2, that is, there exists a homomorphism b "−→ b(A), b ∈ H ∞ (Sν◦ (Cn )), from H ∞ (Sν◦ (Cn )) into L(n) (H(n) ) and there exists Cν > 0 such that 2b(A)2 ≤ Cν 2b2∞ for all b ∈ H ∞ (Sν◦ (Cn )). Moreover, if f is the unique two-sided monogenic function defined on ◦ Sν (Rn+1 ) such that b = f˜, as in Theorem 5.1, and there exists Cν > 0, s > 0 such that |ζ|s , ζ ∈ Sν◦ (Cn ), |b(ζ)| ≤ Cν 1 + |ζ|2s

then f satisfies the bound (6.9) and b(A) = f (A) is given by formula (6.11). 62

Proof. By assumption, the operator T has an H ∞ -functional calculus, so there exists a function ψ ∈ H ∞ (Sν◦ (C)) satisfying conditions (6.3) – (6.6). In [3] a special choice of ψ was made, but we now see that this is not necessary. Following [10, Theorem 6.4.3], our aim is to define b(A) for b ∈ H ∞ (Sν◦ (Cn )), by the formula $ ! ∞# dt ∗ (b(A)u, v) = (bφt )(A)ψt (T )u, ψt (T ) v (6.12) t 0 for all u, v ∈ H(n) . The function φ : Sν◦ (Cn ) −→ C is constructed from ψ by setting φ(ζ) = ψ{iζ} = ψ(|ζ|C )χ+ (ζ) + ψ(−|ζ|C )χ− (ζ), for all ζ ∈ Sν◦ (Cn ), by the functional calculus for multiplication by iζ. Let φt (ζ) = φ(tζ) for all t > 0 and ζ ∈ Sν◦ (Cn ). Let b.) φt be the left monogenic function defined on Sν◦ (Rn+1 ) by Corollary 5.2, that is, the Cauchy-Kowalewski product of the left monogenic counterparts of b and φt . Moreover, the proof of Theorem 5.1 shows that b.) φt satisfies the bound (6.9) with Cθ proportional to 2b2∞ , so (bφt )(A) = b.) φt (A) is defined by formula (6.11) and we have $3 ! ∞ 3# 3 3 ∗ 3 (bφt )(A)ψt (T )u, ψt (T ) v 3 dt 3 3 t 0 21/2 1! ∞ 21/2 1! ∞ ∗ 2 dt 2 dt 2ψt (T ) v2 ≤ sup 2(bφt )(A)2 2ψt (T )u2 t t t>0 0 0 # ≤ C 2b2∞ 2u2 2v2. 4∞ Because 0 ψ 3 (t) dtt = 1, we obtain the desired functional calculus by analogy with [10, Theorem 6.4.3]. The assumptions of Theorem 6.2 are satisfied if the n-tuple A consists of differentiation operators on a Lipschitz surface [9]. The commutativity of the operators is most easily seen from the representation in [9, p. 708].

References [1] D. Albrecht, Integral formulae for special cases of Taylor’s functional calculus, Studia Math. 105 (1993), 51–68. 63

[2] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Research Notes in Mathematics 76, Pitman, Boston/London/Melbourne, 1982. [3] B. Jefferies, Function Theory in Sectors, Studia Math., 163 (2004), 257–287. [4] B. Jefferies, Spectral Properties of Noncommuting Operators, Lecture Notes in Mathematics 1843, Springer-Verlag, Berlin/Heidelberg/New York, 2004. [5] B. Jefferies and A. McIntosh, The Weyl calculus and Clifford analysis, Bull. Austral. Math. Soc. 57 (1998), 329–341. [6] B. Jefferies, A. McIntosh and J. Picton-Warlow, The monogenic functional calculus, Studia Math. 136 (1999), 99–119. [7] T. Kato, Perturbation Theory for Linear Operators, 2nd Ed., SpringerVerlag, Berlin/Heidelberg/New York, 1980. [8] S.G. Krantz, Function Theory of Several Complex Variables, 2nd Ed., Amer. Math. Soc., Providence, 1992. [9] C. Li, A. McIntosh, T. Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Rev. Mat. Iberoamericana 10 (1994), 665–721. [10] C. Li, A. McIntosh, Clifford algebras and H∞ functional calculi of commuting operators, Clifford algebras in analysis and related topics (Fayetteville, AR, 1993), 89–101, Stud. Adv. Math., CRC, Boca Raton, FL, 1996. [11] A. McIntosh, Operators which have an H∞ -functional calculus, in: Miniconference on Operator Theory and Partial Differential Equations 1986, 212–222 Proc. Centre for Mathematical Analysis 14, ANU, Canberra, 1986. [12] A. McIntosh, Clifford algebras, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains, Clifford algebras in analysis and related topics (Fayetteville, AR, 1993), 33–87, Stud. Adv. Math., CRC, Boca Raton, FL, 1996. 64

[13] A. McIntosh and A. Pryde, A functional calculus for several commuting operators, Indiana U. Math. J. 36 (1987), 421–439. [14] V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Operator Theory Adv. Appl. 139, Birkhaüser, BaselBoston-Berlin, 2003. [15] R.M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Graduate Texts in Mathematics 108, Springer-Verlag, New York, 1986. [16] J. Ryan, Complex Clifford analysis and domains of holomorphy, J. Austral. Math. Soc. Ser. A 48 (1990), 413–433. [17] J. Ryan, Cells of harmonicity and generalized Cauchy integral formulae, Proc. London Math. Soc. 60 (1990), 295–318. [18] J.L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970), 1–38. Brian Jefferies, School of Mathematics, UNSW, Kensington 2052 NSW, Australia. [email protected]

65

On an operator-valued T (1) theorem by Hytönen and Weis Cornelia Kaiser Abstract We consider generalized Calderón-Zygmund operators whose kernel takes values in the space of all continuous linear operators between two Banach spaces. In the spirit of the T (1) theorem of David and Journé we prove boundedness results for such operators on vector-valued Riesz potential spaces. This improves and generalizes a result by Hytönen and Weis. MSC (2000): Primary: 42B20, 46E40; Secondary: 46E35. Keywords: vector-valued function spaces, Calderón-Zygmund operators, T (1) theorem. Received 30 July 2006 / Accepted 8 December 2006.

1

Introduction

In this paper we want to study non-convolution type singular integrals of the form ! (T f )(u) = K(u, v)f (v)dv (1.1) RN

(see e.g. [19, 20]). Inspired by the famous T (1) theorem of G. David and J.-L. Journé [4, 5] on the L2 boundedness of operators (1.1) T. Figiel [8] has proved a T (1) theorem for X-valued Lp functions, where X has the UMD-property. The kernels K are still scalar valued, however. Recently T. Hytönen and L. Weis [13] gave a new proof of Figiel’s T (1) theorem and extended it to operator-valued kernels K. On the other hand, various authors (e.g. [9, 11, 18, 21, 23]) obtained results of the same spirit as the T (1) theorem of David and Journé for other scalar-valued function spaces, including homogeneous Besov, TriebelLizorkin and Riesz potential spaces. In [12] T. Hytönen and the author give 66

a new prove of the T (1) theorem for Triebel-Lizorkin spaces, using methods from [13]. And in [15] we show an operator-valued version of the T (1) theorem for vector-valued Besov spaces. In this paper, we prove the following operator-valued T (1) theorem for vector-valued Riesz potential spaces. The definition of these spaces, as well as of the various conditions appearing in the theorem, are given in Section 2. The theorem is then proved in Section 3. Theorem 1.1. Let X, Y be UMD spaces. Suppose that T : S(RN ) → S! (RN , L(X, Y )) is in the class RCZOν for some ν ∈ (0, 1], satisfies the weak R-boundedness property, and T (1) = 0. (a) T extends to a bounded linear operator from H˙ ps (X) to H˙ ps (Y ) for each s ∈ (0, ν) and each p ∈ (1, ∞). (b) If in addition T ! ∈ RCZOν and T ! (1) = 0, then T extends to a bounded linear operator from H˙ ps (X) to H˙ ps (Y ) for each |s| < ν and each p ∈ (1, ∞). Theorem 1.1 contains the Lp result by Hytönen and Weis from [13] as a special case (s = 0). However, we use a weaker version of the weak Rboundedness property, which slightly improves their theorem.

2

Definitions and Notations

Throughout this paper X and Y are complex Banach spaces. The space L(X, Y ) of bounded linear operators from X to Y is endowed with the uniform operator topology. X ! = L(X, C) denotes the dual space of X. All our (possibly vector-valued) functions and distributions will be defined on RN for a fixed positive integer N . Therefore the various function spaces E(RN , X) in this paper are denoted simply by E(X). For example we write Lp (X) for the Bochner-Lebesgue space Lp (RN , X), p ∈ [1, ∞], equipped with its usual norm. N := {0, 1, 2, . . . } is the set of all nonnegative integers. The conjugate exponent p! of p ∈ [1, ∞] is given by p1 + p1! = 1. We write D(RN , X) for the space of all compactly supported smooth functions with values in X. The Schwartz class S(RN , X) is the space of all Xvalued rapidly decreasing smooth functions, endowed with its usual topology. For D(RN , C) and S(RN , C) we also write D(RN ) and S(RN ) respectively. The space S! (RN , X) of all X-valued tempered distributions is defined as the 67

space of all continuous linear operators from S(RN ) to X. The trans# Fourier −iuv N N ϕ(v)dv. form F : S(R ) → S(R ) is defined by (Fϕ)(u) = ϕ(u) " = RN e Let Z(RN , X) be the space of all Schwartz functions ϕ ∈ S(RN , X) such that Dα ϕ(0) " = 0 for all multiindices α ∈ NN . Then Z(RN , X) is a closed subspace of S(RN , X). If Z! (RN , X) denotes the space of all continuous linear operators from Z(RN ) = Z(RN , C) to X, then S! (RN , X)/P(RN , X) and Z! (RN , X) are isomorphic (cf. [22, 5.1.2] and [14, Section 7]). Here P(RN , X) stands for the space of all polynomials on RN with coefficients in X.

Riesz potential spaces Let p ∈ (1, ∞) and s ∈ R. The Riesz potential spaces H˙ ps (X) = H˙ ps (Rn , X) is the space consisting of all f ∈ Z! (RN , X) such that $f $H˙ ps (X) = $F−1 (| · |s f"(·))$Lp (X)

is finite. Note that F−1 (|·|s f"(·)) maps Z! (RN , X) onto itself and that $·$H˙ ps (X) is a norm on H˙ ps (X). Let φ" ∈ D(RN ) be radial, equal to 1 in B(0, 1), and supported in B(0, 2). " Let ϕ " = φ" − φ(2·) and ϕ "j = ϕ(2 " j ·), j ∈ Z. It follows from the results in [16] that, if p ∈ (1, ∞) and X is a UMD space (for a definition see below), then $ $ $f $ ˙ s,p := F (X) 2

%! 1 $& $2 $ $ −js r (t)2 f ∗ ϕ $ j j$ 0

j∈Z

Lp (X)

'1/2 dt .

is an equivalent norm on H˙ ps (X). Here (rj ) is some sequence of distinct Rademacher functions.

The operator T We consider a continuous linear operator T : S(RN ) → S! (RN , L(X, Y )). T can be identified with the continuous bilinear form S(RN ) × S(RN ) → L(X, Y ),

(ϕ, ψ) (→ (T ϕ)(ψ). 68

( ) In place of (T ϕ)ψ we also use the notation ψ, T ϕ . To T we assign an “adjoint” operator ( ) ( )! T ! : S(RN ) → S! (RN , L(Y ! , X ! )), ψ, T ! ϕ := ϕ, T ψ ,

where the latter ! designates the usual Banach adjoint of an operator in L(X, Y ). From T we derive a linear mapping T* : S(RN ) ⊗ X → S! (RN , Y ) : for x ∈ X and ϕ, ψ ∈ S(RN ), we let ( ) ( ) ψ, T*[ϕ ⊗ x] := ψ, T ϕ x ∈ Y.

( ) This makes sense, since ψ, T ϕ ∈ L(X, Y ). So T*[ϕ ⊗ x] is a Y -valued tempered distribution and T* is well-defined on S(RN ) × X. Now we extend T* to S(RN ) ⊗ X by linearity. In the following we will not distinguish between T and T*.

The associated kernel

Suppose now that K : {(u, v) ∈ RN × RN : u *= v} → L(X, Y ) is continuous. We say that T is associated with K if ! ! ( ) ϕ, T φ = ϕ(u) K(u, v)φ(v)dv du (2.1) RN

RN

holds for all ϕ, φ ∈ D(RN ) with suppϕ ∩ suppφ = Ø. This means that, for each φ ∈ D(RN ), the distribution T φ agrees almost everywhere on the # complement of suppφ with the continuous function RN K(·, v)φ(v)dv, defined on the complement of suppφ. It is clear from (2.1) that T ! is associated to K ! given by K ! (u, v) = K(v, u)! for u *= v. Now we assume that K satisfies the standard estimates $ $ + , (SE0 ) sup |u − v|N $K(u, v)$ : u *= v < ∞, . N +ν $K(u, v) − K(u0 , v)$ : |u − v| > 2|u − u0 | > 0 < (SEν ) sup |u − v| |u − u0 |ν ∞. for some ν ∈ (0, 1]. We say that T ∈ CZOν if T is associated with K satisfying (SE0 ) and (SEν ). Note that T ∈ CZOν does not emply that T ! ∈ CZOν . 69

Definition of T (1) The action of T ∈ CZOν is not a priori defined on the constant function 1∈ / S(RN ), but we can make sense of the notion T (1): We will define T (1) as a linear operator acting on ! 0 N N D (R ) := {ϕ ∈ D(R ) : ϕ(u)du = 0}. RN

For doing this we first observe that, if ϕ ∈ D0 (RN ), the distribution T ! ϕ agrees with an integrable function on the exterior of any neighborhood of suppϕ. Now choose ψ ∈ D(RN ) such that ψ ≡ 1 in a neighborhood of suppϕ and define ! ( ) ( ) ! ! 1, T ϕ := ψ, T ϕ + (1 − ψ(u))(T ! ϕ)(u)du. RN

Here the first term is given by the usual pairing between test functions and distributions and the second term exists because T(! ϕ is integrable on the ) ! support of 1 − ψ. One can show that the value of 1, T ϕ is (independent ) of the actual choice of ψ. Now we make the natural definition ϕ, T (1) := ( )! / 1, T ! ϕ /X ∈ L(X, Y ).

The weak boundedness property

The closed ball with center u ∈ RN and radius r > 0 is denoted by B(u, r). We say that ϕ is a normalized bump function associated with the unit ball B(0, 1) if ϕ ∈ D(RN ) with suppϕ ⊆ B(0, 1) and $Dα ϕ$L∞ ≤ 1 for all |α| ≤ M , where M is a large fixed number. φ is a normalized bump function associated with the ball B(u, r) if φ(·) = r−N ϕ(r−1 (· − u)), where ϕ is a normalized bump function associated with the unit ball. The operator T is said to have • the weak boundedness property provided that, for every pair of normalized bump functions ϕ, φ associated with any ball B(u, r) we have $ ( )$ $ φ, T ϕ $ ≤ Cr−N .

• the weak boundedness property at 0 provided that, for every pair of normalized bump ϕ, φ associated with any ball B(0, r) we $( )$ functions −N $ $ have φ, T ϕ ≤ Cr . 70

The following lemma is a refinement of an auxiliary result in [13, Sect. 2]. For a proof see [15]. Lemma 2.1. Let k ∈ N, a > 0, w ∈ RN , and let ϕ, φ ∈ D0 (RN ) be normalized bump functions associated with B(0, a) and B(w, 2k a) respectively. Let T ∈ CZOν satisfy the weak boundedness property at 0. (a) There is a constant C1 < ∞ such that % '−N −ν $( )$ 1 + k |w| $ φ, T ϕ $ ≤ C1 1+ k . (a2k )N a2

(b) If in addition T (1) = 0, then there are constants C2 < ∞ and δ > 0 such that % '−N −δ $( )$ |w| k −N −ν $ ϕ, T φ $ ≤ C2 (a2 ) 1+ k . a2

Notions from Banach space theory

For our result on Riesz potential spaces we will restrict ourselves to Banach spaces that have a certain geometric property, namely the property of Unconditional Martingale Differences (UMD). There are several equivalent definitions for this property (see [2, p.141-142] and the references given there). Here is one of them: Definition 2.1. A Banach space X is a UMD space if and only if the Hilbert transform ! f (v) dv, f ∈ S(R, X), (Hf )(u) = PV − u−v

extends to a bounded linear operator on Lp (R, X) for some (and thus for each) p ∈ (1, ∞). Remark 2.1. (a) It is clear from the definition that each Hilbert space is a UMD space. The dual space and each closed subspace of a UMD space is a UMD space. A UMD space X always has a uniformly convex renorming [1] and therefore is super-reflexive [7]. In particular, '1 is not finitely representable in X. Hence X is B-convex [6, Theorem 13.6]. (b) Let (Ω, Σ, µ) be a σ-finite measure space. If X is a UMD space and p ∈ (1, ∞), then Lp (Ω, µ, X) is also a UMD space [2, p.145]. Next we recall the notion of R-boundedness.

71

Definition 2.2. Let X, Y be Banach spaces. A set of operators τ ⊆ L(X, Y ) is called R-bounded if there is a constant C < ∞ such that for all m ∈ N, all T1 , . . . , Tm ∈ τ and all x1 , . . . xm ∈ X we have that m $& $ $ $ rk Tk xk $ $ k=1

L2 ([0,1],Y )

m $& $ $ $ rk x k $ ≤ C$ k=1

L2 ([0,1],X)

,

(2.2)

where rk is the k-th Rademacher function on [0, 1]. The infimum over all C such that (2.2) holds is called the R-bound of τ and is denoted by R(τ ). It is clear from the definition that R-boundedness implies uniform boundedness. But in general the notion of R-boundedness is stronger than that of uniform boundedness. In fact, G. Pisier proved that every bounded subset of L(X) is R-bounded if and only if X is isomorphic to a Hilbert space (cf. [3]). But R-boundedness is equivalent to uniform boundedness between the so called Rademacher spaces, which we define now. Definition 2.3. For a Banach space X, the Rademacher space RadX is the 0m closure in L2 ([0, 1], X) of the subspace of all finite linear combinations k=1 rk xk , where rk are Rademacher functions and xk are elements of X.

For T1 , . . . , Tm ∈ L(X, Y ), the operator [Tk ]m k=1 : RadX → RadY is defined by m m & & [Tk ]m : r x → ( rk Tk xk . k k k=1 k=1

k=1

With this definition it is immediate that τ ⊆ L(X, Y ) is R-bounded if and only if {[Tk ]m k=1 : m ∈ Z+ , T1 , . . . , Tm ∈ τ } is a bounded subset of L(RadX, RadY ).

Proposition 2.1. [17, Proposition 3.5] Let τ be a R-bounded subset of L(X, Y ). If X is B-convex, then {T ! : T ∈ τ } ⊆ L(Y ! , X ! ) is R-bounded. In particular, if X and Y are UMD spaces, then τ ⊆ L(X, Y ) is Rbounded if and only if {T ! : T ∈ τ } ⊆ L(Y ! , X ! ) is R-bounded.

The class RCZOν For a kernel K : {(u, v) ∈ RN × RN : u *= v} → L(X, Y ) and a real number ν ∈ (0, 1] we consider the standard R-estimates 72

$ $ 1+ ,2 (SRE0 ) R |u − v|N $K(u, v)$ : u *= v < ∞, %.' N +ν $K(u, v) − K(u0 , v)$ (SREν ) R |u−v| : |u−v| > 2|u−u0 | > 0 < |u − u0 |ν ∞. We say that T ∈ RCZOν if T is associated with K satisfying (SRE0 ) and (SREν ). It is clear from the definition that the class RCZOν is contained in CZOν . If X, Y are Hilbert spaces then the two classes coincide.

The weak R-boundedness property For r > 0 and w ∈ RN we define the dilation and translation operators on S(RN ) by δr ϕ = r−N/2 ϕ(r−1 ·), τw ϕ = ϕ(· − w).

Moreover we define the continuous linear operator Twr : S(RN ) → S! (RN , L(X, Y )) by ( ) ( ) φ, Twr ϕ = τw δr φ, T [τw δr ϕ] , ϕ, φ ∈ S(RN ). With this definition we can reformulate the weak boundedness property as follows: The operator T has the weak boundedness property if and only if for bump , functions φ, ϕ associated with the unit ball, the set +( all rnormalized ) φ, Tw ϕ : w ∈ R, r > 0 is bounded. The operator T is said to have the weak R-boundedness property provided that there is a constant C such that, for every pair of normalized1+( bump func) j tions ,2 ϕ, φ associated with the unit ball, we have that supv∈RN R φ, Tv2 ϕ : j ∈ Z ≤ C. If T is a singular integral operator with associated kernel K and φ, ϕ are test functions with disjoint support, then ! ! ) ( ) ( ·−w −N r −N ·−w φ( u−w )K(u, v)ϕ( v−w )dv du φ, Tw ϕ = r φ( r ), T [ϕ( r )] = r r r N N R R ! ! = rN φ(u)K(ru + w, rv + w)ϕ(v)dv du. Rn

RN

Therefore Twr is also a singular integral operator associated with the kernel Kwr given by Kwr (u, v) = rN K(ru + w, rv + w). 73

Corollary 2.1. Let a > 0, k ∈ N, w ∈ RN , and let ϕ, φ ∈ D0 (RN ) be normalized bump functions associated with B(0, a) and B(w, 2k a) respectively. Let T ∈ RCZOν satisfy the weak R-boundedness property. (a) There is a constant C1 < ∞ such that for all v ∈ RN % '−N −ν 1+( ) ,2 |w| 1+k 2j R φ, Tv ϕ : j ∈ Z ≤ C1 1+ k . (a2k )N a2

(b) If in addition T (1) = 0, then there are constants C2 < ∞ and δ > 0 such that for all v ∈ RN , '−N −δ % 1+( ) ,2 |w| 2j k −N −ν . R ϕ, Tv φ : j ∈ Z ≤ C2 (a2 ) 1+ k a2 j

Proof. Let v ∈ Rn be fixed and τv = {Tv2 : j ∈ Z}. For (Sk )m k=1 ⊆ τv consider the continuous linear operator [Sk ] : S(RN ) → S! (RN , L(RadX, RadY )) defined by

(

) 3( )4 ϕ, [Sk ]φ = ϕ, Sk φ ,

ϕ, φ ∈ S(RN ).

Then [Sk ] is in CZOν , with constant not depending on v or the choice of the finite sequence (Sk ). Moreover, [Sk ] satisfies the weak boundedness property at 0, also with independent constant. Finally, if T (1) = 0, then also [Sk ](1) = 0. So we can apply Lemma 2.1.

3

Proof of Theorem 1.1

In the proof of Theorem 1.1, we proceed in a similar way as in [13]. We will decompose our operator T into parts we can handle using Corollary 2.1. For this we use a resolution of unity from [13, Sect. 2]: Take Φ ∈ D0 (RN ) and Ψ ∈ S(RN ) such that Φ is radial and real-valued, both " is supported in " and Ψ " are non-negative, Φ(u) " Φ ≥ 1 for 12 ≤ |u| ≤ 2, Ψ 1 { 2 ≤ |u| ≤ 2}, and & j∈Z

" j u)Ψ(2 " j u) = 1 Φ(2

for all u ∈ RN \ {0}. 74

We write Φj (u) := 2−N j Φ(2−j u), Ψj (u) := 2−N j Ψ(2−j u) and Pj f := Φj ∗ f , Qj f = Ψj ∗ f for f ∈ S! (RN , X). For f ∈ S(RN ) and j, k ∈ Z, 78 5 6! Φk (· − v)f (v)dv (Pj T Pk f )(u) = (Φj ∗ T [Φk ∗ f ])(u) = Φj (u − ·), T ! !R ( ) Kj,k (u, v)f (v)dv = Φj (u − ·), T [Φk (· − v)] f (v)dv = R

R

( ) where Kj,k (u, v) := Φj (· − u), T [Φk (· − v)] ∈ L(X, Y ). (Recall that Φ is radial.) We will consider the operators Tj,k associated with the kernels Kj,k : ! Tj+k,j f = Kj+k,j (u, v)f (v)dv, f ∈ S(RN , X). RN

One can show that Tj,k can be extended to bounded linear operators from Lp (X) to Lp (Y ) for all p ∈ [1, ∞] [15]. Proof. First we consider the case that k ∈ N. Then for g ∈ S(RN ) ⊗ Y ! and f ∈ S(RN ) ⊗ X, we can estimate / / / / /&( / )/ /&( (j+k)s ) ! −(j+k)s / /=/ / g, Q T Q f 2 (T ) Q g, 2 Q f j+k j+k,j j j+k,j j+k j / / / / j∈Z

j∈Z

/ ! 1 5& 8 / & / / (j+k)s ! −(l+k)s = // rj (t)2 (Tj+k,j ) Qj+k g, rl (t)2 Ql f dt// 0

j∈Z

l∈Z

$p! %! 1 $& $ $ (j+k)s ! $ $ ≤ r (t)2 (T ) Q g j j+k,j j+k $ $ 0

j∈Z

Lp! (X ! )

$p %! 1 $& $ $ −(l+k)s $ $ × r (t)2 Q f l l $ $ 0

l∈Z

'1/p! dt

Lp (X)

'1/p dt .

The second factor is bounded by C2−ks $f $H˙ ps by Kahane’s inequality and results from [16]. To estimate the first factor, we observe that ! ! ([Tj+k,j ] Qj+k g)(v) = [Kj+k,j (u, v)]! (Qj+k g)(u)du N !R = 2jN [Kj+k,j (v + 2j u, v)]! (Qj+k g)(v + 2j u)du. RN

75

Since ( )! 2jN [Kj+k,j (v + 2j u, v)]! = 2jN Φj+k (· − v − 2j u), T [Φj (· − v)] ,

and Φj+k (· − v − 2j u) = τv δ2j τu δ2k Φ, Corollary 2.1 (a) is applicable and yields (cf. Proposition 2.1) % '−N −ν 1+ jN ,2 1+k |u| j ! sup R 2 [Kj+k,j (v + 2 u, v)] : j ∈ Z ≤ C 1+ k . (a2k )N a2 v∈RN

On the other hand, by a result of Bourgain ( [10, Lemma 3.5]), $p! %! 1 $& $ $ (j+k)s j $ $ r (t)2 (Q g)(·+2 u) j j+k $ $ 0

Lp! (Y ! )

j∈Z

So

'1/p! dt ≤ C ln(2+2−k |u|)$g$H˙ ps (Y ! ) .

$p! %! 1 $& $ $ (j+k)s ! $ $ r (t)2 (T ) Q g j j+k,j j+k $ $ 0

j∈Z

1+k ≤C (a2k )N

=C(1 + k)

!

!R

N

RN

%

1+

|u| a2k

'−N −ν

Lp! (X ! )

'1/p! dt

ln(2 + 2−k |u|) du $g$H˙ −s ! ! (Y ) p

(1 + |u|)−N −ν ln(2 + a|u|) du $g$H˙ −s ! . ! (Y ) p

Now let −k ∈ N. Then for g ∈ S(RN ) ⊗ Y ! and f ∈ S(RN ) ⊗ X, we can estimate / / / / /&( )/ /&( (j+k)s )/ −(j+k)s / g, Qj+k Tj+k,j Qj f // = // 2 Qj+k g, 2 Tj+k,j Qj f // / j∈Z

j∈Z

$p! %! 1 $& $ $ (j+k)s $ $ ≤ r (t)2 Q g j j+k $ $ 0

Lp! (Y ! )

j∈Z

'1/p! dt

$p %! 1 $& $ $ −(l+k)s $ $ × r (t)2 T Q f l j+k,j l $ $ 0

Lp (Y )

l∈Z

'1/p dt .

Now we proceed in a similar way. We use Corollary 2.1 (b) to show that (j+k)N

sup R(2 u∈RN

Kj+k,j (u, u+2

j+k

|k| −N −ν

v) : j ∈ Z) ≤ C2 (a2 )

%

|v| 1+ |k| a2

'−N −δ

.

76

(Observe that Φj (· − u − 2j v) = τu δ2j+k τv δ2−k Φ and τv δ2−k Φ ∼ B(v, 2|k| a).) Then, using again Bourgain’s result, we obtain $p %! 1 $& $ $ (j+k)s $ $ r (t)2 T Q f j j+k,j j $ $ 0

Lp (Y )

j∈Z

|k| −N −ν −ks

≤C(a2 )

2

|k| −ν −ks

=C(a2 ) 2

!

RN

!

RN

%

|v| 1 + |k| a2

'−N −δ

k∈Z

≤C

j∈Z

0 &

k=−∞

ln(2 + 2k |u|) du $f $H˙ ps (X)

(1 + |v|)−N −δ ln(2 + a|u|) du $f $H˙ ps (X) .

Finally, putting everything together, $ &$ $ $& $ $ $ $ $T f $ ˙ s ≤ Q T Q f j+k j+k,j j $ $ H (Y ) p

'1/p dt

H˙ ps (Y )

2−|k|(n+ν) 2|k|s $f $H˙ ps (X) + C

$ $ ≤ C $f $H˙ s (X) .

∞ & k=1

2−ks (1 + k)$f $H˙ ps (X)

p

Acknowledgements . This research was carried out while the author held a Margarete von Wrangell scholarship at the University of Karlsruhe. The author wants to thank A. McIntosh and the Centre for Mathematics and its Applications at the Australian National University, Canberra, for their kind hospitality. Their support made it possible for me to spend a wonderful month with many interesting and fruitful discussions at Murramarang and Canberra. The author also thanks the organizers of the Research Symposium on Asymptotic Geometric Analysis, Harmonic Analysis and Related Topics for inviting me to give a talk and to contribute to the proceedings.

77

References [1] D. J. Aldous, Unconditional bases and martingales in Lp (F ), Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 117–123. MR 80b:60009 [2] Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston Inc., Boston, MA, 1995, Abstract linear theory. MR 96g:34088 [3] Wolfgang Arendt and Shangquan Bu, The operator-valued Marcinkiewicz multiplier theorem and maximal regularity, Math. Z. 240 (2002), no. 2, 311–343. MR 2003i:42016 [4] Guy David and Jean-Lin Journé, Une caractérisation des opérateurs intégraux singuliers bornés sur L2 (Rn ), C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 18, 761–764. MR 84j:42028 [5]

, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. (2) 120 (1984), no. 2, 371–397. MR 85k:42041

[6] Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. MR 96i:46001 [7] Per Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), vol. 13, 1972, pp. 281–288. MR 49 #1073 [8] Tadeusz Figiel, Singular integral operators: a martingale approach, Geometry of Banach spaces (Strobl, 1989), London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, Cambridge, 1990, pp. 95– 110. MR 94e:42015 [9] M. Frazier, Y.-S. Han, B. Jawerth, and G. Weiss, The T 1 theorem for Triebel-Lizorkin spaces, Harmonic analysis and partial differential equations (El Escorial, 1987), Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 168–181. MR 90j:46035 [10] Maria Girardi and Lutz Weis, Operator-valued Fourier multiplier theorems on Lp (X) and geometry of Banach spaces, J. Funct. Anal. 204 (2003), no. 2, 320–354. MR 2004g:42019 78

[11] Y.-S. Han and Steve Hofmann, T1 theorems for Besov and TriebelLizorkin spaces, Trans. Amer. Math. Soc. 337 (1993), no. 2, 839–853. MR 93h:46038 [12] Tuomas Hytönen and Cornelia Kaiser, New proof of the T (1) theorem for Triebel-Lizorkin spaces, Georgian Math. J. 13 (2006), 485–493. [13] Tuomas Hytönen and Lutz Weis, A T (1) theorem for integral transforms with operator-valued kernel, J. reine angew. Math. 599 (2006), 155–200. [14]

, Singular integrals on Besov spaces, Math. Nachr. 279 (2006), 581–598.

[15] Cornelia Kaiser, Calderon-Zygmund operators with operator-valued kernels on homogeneous Besov spaces, Math. Nachr., to appear. [16] Cornelia Kaiser and Lutz Weis, Wavelet transform for functions with values in UMD spaces, Submitted. [17] N. Kalton, P. C. Kunstmann, and L. Weis, Perturbation and interpolation theorems for the H∞ -calculus with applications to differential operators, Math. Ann. 336 (2006), no. 4, 747–801. [18] Pierre Gilles Lemarié, Continuité sur les espaces de Besov des opérateurs définis par des intégrales singulières, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 4, 175–187. MR 87j:47074 [19] Yves Meyer and Ronald Coifman, Wavelets, Cambridge Studies in Advanced Mathematics, vol. 48, Cambridge University Press, Cambridge, 1997, Calderón-Zygmund and multilinear operators, Translated from the 1990 and 1991 French originals by David Salinger. MR 98e:42001 [20] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. MR 95c:42002 [21] Rodolfo H. Torres, Boundedness results for operators with singular kernels on distribution spaces, Mem. Amer. Math. Soc. 90 (1991), no. 442, viii+172. MR 91g:47044

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[22] Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 86j:46026 [23] Abdellah Youssfi, Continuité-Besov des opérateurs définis par des intégrales singulières, Manuscripta Math. 65 (1989), no. 3, 289–310. MR 91a:47069 Cornelia Kaiser, Institut für Analysis, Universität Karlsruhe, Englerstraße 2, 76128 Karlsruhe, Germany. +49 721 608 8892 [email protected].

80

A remark on the H ∞-calculus Nigel J. Kalton

∗

Abstract If A, B are sectorial operators on a Hilbert space with the same domain and range, and if #Ax# ≈ #Bx# and #A−1 x# ≈ #B −1 x#, then it is a result of Auscher, McIntosh and Nahmod that if A has an H ∞ −calculus then so does B. On an arbitrary Banach space this is true with the additional hypothesis on B that it is almost R-sectorial as was shown by the author, Kunstmann and Weis in a recent preprint. We give an alternative approach to this result. MSC (2000): 47A60. Received 17 August 2006 / Accepted 21 August 2006.

1

Introduction

In [1] the authors showed that if X is a Hilbert space and A, B are sectorial operators with the same domain and range and satisfying estimates #Ax# ≈ #Bx# and

#A−1 x# ≈ #B −1 x#

x ∈ Dom (A) x ∈ Ran (A)

(1.1) (1.2)

then if one of (A, B) admits an H ∞ −calculus then so does the other. Results of this type are useful in applications and were studied in [7] for arbitrary Banach spaces. In that paper, a similar result (Theorem 5.1) is proved under the additional hypothesis that A is almost R-sectorial. In this note we give a rather different approach to this result. We replace the almost R-sectoriality assumption by the technically weaker assumption of almost U-sectoriality, although this is probably not of great significance. However, our approach here is perhaps a little simpler. We also point out ∗

The author was supported by NSF grant DMS-0555670

81

that some additional assumption is necessary in arbitrary Banach spaces; there are examples of sectorial operators A, B satisfying (1.1) and (1.2) but such that only one has an H ∞ −calculus. It is possible to consider estimates on fractional powers and our results can be extended in this direction (as in [7]); however to keep the exposition simple we will not discuss this point. We also point out that our approach is really based on an interpolation method, known as the Gustavsson-Peetre method [5] (see also [4]); but to avoid certain technicalities we have not made this explicit.

2

U-bounded collections of operators

Let X be a complex Banach space. A family T of operators T : X → X is called U-bounded if there is a constant C such that if (xj )nj=1 ⊂ X, (x∗j )nj=1 ⊂ X ∗ , (Tj )nj=1 ⊂ T, n ! j=1

|)Tj xj , x∗j *|

≤ C sup # |aj |=1

n ! j=1

aj xj # sup # |aj |=1

n ! j=1

aj x∗j #.

The best such constant C is called the U-bound for T and is denoted U (T). This concept was introduced in [8]. We recall that T is called R-bounded if there is a constant C such that if (xj )nj=1 ⊂ X, (Tj )nj=1 ⊂ T, (E#

n ! j=1

2 1/2

!j T xj # )

≤ C(E#

n ! j=1

!j xj #2 )1/2 .

Here (!j )nj=1 is a sequence of independent Rademachers. The best such constant C is called the R-bound for T and is denoted R(T). An R-bounded family is automatically U-bounded [8]. We will need the following elementary property: Proposition 2.1. Suppose F : (0, ∞) → L(X) is a continuous function and that T = {F (t) : 0 < t < ∞} is U-bounded with U-bound U (F ). Suppose g ∈ L1 (R, dt/t). Then the family of operators " ∞ dt 0<s 0 such that |f (z)| ≤ C max(|z|δ , |z|−δ ). For f ∈ H0∞ (Σϕ ) where ϕ > ω(A) we can define f (A) by a contour integral, which converges as a Bochner integral in L(X). " 1 f (λ)(λ − A)−1 dλ f (A) = 2πi Γν where Γν is the contour {|t|e−iνsgn t : −∞ < 0 < ∞} and ω(A) < ν < ϕ. We can then estimate #f (A)# by " |dλ| . #f (A)# ≤ Cϕ |f (λ)| |λ| Γν 83

If we have a stronger estimate f ∈ H0∞ (Σϕ )

#f (A)# ≤ C#f #H ∞ (Σϕ )

then we say that A has an H ∞ (Σϕ )−calculus; in this case we may extend the functional calculus to define f (A) for every f ∈ H ∞ (Σϕ ). The infimum of all such angles ϕ is denoted by ωH (A). We will need a criterion for the existence of an H ∞ -calculus. It will be convenient to use the notation fλ (z) = f (λz) and to let u(z) = z(1 + z)−2 so that u ∈ H0∞ (Σϕ ) for all ϕ < π. The following criterion goes back to [2] and [3]. A simple proof is given in [10]. Proposition 3.1. Let A be a sectorial operator and suppose 0 < ϕ < π. Then the following are equivalent: (i) There is a constant C so that " ∞ dt |)uµ (tA)x, x∗ *| ≤ C#x##x∗ # | arg µ| = ϕ, x ∈ X, x∗ ∈ X ∗ . t 0 (ii) A has an H ∞ −calculus with ωH (A) ≤ π − ϕ. Remark. (i) is equivalent by the Maximum Modulus Principle to " ∞ dt | arg µ| ≤ ϕ, x ∈ X, x∗ ∈ X ∗ . |)uµ (tA)x, x∗ *| ≤ C#x##x∗ # t 0

If A is sectorial we can define a closed operator A∗ on X ∗ by A∗ x∗ = x∗ ◦A with domain Dom (A∗ ) consisting of all x∗ such that x → x∗ (Ax) extends to a bounded linear functional on X. Then A∗ need not be sectorial since it need not have dense domain or range. Note that #A∗ x∗ # =

sup %A−1 x%≤1

x∗ ∈ Dom (A∗ )

|)x, x∗ *|

x∈Ran (A)

and

#(A∗ )−1 x# =

sup %Ax%≤1 x∈Dom (A)

|)x, x∗ *|

x∗ ∈ Ran (A∗ ).

Thus if A and B are sectorial operators satisfying (1.1) and (1.2) they will also satisfy Dom (A∗ ) = Dom (B ∗ ), Ran (A∗ ) = Ran (B ∗ ) and #A∗ x∗ # ≈ #B ∗ x∗ #

x∗ ∈ Dom (A∗ )

(3.1) 84

and

#(A∗ )−1 x∗ # ≈ #(B ∗ )−1 x∗ #

x∗ ∈ Ran (A∗ )

(3.2)

If A is a sectorial operator and ϕ > ω(A) we shall that f ∈ H0∞ (Σϕ ) is U-bounded (respectively R-bounded) for A if the family of operators {f (tA) : 0 < t < ∞} is a U-bounded (respectively R-bounded) collection. Proposition 3.2. Suppose A has an H ∞ -calculus and that ϕ > ωH (A). Then for any f ∈ H0∞ (Σϕ ) we have that f is R-bounded (and thus U-bounded) for A. Proof. Suppose ω(A) < ψ < ϕ. Then the map λ → f (λA) is analytic on Σϕ−ψ and extends continuously to the boundary. The operators {f (2k te±i(ϕ−ψ) A)}k∈Z are R-bounded (uniformly in 0 < t < ∞) by Theorem 3.3 of [8] and the result follows by Lemma 3.4 of the same paper. Suppose A is a sectorial operator on X and ϕ > ω(A). We will say that A is almost U-sectorial (respectively almost R-sectorial) if there is an angle ϕ such that the set of operators {λAR(λ, A)2 : | arg λ| ≥ ϕ} is U-bounded (respectively R-bounded). If we define u(z) = z(1+z)−2 this implies that the functions uλ (z) = u(λz) are uniformly U-bounded (respectively uniformly Rbounded) for | arg λ| ≤ π − ϕ. The infimum of such angles is denoted ω ˜ U (A). By Lemma 3.4 of [8] this definition is equivalent to ω ˜ U (A) = π − sup{θ : ue±iθ is U-bounded} or, respectively ω ˜ R (A) = π − sup{θ : ue±iθ is R-bounded}. Proposition 3.3. Suppose A admits an H ∞ -calculus. Then A is almost R-sectorial (and hence almost U-sectorial) and ω ˜ U (A) ≤ ω ˜ R (A) ≤ ωH (A). Proof. This follows from Proposition 3.2. Lemma 3.1. Suppose A is almost U-sectorial and ϕ > ν > ω ˜ U (A). Then ∞ there is a constant C = C(ϕ) so that if f ∈ H0 (Σϕ ) then f is U-bounded for A with U-bound " |dλ| U (f ) ≤ C . |f (λ)| |λ| Γν 85

Proof. Fix ϕ > ψ > ν > ωU (A). We may write f (tA) in the form " 1 f (tλ)λ−1/2 A1/2 (λ − A)−1 dλ. f (tA) = 2πi Γψ Therefore the result follows from Lemma 2.1 once we show that the two families of operators {h(e±iθ tA) : 0 < t < ∞} are U-bounded where θ = π − ψ and h(z) = z 1/2 (1 + z)−1 . Consider g(z) = −i log

1 + iz 1/2 z −π 1/2 1 − iz 1+z

| arg z| < π.

Then g ∈ H0∞ (Σπ ). Furthermore

g ( (z) = z −1/2 (1 + z)−1 − π(1 + z)−2 . Hence ge±iθ ∈ H0∞ (Σψ ). For convenience we consider the case of +θ. Thus if " 1 g(teiθ λ)A(λ − A)−2 dλ Tt = − 2πi Γν

the family of operators {Tt : 0 < t < ∞} is U-bounded, again by Lemma 2.1. Now integration by parts shows that " teiθ ((teiθ λ)−1/2 (1 + teiθ λ)−1 − π(1 + teiθ λ)−2 )λ(λ − A)−1 dλ Tt = 2πi Γν " 1 (h(teiθ λ) − πu(teiθ λ))(λ − A)−1 dλ = 2πi Γν = h(teiθ A) − πu(teiθ A).

Thus it follows that the family {h(teiθ A) : 0 < t < ∞} is U-bounded.

4

The main results

If A is sectorial then the space Dom (A)∩Ran (A) is a Banach space (densely) embedded into X under the norm #Ax#+#A−1 x#+#x#; similarly Dom (A∗ )∩ Ran (A∗ ) is a Banach space embedded into X ∗ under the norm #A∗ x∗ # + #(A∗ )−1 x∗ # + #x∗ #. 86

Theorem 4.1. Suppose A is a sectorial operator. In order that A have an H ∞ -calculus with ωH (A) = ϕ it is necessary and sufficient that: (i) A is almost U-sectorial with ω ˜ U (A) = ϕ. (ii) There exists a constant C1 so that for each x ∈ X there is a continuous function ξ : (0, ∞) → Dom (A) ∩ Ran (A) such that #

N !

k=−N

ak 2jk tj Aj ξ(2k t)# ≤ C1 #x#, j = −1, 0, 1, |ak | ≤ 1, N = 1, 2, . . . , 0 < t < ∞

and

"

∞

dt x∗ ∈ X ∗ . t 0 (iii) There exists a constant C2 so that for each x∗ ∈ X ∗ there is a continuous function ξ ∗ : (0, ∞) → Dom (A∗ ) ∩ Ran (A∗ ) such that ∗

)x, x * =

#

N !

k=−N

)ξ(t), x∗ *

ak 2jk tj (Aj )∗ ξ ∗ (2k t)# ≤ C2 #x∗ #, j = −1, 0, 1, |ak | ≤ 1, N = 1, 2, . . . , 0 < t < ∞

and

∗

)x, x * =

"

0

∞

)x, ξ ∗ (t)*

dt t

x ∈ X.

Proof. Let us assume (i), (ii) and (iii). Suppose |θ| < π − ϕ and #x# ≤ 1, #x∗ # ≤ 1. Let ξ(t), ξ ∗ (t) be chosen according to (ii) and (iii). We define ˜ = tAξ(t) + t−1 A−1 ξ(t) + 2ξ(t), ξ(t)

ξ˜∗ (t) = tA∗ ξ ∗ (t) + t−1 A∗ ξ ∗ (t) + 2ξ ∗ (t).

Thus we have #

N !

˜ k t)# ≤ 3C1 , j = −1, 0, 1, |ak | ≤ 1, N = 1, 2, . . . , 0 < t < ∞ ak 2jk ξ(2

N !

ak 2jk ξ˜∗ (2k t)# ≤ 3C2 , j = −1, 0, 1, |ak | ≤ 1, N = 1, 2, . . . , 0 < t < ∞.

k=−N

and #

k=−N

Note that ξ˜ : (0, ∞) → X and ξ˜∗ : (0, ∞) → X ∗ are both continuous and ˜ ξ(t) = u(tA)ξ(t) 0 ϕ we have " ∞ " ∞" ∞" ∞ dr dt ds dr ∗ | < uµ (rA)x, x > | ≤ |)uµ (rA)ξ(s), ξ ∗ (t)*| r t s r 0 "0 ∞ "0 ∞ "0 ∞ dt ds dr = |)uµ (rtA)ξ(st), ξ ∗ (t)*| t s r 0 0 0

For fixed r, s " ∞ " ∞ dt dt ∗ ˜ |)uµ (rtA)ξ(st), ξ (t)*| = |)uµ (rtA)u(stA)ξ(st), (u(tA))∗ ξ˜∗ (t)*| t t 0 "0 2 ! ˜ j t), ξ˜∗ (2j t)*| dt = |)urµ (2j tA)us (2j tA)u(2j tA)ξ(s2 t 1 j∈Z ≤ 9C1 C2 U (urµ us u) " |dλ| ≤C |u(rµλ)u(sλ)u(λ)| , |λ| Γν

where C is constant independent of x, x∗ . Integrating over r, s gives: % $" %2 $" " ∞ |dλ| dr |dλ| ∗ |u(λ)| . | < uµ (rA)x, x > | ≤ C |uµ (λ)| r |λ| |λ| Γν 0 Γν This estimate shows, by Proposition 3.1, that A has an H ∞ −calculus with ωH (A) ≤ ϕ. Since ω ˜ U (A) ≤ ωH (A) by Proposition 3.3 we have equality. To complete the proof we show that if A has an H ∞ −calculus then (i), (ii) and (iii) hold and that ω ˜ U (A) ≤ ωH (A). To show (ii) and (iii) we observe that " ∞ dt 12 (u(tz))2 = 1. t 0 Note that z j u(z)2 ∈ H0∞ (Σϕ ) for j = −1, 0, 1. It follows easily that if x ∈ X and x∗ ∈ X ∗ then ξ(t) = 12u(tA)2 x,

ξ ∗ (t) = 12(u(tA)2 )∗ x∗

give the required functions. For (i) observe that ω ˜ U (A) ≤ ωH (A) but the first part of the proof shows equality. 88

Theorem 4.2. Suppose A and B are sectorial operators such that Dom (A) = Dom (B), Ran (A) = Ran (B) and for a suitable constant C we have C −1 #Ax# ≤ #Bx# ≤ C#Ax# and

C −1 #A−1 x# ≤ #B −1 x# ≤ C#A−1 x#

x ∈ Dom (A) x ∈ Ran (A).

Suppose A has an H ∞ −calculus. Then the following are equivalent: (i) B has an H ∞ −calculus with ωH (B) = ϕ. (ii) B is almost U-sectorial and ω ˜ U (B) = ϕ. Proof. This is now immediate from Theorem 4.1 using (3.1) and (3.2). If X is a Hilbert space then the assumption that B is almost U-sectorial is redundant and this reduces to the result of Auscher, McIntosh and Nahmod [1]. However, in general this assumption cannot be eliminated. It suffices to take a sectorial operator A with an H ∞ −calculus with ωH (A) > ω(A). Such examples exist [6]; in fact examples are known on subspaces of Lp when 1 < p < 2 [9]. Now fix θ with π − ωH (A) < θ < π − ω(A). Thus e±iθ A are sectorial with ω(e±iθ A) ≤ ω(A)+π−θ. However if both have an H ∞ −calculus we would deduce that for a suitable constant C " ∞ dt x ∈ X, x∗ ∈ X ∗ |)u(te±iθ A)x, x∗ *| ≤ C#x##x∗ # t 0 which would imply that ωH (A) ≤ π − θ. This contradiction implies that at least one of e±iθ A fails to have an H ∞ −calculus. However if B = e±iθ A then (1.1) and (1.2) are trivially satisfied.

References [1] Auscher, P., McIntosh, A., Nahmod, A., Holomorphic functional calculi of operators, quadratic estimates and interpolation, Indiana Univ. Math. J. 46 (1997), 375–403. [2] Boyadzhiev, K., deLaubenfels, R., Semigroups and resolvents of bounded variation, imaginary powers and H ∞ functional calculus, Semigroup Forum, 45 no.3 (1992), 372–384. 89

[3] Cowling, M., Doust, I., McIntosh, A., Yagi, A., Banach space operators with a bounded H ∞ functional calculus, J. Austral. Math. Soc. Ser. A 60 no.1 (1996), 51– 89. [4] Cwikel, M., Kalton, N. J., Interpolation of compact operators by the methods of Calderón and Gustavsson-Peetre, Proc. Edinburgh Math. Soc. 38 no.2 (1995), 261–276. [5] Gustavsson, J., Peetre, J., Interpolation of Orlicz spaces, Studia Math. 60 no. 1 (1977), 33–59. [6] Kalton, N. J., A remark on sectorial operators with an H ∞ -calculus, in “Trends in Banach spaces and operator theory, Memphis, TN, 2001”, Contemp. Math. 321, Amer. Math. Soc. (Providence, RI) 2003, 91–99. [7] Kalton, N. J., Kunstmann, P. C., Weis, L., Perturbation and Interpolation Theorems for the H ∞ -Calculus with Applications to Differential Operators, Math. Ann., to appear. [8] Kalton, N. J., Weis, L., The H ∞ -calculus and sums of closed operators, Math. Ann. 321 (2001), 319–345. [9] Kalton, N. J., Weis, L., Euclidean structures and their applications, in preparation. [10] Kunstmann, P. C., Weis, L., Maximal Lp -regularity for parabolic equations, Fourier multiplier theorems and H ∞ -functional calculus, in “Functional analytic methods for evolution equations”, Lecture Notes in Math. 1855, Springer Verlag 2004, 65–311. Nigel Kalton, Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211. [email protected]

90

Wrapping Brownian motion and heat kernels on compact Lie groups David Maher Abstract The fundamental solution of the heat equation on Rn is known as the heat kernel which is also the transition density of a Brownian motion. Similar statements hold when Rn is replaced by a Lie group. We briefly demonstrate how the results on Rn concerning the heat kernel and Brownian motion may be easily transferred to compact Lie groups using the wrapping map of Dooley and Wildberger. MSC (2000): 22E30, 43A75. Received 24 April 2006 / Accepted 16 January 2007.

1

Introduction

The partial differential equation given on Rn by ∂t u(x, t) = 21 ∆u(x, t),

t ∈ R + , x ∈ Rn ,

(1.1)

where ∆ is the Laplacian, represents the dissipation of heat over a certain time. The fundamental solution of the associated semigroup et∆/2 , known as the heat kernel, pt is given by a unique, strongly continuous, contraction semigroup of convolution operators which may be convolved with the initial data f (x) = u(0, x) to give the solution to the Cauchy problem. That is, ! t∆/2 u(x, t) = e f (x) = (pt ∗ f )(x) = pt (x − y)f (y)dy Rn

The heat kernel may also be expressed as the transition density of a Brownian motion, Bt : pt (x) = E(Bt ),

moreover, (pt ∗ f )(x) = E(f (Bt )) 91

Similar statements hold when Rn is replaced by a Lie group. In this article we will briefly demonstrate how these results may be transferred from the Lie algebra (regarded as Rn ) to a compact Lie group using the so-called wrapping map ( [5]). Additionally, we shall provide the mechanism that allows one to “wrap" a Brownian motion, and then find the heat kernel by taking the expectation of the “wrapped" process and applying a Feynman-Kač type transform. We will also briefly discuss how these results may be extended to compact symmetric spaces and complex Lie groups. Full details and proofs can be found in [11].

2

The wrapping map

The wrapping map was devised by Dooley and Wildberger in [5]. Let G be a compact semisimple Lie group with Lie algebra g. We define the wrapping map, Φ by $Φ(ν), f % = $ν, j f˜% (2.1) where f ∈ C ∞ (G), f˜ = f ◦ exp and j the analytic square root of the determinant of the exponential map. We need to place some conditions on ν for Φ(ν) to be well-defined - this is the case when ν is a distribution of compact support on g, or jν ∈ L1 (g) . We call Φ(ν) the wrap of ν. The principal result is the wrapping formula, given by Φ(µ ∗g ν) = Φ(µ) ∗G Φ(ν)

(2.2)

This formula originated from their previous work on sums of adjoint orbits ( [6]), and can be considered as a global version of the Duflo isomorphism ( [8]). The proof of (2.2) is particularly elegant, using only the Kirillov character formula and some abelian Fourier analysis. Full details are in [5]. What (2.2) shows us is that problems of convolution of central measures or distributions on a (non-abelian) compact Lie group can be transferred to Euclidean convolution of Ad-invariant distributions on g. Thus, since the solution to the Cauchy problem for the heat equation can be written as a convolution between the heat kernel and the initial data, we should be able to wrap the heat kernel on g ∼ = Rn to that on G, and transfer 92

the corresponding solution of the Cauchy problem. Given the remarks in section 1, it is clearly of interest also to consider whether there is a way to wrap Brownian motion to obtain the heat kernel on G.

3

The wrap of Brownian motion

Critical to wrapping a Brownian motion and the heat kernel from g to G is how the infinitesimal generator of the respective process and semigroup - the Laplacian - is affected by wrapping. The Laplacian on g is not quite wrapped to the Laplacian on G - a quantity that may be interpreted as a “curvature" term arises. More precisely, we have: Proposition 3.1. Let G be a compact connected Lie group with Lie algebra g. Then for any Schwartz function, µ on g " # " # Φ Lg(µ) = (LG + (ρ(2 ) Φµ

where Φ is the wrapping map, Lg is the Laplacian on g (regarded as a Euclidean vector space), ρ the half sum of positive roots, and ( · ( the norm given by the Killing form. LG + (ρ(2 is also known as the shifted Laplacian. We shall refer the process and semigroup generated by LG + (ρ(2 as a shifted Brownian motion and a shifted heat kernel, respectively. The actual mechanics of wrapping Brownian motion are not immediately obvious, since the natural objects for the wrapping map to act on are distributions. The wrapping map is a homomorphism from the algebra of Ad-invariant distributions on C ∞ (g) to the algebra of central distributions on C ∞ (G), defined by ϕ )→ ϕι where ι : f )→ j.f ◦ exp. We “wrap Brownian motion” in an analogous way by considering the mapping ι in the context of Itô stochastic differential equations. 93

Very briefly, we may construct a Brownian motion (ζt )t≥0 on g (regarded as the Lie group Rn ) as the solution to the Stratonovich S.D.E.: n $ ∂ζt (i) dζt = ◦ dBt , ∂xi i=1

(3.1)

ζ0 = 0

This is really just a shorthand for the “full" Itô S.D.E.: h(ζt ) = h(0) +

n ! $ i=1

0

t

n ! t 2 $ ∂ h ∂h (i) 1 (ζt )dBt + 2 (ζ )dt 2 t ∂xi ∂x 0 i i=1

(3.2)

where h ∈ C0∞ (Rn ). Likewise, we define our shifted Brownian motion on G as the solution to the S.D.E.: n $

dξt =

i=1

(i)

Xi (ξt ) ◦ dBt + 21 (ρ(2 ξt dt,

(3.3)

ξ0 = e.

" #n where Xi i=1 is an orthonormal basis of the Lie algebra, or in “full" form: f (ξt ) = f (e)+

n ! $ i=1

0

t

(i) (Xi f )(ξt )dBt + 12

n ! $ i=1

0

t

(Xi2 f )(ξt )dt+ 12 (ρ(2

!

t

f (ξt )dt

0

(3.4) where f ∈ C (G). To “wrap of Brownian motion" we replace f ∈ C (G) with j.f ◦ exp ∈ Cc∞ (g), and let j.f ◦ exp = h ∈ C0∞ (g). This can be shown to be n ! t 2 n ! t $ $ ∂ h ∂h (i) 1 (ζ )ds (ζs )dBt + 2 h(ζt ) = h(0) + 2 s ∂x ∂x i 0 0 i i=1 i=1 ∞

∞

which is (3.2). Thus we have

Proposition 3.2. Let ζt be a Brownian motion on g ∼ = Rn . The wrap of ζt is a Brownian motion on G with a potential of (ρ(2 , which we will call ξt . That is, Φ(ζt ) = ξt We may now take expectations of each side to find the law of Brownian motion - the heat kernel - on G:

94

Theorem 3.1. Suppose ξt is the wrap of the Brownian motion on g, ζt . Then the law of ξt may be found by wrapping the law of Brownian motion on its Lie algebra. That is, EX (j.f ◦ exp(ζt )) = Eexp X (f (ξt )) which in law is given by Φ(pt )(exp H) = qtρ (g) where pt (x) is the heat kernel on g = Rn , and qtρ (g) is the heat kernel corresponding to the shifted Laplacian on G The Feynman-Kač theorem can be used to deal with the potential term (ρ(2 to obtain a standard Brownian motion and heat kernel on G. We omit the details, which will be presented in [11].

4

The wrap of the heat kernel

Let pt (x) be the heat kernel on Rn , given by pt (x) = (2πt)−n/2 e−

!x!2 2t

,

and qt (g) is the heat kernel on G, given by $ 2 2 qt (g) = dλ χλ (g)e−(%λ+ρ% −%ρ% )t/2 ,

t ∈ R+ , x ∈ Rn .

(4.1)

t ∈ R+ , g ∈ G.

(4.2)

We write the shifted heat kernel on G as qtρ (g), which is given by $ 2 qtρ (g) = dλ χλ (g)e−%λ+ρ% t/2 , t ∈ R+ , g ∈ G.

(4.3)

λ∈Λ+

λ∈Λ+

Firstly, let’s compute Φ(ν). When ν is suitably nice, it has been shown in [5] we can compute Φ(ν) as a sum over closed geodesics. Let t be the Lie algebra of the maximal torus, T , and let Γ be the integer lattice in t, where Γ = {H ∈ t : exp(H) = e}. We thus have: $% ν & (H + γ), ∀H ∈ t (4.4) Φ(ν)(exp H) = j γ∈Γ 95

or secondly, as sum over highest weights Λ+ : $ dλ ν ∧ (λ + ρ)χλ (g), ∀H ∈ t Φ(ν)(exp H) =

(4.5)

λ∈Λ+

which follows since it can be shown that Φ∧ (ν) = ν ∧ (λ+ρ) (see [5]). Equating these is the Poisson summation formula for a compact Lie group. From the above section, the law of ξt may be found by wrapping the law of Brownian motion on its Lie algebra. We put pt = ν to find the law of the shifted Brownian motion on G: $ 2 Φ(pt )(exp H) = dλ e−%λ+ρ% t/2 χλ (H) (4.6) λ∈Λ+

= (2πt)−d/2

$

e

−!H+n!2 2t

n∈Γ

1 j(H + n)

for all H ∈ t. The first expression follows since pˆt (ξ) = e−%ξ%

5

2 t/2

(4.7) .

Generalisations

The wrapping formula needs some modification to hold for general (compact) symmetric spaces X, equipped with tangent space p, with maximal abelian subalgebra a. This modification is Φ(µ ∗p,e ν) = Φ(µ) ∗X Φ(ν)

(5.1)

where the convolution product on p is “twisted" by a certain function e, which originates in the work of Rouvière [12]. See also [2], [3]. It is well-known in the physics literature that the “sum over classical paths" does not hold for general compact symmetric spaces ( [1], [7]). That is, performing a similar summation to (4.7) to find the heat kernel: $ % pt & (H + γ), ∀H ∈ a j + γ∈Γ

does not yield the (shifted) heat kernel on X. The underlying reason can be easily seen from (5.1) in that we have a twisted convolution on p, which interferes with wrapping the heat convolution semigroup: qt+s = qt ∗X qs = Φ(pt ) ∗X Φ(ps ) = Φ(pt ∗p,e ps ) 96

which is not equal to Φ(pt+s ). It does turn out that we can recover from this situation as the e-function and the j-function are somewhat related. Basically, we need to consider the heat kernel with potentials like j −1 Lpj on p. Even for the 2-sphere this turns out to be difficult - the potential in this case is 1 − cosec2 (H) H2 We have also been able to extend our methods on wrapping Brownian motion and heat kernels to some spaces where we know the wrapping formula holds. A nice example are the complex Lie groups. Instead of having to deal with a maximal torus Tn , as in the case of a compact Lie group, the subgroup corresponding to the Cartan subalgebra is (R+ )n , so instead of summing over a lattice, we just “bend" the heat kernel from g to G by dividing by j, that is, 1 Φ(pt )(exp H) = (2πt)−n/2 exp(−|H|2 /2t), H∈a j(H) We can also wrap other processes - the key is to find how its infinitesimal generator (call it Lg) wraps, that is, " # " # Φ Lg(u) = (LG + C) Φu

6

Further directions

• I am currently proving the wrapping formula for other Lie groups. Once it is then known how to wrap a function, the heat kernel should then be able to be computed. However, this is by no means straightforward - in the case of SL(2, R), the elements are conjugate to a choice of two abelian subgroups, isomorphic to T and R+ . Do we “wrap" or “bend"? Probably both in some suitable fashion. • Wrapping the solutions of other P.D.E.’s. In particular, any phenomena associated to them. For example with the wave equation, what does it mean to “wrap" Huygens’ principle? I should mention that it was for (odd dimensional) compact Lie groups, complex Lie groups, and the symmetric spaces G/K, G complex, that Helgason was able to show that Huygens’ principle holds when the shifted Laplacian is used ( [10]).

97

• We would also like to know the Lp − Lq bounds for a wrapped function. For example, for what p and q do we have (Φ(u)(p ≤ (u(q ? These could then be applied to obtain Lp bounds of solutions of P.D.E.’s on Lie groups. Currently, this is only known when p = q = 1. • These bounds could also be used to examine other behaviour such as convergence of Fourier transforms - if we used the ball multiplier, then in the case of compact Lie groups, our formula for Φ corresponds to the W -invariant polygonal regions of positive weights typically considered for the convergence of Fourier series on compact Lie groups.

References [1] Camporesi, R. Harmonic analysis and propagators on homogeneous spaces, Phys. Rep., 196:1-134, [1990] [2] Dooley, A.H. Orbital convolutions, wrapping maps and e-functions, Proc. CMA, ANU, [2002] [3] Dooley, A.H. Global versions of the e-function for compact symmetric spaces, In preparation. [4] Dooley, A.H. and Wildberger, N.J. Global character formulae for compact Lie groups Trans. Amer. Math. Soc., 351(2):477-495, [1999] [5] Dooley, A.H. and Wildberger, N.J. Harmonic Analysis and the Global Exponential Map for Compact Lie Groups, Funktsional. Anal. i Prilozhen. 27(1):25-32; Eng. Trans., [1993]. Funct. Ana. Appl. 27:21-27; MR 94e:22032, [1993] [6] Dooley, A.H., Repka, J., and Wildberger, N.J. Sums of adjoint orbits, Lin. Multilin. Alg., 36:79-101, [1993] [7] Dowker, J. S. When is the ‘sum over classical paths’ exact? J. Phys. A, 3:451-461, [1970] [8] Duflo, M. Opérateurs différential bi-invariants sur un groupe de Lie, Ann. Sci. École Norm. Sup., 10:265-288, [1977] [9] Gangolli, R. Asymptotic behaiviour of spectra of compact quotients of certain symmetric spaces, Acta. Math. 121:151-192, [1968]

98

[10] Helgason, S. Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, 39, AMS, [1994] [11] Maher, D. G., Brownian motion and heat kernels on compact Lie groups and symmetric spaces, Ph.D thesis, Preprint available at www.maths.unsw.edu.au/∼dmaher [12] Rouvière, F. Invariant analysis and contractions of symmetric spaces, I, II Compositio Math. 73:241-270, [1990], 80:111-136, [1991]

David Maher, School of Mathematics, UNSW, Kensington 2052 NSW, Australia. [email protected].

99

Remarks on the Rademacher-Menshov Theorem Christopher Meaney

∗

Abstract We describe Salem’s proof of the Rademacher-Menshov Theorem, which shows that one constant works for all orthogonal expansions in all L2 -spaces. By changing the emphasis in Salem’s proof we produce a lower bound for sums of vectors coming from bi-orthogonal sets of vectors in a Hilbert space. This inequality is applied to sums of columns of an invertible matrix and to Lebesgue constants. Keywords: Orthogonal expansion, Rademacher-Menshov Theorem, Bessel’s inequality, bi-orthogonal, Lebesgue constants. MSC (2000): 42C15, 46C05. Received 1 August 2006 / Accepted 18 October 2006.

1

Introduction

Here we give an exposition of Salem’s proof [14] of the Rademacher-Menshov Theorem. Although it is more elaborate than some proofs and over sixty years old, Salem’s method makes it clear that one constant works for all orthogonal expansions in all L2 -spaces. Furthermore, some of the inequalities used in the proof lead to a general inequality concerning bi-orthogonal sets of vectors in Hilbert spaces (Proposition 2 in the next section.) In recent work [2] with Leonardo Colzani and Elena Prestini, we used the universal nature of the constant in the Rademacher-Menshov Theorem [13, 11] to produce some almost-everywhere convergence results for inverse Fourier Transforms. Theorem 1 below contains the basic idea used in that work. I am grateful to the organizers of the Murramarang conference for the opportunity to participate. ∗

100

2

Statement of Main Results

Proposition 2.1. There is a positive constant C with the following property. For every positive measure space (X, µ), for every n ≥ 1, and for every finite set {F1 , . . . , Fn } of orthogonal functions in L2 (X, µ), the maximal function ! ! m ! !" ! ! (2.1) Fj (x)! M(x) = max ! 1≤m≤n ! ! j=1

has norm

#M#2 ≤ C log (n + 1)

# n " j=1

#Fj #22

$1/2

.

(2.2)

Suppose H is a Hilbert space, with inner-product written as %v, w&. We say that two sets of vectors {v1 , . . . , vn } and {w1 , . . . , wn } are bi-orthogonal when %vj , wk & = 0, ∀j (= k. Proposition 2.2. There is a positive constant c with the following property. For every Hilbert space H and every pair of bi-orthogonal sets {v1 , . . . , vn } and {w1 , . . . , wn } in H, % % k %" % % % (log n) min |%vk , wk &| ≤ c max #wm # max % vj % . (2.3) 1≤m≤n 1≤k≤n 1≤k≤n % % j=1

These are proved in Section 4. In the next section we give some applications.

3 3.1

Consequences Almost everywhere convergence.

Suppose that (X, µ) is a positive measure space and that L2 (X, µ) = ⊕∞ n=1 Hn is an orthogonal decomposition into closed subspaces Hn . Let Pn be projection onto Hn . Each function f ∈ L2 (X, µ) has an orthogonal expansion ∞ "

Pn f,

n=1

101

which converges to f in norm. The partial sum operators are SN f (x) =

N "

∀N ≥ 1, x ∈ X.

Pn f (x),

n=1

Proposition 2.1 says that % % % % % max |Sm f |% ≤ C log(N + 1) #SN f # , 2 % % 1≤m≤N

2

∀N ≥ 1, f ∈ L2 (X, µ).

Define the maximal function

S ∗ f (x) = sup |SN f (x)| . N ≥1

This is dominated by two pieces, ∗

S f (x) ≤ sup |S2m f (x)| + sup m≥0

m≥0

&

' maxm+1 |Sn f (x) − S2m f (x)| . m

2 ≤n 0, denote by B(x, r) the open ball centered at x with radius r, and by V (x, r) its measure. We call (X, d, µ) a space of homogeneous type if, for all x ∈ X 125

and all r > 0, V (x, r) < +∞ and there exists C > 0 such that, for all x ∈ X and all r > 0, V (x, 2r) ≤ CV (x, r). (1.1)

An easy consequence of (1.1) is that there exist C, D > 0 such that, for all x ∈ X, all r > 0 and all θ > 1, V (x, θr) ≤ CθD V (x, r).

(1.2)

There are of course many examples of spaces of homogeneous type. The simplest one is X = Rn , n ≥ 1, endowed with the Euclidean metric and the Lebesgue measure. Let us describe another example. Let G be a real connected Lie group endowed with a system of left-invariant vector fields X = {X1 , ..., Xk } satisfying the Hörmander condition. If d is the Carathéodory metric associated to X and µ the left-invariant Haar measure, and if, for any r > 0, V (r) denotes the volume of any ball with radius r, then there exists d ∈ N∗ such that V (r) ∼ rd for 0 < r < 1 ( [11]). Moreover, either G has polynomial volume growth, i.e. there exists D ∈ N∗ such that, for all r > 1, V (r) ∼ rD , or G has exponential volume growth, i.e. there exists c1 , C1 , c2 , C2 > 0 such that c1 ec2 r ≤ V (r) ≤ C1 eC2 r for all r > 1 (see [8]). Among the class of Lie groups with polynomial volume growth, there is the strict subclass of nilpotent Lie groups, a strict subclass of which is made of stratified Lie groups. A real connected Lie group with polynomial volume growth is clearly a space of homogeneous type. Another example of space of homogeneous type is the case of connected Riemannian manifolds with nonnegative Ricci curvature (this follows from the Bishop comparison theorem, see [2]). More generally, Riemannian manifolds which are quasi-isometric to a manifold with nonnegative Ricci curvature, or cocompact covering manifolds whose deck transformation group have polynomial growth, are spaces of homogeneous type ( [7]). In discrete settings, assumption (1.1) also plays a fundamental role in analysis on graphs (see for instance [1] and the references therein), and is satisfied for instance on the Cayley graph of a finitely generated group with polynomial volume growth or on some fractal graphs, as the Sierpinsky carpet. Let us now define tent spaces on X. For any α > 0 and any x ∈ X, denote by Γα (x) the cone of aperture α with vertex x ∈ X, namely: Γα (x) = {(y, t) ∈ X × (0, +∞); d(y, x) < αt} . 126

For any closed subset F ⊂ X, let Rα (F ) be the union of all cones with vertices in F : ! Rα (F ) = Γα (x). x∈X

If α > 0 and O is an open subset of X, then the tent over O with aperture α, denoted by Tα (O), is defined by: Tα (O) = (Rα (Oc ))c . Notice that Tα (O) = {(x, t) ∈ X × (0, +∞) ; d(x, Oc ) ≥ αt} .

In the sequel, we write Γ(x) (resp. R(F ) and T (O) instead of Γ1 (x) (resp. R1 (F ) and T1 (O)). For any measurable function f on X × (0, +∞) and any x ∈ X, define Sf (x) =

"# #

Γ(x)

2

|f (y, t)| dt dµ(y) V (x, t) t

$ 21

,

and, for all p > 0, say that f ∈ T p (X) if (f (T p (X) := (Sf (Lp (X) < +∞. We have the following notion of atom (see [4], p. 312): Definition 1.1. Let p ∈ (0, +∞). A measurable function a on X × (0, +∞) is said to be a T p (X) atom if there exists a ball B ⊂ X such that a is supported in T (B) and ## 1 dt . |a(y, t)|2 dµ(y) ≤ 2 t X×(0,+∞) V (B) p −1 It is plain to see that a T p (X)-atom belongs to T p (X) and that its norm is controlled by a constant only depending on X and p. Conversely, when 0 < p ≤ 1, it turns out that any function in T p (X) has an atomic decomposition, and this is the result we prove in the sequel:

127

Theorem 1.1. Let p ∈ (0, 1]. Then, there exists Cp > 0 with the following property: for all f ∈ T p (X), there exist a sequence (λn )n∈N ∈ lp and a sequence of T p (X) atoms (an )n∈N such that f=

∞ %

λ n an

n=0

and

∞ % n=0

2

|λn |p ≤ Cpp (f (pT p (X) .

Proof of the atomic decomposition

The proof of Theorem 1.1, for which we closely follow [4], requires the notion of γ-density (see [4]). Let F be a closed subset of X, and O = F c . Assume that µ(O) < +∞. For any fixed γ ∈ ]0, 1[, say that x ∈ X has global γdensity with respect to F if µ(B ∩ F ) ≥γ µ(B) for any ball B centered at x. The set of all such x’s is denoted by F ∗ . It is a closed subset of F . Define also O∗ = (F ∗ )c . It is clear that O ⊂ O∗ . Moreover, O∗ = {x; M (1O )(x) > 1 − γ} where M denotes the Hardy-Littlewood maximal function. As a consequence,

(2.1)

µ(O∗ ) ≤ Cγ µ(O). The following integration lemma will be used:

Lemma 2.1. Let η ∈ (0, 1). Then, there exists γ ∈ ]0, 1[ and Cγ,η > 0 such that, for any closed subset F of X whose complement has finite measure and any nonnegative measurable function H(y, t) on X × ]0, +∞[, ' ## # &# # H(y, t)V (y, t)dµ(y)dt ≤ Cγ,η H(y, t)dµ(y)dt dµ(x), R1−η (F ∗ )

F

Γ(x)

where F ∗ denotes the set of points in X with global density γ with respect to F. 128

Proof: We claim that there exists c& > 0 such that, for all (y, t) ∈ R1−η (F ∗ ), µ(F ∩ B(y, t)) ≥ c& V (y, t). (2.2)

Assume that this is proved. Write ' # &# # ### H(y, t)dµ(y)dt dµ(x) = H(y, t)1E (x, y, t)dµ(x)dµ(y)dt F

Γ(x)

where E = {(x, y, t) ∈ F × X × (0, +∞); d(y, x) < t} .

One therefore has ' # &# # ## H(y, t)dµ(y)dt dµ(x) = H(y, t)µ(F ∩ B(y, t))dµ(y)dt F Γ(x) # #R(F ) ≥ H(y, t)µ(F ∩ B(y, t))dµ(y)dt #R#1−η (F ∗ ) ≥ c& H(y, t)V (y, t)dµ(y)dt. R1−η (F ∗ )

Let us now prove (2.2). If (y, t) ∈ R1−η (F ∗ ), then there exists x ∈ F ∗ such that d(y, x) < (1 − η)t. One may write µ(F ∩ B(y, t)) ≥ µ(F ∩ B(x, t)) − µ(B(x, t) ∩ (B(y, t))c ). But, since x ∈ F ∗ , µ(F ∩ B(x, t)) ≥ γV (x, t). Moreover, since d(y, x) < (1 − η)t, B(x, ηt) ⊂ B(y, t), so that µ(B(x, t) ∩ B(y, t)) ≥ V (x, ηt) ≥ δV (x, t), where δ = C1 η D and C, D are given by (1.2). It follows that there exists c ∈ (0, 1) only depending on η and the constants in (1.2) such that µ(B(x, t) ∩ B(y, t)c ) ≤ cV (x, t). As a consequence, if 1 > γ > c, one obtains, using (1.1) once more, µ(F ∩ B(y, t)) ≥ (γ − c)V (x, t) ≥ c& V (y, t). 129

This concludes the proof of Lemma 2.1. We now turn to the proof of Theorem 1.1. Let f ∈ T p (X). For any integer k ∈ Z, define ( ) Ok = x ∈ X; Sf (x) > 2k

and let Fk = Okc . The Ok ’s are open subsets of X and, since Sf ∈ L1 (X), µ(Ok ) < +∞ for all k ∈ Z. Fix η ∈ (0, 1) and consider also, for γ given by Lemma 2.1, the set Fk∗ of all the points of global γ- density with respect to Fk , and Ok∗ = (Fk∗ )c . We claim that ! suppf ⊂ T1−η (Ok∗ ). (2.3) k

Indeed, according to Lemma 2.1, for any k ∈ Z, $ ## # "# # 2 dt |f (y, t)| dt |f (y, t)|2 dµ(y) dµ(x) ≤ C dµ(y) t t R1−η (Fk∗ ) Fk Γ(x) V (y, t) # & ≤ C (Sf )2 (x)dµ(x). Fk

When k → −∞, the dominated convergence theorem shows that

0. It follows that

##

T j

R1−η (Fj∗ )

|f (y, t)|2 dµ(y)

#

Fk

(Sf )2 (x)dµ(x) →

dt = 0. t

This shows that f is zero on almost every point of

* j

R1−η (Fj∗ ). In other

words, (2.3) holds. We make use of the following lemma (see [3], Ch 3, Th 1.3; see also [9]): Lemma 2.2. Let Ω be a proper open subset of finite measure of X. For d(x, Ωc ) . Then, there exist an integer M and a any x ∈ X, define r(x) = 10 sequence (xn )n∈N of points in X such that, if rn = r(xn ), ! Ω= B(xn , rn ), n

1 1 i ,= j =⇒ B(xi , ri ) ∩ B(xj , rj ) = Ø, 4 4 ∀n, |{m; B(xn , 5rn ) ∩ B(xm , 5rm ) ,= Ø}| ≤ M. 130

Moreover, there exists a sequence of nonnegative functions (ϕn )n∈N on X such that supp ϕn ⊂ B(xn , 2rn ), ∀x ∈ B(xn , rn ), ϕn (x) ≥ %

1 , M

ϕn = 1Ω .

n

Let k ∈ Z. If Ok∗ is a proper subset of X, apply this lemma with Ω = Ok∗ . The points xn will be denoted by xkn , the radii rn by rnk , the balls B(xkn , rnk ) by Bnk and the functions ϕn by ϕkn , where n ∈ I k and I k is a denumerable set. If Ok∗ = X, then µ(X) < +∞, which forces X to be bounded ( [10]). In this situation, set I k = {1}, and define B1k = X (indeed, X is a ball itself) and ϕk1 (x) = 1 for all x ∈ X. One has, for any (x, t) ∈ X × R∗+ , , , + + % k ∗ ∗ ∗ (x, t) = ϕ (x) 1 − 1 1T1−η (Ok∗ ) − 1T1−η (Ok+1 T1−η (Ok ) T1−η (Ok+1 ) (x, t). ) j j∈I k

∗ Indeed, if (x, t) ∈ T1−η (Ok∗ ) \ T1−η (Ok+1 ), then x ∈ Ok∗ , and the two sides of the identity are equal to 1. Otherwise, they are both equal to zero. From this and (2.3), it follows that , + % ∗ ∗ f (x, t) = f (x, t) 1T1−η (Ok ) − 1T1−η (Ok+1 ) (x, t) k∈Z , + %% ∗ = f (x, t)ϕkj (x) 1T1−η (Ok∗ ) − 1T1−η (Ok+1 ) (x, t). k∈Z j∈I k

Define, for all k ∈ Z and all j ∈ I k , ## , + dt k ∗ µj = |f (y, t)|2 ϕkj (y)2 1T1−η (Ok∗ ) − 1T1−η (Ok+1 , ) (y, t)dµ(y) t + , 1 k 21 − p1 ∗ (µkj )− 2 , akj (y, t) = f (y, t)ϕkj (y) 1T1−η (Ok∗ ) − 1T1−η (Ok+1 ) (y, t)V (Bj ) λkj

1

1

1

= V (Bjk ) p − 2 (µkj ) 2 .

Then f=

%%

λkj akj .

k∈Z j∈I k

131

We claim that, up to a multiplicative constant, the akj ’s are T p (X) atoms. To begin with, notice that (2.4)

supp akj ⊂ T (CBjk )

12 where C := 2 + 1−η . Indeed, this is obvious when Ok∗ = X, since B1k = X in this case. Assume therefore that Ok∗ is a proper subset of X and let (y, t) ∈ T1−η (Ok∗ ) such that ϕkj (y) > 0. Then, d(y, (Ok∗ )c ) ≥ (1 − η)t and y ∈ 2Bjk . We intend to prove that d(y, (CBjk )c ) ≥ t. Let z ∈ (CBjk )c . Then

d(y, z) ≥ d(z, xkj ) − d(y, xkj ) ≥ (C − 2)rjk . Moreover, by definition of rjk , d(xkj , (Ok∗ )c ) = 10rjk . Let ε > 0. There exists u∈ / Ok∗ such that d(xkj , u) < 10rjk + ε. Since u ∈ (Ok∗ )c while d(y, (Ok∗ )c ) ≥ (1 − η)t, one has (1 − η)t ≤ d(y, u) ≤ d(y, xkj ) + d(xkj , u) ≤ 2rjk + 10rjk + ε and, since it is true for every ε > 0, it follows that (1 − η)t ≤ 12rjk . Finally, by the choice of C, one has d(y, z) ≥ t. Thus, (2.4) holds. The very definition of akj implies that ## - k 1 -aj (y, t)-2 dµ(y) dt = 2 t V (Bjk ) p −1 C& ≤ , 2 V (CBjk ) p −1 where the last line is due to (1.2). What remains to be proved is that % % - -p -λkj - ≤ C (Sf (p . p

k∈Z j∈I k

To this purpose, write ## k µj ≤

∗ T (CBjk )∩(T1−η (Ok+1 ))c

|f (y, t)|2 dµ(y)

dt t 132

and apply Lemma 2.1 to |f (y, t)|2 1 H(y, t) = k (y, t) tV (y, t) T (CBj ) and

c F = Fk+1 = Ok+1 .

This yields ##

dt |f (y, t)| dµ(y) ≤ C ∗ t T (CBjk )∩(T1−η (Ok+1 ))c 2

#

If (y, t) ∈ Γ(x) ∩ T (CBjk ), then x ∈ CBjk . It ## dt |f (y, t)|2 dµ(y) ≤ ∗ t T (CBjk )∩(T (Ok+1 ))c ≤ ≤

c Ok+1

"# #

Γ(x)∩T (CBjk )

dt |f (y, t)|2 dµ(y) V (y, t) t

follows that # C (Sf )2 (x)dµ(x) c CBjk ∩Ok+1 k+1 2

C(2 ) V (CBjk ) C & 22k V (Bjk ).

Thus, µkj ≤ C22k V (Bjk ), and, by (1.2), 1

1

1

λkj = V (Bjk ) p − 2 (µkj ) 2 1

≤ C2k V (Bjk ) p k

≤ C2 V

&

1 k B 4 j

' p1

.

Since, for all k ∈ Z, the 14 Bjk are pairwise disjoint for i ∈ I k and included in

133

$

dµ(x).

Ok∗ , one has, by (2.1), % % - -p % -λkj - ≤ C 2kp µ(Ok∗ ) k∈Z j∈I k

k∈Z

&

%

2kp µ(Ok ) k∈Z % .( )/ ≤ Cp (2k−1 )2k(p−1) µ Sf > 2k ≤ C

k∈Z

≤ Cp = Cp

%#

2k

tp−1 µ ({Sf > t}) dt

2k−1

#k∈Z+∞

tp−1 µ ({Sf > t}) dt

0

= C (Sf (pp . The proof of Theorem 1.1 is complete. Acknowledgements: The author would like to thank the referee for an interesting suggestion to improve the paper.

References [1] P. Auscher, T. Coulhon, Gaussian lower bounds for random walks from elliptic regularity, Ann. Inst. Henri Poincaré (B) Probabilités et Statistiques, 35, 5, 605-630, 1999. [2] R. Bishop, R. Crittenden, Geometry of manifolds, Academic Press, N. York, 1964. [3] R. Coifman, Y. Meyer, E. M. Stein, Un nouvel espace adapté à l’étude des opérateurs définis par des intégrales singulières, in: Proc. Conf. Harmonic Analysis (Cortona), Lecture Notes in Math. 992, Springer, Berlin, 1–15, 1983. [4] R. Coifman, Y. Meyer, E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62, 304-335, 1985. [5] R. Coifman, G. Weiss, Analyse harmonique non commutative sur certains espaces homogènes, Lect. Notes Math. 242, Springer Verlag, 1971. 134

[6] R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83, 569-645, 1977. [7] T. Coulhon, L. Saloff-Coste, Variétés Riemanniennes isométriques à l’infini, Rev. Mat. Iberoamericana, 11, 687-726, 1995. [8] Y. Guivarch, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France, 101, 333-379, 1973. [9] R. Macias, C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Advances in Math. 33, 271–309, 1979. [10] J. M. Martell, Desigualdades con pesos en el Análisis de Fourier: de los espacios de tipo homogéneo a las medidas no doblantes, Ph. D., Universidad Autónoma de Madrid, 2001. [11] A. Nagel, E. M. Stein, S. Wainger, Balls and metrics defined by vector fields I: Basic properties, Acta Math. 155, 103-147, 1985. Emmanuel Russ, Université Paul Cézanne, LATP, CNRS UMR 6632, Faculté des Sciences et Techniques, Case cour A, Avenue Escadrille NormandieNiémen, F-13397 Marseille Cedex 20, France. [email protected]

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