Isomorphisms between H1 Spaces

M o n o g r a f i e M a t e m a t y c z n e Instytut Matematyczny Polskiej Akademii Nauk (IMPAN) Volume 66 ( New Ser...

Author: Paul F.X. Müller

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M o n o g r a f i e

M a t e m a t y c z n e

Instytut Matematyczny Polskiej Akademii Nauk (IMPAN)

Volume 66 ( New Series )

Founded in 1932 by S. Banach, B. Knaster, K. Kuratowski, S. Mazurkiewicz, W. Sierpiński, H. Steinhaus

Managing Editor: Przemysław Wojtaszczyk, IMPAN and Warsaw University Editorial Board: Jean Bourgain (IAS, Princeton, USA) Tadeusz Iwaniec (Syracuse University, USA) Tom Körner (Cambridge, UK) Krystyna Kuperberg (Auburn University, USA) Tomasz Łuczak (Poznań University, Poland) Ludomir Newelski (Wrocław University, Poland) Gilles Pisier (Université Paris 6, France) Piotr Pragacz (Institute of Mathematics, Polish Academy of Sciences) Grzegorz Świątek (Pennsylvania State University, USA) Jerzy Zabczyk (Institute of Mathematics, Polish Academy of Sciences)

Volumes 31– 62 of the series Monografie Matematyczne were published by PWN – Polish Scientific Publishers, Warsaw

Paul F.X. Müller

Isomorphisms between H 1 Spaces

Birkhäuser Verlag Basel • Boston • Berlin

Author: Paul F.X. Müller Institute of Analysis Johannes Kepler University Linz Altenbergerstr. 69 Austria e-mail: [email protected]

2000 Mathematics Subject Classification 30D55, 42C10, 46B03, 46B07, 46-99, 47B38, 60G46

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

ISBN: 3-7643-2431-7 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained.

© 2005 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Cover design: Micha Lotrovsky, CH-4106 Therwil, Switzerland Printed on acid-free paper produced of chlorine-free pulp. TCF ∞ Printed in Germany ISBN-10: 3-7643-2431-7 ISBN-13: 978-3-7643-2431-5 987654321

www.birkhauser.ch

To Joanna

Contents Preface

xi

1 The Haar System: Basic Facts and Classical Results 1.1 Bases in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . The Haar basis . . . . . . . . . . . . . . . . . . . . . . . . Khintchine’s inequality . . . . . . . . . . . . . . . . . . . Burkholder’s inequality . . . . . . . . . . . . . . . . . . . The Walsh system . . . . . . . . . . . . . . . . . . . . . . 1.2 Dyadic H 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fefferman’s inequality . . . . . . . . . . . . . . . . . . . . Sharp maximal functions . . . . . . . . . . . . . . . . . . 1.3 Bounded square functions and large deviation inequalities . Square functions in Lp . . . . . . . . . . . . . . . . . . . . Multipliers into SL∞ . . . . . . . . . . . . . . . . . . . . 1.4 Martingales and biorthogonal systems . . . . . . . . . . . . Martingale inequalities . . . . . . . . . . . . . . . . . . . Biorthogonal systems . . . . . . . . . . . . . . . . . . . . 1.5 Basic operators . . . . . . . . . . . . . . . . . . . . . . . . . Multipliers and paraproducts . . . . . . . . . . . . . . . . Rearrangement operators and Calder´ on–Zygmund kernels Orthogonal projections . . . . . . . . . . . . . . . . . . . 1.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 . 1 . 1 . 6 . 13 . 19 . 34 . 35 . 45 . 52 . 52 . 68 . 73 . 74 . 81 . 88 . 88 . 92 . 104 . 112

2 Projections, Isomorphisms and Interpolation 2.1 Complemented subspaces . . . . . . . . Three-valued martingale differences . Rosenthal’s space . . . . . . . . . . . Weighted intersections . . . . . . . . . 2.2 Pelczy´ nski’s decomposition method . . . Infinite direct sums . . . . . . . . . . H 1 with values in 2n . . . . . . . . . .

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117 117 124 130 137 139 140 142

viii

Contents 2.3

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144 148 153 156 159 166

3 Combinatorics of Colored Dyadic Intervals 3.1 The Carleson packing condition . . . . . . . . . . . . . . . . . Generations of dyadic intervals . . . . . . . . . . . . . . . . The Gamlen–Gaudet construction . . . . . . . . . . . . . . 3.2 Orthogonal projections and colored intervals . . . . . . . . . . Jones’s compatibility condition and colored intervals . . . . The first step towards the uniform approximation property 3.3 Rearrangement operators . . . . . . . . . . . . . . . . . . . . Rearrangement operators on BMO . . . . . . . . . . . . . . Rearrangement operators on L1 . . . . . . . . . . . . . . . 3.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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169 169 170 176 181 181 193 196 198 224 228

4 Martingale H 1 Spaces 4.1 Maurey’s isomorphism . . . . . . . . . . Operations on martingale differences . Martingale transform techniques . . . 4.2 Isomorphic classification . . . . . . . . . Classification of martingale H 1 spaces Classification of weighted intersections 4.3 More on subsystems of the Haar system The theorem of Gamlen and Gaudet . Related open problems . . . . . . . . 4.4 Notes . . . . . . . . . . . . . . . . . . .

2.4

Interpolation of operators . . . Calder´ on’s product . . . . . . Pisier’s norm on H 1 . . . . . Dual estimates . . . . . . . . Analytic families of operators Notes . . . . . . . . . . . . . .

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229 229 230 233 242 242 252 257 257 260 265

5 Isomorphic Invariants for H 1 5.1 Complemented copies of Hilbert spaces . . . . Existence, abundance . . . . . . . . . . . . Johnson’s factorization . . . . . . . . . . . Intrinsic description . . . . . . . . . . . . . 5.2 Complemented copies of Hn1 . . . . . . . . . . Dichotomies . . . . . . . . . . . . . . . . . Intrinsic description . . . . . . . . . . . . . H 1 with values in 2 . . . . . . . . . . . . . 5.3 The uniform approximation property of BMO Splitting the Haar support . . . . . . . . . UAP data with large Haar coefficients . . . UAP data with small Haar coefficients . . .

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267 267 267 271 277 282 283 295 301 308 312 314 316

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Contents

5.4

ix

General UAP data . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

6 Atomic H 1 Spaces 1 . . . . . 6.1 Basic similarities between H 1 and Hat Maximal functions and atoms . . . . . . . . . Square functions . . . . . . . . . . . . . . . . 6.2 Carleson’s biorthogonal system . . . . . . . . . Definition . . . . . . . . . . . . . . . . . . . . Carleson coefficients versus Haar coefficients The compensation argument . . . . . . . . . 1 . . . . . . . . . . Unconditional basis in Hat 6.3 Spaces of homogeneous type . . . . . . . . . . . Lipschitz partitions of unity . . . . . . . . . . Estimates for molecules . . . . . . . . . . . . 6.4 Orthogonal projections in atomic H 1 spaces . . The square root of the Gram matrix . . . . . 6.5 Martingale approximation in atomic H 1 spaces 6.6 Notes . . . . . . . . . . . . . . . . . . . . . . .

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347 347 351 355 360 362 366 374 391 397 404 405 407 418 421 430

Bibliography

433

List of Symbols

449

Index

451

Preface The theory of Hardy spaces is firmly rooted in the fields of complex analysis, trigonometric series, and probability. Its history began in the decade between 1906 and 1916. In 1906 appeared P. Fatou’s treatise on power series and trigonometric series in Acta Mathematica. G. H. Hardy proved in 1915 that integral means of power series in the unit disk are log-convex, just in time to be included in the first edition of E. Landau’s “Darstellung und Begr¨ undung einiger neuerer Ergebnisse der Funktionentheorie” (1916). In the same year F. and M. Riesz presented their ¨ treatise “Uber die Randwerte einer analytischen Funktion” to the Scandinavian congress of mathematicians in Stockholm. The classical theory of Hardy spaces is concerned with the boundary behavior of analytic functions on the unit disk and with estimates forthe Fourier coefficients of the limiting function. Given a power series f (z) = an z n , Hardy and 1 Littlewood (1930) proved that on the unit circle the L norm of its non-tangential maximal function is equivalent to the L1 norm of its boundary values. R. E. A. C. Paley (1933) shows that the sequence of lacunary coefficients is square summable provided that the boundary values are integrable on the unit circle, more precisely that 1/2 1 π ≤ |f (eit )|dt. |a2n |2 π −π Kolmogorov’s theorem (1925) asserts that the Hilbert transform is of weak type one-one. These three results, together with the F. and M. Riesz theorem, were crucial in establishing the importance of the Hardy space H 1 (T) of integrable functions on the unit circle for which the harmonic extension to the unit disk is analytic. Probability, in the 1920’s, was confronted with analogous problems of boundary convergence in the work of N. Wiener (construction of Brownian motion), A. Kolmogorov (sums of independent random variables) and J. Khintchine (the law of large numbers) to name just a few. Thus emerged martingales, martingale convergence theorems and J. L. Doob’s maximal function. By 1977 there existed two highly developed classes of H 1 spaces: these were the atomic H 1 spaces which are linked to analytic functions via Fefferman’s duality theorem, and the martingale H 1 spaces consisting of martingales for which Doob’s maximal function is integrable. A remarkably simple object, called dyadic H 1 , appears in both of these classes.

xii

Preface

Several of the most important theorems for atomic H 1 spaces directly correspond to a theorem for the martingale class (atomic decomposition, equivalence of square functions and maximal functions, H 1 − BMO duality, singular integrals correspond to martingale transforms, etc. ) . Given the obvious analogies between the two families of Banach spaces, A. Pelczy´ nski conjectured that there are deeper connections to be uncovered. A. Pelczy´ nski’s conjecture asked specifically whether dyadic H 1 and H 1 (T) (the L1 subspace consisting of boundary values of analytic functions) are isomorphic as Banach spaces. For a family of Banach spaces, whose members are constructed by one and the same rule, the question of its isomorphic classification is inevitable. This applied in particular to the class of H 1 spaces, and clarifying the special role played by dyadic H 1 (with its obvious unconditional basis) posed a problem of particular interest. Its solution by B. Maurey opened a new line of investigation for the Banach space H 1 . In this treatise we give a thorough analysis of B. Maurey’s work and its consequences, including L. Carleson’s construction of an unconditional basis for H 1 (T). This monograph is a study of dyadic H 1 , its isomorphic invariants and its position within the two classes of martingale and atomic H 1 spaces. Simultaneously, we provide a detailed analysis of the Haar system and of the operators that are built from it. These include rearrangement operators, orthogonal projections, paraproducts and Calderon–Zygmund integral operators. This book was written under the conviction that the intensive occupation with the Haar system is fully justified by its impact on wide ranging areas of analysis and probability. The following list has been compiled with the aim of substantiating this point of view. 1. The Haar system is clearly the simplest martingale difference sequence, yet we have B. Maurey’s theorem to the effect that every martingale is conveniently embedded in the partial sums of a Haar series. 2. Calderon–Zygmund operators can be expressed as paraproducts plus an absolutely converging series of basic operators rearranging and splitting the Haar system. This is the content of Figiel’s integral representation. 3. In establishing isomorphic invariants one is clearly free to choose the simplest object among the spaces known to be isomorphic to the Banach space under investigation. For instance, dyadic H 1 is the most unassuming example among the martingale spaces H 1 [(Fn )] and also among the atomic spaces H 1 (X, d, µ). Hence we work in dyadic H 1 to establish isomorphic invariants and simultaneously we search in the classes H 1 [(Fn )] and H 1 (X, d, µ) for the spaces that are isomorphic to dyadic H 1 . This line of development was initiated by the outstanding work of B. Maurey. It led to theresult that H 1 (X, d, µ) is isomorphic to one of the following spaces, H 1 , ( Hn1 )1 and 1 (the same conclusion holds for martingale spaces when the generating σ-algebras are purely atomic).

Preface

xiii

4. Expressing a problem using Haar functions and exploiting their relative simplicity leads us quite often straight to the point, that is, to the problem’s inherent combinatorial difficulty. This occurs for instance in the construction of multipliers into SL∞ , in Johnson’s factorization of operators on Lp , or in the construction of a resolving operator to establish the uniform approximation property for H 1 . 5. The Haar system is a reliable instrument for separating core ideas from their surrounding technicalities. The best known examples illustrating this point are obtained by specializing, to the dyadic setting, the classical theorems of real and harmonic analysis, such as the atomic decomposition, Fefferman’s inequality, Hardy–Littlewood maximal function estimates, good λ inequalities, interpolation and extrapolation of operators etc. Contents. The first chapter introduces the Haar system. In Chapter 1 we present the classical inequalities of Khintchine, Burkholder, Fefferman, Hardy–Littlewood. It contains a discussion of large deviation inequalities, Schechtman’s sign-embeddings and multipliers into SL∞ . We prove that the unconditionality of the Haar basis yields convergence of the Walsh–Paley expansion and estimates for the dyadic gradient in Lp . We study counterexamples pertaining to the Walsh expansion along increasing multiplicities. General martingales and biorthogonal systems are analyzed by exploiting their relation to the Haar system. The first chapter concludes with estimates for paraproducts, Figiel’s representation of singular integral operators and with a discussion of orthogonal projections in H 1 . In Chapter 2 we review basic concepts of functional analysis. We list the known complemented subspaces of H 1 , prove the Banach space decomposition principle of A. Pelczy´ nski and discuss analytic families of operators on H p spaces. The analysis of rearrangement operators and orthogonal projections is predominant in this book. It gives rise to hard problems involving combinatorics of colored dyadic intervals. We address these problems by constructing a coherent set of combinatorial techniques, forming the content of Chapter 3. In Chapter 4 we study martingale H 1 spaces. We present a detailed analysis of Maurey’s isomorphism, thus showing the existence of an unconditional basis. Exploiting the unconditional basis, obtained by Maurey’s isomorphism, yields the classification theorem for martingale H 1 spaces. Chapter 5 covers the isomorphic invariants of H 1 . We establish dichotomies for complemented subspaces of H 1 , prove that H 1 and H 1 (2 ) are not isomorphic as Banach spaces, and that H 1 satisfies the uniform approximation property. Chapter 6 contains the presentation of atomic H 1 spaces and their rela1 in considerable detail, obtaining tion to dyadic H 1 . We treat the example of Hat equivalent norms using the Hilbert transform, the Lusin area function and the non-tangential maximal function. Chapter 6 contains a careful presentation of L. Carleson’s biorthogonal system. We show that it is an unconditional basis for

xiv

Preface

1 1 and thus prove B. Maurey’s theorem that Hat and H 1 are isomorphic Banach Hat spaces. The chapter closes with the classification theorem for atomic H 1 spaces.

Prerequisites. I assume only that the reader knows basic real, complex and functional analysis, and some probability theory, and refer to [76] by C. Goffman and G. Pedrick, to the first chapters of [163] by Z. Nehari, and to the appendix of [61] by R. Durrett. Acknowledgment. My research interests were formed by my collaboration with P. Wojtaszczyk at the Polish Academy of Sciences in Warszawa (1982–1984), with G. Schechtman at the Weizmann Institute of Science in Rehovot (1988–1991) and with P. W. Jones at Yale University in New Haven (1995–1998, and 2000). It is with great pleasure that I acknowledge the profound influence of P. Wojtaszczyk, G. Schechtman and P. W. Jones on my mathematical development and on the topics covered in this monograph. I would also like to thank those mathematicians I discussed with in Linz, M. Bl¨ umlinger, J. B. Cooper, A. Kamont, K. Kiener, B. Kirchheim, V. Pillwein, W. Schachermayer, M. Schmuckenschläger, C. Sch¨ utt, C. Stegall and A. Wakolbinger. I am especially indebted to B. Kirchheim with whom I organized a series of exciting seminars (1992–1995) which focussed intensely on problems of hard analysis. My records of the seminars were the starting point for the work on this book. Special thanks go to V. Pillwein who read the entire manuscript, made many valuable suggestions and detected numerous errors. Financial support was provided by the Austrian Science Foundation (FWF) project P15907-N08.

Chapter 1

The Haar System: Basic Facts and Classical Results This chapter introduces the Haar system. It contains basic inequalities associated with the Haar system and their interpretation as boundedness properties of Haar multipliers, paraproducts, rearrangement operators and orthogonal projections.

1.1

Bases in Lp

In this section we establish that the Haar system is an unconditional basis in Lp (1 < p < ∞). Proving inequalities of Khintchine and Burkholder we obtain a square function characterization of these spaces. We show that unconditionality of the Haar system implies that the Walsh system (in Walsh–Paley order) is a Schauder basis in Lp (1 < p < ∞). We discuss the multiplicity of Walsh functions and study associated approximation problems.

The Haar basis in Lp (1 ≤ p < ∞) An interval I ⊆ [0, 1] is called a dyadic interval if there exists n ∈ N ∪ {0}, and 1 ≤ k ≤ 2n such that k k−1 I= . , 2n 2n Let I1 be the left half of I and I2 be the right half of I. Then I1 , I2 are again dyadic intervals. Explicitly they are given as 2k − 2 2k − 1 2k − 1 2k and I . , = , I1 = 2 2n+1 2n+1 2n+1 2n+1 We call I1 , I2 the dyadic successors of I. Conversely I is called the dyadic predecessor of I1 and I2 . We denote by D the collection of all dyadic intervals, and

2

Chapter 1. The Haar System: Basic Facts and Classical Results

we let Dn denote the finite sub-collections {I ∈ D : |I| ≥ 2−n }. Observe that D is nested in the following sense. If I, J ∈ D are not disjoint, then either I ⊆ J or J ⊆ I. We define the L∞ normalized Haar system indexed by dyadic intervals as follows. For I ∈ D we let ⎧ ⎨ 1 on the left half of I, −1 on the right half on I, hI = ⎩ 0 otherwise. The Haar system is biorthogonal in L2 ([0, 1]). Following are several remarks relating the expansion with respect to the Haar system to a sequence of conditional expectation operators. Let An = {I ∈ D : |I| = 2−n }. Denote by En the conditional expectation with respect to the σ-algebra generated by An . Explicitly En (f ) is given as

1I . (1.1.1) f dt En (f ) = |I| I I∈An

Haar’s identity. The conditional expectation En (f ) and the expansion of f along the Haar system are related through A. Haar’s identity, 1 hJ hJ . En (f ) = f dt + f, (1.1.2) |J| 0 J∈Dn−1

The following remarks verify the basic identity (1.1.2). Let I be a dyadic interval, let I1 be the left half of I, and let I2 be the right half of I. These identities are evident, 2 · 1I1 = 1I + hI , (1.1.3) 2 · 1I2 = 1I − hI . Multiplying the equations of (1.1.3) relations 1 f dt = |I1 | I1 1 f dt = |I2 | I2

by a function f and integrating gives the 1 hI f dt + f, , |I| I |I| (1.1.4) 1 hI f dt − f, . |I| I |I|

Recall that hI = 1 on I1 and hI = −1 on I2 . Hence the two equations of (1.1.4) may be compressed into one line as follows. Let j ∈ {1, 2}, and let x ∈ Ij . Then, 1 hI 1 f dt = f dt + f, hI (x). (1.1.5) |Ij | Ij |I| I |I| Equation (1.1.5) relates the average over a dyadic interval, the Haar coefficient of its dyadic predecessor and the average over the dyadic predecessor. A moment’s

1.1. Bases in Lp

3

reflection shows that the identity expressed in (1.1.5) may be repeatedly applied. Carrying out the iteration shows the following statement. Let K ∈ D and let C(K) = {J ∈ D : J ⊃ K}. (We use J ⊃ K to denote that K is strictly contained in J.) Then for x ∈ K we obtain 1 |K|

K

1

f dt =

f dt + 0

J∈C(K)

f,

hJ hJ (x). |J|

(1.1.6)

Comparing (1.1.6) and (1.1.1) shows that the identity (1.1.2) holds true. The linear span of the indicator functions {1I : I ∈ D} forms a dense subspace of Lp ([0, 1]), 1 ≤ p < ∞. Thus Haar’s identity (1.1.2) implies that augmenting the Haar system {hI : I ∈ D} by the constant function 1[0,1] gives a complete biorthogonal system in L2 ([0, 1]). Partial sum operators associated to the Haar system. In order to form partial sum operators we need to impose a linear ordering on the Haar system {hI : I ∈ D}. We do this by associating an integer to the binary expansion of the left endpoint of a dyadic interval. Let n ∈ N ∪ {0}, and 1 ≤ k ≤ 2n . We form I ∈ D and i ∈ N, as k k−1 and i = 2n + k − 1. , I= 2n 2n Then define hi = h I .

(1.1.7)

By the defining equation (1.1.7) the sequence {hi }∞ i=1 is mapped bijectively onto the Haar system indexed by dyadic intervals {hI : I ∈ D}. We include the constant function by putting h0 = 1[0,1] . Denote by P the partial sum operator with respect to the sequence of Haar functions {hi }∞ i=0 . Thus P (f ) =

f,

i=0

hi hi .

hi 22

(1.1.8)

The representation formula (1.1.9) below implies that the partial sum operators {P }∞ =0 are given by a sequence of conditional expectations. This statement summarizes A. Haar’s original analysis [79] of the orthogonal system that is now named after him. Proposition 1.1.1. For every ∈ N there exists a collection of pairwise disjoint dyadic intervals P so that

1I . (1.1.9) f dt P (f ) = |I| I I∈P

4


Proof. Let ∈ N. Let L ∈ D be the dyadic interval which is uniquely defined by the relation h = h L . Let n ∈ N be such that |L| = 2−n . Let An = {I ∈ D : |I| = 2−n }. Next depending on the position of L we will define a splitting of An into RL and LL . Given L we let RL = {K ∈ An : sup K ≥ sup L + 2−n }. The collection RL contains those dyadic intervals of length 2−n which are strictly to the right of L. Next let LL = {K ∈ An : sup K ≤ sup L}. Thus LL contains the interval L and the dyadic intervals of length 2−n which lie to the left of L. For K ∈ LL we let K1 be the left half of K and K2 be the right half of K. Next define the collection P by taking the union

P = RL ∪

{K1 , K2 }.

K∈LL

We now turn to verifying that with P the identity (1.1.9) holds true. The starting point is the observation that {hi : 1 ≤ i ≤ } = {hI : I ∈ Dn−1 ∪ LL }. Hence with I = Dn−1 ∪ LL we rewrite P (f ) as P (f ) =

1

f dt + 0

I∈I

f,

hI hI . |I|

(1.1.10)

Comparing the right-hand side of (1.1.10) with the identity in (1.1.6) gives (1.1.9). Remarks. 1. The above proof shows that the length of the intervals in P is related to the index by the following two-sided estimate, 2 1 ≤ |I| ≤ 2

for

I ∈ P .

(1.1.11)

2. Let F be the σ -algebra generated by the collection of pairwise disjoint dyadic intervals P . Then F ⊂ F+1 . Hence the partial sum operators of the Haar system are given by conditional expectation operators associated to an increasing sequence of σ -algebras. Schauder’s theorem. The next theorem states that on Lp ([0, 1]) the partial sum operators {P }∞ =0 are uniformly bounded. This is J. Schauder’s contribution to the analysis of the Haar system. The representation formula (1.1.9) provided the starting point for J. Schauder in [185].

1.1. Bases in Lp

5

Theorem 1.1.2. Let 1 ≤ p ≤ ∞, let f ∈ Lp ([0, 1]). Then for every ∈ N,

P f Lp ≤ f Lp .

(1.1.12)

p For 1 ≤ p < ∞, the linear span of {hi }∞ i=0 is dense in L ([0, 1]) and

lim P f − f Lp = 0.

→∞

(1.1.13)

Proof. By Proposition 1.1.1 the partial sum operator can be written as P (f ) =

I∈P

1I , f dt |I| I

(1.1.14)

where P is a collection of pairwise disjoint dyadic intervals. Applying H¨ older’s inequality to the right-hand side of (1.1.14) gives that,

P f pLp

p 1−p = f dt |I| I I∈P ≤ |f |p dt. I∈P

I

The sum appearing in the last line of the above estimate is just f pLp . Thus (1.1.12) holds true for 1 ≤ p < ∞. The case of L∞ follows directly from (1.1.14). The linear span of the indicator functions of dyadic intervals forms a dense subspace of Lp ([0, 1]), 1 ≤ p < ∞. Hence by (1.1.2) the linear span of the Haar system is dense in Lp ([0, 1]), 1 ≤ p < ∞. It remains to verify (1.1.13). Let f ∈ Lp ([0, 1]) and 1 ≤ p < ∞. Given > 0 there exist 0 ∈ N and g ∈ span{hi : i ≤ 0 } such that f − g Lp < . Then for ≥ 0 it is easy to see that g = i=0 bi hi . Hence P (g) = g and the following identity holds, f − P (f ) =f − g + P (g − f ). (1.1.15) The identity (1.1.15) and the norm estimate (1.1.12) imply (1.1.13).

For f ∈ Lp ([0, 1]), 1 ≤ p < ∞, we call the series ∞ i=0

ai h i

where

ai = f,

hi ,

hi 22

the Haar expansion of f and {ai } the sequence of Haar coefficients. By J. Schauder’s Theorem 1.1.2, the Haar expansion converges to f in Lp ([0, 1]) and the Haar coefficients are uniquely determined.

6


Schauder bases. The preceding theorem leads to the notion of bases in Banach spaces. A sequence {xi }∞ i=1 in a Banach space E is called a Schauder basis if for every x ∈ E there exists a unique sequence of coefficients {ai } so that lim x −

n→∞

n

ai xi E = 0.

i=1

Thus associated to a Schauder basis there is the sequence of partial sum operators defined by the equation

∞ n a i xi = a i xi . Pn i=1

i=1

A theorem of St. Banach asserts that for a Schauder basis {xi }∞ i=1 , the partial sum operators are uniformly bounded, i.e., sup Pn : E → E < ∞.

(1.1.16)

n

The supremum in (1.1.16) is called the basis constant of {xi }∞ i=1 in E. We denote by E ∗ the dual Banach space to E consisting of continuous linear functionals on E. A Schauder ∞basis gives rise to continuous biorthogonal functionals as follows. For x = i=1 ai xi , define a linear functional x∗i : E → R by putting x∗i (x) = ai . Thus x∗i (x)xi = (Pi − Pi−1 )(x), hence

x∗i E ∗ xi E ≤ 2 sup Pn : E → E < ∞, n

and x∗i (xj )

=

1 0

if i = j, if i =

j.

By Theorem 1.1.2 for 1 ≤ p < ∞, the Haar system {hi }∞ i=0 is a Schauder p basis in Lp ([0, 1]). The basis constant of {hi }∞ in L ([0, 1]) is equal to 1. The i=0 biorthogonal functional to hi is hi / hi 22 .

Khintchine’s inequality The Khintchine inequality asserts that for series of Rademacher functions the norm in L2 and the norm in Lp , p < ∞, are equivalent. For n ∈ N, we let hI , (1.1.17) rn = {I:|I|=2−n+1 }

1.1. Bases in Lp

7

denote the n-th Rademacher function. The sequence {rn , n ∈ N} is the Rademacher system. It consists of statistically independent functions taking the values {+1, −1}, each with probability one-half. Following is the Khintchine inequality for L1 . Theorem 1.1.3. (Khintchine’s inequality). For every sequence of real numbers {an } and for every N ∈ N,

1 0

N 1/2

N 1 2 an rn (y) dy ≥ √ an . 2 n=1 n=1

(1.1.18)

The constant √12 , appearing on the right-hand side of Khintchine’s inequality is best possible. In this sharp form the inequality (1.1.18) is due to S. Szarek [200]. The Walsh system. While the Khintchine inequality itself involves just the Rademacher functions, the proof relies heavily on certain properties of the Walsh system. We recall the definition of the Walsh system. Let A be a finite set of nonnegative integers. Then the Walsh function wA is defined as a product of Rademacher functions as follows, rk . wA = k∈A

Using the empty set as index we denote the constant function as 1[0,1] = w∅ . The Walsh system {w∅ } ∪ {wA : A ⊂ N, A finite} is complete and orthonormal in L2 ([0, 1]). Hence every X ∈ L2 ([0, 1]) can be expanded in its Walsh series, X= X, wA wA . The Haar system and the Walsh system are related by the basic identity of J.L. Walsh, (1.1.19) span{wA : max A ≤ n} = span{hI : |I| ≥ 2−n+1 }. Hence if X ∈ L2 ([0, 1]) is constant on all dyadic intervals of length = 2−n , then X ∈ span{w∅ } ∪ {wA : max A ≤ n}, and conversely. The dyadic derivative. Given a finite linear combination of Walsh functions X, we define the dyadic derivative of X by the equation D(X) = |A|X, wA wA . Proof of Theorem 1.1.3. Fix real numbers {an }, let N ∈ N and form N an rn (t) . X(t) = n=1

8


Proving Kintchine’s Inequality amounts to showing that the following estimate holds, 1 2 1 2 X (t)dt ≤ 2 X(t)dt . 0

0

N Overview. Below we will see that the function X(t) = | n=1 an rn (t)| is distinguished by satisfying the following two very peculiar properties: If |A|X, wA = 0, then |A| ≥ 2.

(1.1.20)

For every t ∈ [0, 1], we have X(t) ≥ D(X)(t).

(1.1.21)

Let us accept temporarily that X(t) satisfies the properties (1.1.20) and (1.1.21) and let us show how Kintchine’s inequality follows. 1 1 X 2 (t)dt ≥ X(t)D(X)(t)dt 0 0 = |A|X, wA 2 ≥ 2X, wA 2 |A|≥2

1

0

Now subtract 2 −1. This gives

1 0

2

1

X 2 (t)dt − 2

=2

X(t)dt

.

0

X 2 (t)dt on both sides and multiply the resulting inequality by

1 0

2

1

X 2 (t)dt ≤ 2

X(t)dt

,

0

as claimed. Verification of (1.1.20). We prove now that X, wA = 0 for |A| = 1. The Rademacher functions are those Walsh functions wA for which |A| = 1. Hence (1.1.20) is just stating that X, rn = 0, for every Rademacher function. Note that each Rademacher function is odd around the point 1/2. That is for 0 < t ≤ 1/2 we have that 1 1 rn (−t + ) = −rn (t + ). 2 2 By taking absolute values we find that N X(t) = an rn (t) n=1

1.1. Bases in Lp

9

is even around the same point 1/2. The interval [0, 1] is clearly symmetric around the point 1/2. Hence, 1 X(t)rn (t)dt = 0. 0

Verification of (1.1.21). Next we prove the pointwise estimate X ≥ DX. Note that X is constant on all intervals of length = 2−N . In other words, X depends only on (products and sums of) the Rademacher functions r1 (t), . . . , rN (t). Consequently, there exists a function F such that X(t) = F (r1 (t), . . . , rN (t)). The next identity for the dyadic derivative is easy to check for individual Walsh functions wA , hence by linear extension it holds in general. It should be thought of as the dyadic chain rule. 2 · D(X) = F (r1 , . . . , rN ) − F (−r1 , . . . , rN ) + F (r1 , r2 , . . . , rN ) − F (r1 , −r2 , . . . , rN ) .. . + F (r1 , . . . , rN ) − F (r1 , . . . , −rN ). Clearly the N summands on the left column of the above sum are identical and equal to X. Now we let Rj denote the operator acting on the j-th coordinate of F by reversing the sign. That is (Rj F )(r1 , . . . , rj , . . . , rN ) = F (r1 , . . . , −rj , . . . , rN ). We may rewrite the dyadic chain rule for X in the following way, 2 · D(X) = N · X −

N

(Rj F )(r1 , . . . , rN )

j=1

=N ·X −

N

Rj (|a1 r1 + · · · + aN rN |).

j=1

Now we continue by finding a minorization of the sum on the right-hand side. By the triangle inequality we have N N Rj (|a1 r1 + · · · + aN rN |) ≥ Rj ((a1 r1 + · · · + aN rN )) . j=1 j=1 The sum on the right-hand side is clearly a linear combination of the Rademacher functions r1 , . . . , rN . Observe that each Rademacher function appears N −2 times.

10


Hence N

Rj ((a1 r1 + · · · + aN rN )) = (N − 2)(a1 r1 + · · · + aN rN ).

j=1

Combining we observe the following lower bound, N

Rj (|a1 r1 + · · · + aN rN |) ≥ (N − 2)|a1 r1 + · · · + aN rN |.

j=1

It remains to insert this estimate in the identity expressed by the dyadic chain rule. This gives that, 2 · D(X) ≥ N · X − (N − 2) · |a1 r1 + · · · + aN rN | = 2X.

Kahane’s inequality. We may now review the proof of Theorem 1.1.3. We showed that for any choice of scalar coefficients (an ), the function an rn (t) X(t) = satisfies the following surprising estimate, 1 2 X (t)dt ≤ 2 0

2

1

X(t)dt

.

0

J. P. Kahane extended Khintchine’s inequality in the following way. For any normed linear space E and any sequence (xn ) in E the non-negative function Z(t) = xn rn (t) E

satisfies

1

Z (t)dt ≤ C 2

0

2

1

Z(t)dt

,

0

where C is a universal constant. The preceding estimate is known as Kahane’s inequality. We point out that the proof of Theorem 1.1.3 not only shows Kahane’s inequality, but also that the constant in Kahane’s inequality is the same as in the real case. Indeed, we have that 1 2 1 Z 2 (t)dt ≤ 2 Z(t)dt . 0

0

Next we prove Khintchine’s inequality for Lp , where p ≥ 2. The constant we √ obtain in Khintchine’s inequality is of order p. For p → ∞, this gives the right asymptotics.

1.1. Bases in Lp

11

Theorem 1.1.4. Let p ≥ 2. Let {an } be a sequence of real numbers. Then the following inequality holds,

1

| 0

N

1/p p

an rn (t)| dt

1/2

N √ 2 ≤A p an .

n=1

n=1

Proof. Let a ∈ R. Then for each Rademacher function rn the following inequality holds, 1 1 exp{arn }dt = (exp{a} + exp{−a}) 2 0 (1.1.22) 2 a . ≤ exp 2 N Next fix {an } and normalize such that n=1 a2n = 1. Then we form F (t) =

N

an rn (t).

n=1

Let µ > 0. As the Rademacher functions {rn } are independent we may iterate inequality (1.1.22) to obtain that

1

exp{µF } ≤ 0

N

exp{

n=1

µ2 µ2 a2n } = exp{ }. 2 2

Next observe that exp{µ|F (t)|} ≤ exp{µF (t)} + exp{−µF (t)}. Hence,

1

exp{µ|F |} ≤ 2 exp{ 0

µ2 }. 2

By the inequality of Chebyshef, the following estimate holds for the distribution function of |F |. For every α > 0, 1

exp{µ|F (t)|}dt exp α µ2 − α}. ≤ 2 exp{ 2

|{µ|F | > α}| ≤

0

(1.1.23)

We specialize the estimate (1.1.23) by taking α = µ2 , then |{|F | > µ}| ≤ 2 exp{−

µ2 }. 2

(1.1.24)

Integrating inequality (1.1.24) gives an upper bound for ||F ||pp . After a change of variables we find an estimate for ||F ||pp in terms of the Gamma function, for which

12


standard inequalities exist. Indeed,

1

|F (t)| dt = p p

0

∞

sp−1 |{|F | > s}|ds 2 ∞ s p−1 ds s exp − ≤ 2p 2 0 ∞ p−2 p−2 = 2p2 2 r 2 exp{−r}dr 0 p−2 p−2 2 . = 2p2 Γ 2 0

At this point we use a simple estimate for the Gamma function. Namely, p p−2 ≤ (Ap) 2 . Γ 2

Remark. With a surprising trick we obtain from Theorem 1.1.4 the√L1 version of Khintchine’s inequality (albeit with a constant that is larger that 2). Define N X= a n rn . n=1

1/3

2/3

Hölder’s inequality implies that X 2 ≤ X 1 X 4 . Apply Theorem 1.1.4 2/3 1/3 with p = 4. This gives X 4 ≤ 2A X 2 . Cancelling X 2 we get X 2 ≤ 1/3 (2A)2/3 X 1 hence the L1 version of Khintchine’s inequality with constant 4A2 . Basic sequences. In a Banach space E, a sequence {yi }∞ i=1 is called a basic sequence if it is a Schauder basis in its closed linear span. Two basic sequences ∞ {yi }∞ i=1 in E and {zi }i=1 in F are called equivalent if there exist c > 0 and C > 0 so that n n n ai zi F ≤

ai yi E ≤ C

ai zi F , c

i=1

i=1

i=1

for every n ∈ N and every choice of scalars a1 . . . an . Let 1 ≤ p < ∞. Khintchine’s inequality asserts that the Rademacher functions in Lp are equivalent to the unit vector basis in the Hilbert space 2 (in particular the closed linear span of the Rademacher functions is isomorphic to 2 ). If moreover 1 2 and f ∈ Lp we obtain from Theorem 1.1.4 that

∞ 1/2

∞ √ 2 f, rn rn ≤ A p f, rn . n=1

Consequently,

(1.1.25)

n=1

p

√ √

R1 (f ) p ≤ A p f 2 ≤ A p f p .

Next choose 1 < q < 2 and let p be the Hölder conjugate exponent defined by 1/p + 1/q = 1. Then p > 2. Let g ∈ Lq be a finite linear combination of Haar functions. Then R1 (g) is a finite linear combination of Rademacher functions. Determine f ∈ Lp so that f p = 1 and R1 (g) q = R1 (g), f . Using that R1 is self adjoint, we obtain from (1.1.25) that R1 (g), f = g, R1 (f ) ≤ g q R1 (f ) p √ ≤ A p g q .

(1.1.26)

√ Hence R1 (g) q ≤ A p g q , as claimed.

Burkholder’s inequality We give a proof of Burkholder’s inequality, and obtain the square function characterization of Lp by merging the inequalities of Burkholder and Khintchine. Theorem 1.1.5. Let 1 < p < ∞. Let f ∈ Lp ([0, 1]) with

1

f=

f dt + 0

Let I ∈ {−1, 1}, then

g=±

aI h I .

I∈D

1

f dt + 0

I aI hI ,

I∈D

satisfies the estimate ||f ||p ≤ (p∗ − 1)||g||p , where p∗ = max{p, p/(p − 1)}. Proof. By a simple duality argument to prove the case when p > 2. it suffices Then p∗ = p. We assume that f = f + aI hI is a finite linear combination of Haar functions. We will show that ||f ||p ≤ (p − 1)||g||p .

14


With the function v(x, y) = |x|p − (p − 1)p |y|p , we have that

||f ||pp − (p − 1)p ||g||pp =

1

v(f (t), g(t))dt. 0

Hence we have to show that

1

v(f (t), g(t))dt ≤ 0.

(1.1.27)

0

To this end we define Burkholder’s most famous function, 1 u(x, y) = p(1 − )p−1 (|x| + |y|)p−1 (|y| − (p − 1)|x|). p

(1.1.28)

This function became famous because of the following four properties: (B 1) u dominates v, v(x, y) ≤ u(x, y). (B 2) For every a ∈ R and ∈ {−1, 1}, u(x + a, y + a) + u(x − a, y − a) ≤ 2u(x, y). (B 3) u is symmetric, u(x, y) = u(−x, −y). (B 4) u(0, 0) = 0. We denote the mean value of f over the dyadic interval J by fJ . That is, 1 f. fJ = |J| J Correspondingly we denote the mean value of g over J by gJ . Let J1 be the left half of J and let J2 be the right half of J. We claim that for Burkholder’s function u the following inequality holds: |J1 |u(fJ1 , gJ1 ) + |J2 |u(fJ2 , gJ2 ) ≤ |J|u(fJ , gJ ).

(1.1.29)

Note that for t ∈ J1 we have that hJ (t) = 1 whereas for t ∈ J2 , hJ (t) = −1. Hence by (1.1.3) for f the following identities hold, fJ1 = fJ + aJ , fJ2 = fJ − aJ ,

1.1. Bases in Lp

15

and analogously we have for g, gJ1 = gJ + J aJ , gJ2 = gJ − J aJ . Inserting the above identities and using property (B 2) of Burkholder’s function we have that u(fJ1 , gJ1 ) + u(fJ2 , gJ2 ) = u(fJ + aJ , gJ + J aJ ) + u(fJ − aJ , fJ − J aJ ) ≤ 2u(fJ , gJ ). It remains to multiply both sides with |J|/2 to see that the claimed inequality (1.1.29) holds true. Next fix k ∈ N and sum the inequality (1.1.29) over those dyadic intervals for which the length is equal to 2−k . Thus we obtain from inequality (1.1.29) this crucial rescaling estimate, |K|u(fK , gK ) ≤ |J|u(fJ , gJ ). (1.1.30) {K:|K|=2−k }

{J:|J|=2−k+1 }

Note that the right-hand side of the estimate (1.1.30) is a rescaled version of the left-hand side; the difference is that the length of the dyadic intervals we are summing over has increased by a factor of 2. The assertion of (1.1.30) is that by rescaling from intervals of length 2−k to larger intervals of length 2−k+1 we are increasing the value of the sum. Now we turn to verifying (1.1.27). As f and g are finite linear combinations there exists a (large) constant k0 so that f and g are constant on dyadic intervals of length ≤ 2−k0 . Hence, 1 v(f (t), g(t))dt = v(fK , gK )|K|. 0

{K:|K|=2−k0 }

Recall that Burkholder’s function u is a pointwise majorant of v, hence, v(fK , gK ) ≤ u(fK , gK ), and we obtain the following line from which we start an inductive procedure, ||f ||pp − (p − 1)p ||g||pp ≤ u(fK , gK )|K|. (1.1.31) {K:|K|=2−k0 }

Now we exploit the rescaling estimates (1.1.30). We start with the right-hand side of the inequality (1.1.31) where the summation is taken over 2k0 intervals of length = 2−k0 . Call the resulting sum σ(k0 ). Repeatedly applying the rescaling estimate (1.1.30) shows that the following chain of inequalities holds true, σ(k0 ) ≤ σ(k0 − 1) ≤ · · · ≤ σ(1) ≤ σ(0).

16


To form σ(0) we take the sum over one interval of length =1. Thus σ(0) = u(f[0,1] , g[0,1] ). Finally we recall that the Burkholder function u is symmetric, by (B 3), and that u(0, 0) = 0. This gives that u(fK , gK )|K| ||f ||pp − (p − 1)p ||g||pp ≤ {K:|K|=2−k0 }

≤ u(f[0,1] , g[0,1] ) 1 u(f[0,1] , g[0,1] ) + u(−f[0,1] , −g[0,1] ) = 2 ≤ u(0, 0) = 0.

Given f ∈ Lp ([0, 1]) we define the square function of f as follows, S(f ) =

a2I h2I

1/2 ,

where

aI = f,

hI . |I|

Khintchine’s inequality together with Burkholder’s theorem imply a square function characterization of Lp , when 1 < p < ∞. The original proof of Theorem 1.1.6, with different constants is due to R. E. A. C. Paley. Theorem 1.1.6. Let 1 < p < ∞. Let f ∈ Lp ([0, 1]) such that f = 0. Then √ √ (2 2)−1 (p∗ − 1)−1 ||f ||p ≤ ||S(f )||p ≤ 2(p∗ − 1)||f ||p , where p∗ = max{p, p/(p − 1)} denotes the constants appearing in Burkholder’s inequality. Proof. Let f ∈ Lp ([0, 1]), and expand it in its Haar series, f= aI hI . Let {rI } be an enumeration of the Rademacher system indexed by dyadic intervals. For y ∈ [0, 1], we define aI hI rI (y). fy = Then by Burkholder’s inequality, Theorem 1.1.5, for each y ∈ [0, 1], the following inequality holds, 1 1 |fy (t)|p dt ≤ (p∗ − 1)p |f (t)|p dt. 0

0

Now we pass from the above pointwise estimate, holding for every y ∈ [0, 1], to the average over y ∈ [0, 1]. Clearly the average satisfies the same upper bound, that is, 1 1 1 p ∗ p |fy (t)| dt ≤ (p − 1) |f (t)|p dt. 0

0

0

1.1. Bases in Lp

17

Next we fix t ∈ [0, 1] and we apply Minkowski’s inequality followed by Khintchine’s inequality, Theorem 1.1.3, 1 p 1 |fy (t)|p dy ≥ |fy (t)|dy 0 0 p 1 √ S(f )(t) . ≥ 2 It remains to use Fubini’s theorem and string together the above estimates. p 1 1 1 1 √ S p (f )(t)dt ≤ |fy (t)|p dydt 2 0 0 0 1 1 (1.1.32) |fy (t)|p dtdy = 0

0

≤ (p∗ − 1)p

1

|f (t)|p dt. 0

This proves the right-hand side inequality in Theorem 1.1.6. We now turn to the left-hand side. Here we distinguish between the cases p ≥ 2 and p ≤ 2. We start with the latter case, p ≤ 2. We apply Burkholder’s inequality in the following way. For each y ∈ [0, 1], 1 1 p ∗ p |f (t)| dt ≤ (p − 1) |fy (t)|p dt. (1.1.33) 0

0

Consequently the same estimate holds true if we take the average over y ∈ [0, 1] of the right hand side in (1.1.33). Thus 1 1 1 |f (t)|p dt ≤ (p∗ − 1)p |fy (t)|p dtdy. (1.1.34) 0

0

0

Next observe that for p ≤ 2, Minkowski’s inequality gives the estimate

1

p/2

1

|fy (t)|p dy ≤

|fy (t)|2 dy

0

.

(1.1.35)

0

Finally we combine the estimates (1.1.34) and (1.1.35) and apply Fubini’s theorem as follows, 1

1

|f (t)|p dt ≤ (p∗ − 1)p

0

|fy (t)|p dtdy 0

≤ (p∗ − 1)p

0

1

= (p − 1)

p/2

1

|fy (t)|2 dy 0

∗

1

p

0 1

S p (f )(t)dt. 0

dt

18


Summing up we showed that for 1 < p ≤ 2 the left-hand side estimate in Theorem 1.1.6 holds, that is, (p∗ − 1)−1 ||f ||p ≤ ||S(f )||p . Next we turn to the case when p ≥ 2. Here we use duality. Let q ≤ 2 be the H¨ o lder conjugate index to p, thus 1/p + 1/q = 1. Let h ∈ Lq with ||h||q ≤ 2, and h = 0 so that ||f ||p = f, h. By the inequalities of Cauchy–Schwarz and H¨ older we obtain that

1

f, h ≤

S(f )(t)S(h)(t)dt 0

≤ ||S(f )||p ||S(h)||q . We established above (1.1.32). Thereby we proved the right-hand √ side estimate of Theorem 1.1.6. Now we apply (1.1.32) obtaining ||S(h)||q ≤ 2(q ∗ − 1)||h||q . As ||h||q ≤ 2, this gives √ ||f ||p ≤ 2 2(q ∗ − 1)||S(f )||p . Finally we observe that for p ≥ 2, we have q ≤ 2, and also q ∗ − 1 = p − 1 = p∗ − 1. Summing up for p ≥ 2, √ ||f ||p ≤ 2 2(p∗ − 1)||S(f )||p . Unconditional bases. In a Banach space E let {xi }∞ i=1 be a Schauder basis with ∞ biorthogonal functionals {x∗i }∞ . We say that {x } i i=1 is an unconditional basis i=1 for E if the following condition holds: There exists K > 0 so that for each x ∈ E, n ∗ i xi (x)xi ≤ K x E . (1.1.36) sup sup n i ∈{−1,1} i=1

E

The infimum over all constants K > 0 satisfying (1.1.36) is the unconditional basis constant of {xi }∞ i=1 in E. By Schauder’s Theorem 1.1.2 the Haar system is a basis in Lp ([0, 1]), 1 ≤ p < ∞. Burkholder’s inequality asserts that for 1 < p < ∞ the Haar system is actually an unconditional basis and that the unconditional basis constant of the Haar system in Lp is bounded by max{p, p/(p − 1)} − 1. The constant max{p, p/(p − 1)} − 1 is known to be best possible (see for instance Chapter 8 in [165] by I. Novikov and E. M. Semenov). The UMD property of Banach spaces. Before examining further consequences of Burkholder’s Theorem 1.1.5 we will devote a paragraph to the definition of the Banach spaces for which the vector-valued extension of Burkholder’s theorem

1.1. Bases in Lp

19

holds true. A Banach space satisfies the UMD property if for 1 0 so that the following estimate holds, p p 1 1 p I aI hI (t) dt ≤ Cp aI hI (t) dt, (1.1.37) 0 0 I∈H

I∈H

E

E

for every finite collection H of dyadic intervals, every {aI ∈ E : I ∈ H} and every choice of signs {I ∈ {+1, −1} : I ∈ H}. Basic properties of UMD spaces are recorded in B. Maurey’s seminal paper on the Haar system [139]. These include: (a) If E has the UMD property, then every closed subspace of E is reflexive. (b) The observation of G. Pisier that if (1.1.37) holds for some p0 (1 < p0 < ∞) then (1.1.37) holds for all 1 < p < ∞. In particular to verify the UMD property for E it suffices to establish (1.1.37) for L2 . B. Maurey’s work on the Haar system was the starting point for intensive investigations of Banach spaces with the UMD property. We pay close attention to the methods developed for the study of UMD spaces and present in detail the following: (a) B. Maurey’s proof that martingale differences are unconditional in the reflexive Lp spaces (Theorem 1.4.1). (b) J. Bourgain’s proof of E. M. Stein’s martingale inequality (Theorem 1.4.2). (c) T. Figiel’s representation of singular integral operators and T. Figiel’s expansion of biorthogonal systems (Theorem 1.5.3 and Theorem 1.4.3). (d) The proof by A. Naor and G. Schechtman of G. Pisier’s inequality on dyadic gradients (Theorem 1.1.7).

The Walsh system in Lp (1 < p < ∞) Here we use the Haar basis to study the Walsh system. In particular we establish estimates for Walsh series by applying the unconditionality of the Haar basis in Lp . The best known application is Paley’s proof that the Walsh system (in Walsh– Paley order) is a Schauder basis in Lp . A recent connection between the Haar basis and the Walsh system is established in the work of A. Naor and G. Schechtman on Lp estimates for dyadic gradients. Besides expansion in Walsh–Paley order we test a different and natural method of approximation using Walsh functions. The method is briefly described as follows: First project a given function f onto the span of Walsh functions wA with |A| = m. Denote the result by Rm (f ). To recover f use the associated partial sum operators n Rm (f ), m=0

20


and let n tend to infinity. Clearly this method works well for the Hilbert space L2 . Below however we will present the surprising theorem of B. Roider to the effect that for any p = 2 there is f ∈ Lp which is not approximated by the above partial sums. The proof presented here is due to K. Kiener, who builds on core ideas of P. Enflo’s construction of a separable Banach space without the approximation property. The following link between the Walsh system and the Haar system was ob1 served by J. L. Walsh in 1923. Let g ∈ Lp ([0, 1]) with 0 gdt = 0. Let k ∈ N, and put dk = g, wA wA . {A⊂N : max A=k}

By definition, the sequence {dk } is disjointly supported over the Walsh system. Simultaneously, and this is the point of Walsh’s observation, {dk } is also disjointly supported over the Haar system. Indeed, the following identity holds, hI g, hI . dk = |I| −k+1 {I : |I|=2

∞

}

Consequently g = k=1 dk , with convergence in Lp . Moreover with Burkholder’s theorem the observation of Walsh implies that ∞ ±dk ≥ (p∗ − 1)−1 g Lp . (1.1.38) k=1

Lp

The Poincaré inequality for the dyadic gradient We apply the unconditionality of the Haar basis in Lp to prove a dyadic version of Poincaré’s inequality. The dyadic gradient is defined as follows. Let N ∈ N, and 1 fix f so that f − 0 f dt ∈ span{wA : max A ≤ N }. Hence the Walsh expansion of f is of the form 1 f= f dt + cA wA , where cA ∈ R. (1.1.39) 0

A⊆{1,...,N }

Thus f is expressed through the Rademacher functions {r1 , . . . , rN } and can be represented as f (t) = F (r1 (t), . . . , rN (t)). Then the dyadic partial derivative ∂i is defined by the equation 1 [F (r1 (t), . . . , ri (t), . . . , rN (t)) − F (r1 (t), . . . , −ri (t), . . . , rN (t))] . 2 In the course of proving Khintchine’s inequality we observed that the dyadic derivative D(f ), defined through the Walsh expansion of f by Df = |A|wA , f wA , ∂i f (t) =

1.1. Bases in Lp

21

satisfies Df =

N

∂i f.

i=1

This identity holds true since ∂i acts on individual Walsh functions by the relation

if i ∈ A, if i ∈

A.

wA 0

∂i wA =

(1.1.40)

Finally define the dyadic gradient ∇f = (∂i f )N i=1 and put

|∇f | =

N (∂i f )2

1/2 .

i=1

Theorem 1.1.7. Let 1 < p < ∞. Let f ∈ Lp have Walsh expansion (1.1.39). Then

1

f −

f dt Lp ≤ Cp ∇f Lp . 0

Proof. We start by isolating an algebraic identity relating ∂i to the Walsh expansion of f. Let ai ∈ R, and let i ≤ N. Put g=

N

ai ∂i f.

i=1

Equation (1.1.40), the defining relation for ∂i , gives the following useful representation of g. For t ∈ [0, 1],

g(t) =

A⊆{1,...,N }

ai

f, wA wA (t).

(1.1.41)

i∈A

Next we split the Walsh expansion of g using the basic observation of Walsh. For k ≤ N, put

ai f, wA wA (t). (1.1.42) dk (t) = {A:max A=k}

i∈A

By the identity of Walsh, the sequence {dk } is disjointly supported over the Haar system. Hence Burkholder’s Theorem 1.1.5 gives the lower bound N ±dk k=1

≥ (p∗ − 1)−1 g Lp . Lp

(1.1.43)

22


Now we focus on finding estimates for ∇f Lp . Recall that {ri } denotes the Rademacher system. Fix s ∈ [0, 1] and i ≤ n. We will apply the above observations using ai = ri (s). Displaying the dependence on s ∈ [0, 1] we write

ri (s) f, wA wA (t). dk (s, t) = {A:max A=k}

i∈A

Note that the integral identity below holds, since the integrands on the right-hand side and on the left-hand side coincide. p p 1 1 1 1 N N ri (s)∂i f (t) dsdt = dk (s, t) dsdt. (1.1.44) 0 0 0 0 i=1

k=1

Khintchine’s inequality gives pointwise estimates for the length of the dyadic gradient. For t ∈ [0, 1], we have that 1 |∇f (t)|p ≥ (Ap)p/2

1

0

p N ri (s)∂i f (t) ds. i=1

Integrate over t ∈ [0, 1] and invoke the identity (1.1.44). This gives

∇f pLp

1 ≥ (Ap)p/2

1

0

1

0

p N dk (s, t) dsdt.

(1.1.45)

k=1

Next we apply Burkholder’s inequality in the form (1.1.43). Clearly rk (s) = ±1 for each fixed s ∈ [0, 1]. Hence the following lower bound holds, N dk (s, ·)

∗

≥ (p − 1)

−1

Lp

k=1

N rk (s)dk (s, ·)

.

(1.1.46)

Lp

k=1

Integrate the estimate (1.1.46) over s ∈ [0, 1]. By triangle inequality, the integral of the norms ≥ norm of the integral. Thus we arrive at the following lower bound for ∇f Lp . N 1 rk (s)dk (s, ·)ds , (1.1.47) Cp ∇f Lp ≥ 0 Lp

k=1

where Cp = (Ap)1/2 (p∗ − 1). It remains to evaluate the expression appearing on the right-hand side of (1.1.47) . A glance at the definition of dk (s, t) gives that

1

rk (s)dk (s, t)ds = 0

{A:max A=k}

f, wA wA (t).

1.1. Bases in Lp

23

Hence for Cp = (Ap)1/2 (p∗ − 1), we found the lower bound N Cp ∇f Lp ≥ f, w w (t) A A k=1 {A:max A=k}

.

Lp

Finally observe that 1the right-hand side of the above estimate coincides with the norm in Lp of f − 0 f. Pisier’s inequality. Estimates for the dyadic gradient were established first by G. Pisier [178] in the context of vector-valued coefficients cA ∈ E, where E is a Banach space. Following is the statement of G. Pisier’s original inequality. Let

1

f=

f dt + 0

where cA ∈ E.

cA wA ,

(1.1.48)

A⊆{1,...,N }

Equation (1.1.40) defines the dyadic partial derivatives {∂i f } by linear extension, and {ri } denotes the Rademacher functions, then

1

f (x) − 0

0

1

f pE dx ≤ (2e log N )p

1

0

N

ri (t) ∂i f (x) pE dtdx.

(1.1.49)

i=1

In the vector-valued case the constant on the right-hand side depends on N. M. Talagrand [201] shows that 2e log N, appearing in (1.1.49) is of the correct order. Moreover in [201] M. Talagrand proves the scalar case with a constant independent of N. The proof of Theorem 1.1.7 presented above is due to A. Naor and G. Schechtman [162]. Verify, by inspection, that their proof carries over to the case where the coefficients in the Walsh expansion of f belong to a Banach space with the UMD property. Convergence and divergence of Walsh series So far we were concerned with estimates for finite linear combinations of Walsh functions, hence questions of convergence did not arise. Passing from finite sums to infinite series gives rise to interesting and subtle problems of ordering the Walsh system. For the Walsh–Paley enumeration {wj }∞ j=0 we show that for 1 < p < ∞, n ∗ sup sup f, wj wj ≤ p − 1,

f p ≤1 n j=0

(1.1.50)

p

where p∗ = max{p, p/(p − 1)}. Clearly this estimate implies convergence of the corresponding partial sums in Lp . On the other hand, ordering the Walsh system

24


along increasing multiplicity leads to divergence of the corresponding partial sums in Lp for p = 2. We prove below that for 1 < p = 2 < ∞, n sup sup Rm (f ) = ∞, (1.1.51)

f p ≤1 n m=0

p

where Rm denotes the orthogonal projection onto Walsh functions of multiplicity m. The estimate (1.1.50) is due to R.E.A.C Paley, the particular constant p∗ − 1 is a consequence of Burkholder’s Theorem 1.1.5. The lower bound (1.1.51) is a result of B. Roider. The Walsh–Paley order. We now define the Walsh–Paley order in {wA }, and show Paley’s theorem that (thus ordered) the Walsh system is a Schauder basis in Lp , 1 < p < ∞. Let n ∈ N, with dyadic expansion n=

∞

j 2j ,

j=0

where j ∈ {0, 1}. (Clearly only finitely many j are = 0.) Then we define wn =

∞

j rj+1 .

(1.1.52)

j=0

We also define w0 = 1. Note that w1 = r1 , w2 = r2 and that in general w2m = rm+1 . Moreover the sequence {wn }∞ n=1 is an enumeration of the Walsh system {wA : A ⊂ N, A finite }. Paley’s identity. We prepare the proof of Paley’s theorem by stating identities relating the Walsh system to the Haar system. First we have that k 2 −1

j=0

wj =

k

(1 + rn )

n=1

(1.1.53)

= 2k 1[0,2−k [ . Let I be a dyadic interval with |I| = 2−k . The identity (1.1.53) leads to a formula for the Walsh expansion of the Haar function hI . Moreover for t ∈ I and x ∈ [0, 1], 2k+1 −1

wj (t)wj (x) = 2k hI (t)hI (x).

(1.1.54)

j=2k

The above formulas were known to (and used by) Walsh in 1923. Following is Paley’s identity from 1931 which is the key to his proof that the Walsh system

1.1. Bases in Lp

25

p can be ordered ∞to become a Schauder basis in L , 1 < p < ∞ : Let x, t ∈ [0, 1], and let n = k=0 k 2k . Then

wn (x)wn (t)

n−1

wj (x)wj (t) =

j=0

∞

k

k=0

2k hI (x)hI (t).

(1.1.55)

{I:|I|=2−k }

Next we show that the partial sum operators with respect to the Walsh system in Walsh–Paley order are convergent in Lp (1 < p < ∞). By the identity of Walsh it suffices to prove that they form a sequence of uniformly bounded operators. Theorem 1.1.8. Let 1 < p < ∞, and let {wj }∞ j=0 denote the Walsh system in Walsh–Paley order. Then the partial sum operator, Tn (f ) =

n−1

f, wj wj ,

j=0

satisfies the norm estimate

Tn f p ≤ (p∗ − 1) f p ,

(1.1.56)

where p∗ = max{p, p/(p − 1)}. Proof. Fix n ∈ N. Instead of working directly with Tn we consider the operator Tñ : f → wn Tn (wn f ). Clearly in Lp the norms of the operators Tn and Tñ coincide. Hence it suffices to show the estimate (1.1.56) with Tn replaced by Tñ . The following observation is crucial to Paley’s proof. It identifies Tñ as a {0, 1} multiplier on the Haar(!) system. Precisely, we claim that for a given n there exist coefficients δI ∈ {0, 1} so that hI δI f, hI . (1.1.57) Tñ (f ) = |I| I∈D

To verify (1.1.57) we observe that Tñ (f )(x) =

n−1

wn f, wj wj (x)wn (x)

j=0

1

=

f (t)wn (x)wn (t) 0

n−1

(1.1.58) wj (x)wj (t)dt.

j=0

The kernel appearing in the second line of (1.1.58) has been identified in (1.1.55). It is equal to ∞ j 2j hI (t)hI (x), j=0

{I:|I|=2−j }

26


where j ∈ {0, 1} is determined by the dyadic expansion of n. Replacing the kernels and carrying out the integration gives that Tñ (f ) =

∞

j

f,

{I:|I|=2−j }

j=0

hI hI . |I|

Having established (1.1.57) it suffices to apply Theorem 1.1.5 to conclude that

Tñ (f ) p ≤ (p∗ − 1) f p .

The multiplicity of Walsh functions. The multiplicity of a given Walsh function is defined to be the number of Rademacher functions used in its definition. Formally, let A be a finite subset of N. Then the multiplicity of wA = j∈A rj , is the cardinality of A. For w∅ = 1[0,1] we define the multiplicity to be equal to 0. Let Wm denote the Walsh functions of multiplicity m. Let {wm,k }∞ k=1 be any enumeration of the elements in Wm . Let Rm denote the orthogonal projection onto the span of Wm . Thus for f ∈ L2 , Rm (f ) =

∞

wm,k , f wm,k .

(1.1.59)

k=1

For m = m the sets Wm and Wm are disjoint, hence the spaces span{Wm } are orthogonal and ∞

f 22 =

Rm (f ) 22 . m=0

It follows that the partial sums ∞ n

wm,k , f wm,k

(1.1.60)

m=0 k=1

form a convergent sequence in L2 approximating f. Hence elements in L2 are well approximated by adding Walsh functions of increasing multiplicity. We now study this method of approximation in Lp , when p = 2. The starting point is a theorem obtained by A. Bonami [15] and K. Kiener [112] independently. It extends Khintchine’s inequality to Walsh functions of multiplicity m. Theorem 1.1.9. Let 1 0 and 2 Cm,p > 0 so that for every sequence {ak }∞ k=1 ∈ ,

cm,p

∞

k=1

1/2 a2k

∞ 1/2 ∞ 2 ≤ ak wm,k ≤ Cm,p ak . k=1

p

√ m (a) If p > 2, then cm,p = 1 and Cm,p ≤ C p .

k=1

(1.1.61)

1.1. Bases in Lp

27

−2 . (b) If p < 2, then Cm,p = 1 and cm,p ≥ Cm,4

Moreover Rm , the orthogonal projection onto the span of Walsh functions with multiplicity m, is a bounded and self-adjoint operator on Lp , 1 < p < ∞. Proof. First we treat p ≥ 2. The proof proceeds by induction. For m = 1 the theorem holds by Khintchine’s inequality. Next assume that the theorem holds for a given m ∈ N. We will now show that it holds for m + 1. Let Km+1 = {A ⊂ N : |A| = m + 1}. The sets {A ∈ Km+1 : max A = n},

n ∈ N,

form a partition of Km+1 . Let {aA : A ∈ Km+1 } be a sequence where only finitely many entries are = 0. Define h = A∈Km+1 aA wA . Let n ∈ N and put aA wA\{n} rn . (1.1.62) dn = A∈Km+1 max A=n

We get h = dn , and by the identity of Walsh, the sequence {dn } is disjointly supported over the Haar system. The square function characterization of Lp implies that 1/2 .

h p ≤ Cp d2n (1.1.63) p √ (We prove in Theorem 1.3.4 below that Cp ≤ C p for p ≥ 2.) Note that rn is a common factor of each of the Walsh functions appearing in (1.1.62). Hence ⎛ ⎞2 ⎜ ⎟ aA wA\{n} ⎠ . d2n = ⎝ A∈Km+1 max A=n

Since p ≥ 2, the triangle inequality in Lp/2 and the above formula for d2n imply that the right-hand side of (1.1.63) is bounded by ⎛ 2 ⎞1/2 ⎟ ⎜ ⎟ a w (1.1.64) Cp ⎜ ⎠ . A A\{n} ⎝ n∈N A∈Km+1 max A=n

p

Observe that for A ∈ Km+1 and max A = n the Walsh function wA\{n} has multiplicity m. In fact {wA\{n} : A ∈ Km+1 , max A = n} equals {wB : B ∈ Km , max B ≤ n − 1}. Using the induction hypothesis we obtain that 2 2 aA wA\{n} ≤ Cm,p a2A . A∈Km+1 A∈Km+1 max A=n

p

max A=n

28


Recall that the collections {A ∈ Km+1 , max A = n}, n ∈ N form a partition of Km+1 . Thus we obtained that ⎛

h p ≤ Cm+1,p ⎝

⎞1/2 a2A ⎠

.

(1.1.65)

A∈Km+1

Let Rm be the orthogonal projection onto the span of Walsh functions with multiplicity m. Clearly Rm f 2 ≤ f 2 , and (since p ≥ 2) f 2 ≤ f p . Hence (1.1.65) implies that for f ∈ Lp ,

Rm f p ≤ Cm,p Rm f 2 ≤ Cm,p f p .

(1.1.66)

We pass to p ≤ 2 using a well-known trick based on H¨ older’s inequality. Clearly it suffices to consider p = 1. Define h= a A wA . A∈Km

1/3

2/3

Hölder’s inequality implies that h 2 ≤ h 1 h 4 , and by the first part we get 2/3 1/3 2/3 1/3

h 4 ≤ Cm,4 h 2 . Cancelling h 2 gives h 2 ≤ Cm,4 h 1 which finishes the proof for p ≤ 2. Given Theorem 1.1.9, K. Kiener conjectured that the convergence of the partial sums (1.1.60) could be extended to Lp (1 < p = 2 < ∞). This conjecture was disproved by B. Roider (1974) in a letter to K. Kiener. Below we present K. Kiener’s proof [113] of B. Roider’s result that for p = 2 there exists g ∈ Lp which is not approximated by its partial sums ∞ n

wm,k , gwm,k .

m=0 k=1

The proof of K. Kiener [113] starts with the combinatorial identity (1.1.76) which played a central role in the construction of the first Banach space without the approximation property. Specifically (1.1.76) appears in P. Enflo’s classical work [62]. Roider’s example. Note that Rm , the orthogonal projection onto the span of Walsh functions with multiplicity m, is a self-adjoint operator. Hence it suffices to prove (1.1.51) for p < 2. To this end we fix n ∈ N, and consider f = 22n 1[0,2−2n [

(1.1.67)

1.1. Bases in Lp

29

Observe that by (1.1.53) f can be expanded using Walsh functions of multiplicity ≤ 2n. Thus f ∈ span{W0 , . . . , W2n }. Next project f onto the space of Walsh functions with multiplicity ≤ n, and denote the resulting function by X. Thus n

X=

Rm (f ).

(1.1.68)

m=0

Clearly in L2 we have X 2 ≤ f 2 . The next theorem shows that for p = 2 the norm of X in Lp is considerably larger than that of f. Theorem 1.1.10. Let 1 0 is a universal constant independent of n or p. Proof. We begin by defining a partition of the unit interval. (On each set of this partition we will later identify the value of X.) Let 0 ≤ r ≤ 2n. Define Pr ⊂ [0, 1] by the following rule: Let t ∈ [0, 1], then put t ∈ Pr if there exists A ⊆ {1, . . . , 2n} so that |A| = r and so that for each m ∈ {1, . . . , 2n} −1, rm (t) = 1,

if m ∈ A, if m ∈ / A.

(1.1.70)

The proof splits naturally into three parts. First we show that the function X is constant on the sets {Pr } and we represent the value of X on {Pr } by an elegant complex integral formula. This part of the proof is based on Enflo’s identity. Second we parameterize the complex integrals and find closed formulas for their value. Third we use this information to obtain a lower bound for X p . The first two parts of the proof consist in identifying the values of X(t) defined by (1.1.68) on the sets {Pr }. First, by definition, |X(t)| = 1 + 2n 1 + · · · + 2n , for t ∈ P ∪ P . We will show next that for t ∈ P , we have 2|X(t)| = 2n 0 2n 1 n n and that in general for k = 1, 2, . . . n − 1, 2|X(t)| =

−1 2n n 2n n k 2k

for t ∈ P2k ∪ P2k+1 .

(1.1.71)

We obtain the identities (1.1.71) from an elegant formula due to K. Kiener. X(t) =

1 2πi

(1 − z)r−1 (1 + z)2n−r

dz z n+1

for t ∈ Pr ,

where the contour integral is taken over the boundary of the unit disk.

(1.1.72)

30


Applications of Cauchy’s integral formula. We turn to proving (1.1.72). Recall 22n −1 in its Walsh series gives f = j=0 wj . Hence that expanding f = 22n 1[0,2−2n [ for m ≤ 2n, we have Rm (f ) = w∈Wm w. Now fix t ∈ [0, 1] and z ∈ D. Then define 2n z m Rm (f )(t). (1.1.73) F (z, t) = m=0

Verify the following identity by multiplying the factors of the product on the right-hand side of (1.1.74), F (z, t) =

2n

(1 + zrm (t)).

(1.1.74)

m=1

The product representation of F (z, t) yields that for fixed r ≤ 2n and z ∈ D the the function t → F (z, t) is constant on Pr . Indeed for t ∈ Pr , 2n

(1 + rm (t)z) = (1 − z)r (1 + z)2n−r .

(1.1.75)

m=1

We apply the Cauchy integral formula to (1.1.73) and use the equation established in (1.1.75). This gives Enflo’s identity that for t ∈ Pr , Rm (f )(t) =

1 2πi

(1 − z)r (1 + z)2n−r

dz , z m+1

(1.1.76)

where the contour integral is taken over the boundary of the unit disk. Take the sum of the equations (1.1.76) using that n

z −m−1 =

m=0

(1 − z n+1 ) . (1 − z)z n+1

Hence by summing (1.1.76) and Cauchy’s integral formula we find that for t ∈ Pr , n m=0

Rm (f )(t) =

1 2πi

(1 − z)r−1 (1 + z)2n−r

dz . z n+1

(1.1.77)

Thus we verified the complex integral representation (1.1.72). We remark that the representation of X(t) in (1.1.72) differs from that of Rn (f )(t) by a factor of (1−z) in the contour integral.

1.1. Bases in Lp

31

Evaluating complex integrals. Next we determine the complex integral (1.1.77). With the following parametrization we replace (1.1.77) by a standard definite integral whose value is well known. Let z = eiθ . Then (1.1.77) equals 2π dθ 1 (1 − eiθ )r−1 (1 + eiθ )2n−r inθ . (1.1.78) 2π 0 e We simplify the integrand of (1.1.78) using elementary trigonometric identities. e−iθ/2 (1 − eiθ ) = (−2i) sin(θ/2)

and

e−iθ/2 (1 + eiθ ) = 2 cos(θ/2). (1.1.79)

Consequently, e−inθ (1 − eiθ )r−1 (1 + eiθ )2n−r = (−i)r−1 22n−1 e−iθ/2 sinr−1 (θ/2) cos2n−r (θ/2) ! " = (−i)r−1 22n−1 sinr−1 (θ/2) cos2n−r+1 (θ/2) − i sinr (θ/2) cos2n−r (θ/2) . Next replace the integrand in (1.1.78) by the last line of the above equation. Then, by the change of variables ϕ = θ/2 and dθ = 2dϕ, we find that (1.1.78) coincides with the integral " (−i)r−1 22n π ! r−1 sin (ϕ) cos2n−r+1 (ϕ) − i sinr (ϕ) cos2n−r (ϕ) dϕ. 2π 0

(1.1.80)

To derive the pointwise estimates (1.1.71) we evaluate the trigonometric integrals of (1.1.80). This requires us to distinguish between the cases where r is odd and where r is even. Observe that cos(ϕ) is an odd(!) function around the point π/2 and that sin(ϕ) is even(!) around π/2. Clearly if r is an even number, then 2n − r + 1 is odd. Hence π/2 π sinr (ϕ) cos2n−r (ϕ)dϕ = 2 sinr (ϕ) cos2n−r (ϕ)dϕ, 0

0

and

π

sinr−1 (ϕ) cos2n−r+1 (ϕ)dϕ = 0.

0

Similarly if r is odd, then 2n − r is odd, hence π π/2 r−1 2n−r+1 sin (ϕ) cos (ϕ)dϕ = 2 sinr−1 (ϕ) cos2n−r+1 (ϕ)dϕ, 0

and

0

π 0

sinr (ϕ) cos2n−r (ϕ)dϕ = 0.

32


Next recall that for x, y > 0 we have π/2 Γ(x)Γ(y) , (1.1.81) sin2x−1 (ϕ) cos2y−1 (ϕ) = 2 Γ(x + y) 0 ∞ √ where Γ(x) = 0 tx−1 e−t dt. In particular Γ(n + 1) = n! and Γ(1/2) = π. In summary it follows from Kiener’s integral representation (1.1.80) and the identity (1.1.81) that for r = 2k and t ∈ P2k , Γ k + 12 Γ n − k + 12 22n |X(t)| = × . (1.1.82) 2π n! Similarly, for r = 2k + 1 and t ∈ P2k+1 , we obtain from (1.1.80) and (1.1.81) that Γ k + 12 Γ n − k + 12 22n |X(t)| = × . (1.1.83) 2π n! Thus (for different reasons) the value of X(t) on the set P2k coincides with its value on P2k+1 . We replace the expressions involving Gamma functions by more elementary formulas. Recall the so-called duplication formula for the Gamma function. It states that for x > 0, √ π Γ(2x) 1 . Γ(x + ) = 2x−1 2 2 Γ(x) Apply the duplication formula with x = k and write 1/2 = k/(2k). This gives, √ 1 π (2k)! . Γ(k + ) = 2k 2 2 (k)! Apply it to x = n − k and use 1/2 = (n − k)/(2n − 2k) to obtain that √ 1 π (2n − 2k)! . Γ(n − k + ) = 2n−2k 2 2 (n − k)! Substitute into (1.1.82) and (1.1.83), then simplify to obtain (1.1.71) by arithmetic. The lower bound for X p . Now we turn to the last part of the proof where we 1 apply (1.1.71) to obtain a lower bound for the integral 0 |X(t)|p dt. Since X(t) is constant on the sets Pr we denote Xr = X(t) for t ∈ Pr . Note that the disjoint sets Pr , 0 ≤ r ≤ 2n are defined so that their measure is given by |Pr | = 2−2n 2n r . Hence (1.1.71) gives 1 2n 2n |X(t)|p dt = |Xr |p 22n r 0 r=0 (1.1.84)

n−1 2n 2n + . ≥ |X2k |p 2k 2k + 1 k=0

1.1. Bases in Lp

33

The formula for evaluating consecutive binomial coefficients gives 2n 2n 2n + 1 + = . 2k 2k + 1 2k + 1 2n Note that 2n+1 2k+1 ≥ 2k . Inserting the values (1.1.71) for |X2k | we obtain the identities p p −(p−1) 2n 2n n 2n |2X2k |p = . 2k n k 2k Then by (1.1.84) we obtain that

1

|X(t)|p dt ≥

22n+p 0

p n−1 np 2n−(p−1) 2n × . k 2k n

(1.1.85)

k=0

Recall Hölder’s inequality in the form n−1

−(p−1) apk bk

k=0

≥

n−1

ak

p n−1

k=0

−(p−1) bk

.

(1.1.86)

k=0

Apply it with ak = nk and bk = 2n 2k . We continue by giving closed formulas for the two factors appearing on the right-hand side of (1.1.86). n−1 k=0

n k

= 2n − 1 and

n−1 k=0

2n 2k

≤ 22n .

In summary we obtained a lower bound for the right-hand side of (1.1.85). It follows that p 1 2n |X(t)|p dt ≥ × 2−np × 2−p−1 . n 0 √ The remaining calculations on Stirling’s formula, n! ∼ 2πn · nn · e−n . 2n are based It allows us to compute n ∼ 22n n−1/2 . Inserting this value in the previous lower bound for the norm of X in Lp we find 1 |X(t)|p dt ≥ Cn−p/2 2np . 0

Next recall that we put f = 22n 1[0,2−2n [ . Hence f p dt = 22n(p−1) . Consequently the following lower estimate holds true. (1.1.87) |X(t)|p dt ≥ Cn−p/2 2−np+2n f p dt.

34


Recall that Rn denotes the orthogonal projection onto the span of Walsh functions with multiplicity n. We pointed out that the integral representations (1.1.72) for X and (1.1.76) for Rn (f ) are almost identical. With this observation it is easy to see that the proof of Theorem 1.1.10 gives a lower estimate for the operator norm of Rn on Lp . Theorem 1.1.9, on the other hand, provides an upper estimate. In summary the following estimates hold, √ c2cn(1−2/p) ≤ Rn Lp ≤ (C p)n , for p > 2. (1.1.88) Comparing the right and left hand side in (1.1.88) gives rise to the following problem. Problem 1.1.11. Narrow the gap between the lower and upper estimates for the operator norm of Rn on Lp . Notice that the test function (1.1.67) in the proof of Theorem 1.1.10 is very special. Hence it is likely that the lower estimate in (1.1.88) can be improved. Since the appearance of Paley’s work the Walsh system has been continuously investigated from the point of view of harmonic analysis and approximation theory. Moreover the Walsh system plays a remarkable role in the analysis of classical Banach spaces where it is used to exhibit spaces and operators with extremal (and often unexpected) properties. P. Enflo’s original construction of a Banach space without the approximation property is one of the most outstanding examples representing this line of investigation (see the notes for references). Through the work of B. Roider and K. Kiener in 1974 the methods of P. Enflo found an early application to a concrete approximation problem for Walsh functions in Lp . In 1992 J. Bourgain [25] extended B. Roider’s theorem: Denoting by Xnp the subspace of Lp spanned by the Walsh functions {wA : A ⊆ {1, . . . , 2n}, |A| ≤ n}, J. Bourgain [25] proves a trace estimate for any operator T on Lp whose range is contained in Xnp , (1.1.89) tr(T) ≤ B22n(1−γ) T Lp , where B = B(p) and γ = γ(p) > 0 when p = 2. The trace of the orthogonal projection onto Xnp is larger than c22n , hence (1.1.89) gives a lower estimate for the norm of the projection. Thus (1.1.89) implies Roider’s estimate (1.1.51).

1.2

Dyadic H 1

In this book we will work intensively with the following dyadic versions of H 1 and BMO. The space BMO consists of all square integrable functions, having mean zero, bJ hJ h= J∈D

1.2. Dyadic H 1

35

for which

⎞1/2 1 = sup ⎝ b2J |J|⎠ < ∞. |I| I∈D ⎛

||h||BMO

J⊆I

The space H 1 consists of integrable functions f with integrable square functions. For 1 f dt + aJ h J f= 0

J∈D

the square function is given by

S(f )(x) =

1/2 a2J 1J (x)

,

J∈D

and H 1 is the space of integrable functions for which 1 1 f dt + S(f )(x) dx < ∞. ||f ||H 1 = 0

0

Fefferman’s inequality Next we identify BMO as the dual space of H 1 . We begin by showing that each h ∈ BMO induces a bounded linear functional on H 1 . This follows from the next theorem, the content of which is known as Fefferman’s inequality. In this section we present two proofs of Fefferman’s inequality. Theorem 1.2.1. (Fefferman’s inequality) Let f be a finite linear combination of Haar functions, and let h ∈ BMO, then √ f h ≤ 2 2||f ||H 1 · ||h||BMO . Proof. Let h =

bJ hJ be the Haar expansion of h ∈ BMO. Since f hdt = f, hJ bJ ,

we may assume that {J : bJ = 0} is contained in {J : f, hJ = 0}. In particular we assume that h is a finite linear combination of Haar functions. Let I be a fixed dyadic interval. We define S(h | I), the square function localized to I as ⎛ ⎞1/2 S(h | I)(x) = ⎝ b2J 1J (x)⎠ . J⊆I

36


Now fix a point x in the unit interval. Note that S(h | I)(x) < ∞ for every dyadic interval I. Next define I(x) to be the largest dyadic interval containing x such that 1 2 bL |L|. (1.2.1) S 2 (h | I(x))(x) ≤ 2 sup I x |I| L⊆I

Note that, by definition, we have the following upper bound for the square function localized to the interval I(x), √ (1.2.2) S(h | I(x))(x) ≤ 2||h||BM O , x ∈ [0, 1]. Thus Fefferman’s inequality is reduced to showing that the following estimate holds true, f hdx ≤ 2 S(f )(x)S(h | I(x))(x)dx. (1.2.3) To prove the integral estimate (1.2.3) we verify first that for every dyadic interval J, the following measure estimate holds, |{x ∈ J : I(x) ⊇ J}| ≥ |J|/2.

(1.2.4)

To this end we fix J ∈ D and consider the set A = {x ∈ J : I(x) is strictly contained in J}. Then using (1.2.1), we estimate S 2 (h | J)(x) dx ≥ S 2 (h | J)(x) dx J A 1 2 sup bL |L| dx >2 A x∈I |I| L⊆I |A| ≥2 S 2 (h | J)(x) dx. |J| J

(1.2.5)

The last inequality in (1.2.5) used the following pointwise bound that holds for any x ∈ J, 1 1 2 sup bL |L| ≥ S 2 (h | J)(x) dx. |J| J I x |I| L⊆I Finally in (1.2.5) we cancel the factor J S 2 (h | J)(x) dx and obtain that |A| ≤ |J|/2. Taking complements we obtain the measure estimate (1.2.4), that is |{x ∈ J : I(x) ⊇ J}| ≥

1 |J|. 2

We rewrite the estimate (1.2.4), in the following integral form, using indicator functions, 1 1 1J (x) · 1{y:I(y)⊇J} (x) dx ≥ |J|. 2 0

1.2. Dyadic H 1 Now fix f = follows.

37

aJ hJ in H 1 . With the preceding analysis of h, we estimate as f h dx ≤ |aJ bJ ||J| ≤2 |aJ bJ | 1J (x)1{y:I(y)⊇J} (x) dx |bJ aJ |1J (x) dx =2 ≤2

I(x)⊇J

S(f )(x)S(h | I(x)) dx.

Thus we verified that (1.2.3) holds. Recalling the pointwise upper bound (1.2.2) completes the proof of Fefferman’s inequality. Let h be square integrable, with Haar expansion 1 h= hdx + bJ hJ . 0

J∈D

At a given point x ∈ [0, 1] the sharp function of h is defined to be ⎛

⎞1/2 1 b2J |J|⎠ , h (x) = ⎝sup I |I| J⊆I

where the supremum is extended over all dyadic intervals which contain the given point x ∈ [0, 1]. Looking back at the proof of Theorem 1.2.1 we observe that the sharp function played an important role in the proof of Fefferman’s inequality. It appeared on the right-hand side of the estimate (1.2.1). Even more, the sharp function appears implicitly in the definition of BMO. Indeed, by inspection, the space BMO consists of all functions with uniformly bounded sharp functions, and ||h||BMO = sup h (x).

(1.2.6)

x∈[0,1]

Observe that combining (1.2.1) and the integral estimate (1.2.3) we obtain the following inequality, that immediately gives Fefferman’s inequality, 1 √ 1 ≤2 2 f hdx S(f )h dx. (1.2.7) 0

0

It is worth pointing out that the above proof focusses exclusively on the BMO end function h. The fact that f ∈ H 1 , enters only at the √ very 1 when we observe that the right hand side of (1.2.7) is bounded by 2 2 h ∞ 0 S(f )dx. As an application of Khintchine’s inequality we next prove a converse to Fefferman’s inequality. In that way we are showing that Theorem 1.2.1 is sharp.

38


Theorem 1.2.2. For each f ∈ H 1 , with

f = 0, there exists h ∈ BMO such that

1 f, h ≥ √ ||f ||H 1 · ||h||BMO . 2 2 Proof. For the purpose of applying Khintchine’s inequality to the Haar coefficients of a given function f , it is more convenient to first re-index the sequence of Rademacher functions using dyadic intervals. Let {rI : I ∈ D} be any enumeration of the Rademacher functions. Let f ∈ H 1 , and let f= aI h I be its Haar expansion. For y ∈ [0, 1], we define fy by its Haar expansion as fy = aI hI rI (y). Note that fy ∈ H 1 . Applying Khintchine’s inequality we obtain that 1 0

1 0

1 |fy |dy dx ≥ √ 2

a2I h2I (x)

1/2

1 dx = √ ||f ||H 1 . 2

Changing the order of integration one finds at least one point y ∈ [0, 1] such that 1 ||fy ||L1 ≥ √ ||f ||H 1 . 2 Now we choose g ∈ L∞ such that g, fy = ||g||L∞ ·||fy ||L1 . Next we expand g − g in its Haar series, g− g= cI h I , and define h=

I cI hI ,

where I = rI (y). By the biorthogonality of the Haar system we have that f, h = fy , g, and by the definition of the BMO norm we have also, ||h||BMO = ||g − g||BMO ≤ 2||g||L∞ . Combining these observations gives f, h = fy , g = ||fy ||L1 ||g||∞ 1 ≥ √ ||f ||H 1 ||h||BMO . 2 2

1.2. Dyadic H 1

39

By Fefferman’s inequality every h ∈ BMO defines a continuous linear functional on the space H 1 , and moreover the unit ball of BMO norms H 1 by the converse to Fefferman’s inequality. We summarize Theorems 1.2.1 and 1.2.2 as follows. Let f ∈ H 1 with f = 0, then √ 1 √ ||f ||H 1 ≤ sup{ f h dx : ||h||BMO ≤ 1} ≤ 2 2||f ||H 1 . (1.2.8) 2 2 We now show that every continuous linear functional on H 1 , determines a function in BMO. For n ∈ N let fn : [0, 1] → R be integrable. Recall that the sequence 1 2 (fn )∞ n=1 belongs to the space L ( ) if ||(fn )∞ n=1 ||L1 (2 )

1

= 0

∞

1/2 dx < ∞.

fn2

n=1

The dual Banach space of L1 (2 ) is canonically identified with L∞ (2 ), consisting of sequences (hn )∞ n=1 so that

||(hn )∞ n=1 ||L∞ (2 )

= sup x∈[0,1]

∞

1/2 h2n (x)

< ∞,

n=1

where “supx∈[0,1] ” denotes the essential supremum over the unit interval, defining the norm in L∞ . Theorem 1.2.3. For every continuous linear functional L : H 1 → R there exists h ∈ BMO so that ||h||BMO ≤ ||L||, (1.2.9) and so that for every finite linear combination of Haar functions g,

1

hgdt.

L(g) =

(1.2.10)

0

Proof. The idea of the proof comes with the useful observation that H 1 is canonically represented as a closed subspace of L1 (2 ). Hence by the Hahn–Banach theorem every continuous linear functional extends to an element in the dual space 1 2 ∞ 2 1 of L ( ), which is the space L ( ). The details are as follows. Let f ∈ H , with f = 0. For n ∈ N let hI f, hI . (1.2.11) fn = |I| −n−1 {I:|I|=2

}

Then the square function of f satisfies S(f ) =

fn2

1/2 .

40


Thus the linear map f → (fn ) provides the embedding of H 1 to a closed subspace of L1 (2 ), and ||f ||H 1 = ||(fn )∞ n=1 ||L1 (2 ) . With the Hahn Banach theorem extend the linear functional L to a functional on ˜ : L1 (2 ) → R, be an extension satisfying L1 (2 ), with the same norm. Let L ˜ ||L|| = ||L||, ˜ n )), L(f ) = L((f for f ∈ H 1 , and (fn )∞ n=1 defined by (1.2.11). Next recall that the dual space of L1 (2 ) is identified as L∞ (2 ), in the following sense. There exists a sequence (hn ) such that 1/2

˜ = h2n (x) , ||L|| sup x∈[0,1]

and for every (xn ) ∈ L1 (2 ), ˜ n )) = L((x

∞

1

xn hn dt.

(1.2.12)

0

n=1

Next we define h, using the sequence (hn ), by putting h=

∞

hn ,

n=1 {I:|I|=2−n−1 }

hI hI . |I|

We test the equality (1.2.12) on the sequence (fn ) defined by (1.2.11). This gives ˜ n )) = L((f

∞

f,

n=1 {I:|I|=2−n−1 }

hI hI , hn |I|

= f, h. ˜ is given by integration against h, Thus when restricted to (fn ) the action of L

1

˜ n )) = L((f

f hdt. 0

This proves (1.2.10). Now we show that h ∈ BMO, and that

||h||BMO ≤

sup x∈[0,1]

1/2 h2n (x)

.

1.2. Dyadic H 1

41

Evaluate the BMO norm of h by fixing a dyadic interval J with |J| = 2−m−1 . Let (h)J denote the mean value of h over the interval J. By Bessel’s inequality we obtain that, hI |h − (h)J |2 dt = hn , 2 |I| |I| J −n−1 n≥m {I⊆J:|I|=2

≤

∞

J n=1

}

h2n dt.

Dividing by |J| and then taking the supremum over all dyadic intervals J, we find that

1/2 ∞ 2 sup hn (x) ,

h BMO ≤ x∈[0,1] n=1

and consequently, ˜ ||h||BMO ≤ ||L||

which proves (1.2.9).

We will now present a different approach to Fefferman’s inequality. In this approach the proof is carried out by analyzing the functions in H 1 . We will show that every f ∈ H 1 with small enough norm can be decomposed into a convex combination of dyadic atoms. A function a is a dyadic atom if there exists a dyadic interval I such that supp a ⊆ I, 1 a2 (t)dt ≤ |I|−1 , 0

and

1

a(t)dt = 0. 0

We will also say that the constant function 1[0,1] is a dyadic atom. For a given f ∈ H 1 with Haar expansion xI h I , f= I∈D

we define the Haar support of f to be the following collection of dyadic intervals, {I ∈ D : xI = 0}. In the proof of Theorem 1.2.4 we obtain the decomposition of f into atoms by first decomposing the Haar support of f into pairwise disjoint collections of dyadic intervals. At this point we use, for the first time in this book, a stopping time decomposition of the Haar support. Many more applications of this fundamental technique will be given in later chapters.

42


Theorem 1.2.4. For f ∈ H 1 , there exists a sequence of dyadic atoms, (ai ) and a sequence of scalars (ci ), such that 1 ∞ f− f= ci a i , 0

and

∞

i=1

|ci | ≤ 16

1

S(f ). 0

i=1

Conversely, every dyadic atom a belongs to H 1 and satisfies 1 S(a)dt ≤ 1. 0

Before we present the proof we should point out how Theorem 1.2.1 (Fefferman’s inequality) can be deduced from the atomic decomposition of H 1 . Theorem 1.2.4 may be paraphrased by saying that it suffices to prove Fefferman’s inequality only for dyadic atoms a ∈ H 1 . Now fix a dyadic atom a and a dyadic interval I such that 1 supp a ⊆ I, and 0 a2 (t)dt ≤ |I|−1 . Next fix h ∈ BMO. Let (h)I denote the mean 1 value of h over the interval I. Then, using 0 adt = 0 and the Cauchy–Schwarz inequality, we estimate as follows, 1 ahdt = a(h − (h)I )dt 0

I

≤ a L2 1I (h − (h)I ) L2 . Finally, observe that the biorthogonality of the Haar system gives that

1I (h − (h)I ) L2 ≤ |I|1/2 h BMO . Below the following notations are repeatedly used. For a collection of dyadic intervals F we denote by max F the maximal intervals of F. As dyadic intervals are nested, max F consists of pairwise disjoint intervals. For a dyadic interval I we denote by Q(I) the collection of all dyadic intervals which are contained in I, thus Q(I) = {J ∈ D : J ⊆ I}. Proof of Theorem 1.2.4. Let f ∈ H 1 , with Haar expansion f= xJ h J . 1 Thus we assume that 0 f = 0. We will decompose f by splitting its Haar support. We start with the unit interval [0, 1]. Let n([0, 1]) be the smallest integer such that Ω([0, 1]) = {t ∈ [0, 1] : S(f )(t) ≥ 2n([0,1]) }

1.2. Dyadic H 1

43

satisfies |Ω([0, 1])| ≤

1 . 4

Let F([0, 1]) be the collection of dyadic intervals which are contained in Ω([0, 1]). Then define C([0, 1]) = Q([0, 1]) \ F([0, 1]). Note that f[0,1] =

xJ h J

J∈C([0,1])

is defined to satisfy the pointwise estimate S(f[0,1] ) ≤ 2n([0,1]) . Next we define E1 to be the collection of maximal dyadic intervals of F([0, 1]), that is, E1 = max F([0, 1]). We continue the construction inductively. Suppose we have already defined collections of pairwise disjoint dyadic intervals E1 , · · · , Ei . Fix I ∈ Ei . Let n(I) be the smallest integer such that Ω(I) = {t ∈ I : S(f )(t) ≥ 2n(I) } satisfies |Ω(I)| ≤

1 |I|. 4

Let F(I) be the collection of dyadic intervals which are contained in Ω(I). Then let C(I) = Q(I) \ F(I). We define now fI =

xJ h J

J∈C(I)

which satisfies the pointwise estimate S(fI ) ≤ 2n(I) . Moreover the support of S(fI ) is contained in the dyadic interval I. Hence S(fI )2 ≤ |I|22n(I) . Now we let E(I) = max F(I),

44


and we take the union over I ∈ Ei to define Ei+1 = {E(I) : I ∈ Ei }. As Ei consists of pairwise disjoint dyadic intervals, it is easy to see that Ei+1 is a collection of pairwise disjoint dyadic intervals. This completes the induction step. # Finally we define E = Ei . Recall that along with the construction of E we verified that fI aI = |I|2n(I) is a dyadic atom. This suggests a decomposition of f, f=

fI

I∈E

=

|I|2n(I) aI .

I∈E

It remains to show that the coefficients in front of the atoms are in fact summable. We do this by proving that |I|2n(I) ≤ 16 S(f ). (1.2.13) I∈E

We start by observing that the stopping time definition of n(I) implies the measure estimate |I| . |{t ∈ I : S(f )(t) ≥ 2n(I)−1 }| ≥ 4 Then we continue with the proof of the convexity relation (1.2.13),

|I|2n(I) ≤ 4

I∈E

2n(I) |{t ∈ I : S(f ) ≥ 2n(I)−1 }|

I∈E

≤8

2n(I)−1 |{t ∈ I : S(f ) ≥ 2n(I)−1 }|

I∈E

≤8

2k |{S(f ) ≥ 2k }|

k∈N

≤ 16

S(f ).

We find an atomic decomposition of f by rewriting the equation f = f=

I∈E

|I|2n(I)

fI . |I|2n(I)

I∈E

fI as

1.2. Dyadic H 1

45

The converse is actually very easy. We fix an atom a, supported in a dyadic interval I such that a2 ≤ |I|−1 . As a = 0, the dyadic square function S(a) has its support in I as well. Hence by the Cauchy–Schwarz inequality we estimate S(a) ≤ |I|1/2 ( S 2 (a))1/2 . The hypothesis that a is an atom implies of course ( S(a)2 )1/2 ≤ |I|−1/2 . Thus we showed that S(a) ≤ 1. In the course of the above proof we decomposed f by splitting the Haar support into pairwise disjoint collections of dyadic intervals {C(I) : I ∈ E}. Being the result of a stopping time decomposition, the collections C(I) satisfy a very important connectedness property: First, the collection C(I) contains exactly one maximal interval, namely I. Second, the following holds true. Let J ∈ C(I) and let K be contained in the Haar support of f, then J ⊆ K ⊆ I implies K ∈ C(I). The collection C(I) is called a block of dyadic intervals in the Haar support of f. Next the index set E and its construction deserves further # analysis, that provides an important link to BMO. Recall that we defined E = Ei . Reviewing the proof we gave of Theorem 1.2.4 shows that the collections Ei are linearly ordered and shrinking uniformly at a geometric rate. Precisely, the following holds. 1. If I ∈ Ei , J ∈ Ej and I ⊆ J, then i ≥ j. 2. If J ∈ Ej and i ≥ j

|I| ≤ 2−i+j |J|.

{I∈Ei : I⊆J}

As a result we obtain that E satisfies the Carleson packing condition: For any J ∈ E, |I| ≤ 2|J|. (1.2.14) {I∈E:I⊆J}

Summing up: The proof of Theorem 1.2.4 produced a decomposition of the Haar support into blocks of dyadic intervals {C(I) : I ∈ E}, such that the index set E satisfies the Carleson packing condition (1.2.14).

Sharp maximal functions in Lp Renorming Lp with the square function ||S(f )||p and then passing to the limit p → 1 gives H 1 , which is thus a natural endpoint of the Lp scale. We will show that in the opposite direction BMO appears as endpoint of the Lp scale when we renorm

46


with the sharp function and then pass to the limit p → ∞. Let f ∈ L2 ([0, 1]), with Haar expansion f= f+ bI hI . For x ∈ [0, 1], the sharp function is defined as ⎛

⎞1/2 1 f (x) = ⎝sup b2J |J|⎠ , x∈I |I| J⊆I

where the supremum is taken over all dyadic intervals containing x. We observed in (1.2.6), by comparing the definitions of BMO and f , that f ∈ BMO if f ∈ L∞ . Now we show that the sharp function provides an equivalent renorming of Lp when p > 2. Theorem 1.2.5. Let p > 2. If f ∈ Lp ([0, 1]) with f = 0, then (8(p − 1))−1 ||f ||p ≤ ||f ||p ≤ 2

p p−2

1/p ||f ||p .

Proof. Let q = p/(p − 1) be the Hölder conjugate index of p. As p > 2 we have that q < 2. Choose g ∈ Lq with ||g||q ≤ 2 and g = 1, so that

1

||f ||p =

f gdt. 0

Now recall the integral estimate (1.2.3) and the pointwise upper bound (1.2.1) established in the course of the proof of Fefferman’s inequality. They give that

1 0

√ f gdt ≤ 2 2

1

f S(g)dt.

(1.2.15)

0

Now we apply H¨ older’s inequality and Theorem 1.1.6, the square function characterization of Lq , when q < 2.

1 0

f S(g)dt ≤ ||f ||p ||S(g)||q √ ≤ ||f ||p 2 2(p − 1)||g||q √ ≤ 2 2(p − 1)||f ||p .

This shows the left-hand side inequality of Theorem 1.2.5, that is, ||f ||p ≤ 8(p − 1)||f ||p .

1.2. Dyadic H 1

47

We obtain the right-hand side estimate by comparing the sharp function f to the following version of the Hardy–Littlewood maximal function. For x ∈ [0, 1] define M2 (f )(x) = sup x∈I

1 |I|

1/2

f2

,

I

where we take the supremum of all dyadic intervals which contain the point x. A moments reflection shows that pointwise inequalities hold between the sharp function and the maximal function of Hardy and Littlewood, f (x) ≤ M2 (f )(x).

(1.2.16)

For p > 2, the Hardy–Littlewood maximal theorem gives an upper bound for the Lp norm of M2 (f ), in terms of the Lp norm of f. More precisely by Theorem 1.2.6 below, the following holds for p > 2, ||M2 (f )||p ≤ 2

p p−2

1/p ||f ||p .

Assuming the validity of Theorem 1.2.6, we obtain from (1.2.16) the norm estimate for the sharp function, ||f ||p ≤ ||M2 (f )||p ≤ 2

p p−2

1/p ||f ||p .

Completing the proof that the sharp function provides an equivalent norm on Lp we show the Lp estimate for the Hardy–Littlewood maximal function. Besides M2 we will encounter different versions of the Hardy–Littlewood maximal function. Hence we will prove the required estimate in a more general form. Theorem 1.2.6. Fix p, q, such that 1 ≤ q < p. Let f ∈ Lp ([0, 1]), then Mq (f )(x) =

1/q 1 sup |f |q x∈I |I| I

satisfies the norm estimate ||Mq (f )||p ≤ 2

p p−q

1/p ||f ||p .

Proof. Fix f ∈ Lp ([0, 1]). Without loss of generality we assume that f ≥ 0. Let t > 0 and define the following function h by truncating f at the places where f (x) ≤ t/2, f (x) if f (x) > t/2, h(x) = 0 otherwise.

48


Put g = f −h. Then g(x) ≤ t/2, hence f (x) ≤ h(x)+t/2, and Mq (f ) ≤ Mq (h)+t/2. Consequently {Mq (f ) > t} ⊆ {Mq (h) > t/2}. (1.2.17) By Lemma 1.2.7 below, the maximal function Mq satisfies a weak type q − q estimate. We apply Lemma 1.2.7 to h and recall that the support of h is contained in the set E = {x : f (x) > t/2}. This gives tq |{Mq (h) > t/2}| ≤ 2q ||h||qq q ≤2 f q dx.

(1.2.18)

E

The pieces of information obtained so far are the inclusion relation (1.2.17) and the weak type estimate (1.2.18) Now we combine them and reach an upper bound for the distribution function 2q |{Mq (f ) > t}| ≤ q f q (x)1{f >t/2} (x)dx. (1.2.19) t Finally we obtain estimates for ||Mq (f )||p by integration from (1.2.19) in the following way, ∞ tp−1 |{Mq (f ) > t}|dt ||Mq (f )||pp = p 0 ∞ q ≤2 p tp−1 t−q f q (x)1{f >t/2} (x)dxdt 0

q

1

=2 p

f (x) 0

= 2q

p p−q

(1.2.20)

2f (x)

q

t

p−q−1

dtdx

0

1

f q (x)(2f )p−q (x)dx. 0

It remains to collect terms and to take the p-th root to obtain that ||Mq (f )||p ≤ 2

p p−q

1/p

f p .

Now we prove the fact that the maximal function Mq satisfies a weak type q − q estimate. Lemma 1.2.7. For h ∈ Lq , and t > 0 define Ωt = {x : Mq (h)(x) > t}. Then tq |Ωt | ≤ ||h||qq .

1.2. Dyadic H 1

49

Proof. For x ∈ Ωt there exists a uniquely determined maximal dyadic interval Ix such that |h|q > tq |Ix |. Ix

Let M = {Ix : x ∈ Ωt }. Note that M consists of pairwise disjoint dyadic intervals which cover the set Ωt . Hence |I| = |Ωt |. I∈M

For I ∈ M we have that

|h|q > tq |I|. I

We sum this estimate over all I ∈ M and use that M consists of pairwise disjoint dyadic intervals. This gives the weak type q − q estimate as follows, q |h|q ||h||q ≥ I

I∈M

≥ =

t |I| q

I∈M tq |Ωt |.

For p > 1 we use H p to denote Lp equipped with the square function norm

g H p

=

0

1

1/p 1 2 2 p/2 gdt + ( aI h I ) 0

where aI = g,

I

hI . |I|

Assume that q > 2 and that f : [0, 1] → R satisfies ||f ||Lq < ∞. Assume that g is a finite linear combination of Haar functions. The inequality

√ f gdt ≤ 2 2

1 0

1

f S(g)dt

0

implies that f induces a bounded linear functional on H p where 1/p + 1/q = 1. The next theorem states that the converse holds as well: Every bounded linear functional on H p induces f : [0, 1] → R such that ||f ||Lq < ∞. Theorem 1.2.8. Let 1 ≤ p < 2, and let q be the H¨ older conjugate exponent to p satisfying 1/p + 1/q = 1. For every continuous linear functional L : H p → R there exists f : [0, 1] → R so that f ∈ Lq , with ||f ||

Lq

≤

√

2

q q−2

1/q ||L||,

50


and so that for every finite linear combination of Haar functions g,

1

f gdt.

L(g) = 0

Proof. Let g be a finite linear combination of Haar functions. For n ∈ N denote by gn the orthogonal projection of g onto the span{hI : |I| = 2−n }. That is,

gn =

g,

{I:|I|=2−n−1 }

hI hI . |I|

(1.2.21)

The following identity holds for the square function of g, S(g) = ( gn2 )1/2 . We represent H p as a closed subspace of Lp (2 ). Let g˜ = (gn )∞ n=1 . Then ||g||H p = ||˜ g ||Lp (2 ) . By the Hahn–Banach theorem we extend the linear functional L to a functional ˜ : Lp (2 ) → R, be an extension satisfying on Lp (2 ), with the same norm. Let L ˜ ||L|| = ||L||, and ˜ g ) for g ∈ H p . L(g) = L(˜ Next recall that the dual space of Lp (2 ) is identified as Lq (2 ), in the following sense. There exists a sequence (fn ) such that ˜ = ||( ||L|| fn2 )1/2 ||Lq , (1.2.22) and for every (hn ) ∈ Lp (2 ), ˜ n )) = L((h

∞

fn hn .

(1.2.23)

n=1

Next use the sequence (fn ), to define f. We put f=

∞

fn ,

n=1 {I:|I|=2−n−1 }

hI hI . |I|

Testing the identity (1.2.23) on the sequence (gn ) defined by (1.2.21) we obtain that ∞ hI ˜ n )) = g, hI , fn L((g |I| −n−1 n=1 {I:|I|=2

= g, f .

}

1.2. Dyadic H 1

51

˜ is given by integration as Thus restricted to (gn ) the action of L

1

˜ n )) = L((g

gf dt. 0

Now we show that f ∈ Lq , and that √ ||f ||q ≤ 2

q q−2

1/q ||(

fn2 )1/2 ||q .

At a given point x ∈ [0, 1], we evaluate the sharp function of f by fixing a dyadic interval J such that x ∈ J. Choose m ∈ N such that |J| = 2−m−1 . By Bessel’s inequality, we obtain hI |f − (f )J |2 dx = fn , 2 |I| |I| J −n−1 n≥m {I⊆J:|I|=2

≤

∞

J n=1

}

fn2 dx.

Divide by |J| and take the supremum over all dyadic intervals containing the fixed point x ∈ [0, 1]. Thus we restate the last estimate as a pointwise inequality between the sharp function and the Hardy–Littlewood maximal function M1 . For every x ∈ [0, 1],

1/2 ∞ fn2 ) (x). (1.2.24) f (x) ≤ M1 ( n=1

By our assumption q > 2, hence by Theorem 1.2.6 the maximal function M1 acts boundedly on the space Lq/2 . Moreover the following norm estimate holds true,

M1 (h) q/2 ≤ 2

q q−2

2/q

h q/2 .

This allows us to finish the proof as follows. By (1.2.24) and by Theorem 1.2.6, ||f ||q ≤ M1 ( √ ≤ 2

n

q q−2

Invoking (1.2.22) gives that ||f ||q ≤

1/2

fn2 ) q/2 1/q

||(

fn2 )1/2 ||q .

n

√ q 1/q 2 q−2

L , as claimed.

52

1.3


Bounded square functions and large deviation inequalities

We return to square functions in Lp , 1 < p < ∞. In Theorem 1.1.6 we combined the inequalities of Burkholder, and Khintchine to prove the square function characterization of Lp asserting that for f ∈ Lp , with f = 0, √ √ (2 2)−1 (p∗ − 1)−1 ||f ||p ≤ ||S(f )||p ≤ 2(p∗ − 1)||f ||p . In this section we show that for 2 ≤ p < ∞ the following equivalence holds, √ p 1 √ ||f ||p ≤ ||S(f )||p ≤ √ ||f ||p , 16 p 2 √ when f = 0. The constants of the above inequality which are of order p, are known to represent the correct asymptotic behavior as p → ∞. Along with the proof we introduce important tools of general interest in real analysis. These include a detailed study of the exponential function, 1 exp{f − S 2 (f )}, 2 large deviation inequalities, stopping time arguments, and the use of so-called good λ inequalities. We apply these methods to study Schechtman’s sign-embedding of high-dimensional Euclidean subspaces into 1n . We also show how to obtain non-trivial multipliers into SL∞ (the Banach space consisting of functions with bounded square functions).

Square functions in Lp (2 < p < ∞) We prove now a strong upper bound for the expectation of the exponentials exp{f − 12 S 2 (f )}. The following theorem has wide ranging applications. These include the law of the iterated logarithms, eigenvalue estimates for the Laplacian, embedding theorems for fractional Sobolev spaces, the existence of high-dimensional Euclidean sections in convex bodies and the square function inequality, √ ||f ||p ≤ 16 p||S(f )||p , for p ≥ 2, where f ∈ Lp and f = 0. The notes at the end of the chapter provide the references to the applications of our next theorem. 1 Theorem 1.3.1. Let f ∈ L2 ([0, 1]) with 0 f dt = 0. Then for every µ > 0,

1

exp{µf − 0

µ2 2 S (f )} ≤ 1. 2

1.3. Bounded square functions and large deviation inequalities Proof. As

1 0

53

f dt = 0 we expand f in its Haar series as f=

aI hI .

We fix a dyadic interval I. The following estimate is basic and elementary. It controls the mean values of exp{aI hI } over I. |I| (exp(aI ) + exp(−aI )) exp{aI hI } = 2 I 2 (1.3.1) aI . ≤ |I| exp 2 Now we split the Haar series of f according to the levels {J : |J| = 2−k }. Write

dk =

aJ h J .

{J:|J|=2−k }

We restate the estimate (1.3.1) in the following way. For every dyadic interval I of length |I| = 2−k , d2k a2 exp{dk − } = exp{aI hI − I h2I } 2 2 I I ≤ |I|.

For future reference it is convenient to rewrite the last estimate using conditional expectations. Recall that Ek denotes the conditional expectation with respect to the σ-algebra generated by the dyadic intervals of length = 2−k . We showed so far that d2 Ek (exp{dk − k }) ≤ 1. (1.3.2) 2 Next let n ∈ N. For k ≤ n − 1, the conditional expectation En reproduces the functions dk and d2k . Therefore we have the following straightforward identities,

1

exp 0

n k=0

d2 dk − k 2

$

1

dt =

exp 0

k=0

1

=

exp 0

n−1 n−1 k=0

d2 dk − k 2 d2 dk − k 2

$ (exp{dn − $

d2n })dt 2

En (exp{dn −

The basic inequality (1.3.2) and the above identities imply that $ $ n n−1 1 1 d2k d2k dt ≤ dt. exp dk − exp dk − 2 2 0 0 k=0

k=0

d2n })dt. 2

(1.3.3)

54


The net effect of (1.3.3) is that we increased the value of the integral while we reduced the upper limit of summation from n to n − 1. Now we repeat this procedure to reduce the upper limit of summation from n − 1 to n − 2, and we continue in this fashion until we reach the level k = 0. This gives in summary the inequality $ n 1 1 d2k d2 dt ≤ exp dk − exp{d0 − 0 }dt 2 2 (1.3.4) 0 0 k=0

≤ 1. By the lemma of Fatou, in the estimate (1.3.4) we may pass to the limit as n → ∞. As ∞ $ d2k S 2 (f ) = exp dk − exp f − 2 2 k=0 ! " we proved that exp f − 12 S 2 (f ) ≤ 1. Finally we fix µ > 0, and replace f by (µf ). This gives that 1 µ2 2 S (f )} ≤ 1. exp{µf − 2 0 The space SL∞ is the class of functions for which the square function is uniformly bounded. Thus f ∈ SL∞ iff S(f ) ∈ L∞ . Our first corollary to Theorem 1.3.1 derives a strong upper bound for the distribution of functions f ∈ SL∞ . We show a square-exponential estimate for the Hardy–Littlewood maximal function M1 (f ). To prove this statement we combine Theorem 1.3.1, with the weak type 1 : 1 estimate of Lemma 1.2.7 and Jensen’s inequality. 1 Corollary 1.3.2. Let f ∈ L2 ([0, 1]) with 0 f dt = 0. If S(f ) ∈ L∞ , then t2 . |{x : M1 (f )(x) > t}| ≤ 2 exp − 2||S(f )||2∞ Proof. We assume that ||S(f )||∞ = 1. Write (|f |)I for the mean value of |f | over the interval I. Analogously we let (exp{|f |})I denote the mean value of exp{|f |} over I. By Jensen’s inequality the mean values are related as follows, exp{(|f |)I } ≤ (exp{|f |})I .

(1.3.5)

The inequality (1.3.5) translates into a pointwise estimate for the Hardy–Littlewood maximal function M1 , exp{M1 (f )} ≤ M1 (exp{|f |}). By Lemma 1.2.7, the weak type 1 : 1 estimate for the maximal function M1 we have that exp{µ|f |} , (1.3.6) |{M1 (exp{µ|f |}) ≥ exp α}| ≤ exp α

1.3. Bounded square functions and large deviation inequalities

55

for every µ, α > 0. Theorem 1.3.1 and the hypothesis that ||S(f )||∞ = 1 imply that exp{µ|f |} ≤ exp{µf } + exp{−µf } (1.3.7) µ2 ≤ 2 exp{ }. 2 Inserting the integral estimate (1.3.7) into the right-hand side of the measure estimate (1.3.6) gives that 2 exp{µ|f |} µ ≤ 2 exp −α . |{M1 (exp{µ|f |}) ≥ exp α}| ≤ exp α 2 Summing up we find the following upper bound for the distribution function of M1 (f ). Let α > 0, then |{µM1 (f ) > α}| = |{exp{µM1 (f )} > exp α}| ≤ |{M1 (exp(µ|f |) > exp α}| 2 µ ≤ 2 exp −α . 2 √ Specializing this estimate to µ = α = t, gives that 2 t , |{M1 (f ) > t}| ≤ 2 exp − 2 provided that ||S(f )||∞ = 1. We get rid of the restriction ||S(f )||∞ = 1 by applying the above special case to f /||S(f )||∞ . We will next discuss rescaled versions of Corollary 1.3.2. Fix a dyadic interval Q . We now introduce the Hardy–Littlewood maximal function localized to the interval Q. Here we take the supremum only over those intervals which are contained in Q. For x ∈ Q we put, 1 |f |. M1,Q (f )(x) = sup I⊆Q, x∈I |I| I Let f ∈ L2 . We denote the mean value of f over the interval Q by fQ , that is, 1 fQ = f dx. |Q| Q Then by rescaling Corollary 1.3.2 from the unit interval to the interval Q, we obtain the following renormalized distributional estimate: If S(f ) ∈ L∞ , then t2 |{x ∈ Q : M1,Q (f − fQ )(x) > t}| ≤ 2|Q| exp − . (1.3.8) 2||S(f )||2∞

56

Chapter 1. The Haar System: Basic Facts and Classical Results Our aim is still the square function estimate ||f ||p ≤ 16p1/2 ||S(f )||p .

(1.3.9)

The essential step in the proof is the next proposition which gives an estimate for the distribution of M1 (f ) over the set where S(f ) satisfies a pointwise upper bound. The distributional estimates of Proposition 1.3.3 are traditionally called good λ inequalities, or more descriptively, relative distributional estimates. We 1/2 in the square function estimate remark that the appearance of the constant " ! 116p on the right hand side of the good (1.3.9) can be traced to the factor exp − 2 4 " ! λ inequality, below. The factor exp − 412 has in turn its origin in the integral estimate of the exponential 1 exp{f − S 2 (f )} ≤ 1. 2 1 Proposition 1.3.3. Let f ∈ L2 ([0, 1]) with 0 f dt = 0. Then for every 0 < ≤ √ 1 − 1/ 2, and t > 0 the following distributional inequality holds, 1 |{M1 (f ) > 2t, S(f ) < t}| ≤ 2 exp − 2 |{M1 (f ) > t}|. 4 Proof. Fix t > 0, and define Ωt = {x : M1 (f ) > t}. For every x ∈ Ωt , there exists a maximal dyadic interval Ix such that 1 |f | > t. |Ix | Ix Let M = {Ix : x ∈ Ωt }. The collection M covers the set Ωt . By the maximality condition, M consists of pairwise disjoint dyadic intervals, hence, |Q| = |{M1 (f ) > t}|. {Q∈M}

Maximality gives also the following implication: If Q ∈ M, and if Q∗ denotes the dyadic predecessor of Q, then 1 f ≤ t. |Q∗ | Q∗ Let Q ∈ M and assume that there exists x ∈ Q such that M1 (f )(x) > 2t. Observe that then the relevant contribution to the value of the maximal function M1 (f ) at the point x necessarily comes from the intervals which are contained in Q (and not from those containing Q). Precisely, if Q ∈ M, and M1 (f )(x) > 2t, then the following equality holds at the point x, M1 f (x) = M1,Q f (x).

(1.3.10)


57

Expand f in its Haar series, f=

aI hI .

Define L to be the collection of dyadic intervals I such that the following estimate holds true, a2J h2J (x) ≥ 2 t2 , J⊇I

for some (or equivalently for all) x ∈ I. Let L be the set of points which is covered by the collection L, thus L= I. I∈L

Define S = D \ L, and put ft = J∈S aJ hJ . The reason for our choice of L and S is that now, by definition, ft has uniformly bounded square function and satisfies ||S(ft )||∞ ≤ t.

(1.3.11)

ft (x) = f (x),

(1.3.12)

Observe that if x ∈ / L, then and furthermore, if J is a dyadic interval containing x we have that ft = f. J

(1.3.13)

J

So far we defined two important collections of dyadic intervals, M and L and we obtained several consequences encoded in their very definitions. Now we analyze the interplay between M and L. We fix Q ∈ M, and we assume that there exists x ∈ Q such that x ∈ / L. Then we claim that the following integral estimate holds, 1 f ≤ (1 + )t. (1.3.14) |Q| Q To verify this claim we consider the dyadic predecessor of Q and call it Q∗ . Note that the coefficient aQ∗ appearing in the Haar expansion of f is obtained from the mean values of f over Q and over Q∗ as follows, 1 1 f = |aQ∗ |. (1.3.15) |Q∗ | ∗ f − |Q| Q Q Recall that for Q ∈ M, the mean value of |f | over Q∗ is bounded by t, hence 1 1 f ≤ |f | ≤ t. (1.3.16) |Q∗ | Q∗ |Q∗ | Q∗ Furthermore if x ∈ Q, and x ∈ / L, then |aQ∗ | ≤ t.

(1.3.17)

58


Inserting the estimates (1.3.17) and (1.3.16) into the identity (1.3.15) gives the upper bound 1 f ≤ (1 + )t, |Q| Q hence inequality (1.3.14) holds true. Next we claim that for Q ∈ M, the following estimate holds, 1 |{x ∈ Q : M1 (f )(x) > 2t, S(f )(x) < t}| ≤ 2|Q| exp − 2 . 4

(1.3.18)

We start proving this claim by first rewriting the set whose measure we wish to estimate. Using first (1.3.10) then (1.3.13) we obtain that, / L} {x ∈ Q : M1 (f )(x) > 2t, S(f )(x) < t} = {x ∈ Q : M1,Q (f )(x) > 2t, x ∈ / L}. = {x ∈ Q : M1,Q (ft )(x) > 2t, x ∈ /L We write (ft )Q for the mean value of ft over Q. By (1.3.13), for x ∈ Q and x ∈ the mean values (ft )Q and (f )Q coincide. Invoking (1.3.14) gives that |(ft )Q | ≤ (1 + )t. By inserting (ft )Q we obtain the following inclusion of sets, which incidentally is a one line summary of the proof given so far. / L} ⊆ {x ∈ Q : M1,Q (ft − (ft )Q ) (x) > (1 − )t} . {x ∈ Q : M1,Q (ft )(x) > 2t, x ∈ By definition ft has uniformly bounded square function with ||S(ft )||∞ ≤ t. Hence (1.3.8) gives the measure estimate (1 − )2 t2 . |{x ∈ Q : M1,Q (ft − (ft )Q ) (x) > (1 − )t}| ≤ 2|Q| exp − 22 t2 Now observe that inside the exponential the factors t2 cancel and that for 0 < ≤ √ 1−1/ 2, we have clearly (1−)2 /2 ≥ 1/4. Thus we verified the claimed inequality (1.3.18). Finally recall that M is a collection of pairwise disjoint dyadic intervals, covering the set {M1 (f ) > t}. Now we sum over Q ∈ M, thereby finishing the proof of Proposition 1.3.3: |{x ∈ Q : M1 (f )(x) > 2t, S(f )(x) < t}| |{M1 (f ) > 2t, S(f ) < t}| = Q∈M

1 ≤ 2 exp − 2 |Q| 4 Q∈M 1 = 2 exp − 2 |{M1 (f ) > t}|. 4


59

We now prove a square function characterization of Lp with the correct asymptotic behavior of the constants as p → ∞. 1 Theorem 1.3.4. Let p ≥ 2, and let f ∈ Lp ([0, 1]) with 0 f dt = 0. Then √ ||f ||p ≤ ||M1 (f )||p ≤ 16 p||S(f )||p . Proof. We wish to evaluate ||M1 (f )||pp . Fix t > 0 and > 0. We start the proof with the triangle inequality, |{M1 (f ) > 2t}| ≤ |{M1 (f ) > 2t, S(f ) < t}| + |{S(f ) ≥ t}|.

(1.3.19)

The effect of the second summand |{S(f ) ≥ t}| is easy to evaluate. We have ∞ tp−1 |{S(f ) ≥ t}|dt = −p ||S(f )||pp . (1.3.20) p 0

The relative distributional estimate of Proposition 1.3.3 controls the contribution of the first summand. It gives 1 (1.3.21) |{M1 (f ) > 2t, S(f ) < t}| ≤ 2 exp − 2 |{M1 (f ) > t}|. 4 We point out that it is the presence of factor exp{−(1/42 )} that allows us to compensate the effect caused by −p appearing in (1.3.20). Combining the estimates (1.3.19) and (1.3.20) with the relative distributional estimate (1.3.21) gives the following bound for ||M1 (f )||pp . 1 2p

1 p

∞

M1 (f ) (x) = p 0

0

tp−1 |{M1 (f ) > 2t}|dt

∞

tp−1 |{M1 (f ) > 2t, S(f ) < t}|dt + −p ||S(f )||pp 0 ∞ 1 ≤ 2p exp{− 2 } tp−1 |{M1 (f ) > t}|dt + −p ||S(f )||pp 4 0 1 = 2 exp{− 2 }||M1 (f )||pp + −p ||S(f )||pp . 4 ≤p

Now subtract exp{−(1/2 )}||M1 (f )||pp on both sides. This gives 1 1 − 2 exp{− } ||M1 (f )||pp ≤ −p ||S(f )||pp . 2p 2

(1.3.22)

Note that (1.3.22) can become a meaningful estimate only if is chosen so small that the factor in front of ||M1 (f )||pp is a positive number. For p ≥ 2 define =

1 . (4 ln 2)1/2 (p + 2)1/2

60

Chapter 1. The Haar System: Basic Facts and Classical Results p

Then exp{−(1/42 )} ≤ 2−p−1 , and −p ≤ (8 ln 2)p/2 p 2 . This gives p

||M1 (f )||pp ≤ 16p p 2 ||S(f )||pp . It remains to raise both sides to the power 1/p. For f ∈ Lp with f = 0 we established the square function estimate √ ||f ||p ≤ 16 p||S(f )||p

for

2 ≤ p < ∞.

Now we turn to proving the converse inequality. In the proof below we present a method due to A. M. Garsia [75]. Theorem 1.3.5. Let p > 2 and f ∈ Lp . Then, √ p ||S(f )||p ≤ √ ||f ||p . 2 Proof. To start the proof we expand f in its Haar series, f = f + aI hI . Then we introduce the following notation to denote partial sum approximations to S 2 (f ), 2 S−1 = 0,

and

Sn2 (t) =

a2I h2I (t) for n ∈ N0 .

{I:|I|≥2−n }

We also abbreviate the notation for the square function of f, by writing S = S(f ). Below we state two crucial identities, and an inequality which uses that for p > 2, the function r p/2−1 is increasing on the positive real axis. 2 p S (f )(t) = p = ≤

S 2 (t)

r p/2−1 dr

0

∞ n=0 ∞

2 Sn (t)

r p/2−1 dr.

2 (t) Sn−1

(1.3.23)

! " 2 Snp−2 (t) Sn2 (t) − Sn−1 (t) .

n=0

For brevity it is convenient to introduce more notation. Let n ∈ N0 , then put p−2 . mn = Snp−2 − Sn−1

Using the summation by parts formula we next rewrite the last line of (1.3.23), ∞ n=0

∞ ! " 2 2 Snp−2 (t) Sn2 (t) − Sn−1 (t) = mn {S 2 (t) − Sn−1 (t)}. n=0

(1.3.24)


61

Next observe that En , the conditional expectation with respect to the σ-algebra generated by the dyadic intervals of length = 2−n reproduces mn . That is En (mn ) = mn . Integrating the estimate (1.3.23) over the unit interval and taking into account the identity (1.3.24) we find ∞ 1 2 1 p 2 S (f )(t)dt ≤ mn En (S 2 (t) − Sn−1 (t))dt. (1.3.25) p 0 n=0 0 2 )(t) satisfy the pointwise Observe that the conditional expectations En (S 2 − Sn−1 estimate, 2 ) ≤ En (f 2 ). (1.3.26) En (S 2 − Sn−1

Inserting the estimate (1.3.26) into the right-hand side of (1.3.25) and using again that En (mn ) = mn , allows us to continue the right-hand side of (1.3.25) as follows, ∞ 1 ∞ 1 2 mn En (S 2 − Sn−1 )dt ≤ mn f 2 dt. (1.3.27) n=0

0

0

n=0

Finally we change the order of integration and summation, and sum the resulting ∞ telescoping series k=0 mk = S p−2 . This gives that 1 ∞ 1 2 mn f dt = S p−2 f 2 dt. (1.3.28) n=0

0

0

Thus starting from (1.3.23) we showed with (1.3.25), (1.3.27) and (1.3.28), this crucial inequality, 1 2 1 p S (f )(t)dt ≤ S p−2 f 2 dt. p 0 0 Now we use H¨ older’s inequality with a = p/(p − 2) and b = p/2. This gives 1/a 1 1/b 1 1 p−2 2 (p−2)a 2b S f dt ≤ S f 0

0

=

(p−2)/p

1

S

p

0

0 1

2/p |f | . p

0

Summing up we have that 1 (p−2)/p 1 2/p 2 1 p p p S ≤ S |f | . p 0 0 0 1 Cancelling the factor ( 0 S p )(p−2)/p on both sides gives that 2 ||S(f )||2p ≤ ||f ||2p . p

62


In the following paragraphs we will discuss three different applications of the square function characterization of Lp and the SL∞ estimates that formed the basis of its proof. First, we compare sharp functions with square functions in Lp and let p → ∞, second we use SL∞ estimates to produce Euclidean subspaces in √ 1n , and third we exploit the p dependence in Theorem 1.3.5 to prove that the square function of a bounded function is square-exponentially integrable. Endpoints of the Lp scale. Let f ∈ Lp with f = 0. Combining Theorem 1.3.4 and Theorem 1.3.5 gives that for p ≥ 2, √ √ √ (1.3.29) ( 2/ p) S(f ) p ≤ f p ≤ 16 p S(f ) p . Recall that a function is in SL∞ if its square function is uniformly bounded. Thus SL∞ appears as an endpoint of the scale of Lp spaces for p → ∞. Compare this to the renorming obtained in the previous section. For p ≥ 2, (p − 2)/(2p) f p ≤ f p ≤ 8(p − 1) f p .

(1.3.30)

This gives a different endpoint to the scale of Lp spaces for p → ∞, namely BMO. A glance at the definitions of square functions and sharp maximal functions shows that SL∞ is contained in BMO. However much more is true. A result of J. Garnett and P. W. Jones [73] displays the relation between the three spaces L∞ , BMO and SL∞ that appeared as endpoints of the Lp scale. It asserts that for any f ∈ SL∞ with S(f ) ≤ 1 and > 0 there exists g ∈ L∞ satisfying

f − g BMO ≤ and

g ∞ ≤ A−1 ,

(1.3.31)

where A > 0 is a universal constant. In particular the closure of L∞ in BMO contains SL∞ . Applying the Garnett–Jones result, J. Bourgain [18] obtains a remarkable estimate for the Rademacher projection on H 1 . Recall that {rm } denotes the Rademacher system consisting of {+1, −1} valued, independent random variables. J. Bourgain’s inequality states that ∞

h, rm 2 ≤ C h L1 h H 1 ,

for h ∈ H 1 .

(1.3.32)

m=1

The interesting aspect of J. Bourgain’s inequality is the appearance of the L1 norm on the right-hand side. Replacing h L1 h H 1 by h 2H 1 would turn it into a trivial estimate. On the other hand replacing h L1 h H 1 by the smaller number

h 2L1 would lead to a false statement. To prove (1.3.32) put am = h, rm (

h, rn 2 )−1/2 ,

1.3. Bounded square functions and large deviation inequalities and let f = define

am rm . Since

63

a2m = 1, we have f ∈ SL∞ with S(f ) = 1. Next −1/2

1/2

= h L1 h H 1 . By the Garnett–Jones result there exists g ∈ L∞ so that

f − g BMO ≤ and

g ∞ ≤ A−1 .

(1.3.33)

With the inequalities of Fefferman and H¨ older we obtain that 1/2 = h, f h, rm 2 = h, f − g + h, g √ ≤ 2 2 h H 1 f − g BMO + h L1 g ∞ .

(1.3.34)

Use the estimates (1.3.33) in the last line of (1.3.34). Then our choice of > 0 gives J. Bourgain’s inequality (1.3.32). Schechtman’s sign-embedding. In this section we show how SL∞ estimates lead to Euclidean subspaces of proportional dimension in 1n . The resulting embedding — Schechtman’s sign-embedding — turns out to be quite explicit. We write 1n to denote Rn equipped with the norm of the sequence space 1 . Likewise we denote by 2k the space Rk equipped with the norm of the Hilbert space 2 . A straightforward exercise with Khintchine’s inequality yields a subspace X ⊆ 1n and a linear bijection i : 2k → X so that k = [log2 n]

and

i · i−1 ≤ 16.

It is a much harder problem to find such an embedding i : 2k → X under the hypothesis that the algebraic dimension of the subspace X is proportional to the dimension of the ambient space 1n . That is, under the hypothesis that k = [δn] . Here, of course, we demand that the factor of proportionality n/k ∼ δ −1 is an absolute constant, not depending on n. In the seminal paper [186] G. Schechtman develops an approach to these problems based integral estimates for the exponentials exp{f − 12 S 2 (f )}. Specifically, G. Schechtman’s construction of the embedding is based on Khintchine’s inequality and Corollary 1.3.2 asserting that |f −

1 0

t2 . f | > t ≤ 2 exp − 2||S(f )||2

(1.3.35)

∞

Subsequently, martingale methods based on (1.3.35) found numerous applications to related problems in the local theory of Banach spaces. See the notes for references that give an overview of this development.

64


Next we turn to defining G. Schechtman’s embedding. Let {ei : i ≤ n} denote the unit vector basis in 1n . Thus for coefficients bi ∈ R we get n bi ei i=1

=

n

|bi |.

i=1

1n

Recall that {rl } denotes the Rademacher system. Fix k ≤ n and define 1 rik+j (t)ei , n i=1 n

xj (t) =

j≤k

where

and

t ∈ [0, 1].

Thus xj (t) is formed by attaching random signs {±1} to the unit vectors of 1n and averaging. Note that for each t ∈ [0, 1] the vectors xj (t) are of norm 1 in 1n . The next theorem asserts that for some t ∈ [0, 1], they are equivalent to the unit vector basis of 2k provided that k ∼ δn. Theorem 1.3.6. There exists δ > 0 so that the following holds true. For each n ∈ N and k ≤ δn there exist t ∈ [0, 1] so that ⎞1/2 ⎛ k k 1 ⎝ 2 ⎠ aj ≤ aj xj (t) 4 j=1 j=1

⎛ ⎞1/2 k ≤ 4⎝ a2j ⎠ ,

1n

(1.3.36)

j=1

for every choice of coefficients a = (aj ) ∈ Rk . Proof. Fix coefficients a = (aj ) ∈ Rk , and assume that kj=1 a2j = 1. The first part of the proof shows that with a = (aj ) fixed the estimates (1.3.36) hold for a very large set of t ∈ [0, 1]. The purpose of the second part is to exploit this fact and extend (1.3.36) from one set of coefficients to the entire unit sphere of 2k . Part 1. For t ∈ [0, 1], we first define the function f (t) and then unwind the definition of the random vectors xj (t). Thus, k f (t) = a x (t) j j j=1 1 n n k 1 = aj rik+j (t) . n i=1 j=1

(1.3.37)

Next we estimate the Haar coefficients of f, its square function and its mean value over the interval [0, 1]. Starting with the Haar coefficients we fix i ≤ n and


65

m ≤ k. Let I be a dyadic interval. The statistical independence of the Rademacher functions gives the following implication: k aj rik+j (t) hI (t)dt ≤ |I| · |am |. (1.3.38) If hI , rik+m = 0, then j=1 The remaining Haar coefficients vanish. By (1.3.38) the square function of f satisfies the following upper bound. S 2 (f )(t) ≤

k n 1 2 a n2 i=1 m=1 m

(1.3.39)

1 2 ≤ a . n j=1 m k

Using Khintchine’s inequality (for the left-hand side of (1.3.40)) and the biorthogonality of the Rademacher functions (for the right-hand side of (1.3.40)) we obtain that ⎛ ⎛ ⎞1/2 ⎞1/2 1 k k k √ −1 a2j ⎠ ≤ aj rik+j (t) dt ≤ ⎝ a2j ⎠ . (1.3.40) ( 2) ⎝ 0 j=1 j=1 j=1 Recall now that we imposed that √ −1 ( 2) ≤

a2j = 1. In summary we obtained that

1

f (t)dt ≤ 1 and

S 2 (f )(t) ≤

0

1 . n

(1.3.41)

It follows from (1.3.41) and (1.3.35) that this deviation inequality holds, |f −

1 0

2 s n . f | > s ≤ 2 exp − 2

(1.3.42)

√ √ 1 By (1.3.41) we get {f ≥ 2 or f ≤ (2 2)−1 } ⊆ {|f − 0 f | > (2 2)−1 }. Taking √ complements and using (1.3.42) with s = (2 2)−1 gives that % √ & % n& . (1.3.43) (2 2)−1 ≤ f ≤ 2 ≥ 1 − 2 exp − 16 For obvious reasons, (1.3.43) is frequently called the concentration of measure inequality for f. Summing up the first part of the proof, the function f, formed by a single set of coefficients a = (aj ) is bounded (from above and below) on an extremely large subset of the unit interval.

66


Part 2. Next we pass from a fixed set of coefficients a = (aj ) to a sufficiently large collection of coefficients in the unit sphere of 2k . It is at this point of the argument that we will impose a restriction on the size of k. Let 0 > 0. Denote by S k−1 the unit sphere of 2k . Choose a discrete set M ⊆ S k−1 — with minimal cardinality — so that for each b ∈ S k−1 there exists a ∈ M satisfying |a − b| ≤ 0 . Verify, by counting, that the cardinality of M is bounded by exp{+2k/0 }.

(1.3.44)

(See, for instance, Proposition 10.II.E in [213].) Now choose 0 > 0 small enough so that this implication holds: If k √ −1 ≤ 2 for all (aj ) ∈ M, (2 2) ≤ a x (t) (1.3.45) j j j=1 1 n

then

k 1 ≤ aj xj (t) 4 j=1

≤ 4 for all (aj ) ∈ S k−1 .

(1.3.46)

1n

An elementary iteration argument allows one to choose 0 > 0 satisfying the above property, and 0 does not depend on the value of k. (The details of the iteration are worked out in [148].) Next apply the first part of the proof to each a = (aj ) ∈ M, and define ⎧ ⎫ ⎪ k ⎪ ⎨ ⎬ + √ −1 and E = aj xj (t) ≤ 2 Ea . Ea = t : (2 2) ≤ ⎪ ⎪ j=1 1 ⎩ ⎭ a∈M n

Inequality (1.3.43) states that |Ea | > 1 − 2 exp{−n/16}. By (1.3.44), the measure of the intersection E is therefore larger than +2k −n exp . 1 − 2 exp 16 0 Hence, for k ≤ (n/32 − 2)0 say , the set E is non-empty. For each t ∈ E, by its definition, the estimates (1.3.45) hold true. Passing from (1.3.45) to (1.3.46) completes the proof. We emphasize that (1.3.44) and (1.3.43) are the central and competing inequalities of the preceding proof. √ The square function of bounded functions. We exploit the constants p appearing in Theorem 1.3.5 to prove that the square function of a bounded function is square-exponentially integrable.


67

Corollary 1.3.7. Let f ∈ L∞ . Then for 0 < α < 1/e,

1

exp{α 0

S 2 (f ) }dt < ∞. ||f ||2L∞

Proof. We use the normalization ||f ||L∞ = 1. Expanding the exponential in its Taylor series gives ∞ αn S(f )2n . exp{αS 2 (f )} = n! n=0 Apply Theorem 1.3.5, with p = 2n to f. This gives

1

S(f )2n dt ≤ nn ||f ||2n 2n . 0

Note that nn /n! ≤ en , since nn /n! is just one of the summands in the Taylor expansion of en . Then we have

1

exp{αS 2 (f )}dt = 0

≤ ≤

∞ αn 1 S(f )2n dt n! 0 n=0 ∞ αn n n n! n=0 ∞

αn en .

n=0

For α < 1/e the geometric series converges.

Quadratic exponential integrability is the best estimate available for S(f ) when f is uniformly bounded. To see this consider the following example. Let In = [0, 4−n ). The sequence In is decreasing, In ⊃ In+1 . Next define the following sequence of pairwise disjoint intervals , 3 Kn = [ 4−n , 4−n ), 4 and let, f=

∞

1Kn .

(1.3.47)

n=0

As the intervals Kn are pairwise disjoint, f is bounded by 1. Now we compute the Haar coefficients. We have −|In | ; hIn f dt = 6

68


the other coefficients in the Haar expansion of f vanish. Hence, S 2 (f )(t) = It follows that S 2 (f )(t) =

n 36

∞ 1 1I n . 36 n=0

for t ∈ In \ In+1 .

This shows that S(f ) is unbounded and the distribution function of S(f ) satisfies the lower estimate |{S(f ) > t}| ≥ A0 exp{−a0 t2 }. Clearly this shows that quadratic exponential integrability is the best one can get for the square function of f, even when f is uniformly bounded. Observe however that for the example (1.3.47) and m(t) = t the product mf is non-trivial in the sense that mf > 0 and mf ∈ SL∞ . In the next subsection we will find such a multiplier for every bounded nonnegative function f.

Multipliers into SL∞ To prove the good-λ inequality in Proposition 1.3.3 we produced functions in SL∞ by deleting certain coefficients in the Haar expansion of a given function. Thereby we contracted the Haar support and lost any information about its pointwise support. In this subsection we present a method to produce elements in SL∞ by contracting simultaneously the Haar coefficients and the pointwise support of a given function. Let f ∈ L∞ ([0, 1]) be non-negative. Let its Haar expansion be

1

f+

f=

aI h I .

0

In the next theorem we define a multiplier m : [0, 1] → [0, 1], such that h = m · f, is non-trivial and has uniformly bounded square function. We determine m so that the operation f −→ mf is a pointwise contraction on f that also contracts each individual Haar coefficient of f to the extent that h has a bounded square function. The product mf is non-trivial in the sense that mf ≥ exp(− f ∞ ) f. Replacing f with f / f ∞ allows us to restrict ourselves to the case where f is normalized in L∞ ([0, 1]).


69

Theorem 1.3.8. Let f ∈ L∞ ([0, 1]), and assume that 0 ≤ f ≤ 1. Let fI be the mean value of f over the interval I. Then there are constants mI ∈ [1/6, 6] so that the multiplier m : [0, 1] → [0, 1], defined by 1 2 1 2 aI mI ( + )hI m = exp(f − f ∞ ) exp − 2 fI satisfies the following conditions: The product h = m · f is non-trivial, and 1 1 h ≥ exp(− f ∞ ) f. 0

0

The Haar coefficients of h = m · f satisfy the upper bound ⎛ ⎞ |h, hI | ≤ A|I| exp ⎝− a2J ⎠ |aI |. J⊇I

Consequently the dyadic square function of h is uniformly bounded, , hI -2 h, h2I ≤ CA2 . |I| I

2

Proof. Clearly e−S (f ) is the first candidate for a multiplier to satisfy the conclusions of our theorem. However there are serious obstacles that prevent one 2 from getting satisfactory estimates for the Haar coefficients f e−S (f ) , hI . The construction of the multiplier ϕ below is in direct response to these difficulties; indeed the combinatorial identity (1.3.48) combined with the estimates (1.3.51) are crucial in proving the main inequality (1.3.52) for the Haar coefficients of f eϕ . Overview. Given 0 ≤ f ≤ 1, we write down its Haar expansion, 1 f+ aI h I . f= 0

First we show that for every dyadic interval there exists a constant mI satisfying 1 ≤ mI ≤ 6, 6 and such that the following identity holds, 1 1 2 2 + mI hI dt = f. f exp aI hI − aI 2 fI I I Once the sequence {mI } satisfying (1.3.48) is determined we put 1 1 2 + h2I , aI m I ϕ = f − f ∞ − 2 fI I

(1.3.48)

70


and define the multiplier as m = eϕ . We will verify below that the function h = mf satisfies the conclusions of Theorem 1.3.8. Verification of (1.3.48). Let I be a fixed dyadic interval. We begin by evaluating the integral f eaI hI dt. I

Define R(t) to be the remainder term of order 3 in the Taylor expansion of the exponential exp(aI hI (t)), that is 1 R(t) = exp(aI hI (t)) − 1 − aI hI (t) − a2I h2I (t). 2

Define also

f (t)R(t)dt.

Q= I

Then, |Q| ≤ |a3I | I f (t)dt. Next we rewrite , 1 f (t)eaI hI (t) dt = f (t) 1 + aI hI (t) + a2I h2I (t) + R(t) dt 2 I I 2 a f (t)dt + Q = f (t)dt + a2I |I| + I 2 I I Q a2 |I| a2 + I + . = f (t)dt 1 + I 2 f dt f dt I I I Now recall that we write fI for the mean value of f over the interval I. Then, clearly, we have the identity Q 1 Q a2 1 a2 |I| + + I + = 1 + a2I + . 1+ I 2 2 fI |I|fI f dt f dt I I Next we determine constants mI so that the following equation holds. 1 1 Q 1 1 2 2 1 + aI + + + mI . = exp aI 2 fI |I|fI 2 fI

(1.3.49)

As |Q| ≤ |a3I ||I|fI and |aI | ≤ fI , the solutions mI of (1.3.49) satisfy the estimates 1 < mI < 6. 6 Summing up we showed that 1 1 aI hI 2 + mI . fe dt = f dt exp aI 2 fI I I

(1.3.50)


71

Dividing (1.3.50) by exp(a2I ( 21 + f1I )mI ) gives the identity (1.3.48). Thus we chose a constant mI ∈ [1/6, 6] so that the functions f and f exp(aI hI −a2I ( 21 + f1I )mI h2I ), have identical mean value over the interval I. Estimates for f eaI hI , hI . Next we show that the Haar coefficient f eaI hI , hI , admits the upper bound f eaI hI hI ≤ 3|aI | |I|. (1.3.51) I

To verify this we start again with the Taylor expansion of the exponential, exp(aI hI (t)) = 1 + aI hI (t) + E(t). Then |E(t)| ≤ a2I . In the integral (1.3.51) we replace eaI hI (t) by 1 + aI hI (t) + E(t) and estimate as follows, using |E(t)| ≤ a2I . f eaI hI hI = f (1 + aI hI + E) hI I I ≤ f hI + |aI | f + a2I f I

I

I

≤ 2|aI | · |I| + a2I |I|. As 2|aI | · |I| + a2I |I| ≤ 3|aI | |I|, we have (1.3.51). The main inequality. coefficient of eϕ f,

We are now ready to prove a crucial estimate for the Haar

⎛ ⎞ 1 1 eϕ f hI dt ≤ 3|aI | |I| exp ⎝− a2J ( + )mJ ⎠ . 2 fJ

(1.3.52)

J⊇I

To this end we split ϕ into three pieces. For x ∈ I we write 1 1 ϕ(x) = vI + aI hI (x) − a2I ( + )mI h2I (x) + zI (x), 2 fI where vI is defined to be the number 1 1 1 2 + h2J . f dt + aJ h J − aJ m J vI = − f ∞ + 2 fJ 0 J⊃I

(1.3.53)

J⊃I

Recall that by writing J ⊃ I we denote the summation over all dyadic intervals J that are strictly containing I. Note that zI is the function given by 1 1 2 + h2J , aJ h J − aJ m J zI = 2 fJ J⊂I

J⊂I

72


here J ⊂ I denotes all dyadic intervals that are strictly contained in I. Now let I1 be the left half of I and let I2 be the right half of I. Inductively applying (1.3.48) we obtain the identities ezI f = f, I1 I1 (1.3.54) ezI f = f. I2

I2

The identities (1.3.54) are crucial in reducing estimates for the Haar coefficient ϕ f, hI , to the estimate for f eaI hI , hI in (1.3.51). Indeed as eaI hI is a constant function on both intervals I1 and I2 , we observe that with (1.3.54), eaI ezI f dt − e−aI ezI f dt eaI hI ezI f hI dt = I1 I2 = eaI f dt − e−aI f dt (1.3.55) I1 I2 = eaI hI f hI dt. By (1.3.53) we observe that ⎛ vI

e

≤ exp ⎝−

J⊇I

⎞ 1 + )mJ ⎠ . 2 fJ

1 a2J (

(1.3.56)

The identity (1.3.55) and the estimates (1.3.51) and (1.3.56) imply the following upper bound for the coefficient eϕ f, hI , eϕ f hI dt = evI exp −a2I ( 1 + 1 )mI eaI hI ezI hI f dt 2 fI ⎛ ⎞ 1 1 2 aI hI ⎝ ⎠ aJ ( + )mJ e f hI dt = exp − (1.3.57) 2 fJ J⊇I ⎛ ⎞ 1 1 a2J ( + )mJ ⎠ |aI |. ≤ 3|I| exp ⎝− 2 fJ J⊇I

Upper bounds for the square function of eϕ f . Next we shorten notation and abbreviate, ⎞ ⎛ a2 J⎠ . pI = exp ⎝− 12 J⊇I

Note that pI is clearly bounded by 1. The sum {I:x∈I} pI a2I is a lower Riemann ∞ s sum for 0 exp(− 12 )ds, where the partition is given by the partial sums of the

1.4. Martingales and biorthogonal systems series

{I:x∈I}

73

a2I . Thus for every x ∈ [0, 1],

pI a2I ≤

{I:x∈I}

∞

exp(− 0

s )ds = 12. 12

It is now quite easy to estimate the square function of eϕ f. Expand eϕ f = eϕ f + cI h I in its Haar series. Then by (1.3.57) and (1.3.56), |cI | ≤ pI aI . At each point x ∈ [0, 1] we have therefore the uniform bound, S 2 (eϕ f )(x) ≤ 9

pI a2I

{I:x∈I}

≤ 108. The mean value of eϕ f . Next we observe that by iteratively applying equation (1.3.48) we arrive at the identity 0

1

eϕ f dt = exp(−f +

0

1

f )

1

f dt,

∞

0

that implies the lower bound

1

e f dt ≥ exp(−f ) ∞ ϕ

0

1

f dt. 0

Finally, as ϕ ≤ 0 the multiplier m = eϕ , satisfies 0 ≤ m ≤ 1.

1.4

Martingales and biorthogonal systems

The Haar basis is at the same time the simplest martingale difference sequence and the simplest biorthogonal system. In this section we extend the content of Burkholder’s Theorem 1.1.5 to all martingale difference sequences and to a large class of biorthogonal systems, including the Franklin system and the wavelet family. The results of this section illustrate the use of the Haar system in the solution of problems arising in probability and analysis.

74


The martingale inequalities of D. Burkholder and E. M. Stein We define first conditional expectations and recall the notions of martingales and martingale difference sequences. The following summary is based on the appendix in [61] by R. Durrett. We refer to this book for the proofs. Let (Ω, F, | . |) be a probability space. Let G ⊆ F be a sub σ-algebra of F. The theorem of Radon and Nikodym implies that for every f ∈ L1 (Ω, F, | . |) there exists an (essentially unique) g : Ω → R which is measurable with respect to G and satisfies f= g for A ∈ G. A

A

The function g is then called the conditional expectation of f with respect to G, for which we use the notation E(f |G) = g. Taking conditional expectations defines a contraction on Lp (Ω) (1 ≤ p ≤ ∞). Let {Fn } be a sequence of increasing σ-algebras contained in F. Assume that F coincides with the σ-algebra generated by F0 ∪ · · · ∪ Fn ∪ . . . . Let {fn } be a sequence in L1 (Ω, F), such that fn = E(fn+1 |Fn ). Then {fn } is called a martingale and dn = fn − fn−1 is called a martingale difference sequence. Examples of martingales are most easily obtained by fixing f ∈ L1 (Ω, F) and forming the conditional expectations, fn = E(f |Fn ). The partial sum operators of the Haar basis {hi }∞ i=0 are given by conditional expectation with respect to an increasing sequence of σ-algebras. See Proposition 1.1.1 and itsproof. The same holds for the partial sums of the Rademacher assertion ∞ a2i < ∞. series i=1 ai ri , when Burkholder’s martingale inequality For f ∈ Lp (Ω) the conditional expectations fn = E(f |Fn ) form a convergent sequence in Lp (Ω) (1 ≤ p < ∞). Indeed we have

f − fn Lp (Ω) → 0 for n → ∞. Hence with dn = fn − fn−1 the martingale difference series ∞ n=1 dn is convergent in Lp (Ω). The next result extends Burkholder’s theorem to martingales. It asserts that in Lp (Ω), 1 < p < ∞, the decomposition of f in its martingale differences f = E(f |F0 ) +

∞ n=1

dn

1.4. Martingales and biorthogonal systems

75

is unconditionally converging. We present the proof of B. Maurey [139], which consists of reducing estimates for general martingale difference sequences to the case of Haar functions. Theorem 1.4.1. Let f ∈ Lp (Ω, F), 1 < p < ∞. Let {Fn } be a sequence of increasing σ-algebras contained in F. Let dn = E(f |Fn ) − E(f |Fn−1 ). Then for every choice of n ∈ {+1, −1}, the following estimate holds, ∞ n dn ≤ (p∗ − 1) f Lp (Ω) . E(f |F0 ) + p n=1

(1.4.1)

L (Ω)

Proof. In this proof, we find it convenient to use systematically the following notation. For a collection of pairwise disjoint dyadic intervals C we denote Q(C) = {J ∈ D : J ⊆ K}. K∈C

Thus Q(C) is the collection of intervals which are contained in one of the intervals of C. For a single dyadic interval I we write Q(I) = {J ∈ D : J ⊆ I}. Overview. Given > 0 we will construct a sequence of functions hn : [0, 1] → R with the following three properties. First, their Haar support, given by Hn = {I : hn , hI = 0}, forms a sequence of pairwise disjoint collections of dyadic intervals, second, ∞ hn ≤ (1 + ) f Lp (Ω) , (1.4.2) h0 + p n=1

L ([0,1])

and third, for every choice of n ∈ {+1, −1} the following estimate holds, ∞ ∞ n dn ≤ (1 + ) h0 + n hn . (1.4.3) E(f |F0 ) + p p n=1

n=1

L (Ω)

L ([0,1])

Once this sequence is obtained we get (1.4.1) from Burkholder’s inequality. Indeed, using (1.4.3), Theorem 1.1.5 and (1.4.2) gives that ∞ ∞ n dn ≤ (1 + ) h0 + n hn E(f |F0 ) + p p n=1 n=1 L (Ω) L ([0,1]) ∞ ∗ ≤ (p − 1)(1 + ) h0 + hn p n=1

∗

≤ (p − 1)(1 + ) f Lp (Ω) . 2

L ([0,1])

76


The basic observation. Below we will construct the sequence {hn } satisfying (1.4.2) and (1.4.3) by iterating an elementary observation. Let C be # a collection of pairwise disjoint dyadic intervals covering the unit interval. Thus I∈C I = [0, 1[. Assume that C is decomposed into pairwise disjoint collections C1 , . . . , Cm . Let Ci be the point-set covered by the collection Ci . The sets Ci are pairwise disjoint and cover the unit interval. Choose now scalars ai ∈ R so that m ai |Ci | = 0, (1.4.4) i=1

and form the function h : [0, 1] → R by putting h=

m

ai 1Ci .

(1.4.5)

i=1

Thus we defined the function h to be measurable with respect to the σ-algebra generated by the collection C. By (1.4.4) we have h = 0. Next put H = Q([0, 1]) \ Q(C). Since h is measurable with respect to the σ-algebra generated by C and the Haar support of h is contained in H. Thus hJ hJ . h, h= |J|

(1.4.6) h = 0,

(1.4.7)

J∈H

Summing up for a function h, given by (1.4.5) and (1.4.4), the Haar support is contained in H, defined by (1.4.6). Perturbation, preparation. Now, let f ∈ Lp (Ω, F), 1 < p < ∞. Let {Fn } be a given sequence of increasing σ-algebras contained in F. We assume for simplicity that F0 = {Ω, ∅}. Let dn = E(f |Fn ) − E(f |Fn−1 ). Furthermore we assume that each of the martingale differences dn is a step function. Measured in Lp , the error of approximating by a step function can be made arbitrarily small. Consequently we may assume that each of the σ-algebras {Fn } is generated by a finite set of atoms. Let An denote the atoms of {Fn }. The collections {An } are linearly ordered in the following sense. If A ∈ An and # B ∈ An+1 so that A ∩ B = ∅, then B ⊆ A. Taking the union and forming A = An , gives a collection of sets which satisfies the following condition. If A, B ∈ A, and A ∩ B = ∅, then A ⊆ B or B ⊆ A.

(1.4.8)

Next we will reproduce the intersection pattern of A, by using collections of dyadic intervals. Precisely we claim that for each A ∈ A there #exists a collection of pairwise disjoint dyadic intervals CA so that with CA = I∈CA I the following conditions hold.


77

1. CΩ = [0, 1], and |CB | = |B| for B ∈ A. 2. If A, B ∈ A, and CA ∩ CB = ∅, then CA ⊆ CB or CB ⊆ CA . 3. CB ⊆ CA iff B ⊆ A. 4. If B ⊆ A, and I ∈ CA , then |B| |I ∩ CB | = . |I| |A|

(1.4.9)

5. If I ∈ CB , J ∈ CA and I ⊆ J then B ⊆ A. Collections {CA : A ∈ A} with the above properties are constructed inductively in the course of the proof of Proposition 2.1.4 in Chapter 2. Before we continue we should comment on condition (1.4.9): Let A ∈ An and I ∈ CA . Then by (1.4.9), conditioned to I the measures of the sets {I ∩ CB : B ∈ An+1 , B ⊆ A} are the same as the measures of {B ∈ An+1 : B ⊆ A} conditioned to A. # The induction argument. Define Cn = B∈An CB . Note that Cn is a collection of pairwise disjoint dyadic intervals. We are going to construct now a sequence of functions {hn } with Haar support contained in Hn = Q(Cn−1 ) \ Q(Cn ) so that the joint distribution of {hn } is the same as the joint distribution of the martingale difference sequence {dn }. Recall that F0 = {Ω, ∅}. Hence E(f |F0 ) is a constant function such that f. We let h0 : [0, 1] → R be the constant function defined by h0 (t) = E(f |F 0) = f. Now we define h1 : [0, 1] → R. For d1 : Ω → R there exist coefficients d(B), B ∈ A1 so that d1 = d(B)1B and d(B)|B| = 0. B∈A1

B∈A1

Next we modify d1 replacing the indicator function 1B : Ω → {0, 1} by 1CB : [0, 1] → {0, 1}. By doing this we obtain a copy of d1 with the same distribution function. We put d(B)1CB . h1 = B∈A1

Note that h1 is measurable with respect to the σ-algebra generated by the collection C1 , and h1 = 0. Consequently with H1 = Q([0, 1]) \ Q(C1 ),

78


the Haar expansion of h1 assumes the form h1 =

h1 ,

I∈H1

hI hI . |I|

By construction the martingale differences {d0 , d1 } and the newly constructed functions {h0 , h1 } are identically distributed. Suppose now that {h0 , h1 , . . . hn } are defined so that for i ≤ n, hi =

I∈Hi

hi ,

hI hI , |I|

where

Hi = Q(Ci−1 ) \ Q(Ci ),

and so that the joint distribution of {d0 , d1 , . . . , dn } and {h0 , h1 , . . . , hn } are identical. Next we determine hn+1 . Let I ∈ Cn . There exists a uniquely determined A ∈ An such that I ∈ CA . Observe that restricting h0 , h1 , . . . , hn to the interval I gives a constant function. Similarly the restriction of d0 , d1 , . . . , dn to A is a constant. Next fix B ∈ An+1 such that B ⊆ A. Then form the collection of dyadic intervals CB,I = {J ⊆ I : J ∈ CB }, and let CB,I denote the point set covered by CB,I . Next we consider the restriction of dn+1 to the atom A ∈ An . For B ∈ An+1 , with B ⊆ A, there exist coefficients d(B) such that d(B)1B . 1A dn+1 = B∈An+1 ,B⊆A

Note that E(dn+1 |Fn ) = 0 implies that A dn+1 = 0. Next we replace the indicator functions 1B : A → {0, 1} by 1CB,I : I → {0, 1}. By (1.4.9), we obtain a copy of 1A dn+1 , by defining hn+1,I = d(B)1CB,I . B∈An+1 ,B⊆A

Observe that hn+1,I is measurable with respect to the σ-algebra generated by {J ∈ Cn+1 : J ⊆ I}. Moreover I hn+1,I = 0. It follows that the Haar support of hn+1,I is contained in Hn+1,I = Q(I) \ Q({J ∈ Cn+1 : J ⊆ I}). Finally define hn+1 by summing over I ∈ Cn . Thus, hn+1 = hn+1,I . I∈Cn


79

With this definition of hn+1 the functions {d0 , . . . , dn+1 } and {h0 , . . . , hn+1 } are identically distributed. Finally for Hn+1 = Q(Cn ) \ Q(Cn+1 ), the Haar expansion of hn+1 is given by hI hn+1 = hn+1 , hI . |I| I∈Hn+1

This completes the inductive construction of the sequence {hn }.

Consequences. Combining Theorem 1.4.1 with Khintchine’s inequality gives a large class of square function characterizations for Lp (Ω, F), 1 < p < ∞. We let {Fn } be a sequence of increasing σ-algebras contained in F. Assume that F is the # σ-algebra generated by Fn . For f ∈ Lp (Ω, F) define d0 = E(f |F0 ),

and

dn = E(f |Fn ) − E(f |Fn−1 ),

for n ∈ N.

∞ We call (d20 + n=1 d2n )1/2 the martingale square function of f. By Theorem 1.4.1 the decomposition of f into martingale differences f = d0 +

∞

dn

n=1

converges unconditionally in Lp , 1 < p < ∞, and by Khintchine’s inequality we have the following characterization of Lp (Ω) by martingale-square-functions 1/2 ∞ 2 2 d cp f Lp (Ω) ≤ + d ≤ Cp f Lp (Ω) . (1.4.10) n 0 p n=1 L (Ω)

For 1 < p < ∞ we denote by H p [(Fn )] the space of all f ∈ Lp (Ω, F) for which 1/2 ∞ 2 2

f H p [(Fn )] = E(f |F0 ) + (E(f |Fn ) − E(f |Fn−1 )) < ∞. p n=1 L (Ω)

By (1.4.10) the identity operator Id : Lp (Ω) → H p [(Fn )] is an isomorphism. The importance of this result lies in its generality: One is completely free to choose the σ-algebras {Fn } to decompose a given f ∈ Lp (Ω). Stein’s martingale inequality A very surprising application of Burkholder’s martingale inequality is J. Bourgain’s proof of E. M. Stein’ s martingale inequality.

80


Theorem 1.4.2. Let (Ω, F, | · |) be a probability space. Let F1 ⊆ · · · ⊆ Fm ⊆ F be σ algebras in Ω. Then, for any choice of f1 , . . . , fm ∈ Lp (Ω) the following estimate holds true, p p 1 1 m m ∗ p rn (t)E(fn |Fn ) dt ≤ (p − 1) rn (t)fn dt, (1.4.11) p 0 0 p n=1

n=1

L (Ω)

L (Ω)

∗

where p = max{p, p/(p − 1)}, and where r1 , . . . , rm are Rademacher functions. Proof. For t ∈ [0, 1], and x ∈ Ω define F (t, x) =

m

rn (t)fn (x).

n=1

On the product space [0, 1] × Ω define the increasing σ-algebras Gn = σ{r1 , . . . , rn } ⊗ Fn ,

n ≤ m.

Observe that taking the conditional expectation with respect to Gn corresponds to forming partial sums as follows, ri (t)E(fi |Fn )(x). (1.4.12) E(F | Gn )(t, x) = i≤n

Let F0 = 0, put Fn = E(F | Gn ), for 1 ≤ n ≤ m, and dn (fi ) = E(fi | Fn ) − E(fi | Fn−1 ), for 2 ≤ n ≤ m. Note that by (1.4.12), we get ri dn (fi ), for 1 ≤ n ≤ m. Fn − Fn−1 = rn E(fn |Fn ) +

i 0 and α > 0 are independent of I ∈ D. Then the system {fI : I ∈ D} is dominated by the Haar system. Proof. Fix a dyadic interval I, with |I| = 2−i . Let m ∈ Z. Assume that I + m|I| is contained in the unit interval. By (1.4.34) every x ∈ I + m|I| satisfies −1−α

|fI (x)| ≤ A {1 + |m|}

.

Integrating over x ∈ I + m|I| gives the claimed estimate of the mean value of fI over the shifted interval I + m|I|, fI (x)dx ≤ A|I|{1 + |m|}−1−α . I+m|I| Next we turn to estimating the Haar coefficients of fI . Let J be a dyadic interval such that, J ⊆ I + m|I| and |J| = 2−l |I|. If x, y ∈ J, then |x − y| ≤ 2−i−l . Hence the Lipschitz estimate of (1.4.35) implies that for x, y ∈ J and J ⊆ I + m|I| we have −1−α . (1.4.36) |fI (x) − fI (y)| ≤ A2−l {1 + |m|} Let mJ denote the midpoint of the interval J. By (1.4.36), we obtain the following estimate for the Haar coefficients, 1 1 fI (y)hJ (y)dy = {fI (y) − fI (mJ )}hJ (y)dy 0

0

≤ |J| sup |fI (y) − fI (mJ )| y∈J

≤ A|J|2−l {1 + |m|}−1−α .

88


The books by B. Kashin and A. Saakyan [111], by Y. Meyer and R.R. Coifman [147], or by G. David [58] discuss in detail classes of orthonormal systems for which the hypothesis of Theorem 1.4.4 and Proposition 1.4.5 are satisfied.

1.5

Basic operators

We present Figiel’s representation of integral operators using paraproducts, rearrangements and Haar multipliers. We discuss orthogonal (averaging) projections acting on the Haar system, and introduce a basic criterion implying their boundedness on H 1 .

Multipliers and paraproducts First we discuss multiplier operators acting on the Haar system. Let (xI ) be a sequence in R. We define the multiplier operator X to be the linear extension of the map hI → xI hI . Thus for f with finite Haar expansion, f= aI h I , I

X(f ) is defined by X(f ) =

xI a I h I .

(1.5.1)

I

So far the multiplier operator X has been defined just algebraically. Now we study its boundedness properties. We show that Burkholder’s inequality implies that for bounded sequences (xI ) the induced multiplier is a bounded operator on Lp , 1 < p < ∞. The proof given below works for any unconditional basis in any Banach space. Theorem 1.5.1. Let 1 < p < ∞. Let (xI ) be a bounded sequence. Then the multiplier defined by (1.5.1) satisfies the following norm estimate on Lp , ∗

X(f ) p ≤ (p − 1) sup |xI | f p , I

where p∗ = max{p, p/(p − 1)}. Proof. Fix 1 < p < ∞ and let q denote the H¨ older conjugate index to p. Choose g ∈ Lq satisfying g q = 1, and

X(f ) p = X(f ), g.

(1.5.2)

1.5. Basic operators

89

With the Haar system we expand g = gdt + bI hI . Then by biorthogonality, xI aI bI |I| X(f ), g = I

≤ sup |xI | I

|aI bI | |I|.

(1.5.3)

I

Next choose I ∈ {±1} such that I aI bI = |aI bI |. Again by biorthogonality, I aI bI |I| = I aI hI , g. (1.5.4) I

I

With (1.5.3) and (1.5.4) H¨ older’s inequality gives that X(f ), g ≤ sup |xI | I aI hI g q . I I

(1.5.5)

p

Burkholder’s inequality provides the upper bound I aI hI p ≤ (p∗ − 1) f p .

I

With (1.5.2) and (1.5.5) the last line finishes the proof.

Next we define paraproduct operators and study their boundedness on the Lp spaces. Let a ∈ BMO and let f ∈ Lp with 1 < p < ∞. The paraproduct Pa is then defined by, hI hI a, f, 1I . (1.5.6) Pa (f ) = |I| |I| I

The operator Pa is thus defined to be the linear extension of the map hI → a,

hI 1I . |I|

Paraproducts appear quite naturally in connection with Figiel’s method considered in the previous section. There we fixed a family of functions {fI } and asked for conditions which imply that the map T : hI → fI extends to a bounded linear operator on Lp . Following Figiel’s approach we demanded that the functions fI satisfy the mean-value conditions (1.4.17) and estimates for their Haar coefficient expressed in (1.4.18). Recall the following expansion which marked the starting point for Figiel’s work: fI = aI 1I +

m

bIm Um (hI ) +

∞ m l=0

d(m,l) (fI ),

(1.5.7)

90


where bIm is defined in (1.4.22), d(m,l) (fI ) is defined in (1.4.25) and where

1

aI =

fI dt. 0

For a finite linear combination g = I xI fI , with xI ∈ R the representation (1.5.7) allows us to estimate the norm of g as follows, xI aI 1I

g Lp ≤ p I L (1.5.8) ∞ I + xI bm Um (hI ) + xI d(m,l) (fI ) . p m

I

m l=0

L

I

Lp

We saw in the proof of Theorem 1.4.3 that (1.4.17) gives a nice upper bound for the second series appearing on the right-hand side of (1.5.8). The hypothesis and (1.4.18) takes care of the third sum in (1.5.8). So, by reviewing the proof of Theorem 1.4.3 we made the following observation. If the family {fI } satisfies (1.4.17) and (1.4.18), then T : hI → fI extends boundedly to Lp , provided that

xI aI 1I Lp ≤ Cp

I

xI hI Lp

(1.5.9)

I

where

1

aI =

fI dt. 0

But clearly the estimate (1.5.9) states that the paraproduct Pa is bounded on on– Lp . Simultaneously, the paraproduct is one of the basic operators of Calder´ Zygmund theory, since it plays a crucial role in the proof of the famous T (1) theorem of G. David and J.L. Journé [59] Theorem 1.5.2. Let a ∈ BMO, then Pa defines a bounded operator on Lp , for 1 < p < ∞, and satisfies the estimate ||Pa (f )||p ≤ Cp ||a||BMO ||f ||p , for every f ∈ Lp with 1 < p < ∞. The constant Cp can be chosen as follows Cp =

32(p − 1)−1 4p3/2

if 1 < p ≤ 2, if 2 ≤ p < ∞.

(1.5.10)


91

Proof. Let f ∈ Lp and let g ∈ Lq , where q denotes the Hölder conjugate index to p. We will show that |Pa (f ), g| ≤ Cp ||a||BMO ||f ||p ||g||q .

(1.5.11)

The essential step towards the proof is the estimate 1 √ |Pa (f ), g| ≤ 2 2||a||BMO M1 (g)(x)S(f )(x)dx,

(1.5.12)

0

where M1 (g) denotes the Hardy–Littlewood maximal function introduced in Section 1.1 in Theorem 1.2.6. We use the notation gI to denote the mean value of g over the interval I, 1 g. gI = |I| I We start with the identity hI a, hI f, gI |I| I

1 hI = a(x) f, gI hI (x) dx. |I| 0

Pa (f ), g =

(1.5.13)

I

To the integral appearing in the last line of (1.5.13) we apply Fefferman’s inequality. This gives that √ hI |Pa (f ), g| ≤ 2 2||a||BMO f, gI hI . (1.5.14) 1 |I| I

H

We estimate the H 1 factor by analyzing its square function pointwise. For x ∈ [0, 1] we have that

f,

I

hI 2 2 2 hI gI hI (x) ≤ sup gI2 · f, 2 h2I (x), |I| |I| I x I

where the supremum is taken over all dyadic intervals containing the point x. 1/2 Note that supI x gI2 ≤ M1 (g)(x), where M1 (g) denotes the Hardy–Littlewood maximal function introduced in Theorem 1.2.6. Thus we obtain that

I

hI f, 2 gI2 h2I (x) |I|

1/2 ≤ M1 (g)(x)S(f )(x).

(1.5.15)

Integrating (1.5.15) gives (1.5.12), and by H¨ older’s inequality we obtain that √ |Pa (f ), g| ≤ 2 2||a||BMO ||M1 (g)||q ||S(f )||p .

(1.5.16)

92


The factors on the right-hand side of (1.5.16) are easily estimated with the upper bound for the Hardy–Littlewood maximal function, Theorem 1.2.6, and the square function characterization of Theorem 1.1.6. Thus (1.5.16) implies that |Pa (f ), g| ≤ Cp ||a||BMO ||g||q ||f ||p ,

(1.5.17)

where Cp = 8p(p∗ − 1) and p∗ = max{p, p/(p − 1)}. Taking the supremum (1.5.17) over all g ∈ Lq with ||g||q ≤ 1 gives that ||Pa (f )||p ≤ Cp ||a||BMO ||f ||p . A more careful use of square function estimates is possible by distinguishing between the cases 1 < p < 2 and 2 < p < ∞. If we use Theorem 1.1.6 for 1 < p < 2, and Theorem 1.3.5 for 2 < p < ∞, then we obtain improved constants Cp as follows, 32(p − 1)−1 if 1 < p < 2, (1.5.18) Cp = 4p3/2 if 2 < p < ∞.

Rearrangement operators and Calder´ on–Zygmund kernels Next we introduce rearrangement operators. In combination with paraproducts they are an important tool in the study of integral operators. Most notably in the analysis of Calder´ on–Zygmund integral operators. In this section we present Figiel’s representation of integral operators, to illustrate this point. For m ∈ Z, we define the operators Tm and Um by their action on the Haar basis. Let I be a dyadic interval such that I + m|I| ⊆ [0, 1]. Then we put Tm (hI ) = hI+m|I| , Um (hI ) = 1I+m|I| − 1I . For the remaining intervals we simply put Tm (hI ) = Um (hI ) = 0. Defined that way, the operators Tm and Um are bounded operators on Lp when 1 < p < ∞. We will give the precise estimates later. Now we discuss their connection to integral operators in detail. We start with integral operators for which the kernel, a priorily, belongs to the space L2 ([0, 1] × [0, 1]). To analyze the operator we expand the kernel using the following biorthogonal system in L2 ([0, 1] × [0, 1]). ! " hI+m|I| (x) · hI (y) : I + m|I| ⊆ [0, 1], m ∈ Z, I ∈ D ∪ " ! 1I+m|I| (x) · hI (y) : I + m|I| ⊆ [0, 1], m ∈ Z, I ∈ D ∪ " ! hI+m|I| (x) · 1I (y) : I + m|I| ⊆ [0, 1], m ∈ Z, I ∈ D .


93

This system is complete in L2 ([0, 1] × [0, 1]), provided that we augment it with the constant function 1. Now we fix an integral kernel in K ∈ L2 ([0, 1] × [0, 1]). We will now expand K using the basis introduced above. This is done in three steps corresponding to the three groups of basis functions introduced. We fix dyadic intervals I, J, with |I| = |J|, then we define the Haar coefficients 1 1 K(x, y)hI (x)hJ (y)dxdy, (K, hJ ⊗ hI ) =

0

0

0

1

1

(K, 1J ⊗ hI ) =

K(x, y)hI (x)1J (y)dxdy,

(1.5.19)

0

1

1

(K, hJ ⊗ 1I ) =

K(x, y)1I (x)hJ (y)dxdy. 0

0

Next we introduce three kernels which also correspond to the three groups of the basis. hI+m|I| (x) hI (y) , K, hI+m|I| ⊗ hI K1 (x, y) = |I| |I| m∈Z {I:I+m|I|⊆[0,1]}

K2 (x, y) =

K, 1I+m|I| ⊗ hI

1I+m|I| (x) hI (y) , |I| |I|

K, hI+m|I| ⊗ 1I

hI+m|I| (x) 1I (y) . |I| |I|

m∈Z {I:I+m|I|⊆[0,1]}

K3 (x, y) =

m∈Z {I:I+m|I|⊆[0,1]}

By definition, our original kernel K can be recovered from K1 , K2 , K3 , as follows, 1 1 K= K(x, y)dxdy + K1 + K2 + K3 . (1.5.20) 0

0

Following is the definition of three Haar multipliers. Their defining sequences come from the Haar coefficients (1.5.19). We let m ∈ Z and let I be a dyadic interval, then we define (K, hI+m|I| ⊗ hI ) . |I| For fixed m the sequence (am (I)) defines the multiplier hI am (I)f, hI . Am (f ) = |I| am (I) =

(1.5.21)

(1.5.22)

{I:I+m|I|⊆[0,1]}

Theorem 1.5.1 asserts that if (am (I)) is bounded, then the Haar multiplier Am is a bounded operator on Lp and satisfies ∗

Am (f ) p ≤ (p − 1) sup |am (I)| f p . I

94


The Haar multiplier Am corresponds to the kernel K1 . Now we define the multiplier corresponding to K2 . Let I be a dyadic interval, then we define bm (I) =

(K, 1I+m|I| ⊗ hI ) . |I|

For fixed m the sequence (bm (I)) defines the multiplier

Bm (f ) =

bm (I)f,

{I:I+m|I|⊆[0,1]}

hI hI . |I|

By Theorem 1.5.1, the Haar multiplier Bm satisfies the norm estimates ∗

Bm (f ) p ≤ (p − 1) sup |bm (I)| f p . I

It remains to define the Haar multiplier associated to the kernel K3 . Let I be a dyadic interval, then define the coefficients cm (I) =

(K, hI ⊗ 1I+m|I| ) , |I|

which in turn define the multiplier Cm ,

Cm (f ) =

cm (I)f,

{I:I+m|I|⊆[0,1]}

hI hI . |I|

Theorem 1.5.1 gives that

Cm (f ) p ≤ (p∗ − 1) sup |cm (I)| f p . I

The above discussion shows that in Lp the operator norms of the multipliers Am , Bm , Cm , are bounded by the quantity |K|m = (p∗ − 1) sup{|am (I)| + |bm (I)| + |cm (I)|}, I

where the supremum is extended over all dyadic intervals I for which I + m|I| is contained in the unit interval. We are now prepared to formulate Figiel’s representation of integral operators. This is a powerful theorem which expresses a given integral operator as a sum of paraproducts, Haar multipliers and rearrangement operators. The classical T (1) theorem of G. David and J.L Journé [59] follows easily from Figiel’s representation, thereby showing the importance of these basic operators.


95

Theorem 1.5.3. Let K ∈ L2 ([0, 1] × [0, 1]). Let R be the integral operator defined by the kernel K, 1 R(f )(x) = K(x, y)f (y)dy. 0

11 Let a = R(1), b = R (1) and put K0 = 0 0 K(x, y)dxdy. Then R admits the representation Tm (Am (f )) + Um (Bm (f )) + (Um Cm )∗ (f ), R(f ) = K0 + Pa (f ) + Pb∗ (f ) + ∗

m∈Z p

and its norm on L satisfies the estimate ||R||p ≤ K0 + 8p(p∗ − 1) (||a||BMO + ||b||BMO ) + Cp

log2 (4 + |m|)|K|m .

m∈Z

Here p∗ = max{p, p/(p − 1)}. We formulated Figiel’s representation theorem for integral operators under the initial hypothesis that K ∈ L2 ([0, 1] × [0, 1]). However the L2 norm of the kernel does not appear in the conclusion of Theorem 1.5.3, that is the norm estimate of the integral operator 1

K(x, y)f (y)dy.

R(f )(x) = 0

It is therefore possible to extend the scope of the representation Theorem 1.5.3 to singular integral operators, using simple limiting procedures. Recall from (1.5.20), that the kernel K appearing in Theorem 1.5.3 is decomposed as 1 1 K= K(x, y)dxdy + K1 + K2 + K3 . 0

0

For i ∈ {1, 2, 3} let Ri denote the integral operator corresponding to the kernel Ki , 1 Ki (x, y)f (y)dy. Ri (f )(x) = 0

The following three propositions give the respective representations for the integral operators Ri , i ∈ {1, 2, 3}. Put together they give Theorem 1.5.3. We begin with the simplest case which is the operator R1 . Proposition 1.5.4. Let f ∈ Lp . Let Am be the multiplier given by (1.5.21) and (1.5.22). Then Tm (Am (f )), (1.5.23) R1 (f ) = m∈Z

and ||R1 ||p ≤

m∈Z

||Tm ||p Am p ,

96


where

Am p ≤ (p − 1) sup |am (I)| . ∗

I

Proof. Once we have shown the representation (1.5.23), the norm estimate for R1 follows easily. Indeed by (1.5.22), Theorem 1.5.1 implies that ||Am ||p ≤ (p∗ − 1) sup |am (I)| . I

We turn to the representation of R1 . Note that the action of Tm on the Haar function hI is given by Tm (hI ) = hI+m|I| . The claimed representation for the integral operator R1 we find by unwinding the definitions involved. The first identity below follows from the definition of the kernel K1 , for the second one we use the definition of the rearrangement Tm , for the last identity we invoke the definition of the multiplier Am . In summary we find that

1

K1 (x, y)f (y)dy = 0

m

=

m

=

hI hI+m|I| (x) |I| |I|

(K, hI+m|I| ⊗ hI )f,

hI Tm (hI )(x) |I| |I|

I

(K, hI+m|I| ⊗ hI )f,

I

Tm (Am (f ))(x).

m

1 Proposition 1.5.5. Let f ∈ Lp . Let a(x) = 0 K(x, y)dy, and let Pa be the paraproduct defined by a. Then R2 has the representation R2 (f ) = Pa (f ) + Um (Bm (f )). m∈Z

Its operator norm admits an upper bound, ||R2 ||p ≤ 8p(p∗ − 1)||a||BMO +

||Um ||p Bm p ,

m∈Z

and

Bm p ≤ (p − 1) sup |bm (I)| . ∗

I

Proof. Here we start by evaluating the integral operator R2 . The definition of the kernel K2 gives that

1

K2 (x, y)f (y)dy = 0

m∈Z I

hI 1I+m|I| (x) , K, 1I+m|I| ⊗ hI f, |I| |I|

(1.5.24)


97

where the inner sum is extended over all dyadic intervals I for which I + m|I| ⊆ [0, 1]. Next we focus on the function 1I+m|I| (x) appearing in the above sum. Using the operator Um we rewrite 1I+m|I| (x) as 1I+m|I| (x) = Um (hI )(x) + 1I (x).

(1.5.25)

Inserting the identity (1.5.25) into (1.5.24) defines a splitting of the operator R2 into two parts. A first part which involves the operators Um , and a remaining part which leads to the appearance of the paraproduct Pa . We isolate first the paraproduct. By changing the order of summation we obtain the identity

m∈Z {I:I+m|I|⊆[0,1]}

=

⎛

⎝

{m:I+m|I|⊆[0,1]}

I

1I hI f, |I| |I| ⎞ 1I hI K, 1I+m|I| ⊗ hI ⎠ f, . |I| |I|

K, 1I+m|I| ⊗ hI

We continue by identifying the sum appearing inside the brackets. We will show 1 that it is just the Haar coefficient a, hI , where a(x) = 0 K(x, y)dy. We begin by evaluating

K, 1I+m|I| ⊗ hI =

{m:I+m|I|⊆[0,1]}

1

1

K(x, y)hI (x)dxdy. 0

(1.5.26)

0

Next observe that the 1double integral on the right-hand side is a Haar coefficient, namely with a(x) = 0 K(x, y)dy,

1

1

K(x, y)hI (x)dxdy = a, hI . 0

(1.5.27)

0

Recall next the definition of the paraproduct operator Pa . It is given by the equation hI 1I (x) Pa (f )(x) = . a, hI f, |I| |I| I

Combining (1.5.26) with (1.5.27), and inserting the value of a, hI we arrived at the following identity for the paraproduct, Pa (f )(x) =

m∈Z {I:I+m|I|⊆[0,1]}

hI 1I K, 1I+m|I| ⊗ hI f, . |I| |I|

(1.5.28)

98

Chapter 1. The Haar System: Basic Facts and Classical Results Next we unwind the definition of the multipliers Bm , and thus we find that

Um (Bm (f ))

m∈Z

=

Um

m∈Z

=

m∈Z I

I

hI Bm (f ), hI |I|

hI Um (hI ) , K, 1I+m|I| ⊗ hI f, |I| |I|

where we extended the inner sum over all dyadic intervals I such that I + m|I| ⊆ [0, 1]. It remains to combine the last identity with (1.5.24), (1.5.25) and (1.5.28). This gives that

1

K2 (x, y)f (y)dy = Pa (f )(x) + 0

Um (Bm (f ))(x),

m∈Z

which is the claimed representation for R2 . The resulting upper bound for the norm comes from Theorem 1.5.2 that controls the norm of the paraproduct, ||Pa ||p ≤ 8p(p∗ − 1)||a||BMO , and Theorem 1.5.1 that gives norm estimates of the multiplier Bm . Hence ||Bm ||p ≤ (p∗ − 1) sup |bm (I)| . I

Following is the representation of R3 . Up to transposition and duality the the proof below is the same as the preceding proof of Proposition 1.5.5. 1 Proposition 1.5.6. Let g ∈ Lp . Let b(y) = 0 K(x, y)dx, and let Pb be the paraproduct defined by b. Then the adjoint operator to R3 has the representation R3∗ (g) = Pb (g) +

Um (Cm (g)),

m∈Z

and ||R3 || ≤ 8p(p∗ − 1)||b||BMO +

||Um ||p Cm p ,

m∈Z

where

||Cm ||p ≤ (p∗ − 1) sup |cm (I)| . I


99

Proof. Fix f ∈ Lp , and g ∈ Lq . Then we evaluate R3 (f ), g. We invoke the definition of the kernel K3 , then we change the order of summation and relabel, 1 1 K3 (x, y)f (y)g(x)dxdy 0

= =

0

m

I

J

n

hI+m|I| 1I (y) f, K, hI+m|I| ⊗ 1I g, |I| |I| 1J+n|J| hJ f, . K, hJ ⊗ 1J+n|J| g, |J| |J|

Next we look carefully at the last factor in the above sum. That is f, rewrite it using the operator Un as f,

1J+n|J| |J| .

We

1J+n|J| Un (hJ ) 1J = f, + f, . |J| |J| |J|

Now the proof proceeds parallel to that of Proposition 1.5.5. We identify the 1 paraproduct first. Recall that we put b(y) = 0 K(x, y)dx. One finds as in the proof of Proposition 1.5.5 that K, hJ ⊗ 1J+n|J| , b, hJ = n

where we extend the sum over those n ∈ Z for which J + n|J| ⊆ [0, 1]. The paraproduct Pb (g) is defined by h J 1J . b, hJ g, Pb (g) = |J| |J| J

Consequently, f, Pb (g) =

J

n

hJ 1J f, . K, hJ ⊗ 1J+n|J| g, |J| |J|

Next we recall the definition of the multiplier Cn , and note that f, Un (Cn (g)) n

=

hJ Un (hJ ) f, . K, hJ ⊗ 1J+n|J| g, |J| |J| n J

It follows that f, R3∗ (g) = R3 (f ), g = f, Pb (g) +

f, Un (Cn (g)),

n∈Z

which proves the representation of the operator R3 . The norm estimates are obtained as in the proof of Propositions 1.5.4 and 1.5.5.

100


Proof of Theorem 1.5.3. We apply Propositions 1.5.4–1.5.6, and use the following upper bound for the norms of the operators Tm and Um in Lp . ||Tm ||p ≤ Cp log2 (4 + |m|), ||Um ||p ≤ Cp log2 (4 + |m|). These estimates will be proven in Chapter 3, Theorem 3.3.9. With this remark the proof of Theorem 1.5.3 is complete. The Hilbert transform is our first example illustrating Theorem 1.5.3. For x, y ∈ [0, 1] and x = y we define H(x, y) = cot Let > 0 and put

H (x, y) =

H(x, y) 0

(x − y) . π if |x − y| > , otherwise.

The kernel H is a bounded function defined on [0, 1] × [0, 1]. It gives rise to an integral operator 1 R f (x) = H (x, y)f (y)dy. 0

Applying Theorem 1.5.3 we will show that the integral operator R is bounded in Lp (1 0. Corollary 1.5.7. Let 1 < p < ∞. The Hilbert transform defined as the limit R(f ) = lim R (f ) →0

p

is a bounded operator on L . Proof. We will show that for f ∈ Lp ([0, 1]), ||R f ||p ≤ Ap ||f ||p ,

(1.5.29)

and that lim→0 R (f ) exists in Lp . To prepare for the application of Theorem 1.5.3 we recall that the kernel H belongs to L2 ([0, 1] × [0, 1]). Furthermore 1 1 H (x, y)dy = 0 and H (x, y)dx = 0. 0

0

Consequently, the paraproducts in Figiel’s representation of R are vanishing operators. Using integration by parts we easily obtain that there exists A > 0 so that for m ∈ Z, the following estimates hold, |(H , hI ⊗ hI+m|I| )| ≤ A(1 + |m|)−2 |I|, |(H , 1I ⊗ hI+m|I| )| ≤ A(1 + |m|)−2 |I|, |(H , hI ⊗ 1I+m|I| )| ≤ A(1 + |m|)−2 |I|.


101

The constant A does not depend on I, m or > 0. Hence Theorem 1.5.3 gives (1 + |m|)−2 log2 (4 + |m|) ||R ||p ≤ Cp A m∈Z

≤ Ap . In this upper bound the L2 norm of the kernel does not appear. In particular, the norm of R in Lp is bounded independent of . Next we show convergence of R (f ) when f has bounded derivative. As H (x, y)dy = 0, we may insert the value f (x) in the integral defining R f (x). Hence 1

H (x, y){f (y) − f (x)}dy.

R f (x) = 0

Fix 0 < δ < , and estimate the difference 1 |Rδ f (x) − R f (x)| = {Hδ (x, y) − H (x, y)}{f (y) − f (x)}dy 0 (x − y) ≤ · |f (y) − f (x)|dy. cot π δ≤|x−y|≤ Recall that f is assumed to have bounded derivative, hence |f (y) − f (x)| ≤ ||f ||∞ |x − y|. Furthermore, the product |(x − y) cot(x − y)| is bounded. It follows that |Rδ f (x) − R f (x)| ≤ C||f ||∞ ( − δ). Summing up, R f is a Cauchy sequence in Lp , when f has bounded derivative. By (1.5.29) and the Banach–Steinhaus theorem, the limit defined by R(f ) = lim R (f ) →0

induces a bounded linear operator on Lp .

Now we apply Theorem 1.5.3 to a wide range of singular kernels, and prove a version of the T (1) theorem of G. David and J. Journé [59]. A kernel K(x, y) defined for x = y is called a Calderón–Zygmund kernel if there exists δ > 0 and C > 0 such that for x = y the following estimates hold, |K(x, y)| ≤ C|x − y|−1 , |∂x K(x, y)| + |∂y K(x, y)| ≤ C|x − y|−1−δ , K(x, y) = −K(y, x). We will now determine when a Calder´ on–Zygmund kernel is associated to a bounded linear operator T : Lp → Lp , in the following sense: 1 1 K(x, y)f (y)g(x)dxdy = T (f ), g, 0

0

102


whenever f ∈ Lp ([0, 1]), g ∈ Lq ([0, 1]) and supp (f ) ∩ supp (g) = ∅. Fix a Calder´ on–Zygmund kernel K. We introduce now a canonical approximation procedure for K. For a dyadic interval I define the following collection of dyadic intervals, AI = {J : |J| = |I|, dist(I, J) ≥ |I|}. Fix n ∈ N. We define the approximating kernel K (n) by setting K(x, y)1I (x)1J (y). K (n) (x, y) = {I:|I|=2−n } J∈AI

Note that the approximation is obtained by deleting a thin strip around the diagonal x = y, hence K (n) ∈ L2 ([0, 1] × [0, 1]). By the defining properties for the Calder´ on–Zygmund kernel K there exists a constant Aδ > 0 such that for every n ∈ N the following estimates hold, (n) K ≤ Aδ |m|−1−δ . m

The L2 kernel K (n) defines the integral operator R(n) (f )(x) =

1

K (n) (x, y)f (y)dy. 0

Let an = R(n) (1), and bn = (R(n) )∗ (1). Theorem 1.5.3 gives the following estimates for the operator norm of R(n) : Lp → Lp . Aδ (log2 (4 + |m|))|m|−1−δ ||R(n) ||p ≤ Cp (||an ||BMO + ||bn ||BMO ) + Cp m∈Z

≤ Cp (||an ||BMO + ||bn ||BMO ) + Ap,δ . We point out that the constants Ap,δ appearing in the norm estimate for R(n) are actually independent of n. Theorem 1.5.8. If supn ||an ||BMO < ∞ and supn ||bn ||BMO < ∞, then there exists a bounded linear operator T : Lp → Lp , 1 < p < ∞ such that 1 1 K(x, y)f (y)g(x)dxdy, T (f ), g = 0

0

whenever f ∈ L ([0, 1]), g ∈ L ([0, 1]) and supp (f ) ∩ supp (g) = ∅. p

q

Proof. First observe that we defined the approximating kernels by truncating so that for x = y, K(x, y) = lim K (n) (x, y). n→∞

As supn ||an ||BMO < ∞ there exists a subsequence of (an ) which converges, weak∗ in BMO. Denote it again by (an ). Let f have bounded derivative. Let g ∈ Lq . We


103

show next that (g, R(n) (f )) is a Cauchy sequence. Let n ∈ N and let m ≥ n. Then g, (R(n) − R(m) )(f ) 1 1 (K (n) (x, y) − K (m) (x, y))f (y)g(x)dxdy =

0

0

1

= 0

1

K (n) (x, y) − K (m) (x, y) (f (y) − f (x))g(x)dxdy

0

+ an − am , (gf ). The definition of the approximating kernels give that |(K (n) (x, y) − K (m) (x, y))| · |f (y) − f (x)| ≤ C|K (n) (x, y) − K (m) (x, y)| · |x − y| ≤ C. Moreover supp {(K (n) −K (m) )} ⊆ {(x, y) : |x−y| ≤ 2−n }. Thus for each x ∈ [0, 1], 1 |(K (n) (x, y) − K (m) (x, y))(f (y) − f (x))|dy ≤ C2−n . 0

It follows that g, (R(n) − R(m) )(f ) is a Cauchy sequence, and for a Lipschitz function f, the following limit exists in Lp , lim R(n) f.

n→∞

By Theorem 1.5.3 and the hypothesis we find sup ||R(n) ||p ≤ sup Cp (||an ||BMO + ||bn ||BMO ) + Ap, δ n

n

< ∞. As Lipschitz functions are dense in Lp , the theorem of Banach and Steinhaus implies the existence of a bounded linear operator T : Lp → Lp , such that T (f ) = lim R(n) (f ), n→∞

where the limit exists in Lp ; moreover, ||T ||p ≤ sup ||R(n) ||p . n

Finally note that if supp (f ) ∩ supp (g) = ∅, then dist( supp (f ), supp (g)) > 0. Hence the identity K(x, y) = lim K (n) (x, y), n→∞

holding for x = y, implies that 1 1 K(x, y)f (y)g(x)dxdy = lim 0

0

n→∞

0

1

1

K (n) (x, y)f (y)g(x)dxdy, 0

104


provided that f ∈ Lp ([0, 1]), g ∈ Lq ([0, 1]) and supp (f ) ∩ supp (g) = ∅. Summing up we found a bounded linear operator T : Lp → Lp , such that, T (f ), g = lim R(n) (f ), g n→∞ 1 1 K (n) (x, y)f (y)g(x)dxdy = lim n→∞

1

0 1

0

K(x, y)f (y)g(x)dxdy,

= 0

0

whenever f ∈ Lp ([0, 1]), g ∈ Lq ([0, 1]) and supp (f ) ∩ supp (g) = ∅.

Orthogonal projections We introduce now a class of orthogonal projections which plays a very important role in this book. We will use them for three main reasons. First we construct several families of complemented subspaces in H 1 and BMO. Second their boundedness properties will be crucial in finding isomorphic invariants of H 1 , such as the uniform approximation property, the dimension conjecture or dichotomies for complemented subspaces. Third, orthogonal projections yield canonical examples of operators by which we will test (and refute) several natural conjectures. From now on it is convenient to use systematically the following notation. Let H be a collection of dyadic intervals. Let f = aI hI . We define S(f |H) by the equation S 2 (f | H) =

a2I h2I ,

I∈H

and call it the square function of f restricted to H. If J is a dyadic interval and if we put Q(J) = {I : I ⊆ J}, then a2I |I|. S 2 (f | Q(J))dt = I⊆J

Hence

f 2BMO = sup J

S 2 (f | Q(J))

dt . |J|

We let C1 , . . . , CN be disjoint collections of dyadic intervals. We form linear combinations of Haar functions by putting bi = {hI : I ∈ Ci }, i ≤ N, (1.5.30) and we normalize in L2 , ki = bi /||bi ||2 .

(1.5.31)


105

The orthogonal projection onto span{bi }N i=1 is given by Pf =

N

f, ki ki .

(1.5.32)

i=1

Clearly, by the disjointness of the collections C1 , . . . , CN the operator P is of norm 1 in L2 ([0, 1]). Now we introduce a condition on C1 , . . . , CN which implies that the operator norm of P on BMO is bounded independent of N . We let Ci denote the point-set covered by the collection Ci , that is, we let Ci =

I.

I∈Ci

Disjoint collections of dyadic intervals C1 , . . . , CN are said to satisfy Jones’s compatibility condition (J) if the following conditions (J 1)–(J 4) are satisfied. (J 1) Each Ci is a collection of pairwise disjoint dyadic intervals. (J 2) There exists a constant B, so that for each J ∈ Cj and each i ≥ j the following estimate holds, |I| |Cj ∩ Ci | ≤B . |J| |Cj | I∈Ci ,I⊆J

(J 3) The collections {Ci : 1 ≤ i ≤ N } are ordered in the following sense: If there exists I ∈ Ci and J ∈ Cj , such that I ⊆ J, then i ≥ j. Conversely, if i ≥ j, if I ∈ Ci and J ∈ Cj , such that I ∩ J = ∅, then I ⊆ J. (J 4) If j ≤ i and if Cj ∩ Ci = ∅, then Ci ⊆ Cj . In particular the sets {Ci : 1 ≤ i ≤ N } are nested; if Cj ∩ Ci = ∅, then either Ci ⊆ Cj or Cj ⊆ Ci . If the compatibility condition (J) is satisfied we will show that the BMO norm of the orthogonal projection defined by (1.5.32) is bounded independent of N. √ Precisely we prove below that the norm of P in BMO is bounded by B, where B is the constant appearing in condition (J 2). Following is a basic criterion for the boundedness of orthogonal projections in BMO. Theorem 1.5.9. If C1 , . . . , CN are pairwise disjoint collections of dyadic intervals satisfying Jones’s compatibility condition (J), then, on BMO, the orthogonal projection N f, ki ki Pf = has norm bounded by

√

i=1

B.

106


Proof. Let f ∈ BMO with Haar expansion f = cI hI . We expand P f in its Haar series Pf = αI hI , #N where αI = f, ki |Ci |−1/2 , for I ∈ Ci , and where αI = 0 for I ∈ i=1 Ci . Now we fix k ≤ N and a dyadic interval J0 ∈ Ck . Then the following identity holds, 1 2 1 S 2 (P f | Q(J0 )) = αI |I|. |J0 | |J0 | I⊆J0

We will now show that the following estimates holds, 1 2 αI |I| ≤ B||f ||2BMO . |J0 | I⊆J0

By condition (J 3), if I ⊆ J0 and if I ∈ Ci , then i ≥ k. Therefore by changing the order of summation we have the identity I⊆J0

αI2 |I| =

N

f, ki 2

i=k

I∈Ci ,I⊆J0

|I| . |Ci |

(1.5.33)

We will now estimate the right-hand side of (1.5.33) using the conditions (J 1)–(J 4). Recall that condition (J 2) states that for J ∈ Ck and i ≥ k, the following estimate holds, |I| |Ci ∩ Ck | ≤B . (1.5.34) |J0 | |Ck | I∈Ci ,I⊆J0

Thus conditions (J 2) and (J 3) give the following upper bound for E, 1 |J0 |

B |Ci ∩ Ck | . f, ki 2 |Ck | |Ci | N

S 2 (P f | Q(J0 )) ≤

(1.5.35)

i=k

Now we use condition (J 4) to continue. It follows from (J 4) that if Ci ∩ Ck = ∅ and i ≥ k, then Ci ⊆ Ck , and |Ci ∩ Ck | = |Ci |. Moreover if i ≥ k and Ci is not contained in Ck , then Ci ∩ Ck = ∅. Hence the estimate (1.5.35) implies 1 B S 2 (P f | Q(J0 )) ≤ f, ki 2 . (1.5.36) |J0 | |Ck | {i:Ci ⊆Ck }


107

Next we use the converse part of condition (J 3), Bessel’s inequality and the definition of BMO to obtain f, ki 2 ≤ c2K |K| {i:Ci ⊆Ck }

J∈Ck K⊆J

≤ ||f ||2BMO

(1.5.37)

|J|.

J∈Ck

Finally recall condition (J 1) that Ck is a collection of pairwise disjoint dyadic intervals. Thus |J| = |Ck |. (1.5.38) J∈Ck

Inserting identity (1.5.38) and the estimate (1.5.37) into (1.5.36) gives 1 S 2 (P f | Q(J0 )) ≤ B||f ||2BMO , |J0 |

as claimed. 1

By duality the orthogonal projection P is bounded on H if and only if it is bounded on BMO. Theorem 1.5.10. Let C1 , . . . , CN be collections of dyadic intervals satisfying Jones’s compatibility condition (J). Let ki be defined by (1.5.31). Then the orthogonal projection N f, ki ki Pf = i=1

√ defines a bounded operator on H with norm ≤ 8 B. 1

Proof. We use Fefferman’s duality theorem, the orthogonality of P and Theorem 1.5.9. This gives the following bound for the norm of P, √ ||P f ||H 1 ≤ 2 2 sup{P f, g : ||g||BMO ≤ 1} √ = 2 2 sup{f, P g : ||g||BMO ≤ 1} √ √ ≤ 2 2 B sup{f, h : ||h||BMO ≤ 1} √ ≤ 8 B||f ||H 1 . The glueing process. Below we need to be able to reduce the rank of a given orthogonal projection. For this purpose we glue together families of orthogonal projections acting locally, into one large orthogonal projection acting globally. We describe now the glueing process in detail, and study boundedness properties of orthogonal projections obtained by this method. We recall first the notion of a block of intervals. It was introduced in Section 1.2. Let K be a collection of dyadic intervals, and let B ⊆ K. We say that B is a block of intervals in K, if the following conditions are satisfied.

108


1. B contains exactly one maximal interval, say I. 2. For every J ∈ B and K ∈ K the following “connectedness” condition holds true: If J ⊆ K ⊆ I, then K ∈ B. Given a block of intervals B we will systematically use the notation B = B(I) to indicate that I is the unique maximal interval of the block B. Assume now that {B(I) : I ∈ A} is a collection of disjoint blocks in K. Fix N, B ∈ N. Assume that for each I ∈ A the block B(I) contains collections of dyadic intervals, Ci (I), i ≤ N so that C1 (I), . . . , CN (I) satisfy Jones’s compatibility condition (J).

(1.5.39)

Let the constant appearing in the compatibility condition be the fixed number B ∈ N. For i ≤ N define bi = {hK : K ∈ Ci (I)}, (1.5.40) I∈A

and let P denote the orthogonal projection onto span{bi }, that is, P (f ) =

N

f,

i=1

bi bi . ||bi ||2 ||bi ||2

(1.5.41)

Applying Theorem√1.5.9 to the projection defined by (1.5.41) gives an upper bound for P BMO by B × (the cardinality of A). For the applications to follow we now isolate a further structural condition ensuring that the norm of P is bounded by a function depending only on B and [[A]], where 1 I∈A |I|

[[A]] = sup

|J|.

J∈A, J⊆I

This is the purpose of Theorem 1.5.11. For m ≤ N and J ∈ A let Cm (J) = K. K∈Cm (J)

Loosely speaking the hypothesis of Theorem 1.5.11 states that for I, J ∈ A the collections {Cm (J) : m ≤ N } and {Cm (I) : m ≤ N } are rescaled versions of each other. Precisely, assume that {Cm (J) : m ≤ N } and {Cm (I) : m ≤ N } are linked through the following conditions: For I, J ∈ A and m, n ≤ N,

and

Cm (I) ⊆ Cn (I) if Cm (J) ⊆ Cn (J),

(1.5.42)

|Cm (I)| |Cm (J)| ≤2 . |Cn (I)| |Cn (J)|

(1.5.43)


109

Remarks. 1. Note that condition (1.5.42) states that for I, J ∈ A, the intersection pattern of {Cm (J) : m ≤ N } coincides with that of {Cm (I) : m ≤ N }. 2. Below we will apply also the following averaged version of condition (1.5.43). Multiply both sides of (1.5.43) by |Cn (J)|, and sum the resulting estimate over J ∈ A. This gives that . / |Cm (I)| ≤2 |Cn (J)| |Cm (J)|. |Cn (I)| J∈A

Dividing further by For every I ∈ A,

J∈A

J∈A

|Cn (J)| yields the averaged version of condition (1.5.43): |Cm (J)| |Cm (I)| ≤ 2 J∈A . |Cn (I)| J∈A |Cn (J)|

Theorem 1.5.11. Assume that (1.5.39) and the rescaling conditions (1.5.42) and (1.5.43) are satisfied. Then for the projection P, defined by (1.5.41) the following estimates hold: Let I0 ∈ A, and J0 ∈ B(I0 ), then S 2 (P (f )|L) ≤ 2B f 2BMO |J0 |, (1.5.44) where L = {J ∈ B(I0 ) : J ⊆ J0 }. Consequently, 1

||P ||BMO ≤ (2B[[A]]) 2 .

(1.5.45)

Proof. The proof is divided into two separate parts. First we show how the norm estimate for P follows from (1.5.44). The second part consists in proving (1.5.44). The norm estimate (1.5.45) follows from (1.5.44). Let f ∈ BMO. Let H be the Haar support of P (f ). For J0 ∈ H there exists I0 ∈ A, such that J0 ∈ B(I0 ). Next observe that the following inclusion holds, {J ∈ H : J ⊆ J0 } ⊆ {J ∈ B(I0 ) : J ⊆ J0 } ∪ B(I). I∈A, I⊂I0

Introducing the following abbreviations simplifies considerably our notation. J0 ∩ H = {J ∈ H : J ⊆ J0 }, J0 ∩ B(I0 ) = {J ∈ B(I0 ) : J ⊆ J0 }. Now we rewrite the square function S 2 (P (f )|J0 ∩ H) as S 2 (P (f )|J0 ∩ H) = S 2 (P (f )|J0 ∩ B(I0 )) +

I∈A, I⊂I0

S 2 (P (f )|B(I)).

110


We conclude that the norm estimate (1.5.45) for P follows from the inequalities S 2 (P (f )|J0 ∩ B(I0 )) ≤ 2||f ||2BMO B|J0 |, S 2 (P (f )|B(I)) ≤ 2||f ||2BMO B|I|. In order to prove (1.5.44), it suffices to verify the first of the above estimates; once this is done we specialize to I0 = J0 and observe that I0 was an arbitrary interval from A. Verification of (1.5.44). Recall that H is the Haar support of P (f ). Restricted to the block B(I0 ) the Haar support of P (f ) can be written as B(I0 ) ∩ H = C1 (I0 ) ∪ · · · ∪ CN (I0 ). Hence there exists m0 ≤ N such that J0 ∈ Cm0 (I0 ). Now define the set of integers Um0 = {m : Cm (I0 ) ⊆ Cm0 (I0 )}. By the hypothesis (1.5.42) the set Um0 does not depend on the choice of the interval I0 . Indeed, if for m0 and m we have that Cm (I0 ) ⊆ Cm0 (I0 ), then for any other I ∈ A the corresponding inclusion holds, namely, Cm (I) ⊆ Cm0 (I). (For Um0 to be independent of I0 , we used the hypothesis that the intersection pattern of {Cm (I) : m ≤ N } is independent of I.) Now we rewrite, using the index set Um0 , S 2 (P (f )|J0 ∩ B(I0 )) =

m∈Um0

f

bm 2 S 2 (bm |J0 ∩ B(I0 )) . ||bm ||2 ||bm ||22

(1.5.46)

Fix m ∈ Um0 , and focus on the factor S 2 (bm |J0 ∩ B(I0 )). We claim that for m ∈ Um0 , the following uniform estimate holds, 2 ||bm ||22 S (bm |J0 ∩ B(I0 )) ≤ 2B . (1.5.47) |J0 | ||bm0 ||22 Observe that the right-hand side of (1.5.47) depends on m0 and m only. In particular it is independent of I0 ∈ A and J0 ∈ B(I0 ). We turn to verifying (1.5.47). Observe that (1.5.40), the definition of bm , gives |K|. (1.5.48) S 2 (bm |J0 ∩ B(I0 )) = K∈J0 ∩Cm (I0 )

Next we exploit that Jones’s compatibility condition holds for {Cm (I0 ) : m ≤ N }. Condition (J2) states that K∈J0 ∩Cm (I0 )

|Cm (I0 )| |K| ≤B . |J0 | |Cm0 (I0 )|

(1.5.49)


111

Note that the upper bound on the right-hand side of (1.5.49) is (now) independent of J0 . Next we remove also the dependence on the particular interval I0 . By the rescaling hypothesis (1.5.43) the value of the ratios |Cm (I)|/|Cm0 (I)| is independent of I ∈ A. Hence by averaging we obtain the inequality |Cm (I)| |Cm (I0 )| ≤ 2 I∈A . (1.5.50) |Cm0 (I0 )| |C m0 (I)| I∈A The identity (1.5.48) followed by the estimates (1.5.49) and (1.5.50) proves the claim made in (1.5.47) since, |Cm (I)|. ||bm ||22 = I∈A

In summary, using locally the compatibility assumption (J) together with the rescaling hypotheses (1.5.42) and (1.5.43) we passed from a small scale quantity to a large scale estimate; we showed that integrating (1.5.46) and invoking (1.5.47) gives that 2 bm 2 2B S (P (f )|J0 ∩ B(I0 )) ≤ f, . (1.5.51) 2 |J0 | ||bm0 ||2 ||bm ||2 m∈Um0

Finally we use Bessel’s inequality, and the fact that index set Um0 is actually independent of I0 to conclude that bm 2 f, ≤ |Cm0 (I)| · ||f ||2BMO ||bm ||2 (1.5.52) m∈Um0 I∈A = ||bm0 ||22 ||f ||2BMO . Combining (1.5.51) and (1.5.52) implies (1.5.44).

The pigeon hole principle. Often we use a very simple pigeon hole principle to realize the hypothesis of Theorem 1.5.11. As before we are given disjoint blocks of dyadic intervals, {B(I) : I ∈ A}. Let B(I) contain collections of dyadic intervals C1 (I), . . . , CN (I), which satisfy Jones’s compatibility condition (J). Moreover, we assume that the number of families N, and the constant B appearing in the compatibility condition remain fixed, as I ranges over A. We let J. Ci (I) = J∈Ci (I)

Now we assume that a lower bound holds for the density of Ci (I) in I : There exists δ > 0 such that |Ci (I)| > δ|I|,

for i ≤ N and I ∈ A.

(1.5.53)

112


Then, by the pigeon hole principle, there exists K = K(δ) and a decomposition of A, A = A 1 ∪ · · · ∪ AK , so that each Ak satisfies the hypothesis of Theorem 1.5.11. That is, if I, J ∈ Ak and m, n ≤ N, then Cm (I) ⊆ Cn (I) if Cm (J) ⊆ Cn (J), and

|Cm (J)| |Cm (I)| ≤2 . |Cn (I)| |Cn (J)|

Let k ≤ K. Define Pk to be the orthogonal projection onto the span of bi = {hK : K ∈ Ci (I)}, i ≤ N. I∈Ak

Then, by Theorem 1.5.11 each of the projections Pk has norm on BMO bounded by 4(B[[Ak ]])1/2 . The sum satisfies the norm estimate K 1 Pk ≤ 4K(B[[A]]) 2 , (1.5.54) k=1

BMO

where A = A1 ∪ · · · ∪ AK , and where K = K(δ).

1.6

Notes

The work of A. Haar [79] and J. Schauder [185] is represented by Proposition 1.1.1 and Theorem 1.1.2. See [76] by C. Goffman and G. Pedrick for a presentation of the basic properties of Schauder bases, in particular for a proof of St. Banach’s theorem mentioned in the text. Theorem 1.1.3 is the result of S. Szarek [200]. The text presents the method introduced by R. Latala and K. Oleszkiewicz in [124], with simplifications given in [180]. The proof of Theorem 1.1.4 is taken from the appendix D of [192] by E. M. Stein which also contains the trick by which we transferred Khintchine’s inequality from L4 to L1 . Proofs of Burkholder’s Theorem 1.1.5 are contained in [33] and [35]. Theorem 1.1.6, the square function characterization of Lp , is originally due to R.E.A.C. Paley [167]. The constants of equivalence in Paley’s proof are different from the ones obtained in Theorem 1.1.6. Applying the unconditionality of the Haar system in Lp (1 < p < ∞) we obtain estimates for the Walsh system. The proof given for Pisier’s inequality in the scalar case (that is Theorem 1.1.7 referred to as the dyadic version of Poincaré’s inequality) is due to A. Naor and G. Schechtman [162]. It extends without changes to UMD valued Lp spaces. For a different proof see M. Talagrand [201]. Theorem 1.1.8, asserting that the Walsh system in Walsh–Paley order forms a Schauder

1.6. Notes

113

basis for Lp (1 < p < ∞) is due to R. E. A. C. Paley [167]. Theorem 1.1.9, the extension of Khintchine’s inequality to Walsh functions of finite multiplicity, is obtained by A. Bonami [15] and by K. Kiener [112] independently. Theorem 1.1.10 is a result of B. Roider. The proof in the text is based on K. Kiener’s work in [113] and on additional, unpublished manuscripts of K. Kiener. In [25] J. Bourgain gives an extension of B. Roider’s Theorem 1.1.10. For a systematic treatment of the Walsh system consult the monograph [77] by B. Golubov, A. Efimov and V. Skvortsov. There, in particular, one finds the identities of Walsh and Paley used in the text. The Walsh system is used as a model providing insight into problems concerning the trigonometric system. The work of R. Hunt in [87] and [88] that analyzes the Walsh model is an excellent preparation for the study of L. Carleson’s treatise [44] on almost everywhere convergence of Fourier series. In [203] Ch. Thiele forms the notion of a quartile operator in the context of the Walsh system. Subsequently quartile operators played a crucial role in the work of M. Lacey and Ch. Thiele [121], [122] on the boundedness of the bilinear Hilbert transform. In Banach space theory Walsh functions are used for the dual purpose of constructing operators (Walsh matrices) and spaces (spanned by subsets of the Walsh system). Typical applications include: (a) The construction of operators with unexpected or extremal properties in the work of P. Enflo [62], A. Szankowski [197], [199], A. Pelczy` nski and C. Sch¨ utt [173], C. Sch¨ utt [189], V.I. Gurarii, M. Kadec and V.I. Macaev [78]. (b) The construction of complemented subspaces in Lp (1 < p < ∞) by J. Bourgain, H. Rosenthal and G. Schechtman [29], and subspaces in L1 with extremal properties obtained by J. Bourgain [16]. (c) The proof of the existence of a fundamental, total and bounded biorthogonal sequence in every separable Banach space by R. Ovsepian and A. Pelczy` nski [166], [170]. (d) The study of type, cotype and K-convexity in the work of G. Pisier [177], [178]. The proof of Theorem 1.2.1 (Fefferman’s Inequality) is taken from the classic paper [64] by C. Fefferman and E. M. Stein. Theorem 1.2.2 is probably a folk result, Theorem 1.2.3 is from A.M. Garsia [75]. The direct approach to the atomic decomposition of H 1 is due to R. Coifman [55]. The proof of Theorem 1.2.4 given in the text is due to S. Janson and P.W. Jones [90]. The sharp function characterization of Theorem 1.2.5 is due to C. Fefferman and E. M. Stein [64]. The proof in the text follows A.M. Garsia [75]. Theorem 1.2.6 is the dyadic version of the Hardy–Littlewood maximal theorem, for its proof we used the presentation in [192]. For Theorem 1.2.8 see again A.M. Garsia [75]. Large deviation inequalities as considered in Section 1.3 originate with the work of W. Hoeffding [85] and K. Azuma [5]. The proof of Theorem 1.3.1 and

114


Corollary 1.3.2 are taken from [49] by S. Y. A. Chang, M. Wilson and Th. Wolff. The proof of Theorem 1.3.4 based on Proposition 1.3.3 was compiled using [6] by R. Banuelos together with [49] by S. Y. A. Chang, M. Wilson and Th. Wolff and [7] by R. Banuelos and C. Moore. A different approach to Theorem 1.3.4 was developed by J. Pipher [174]. The proof of Theorem 1.3.5 and Corollary 1.3.7 are from Garsia’s book [75]. Theorem 1.3.6 is due to G. Schechtman [186]. Its proof introduces martingale methods, in particular SL∞ estimates, to obtain Euclidean subspaces of proportional dimension in 1n . For an overview of the development that followed, we refer to the survey articles by G. Schechtman [187] and W. B. Johnson and G. Schechtman [97] as well as to the books by V. Milman and G. Schechtman [148] and G. Pisier [179] . Historically the first application of large deviation inequalities to a problem in Banach spaces appears in the note [141] by B. Maurey. T. Figiel, J. Lindenstrauss and V. Milman [68], B. Kashin [110], and G. Schechtman [186] present distinct solutions to the problem of finding Euclidean subspaces of proportional dimension in 1n . Multipliers into SL∞ as presented in Theorem 1.3.8 are constructed in [104]. The monograph [148] by V. Milman and G. Schechtman contains far-reaching applications to Banach space theory of large deviation inequalities such as Theorem 1.3.1. In a completely different direction N. Makarov [133] applies Theorem 1.3.1 to study the boundary behavior of conformal maps, thereby N. Makarov obtains important estimates of harmonic measure. Recently, J. Bourgain, H. Brezis and P. Mironescu [26], obtained surprising, new applications of large deviation in√ equalities and the p dependence in Theorem 1.3.4 to embedding theorems of fractional Sobolev spaces. See also the survey article [30] by H. Brezis. Theorem 1.3.1 provides the basis for an algorithm that determines the rectifiable pieces of a given data set in Rn . Thus the work of G. Lerman [126] — that builds on the travelling salesman theorem of P. Jones [103] — gives unexpected applications of SL∞ estimates to geometric measure theory. The theorem of J. Garnett and P. W. Jones [73] is presented with a stochastic proof in Durrett’s book [61]. Both the original proof in [73] and the stochastic proof in [61] are easily specialized to the context of dyadic martingales. The estimate (1.3.32) for the Rademacher projection H 1 is a result of J. Bourgain [18]. J. Bourgain’s estimate (1.3.32), is related (in spirit) to R. E. A. C. Paley’s inequality on lacunary coefficients [168] and to the interpolatory inequalities for Fourier multipliers, appearing in the work of S. Kislyakov [115] and [116]. (The basis of the relationship is the Lusin square function characterization presented in Section 6.1.) The proof of Theorem 1.4.1 is due to B. Maurey [139] (see also [144]). The proof of Stein’s inequality (Theorem 1.4.2) is due to J. Bourgain and applied in 1983/4 [24]. It extends without changes to the case of Lp spaces of functions with values in a UMD space. The only published account of J. Bourgain’s proof seems to be in [69] by T. Figiel and P. Wojtaszczyk. Theorem 1.4.3, Theorem 1.4.4 and Proposition 1.4.5 are results of T. Figiel [65].

1.6. Notes

115

The proof of Theorem 1.5.1 is obtained (by specialization) from [130], Proposition 2.c.17. Theorem 1.5.2 is taken from [147] by Y. Meyer and R. Coifman. The proof in the text is strictly limited to the case of real-valued coefficients and not extending to coefficients in a UMD space. However there exist paraproduct estimates due to J. Bourgain that extend to the UMD case. See [69] for the only published proof. Theorem 1.5.3 and the proof presented in the text are due to T. Figiel [66]. The survey [69] by T. Figiel and P. Wojtaszczyk contains a very informative summary of Figiel’s work on singular integral operators. Our presentation incorporates also lecture notes of talks given by T. Figiel at a Conference on the Geometry of Banach spaces in Strobl (1989), and at the Weizmann Institute in Rehovot (1988). Z. Ciesielski [53] gives a detailed presentation of the Haar system on [0, 1] × [0, 1] indexed by dyadic squares. J. O. Stromberg’s construction [196] of a modified Franklin system, is based on the same principles. T. Figiel’s use of the square indexed Haar system in the expansion of a kernel on [0, 1] × [0, 1] resulted in the representation of integral operators as paraproducts, (simple) rearrangements and multipliers. Using the standard Haar system on [0, 1] × [0, 1] indexed by dyadic rectangles leads to a different representation of integral operators. It was invented by Y. Meyer [146] to give the first wavelet proof of the T (1) theorem of G. David and J.L Journé [59] . Comparing Y. Meyer’s proof [146] of the T (1) theorem to that of T. Figiel [66] the following attracts attention: The norm estimates for rearrangement operators in [66] are replacing the applications of Cotlar’s lemma in [146]. Having freed the proof of the T (1) theorem from Cotlar’s lemma, T. Figiel [66] extended the T (1) theorem beyond Hilbert spaces to the UMD class (being able to do this provided the main stimulus for T. Figiel’s work). Simultaneously and independent of the work of T. Figiel the advantages of expanding integral kernels along the Haar system indexed by dyadic squares were understood by G. Beylkin, R. Coifman and V. Rokhlin [10] while developing numerical algorithms for the evaluation of integral operators. The boundedness criteria for orthogonal projections as formulated in Theorem 1.5.9–Theorem 1.5.11 are due to P. W. Jones [102]. The use of orthogonal projections has led to deep results in function spaces other than H 1 and BMO. For instance, the results on complemented subspaces of rearrangement invariant function spaces in the monograph [93] by W. Johnson, B. Maurey, G. Schechtman, L. Tzafriri.

Chapter 2

Projections, Isomorphisms and Interpolation This chapter reviews three basic concepts of functional analysis. We discuss comnski, plemented subspaces (mostly in H 1 ), the decomposition method of A. Pelczy´ and elements of complex interpolation theory.

2.1

Complemented subspaces

In this section we study Banach spaces which are isomorphic to a complemented subspace of H 1 . To this end we introduce three families of Banach spaces closely related to H 1 . These are atomic H 1 over spaces of homogeneous type, martingale H 1 associated to an increasing sequence of σ algebras, and Banach spaces spanned by nested, three-valued martingale difference sequences. Furthermore we discuss the Rosenthal space and related Banach spaces spanned by independent random variables. Standard concepts and terminology We recall standard concepts associated with the study of Banach spaces. We define projections, isomorphisms, well isomorphic sequences of finite-dimensional spaces, complemented copies, infinite direct sums, and the finite-dimensional building blocks of the spaces H 1 , Lp , and BMO. Projections. A projection P on a Banach space E is a bounded linear operator P : E → E such that P 2 = P . The range of a projection is called a complemented subspace. Clearly the orthogonal projections discussed in Section 1.5 satisfy P 2 = P.

118

Chapter 2. Projections, Isomorphisms and Interpolation

Isomorphisms. Two Banach spaces E and F are isomorphic if there exists a bijective linear operator T : E → F so that

T · T −1 < ∞.

(2.1.1)

In that case T : E → F is called an isomorphism between E and F. The infimum of (2.1.1) taken over all isomorphisms between E and F is called the Banach–Mazur distance between E and F. If two Banach spaces E and F are isomorphic, then we say that E is a copy of F and we write E ∼ F. A property of a Banach space which is preserved by isomorphisms is called an isomorphic invariant. Well isomorphic sequences. Given two sequences of finite-dimensional Banach spaces {En } and {Fn }. We say that {En } is well isomorphic to {Fn } if there exist bijective linear operators Tn : En → Fn such that sup Tn · Tn−1 < ∞.

n∈N

Complemented copies. Below we will be concerned with Banach spaces that are isomorphic to a complemented subspace of H 1 . Observe that a Banach space E is isomorphic to a complemented subspace of a Banach space F if there exist linear operators M : E → F and N : F → E so that E M

Id −→

E N

and

M · N < ∞.

(2.1.2)

F Indeed, M (E) is a subspace of F that is isomorphic to E, and P = M ◦ N is a projection on F whose range equals M (E). If (2.1.2) holds, then we say that F contains a complemented copy of E. Infinite direct sums. Among the simplest procedures to generate new Banach spaces from old ones is the formation of infinite direct sums. Let En , n ∈ N be a sequence of Banach spaces, let 1 ≤ p ≤ ∞. Then ( En )p is defined as the space of sequences (xn )∞ n=1 , where xn ∈ En , equipped with the norm 1/p when 1 ≤ p < ∞, ||xn ||pEn (2.1.3) sup ||xn ||En when p = ∞. For 1 ≤ p < ∞ let q be the H¨ older conjugate index to p, satisfying 1/p + 1/q = 1. The dual space to ( En )p is naturally identified as ∗ En = En∗ , p

where En∗ denotes the dual space to En .

q

2.1. Complemented subspaces

119

Finite-dimensional building blocks. We encounter infinite direct sums most frequently in connection with the following examples. Define the algebraic vector space Vn = span{hJ : |J| ≥ 2−n }. The algebraic dimension of Vn is equal to 2n+1 − 1. Equipping Vn with the norm of H 1 , Lp or BMO gives the finite-dimensional building blocks of H 1 , Lp or BMO, Hn1 = (Vn , || · ||H 1 ) ,

Lpn = (Vn , || · ||Lp ) ,

BMOn = (Vn , || · ||BMO ) .

Clearly Hn1 is complemented in H 1 ; it is the range of the norm 1 projection,

P (f ) =

f, hJ

{J:|J|≥2−n }

hJ . |J|

The projection P is also a norm 1 operator on Lp and on BMO. Hence Lpn and BMOn are one-complemented subspaces of Lp respectively BMO. The following proposition is simple and useful. Its proof shows how infinite direct sums are represented as subspaces of H 1 . Proposition 2.1.1. The space ( Hn1 )1 is isomorphic to a complemented subspace of H 1 . Proof. Let (In ) be a sequence of pairwise disjoint dyadic intervals. Let fn : [0, 1] → In be an affine function mapping [0, 1] bijectively onto the interval In . The slope of fn is |In |. Define the following collections of dyadic intervals contained in In , Cn = {J ⊆ In : |J| ≥ 2−n |In |}. Let Yn be the algebraic vector space Yn = span{hJ : J ∈ Cn }. The composition operator Tn defined by Tn : g → |In |g ◦ fn , is an isometry from (Yn , || · ||H 1 ) onto Hn1 . Note that for yn ∈ Yn we have the identity ||yn ||H 1 . (2.1.4)

yn H 1 = Let Y be the closure H 1 of span{Yn : n ∈ N}. By the above identity (Y, || · ||H 1 ) in 1 is isometric to ( Hn )1 . Moreover Y is a complemented subspace of H 1 . Indeed let P be the orthogonal projection defined by P (f ) =

∞ n=1 J∈Cn

Then ||P ||H1 = 1 and Y is the range of P.

f, hJ

hJ . |J|

120


space. With obvious modificaLet 2n denote the n-dimensional Euclidean tions the proof of Proposition 2.1.1 shows that ( 2n )1 is isomorphicto a complemented subspace of H 1 . By duality the spaces ( BMOn )∞ and ( 2n )∞ are isomorphic to complemented subspaces of BMO. In the next section we will considerably strengthen this observation and prove a theorem of P. Wojtaszczyk that ( BMOn )∞ is isomorphic to BMO. This provides us with a useful alternative representation of the space BMO. Consequences of this isomorphism will be drawn in Chapter 5 where we prove isomorphic invariants of H 1 and BMO. Martingale H 1 spaces Martingale H 1 spaces provide a class of complemented subspaces of H 1 . Let (Ω, F, | · |) be a probability space and let F0 ⊆ · · · ⊆ Fn ⊆ · · · F

# be a sequence of increasing σ-algebras on Ω so that F is generated by Fn . For an integrable function f : Ω → R, the induced square function is defined to be 1/2

2 |E(f |Fn ) − E(f |Fn−1 )| , n

where E(f |Fn ) denotes the conditional expectation of f with respect to Fn . Then H 1 [(Fn )] is the space of integrable functions f : Ω → R, with integrable square function. The norm on H 1 [(Fn )] is defined by 1/2 2 2 E(f |F0 ) + |E(f |Fn ) − E(f |Fn−1 )| . ||f ||H 1 [(Fn )] = Ω

n

The space BMO[(Fn )] consists of all functions g ∈ L2 (Ω) so that

g BMO[(Fn )] = sup E(|g − E(g|Fn−1 )|2 Fn ) < ∞. n∈N

The space BMO[(Fn )] is (identifiable with) the dual space of H 1 [(Fn )] as follows. Let f, g ∈ L2 (Ω), then √ |E(f · g)| ≤ 2||f ||H 1 [(Fn )] g BMO[(Fn )] . Thus every g ∈ BMO[(Fn )] induces a bounded linear functional on H 1 [(Fn )], by L(f ) = lim E(fn · gn ), n→∞

fn = E(f |Fn ) and gn = E(f |Fn ).

Conversely for every bounded linear functional L : H 1 [(Fn )] → R there exists g ∈ BMO[(Fn )] so that L(f ) = lim E(fn · gn ) n→∞

for f ∈ H 1 [(Fn )].


121

The book of A. M. Garsia [75] is the standard reference to martingale inequalities, H 1 [(Fn )] and BMO[(Fn )]. Dyadic H 1 is in the class of H 1 [(Fn )] spaces. To see this let Fn be the σalgebra generated by dyadic intervals of length ≥ 2−n . This observation, and many other analogies between general martingale H 1 space and the dyadic H 1 , suggest strong ties between these spaces. On the other hand also the Banach spaces 1 and L1 ([0, 1]) are in this class. To see how L1 ([0, 1]) arises, let F0 = {∅, [0, 1]} and define F1 = F2 · · · to be the Lebesgue σ-algebra on the unit interval [0, 1]. For this filtration H 1 [(Fn )] coincides with L1 ([0, 1]). The Banach spaces L1 , 1 , and H 1 share very few isomorphic invariants. In particular, by a result of A. Pelczy´ nski, L1 ([0, 1]) is not isomorphic to a subspace of a Banach space with an unconditional basis. We conclude that without putting restrictions on the filtration (Fn ) the ties between the spaces in the class H 1 [(Fn )] and dyadic H 1 are not very strong. However, after imposing the simplest condition ruling out the case of L1 , B. Maurey [144] obtained this truly remarkable theorem: Theorem 2.1.2. If the σ-algebras Fn are purely atomic, then H 1 [(Fn )] is isomorphic to a complemented subspace of H 1 . It takes up almost all of Chapter 4 to prove B. Maurey’s result. In Chapter 4 we present also two applications to Maurey’s theorem. First we use it to obtain the isomorphic classification of martingale H 1 spaces when the underlying filtration consists of purely atomic σ-algebras. Separately we apply its proof to show that the span of finitely many Haar functions in Lp is well isomorphic to pn . Atomic H 1 spaces The next examples of complemented subspaces in H 1 are the atomic H 1 spaces appearing in analysis. On a set X let d(., .) be a quasi-metric, that is, d(x, y) = 0 if and only if x = y, d(x, y) = d(y, x), d(x, z) ≤ K(d(x, y) + d(y, z)) for x, y, z ∈ X. The balls of radius r in X are the following sets, B(x, r) = {y ∈ X : d(x, y) < r}. We let µ be a probability measure on X satisfying the doubling condition, µ(B(x, 2r)) ≤ Aµ(B(x, r)), for some A and all x ∈ X, r > 0. Following R. Coifman and G. Weiss [56] we call the triple (X, d, µ) a space of homogeneous type. An atom for (X, d, µ) is either a constant function on X or a function a : X → R such that adµ = 0,

122 and


a2 dµ ≤ µ(B)−1 ,

where B is a ball in X such that supp a ⊆ B. We define the associated atomic H 1 space to be the space of functions f : X → R which admit a decomposition into atoms, f= c i ai , where ai are atoms for (X, d, µ), and where |ci | < ∞. We denote this space by H 1 (X, d, µ). The norm of f ∈ H 1 (X, d, µ) is |ci |}, ||f ||H 1 (X,d,µ) = inf{ where the infimum is taken over all decompositions of f into atoms. The following two examples appear most frequently. 1. Historically the first example of a space of homogeneous type is the boundary of the unit disk T = {z ∈ D : |z| = 1}, where the metric is given by the Euclidean distance, and the measure is the Lebesgue measure on T. The 1 . In Chapter 6 we present a thorough space H 1 (T, dt, | · |) is denoted Hat 1 discussion of the space Hat . Section 6.1 covers the basic theorems about 1 , relating it to functions for which the Hilbert transform and the Lusin Hat area function are integrable; Section 6.2 contains L. Carleson’s proof of B. 1 is isomorphic to H 1 . Maurey’s theorem that Hat 2. By Theorem 1.2.4 we identified the space of homogeneous type (X, d, µ) for which the associated atomic H 1 space coincides with dyadic H 1 . Indeed, let X be the unit interval [0, 1], let µ be Lebesgue measure on [0, 1], and let d(x, y) be the length of the smallest dyadic interval containing both x and y. Then (X, d, µ) is a space of homogeneous type. The following example sheds some light on this metric. We fix numbers n, q ∈ N such that n ≥ q. Consider Iq−1 a dyadic interval of length 2−q+1 . Let m be the midpoint of Iq−1 . For each q ≤ j ≤ n define Jj = [m − 2−j , m),

and

Ij = [m, m + 2−j ).

Observe that Jj and Ij are adjacent dyadic intervals, and that Jq ∪ Iq = Iq−1 .

(2.1.5)


123

The interval Iq−1 is then the smallest dyadic interval containing Ij and Jj . Hence for q ≤ j ≤ n, x ∈ Ij and y ∈ Jj we obtain d(x, y) = |Iq−1 |. Given x ∈ [0, 1], the ball B(x, r) of radius r in (X, d, µ) is just the dyadic interval I for which x ∈ I and r/2 ≤ |I| < r. Hence a function a is an atom for (X, d, µ) if there exists a dyadic interval I such that supp a ⊆ I, a2 (t)dt ≤ |I|−1 , and

a(t)dt = 0.

By Theorem 1.2.4 for f ∈ H 1 , there exists a decomposition f− f= ci ai , where ai are atoms for (X, d, µ), and ci are scalars such that |ci | ≤ 16 S(f ). Conversely, any atom a for (X, d, µ) satisfies S(a) ≤ 1. In Section 6.5 we will show that atomic H 1 spaces are isomorphic to complemented subspaces in H 1 , for each choice of (X, d, µ). More precisely we will prove the following Theorem [155]. Theorem 2.1.3. For every space of homogeneous type, H 1 (X, d, µ) is isomorphic to a complemented subspace of H 1 . Moreover H 1 (X, d, µ) and H 1 are isomorphic if (and only if ) µ{x ∈ X : µ({x}) = 0} > 0. We close this section by describing the dual space of H 1 (X, d, µ). For x ∈ X and r > 0 let B(x, r) = {y ∈ X : d(x, y) < r}. Let h ∈ L2 (X) and B = B(x, r), then we write mB (h) to denote the mean value of h over B, thus, dµ . mB (h) = h µ(B) B

124


A function h ∈ L2 (X) belongs to BMO(X, d, µ) if |h − mB (h)|2

h BMO(X,d,µ) = sup B

B

dµ µ(B)

1/2 < ∞,

(2.1.6)

where the supremum is taken over all balls B = B(x, r), x ∈ X, and r > 0. The classical survey article [56] by R. R. Coifman and G. Weiss is our standard reference to spaces of homogeneous type, H 1 (X, d, µ) and BMO(X, d, µ). The duality results described below are proved there in detail. The dual space of H 1 (X, d, µ) is characterized as BMO(X, d, µ). This characterization comprises two statements. First, each h ∈ BMO(X, d, µ) induces a bounded linear functional on H 1 (X, d, µ). For f ∈ H 1 (X, d, µ), with atomic decomposition f= λi ai and |λi | < ∞, the mapping f → lim

N →∞

N i=1

λi

hai dµ ,

(2.1.7)

X

is a well-defined linear functional on H 1 (X, d, µ). Conversely, if x∗ is in the dual space of H 1 (X, d, µ) then there exists a unique h ∈ BMO(X, d, µ) so that

h BMO(X,d,µ) ≤ C x∗ and so that for f ∈ H 1 (X, d, µ) the value of x∗ (f ) is given by the mapping (2.1.7). Three-valued martingale differences Dyadic intervals are distinguished by the following properties: First for a dyadic interval I there exists n ∈ N such that |I| = 2−n . Second, if I, J are dyadic intervals and I ∩ J = ∅, then either I ⊆ J or J ⊆ I. We now consider arbitrary collections of measurable sets which satisfy the second property but not necessarily the first. Let E be a countable collection of measurable subsets in the measure space [0, 1], equipped with Lebesgue measure. We say that E is a nested collection (of measurable sets) if the elements of E satisfy the following condition, if A, B ∈ E and A ∩ B = ∅, then A ⊆ B or B ⊆ A. We let E play the role of dyadic intervals and define spaces of three-valued martingale difference sequences. As soon as E is fixed the construction of these spaces is in complete analogy to that of H 1 and BMO. A nested collection E gives rise to spaces X[E] and Y [E] of three-valued martingale difference sequences. We start with an analog to the Haar system. Let {rA : A ∈ E} be any enumeration of the sequence of Rademacher functions. Then for A ∈ E, put dA (w, t) = 1A (w)rA (t).


125

Note that the functions {dA } assume only the values {+1, −1, 0}, they form a martingale difference sequence. The support of dA is the set A × [0, 1]. Hence the support sets of the functions {dA : A ∈ E} form a nested collection of measurable sets. We define the space X[E] to consist of those f = A∈E aA dA for which ||f ||X[E] =

1/2 a2A 1A

< ∞.

A∈E

This norm makes X[E] a Banach space, and {dA : A ∈ E} an unconditional basis for X[E]. We define the space Y [E] to consist of all functions h = A∈E bA dA , for which ⎛ ⎞1/2 1 b2B |B|⎠ < ∞. ||h||Y [E] = sup ⎝ |A| A∈E B∈E, B⊆A

Repeating the proof of Theorem 1.2.1 (Fefferman’s inequality) gives the integral estimate √ f h dx ≤ 2 2||f ||X[E] ||h||Y [E] (2.1.8) for any finite linear combination f = aA dA . Moreover Y [E] can be identified with the dual space of X[E]. By choosing different collections E we should be able to produce several different spaces in the class X[E]. Below we find that the following list of Banach spaces can be obtained by choosing E appropriately, 2n )1 , ( 2n )1 ⊕ 2 , ( 2 )1 , 1 , 2 , 1 ⊕ 2 , ( (2.1.9) Hn1 )1 , ( ( Hn1 )1 ⊕ 2 , ( Hn1 )1 ⊕ ( 2 )1 , H 1 . It was pointed out by P. Wojtaszczyk [211] that the spaces in (2.1.9) are all different. By that we mean that no two of them are isomorphic. We will now show how to select nested collections E so that X[E] is isomorphic to a given space from the list (2.1.9). 1. Let C be a collection of dyadic intervals. Then clearly C is a nested collection of measurable sets and for E = C, the unconditional basis for X[E], dI (w, t) = 1I (w)rI (t),

I ∈ C,

is equivalent to the Haar functions {hI : I ∈ C} in H 1 . Thus the class of spaces X[E] contains the class of all subspaces in H 1 which are spanned by subsequences of the Haar system. Clearly H 1 can be obtained in that way. In the course of proving Proposition 2.1.1 we showed how to choose a subsequence of the Haar system so that its closed linear span is isometric to ( Hn1 )1 . Finally we obtain the space 1 by letting C be a collection of pairwise disjoint dyadic intervals.

126


2. Here we construct E such that X[E] is isomorphic to 2 . Let I0 = [0, 1] and for n ∈ N let In = [0, 1 − 4−n ]. Then the collection E = {In : n ∈ N0 } is a nested collection and X[E] is isomorphic to 2 . Indeed the unconditional basis for X[E] is dn (w, t) = 1In (w)rn (t). For this system we have

∞ 1/2 ∞ 1 2 an ≤ dn an 2 n=0 n=0

≤

X[E]

∞

1/2 a2n

.

n=0

3. Now we define a collection E such that X[E] is isomorphic to ( 2 )1 . For n ∈ N, let In be a sequence of pairwise disjoint dyadic intervals. Then for m ≥ 1 choose In,m ⊆ In such that |Im,n | = |In |(1 − 4−m ). Now consider E = {In,m : m, n ∈ N}, and define the normalized unconditional basis for X[E], dn,m (w, t) = |In,m |−1 1In,m (w)rn,m (t), where rn,m denotes independent Rademacher functions. Let {an,m } be a sequence of scalars. For each n fixed we have an 2 estimate,

∞ 1/2 ∞ 1/2

∞ 1 2 2 a ≤ am,n dn,m ≤ an,m . 2 m=1 n,m m=1 m=1 X[E]

Next we observe that for n = n and m, m ∈ N, the intervals In,m and In ,m are disjoint. Hence,

∞ 1/2 ∞ ∞ ∞ 1 2 an,m ≤ an,m dn,m 2 n=1

m=1

n=1 n=1

≤ In other words the unit vectors of (

∞ n=1

∞

1/2 a2n,m

X[E]

.

m=1

2 )1 are equivalent to {dn,m : n, m ∈ N}.

4. In our final example we describe an entire class of collections E for which X[E] is isomorphic to 1 . We consider now collections E which satisfy the Carleson packing condition, 1 |B| < ∞. [[E]] = sup A∈E |A| B⊆A, B∈E

We claim that then X[E] is isomorphic to 1 . To see this we consider the unconditional basis, dA (w, t) = 1A (w)rA (t), A ∈ E.


127

We will show that {|A|−1 dA : A ∈ E} is equivalent to the unit vector basis of 1 . To this end, we fix aA ∈ R. Let f= aA dA (w, t), and define h=

sign (aA )dA (w, t).

1/2 |aA ||A| = A moment’s reflection shows that ||h||Y [E] ≤ [[E]] , and that hf. Applying (2.1.8), that is Fefferman’s inequality for nested martingale differences, we obtain the upper bound √ (2.1.10) |aA ||A| ≤ 2 2[[E]]1/2 ||f ||X[E] . Next we show that the converse estimate holds true as well. Indeed, note that ||dA ||X[E] = |A|, and apply the triangle inequality to f . This gives the corresponding lower bound, (2.1.11) |aA ||A| ≥ ||f ||X[E] . Combining (2.1.10) and (2.1.11) we find that {|A|−1 dA : A ∈ E} is equivalent to the unit vector basis of 1 . It is now easy to combine these constructions and to obtain the remaining members of the list (2.1.9). We remarked already that the spaces appearing in (2.1.9) are known to be isomorphically different (see [211]). Next we show that each of the spaces X[E] is isomorphic to a complemented subspace of H 1 . Thus we have at once ten different examples of complemented subspaces of H 1 . Nested collections can be decomposed naturally in a sequence of generations. This is done as follows. Given A ∈ E we denote by G1 (A, E) ⊆ E the collection of maximal elements of E which are strictly contained in A. This is called the first generation of A. As E is nested, G1 (A, E) is a collection of pairwise disjoint sets. Now we define the following generations by induction, using the first step as a model. Suppose that the first p − 1 generations, G1 (A, E), . . . , Gp−1 (A, E), are already defined. Then we put H = {B ∈ E : B ⊂ A} \ {G1 (A, E) ∪ · · · ∪ Gp−1 (A, E)}, and define the p-th generation of A as Gp (A, E) = G1 (A, H). Let G0 (E) denote the collection of maximal elements E. Inductively, for p ∈ N, we define Gp (E) = G1 (A, E). A∈Gp−1 (E)

128


Proposition 2.1.4. For any nested family of measurable sets E, the space X[E] is isometric to a complemented subspace of H 1 . Proof. To simplify the presentation of the proof we make the convenient but inessential assumption that for each A ∈ E, G1 (A, E) is a finite set. We will construct collections {CA : A ∈ E} of pairwise disjoint dyadic intervals which provide a model of the collection E, since {CA } will be defined to satisfy the following properties. 1. Let CA be the point-set covered by the collection CA . Then {CA : A ∈ E} is a nested family of subsets of [0, 1] and |A| = |CA |. 2. CB ⊆ CA iff B ⊆ A. 3. If B ⊆ A, and I ∈ CA , then |B| |I ∩ CB | = . |I| |A| 4. If I ∈ CB , J ∈ CA and I ⊆ J, then B ⊆ A. Now we construct these collections by induction. We fix a maximal element in E, say A. As |A| ≤ 1, there exists a collection CA of pairwise disjoint dyadic intervals so that I CA = I∈CA

satisfies |CA | = |A|. It is easy to ensure that for A, A ∈ G0 (E), with A = A we have CA ∩CA = ∅. Thus we define CA for each A in the generation G0 (E). Next we turn to CB for the sets B in the first generation of E. Let A ∈ G0 (E). Recall that G1 (A, E) consists of pairwise disjoint elements from E. For each I ∈ CA and each B ∈ G1 (A, E) we find easily a collection CB,I of pairwise disjoint dyadic intervals with the following properties: Let CB,I be the point-set covered by the collection CB,I , then CB,I ⊆ I and |B| |CB,I | = . |I| |A| Furthermore we arrange also for the following condition to hold: If B1 , B2 ∈ G1 (A, E) are different (and hence disjoint sets), then CB1 ,I ∩ CB2 ,I = ∅. Now we define CB for B ∈ G1 (A, E) by taking the union, CB,I . CB = I∈CA

Having defined CB for B ∈ G0 (E), . . . Gp (E) we continue as follows. Fix A ∈ Gp (E). Then G1 (A, E) consists of pairwise disjoint elements from E. Hence for each I ∈ CA


129

and each B ∈ G1 (A, E) there exists a collection of pairwise disjoint dyadic intervals CB,I such that CB,I ⊆ I, and |B| |CB,I | = . |I| |A| Now we obtain CB for B ∈ G1 (A, E) by taking unions, CB,I . CB = I∈CA

The resulting collections {CA : A ∈ E} satisfy the above conditions 1) − 4). Consequently they satisfy Jones’ compatibility condition (J). Now we form the linear combinations {hI : I ∈ CA }, bA = and put kA = bA /||bA ||2 . By Theorem 1.5.9, the orthogonal projection f, kA kA Pf = A∈E

is a bounded operator on H 1 with norm ≤ 1. The range of P is the closed linear span of {bA : A ∈ E}. Thus it remains to prove that in H 1 , the closure of span{bA : A ∈ E} is isomorphic to X[E]. To show this let dA (ω, t) = 1A (ω)rA (t),

A ∈ A,

R. Observe that the be the unconditional basis for X[E] and fix a sequence aA ∈ distribution of A∈E a2A d2A is the same as the distribution of A∈E a2A b2A . Thus it follows that aA dA = aA bA . A∈E

X[E]

A∈E

H1

So far we used the spaces X[E] to produce examples of complemented subspaces in H 1 . They play a second, even more important role in the isomorphic classification of martingale H 1 spaces. The link to the martingale H 1 spaces is provided by Maurey’s theorem. Following is a more detailed formulation of Theorem 2.1.2. Theorem 2.1.5. Let (Fn ) be a sequence of increasing and purely atomic σ-algebras. There exists a nested collection E, satisfying A, B ∈ E and A ⊂ B implies |A| ≤ |B|/2, such that H 1 [(Fn )] is isomorphic to X[E]. Consequently H 1 [(Fn )] is a space with an unconditional basis.

130


Maurey’s result is the main step in the classification of martingale H 1 spaces. We will prove Theorem 2.1.5 in Chapter 4 and complement it with the following result. Theorem 2.1.6. Let E be a nested collection of measurable sets satisfying A, B ∈ E and A ⊂ B implies |A| ≤ |B|/2. Then X[E] is isomorphic to one of the spaces 1 , ( Hn1 )1 , or H 1 . Above we found ten different Banach spaces in the class X[E]. It is not possible to obtain more spaces by this method, since the isomorphic types of the family X[E] are classified in [151]: For every choice of E, the space X[E], is isomorphic to one of the ten spaces listed in (2.1.9) above. The Rosenthal space and independent random variables Next we use independent random variables to construct complemented subspaces in H 1 , respectively BMO. We present the Rosenthal space, its predual, the independent sum of BMOn and the intersection of weighted L2 with BMO. The Rosenthal space. We define the intersection of ∞ with a weighted 2 space. More precisely, let (wi ) be a bounded sequence of positive real numbers such that and lim wi = 0. wi = ∞, Then the sequence space R = R({wi }) consists of all bounded sequences (ai ) for which the following norm is finite, 1/2 a2i wi . ||(ai )||R = sup a2i + The space R that is now called the Rosenthal space was introduced and studied first by H. Rosenthal in [181]. Rosenthal showed in particular that the isomorphic type of R = R({wi }) is determined by the following two conditions on the weights, wi = ∞, and lim wi = 0. vi = ∞ and Precisely, if {vi } is a sequence of positive real numbers so that lim vi = 0, then R = R({vi }) is isomorphic to R = R({wi }). The proof shownski’s ing the existence of an isomorphism between R and R is based on A. Pelczy´ decomposition method. It was observed in [158] that the Rosenthal space is isomorphically different from the spaces Y [E] that are generated by nested three-valued martingale difference sequences. Proposition 2.1.7. The Rosenthal space R is isomorphic to a complemented subspace of BMO.


131

Proof. The proof is divided into three basic steps. We first define a subspace of BMO, by constructing the sequence (bi ) in (2.1.12). Then we show that in BMO the sequence (bi ) is equivalent to the unit vector basis of the Rosenthal space. In a third step we verify that the orthogonal projection onto the linear span of (bi ) is bounded on BMO. Combined, these three steps imply that the Rosenthal space is isomorphic to a complemented subspace of BMO. Step 1. We are given a sequence (wi ) of positive real numbers ≤ 1. Without loss of generality we assume that the sequence (wi ) is sorted in decreasing order. First we group them according to their value, as follows. For n ≥ 1 define the index set In = {i : 2−n ≤ wi ≤ 2−n+1 }. Let in denote the cardinality of the set In . Next we select pairwise disjoint intervals of natural numbers Kn , and Mn such that |Kn | = n , |Mn | = in , and so that sup Kn < inf Mn ≤ sup Mn < inf Kn+1 . We use the index set Kn to define the indicator functions Xn =

1 (1 + rs ). 2

s∈Kn

The supports of the functions Xn form a sequence of independent sets, the measure of the support of Xn equals 2−n . Next we attach the following frequencies to the indicator function Xn . For j ∈ Mn , we form bn,j = Xn rj . Now transform the lexicographic order on the set {(n, j) : n ∈ N, j ∈ Mn } into the linear order on the natural numbers, and accordingly we relabel the sequence {bn,j }, as hI , i ∈ N. (2.1.12) bi = I∈Ci

Let Ci be the point-set covered by the collection Ci . Then by construction the measure of Ci is related to the weights wi as follows, wi /2 ≤ |Ci | ≤ wi .

132


Step 2. We will now show that in BMO the sequence {bi } spans the Rosenthal space. Fix a bounded sequence ai ∈ R and put f=

ai bi .

We verify the equivalence of norms ||f ||2BMO ≤ max a2i +

∞

i∈N

a2i wi ≤ 2||f ||2BMO .

(2.1.13)

i=1

We show first the left-hand side estimate in (2.1.13). To this end we expand f in its Haar series and let let αI hI be the Haar expansion of f. Next we fix n ∈ N and I ∈ Cn . Note that for i ≥ n + 1 the sets I and Ci are independent. Hence |bi |2 = |I ∩ Ci | = |I| · |Ci |. I

Therefore we observe the identities

αJ2 |J| = |I|a2n +

J⊆I

= |I|a2n +

∞ i=n+1 ∞

a2i

|bi |2 I

a2i |I||Ci |.

i=n+1

Dividing by |I|, and taking the supremum over all dyadic intervals I gives $ ∞ 1 2 2 2 sup αJ |J| ≤ sup an + ai |Ci | n I |I| i=n+1 J⊆I

≤ sup a2i + i

∞

a2i |Ci |.

i=1

As |Ci | ≤ wi we have verified the left-hand side of (2.1.13). To prove the reverse estimate, observe that {bi } is disjointly supported over the Haar system. Hence for f = ai bi we have ||f ||BMO ≥ sup |ai |. The norm of f in L2 is bounded by its BMO norm, hence, ||f ||2BMO ≥ ||f ||2L2 ∞ = a2i |Ci |. i=1

Combining this with |Ci | ≥ wi /2 we obtain the right-hand side of (2.1.13).


133

Step 3. The orthogonal projection onto span{bi } is defined by P (f ) =

∞ i=1

f,

bi bi . ||bi ||2 ||bi ||2

By (2.1.13) boundedness of P on BMO is easy to check. Indeed (2.1.13) gives that ||P (f )||2BMO

∞ bi 2 bi 2 ≤ sup f, f, 2 + 2 . ||b || ||b i i 2 i ||2 i=1

Next observe that ||bi ||22 = |Ci |. Hence by Bessel’s inequality f,

bi 2 ≤ ||f ||2BMO . ||bi ||22

Moreover we have the L2 estimate ∞

f,

i=1

bi 2 ≤ ||f ||22 ||bi ||22 ≤ ||f ||2BMO .

In summary we showed that ||P (f )||2BMO ≤ 2||f ||2BMO , thus proving that the Rosenthal space is isomorphic to a complemented subspace of BMO. The predual of Rosenthal’s space. Let R∗ be the space of all sequences (ai ) for which

∞ 1/2 2 −1 min{|ai |, ai wi } < ∞.

(ai ) R∗ = i=1

This space can be identified — canonically — with the predual of Rosenthal’s space. By Proposition 2.1.7 and duality the space R∗ is isomorphic to a complemented subspace of H 1 . A direct proof of this fact, not using duality, follows from work of W. B. Johnson and G. Schechtman [95]. It was verified in [158] that R∗ is not isomorphic to one of the spaces listed in (2.1.9). The independent sum of BMOn . The next example is very close to Rosenthal’s space. Recall that L2n ⊂ L2 and BMOn ⊂ BMO denote the subspaces spanned by the Haar functions {hJ : |J| ≥ 2−n }. Define the intersection BMOn ∩ L2n ∞

2

equipped with the norm

||(xn )|| =

max ||xn ||2BMO + n∈N

∞ n=1

1/2 ||xn ||22

.

134


This space iscommonly called the independent sum of the spaces BMOn , and denoted by ( BMOn )ind . The name comes from the embedding defined below. There we use statistically of BMOn as building blocks of a independent copies subspace isomorphic to ( BMOn )∞ ∩ ( L2n )2 . We point out that implicitly we actually determine a subsequence of the Walsh system that spans a copy of the independent sum ( BMOn )ind . Theorem 2.1.8. The independent sum ( BMOn )ind is isomorphic to a complemented subspace of BMO. Proof. We start by defining a subspace of BMO which is isomorphic to the independent sum ( BMOn )ind . First we fix i ∈ N and let N = {i, . . . , i + n}. Now we define a copy of BMOn using the Rademacher functions {rj : j ∈ N }. We start by defining b[0,1] = ri . Now we suppose that the functions bI have been constructed for all dyadic intervals of length ≥ 2− . Then fix a dyadic interval J of length 2− , and split J into its left half J1 , and its right half J2 . Then we put bJ1 = ri++1 1{bJ =1} , bJ2 = ri++1 1{bJ =−1} . In this way we defined bI for every dyadic interval I with |I| ≥ 2−n . We let G(N ) = span{bI : |I| ≥ 2−n }. Clearly, when equipped with the BMO norm, the space G(N ) is isometric to BMOn . Incidentally we defined G(N ) in such a way that it coincides with the span of Walsh functions {wA : A ⊆ {i, . . . , i + n}}. Moreover G(N ) is the range of a norm 1 projection. To see this, we renormalize in L2 , and put kI = bI /||bI ||2 . Then by Theorem 1.5.9 the orthogonal projection PN (f ) = f, kI kI is bounded on BMO with norm ≤ 1. The range of PN is G(N ). Next we produce a sequence of independent copies of BMOn . We fix a sequence of disjoint intervals Nn ⊂ N, such that |Nn | = n and sup Nn < inf Nn+1 . Define Gn = G(Nn ) using the above construction. Note that if xn ∈ Gn and if xm ∈ Gm , then xn and xm are independent functions, for m = n. Repeating the proof of the estimates (2.1.13) shows that for any sequence of functions xn ∈ Gn , the following equivalence of norms holds true, max ||xn ||2BMO +

2 ||xn ||22 ≤ xn

BMO

≤ 2 max ||xn ||2BMO + 2

||xn ||22 . (2.1.14)


135

Note that the orthogonal projection onto span{Gn , n ∈ N} is given by the operator P = PNn . By the equivalence of norms (2.1.14) one obtains that P is a bounded operator on BMO. Indeed, let f ∈ BMO, then PNn (f ) ∈ Gn , hence

P (f ) 2BMO =

PNn (f ) 2BMO ≤ 2 max PNn (f ) 2BMO + 2

PNn (f ) 22 ≤ 2 f 2BMO + 2 f 22 . As f 2 ≤ f BMO , this shows that P (f ) BMO ≤ 2 f BMO . Summing up we proved our claim that ( BMOn )ind is isomorphic to a complemented subspace of BMO. It is not known whether the Rosenthal space and the independent sum of BMOn are isomorphic Banach spaces. Problem 2.1.9. Prove that R is not isomorphic to ( BMOn )ind . In Chapter 5 we study structural results on complemented subspaces of H 1 . One of them states the following dichotomy: A complemented subspace of H 1 2 either contains a copy of , or it is isomorphic to a complemented subspace of ( Hn1 )1 . To arrive at that result we use that the members of the following family of subspaces are well complemented in BMO. We fix p < q and let S(p, q, M ) = span{hJ : 2−q ≤ |J| < 2−p }. We define the unit ball in S(p, q, M ) as the intersection of the unit ball of BM O and 1/M times the unit ball of L2 . This is achieved using the norm max{M ||x||L2 , ||x||BMO }. The dual space of S(p, q, M ) consists of T (p, q, M ) = span{hJ : 2−q ≤ |J| < 2−p }, where the norm of x ∈ T (p, q, M ) is defined as |x|T (p,q,M ) = inf{||y||H 1 + ||z||L2 M −1 : x = y + z}. Proposition 2.1.10. If M ≤ 2p/2 , then the space S(p, q, M ) is isomorphic to a complemented subspace of BMO. Embedding and projection have norm bounded by 8. By duality T (p, q, M ) is isomorphic to a complemented subspace of H 1 , again embedding and projection have norm bounded by 8.

136


Proof. By Theorem 2.1.8 it suffices to construct embedding and projection between S(p, q, M ) and ( BMOn )ind . We let A = {J : J dyadic |J| = 2−p }. Then for x ∈ S(p, q, M ) we have the simple and useful identity max{M ||x||L2 , ||x||BMO } = max

⎧ ⎨ ⎩

1/2 M 2 ||x · 1J ||2L2

J∈A

,

max ||x1J ||BMO J∈A

⎫ ⎬ ⎭

.

Recall that we chose M 2 ≤ 2p , and that the support of x · 1J has measure ≤ 2−p . Therefore we may find a linear map of slope ≤ M 2 mapping J into [0, 1]. This linear change of variables maps the function x · 1J into an element zJ ∈ BMOn such that, ||zJ ||L2 = M ||x · 1J ||L2 , ||zJ ||BMO = ||x · 1J ||BMO . In this way we obtain an embedding T : S(p, q, M ) →

BMOn

, ind

x → (zJ )J∈A

for which max{M ||x||L2 , ||x||BMO } ≤ ||T x||( BMOn )ind ≤ 2 max{M ||x||L2 , ||x||BMO }. Note that the orthogonal projection onto the range of T is (trivially) bounded on ( BMOn )ind with norm = 1. Thus by Theorem 2.1.8, S(p, q, M ) is isomorphic √ to a complemented subspace of BMO. The norm of the embedding is ≤ 2 2 the norm of the projection is bounded by 2. Sequences of uniformly complemented subspaces. Given a sequence of finite dimensional Banach spaces {En }, we say that {En } is well isomorphic to uniformly complemented subspaces of F if there exist linear operators An : En → F and Bn : F → En so that Id −→ En En An Bn F

and

sup An · Bn < ∞.

n∈N

Equivalently we say that F contains uniformly complemented copies of {En }. Let {pn }, {qn } and {Mn } be sequences of natural numbers so that pn ≤ qn and Mn ≤ 2pn /2 . Then, by Proposition 2.1.10 the spaces {T (pn , qn , Mn )} are well isomorphic to uniformly complemented subspaces of H 1 . (Or equivalently, H 1 contains uniformly complemented copies of {T (pn , qn , Mn )}.)


137

Weighted intersections The next class of examples is obtained by intersecting ∞ with a weighted version of BMO. The weight is given by a fixed sequence of scalars, 0 ≤ xI ≤ 1. We define the space S ∞ to consist of all bounded sequences (aI ) for which the following norm is finite, ||(aI )||S ∞ = || xI aI hI ||BMO + sup |aI |. The norm in S ∞ depends of course explicitly on the chosen weight. For example, when xI = 1 for all dyadic intervals I, then the norm of S ∞ is equivalent to the norm of BMO. When the weight disappears, that is when xI = 0, for all I, then the norm of S ∞ is equivalent to that of ∞ . When xI ∈ {0, 1}, then S ∞ is the direct sum of ∞ and the span of a subsequence of the Haar basis. Our notation, however, suppresses this dependence. In Chapter 4 we will give the isomorphic classification of the spaces S ∞ . It turns out that BMO and ∞ are the only isomorphic types which appear in this class. The first step in this direction is the following result. Theorem 2.1.11. The space S ∞ is isomorphic to a complemented subspace of BMO. The norm of embedding and projection are independent of the choice of scalars, 0 ≤ xI ≤ 1. Proof. The proof consists of two steps. It starts by showing that for every choice of scalars 0 ≤ xI ≤ 1, there exists a sequence gI ∈ BMO which spans a subspace of BMO that is isomorphic to S ∞ . Here the construction of the functions gI reflects the given sequence of weights {xI }, so that the norm of the isomorphism is independent of the given weight. In the second step we show that the orthogonal projection gI gI f, P (f ) = ||gI ||2 ||gI ||2 I

is bounded in BMO. For convenience we assume that each of the numbers xI is a negative power of 2. Then depending on the weights {xI } we choose a rapidly increasing sequence of natural numbers 1 0 in (2.3.24) are independent of p. The following theorem asserts that the left-hand side estimate in (2.3.24) holds true. Theorem 2.3.7. Let 1 ≤ p ≤ 2 and θ = 2 − 2/p. For u ∈ H p with u = 0, there exist v ∈ BMO, w ∈ L2 , with ||v||BMO ≤ 1 and ||w||2 ≤ 1, so that their Haar coefficients v(I), w(I) give ||u||H p ≤ C u v(I)1−θ w(I)θ hI dt , where C > 0 is independent of p.

2.3. Interpolation of operators

157

The proof of Theorem 2.3.7 uses the stopping time argument of Theorem 2.3.3 applied to the sharp function in place of the square function. Assume first that 1 ≤ p < 4/3 and let q denote the H¨ older conjugate index to p. In Section 1.2 we proved that for u ∈ H p there exists f ∈ Lq so that 1 uf dt and f q ≤ 1. ||u||H p ≤ C 0

Next let 4/3 < p ≤ 2. By the square function characterization of Lp there exists f ∈ Lq so that 1 uf dt and S(f ) q ≤ 1. ||u||H p ≤ C 0

In either case f admits a representation as f= v(I)1−θ w(I)θ hI , where v = v(I)hI and w = w(I)hI satisfy ||v||BMO ≤ C and ||w||2 ≤ C. The reader is invited to supply the details by adapting the proof of Theorem 2.3.3. The right-hand side estimate in (2.3.24) is a corollary to Theorem 2.3.4 and to Fefferman’s inequality as stated in Theorem 2.3.1. Corollary 2.3.8. Let 1 < p < 2, and θ = 2 − 2/p. For v ∈ BMO, w ∈ L2 , and u ∈ H p , the following estimate holds, 1 √ 1−θ θ 6 θ u( v(I) w(I) h ) 2||u||H p ||v||1−θ I ≤ 2 BMO ||w||2 . 0

Proof. By the decomposition Theorem 2.3.4, u ∈ H p can be represented as u= x(I)1−θ y(I)θ hI , where θ 5 p ||x||1−θ H 1 ||y||2 ≤ 2 ||u||H .

Inserting the representation of u and applying Theorem 2.3.1 we obtain that 1 √ 6 1−θ θ θ u( v(I) w(I) h ) 22 ||u||H p ||v||1−θ I ≤ BMO ||w||2 , 0

as claimed.

Summing up, the renorming of H p (2.3.2) follows from Theorem 2.3.4 and Proposition 2.3.2. Its dual version (2.3.24) we obtain by combining Theorem 2.3.7 and Corollary 2.3.8. In the next sub-section we combine (2.3.2) and (2.3.24) to prove a basic boundedness principle for analytic families of operators.

158


The duality proof of Theorem 2.3.5. It is instructive to see how G. Pisier’s original proof [176] — via its interpretation by M. Cwikel, P. Nilsson and G. Schechtman [57] — specializes to the context of H 1 . To this end we give a second proof for the left-hand side of (2.3.16) using these sources. Fix g ∈ H 1 , w ∈ L2 and consider u= g(I)1−θ w(I)θ hI . By Corollary 2.3.8 the following lower bound holds, √ 6 1−θ θ p u 2 2||u||H ≥ sup z(I) y(I) hI dt : ||y||2 ≤ 1, ||z||BMO ≤ 1 & % = sup g(I)1−θ z(I)1−θ w(I)θ y(I)θ |I| : ||y||2 ≤ 1, ||z||BMO ≤ 1 . Inserting the preceding lower bound gives an expression displaying a high degree of symmetry, % & √ 26 2 sup g(I)1−θ w(I)θ hI p : ||w||2 ≤ 1 H & % 1−θ 1−θ θ ≥ sup g(I) z(I) w(I) y(I)θ |I| : ||y||2 ≤ 1, ||w||2 ≤ 1, ||z||BMO ≤ 1 . Expressing the self duality of Hilbert spaces we write % & % & w(I)θ y(I)θ |I| : ||y||2 ≤ 1, ||w||2 ≤ 1 = µ(I)θ |I| : |µ(I)||I| ≤ 1 . Inserting this identity and thereby reintroducing a different form of asymmetry, we find that & % sup g(I)1−θ z(I)1−θ w(I)θ y(I)θ |I| : ||y||2 ≤ 1, ||w||2 ≤ 1, ||z||BMO ≤ 1 % & = sup |µ(I)||I| ≤ 1, ||z||BMO ≤ 1 . g(I)1−θ z(I)1−θ µ(I)θ |I| : 1

1

Finally we invoke duality between the sequence spaces θ and 1−θ in the following way. For any sequence of scalars a(I), % & 1−θ 1 sup |µ(I)||I| ≤ 1 = . a(I)µ(I)θ |I| : |a(I)| 1−θ |I| With a(I) = g(I)1−θ z(I)1−θ , the preceding identity gives & % |µ(I)||I| ≤ 1, ||z||BMO ≤ 1 sup g(I)1−θ z(I)1−θ µ(I)θ |I| : 1−θ = sup : ||z||BMO ≤ 1 |g(I)z(I)||I| = sup

gz : ||z||BMO

1−θ ≤1 .


159

In summary we showed the following minorizations, thereby completing the proof of the theorem. & % sup g(I)1−θ w(I)θ hI p : ||w||2 ≤ 1 H 1−θ √ gz : ||z||BMO ≤ 1 ≥ (26 2)−1 sup ≥ 2−8 ||g||1−θ H1 .

Analytic families of operators Now we open a second source of strong estimates. Following is the three lines theorem. This is a maximum principle for analytic functions in the vertical strip V. Its applications form the core results of complex interpolation theory. Theorem 2.3.9. Let F (z) be analytic in V = {x + iy : x ∈ (0, 1), y ∈ R}. Assume that F extends continuously to the boundary of V, and that there exists a polynomial P such that log |F (x + iy)| ≤ P (y), x + iy ∈ V. If supt |F (it)| ≤ R0 and supt |F (1 + it)| ≤ R1 , then for every θ ∈ (0, 1), |F (θ)| ≤ R01−θ R1θ . First we slightly change the appearance of Theorem 2.3.9. Thus the above maximum principle becomes directly applicable to show boundedness of operators. For z ∈ V we let Jz be a linear operator defined a priorily on the set of finite linear combinations of Haar functions. For f, g finitely supported over the Haar system, we assume that the map F (z) = Jz (g)f is analytic in the vertical strip V, and extends continuously the boundary. Furthermore we assume that there exists a polynomial P, depending on the choice of f, g, such that log |F (x + iy)| ≤ P (y),

where

x + iy ∈ V.

On the boundary of V we have Jit and J1+it implicitly defined as follows. We put Jit (g, f ) = lim Jz (g)f, z→it Jz (g)f, J1+it (g, f ) = lim z→1+it

thereby defining linear operators Jit , J1+it , which act initially on finite linear combinations of Haar functions. Under these conditions we call {Jz : z ∈ V } an analytic family of operators.

160


Throughout this sub-section we assume that for t ∈ R, the operators Jit , and J1+it have bounded extensions to H 1 and L2 respectively, and that they satisfy the norm estimates ||Jit ||H 1 ≤ R0

||J1+it ||L2 ≤ R1 .

and

(2.3.25)

Below we prove that for analytic families of operators the hypothesis (2.3.25) implies that for p ∈ (1, 2) and θ = 2 − 2/p, the operator Jθ satisfies ||Jθ g||H p ≤ CR01−θ R1θ ||g||H p , where C > 0 is independent of p. The first step in this direction is the following boundedness principle for analytic families of operators. Its power comes from the three lines theorem and the Fefferman inequality. Theorem 2.3.10. Let x ∈ H 1 , v ∈ BMO, and let w, y, ∈ L2 . Assume that x, v, w, y, are finitely supported over the Haar system. For θ ∈ (0, 1), form the functions g=

x(I)1−θ y(I)θ hI ,

and

f=

v(I)1−θ w(I)θ hI .

Then, the operator Jθ satisfies the estimate 1 1−θ √ θ ≤ 2 2R0 ||x||H 1 ||v||BMO (J g) f (R1 ||y||2 ||w||2 ) . θ 0

Proof. Let x ∈ H 1 , v ∈ BMO, and let w, y, ∈ L2 . For z ∈ V we form the functions x(I)1−z y(I)z hI , and fz = v(I)1−z w(I)z hI . gz = For z ∈ V the map

F (z) =

1

(Jz gz ) fz 0

is analytic in V with continuous boundary values. Since the functions x, y, v, w are finite linear combinations of Haar functions there exists a constant C = C(x, y, v, w) such that for z ∈ V, log |F (z)| ≤ C log(4 + |z|). On the boundary of V the following identities hold for the norm of fz , ||fit ||BMO = ||v||BMO ,

and

||f1+it ||2 = ||w||2 .

Moreover by assumption we have for gz , ||Jit (git )||H 1 ≤ R0 ||x||H 1 ,

and

||J1+it (g1+it )||2 ≤ R1 ||y||2 .


161

Fefferman’s inequality implies that 1 (Jit git ) fit |F (it)| = 0 √ ≤ 2 2R0 ||x||H 1 ||v||BMO . The inequality of Cauchy–Schwarz gives 1 |F (1 + it)| = (J1+it g1+it ) f1+it 0

≤ R1 ||w||2 ||y||2 . Applying the three lines theorem we obtain that 1−θ √ (R1 ||y||2 ||w||2 )θ . |F (θ)| ≤ 2 2R0 ||x||H 1 ||v||BMO This completes the proof, since

1 0

(Jθ g) f = |F (θ)|.

The next results are derived by combining the decomposition Theorems 2.3.4 and 2.3.7 with the estimate of Theorem 2.3.10. These tools complement each other very well, despite their origin in well separated branches of analysis. Following is the main step towards the proof of the boundedness theorem for analytic families of operators. Corollary 2.3.11. Let1 ≤ p ≤ 2, and θ = 2 − 2/p. Let x ∈ H 1 and y ∈ L2 with Haar expansion x = x(I)hI and y = y(I)hI ; then θ x(I)1−θ y(I)θ hI ≤ CR01−θ R1θ ||x||1−θ Jθ H 1 ||y||2 , p H

where C > 0 is independent of p. Proof. Fix x ∈ H 1 , and let y ∈ L2 , be finitely supported over the Haar system. Let x(I)1−θ y(I)θ hI . (2.3.26) u = Jθ By Theorem 2.3.7 there exist v ∈ BMO and w ∈ L2 , such that ||v||BMO ≤ 1, ||w||2 ≤ 1, and 1 ||u||H p ≤ C u v(I)1−θ w(I)θ hI . 0

Inserting the equation (2.3.26) for u and applying Theorem 2.3.10 gives √ θ ||u||H p ≤ C2 2R01−θ R1θ ||x||1−θ H 1 ||y||2 .

162


Recall that we are working with an analytic family of operators {Jz : z ∈ V } such that Jit , and J1+it have bounded extensions to H 1 and L2 respectively, and ||Jit ||H 1 ≤ R0 and ||J1+it ||H 1 ≤ R1 . The next corollary shows that the norms of analytic families satisfy a convexity estimate. In particular it shows that Jθ is bounded on H p when θ = 2 − 2/p. The proof below merges the three lines theorem and the decomposition Theorem 2.3.4 into one single estimate. Corollary 2.3.12. For p ∈ (1, 2) and θ = 2 − 2/p, the operator Jθ has a bounded extension to H p , satisfying the norm estimate ||Jθ ||H p ≤ CR01−θ R1θ , where C > 0 is independent of p. Proof. Let u ∈ H p with u = 0. By Theorem 2.3.4 there exists x ∈ H 1 and y ∈ L2 , such that u= x(I)1−θ y(I)θ hI , and θ 5 p ||x||1−θ H 1 ||y||2 ≤ 2 ||u||H .

With Corollary 2.3.11 this gives θ ||Jθ u||H p ≤ CR01−θ R1θ ||x||1−θ H 1 ||y||2

≤ CR01−θ R1θ ||u||H p .

Next we present a typical application of Corollary 2.3.12. We give a second proof that the norm of rearrangement operators satisfies a convexity estimate. Recall that hτ (I) hI , Tp : 1/p → |I| |τ (I)|1/p and that we denote ||Tp ||p = ||Tp : H p → H p ||. Using Corollary 2.3.12 we prove that for 1 ≤ p ≤ 2 and θ = 2 − 2/p, ||Tp ||p ≤ C||T1 ||1−θ , 1 √ where C = 26 2. The plan is to embed the rearrangement operators {Tp : p ∈ (1, 2)} into an analytic family of operators as follows. For z ∈ V define operators Jz by their action on single Haar functions, Jz (hI ) = hτ (I)

|I| |τ (I)|

1−(z/2) .


163

Observe that for t ∈ R, (1 −

(1 + it) 1 (it) ) = 1, and (1 − )= . 2 2 2

Moreover as θ = 2 − 2/p, we have 1−

1 θ = . 2 p

Consequently, Jθ = Tp , and for every function f finitely supported over the Haar system we have ||Jit (f )||H 1 = ||T1 (f )||H 1 ,

and

||J1+it (f )||2 = ||T2 (f )||2 .

Applying Corollary 2.3.12, the maximum principle for analytic families of operators, gives ||T2 ||θ2 ||f ||H p . ||Jθ (f )||H p ≤ C||T1 ||1−θ 1 As ||T2 ||2 = 1 and Jθ = Tp , the preceding inequality proves that ||Tp ||p ≤ . C||T1 ||1−θ 1 The three lines theorem. We review quickly basic notions of potential theory necessary to understand the three lines theorem, in particular the concept of harmonic measure in its most elementary form. We let D be a simply connected domain, bounded by Jordan curves. (In the applications to follow the domain D is the vertical strip V bounded by two vertical lines.) Then for every z ∈ D the Riemann mapping theorem provides an analytic and bijective map onto the unit disk ϕ : D → D such that ϕ(z) = 0. For domains bounded by Jordan curves the Riemann map ϕ can be extended continuously to the boundary; for every ζ ∈ ∂D the following limit exists, ϕ(ζ) = lim ϕ(w). w→ζ

For any measurable set E ⊆ ∂D, we define the harmonic measure of E in D evaluated at z by the equation z (E) = |ϕ(E)|. ωD z (E) the harmonic measure of E in D. The dependence Often we simply call ωD z z → ωD (E), defines a harmonic function in D. If, moreover, the domain D is bounded, then every harmonic function u : D → R with continuous extension to the boundary can be obtained from its boundary values and harmonic measure of the domain D. Indeed, for z ∈ D, z u(w)dωD (w). u(z) = ∂D

164


In the case of analytic functions defined on bounded domains f : D → C, the logarithm log |f (z)| satisfies the following subharmonicity estimate: Assume that f extends continuously to the boundary. Then for n ∈ N and z ∈ D we have z max{−n, log |f (w)|}dωD (w). max{−n, log |f (z)|} ≤ ∂D

This is a very interesting result that incidentally distinguishes analytic functions from harmonic functions. It becomes really important when it is combined with estimates of harmonic measure which are derived from the geometry of the domain D and its boundary. We point out that in general subharmonicity estimates by their very nature imply maximum principles. In the context of interpolation, we apply the above facts in very simple domains that are obtained by truncating the vertical strip at a certain height M. We fix M > 0 and define VM = {x + iy : x ∈ (0, 1) , |y| < M }, OM = {x + iy : x ∈ (0, 1) , y = M }, UM = {x + iy : x ∈ (0, 1) , y = −M }. The sets OM and UM denote the upper and lower part of the truncated vertical strip. When evaluated at x ∈ (0, 1) the harmonic measure in VM puts very little mass on the sets OM ∪ UM . It turns out that the harmonic measure of OM ∪ UM decreases exponentially as M increases; this is of crucial importance; not only for the proof of the maximum principle below but also for many other important results of potential theory. Specifically the following inequality holds, ωVxM (OM ∪ UM ) ≤ e−M a , x ∈ (0, 1). We will now show that under very mild assumptions the subharmonicity estimate for analytic functions can be extended to unbounded domains such as the vertical strip. Thus we obtain the generalization of the maximum principle that is called the three lines theorem. Theorem 2.3.13. Let F (z) be analytic in V = {x + iy : x ∈ (0, 1), y ∈ R}. Assume that F extends continuously to the boundary of V, and that there exists a polynomial P such that log |F (x + iy)| ≤ P (y), x + iy ∈ V. If supt |F (it)| ≤ R0 and supt |F (1 + it)| ≤ R1 , then for every θ ∈ (0, 1), |F (θ)| ≤ R01−θ R1θ . Proof. Let w ∈ ∂V, then by hypothesis, |F (w)| ≤ max{R0 , R1 }. Hence log+ |F (w)|dωVz (w) < ∞. ∂V

(2.3.27)


165

It follows from (2.3.27) that in (2.3.28) below the integral on the left-hand side is well defined as the limit appearing on the right-hand side. We define log |F (w)|dωVz (w) = lim max{−n, log |F (w)|}dωVz (w). (2.3.28) n→∞

∂V

∂V

The value of the limit may be finite or equal to −∞. We shall show below that the assumption log |F (x + iy)| ≤ P (y), guarantees the following subharmonicity estimate. For z ∈ V, log |F (w)|dωVz (w). (2.3.29) log |F (z)| ≤ ∂V

We postpone the proof of (2.3.29) and derive first one of its consequences: In C, the boundary of V consists of the components, {it : t ∈ R} and {1 + it : t ∈ R}. For θ ∈ (0, 1) we obtain with (2.3.29) that θ log |F (it)|dωV (it) + log |F (1 + it)|dωVθ (1 + it) log |F (θ)| ≤ R R θ ≤ log R0 dωV (it) + log R1 dωVθ (1 + it) R R θ = log R0 dωV (it) + log R1 1 − dωVθ (it) . R

R

θ We proceed by identifying the value of the integral R dωV (it). For z ∈ V, consider now f (z) = R dωVz . Note that f is harmonic in V. Moreover by the geometry of the vertical strip, the value of the function f (x + iy) does not depend on y ∈ R. Hence f (x) is linear in x ∈ (0, 1). Moreover f (0) = 1, f (1) = 0, and f (x) = 1 − x. Summing up, we proved that dωVθ (it) = 1 − θ. R

Inserting we obtain log |F (θ)| ≤ (1 − θ) log R0 + θ log R1 . Exponentiating gives that |F (θ)| ≤ R01−θ R1θ . Now we turn to the proof of the crucial subharmonicity estimate (2.3.29) for the unbounded domain V. With two approximation procedures we reduce the estimate (2.3.29) to the case of bounded subharmonic functions in bounded domains. By hypothesis, sup

w∈OM ∪UM

log |F (w)| ≤ max{P (M ), P (−M )}.

166


The upper bound of the harmonic measure of OM ∪ UM in VM gives max{−n, log |F (w)|}dωVθ M (w) ≤ e−M a max{P (M ), P (−M )}. (2.3.30) OM ∪UM

Now we keep n fixed and let M tend to infinity. The estimate (2.3.30) implies that θ max{−n, log |F (w)|}dωV (w) = 0. lim M →∞

OM ∪UM

Consequently, lim max{−n, log |F (w)|}dωVθ M (w) = M →∞

∂VM

∂V

max{−n, log |F (w)|}dωVθ (w).

Invoking the subharmonicity estimate for the bounded domain VM gives max{−n, log |F (w)|}dωVz M (w), max{−n, log |F (z)|} ≤ ∂VM

and letting M tend to infinity gives max{−n, log |F (z)|} ≤ ∂V

max{−n, log |F (w)|}dωVθ (w).

The assumed upper bound of log |F (w)| on ∂V allows us to take the limit n → ∞, which gives that log |F (θ)| ≤ ∂V

2.4

log |F (w)|dωVθ (w).

Notes

The book by C. Goffman and G. Pedrick [76] is an introduction to basic functional analysis treating also some of the advanced topics like complemented subspaces, Schauder bases and the biorthogonal systems of Haar, Rademacher and Walsh. Our sources for concepts in Banach space theory are the books by J. Lindenstrauss and L. Tzafriri [130] and P. Wojtaszczyk [213]. The proof of Proposition 2.1.1 is taken from B. Maurey [144]. The book of A. M. Garsia [75] is the standard reference to martingale H 1 spaces. Theorem 2.1.2 is due to B. Maurey [144]. The importance of atomic H 1 spaces is established in the classic paper [56] by R. Coifman and G. Weiss. Recent developments, in particular the T(1) theorem for operators in spaces of homogeneous type, are presented by M. Christ [50]. Theorem 2.1.3 is from [155]. Three-valued martingale difference sequences are basic tools in the construction of complemented subspaces in Lp (1 < p < ∞). See J. Bourgain, H. Rosenthal

2.4. Notes

167

and G. Schechtman [29], J. Bourgain [16] and W. B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri [93]. The proof of Proposition 2.1.4 uses the methods introduced by B. Maurey in [139] and in Section 4 of [144]. Theorem 2.1.5 asserting that martingale H 1 spaces generated by sequences of finite σ algebras have an unconditional basis is due to B. Maurey [140]. Theorem 2.1.6, providing the isomorphic classificaton of the spaces X[E], appears in [151]. In [158] the Rosenthal space and the independent sum of BMOn are shown to be isomorphic to a complemented subspace of BMO. Proposition 2.1.7, Theorem 2.1.8 and Proposition 2.1.10 are from [158]. The space S ∞ appears as canonical endpoint to a class of spaces studied by D. Kleper and G. Schechtman in [117]. Theorem 2.1.11, showing that S ∞ is isomorphic to a complemented subspace of BMO, is proven in [157]. The decomposition method was introduced by A. Pelczi´ nski in [169]. It is reproduced in many books, for instance in [213] and [130]. Significant finite dimensional refinements of the decomposition method were obtained by G. Bennett, L.E. Dor, V. Goodman, W.B. Johnson, and C.N. Newman [9], W.B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri [93] and by W.B. Johnson and G. Schechtman [96]. The proof of Wojtaszczyk’s Theorem 2.2.3 is taken from Chapter II.E of [213]. The representation of the sequence {H 1 (2n )} as copies of uniformly complemented subspaces of H 1 appears in [154]. M. Schmuckenschläger informed me about the elementary proof of the Riesz convexity theorem by which we introduced the section on interpolation. He pointed out that it is a widely known folklore construction. Real-variables methods for the characterization of the complex interpolation spaces between H 1 and Lp (1 < p < ∞) are due to C. Fefferman and E. M. Stein. In particular Corollary 2.3.12, the interpolation theorem for analytic families of operators appears in [64]. The stopping time argument in Theorem 2.3.3 is due to S. Janson and P. W. Jones [90]. The proofs for Theorem 2.3.1, Proposition 2.3.2, Theorem 2.3.4, Theorem 2.3.7 and Corollary 2.3.8 use ingredients extracted from Section 8 of [70] by M. Frazier and B. Jawerth. The survey paper [99] by P. W. Jones contains information about history, methods and results on interpolation problems, H 1 and H ∞ . Theorem 2.3.4 shows that the Calderón product of the Banach lattices H 1 and H 2 coincides with H p . The Calder´ on product is introduced by A. Calder´ on in [38]. See the monograph [119] by S. Krein, J. Petunin and E. M. Semenov for further information on the Calder´ on product. The norm for H 1 appearing in Theorem 2.3.5 was invented by G. Pisier in [175] and [176]. In Chapter 3 of [57] by M. Cwikel, P. Nilsson and G. Schechtman, the norm devised by G. Pisier is shown to express a uniqueness result for the endpoints of complex interpolation scales of Banach lattices. Specializing the uniqueness result to the context of H 1 , gives Theorem 2.3.5. The proof on page 154 for the left-hand side estimate in (2.3.16) emerged in conversations with V. Pillwein. The duality proof of Theorem 2.3.5 is obtained by specialization from G. Pisier [176] and M. Cwikel, P. Nilsson and G. Schechtman [57]. The content of Chapter 3 in [57] has circulated in preprint form for some time. Early references to

168


[57] and the uniqueness result for complex interpolation scales include the papers by N. Kalton [105] and [106]. See also the related work by P. Nilsson [164]. I don’t know who is to be credited for the proof of Theorem 2.3.6 presented in the text. The original proof, however, of Theorem 2.3.6, is in [156] and will be discussed in Chapter 3 below. For the three lines theorem, analytic families of operators and related results see the books by E. M. Stein and G. Weiss [195], C. Bennett and R. Sharpley [8] or [119] by S. Krein, J. Petunin and E. M. Semenov.

Chapter 3

Combinatorics of Colored Dyadic Intervals In this chapter we establish the combinatorial connections between the Carleson packing condition and the boundedness of orthogonal projections in BMO. We introduce the Gamlen–Gaudet construction and prove the uniform approximation property for BMO in a special case. Rearrangement operators of the Haar basis that preserve BMO and Lp are characterized.

3.1

The Carleson packing condition

In the unit interval [0, 1], the dyadic subintervals are those of the form

k−1 k , , 2n 2n

where n ∈ N and 1 ≤ k ≤ 2n . We let D denote the collection of all dyadic intervals. Note that D is a nested family of sets. It satisfies the following property. If I, J ∈ D are not disjoint, then either I ⊆ J or J ⊆ I. The maximal dyadic intervals of any collection C ⊆ D are pairwise disjoint. To denote the maximal dyadic intervals of C we use the notation G0 (C), and also max C. The collection G0 (C), is called the founding generation of C. Note that G0 (C) covers the same set as C. Later generations are defined inductively. Let p ∈ N, and assume that G0 (C), . . . , Gp−1 (C) are already defined, then we put H = C \ G0 (C) ∪ · · · ∪ Gp−1 (C) and define Gp (C) = G0 (H).

170

Chapter 3. Combinatorics of Colored Dyadic Intervals

Following standard notation we let J ∩ C = {I ∈ C : I ⊆ J}. A localized version of the above construction is obtained as follows. We let J ∈ D and for p ∈ N we denote Gp (J, C) = Gp (J ∩ C). Note that for J ∈ C we have G0 (J, C) = {J}, and G1 (J, C) consists of the maximal dyadic intervals of {I ∈ C : I ⊂ J}. ∗ Throughout this book we denote by # C the point-set which is covered by ∗ the collection C, that is, we put C = I∈C I. Often we measure the size of C by considering lim sup C, the subset of [0, 1] which is covered by infinitely many intervals of C. Note that Gn−1 (C)∗ is the subset of [0, 1[ which is covered by at least n intervals in C. Hence, lim sup C =

∞ +

Gn (C)∗ .

n=0

The next proposition provides a link between the generations Gn (I, C), defined above, and the measure of the set lim sup C. Proposition 3.1.1. If C is a collection of dyadic intervals such that | lim sup C| = 0, then for any dyadic interval I there exists N = N (I) such that |GN (I, C)∗ | ≤

1 |I|. 4

Proof. Suppose that the proposition fails. Then there exists I ∈ C such that |Gn (I, C)∗ | > |I|/4, for every n ∈ N. Note that each point in Gn (I, C)∗ is contained in more than n intervals of C. Moreover, observe that Gn+1 (I, C)∗ ⊆ Gn (I, C)∗ for any n ∈ N. This implies that 1 + Gn (I, C)∗ ≥ |I|, 4 n∈N

and hence | lim sup C| ≥ |I|/4. This contradicts the assumption that lim sup C has vanishing Lebesgue measure. The Carleson constant and generations of dyadic intervals A quantitative and scale invariant measure for the size of C is the Carleson packing condition. Recall that a collection C of dyadic intervals satisfies the Carleson packing condition if, the following expression is finite, [[C]] = sup I∈C

1 |J|. |I| J∈I∩C

3.1. The Carleson packing condition

171

The number [[C]] is then called the Carleson constant of the collection C. If there exists M > 0 such that [[C]] ≤ M , then we say that C satisfies the M -Carleson condition. Observe that | lim sup C| > 0 implies that [[C]] = ∞, and conversely [[C]] < ∞ implies | lim sup C| = 0. The next proposition is similar in spirit to the previous one. Note however, that the qualitative consideration in the proof of Proposition 3.1.1 is replaced by a simple estimate. Proposition 3.1.2. Let C be a collection of dyadic intervals which satisfies the M Carleson condition. Let A be the largest integer which is less than 4M + 1. Then for every I ∈ C, 1 |GA (I, C)∗ | ≤ |I|. 4 Proof. Suppose that the proposition fails. Then there exists I ∈ C such that |GA (I, C)∗ | > |I|/4. Clearly, the following inclusions hold between generations, Gs (I, C)∗ ⊇ GA (I, C)∗ , for s ≤ A. Consequently the following lower bound holds true for any s ≤ A. 1 |J| ≥ |GA (I, C)∗ | > |I|. 4 J∈Gs (I,C)

Summing over s ≤ A gives a lower bound for the Carleson constant of C as follows, 1 |I| s=0 A

[[C]] ≥

|J|

J∈Gs (I,C)

1 |G4M (I, C)∗ | |I| s=0 A

≥

> M. This contradicts our assumption that C satisfies the M -Carleson condition.

We will now discuss two basic techniques to decompose collections of dyadic intervals. We begin by showing that the Carleson constant [[C]] satisfies a property resembling the positive homogeneity of norms: Each collection C of dyadic intervals can be decomposed into 4[[C]] + 1 subcollections Ci such that [[Ci ]] ≤ 4/3. We obtain this by considering arithmetic progressions of the generations of C. Lemma 3.1.3. Suppose that C satisfies the M -Carleson condition. Let A be the largest integer less than 4M + 1. Then C can be decomposed into C0 , . . . , CA−1 such that each Ci satisfies the 4/3-Carleson condition. Proof. First we split C into its generations G0 (C), G1 (C), . . . , Gn (C), . . . . Then for k = 0, 1, . . . , A − 1 we put ∞ GjA+k (C). Ck = j=0

172


To obtain the claimed upper bound for the Carleson constant of Ck , we choose a dyadic interval I ∈ Ck . It follows from the previous Proposition 3.1.2 that |GA (I, C)∗ | ≤

|I| . 4

Note that the first generation of I in Ck , coincides with generation A of I in C, so G1 (I, Ck ) = GA (I, C). Applying this observation repeatedly gives a geometrically decreasing estimate for the point-set covered by the generations Gj (I, Ck ). Indeed we obtain that |Gj (I, Ck )∗ | ≤ 4−j |I|. Summing over j ≥ 0 we arrive at the upper estimate |J| = |I| + |Gj (I, Ck )∗ | J∈Ck ∩I

⎛

j≥1

≤ |I| ⎝1 +

⎞ 4−j ⎠ .

j≥1

This gives [[Ck ]] ≤ 4/3.

Next we discuss the so-called condensation lemma. This is a widely used result asserting that if the Carleson constant of a collection C is very large, then somewhere inside this collection the Carleson constant is as large as possible, thus condensation has occurred and it can be measured with the estimate of Lemma 3.1.4. Thereby we obtain a natural converse to Proposition 3.1.2. Lemma 3.1.4. Let > 0, n ∈ N, and let C be a collection of dyadic intervals such that n 1 |J| > . sup I∈C |I| J∈I∩C

Then there exists K ∈ C such that |Gn (K, C)∗ | > (1 − )|K|. We point out that the conclusion |Gn (K, C)∗ | > (1 − )|K| implies, of course, that for each of the generations 1 ≤ s ≤ n the same measure estimate holds, namely (3.1.1) |Gs (K, C)∗ | > (1 − )|K|. Thus the condensation lemma asserts that everywhere between the interval K and the n-th generation of K the density of the collection C is larger than 1 − .


173

Proof of Lemma 3.1.4. Let > 0 and n ∈ N be given, then by assumption we have [[C]] > n/. Now we suppose that the lemma fails, that is, |Gn (J, C)∗ | ≤ (1 − )|J| for any J ∈ C. Next we will use this condition to obtain an upper estimate for the Carleson constant of C. We fix I ∈ C. Under the assumption that |Gn (J, C)∗ | ≤ (1 − )|J|, for J ∈ C we obtain by induction that |Gnm+k (I, C)∗ | ≤ (1 − )m |I|

(3.1.2)

for any m ∈ N and any 0 ≤ k ≤ n − 1. Summing the estimates (3.1.2) gives an upper bound for the Carleson constant of C that contradicts the hypothesis. Indeed we have n−1 ∞ |J| = |Gmn+k (I, C)∗ | k=0 m=0

J∈C∩I

≤

∞ n−1

(1 − )m |I|

k=0 m=0

=

n |I|.

As I ∈ C was arbitrary we have thus obtained the following upper bound for the Carleson constant of C, [[C]] ≤ n/. This contradicts our assumption that [[C]] > n/.

The Proposition 3.1.2 and Lemma 3.1.3 are especially useful in combination with Proposition 3.1.5, below. Estimates similar to those of Proposition 3.1.5 yielded the atomic decomposition of H 1 (Theorem 1.2.4) and also the closely related decomposition theorems in Section 2.3. Recall that for f ∈ Lp the Haar support of f is the following collection of dyadic intervals {I : f, hI = 0}. Proposition 3.1.5. Let 1 ≤ p ≤ 2, and let {fi } be a sequence in H p . Let Fi denote the support of the square function S(fi ). Assume that any dyadic interval J in the Haar support of fi satisfies ∞

|J ∩ Fj | ≤

j=i+1

Then for a linear combination h=

ai fi ,

|J| . 2

(3.1.3)

174


the following estimates hold true, |ai |p ||S(fi )||pp ≤ 2||S(h)||pp . ||S(h)||pp ≤

(3.1.4)

Moreover if {fi } is a sequence in BMO, then ||h||BMO ≤ 16 sup |ai |||fi ||BMO .

(3.1.5)

i

Proof. We consider the H p case first, and start proving the left-hand inequality. Note that assumption (3.1.3) implies that the sequence {fi } is disjointly supported over the Haar system. Hence S(h) = ( a2i S 2 (fi ))1/2 . As 1 ≤ p ≤ 2, we apply Hölder’s inequality to obtain |ai |p S p (fi ). ( a2i S 2 (fi ))p/2 ≤

(3.1.6)

Integrating the pointwise inequality (3.1.6) gives ||S(h)||pp ≤ |ai |p ||S(fi )||pp . Thus we verified the left-hand side of (3.1.4). Now we turn to the right-hand side estimate. Recall that we denote the support set of S(fi ) by Fi . Next we let Di = Fi \

∞

Fj .

j=i+1

By definition the difference sets Di are pairwise disjoint. Our hypothesis (3.1.3) gives the following lower bound |I ∩ Di | ≥ 1/2|I|, for every interval I in the Haar support of fi . Consequently, S(fi )p ≤ 2 S(fi )p 1Di . Summing up we obtained the lower bound p S(h) ≥ ( |ai |2 S 2 (fi )1Di )p/2 = |ai |p S(fi )p 1Di 1 p S(fi )p . ≥ |ai | 2


175

The BMO case follows from the above estimate with p = 1, and the H 1 − BMO duality theorem. By Fefferman’s inequality and Theorem 1.2.3 there exists g ∈ H 1 with √

g H 1 ≤ 2 2, such that h, g = h BMO . Let Fi be the Haar support of the function fi . Using the collections Fi we split g ∈ H 1 as follows, hI g, hI . gi = |I| I∈Fi

Then of course fi , g = fi , gi . The sequence (gi ) satisfies the assumption (3.1.3). Hence (3.1.4) with p = 1 holds for the sequence (gi ). Thus we obtain that √

gi H 1 ≤ 2 g H 1 ≤ 4 2. i

Summing up, we estimate

h BMO = h, g ai fi , gi = i

√ ≤ 2 2 sup |ai | fi BMO

gi H 1 i

i

≤ 16 sup |ai | fi BMO . i

Thus (3.1.5) is verified.

Remarks. 1. One obtains (3.1.5) also directly, without using duality. (The direct proof sketched below yields also better constants than the # duality argument.) The starting observation is that (3.1.3) implies that F = i Fi . satisfies the 2Carleson condition. Having made this observation it is then straightforward to estimate h BMO so that (3.1.5) holds true. 2. Assume that the hypothesis (3.1.3) is replaced by the weaker condition ∞ j=i+2

|J ∩ Fj | ≤

|J| . 2

(3.1.7)

Then by considering separately the sequences {a2i f2i } and {a2i+1 f2i+1 } we obtain from (3.1.4) that (3.1.8) |ai |p ||S(fi )||pp ≤ 4||S(h)||pp .

176


Large Carleson constants and the Gamlen–Gaudet construction Following is an application of the condensation Lemma 3.1.4. It is selected to illustrate the relationship between large Carleson constants, the Gamlen–Gaudet construction and Jones’s compatibility condition (J). Let C1 , . . . , CN be disjoint collections of dyadic intervals. Let Ci denote the point-set covered by the collection Ci , that is, Ci = I. I∈Ci

Recall from Section 1.5 that disjoint collections of dyadic intervals C1 , . . . , CN are said to satisfy Jones’ compatibility condition (J) if the following conditions (J 1)– (J 4) are satisfied. (J 1) Each Ci is a collection of pairwise disjoint dyadic intervals. (J 2) There exists a constant B, so that for each J ∈ Cj and each i ≥ j the following estimate holds, |I| |Cj ∩ Ci | ≤B . |J| |Cj | I∈Ci ,I⊆J

(J 3) The collections {Ci : 1 ≤ i ≤ N } are ordered in the following sense: If there exists I ∈ Ci and J ∈ Cj , such that I ⊆ J, then i ≥ j. Conversely, if i ≥ j, if I ∈ Ci and J ∈ Cj , such that I ∩ J = ∅, then I ⊆ J. (J 4) If j ≤ i and if Cj ∩ Ci = ∅, then Ci ⊆ Cj . In particular the sets {Ci : 1 ≤ i ≤ N } are nested; if Cj ∩ Ci = ∅, then either Ci ⊆ Cj or Cj ⊆ Ci . Fix a collection of dyadic intervals K and assume that the Carleson constant of K is bounded below by a large constant. Our aim is to find functions {kI : |I| ≥ 2−n } for which the Haar support is contained in K and which behave like the first 2n+1 Haar functions. This can be most easily achieved through the construction of Gamlen and Gaudet which we present now in its simplest form. Proposition 3.1.6. Let n ∈ N. Let K be a collection of dyadic intervals such that sup I∈K

J∈K,J⊆I

|J| ≥ n4n . |I|

Then there exists K ∈ K and a family of pairwise disjoint collections of dyadic intervals {CI : |I| ≥ 2−n } so that the following properties hold. 1. Each of the collections CI , is contained in {J ∈ K : J ⊆ K}. 2. The family {CI : |I| ≥ 2−n } satisfies the compatibility condition (J). 3. Let kI = J∈CI hJ . Then relative to the interval K the joint distribution of the functions {S 2 (kI ) : |I| ≥ 2−n } is equivalent to the joint distribution of the Haar functions {1I : |I| ≥ 2−n }. The constant of equivalence is bounded by 2.


177

4. In BMO, {kI : |I| ≥ 2−n } is equivalent to the Haar functions {hI : |I| ≥ 2−n } so that for any choice of scalars {xI : |I| ≥ 2−n }, 1 xI hI BMO ≤

xI kI BMO ≤ 2

xI hI BMO . 2 Proof. Let K be a collection of dyadic intervals such that [[K]] > n4n . Applying the condensation Lemma 3.1.4 we obtain K ∈ K such that |G∗n (K, K)| > (1 − 4−n )|K|.

(3.1.9)

Note that the estimate (3.1.9) implies that the generations prior to G∗n (K, K) are covering a set of measure > (1 − 4−n )|K|. We have |G∗k (K, K)| > (1 − 4−n )|K|,

for k ≤ n .

(3.1.10)

Now we describe the inductive construction of the families {CI : |I| ≥ 2−n }. We start by putting C[0,1) = {K}. Let I1 = [0, 12 ), and I2 = [ 12 , 1). In the next step we define CI1 and CI2 . Recall that C[0,1) contains only one interval, namely K. Let K1 be the left half of K and K2 be the right half of K. Then define CI1 = {L ∈ G1 (K, K) : L ⊆ K1 }

and

CI2 = {L ∈ G1 (K, K) : L ⊆ K2 }.

Let k ≤ n. Assume that the collections CI have been constructed for all dyadic intervals of length ≥ 2−k+1 . Then we fix a dyadic interval I of length |I| = 2−k+1 . Let I1 be the left half of I, and let I2 be the right half of I. Next pick any interval J ∈ CI . Splitting J in the same way we let J1 be the left half of J, and J2 be the right half of J. Then we form the auxiliary collections, CI1 (J) = {L ∈ Gk (K, K) : L ⊆ J1 }

and

CI2 (J) = {L ∈ Gk (K, K) : L ⊆ J2 }.

Completing the induction step we take the union over all intervals J ∈ CI thus forming, CI1 (J) and CI2 = CI2 (J). CI1 = J∈CI

J∈CI

Having completed the Gamlen–Gaudet construction # of the collections {CI : |I| ≥ 2−n } we analyze their properties. We let CI = L∈CI L. Thus {CI : |I| ≥ 2−n } is a nested collection of sets satisfying the same intersection pattern as the collection of dyadic intervals. That is, CI ⊆ CJ

iff

I ⊆ J.

178


Observe that by (3.1.10) the following condition holds true. If L ∈ CJ and I ⊆ J, then |L ∩ CI | 2|I| |I| ≤ ≤ . (3.1.11) 2|J| |L| |J| By (3.1.11) the collection {CI : |I| ≥ 2−n } satisfies Jones’s compatiblity condition (J). Next we analyze the block bases kI = L∈CI hL and the joint distribution of {S 2 (kI )}. Given coefficients {aI : |I| ≥ 2−n } we form aI kI . g= aI hI and f = Since {CI : |I| ≥ 2−n }} are disjoint collections consisting of pairwise disjoint dyadic intervals we obtain S(kI ) = 1CI and 1/2 S(f ) = . a2I 1CI Hence it follows from (3.1.11) that for t > 0, |{S(f ) > t}| 1 |{S(g) > t}| ≤ ≤ 2|{S(g) > t}|. 2 |K| The BMO part follows by restriction and re-normalization from (3.1.12).

(3.1.12)

Consequences. Next we draw some consequences of proposition 3.1.6. Using the functions {kI : |I| ≥ 2−n } obtained by the Gamlen–Gaudet construction we define the orthogonal projection kI kI f, . P (f ) =

kI 2 kI 2 −n {I:|I|≥2

}

By Theorem 1.5.9 the projection P is a bounded operator on BMO, with norm estimates depending only on the fact that the collections {CI : |I| ≥ 2−n } satisfy the compatibility condition (J). Hence P BMO ≤ C, where C is independent of n. The assertions of Proposition 3.1.6 show that the linear extension of the map E : hI → kI ,

where

|I| ≥ 2−n ,

defines an isomorphism from BMOn onto its range, so that

E · E −1 ≤ 4. Thus the range of P in BMO is well isomorphic to BMOn . The H 1 − BMO duality implies the analogous boundedness of P in H 1 , and

P H 1 ≤ C. Moreover also in H 1 the map E : hI → kI ,

where

|I| ≥ 2−n ,

defines an isomorphism from Hn1 onto its range.


179

Small Carleson constants and Jones’s compatibility condition Next we give an application of Proposition 3.1.2. Recall the following convenient notation. For a collection of pairwise disjoint dyadic intervals C we denote Q(C) =

{J ∈ D : J ⊆ K}.

K∈C

Thus Q(C) is the collection of intervals which are contained in one of the intervals of C. For a single dyadic interval I we write Q(I) = {J ∈ D : J ⊆ I}. Next recall the notion of a block of intervals which was introduced in Chapter 1 following the atomic decomposition of H 1 . Given a collection K of dyadic intervals, and a subcollection L ⊆ K. We say that L is a block of intervals in K if the following conditions are satisfied. 1. L contains exactly one maximal interval, say I0 . 2. For every J ∈ L and K ∈ K the following implication holds true. If J ⊆ K ⊆ I0 , then K ∈ L. Let L ⊆ K be a block of intervals in K. Fix now one interval, say I ∈ L. Then the collection L \ {I} need not be a block of intervals. However if we remove Q(I) from the block L, then the result is again a block. Expressed formally, we have the following implication. If L is a block in K, then also L \ Q(I) is a block in K.

(3.1.13)

We continue by giving an illustration of Proposition 3.1.2. Let K be a collection of dyadic intervals K which satisfies the Carleson packing condition. Assume that K has just one maximal interval, call it I0 . Suppose we wish to decompose K into families C0 , . . . , CM satisfying Jones’s compatibility condition (J). Can one do that with a good upper bound for M ? Not in general, but the following is true: Proposition 3.1.7. Let K be a collection of dyadic intervals K, such that max{K} = {I0 }. Assume that K satisfies the Carleson packing condition, [[K]] = sup I∈K

J∈K,J⊆I

|J| < ∞. |I|

Then there exists A ≤ 4[[K]], so that K contains disjoint subcollections C0 , . . . , CA satisfying Jones’s compatibility condition, and so that the following holds.

180


1. The union L = C0 ∪ · · · ∪ CA is a block of intervals such that |{K \ L}∗ | ≤ 2. Each of the Ci satisfies

|I| ≥

I∈Ci

1 |I0 |. 4

1 |I0 |. 4

Proof. We construct L by a stopping time argument, where the size of the generations Gm (I0 , K) determines the stopping rule. Denote Cm = Gm (I0 , K) and let Cm be the point-set covered by the collection Cm . Define A ∈ N by the following rule: Let A be the largest integer for which |CA | >

1 |I0 |. 4

(3.1.14)

Proposition 3.1.2 implies the following upper bound holds for A, A ≤ 4[[K]].

(3.1.15)

Moreover the stopping time definition of A implies that |CA+1 | ≤

1 |I0 |, 4

(3.1.16)

since otherwise we would have chosen A + 1 and not A. Define now L=

A

Cj .

j=0

It is easy to see that L is a block of dyadic intervals. By (3.1.14) the linearly ordered collections {C0 , . . . , CA } satisfy Jones’s compatibility condition. By (3.1.16) the block L is large in the following sense, |{K \ L}∗ | ≤

1 |I0 |, 4

(3.1.17)

and by (3.1.15) we needed less than 4[[K]] generations to achieve (3.1.17). In that sense {C0 , . . . , CA }, is an efficient decomposition of L.

3.2. Orthogonal projections and colored intervals

3.2

181

Orthogonal projections and colored intervals

A Banach space X satisfies the uniform approximation property if there exists λ > 0 so that for every n ∈ N there exists f (n) ∈ N so that for every x1 , . . . , xn ∈ X there exists a linear operator T : X → X satisfying ||T || ≤ λ, T x i = xi , rank T ≤ f (n). In Chapter 5 we prove the deep result by P. W. Jones that BMO satisfies the uniform bounded approximation property. As a preparation we treat in Theorem 3.2.3 the special case when xi =

{hI : I ∈ Bi }.

To be able to do this we need to establish the combinatorial connection between the Carleson constant of collections of dyadic intervals and Jones’s compatibility condition (J). The chief difficulties are addressed in Theorem 3.2.1. We view Theorem 3.2.1 as a multicolored version of Proposition 3.1.7. Formally, colors are attached to intervals in a collection K as follows. Let L ∈ N. Let B1 , . . . , BL be disjoint collections of dyadic intervals such that K = B1 ∪ · · · ∪ BL .

(3.2.1)

Then the color of J ∈ K is defined to be j ∈ {1, . . . , L} if J ∈ Bj . Accordingly, on K we define the function color : K → {1, . . . , L} (3.2.2) by the relation color(J) = j

iff

J ∈ Bj .

(3.2.3)

Let C ⊆ K. We say that C is monochromatic if color(J) = color(I)

for all

I, J ∈ C.

(3.2.4)

Jones’s compatibility condition and colored intervals Next we obtain a decomposition of a colored collection into monochromatic subcollections satisfying Jones’s compatibility condition. The following theorem plays an important role in proving a special case of the uniform approximation property for BMO (Theorem 3.2.3 below). It also constitutes a crucial component in the proof of the general case.

182


Theorem 3.2.1. Let L ∈ N. Let K be a collection of dyadic intervals with one maximal interval, I0 ; each I ∈ K has attached to it one of L different colors. Assume that K satisfies the Carleson packing condition, sup I∈K

J∈K,J⊆I

|J| ≤ K. |I|

Then K contains disjoint subcollections C1 , · · · , CM satisfying Jones’s compatibility condition and also the following properties: 1. The union L = C1 ∪ · · · ∪ CM is a block of intervals such that |{K \ L}∗ | ≤ 2. Each of the collections Ci satisfies

1 |I0 |. 2

|I| ≥ δ|I0 |.

I∈Ci

3. Each of the collections Ci is monochromatic (with respect to the colors attached to the intervals of K). 4. There exist upper and lower bounds M ≤ M (K, L), and δ ≥ δ(K, L). Proof. The proof is divided into four separate parts. The first part contains a summary and a description of the main steps in the argument. Part 1. Preparation and overview. For the generations of K we use the abbreviation Gm = Gm (I0 , K). Define s ∈ N to be the largest integer for which |G∗s | >

1 |I0 |. 4

Since K satisfies the Carleson packing condition with constant ≤ K, Proposition 3.1.2 implies that the following upper bound holds for s, s ≤ 4K.

(3.2.5)

In the course of the proof we will define the block L to be a subcollection of # s m=0 Gm (I0 , K). The construction starts by considering first the intervals in Gs and later the intervals in Gs−1 until the generation G1 is reached. Recall that we have attached one of L colors to the intervals in K. Put Ls = L. For k ≤ Ls define the following subcollection of Gs , O(s,k) = {I ∈ Gs : color(I) = k}.


183

Now let 1 ≤ m < s. In parts 2 and 3 of the proof we find Lm ∈ N, and monochromatic, disjoint subcollections of Gm O(m,n)

where n ≤ Lm ,

so that the following conditions hold. 1. There exists L0 = L0 ([[K]], L), depending only on the Carleson constant [[K]] and the number of colors L, so that Lm ≤ L0 . 2. The union L = {I0 } ∪

s L m

O(m,n)

m=1 n=1

forms a block of dyadic intervals in K so that |{K \ L}∗ | ≤

1 |I0 |. 2

3. Each of the collections O(m,n) , n ≤ Lm is monochromatic with respect to the colors in K. 4. For 1 ≤ m < s the collections {O(m,n) : n ≤ Lm } and {O(m+1,k) : k ≤ Lm+1 } are related by the following (extremely restrictive) homogeneity condition: For n ≤ Lm and k ≤ Lm+1 there exists A = A(m, n, k) so that for each I ∈ O(m,n) , 1 |J| 1s ≤ 2 A. (3.2.6) 2− s A ≤ |I| J∈I∩O(m+1,k)

The hard part of the proof consists of constructing the block L and the collections O(m,n) where n ≤ Lm and 1 ≤ m ≤ s. This is done in parts 2 and 3 below. In part 4 we further split the monochromatic collections {O(m,n) : n ≤ Lm }, ∗ into {F(m,k) : k ≤ Nm }, in such a way that {F(m,k) : k ≤ Nm , m ≤ s}, is a nested collection of sets. (This is done by a straightforward procedure.) After that we relabel {F(m,k) }, as {Cj } transforming lexicographic order into linear order. We will easily see that {Cj } satisfies the compatibility conditions (J 1), (J 3), and (J 4). Exploiting (3.2.6) we will verify that also (J 2) holds true. Part 2. The construction of the block L. Step s. Recall that L is the number of colors in K. Put Ls = L. At step s we are concerned with the intervals in Gs . Each interval I ∈ Gs has attached to it one of Ls colors. Then for k ≤ Ls we define O(s,k) = {I ∈ Gs : color(I) = k}.

184


The combinatorial idea of the proof (as described below) is encoded in the following construction which successively selects in Gm (m ≤ s) certain subcollections O(m,n) . The first appearance of O(m,n) is at step m of the construction. At each of the following steps (that is at the steps m − 1, m − 2, . . . , 1) it may become necessary to modify the already existing collection O(m,n) , by deleting some of its intervals. The following rule is applied throughout: If at some point we decide to discard an interval I from the collection O(m,n) , then simultaneously we delete from O(m+j,n ) the collections I ∩ O(m+j,n ) ,

for

n ≤ Lm+j

and

1 ≤ j ≤ s − m.

This rule gives that the collection L defined by (3.2.15) is a block of intervals. We will make sure that the cumulative effect of updating O(m,n) remains small compared to |I0 |. Step s − 1. At step s − 1 we define Ls−1 ∈ N and subcollections of Gs−1 denoted O(s−1,n) , where n ≤ Ls−1 . We start by modifying the already existing collections {O(s,k) : k ≤ Ls }. Fix I ∈ Gs−1 . For each k ≤ Ls we now determine the density of O(s,k) in I. Let O(s,k) be the point-set covered by O(s,k) . If |I ∩ O(s,k) | 1 ≤ , |I| 8sLs

(3.2.7)

then we define the collection I ∩ E(s,k) by putting I ∩ E(s,k) := I ∩ O(s,k) . Applying this rule to each of the collections {O(s,k) , k ≤ Ls } defines I ∩ Es =

Ls

I ∩ E(s,k) .

k=1

Having created the collections I ∩ Es for each I ∈ Gs−1 , we take the union and form I ∩ Es . Es = I∈Gs−1

Next we modify O(s,k) by renaming O(s,k) := O(s,k) \ Es .

(3.2.8)

Let I ∈ Gs−1 , and let k ≤ Ls . After the modifications defined by (3.2.8) the following implication holds. If

I ∩ O(s,k) = ∅,

then

|I ∩ O(s,k) | 1 > . |I| 8sLs


185

Consequently for each k ≤ Ls and each I ∈ Gs−1 , there exists an integer n(k) with n(k) ∈ {1, . . . , s log2 (8sLs )} ∪ {∞} and such that 2−

n(k)+1 s

≤

n(k) |I ∩ O(s,k) | ≤ 2− s . |I|

In summary, for a fixed I ∈ Gs−1 and for each k ≤ Ls we determined n(k) ∈ N satisfying n(k) ≤ s log(8sLs ) or n(k) = ∞ so that the vector α = (n(1), . . . , n(L)) records the densities of O(s,k) in I. Now we change the point of view (thereby testing our understanding of the above) . We fix a color k ≤ L. Then we collect all intervals I ∈ Gs−1 , for which the color is k and which produce the same vector of indices when the densities of O(s,k) in I are computed. Precisely, given k ≤ L and α = (n(1), . . . , n(L)) where n(k) ≤ s log(8sL), or n(k) = ∞, we define O(s−1,k ,α) to be the collection of all dyadic intervals I ∈ Gs−1 , which have the color k attached to them and for which the following two-sided estimates hold, 2−

n(k)+1 s

0 and choose a dyadic interval J such that (1 − )||T x||2BMO ≤

1 2 xI |τ (I)|. |J|

(3.3.4)

τ (I)⊆J

The idea of the proof is to define collections of dyadic intervals E1 , . . . , EK , with the following properties: Each of the Ei satisfies the 3-Carleson condition,

3.3. Rearrangement operators

199

τ (Ei )∗ is contained in J, and the following crucial convexity relation holds true, K 1 1 kI |τ (I)| = |τ (I)|. K K i=1

(3.3.5)

I∈Ei

τ (I)⊆J

Now we will carry out the construction of the collections Ei . Let n ∈ N. Let vn be the vector whose entries are dyadic intervals I of length 2−n such that τ (I) ⊆ J. Moreover in the vector vn we let I appear consecutively kI times, where we defined kI in (3.3.3). Now we define the rule which determines when a dyadic interval belongs to the collection Ei . Fix I ∈ D and suppose that |I| = 2−n . Choose i ≤ K. Then we define I to be in Ei iff I occupies the p-th position in vn , and p = i mod K. Note that since kI ≤ K, the entries of the vectors vn are bijectively distributed among the collections E1 , . . . , EK , by the above rule. It is also easy to see from that definition that the convexity relation (3.3.5) is satisfied. The decisive property of our definition is however that simultaneously with (3.3.5) each of the collections Ei satisfies the 3-Carleson condition. To verify this claim we fix Ei . Then we choose n ∈ N and I0 ∈ D. Let An,i be the cardinality of the collection {I ⊆ I0 : I ∈ Ei , |I| = 2−n }. Taking another look at the defining properties of Ei lets us observe the estimate An,i ≤ 1 +

1 K

kI .

I⊆I0 ,|I|=2−n

Having made these observations we verify the claim that Ei satisfies the 3-Carleson condition. ∞ |I| = 2−n An,i I∈Ei ∩I0

n=− log2 |I0 |

≤ 2|I0 | +

1 kI |I| K I⊆I0

≤ 3|I0 |. So, [[Ei ]] ≤ 3. Now we finish the proof by showing that ||T x|| ≤ (3M )1/2 . Recall that in (3.3.4) the interval J was chosen so that (1 − )||T x||2 ≤

1 2 xI |τ (I)|. |J|

(3.3.6)

τ (I)⊆J

Recall also that by (3.3.3) the Haar coefficients of x satisfy x2I = kI K −1 . With the convexity relation (3.3.5) we may rewrite the sum in the right-hand side of (3.3.6) as follows, K 1 x2I |τ (I)| = |τ (I)|. (3.3.7) K i=1 τ (I)⊆J

I∈Ei

200


Next we give an obvious upper bound for the inner sum appearing in (3.3.7) above, |τ (I)| ≤ [[τ (Ei )]]|τ (Ei )∗ |. (3.3.8) I∈Ei

We showed already that the collection Ei satisfies the 3-Carleson condition, and by hypothesis the rearrangement τ preserves the Carleson condition. So we have [[τ (Ei )]] ≤ 3M. Next recall that, by construction, |τ (Ei )∗ | ≤ |J|. Inserting into the right-hand side of (3.3.8) we showed that the right-hand side of (3.3.6) admits the upper bound 1 2 xI |τ (I)| ≤ 3M. |J| τ (I)⊆J

This proves that ||T x||BMO ≤ (3M )1/2 . The converse follows from the observation that for special functions of the form x = I∈C hI , the norm in BMO satisfies ||x||BMO = [[C]]1/2 , and that T x =

I∈C

hτ (I) . Hence ||T x||BMO = [[τ (C)]]1/2 .

Theorem 3.3.2 raises the question if for general operators on BMO the norm can be tested on special functions of the form x= hI . (3.3.9) I∈C

Equivalently we ask if in some renorming of BMO the simple functions (3.3.9) are the extreme points of the unit ball. That this is not so can be seen by looking at the class of orthogonal projections discussed in Section 1.5. Recall that pairwise disjoint collections {Sj } determine an orthogonal projection in the following way. Pf =

n f, kj kj , j=1

where kj = bj /||bj ||2 and where bj =

{hI : I ∈ Sj }.

Below we will select pairwise disjoint collections {Sj : 1 ≤ j ≤ n}, such that ||P ||BMO ≥

3√ n, 4

and so that for functions of the form x = I∈C hI , the following upper bound holds true, ||P x||BMO ≤ [[C]].


201

To begin with, we fix n ∈ N. Then define Ij = [0, 4−j [, where 1 ≤ j ≤ n.

(3.3.10)

We attach frequencies to the intervals Ij as follows. We start with the smallest interval In . By our choice |In | = 4−n . For 1 ≤ k ≤ n, we define the collection Bn,k to consist of all dyadic intervals with length = 2−k 4−n , which are contained in In . Thus, we form Bn,k = {J ⊆ In : |J| = 2−k 4−n }. To the remaining intervals Ij , 1 ≤ j ≤ n − 1 we attach frequencies in a similar fashion. We define Bj,k = {J ⊆ Ij : |J| = 2−k 4−n(n−j+1) }. Next we fix j and take the union over k ≤ n, thus defining Sj = Bj,1 ∪ · · · ∪ Bj,n . Note that the collections {Sj : 1 ≤ j ≤ n} are pairwise disjoint and that Sj is an n-fold cover of the interval Ij . Moreover we observe the following crucial relation among the collections {Sj : 1 ≤ j ≤ n}. If K ∈ Sj and L ∈ Sj+1 , then |K| ≤

1 |L|. 2

(3.3.11)

The collections {Sj : 1 ≤ n}, are thus inversely ordered in the following sense: The larger Ij is, the shorter are the intervals in the collection Sj . Now we form the block bases {hI : I ∈ Sj }. bj = It is straightforward to evaluate the square function of bj , We get S(bj ) = n1/2 1Ij . Hence for the norm of bj in the spaces H 1 , L2 and BMO, we obtain that ||bj ||H 1 = n1/2 |Ij |,

||bj ||2 = n1/2 |Ij |1/2

and

||bj ||BMO = n1/2 .

Let kj = bj /||bj ||2 and define the orthogonal projection Pf =

n f, kj kj . j=1

√ n/4. For a collection Proposition 3.3.3. The norm of P on BMO is larger than 3 of dyadic intervals C, the following estimate holds when x = J∈C hJ , ||P x||BMO ≤ [[C]].

202


Proof. First we show the lower bound on the norm of P on BMO. We consider the collections of dyadic intervals Lj = {J ∈ Sj : J ∩ Ij+1 = ∅}.

(3.3.12)

Observe that L∗j = Ij \ Ij+1 . Now we define, {hI : I ∈ Lj }. fj = n−1/2 Note that ||fj ||BMO ≤ 1, since the square functions S(fj ) are pointwise ≤ 1. Moreover the support of S(fj ) equals Ij \ Ij+1 . Hence the sum f=

n

fj

j=1

satisfies the upper bound ||f ||BMO ≤ 1. Now we give a lower estimate for ||P f ||BMO . Observe that, by (3.3.12), we have the identities f, kj = and therefore,

|Ij \ Ij+1 | 3 = |Ij |1/2 , 4 |Ij |1/2

3 kj |Ij |1/2 . 4 j=1 √ ≥ 3 n/4. To this end we expand P f in its Haar n

Pf = Next we prove that ||P f ||BMO series. This gives

Pf = #

αI hI ,

where αI = 3n−1/2 /4, for I ∈ Sj , and αI = 0 otherwise. Next we fix an interval J ∈ Sn . By the condition of order inversion that is by (3.3.11) it follows that n 16 2 αI |I| = ||kj ||22 |J| 9 j=1 I⊆J

(3.3.13)

= n|J|. √ Clearly the estimate (3.3.13) implies that ||P f ||BMO ≥ 3 n/4. Recall that the norm of f in BMO is √ bounded by 1. Hence the operator norm satisfies the lower bound ||P ||BMO ≥ 3 n/4. Next we prove an upper bound for ||P x||BMO , when x = J∈C hJ . It hinges on the following estimate for the coefficients x, kj . As ||bj ||2 = n1/2 |Ij |1/2 , and kj = bj /||bj ||2 obtain, x, kj = n−1/2 |Ij |−1/2 |J| J∈C∩Sj (3.3.14) ≤ [[C]]n−1/2 |Ij |1/2 .


203

Next we observe that kj |Ij |1/2 = bj n−1/2 , and that bj BMO = n1/2 . Recall also that the Haar support of kj |Ij |1/2 is pairwise disjoint. Thus ||

n

kj |Ij |1/2 ||BMO ≤ n1/2 .

(3.3.15)

j=1

Now we combine the upper bounds in (3.3.15) and (3.3.14). This causes the factors n−1/2 in (3.3.14) and n1/2 in (3.3.15) to cancel. We find,

n

x, kj kj ||BMO ≤ [[C]]n−1/2 ||

j=1

n

kj |Ij |1/2 ||BMO

j=1

≤ [[C]]. Thus ||P x||BMO ≤ [[C]], as claimed.

Now we continue the analysis of rearrangement operators by merging the conclusion of Proposition 3.3.2 with the results on complex interpolation. This gives a criterion for the boundedness of rearrangement operators on Lp (2 0 so that [[τ C]] ≤ M [[C]], for every collection of dyadic intervals C. Theorem 3.3.4. Suppose that τ is a rearrangement that preserves the Carleson condition. Then for 2 ≤ p < ∞, Tp :

hτ (I) hI → , 1/p |I| |τ (I)|1/p

extends to a bounded operator on Lp . Proof. The operator Tp coincides with the operator hI → hτ (I)

|τ (I)| |I|

−1/p .

The rearrangement operators {Tp : 2 ≤ p < ∞} are embedded in the analytic family of operators (z−1)/2 |τ (I)| , Rz : hI → hτ (I) |I| since Tp = R1−2/p . By the BMO result, Proposition 3.3.2, and the interpolation Corollary 2.3.12 for analytic families of operators, the conclusion of Theorem 3.3.4 holds. The duality Theorem 1.2.8 and Corollary 2.3.12 apply here since for rearrangements the transposed operator coincides with the inverse operator.

204


We have proven so far that if τ preserves the Carleson condition, then permuting the Lp -normalized Haar system by τ leads to a bounded operator on Lp whenever 2 < p < ∞. We now turn to the converse of the interpolation result in Theorem 3.3.4. We wish to prove that if a rearrangement of the Lp normalized Haar system is bounded in Lp , then the same rearrangement of the L∞ normalized Haar system is bounded in BMO. Thus we intend to establish an extrapolation result from Lp to BMO. We saw already a proof of this as a corollary to G. Pisier’s renorming of H 1 . Different in spirit is the proof given below. Theorem 3.3.5. If there exists p > 2 such that the linear extension of Tp :

hτ (I) hI → 1/p |I| |τ (I)|1/p

defines a bounded operator on Lp , then τ preserves the Carleson condition, and T∞ : hI → hτ (I) extends linearly to a bounded operator on BMO satisfying 1−2/p

||T∞ ||BMO ≤ C||Tp ||p . Proof. We assume that τ fails to preserve the Carleson constant. We will show that this assumption implies that the operator Tp maps the unit vector basis of p 2 into the unit vector basis of lN , for any N ∈ N. However for p > 2, and N large lN enough, this contradicts the hypothesis that Tp is bounded on Lp . Now we fix N ∈ N. We assume that there exists a collection of dyadic intervals C such that [[τ (C)]] > 16N [[C]]. (3.3.16) In a first step we show how (3.3.16) implies that there exists a collection of dyadic intervals A, such that [[A]] ≤ 2,

and

[[τ (A)]] ≥ 2N.

(3.3.17)

Assuming that [[C]] is an integer, we let A = 4[[C]]. By the positive homogeneity of Carleson constants, as expressed in Lemma 3.1.3, there exist pairwise disjoint collections of dyadic intervals C1 , . . . , CA , such that C = C1 ∪ · · · ∪ CA , and [[Cj ]] ≤ 2,

for j ≤ A.

Next we let M ≥ 1 be such that [[τ (Cj )]] ≤ M [[Cj ]],


205

for j ≤ A. We claim that this is possible only if M > 2N. To see that M has to be larger that 2N we split τ (C) and estimate as follows, [[τ (C)]] ≤

A

[[τ (Cj )]]

j=1

≤M

A [[Cj ]]

(3.3.18)

j=1

≤ 8M [[C]]. On the other hand we assume that (3.3.16) holds, that is [[τ (C)]] > 16N [[C]]. Comparing the inequalities (3.3.16) and (3.3.18) gives that M > 2N. Thus we showed the following statement: If τ fails to preserve the Carleson condition, then for any N ∈ N, there exists a collection of dyadic intervals A for which, [[A]] ≤ 2, and [[τ (A)]] ≥ 2N. Now we apply the condensation lemma to the collection τ (A). By Lemma 3.1.4, the condensation lemma, there exists K ∈ τ (A), such that |GN (K, τ (A))∗ | >

1 |K|. 2

(3.3.19)

Combining the estimate (3.3.19) with the fact that A satisfies the 2-Carleson condition, we will next construct f ∈ Lp , with Haar support in A, such that ||S(Tp f )||p ≥ ||S(f )||p

1 1/2−1/p N . 32

Clearly for p > 2 and N large enough, this estimate contradicts the boundedness of ||Tp ||p . For i ≤ N we define Ei = Gi (K, τ (A)) and Fi = τ −1 (Ei ).

(3.3.20)

For i ≤ N, and J ∈ Fi we define coefficients as bJ = |τ (J)|1/p . Then let f=

N i=1 J∈Fi

bJ

hJ . |J|1/p

(3.3.21)

206


Applying the rearrangement operator Tp to f gives Tp f =

N

bJ

i=1 J∈Fi

hτ (J) . |τ (J)|1/p

We continue with an upper bound for ||S(f )||p . Note that the Haar support of # f is the collection N i=1 Fi which is contained in A. Hence as [[A]] ≤ 2, we apply Proposition 3.1.2 (the decomposition lemma) and Proposition 3.1.5 to obtain 1/p

N p ||S(f )||p ≤ 16 |bJ | . (3.3.22) i=1 J∈Fi

Next recall that the coefficients are defined by |bJ |p = |τ (J)|. Hence for each i the inner sum in (3.3.22) is bounded as follows, |bJ |p = |τ (J)| J∈Fi

J∈Fi

≤ |K|. Inserting this bound in the inequality (3.3.22) gives an upper bound for the norm of f as N ||S(f )||p ≤ 16( 1)1/p |K|1/p (3.3.23) i=1 = 16N 1/p |K|1/p . Next we give a lower estimate for ||S(Tp f )||p . Here we will exploit (3.3.19) which followed from the condensation lemma. By (3.3.21) we obtain the following identities for S(Tp f ), S 2 (Tp f ) =

N

|bJ |2 |hτ (J) |2 |τ (J)|−2/p

i=1 J∈Fi

=

N

(3.3.24)

|hI | . 2

i=1 I∈Ei

By (3.3.20) we have Ei = Gi (K, τ (A)). Hence with (3.3.24) and (3.3.19) we obtain the lower bound N 1 1K )1/2 ||p ||S(Tp f )||p ≥ ||( 4 i=1 =

N 1 1/2 ( 1) |K|1/p 4 i=1

=

1 1/2 N |K|1/p . 4

(3.3.25)


207

Clearly for N big enough and p > 2, the estimates (3.3.23) and (3.3.25) contradict the boundedness of Tp . Thus we showed that if a rearrangement τ does not preserve the the Carleson condition, then the induced rearrangement operator Tp is unbounded on Lp . Let us now review the preceding argument to obtain extrapolation formulas for the norm, that is 1−2/p ||T∞ ||BMO ≤ C||Tp ||p . Let N ∈ N be a large integer. Combine Proposition 3.3.2 and (3.3.17) to see that ||T∞ ||BMO ≥ CN 1/2 implies the existence of a collection A with [[A]] ≤ 2 and [[τ (A)]] ≥ N. Using the collection A we defined f ∈ Lp so that ||S(Tp f )||p ≥ ||S(f )||p

1 1/2−1/p N . 32

Consequently the hypothesis that ||T∞ ||BMO ≥ CN 1/2 implies that ||Tp ||p ≥ N 1/2−1/p . As N was arbitrary we have 2(1/2−1/p)

||Tp ||p ≥ c||T∞ ||BMO

.

Rearrangements with Property P Now we introduce Property P. We prove that a rearrangement preserves the Carleson packing condition if and only if it satisfies Property P. We say that a rearrangement τ satisfies the Property P if there exists M > 0 so that for every B ⊂ D and J ∈ D, the collection τ (B) ∩ J can be decomposed as a disjoint union ∞ ∞ τ (Li ) ∪ Ei , i=1

so that:

∞

Ei

i=1

satisfies the M -Carleson condition.

(P 1)

i=1

For every I ∈ Li , ∞ i=1

|τ (Li )∗ | + |Ei∗ | |τ (I)| ≤M . |I| |L∗i |

|τ (Li )∗ | ≤ M |J| sup [[τ −1 (max τ (Li ))]].

(P 2) (P 3)

i

We start with the simple implication that a rearrangement preserves the Carleson packing condition provided that it satisfies Property P.

208


Proposition 3.3.6. If τ satisfies Property P, then τ preserves Carleson’ s condition. Proof. We let B be a collection of dyadic intervals satisfying Carleson’s packing condition, and we assume that τ satisfies Property P. We will show that then τ (B) also satisfies Carleson’s packing condition. To this end we fix an arbitrary dyadic interval J, and we seek good estimates for the sum |τ (I)|. S= τ (I)∈τ (B)∩J

Now, by Property P we may write down the decomposition τ (B) ∩ J =

∞

τ (Li ) ∪

i=1

∞

Ei ,

i=1

so that the conditions (P 1)–(P 3) are satisfied. Note that the collection τ (B)∩J is the index set of the sum S. Thus the splitting of τ (B) ∩ J induces a corresponding splitting of S as S = S 1 + S2 , where

S1 =

i

and S2 =

|τ (I)|

I∈Li

i

|K|.

K∈Ei

We first focus on estimating S1 . We begin by rewriting the homogeneity condition (P 2) as |τ (Li )∗ | + |Ei∗ | |I|. |τ (I)| ≤ M |L∗i | Inserting this estimate in the sum defining S1 , we obtain the upper bound S1 ≤ M

∞ |τ (Li )∗ | + |E ∗ | i=1

|L∗i |

i

|I|.

(3.3.26)

I∈Li

Next we will further analyze the inner sums of the form I∈Li |I| appearing on the right-hand side of the estimate (3.3.26) for S1 . Note that the obvious inclusion Li ⊆ B gives the upper estimate |I| ≤ [[B]]|L∗i |. (3.3.27) I∈Li

Next we insert the bounds (3.3.27) into the estimate (3.3.26). This gives that S1 ≤ M [[B]]

∞ i=1

|τ (Li )∗ | + M [[B]]

∞ i=1

|Ei∗ |.

(3.3.28)


209

We continue by noting the following obvious inclusions τ −1 (max τ (Li )) ⊆ Li ⊆ B. This gives the upper bounds [[τ −1 (max τ (Li ))]] ≤ [[B]], for any i. We invoke condition (P 3) to obtain, ∞

|τ (Li )∗ | ≤ M |J|[[B]].

(3.3.29)

i=1

Next we observe that the sets Ei∗ are contained in J. Thus for E = by condition (P 1), ∞ |Ei∗ | ≤ [[E]]|J|

#

Ei we have

(3.3.30)

i=1

≤ M |J|. Inserting the estimates (3.3.29) and (3.3.30) into (3.3.28), we obtain the upper bound for S1 , ∞ S1 ≤ M [[B]] |τ (Li )∗ | + |Ei∗ | i=1

≤ M 2 [[B]]([[B]] + 1)|J|. Now we turn to the estimate for S2 . Note, that with E =

#

Ei we have,

S2 ≤ [[E]]|J| ≤ M |J|. Now we combine the estimates for S1 and S2 . We observe that [[B]] ≥ 1 whenever B is not empty, hence we can combine the estimates for S1 and S2 to obtain 2

S1 + S2 ≤ 3M 2 [[B]] |J|. As J ∈ D was an arbitrary dyadic interval, we obtained that 2

[[τ (B)]] ≤ 3M 2 [[B]] .

(3.3.31) 2

To complete the proof it remains to remove the quadratic term [[B]] from the estimate (3.3.31) and replace it by the correct linear term 8[[B]]. To do so we use Lemma 3.1.3 asserting that the Carleson constant measures the size of collections like a positively homogeneous gauge function. We assume that [[B]] is an integer and put A = 4[[B]]. By Lemma 3.1.3 we can decompose B into B1 , . . . , BA such that [[Bi ]] ≤ 2, for i ≤ A. Now recall that in the first part of 2 the proof we showed the estimates [[τ (Bi )]] ≤ 3M 2 [[Bi ]] . Taking the sum we obtain finally that A [[τ (B)]] ≤ [[τ (Bi )]] i=1

≤ 48M 2 [[B]].

210


Now we will continue with a detailed analysis of rearrangements which preserve the Carleson condition. We show that for a rearrangement τ, Property P is equivalent to preserving the Carleson condition. Theorem 3.3.7. Let τ be a rearrangement of dyadic intervals. Suppose that there exists M ≥ 1 such that [[τ (E)]] ≤ M [[E]], for any collection of dyadic intervals E. Let B be a fixed collection of dyadic intervals, and let J be a fixed interval. Then, the collection τ (B) ∩ J can be decomposed as ∞ ∞ τ (Li ) ∪ Ei , i=1

i=1

so that the following conditions hold. Ei satisfies the 2M-Carleson condition. |τ (I)| |τ (Li )∗ | + |Ei∗ | ≤ 2M , |I| |L∗i | ∞

(3.3.32)

for I ∈ Li .

(3.3.33)

|τ (Li )∗ | ≤ |J|2M sup [[τ −1 (max τ (Li ))]].

(3.3.34)

i

i=1

The natural approach to prove this decomposition of τ (B) ∩ J is by iterating the following stopping time rule: “Select the largest block of intervals L on which τ acts homogeneously.” After a moments reflection we observe however that our rule formulates two competing (if not conflicting) conditions. Hence in the actual implementation of this idea we are confronted with combinatorial difficulties which we have to address first. We isolate the combinatorial aspects of the proof in Lemma 3.3.8. We recall first the following notation introduced earlier. For a collection of pairwise disjoint dyadic intervals C we let {J ∈ D : J ⊆ K}. Q(C) = K∈C

This is the collection of intervals which are contained in (at least) one of the intervals of C. For a single dyadic interval I we let Q(I) = {J ∈ D : J ⊆ I}. Lemma 3.3.8. Let τ be a rearrangement of dyadic intervals. Suppose that there exists M ≥ 1 such that [[τ (E)]] ≤ M [[E]], for any collection of dyadic intervals E. Let B be a fixed collection of dyadic intervals, and let I0 be a fixed interval. Then there exists a collection of pairwise disjoint dyadic intervals C ⊆ B ∩ I0 so that the following conditions hold:

3.3. Rearrangement operators 1.

211

|L| ≤

L∈C

|I0 | . 2

2. Let L = B ∩ [Q(I0 ) \ Q(C)], then for I ∈ L,

|τ (L)∗ | + |τ (C)∗ | |τ (I)| ≤ 2M . |I| |I0 |

Proof. We will first construct an auxiliary collection K of colored dyadic intervals. In each step of the construction we modify K by either changing the color of a particular I ∈ K, or by adding a new interval to K. We write K = K ∪ {L} when we decide to add the dyadic interval L to the already existing collection K. We shall now define two construction rules and a stopping rule. Following these rules we will eventually construct K and C. Rule 1. Suppose there exists a green interval I ∈ K such that at least one of the intervals in G1 (I, B) is not contained in K. Call it I1 . Now we define the color of I1 as follows: If |τ (K ∪ {I1 })∗ | |τ (I1 )| ≤ 2M , |I1 | |I0 | then we define I1 to be a green interval and put K = K ∪ {I1 }. If |τ (K ∪ {I1 })∗ | |τ (I1 )| > 2M , |I1 | |I0 | then we define I1 to be a red interval and put K = K ∪ {I1 }. Rule 2. If

Suppose I ∈ K is a red interval. |τ (K)∗ | |τ (I)| ≤ 2M , |I| |I0 |

then we redefine the color of I from red to green. (Otherwise we don’t change the color of I.) Rule 3.

Suppose that for each red interval I ∈ K we have |τ (I)| |τ (K)∗ | > 2M , |I| |I0 |

then we define C to be the collection of red intervals in K. Next we decree how these rules are to be applied.

212


We start by defining I0 to be a green interval and by putting K := {I0 }. Then we apply Rule 1 until for every green interval I ∈ K, the following condition holds true G1 (I, B) ⊆ K. Clearly in this case Rule 1 can’t be applied any longer. We now apply Rule 2 to the red intervals of K. If a new green interval is created by applying Rule 2, then we apply Rule 1, as above, and thereafter Rule 2 again, etc. If we don’t create new green intervals by applying Rule 2 to the red intervals of K, then the stopping criterion of Rule 3 holds true. Then finally we define C to be the red intervals in K. Let C ⊂ K ⊆ B ∩ I0 be the collections constructed by applying our three rules in the way described above. Now we will verify that C is a collection of pairwise disjoint intervals satisfying |I0 | . |L| ≤ 2 L∈C

We begin by checking that C is in fact a collection of pairwise disjoint dyadic intervals: Recall that during the construction, each interval in K has been colored. Note that only subintervals of green intervals can be placed into K by Rule 1. Hence, if I ∈ K is red and K is strictly contained in I, then K ∈ K. Therefore any two red intervals L, K ∈ K are necessarily disjoint. Hence, C is a collection of pairwise disjoint intervals. Next we show that C covers only a small fraction of I0 . First we note that by summing over intervals for which the stopping condition of Rule 3 is satisfied, we obtain the upper bound 1 1 |I| ≤ |τ (I)|. |I0 | 2M |τ (K)∗ | I∈C

I∈C

Second we recall that τ preserves the Carleson condition. Hence the collection τ (C) satisfies the M -Carleson condition, since C satisfies the 1-Carleson condition. Therefore using that τ (K) ⊇ τ (C), we obtain the estimate 1 1 |τ (I)| ≤ |τ (I)| ≤ M. ∗ ∗ |τ (K) | |τ (C) | I∈C

I∈C

Combining the last two estimates we obtain that I∈C

|I| ≤

|I0 | . 2

Next we verify that the collection of green intervals of K coincides with the collection L = B ∩ [Q(I0 ) \ Q(C)]. Recall that K ∈ K when K is strictly contained


213

in I ∈ C. Recall also that I0 ∈ K and if K ∈ K, then every interval L satisfying L ∈ B and I0 ⊇ L ⊇ K is also an element of K. Summing up we showed that B ∩ [Q(I0 ) \ Q(C)] = K \ C, or equivalently that L consists of the green intervals of K. Recall that I ∈ K is a green interval when |τ (K)∗ | |τ (I)| ≤ 2M . |I| |I0 | Clearly, |τ (K)∗ | ≤ |τ (L)∗ | + |τ (C)∗ |. Therefore we arrived at the following estimate holding for intervals I ∈ L, |τ (L)∗ | + |τ (C)∗ | |τ (I)| ≤ 2M . |I| |I0 |

This completes the proof of the lemma. Repeatedly applying Lemma 3.3.8 gives the proof of Theorem 3.3.7.

Proof of Theorem 3.3.7. Fix J ∈ D. Let G0 be the maximal intervals of τ −1 {τ (B) ∩ J}. Clearly G0 is a collection of pairwise disjoint intervals. Now starting with G0 we construct a collection G1 as follows. We fix I ∈ G0 , and apply Lemma 3.3.8 to the collection B ∩ I. This gives a collection of pairwise disjoint dyadic intervals CI , and LI = B ∩ [Q(I) \ Q(CI )], which satisfy the conditions

|L| ≤

L∈CI

and

|I| 2

|τ (LI )∗ | + |τ (CI )∗ | |τ (K)| ≤ 2M , |K| |I|

for K ∈ LI . Now we let G1 (CI , B) be the collection of maximal intervals of B which lie underneath CI . That is we put G1 (L, B). G1 (CI , B) = {L∈CI }

Clearly, we defined G1 (CI , B) to be a collection of pairwise disjoint intervals. Hence also the following is a collection of pairwise disjoint intervals, G1 (CI , B), G1 = I

here I ranges over G0 . This completes the first step of the proof.

214


Now we repeat the first step and construct G2 from G1 . We choose now K ∈ G1 , and we let CK , respectively LK , satisfy the conclusions of Lemma 3.3.8. Now we define G1 (L, B), G1 (CK , B) = {L∈CK }

and G2 =

G1 (CK , B),

K

where K ranges over G1 . Note that G2 is a collection of pairwise disjoint dyadic intervals, since G1 was. Continuing now inductively we construct the sequence {Gk }, and simultaneously, we obtain the decomposition τ −1 {τ (B) ∩ J} = LI ∪ CI , where I ranges over G = matically the identity

#∞

k=0 Gk .

Note that our recursive definitions give auto-

G1 (CI , B) =

I∈G

∞

Gk .

k=1

We will now verify that G satisfies the 2-Carleson # condition. Recall that Gk is a collection of disjoint intervals and that Gk+1 = G1 (CI , B) where I ranges over Gk . Hence, Gk+1 ⊂ Q(Gk ). (3.3.35) Note that by Lemma 3.3.8, for I ∈ Gk , the following estimate holds,

|K| ≤

|I| . 2

|K| ≤

|I| . 2l

K∈Gk+1 ∩I

Hence, by induction, for I ∈ Gk , K∈Gk+l ∩I

(3.3.36)

From these properties we can easily evaluate the Carleson constant of G. Indeed, by (3.3.36) and summing a geometric series we obtain that [[G]] ≤ 2. We will show next that the collections {LI : I ∈ G} satisfy the estimate 1 |τ (LI )∗ | ≤ 2M sup [[τ −1 (max τ (LI ))]]. |J| {I∈G} {I∈G}

To see why this holds we will examine the inductive construction of the collections {LI : I ∈ G}. Recall that LI ⊆ Q(I) \ Q(CI ). Hence LI ⊆ Q(Gk ) \ Q(Gk+1 ). (3.3.37) I∈Gk


215

Recall also that L∗I ∩ L∗K = ∅, when I, K ∈ Gk and clearly the following inclusion holds true, τ −1 (max τ (LI )) ⊆ LI . Now let Vk =

τ −1 (max τ (LI )),

where the union is taken over I ∈ Gk . By (3.3.35) – (3.3.37), the hypothesis of Proposition 3.1.5 is satisfied. Its BMO assertion gives [[ Vk ]] ≤ 2 max[[Vk ]]. Consequently we obtain that .. // −1 τ (max τ (LI )) ≤ 2 sup[[τ −1 (max τ (LI ))]]. I∈G

I∈G

With these observation we now obtain the estimates 44 55 1 |τ (LI )∗ | ≤ max τ (LI ) |J| I∈G 55 44 ≤M τ −1 (max τ (LI )) ≤ 2M sup [[τ −1 (max τ (LI ))]]. I

We now let EI = τ (CI ). We prove next that the union EI E= I∈G

satisfies # the 2M -Carleson condition. To do so we only need to combine the fact that CI is a subcollection of G, and the fact G satisfies the 2-Carleson condition. Indeed, as τ preserves the Carleson condition, we have the following string of estimates. [[E]] = [[τ (G)]] ≤ M [[G]] ≤ 2M. Finally we relabel the collections {CI , LI : I ∈ G} as {Ci , Li : i ∈ N}. We also let Ei = τ (Ci ), where the actual enumeration is not at all important. We have thus verified that the decomposition τ (B) ∩ J =

∞

τ (Li ) ∪

i=1

satisfies the conclusions of Theorem 3.3.7.

∞

Ei ,

i=1

216


Norm estimates for Um and Tm In Section 1.4 and Section 1.5 we presented Figiel’s work on biorthogonal systems and integral operators. We used there (without proof) that on Lp , 1 < p < ∞ the rearrangement operator Tm and the shift operator Um have norm bounded by Cp log2 (4 + |m|). In this subsection we verify these estimates and thereby complete the proof of Theorem 1.5.3. Let m ∈ Z, and define Rm = {I ∈ D : I + m|I| ⊆ [0, 1]}. In Figiel’s work the analysis of integral operators is based on the rearrangement respectively splitting operators defined by the equations Um (hI ) = 1I+m|I| − 1I

and

Tm (hI ) = hI+m|I| ,

for I ∈ Rm . The logarithmic dependence on m in the estimates for Tm and

Um was crucial in applying T.Figiel’s integral representation to the kernels of Calderon–Zygmund operators. The norm estimate for Tm as stated in Theorem 3.3.9 follows directly from E. M. Semenov’s criterion (3.3.1) on boundedness of rearrangement operators. We present an independent proof below for which we use just Burkholder’s Theorem 1.4.1 and E. M. Stein’s martingale inequality (Theorem 1.4.2). Thus we show that 1 1

xI Tm (hI ) pE ≤ Cpp log2 (4 + |m|)p

xI hI pE , (3.3.38) 0

0

for xI ∈ E whenever E is a Banach space with the UMD property. This conclusion can’t be drawn from the methods of the previous subsections, since they are strictly(!) limited to the scalar case. Estimates analogous to (3.3.38) hold with Tm replaced by Um . Theorem 3.3.9. On Lp (1 < p < ∞) the following norm estimates hold true,

Um Lp ≤ Cp log2 (4 + |m|)

and

Tm Lp ≤ Cp log2 (4 + |m|).

We prepare the proof by formulating a general criterion on τ : A → D ensuring that the linear extension of Uτ (hI ) = 1τ (I) − 1I ,

I ∈ A,

respectively Tτ (hI ) = hτ (I) , defines a bounded operator on Lp . Let A be a collection of dyadic intervals. We say that an injective map τ : A → D satisfies Figiel’s compatibility condition (F ) if the following three conditions hold: |τ (J)| = |J| for J ∈ A.

(F 1)

If I ∈ A, then τ (I) ∈ A.

(F 2)

C = {I, τ (I), I ∪ τ (I) : I ∈ A} is a nested collection of sets.

(F 3)


217

Lemma 3.3.10. Let τ : A → D satisfy Figiel’s compatibility condition (F ). On Lp (1 < p < ∞) the following norm estimates hold,

Uτ p ≤ Bp

and

Tτ p ≤ Kp ,

where Bp = C2(p∗ − 1)3 and Kp = C2(p∗ − 1)3 . Proof. Since, by hypothesis, the family C = {I, τ (I), I ∪ τ (I) : I ∈ A} forms a nested collection of sets, we may split it canonically into generations. Let Gn (C) denote the n-th generation of C. We define first a sequence of σ-algebras using the generations of the nested collection C as follows. For n ∈ N0 let Fn denote the σ-algebra generated by G0 (C) ∪ · · · ∪ Gn (C). Observe that for every I ∈ A there exists a uniquely determined n ∈ N so that the union I ∪ τ (I) is an atom for the σ-algebra Fn . Thus we define Kn = {I ∈ A : I ∪ τ (I) is an atom in Fn }. Hence for every I ∈ Kn , we obtain the identity 2E(1I |Fn ) = 1I + 1τ (I) .

(3.3.39)

Next we prove the estimate for ||Uτ ||p . Let {aI } be a sequence of scalars. Let {rI } be an enumeration of the Rademacher sequence. The assumption that C = {I, τ (I), I ∪ τ (I) : I ∈ A} is a nested collection implies that the functions {1τ (I) − 1I : I ∈ A} can be relabelled to become a martingale difference sequence. Since martingale differences are unconditional in Lp , by Burkholder’s Theorem 1.4.1, we get p p 1 ∗ p aI (1τ (I) − 1I ) ≤ (p − 1) aI rI (t)(1τ (I) − 1I ) dt. (3.3.40) 0 I∈A

I∈A

Lp

Lp

Recall (3.3.39) that 2E(1I |Fn ) = 1I + 1τ (I) . Hence 1τ (I) − 1I = 2E(1I |Fn ) − 2 · 1I .

(3.3.41)

The purpose of rewriting 1τ (I) −1I in (3.3.41) is that the conditional expectation of 1I appears on the right-hand side. Hence by Bourgain’s version of Stein’s inequality the right-hand side of (3.3.40) is bounded by p 1 aI rI (t)1I dt, (3.3.42) Cpp 0 I∈A

Lp

where Cp = 4(p∗ − 1)2 . Combining (3.3.40)–(3.3.42) and applying again Burkholder’s inequality we obtain that ||Uτ ||p ≤ Bp .

218


Next we turn to estimating ||Tτ ||p . Applying the same method of proof we relate Tτ (hI ) to a conditional expectation of 1I . We start with the Burkholder inequality (Theorem 1.1.5). p p 1 ∗ p aI hτ (I) ≤ (p − 1) aI rI (t)1τ (I) dt. (3.3.43) 0 I∈A

I∈A

Lp

Lp

By equation (3.3.39), the integral on the right-hand side of (3.3.43) is bounded by p p 1 1 ∞ p 2 aI rI (t)E(1I |Fn ) dt + aI rI (t)1I dt. (3.3.44) 0 0 n=0 I∈Kn

I∈A

Lp

Lp

Next, Bourgain’s version of Stein’s inequality (Theorem 1.4.2) asserts that p p 1 1 ∞ aI rI (t)E(1I |Fn ) dt ≤ (p∗ − 1)p aI rI (t)1I dt. 0 0 n=0 I∈Kn

I∈A

Lp

Summing up, the estimates (3.3.43)–(3.3.45) imply that ||Tτ ||p ≤ Kp .

Lp

(3.3.45)

By combinatorial considerations we reduce the norm estimates of Theorem 3.3.9 to those of the above lemma. This is done by decomposing Rm into roughly log(4 + m) families of dyadic intervals, in such a way that on each sub-family Figiel’s compatibility condition is satisfied for rearrangements closely related to the original one given by τ (I) = I + m|I|. Proof of Theorem 3.3.9. Let m ∈ Z. We first treat the case where |m| ≥ 2. By symmetry it suffices to consider only m ≥ 2. Below we give a separate and simple argument for the case |m| = 1. Let Rm = {I ∈ D : I + m|I| ⊆ [0, 1[}. We study in detail the action of the rearrangement τ (I) = I + m|I|, for I ∈ Rm . For n ∈ N let Ln = {I ∈ Rm : |I| = 2−n }. Next split Ln into four collections (1) (4) (i) Kn ,. . . , Kn so that for i ∈ {1, . . . , 4}, the following conditions hold. Let Kn (i) be the point-set covered by the collection Kn . I ∈ Kn(i)

dist(τ (I), Kn(i) ) ≥ |I|,

implies

and I, J ∈ Kn(i)

implies

dist(I ∪ τ (I), J ∪ τ (J)) ≥ |I|.

Since τ is of the simple form τ (I) = I + m|I|, it is straightforward to obtain the splitting of Ln into sub-collections with the above property. Define k ∈ N by the relation k − 1 ≤ log2 (m) ≤ k. Next we fix 0 ≤ j ≤ k, and take the union (i)

Ej =

∞ n=1

(i)

Kn(k+1)+j .


219 (i)

Thus we decomposed Rm into C log(4 + m) families of the form Ej . We drop the subscript j and superscript (i) and put (i)

E = Ej . We aim to show that the linear extension of the maps Um (hI ) = 1I+m|I| − 1I

and

Tm (hI ) = hI+m|I|

for I ∈ E,

are defining bounded operators on Lp with norm independent of m. Now we examine closely the collection E. Let I ∈ E, then |I| ≤ 2−k . By I (k) we denote the dyadic interval satisfying I ⊂ I (k) and |I (k) | = 2k |I|. Depending on the position of I in the unit interval the following cases may hold for τ (I) = I + m|I| : 1. τ (I) ⊆ I (k) . 2. τ (I) ⊆ I (k) . If this holds, then the following three, mutually exclusive subcases may occur: (a) Neither τ (I) nor I are touching the right endpoint of I (k) . In that case τ (I) and I are called good intervals. (b) I touches the right endpoint of I (k) . In that case we call I a bad interval. (c) τ (I) touches the right endpoint of I (k) . In that case we call τ (I) a bad interval. Note that for I ∈ E precisely one of the above conditions (1), (2.a), ( 2.b) or (2.c) are satisfied. Accordingly the collection E splits into the following four disjoint subcollections. A = {I ∈ E : τ (I) ⊆ I (k) } G = {I ∈ E : τ (I) ⊆ I (k) , and τ (I) and I are good intervals.} B1 = {I ∈ E : τ (I) ⊆ I (k) , I is a bad interval.} B2 = {I ∈ E : τ (I) ⊆ I (k) , τ (I) is a bad interval.} Case 1. Observe that by our definition of E and A every I ∈ A satisfies Figiel’s compatibility condition (F ). Hence, by Lemma 3.3.10, the linear extension of the maps Um (hI ) = 1I+m|I| − 1I

and

Tm (hI ) = hI+m|I|

for I ∈ A,

define bounded operators on Lp with norm independent of m.

220


Case 2a. Next we turn to the analysis of G1 and G2 so that the restriction of I → Figiel’s compatibility condition. With the and Lemma 3.3.10 we thus obtain that the Um (hI ) = 1I+m|I| − 1I

and

the collection G. We will split G into I + m|I| to Gi , i ∈ {1, 2}, satisfies unconditionality of the Haar system linear extension of

Tm (hI ) = hI+m|I|

for I ∈ Gi ,

define bounded operators on Lp . Now we turn to defining the splitting G = G1 ∪ G2 . Fix K ∈ G ∪ τ (G) and J ∈ G with |J| < |K|. Denote by co[J, τ (J)] the convex hull of the intervals J and τ (J). We let L(K) be those intervals J ∈ G for which co[J, τ (J)] intersects both K and the complement of K. Thus L(K) = {J ∈ G : |J| < |K|, co[J, τ (J)] ∩ K = ∅, and co[J, τ (J)] ∩ ([0, 1] \ K) = ∅}. Finally we let C(K) = {J, τ (J) : J ∈ L(K)}. Due to the defining properties of G the collection C(K) is a collection of pairwise disjoint dyadic intervals. Moreover it satisfies the following remarkable properties. (a) For each J ∈ G ∪ τ (G) there exists at most one K ∈ G ∪ τ (G) so that J ∈ C(K). Thus each J ∈ C(K) determines uniquely K and hence C(K). (b) If L ∈ G ∪ τ (G) and L ∈ C(K), then L ∪ τ (L) ⊆ K, or L ∪ τ (L) ⊆ [0, 1] \ K.

(3.3.46)

Exploiting the above properties of C(K) allows us to define a two-coloring of G ∪ τ (G) so that the following conditions hold. (a) For each K ∈ G ∪ τ (G) the collection C(K) is monochromatic. (b) If the color of K ∈ G ∪ τ (G) is already determined, then each J ∈ C(K) satisfies color(K) = color(J). (c) For each K ∈ G, color(K) = color(τ (K)). Assume that we used the colors black and white to describe the two-coloring of G. Then we define G1 = {K ∈ G : color(K) = white } and G2 = {K ∈ G : color(K) = black }.


221

Thus we obtained a splitting G = G1 ∪ G2 so that G1 and G2 satisfy Figiel’s compatibility condition. By Lemma 3.3.10 for i ∈ {1, 2}, the linear extension of hI → 1I+m|I| − 1I

I ∈ Gi ,

for

and hI → hI+m|I|

for

I ∈ Gi ,

p

gives bounded operators on L . Case 2b. Next we turn to the analysis of the collection B1 . By reduction to the previous case we will show that the linear extension of the map Um (hI ) = 1I+m|I| − 1I

for

I ∈ B1 ,

defines a bounded operator on Lp . Let I ∈ B1 . Recall that then, by definition, the right endpoint of I coincides with the right endpoint of I (k) , and that the interval τ (I) is well inside of τ (I)(k) , satisfying dist(τ (I), [0, 1] \ τ (I)(k) ) ≥ |I|. Next we let J be the dyadic predecessor of I. Thus J is a dyadic interval such that J ⊇ I and |J| = 2|I|. Let J1 be the left half of J and let J2 be the right half of J. Our assumption that the right endpoint of I coincides with the right endpoint of I (k) , implies that the interval I is the right half of its dyadic predecessor. Thus I = J2 . Next we define an injective map σ by interchanging J2 and J1 . Hence, σ(I) = σ(J2 ) = J1

and

σ(J1 ) = J2 = I.

Observe that σ : B1 → D satisfies Figiel’s compatibility condition (F ). Hence by Lemma 3.3.10 the linear extension of the map S(hI ) = hσ(I)

for

I ∈ B1

defines a bounded operator in L . Let R = σ(B1 ), and define ρ = τ ◦ σ. It is easy to see that for K ∈ R the intervals ρ(K) and K are good intervals with respect to the map ρ = τ ◦ σ. Consequently, by case 2a, the linear extension of p

R(hK ) = 1ρ(K) − 1K

for

K∈R

is bounded on Lp . Hence the composition RS is a well-defined and bounded linear operator on Lp . As σ(σ(I)) = I the composition RS coincides with the linear extension of the map Um (hI ) = 1I+m|I| − 1I

for

I ∈ B1 .

A moment’s reflection yields that simultaneously we proved that the same holds for the map Tm (hI ) = hI+m|I| for I ∈ B1 .

222


Case 2c. It remains to analyze the contribution of the collection B2 . Again we proceed by reduction to case 2a. Recall that I ∈ B2 if τ (I) ⊆ I (k) , so that τ (I) touches the right endpoint of I (k) . Precisely, the right endpoint of I (k) coincides with the left endpoint of τ (I). Define a new rearrangement by shifting the interval τ (I) to the right by an amount of |I|. Thus for I ∈ B2 we let τ1 (I) = τ (I) + |I|. Note that for I ∈ B2 the union of τ (I) and τ1 (I) is a dyadic interval. Define P (τ (I)) to be the dyadic predecessor of τ (I). Note that for I ∈ B2 we have that P (τ (I)) = τ (I) ∪ τ1 (I). Observe that τ1 (I) is well inside of τ (I)(k) . That is, dist(τ1 (I), [0, 1] \ τ (I)(k) ) ≥ |I|. On the other hand for I ∈ B2 we have that, by definition, dist(I, [0, 1] \ I (k) ) ≥ |I|. Thus for I ∈ B2 the intervals τ1 (I) and I are good intervals with respect to the rearrangement τ1 . By the estimate of case 2a, the linear extension of the shift hI → 1τ1 (I) − 1I

I ∈ B2

for

extends to a bounded operator on Lp , 1 < p < ∞. For the same reason the rearrangement operator defined by hI → hτ1 (I)

for

I ∈ B2

extends boundedly to Lp , 1 < p < ∞. Notice that the map hτ1 (I) → hP (τ (I)) for I ∈ B2 extends to a bounded linear operator on Lp . Next observe that for I ∈ B2 the following identity holds, 1τ (I) = 1τ1 (I) + hP (τ (I)) , and consequently

" ! 1τ (I) − 1I = 1τ1 (I) − 1I + hP (τ (I)) .

(3.3.47)

We observed that the two terms on the right-hand side of (3.3.47) represent maps that extend to a bounded linear operator on Lp , 1 < p < ∞. Thus we showed that Um (hI ) = 1τ (I) − 1I

for

I ∈ B2

defines a bounded linear operator on Lp , 1 < p < ∞.


223

The case |m| = 1. We now analyze Tm and Um when |m| = 1. We present also the estimates for the rearrangement operators since we are still interested in having a proof for vector-valued coefficients taken from a UMD space. The scalar case is of course covered by E. M. Semenov’s criterion (3.3.1). By symmetry it suffices to consider m = 1. Let R1 = {I ∈ D : I + |I| ⊆ [0, 1]}. For I ∈ R1 let I (1) denote the dyadic predecessor of I. Next split R1 as follows, A = {I ∈ R1 : I + |I| ⊆ I (1) } and

B = R1 \ A.

Observe that the rearrangement I → I + |I| for I ∈ A satisfies Figiel’s compatibility condition, hence the linear extension of the maps T1 (hI ) = hI+|I|

and

U1 (hI ) = 1I+|I| − 1I

for I ∈ A

define bounded operators on Lp (1 < p < ∞). Now we consider the collection B = R1 \ A. Define the map σ by shifting the intervals of B to the left by one unit, thus σ(I) = I − |I|. Note that σ : B → D satisfies Figiel’s compatibility condition, hence the linear extension of S(hI ) = hσ(I) for I ∈ B defines a bounded operator on Lp . Moreover the following identity holds, T1 (hI ) = T2 (S(hI )) for I ∈ B. Consequently T1 extends to a bounded operator since T2 is bounded and S extends. Finally we analyze U1 on B. The starting point is the identity 1I+|I| = 1I+2|I| + hI (1) +|I (1) |

for I ∈ B,

(3.3.48)

where I (1) denotes the dyadic predecessor of I. Recall that U1 (hI ) = 1I+|I| − 1I and U2 (hI ) = 1I+2|I| − 1I . Next subtract on both sides of (3.3.48) the function 1I . Then rewrite (3.3.48) as U1 (hI ) = U2 (hI ) + hI (1) +|I (1) |

for I ∈ B.

(3.3.49)

Finally hI (1) +|I (1) | = T1 (hI (1) ), hence U1 (hI ) = U2 (hI ) + T1 (hI (1) )

for I ∈ B.

(3.3.50)

The right-hand side of (3.3.50) consists of maps that extend to bounded linear operators on Lp , hence the same is true for the left-hand side.

224


Rearrangement operators on L1 We close this section with a discussion of rearrangement operators acting in L1 . We begin with an example showing that boundedness in L1 requires a much more restrictive condition on the underlying permutation than boundedness in Lp , 1 < p < ∞. Let In = [1/2, 1/2 + 2−n ), n ∈ N. We define a rearrangement τ first on the even numbered intervals {I2n : n ∈ N}, 1 1 τ (I2n ) = [ , − 2−2n ). 2 2 Thus τ reflects the even numbered intervals at the point 1/2. Next we define that τ leaves invariant the dyadic intervals J ∈ / {I2n : n ∈ N}, that is J∈ / {I2n : n ∈ N}.

τ (J) = J,

Now we show that τ defines an unbounded operator on L1 . Let M ∈ N and put f=

M

hIn |In |−1 .

n=1

We have ||f ||1 ≤ 2. On the other hand a moment’s reflection shows that ||

M

hτ (In ) |In |−1 ||1 ≥ M/2.

n=1

Thus the induced rearrangement operator Tτ is unbounded on L1 . By contrast, Theorem 3.3.4 shows clearly that ||Tτ ||p ≤ Cp , when 1 < p < ∞. It would be interesting to analyze the L1 case more closely and to describe the bounded rearrangement operators on L1 . As an introduction to this open problem we discuss now the work of N. Kislovskaya and V. Osipov [114] who solved a special case. We consider rearrangements τ which satisfy the additional hypothesis that |τ (I)| = |I|. Their result shows that there are just two possibilities for how a rearrangement satisfying |τ (I)| = |I| can induce a bounded operator on L1 . Roughly speaking they obtain that either τ is a reflection or τ is a translation. In the analysis given below these cases are represented by green, respectively red, colors attached to the intervals in D. Black intervals also appear and they indicate that the rearrangement had done something very bad there. With the theorem of N. Kislovskaya and V. Osipov [114] we may read off the correct upper and lower estimates for ||Tτ ||L1 , from the number of non-empty generations of black intervals obtained by the following coloring process. Before we define the coloring algorithm in detail we will summarize its essential properties: Let B denote the collection of black intervals obtained, and let Bn = Gn (B)


225

# be the the n-th generation of B. Let Q(Bn ) = K∈Bn {J ∈ D : J ⊆ K}. Then define Gn = Q(Bn ) \ Q(Bn+1 ). Note that the maximal intervals of Gn is just the collection Bn . Removing Bn from Gn we form the collection Ln , that is we put Ln = Gn \ Bn . The coloring will be such that each of the collections Ln is monochromatic: Either all intervals in Ln are red or all intervals in Ln are green. Next we point out that by the definition (given below) of the coloring algorithm, the restriction of the rearrangement operator to Ln is a bounded operator. We start by calling the unit interval [0, 1] a black interval. Now suppose that J ∈ D has been colored. Black, red or green are the colors possible for J. We let J1 be the left half of J, and J2 be the right half of J. We assume that J1 or J2 have not yet been colored. First we formulate a simple rule which attaches the color black to the interval Ji : If τ (Ji ) is not contained in τ (J) then Ji is a black interval. We call this rule simple, since the color of Ji is determined independently of the color of J. The following rules define the color of J1 , J2 depending on the color of J. There are three possible colors for J, hence there are three set of rules. Note that by the simple rule we only have to consider the cases when τ (Ji ) ⊆ τ (J). (Since otherwise the color of Ji is already determined to be black.) The following rules determine the color of Ji when τ (Ji ) ⊆ τ (J). The color depends on the position of τ (Ji ) inside τ (J). The only possibilities are that τ (Ji ) is the right half of τ (J), or that τ (Ji ) is the left half of τ (J). Case 1. J is a black interval: In this case the colors of J1 , J2 can be green or red. We start with the color of J1 , using the following set of rules. 1. If τ (J1 ) is the left half of τ (J), then J1 is a green interval. 2. If τ (J1 ) is the right half of τ (J), then J1 is a red interval. Now we define the set of rules which determine the color of J2 . 1. If τ (J2 ) is the left half of τ (J), then J2 is a red interval. 2. If τ (J2 ) is the right half of τ (J), then J2 is a green interval. Case 2. J is a green interval: In this case the colors of J1 , J2 can be green or black. We start with the color of J1 , using the following set of rules. 1. If τ (J1 ) is the left half of τ (J), then J1 is a green interval.

226


2. If τ (J1 ) is the right half of τ (J), then J1 is a black interval. Now we define the set of rules which determine the color of J2 . 1. If τ (J2 ) is the left half of τ (J), then J2 is a black interval. 2. If τ (J2 ) is the right half of τ (J), then J2 is a green interval. Case 3. J is a red interval: In this case the colors of J1 , J2 can be red or black. We start with the color of J1 , using the following set of rules. 1. If τ (J1 ) is the left half of τ (J), then J1 is a black interval. 2. If τ (J1 ) is the right half of τ (J), then J1 is a red interval. Now we define the set of rules which determine the color of J2 . 1. If τ (J2 ) is the left half of τ (J), then J2 is a red interval. 2. If τ (J2 ) is the right half of τ (J), then J2 is a black interval. Having defined the color of each dyadic interval, we let B be the collection of all black intervals, and we let Bn = Gn (B) be the generations of B. Now we can decide whether the operator Tτ : L1 → L1 is bounded or not. Theorem 3.3.11. Let τ : D → D be a rearrangement satisfying |τ (I)| = |I|. The rearrangement operator Tτ : L1 → L1 is bounded if the number of non-empty generations of B is finite. Moreover N −1 ≤ ||Tτ || ≤ 2N, 2 where N is the smallest number such that GN (B) = ∅. Proof. Let n ≤ N. Fix J ∈ Bn . Then define a block of intervals, G(J) = Q(J) \ Q(Bn+1 ∩ J). Next observe that once we remove the black interval J from the block G(J), then the entire remaining collection is monochromatic. In other words the intervals G(J) \ J are either all red or all green. If they are all green, the collection τ (G(J)) is again a block of intervals and more importantly, on G(J), the rearrangement τ acts by translation, and there exists a ∈ R, such that τ (I) = a + I,


227

for I ∈ G(J). Now consider the case when the intervals in G(J) \ J are all red. In that case τ acts by first translating, then reflecting and then translating again. In either case we have the identity hτ (I) aI ||L1 = || hI aI ||L1 . || G(J)

G(J)

Next, as G(J) is a block of intervals, we obtain by conditional expectation, ||

hI aI ||L1 ≤ 2||f ||L1 .

J∈Bn G(J)

Now we show the boundedness of Tτ . Let f ∈ L1 with Haar expansion f = Then we estimate as follows, ||Tτ (f )||

L1

≤

N

||

=

||

n=1 J∈Bn

=

N n=1

||

aI h I .

hτ (I) aI ||L1

G(J)

n=1 J∈Bn N

hI aI ||L1

G(J)

hI aI ||L1

J∈Bn G(J)

≤ 2N ||f ||L1 . Next we show that ||Tτ ||L1 ≥ (N − 1)/4. Here we will find f ∈ L1 , with ||f ||1 = 2 such that ||Tτ (f )||L1 ≥ (N − 1)/2. Fix J ∈ BN −1 . By assumption BN −1 is nonempty. Next we let U(J) be the collection of dyadic intervals which contain the interval J, that is U(J) = {K ∈ D : K ⊇ J}. We let {In } be an enumeration of the intervals in U(J) such that |In | = 2−n . Define f= hIn 2n sn , where sn ∈ {−1, 1} is given as follows. If In is the left half of In−1 , then sn = 1, otherwise sn = −1. With this choice of sn there is cancellation in the sum defining f and ||f ||L1 ≤ 2. Now we analyze the action of τ on U(J). First note that for any 1 ≤ k ≤ N − 1, the intersection U(J) ∩ Bk consists of exactly one interval. Next we write Jk = U(J) ∩ Bk . Then we define Uk to be the collection of dyadic intervals I satisfying, Jk ⊂ I ⊆ Jk+1 . Observe that Uk \ {Jk+1 } is again monochromatic, and, incidentally, the

228


collections {Uk } define the largest monochromatic decomposition of U(J). Now let hI fk = f, hI . |I| I∈Uk Then ||Tτ (fk )||L1 = ||fk ||L1 ≥ 1, and f = fk . And moreover we have the following crucial lower bound, ||Tτ (f )||L1 ≥

N −1 1 ||Tτ (fk )||L1 . 2 k=1

(The best way to see this lower bound is to reconsider the example which marked the beginning of our discussion on rearrangement operators in L1 .) Summing up our observations we obtain the following lower bound for ||Tτ (f )||L1 . ||Tτ (f )||L1 ≥

N −1 1 ||Tτ (fk )||L1 2 k=1

≥ ≥

3.4

1 2

N −1

||fk ||L1

k=1

1 (N − 1). 2

Notes

Sources for Section 3.1 are L. Carleson and J. Garnett [46] and J. Garnett and P.W. Jones [74]. See also Chapter VII in the book by J Garnett [72]. Proposition 3.1.6 is from [149]. Section 3.2 is based entirely on the work of P.W. Jones in [102]. The systematic investigation of rearrangement operators started with E. M. Semenov [190] who treated rearrangements satisfying |τ (I)| = |I|. Further extensions were obtained by E. M. Semenov and B. St¨ ockert [191]. These results are presented in detail in the monograph [165] by I. Novikov and E. M. Semenov. In [188] F. Schipp obtained E. M. Semenov’s theorem from the corresponding BMO result and interpolation. In Section 3.3 the treatment of general rearrangements in Lp and BMO follows [156]. The proof of Proposition 3.3.2 is extracted from P.W. Jones’s work [98] on Carleson measures and the Fefferman–Stein decomposition of BMO(R). The norm estimates for the operators Um and Tm in Theorem 3.3.9 are due to T. Figiel [65]. The use of Stein’s martingale inequality in the proof of Theorem 3.3.9 replaces two algebraic identities of T. Figiel’s original proof presented in Section 6 of [65]. Rearrangements in L1 are treated by N. Kislovkaya and V. Osipov in [114]. In particular Theorem 3.3.11 is taken from [114]. Our discussion follows their approach.

Chapter 4

Martingale H 1 Spaces On the probability space (Ω, F, | · |) we are given an increasing sequence of σalgebras F0 , . . . , Fn , . . . which generate the σ-algebra F. Then the martingale space, H 1 [(Fn )], consists of integrable f : Ω → R, with integrable square function, that is, 1/2 2 2 E(f |F0 ) + |E(f |Fn ) − E(f |Fn−1 )| < ∞.

f H 1 [(Fn )] = n

In this chapter we will be treating the case when each Fn is generated by a finite collection of pairwise disjoint sets An . Under this hypothesis the resulting family of Banach spaces H 1 [(Fn )] will be classified. We show that there are exactly three different isomorphic representatives for this family. In Section 4.1 we present important work of B. Maurey who established the isomorphism between the martingale H 1 [(Fn )] spaces and the class of X[E] spaces, consisting of three-valued martingale differences. The method presented there is quite explicit. In Section 4.2 we analyze the spaces X[E] and show that their isomorphic representation is given by the Banach spaces H 1 , ( Hn1 )1 or 1 .

4.1

Maurey’s isomorphism

Maurey’s Isomorphism provides the equivalence of the martingale spaces H 1 [(Fn )], to the spaces X[E]. We recall the definition of X[E] as given in Chapter 2. We say that E is a nested collection of measurable sets if for A, B ∈ E, A ∩ B = ∅ implies A ⊆ B or B ⊆ A. Let {rA : A ∈ E} be an enumeration of the independent Rademacher functions, and form the three-valued basis functions dA (w, t) = 1A (w)rA (t),

(4.1.1)

Chapter 4. Martingale H 1 Spaces

230

where A ∈ E. Then X[E] is the space of functions f = aA dA for which 1/2 ||f ||X[E] = a2A 1A < ∞. The aim of this section is to prove the following theorem of B. Maurey [144]. Theorem 4.1.1. Let {Fn : n ∈ N} be a sequence of increasing σ-algebras, where each Fn is generated by finitely many atoms. Then there exists a nested collection of measurable sets E such that H 1 [(Fn )] and X[E] are isomorphic Banach spaces. Moreover the nested collection E satisfies the following condition. If A, B ∈ E and if A is strictly contained in B, then |A| ≤ |B|/2. As a corollary to Maurey’s theorem we obtain that H 1 [(Fn )] has an unconditional basis. The isomorphism between H 1 [(Fn )] and X[E], will be explicitly constructed below. It will provide us with a detailed pictorial description of the unconditional basis in H 1 [(Fn )]. The proof of Maurey’s theorem blends a hard combinatorial construction and analytic estimates based on martingale inequalities. Combinatorial problems arise when we construct a local isomorphism that acts on the martingale differences between Fn and Fn−1 , for a fixed n ∈ N. Addressing these problems forms the first step in the construction of Maurey’s isomorphism. In the second step we will sum the local operators, and then we use martingale transform techniques to show that the resulting series defines an isomorphism between H 1 [(Fn )] and X[E].

Operations on martingale differences We begin with the construction of the local isomorphism. Clearly if we restrict the attention to a fixed atom of Fn−1 , then the martingale differences at that level appear as Fn measurable functions with mean zero. Let An−1 denote the atoms of Fn−1 . Let A ∈ An−1 , and let L◦ (A) denote the finite dimensional space of all functions f : A → R which are measurable with respect to Fn ∩ A, and for which E(f ) = 0. Let A∗ ⊂ A be an atom in An , satisfying |A∗ | =

max

B∈An B⊆A

|B|.

Then L∗ (A) denotes the space of functions f : A → R which are Fn ∩A measurable, and which vanish on A∗ . Note that L◦ (A) and L∗ (A) are finite dimensional and of the same algebraic dimension. Below we need a linear bijection TA : L∗ (A) → L◦ (A), which is an isomorphism for both the L1 -norm and the L2 -norm (with constants independent of the dimensions). The existence of such an isomorphism is the content of the next proposition. Proposition 4.1.2. There exists a linear bijection between TA : L∗ (A) → L◦ (A) which satisfies 1 ||f ||1 ≤ ||TA f ||1 ≤ 6||f ||1 16

4.1. Maurey’s isomorphism and

231

1 ||f ||2 ≤ ||TA f ||2 ≤ 6||f ||2 , 16

for f ∈ L∗ (A). Proof. Ones first guess for such an operator might be T f = f − E(f )

1 A∗ . |A∗ |

T maps L∗ (A) bijectively onto L◦ (A) and satisfies the correct L1 estimates. However the L2 bounds for T depend on the ratio |A|/|A∗ |, and hence for the case where |A∗ | |A| the operator T is just not the right one. A very careful adjustment is required. We proceed by defining the operators TA . The first step consists in defining an auxiliary shift operator DA and a carefully chosen sigma algebra CA . Defining the shift operator. Assume that |A∗ | ≤ |A|/4. Then there exist pairwise disjoint sets F1 , . . . , Fm in Fn ∩ A such that F1 ∪ · · · ∪ Fm = A \ A∗ and so that |A|/2 ≤ |F1 | ≤ |A|, |Fi |/2 ≤ |Fi+1 | ≤ |Fi |, |Fm |/4 ≤ |A∗ | ≤ 2|Fm |.

for i ≤ m − 1,

Next put Fm+1 = A∗ and let CA be the sub σ-algebra of Fn ∩ A generated by F1 , . . . , Fm+1 . For a function g, measurable with respect to CA , we denote its value on Fi by g(Fi ). Next we define the L1 normalized shift operator DA as follows. DA (g)(t) = 0 for t ∈ F1 , DA g(t) = g(Fi−1 )

|Fi−1 | |Fi |

for t ∈ Fi

and 2 ≤ i ≤ m + 1.

Note that DA maps CA measurable functions into CA measurable functions. Defining TA . For f ∈ L∗ (A), we define TA (f ) depending on the ratio |A∗ |/|A|. If |A∗ |/|A| ≥ 14 , then 1A ∗ TA f = f − E(f ) ∗ . |A | If |A∗ |/|A| ≤ 14 , define TA f = f − E(f |CA ) + DA E(f |CA ) − E(f )

1F1 . |F1 |

By definition, the operator TA is a linear bijection from L∗ (A) onto L◦ (A). To prove the norm estimates we distinguish between the cases |A∗ |/|A| ≥ 1/4 and |A∗ |/|A| ≤ 1/4.


232

Norm estimates for TA . In the case when |A∗ |/|A| ≥ 1/4 we only need to observe that f ∈ L∗ (A) is supported on A \ A∗ and that E(f )1A∗ /|A∗ | is supported on A∗ . Now we turn to the case |A∗ |/|A| ≤ 1/4. We analyze the operator TA in two steps. First we consider the main part of TA , namely U f = f − E(f |CA ) + DA (E(f |CA )). Note that E(U f |CA ) = DA (E(f |CA )) and that ||E(f |CA )||p ≤ 2||DA (E(f |CA ))||p , for 1 ≤ p ≤ 2. Then we write f = U f + E(f |CA ) − DA (E(f |CA )). The above observations give that ||f ||p ≤ ||U f ||p + 3||DA (E(f |CA ))||p ≤ ||U f ||p + 3||E(U f |CA )||p ≤ 4||U f ||p .

(4.1.2)

On the other hand we have, by the triangle inequality, ||U f ||p ≤ 2||f ||p + ||DA E(f |CA )||p ≤ 4||f ||p .

(4.1.3)

Now we study the interaction between U f and E(f )1F1 /|F1 |. We observe that by construction the mean value of U f over the set F1 vanishes, that is F1 U f = 0. We will now give a lower bound for the Lp norm of TA f = U f − E(f )1F1 /|F1 |. Fix p ∈ [1, 2] and choose h ∈ Lq with ||h||q = 1, such that ||U f ||p = E(U f · h). Next we let 1F g = h − E(h1F1 ) 1 , |F1 | then ||g||q ≤ 4||h||q , E(U f · g) = E(U f · h) and E(g1F1 ) = 0. It follows that 1 E(TA (f ) · g) 4 1 = (E(U f · g) + E(g · 1F1 )E(f )/|F1 |) 4 1 = E(U f · h). 4

||TA f ||p ≥

Thus ||TA (f )||p ≥ (1/4)||U (f )||p , and by estimates (4.1.2) for the operator U we have ||TA f ||p ≥ ||U f ||p /4 ≥ ||f ||p /16. On the other hand by (4.1.3) the following upper bound holds for the Lp norm of TA , ||TA f ||p ≤ ||U f ||p + 2||f ||p ≤ 6||f ||p .

4.1. Maurey’s isomorphism

233

Martingale transform techniques Next we define a nested collection of measurable sets E, and construct the isomorphism between X[E] and H 1 [(Fn )]. Let us fix A ∈ An−1 and let A∗ denote an atom in An such that A∗ ⊂ A and each B ∈ An , with B ⊂ A satisfies |B| ≤ |A∗ |. Clearly such an A∗ ∈ An exists for each A ∈ An−1 , but in general it is not unique. We let (4.1.4) E(A) = {B ⊂ A : B ∈ An } \ A∗ , and define the nested collection of sets E=

∞

E(A).

n=1 A∈An−1

Note that the nested collection E satisfies the following property which becomes crucial later, when we determine the isomorphic type of X[E]. If A, B ∈ E and A ⊂ B, then |A| ≤ |B|/2. Now we construct the isomorphism between X[E] and H 1 [(Fn )]. Let T be defined by its action on the unconditional basis {dB : B ∈ E} as follows: For n ∈ N, A ∈ An−1 and B ∈ E(A), put T (dB ) = TA (1B ). The next theorem specifies the isomorphism whose existence is claimed in Theorem 4.1.1. Theorem 4.1.3. The isomorphism between X[E] and H 1 [(Fn )] is given by the operator Tf = aB TA (1B ), n∈N A∈An−1 B∈E(A)

where f=

aB dB .

n∈N A∈An−1 B∈E(A)

Proof. The proof of Maurey’s theorem is long and hard. We therefore break it up into six connected components of manageable size. We begin by isolating special properties of T. Local L1 estimates. We fix n ∈ N and consider the n-th level in the basis expansion of f ∈ X[E]. For A ∈ An−1 let E(A) be given by (4.1.4). Then put δn =

A∈An−1 B∈E(A)

aB dB .


234 On δn the operator T acts as follows, T δn =

⎛

TA ⎝

A∈An−1

⎞ aB dB ⎠ .

B∈E(A)

Observe that the family of support sets ⎫ ⎧ ⎛ ⎞ ⎬ ⎨ aB dB ⎠ : A ∈ An−1 supp TA ⎝ ⎭ ⎩ B∈E(A)

is a collection of pairwise disjoint, Fn -measurable sets. This holds since the support of TA (dB ) is contained in A, for each B ∈ E(A). Hence we obtain the following identity for the L1 norm of T δn , ⎛ ⎞ TA ⎝ ⎠ a d ||T δn ||1 = B B . A∈An−1 B∈E(A) 1

By Proposition 4.1.2, the operator TA is an L1 isomorphism (from its domain onto its range), and therefore 1 ||T δn ||1 ≤ a d B B ≤ A1 ||T δn ||1 . A1 A∈An−1

B∈E(A)

1

# # Here A1 = 16. It remains to observe that the collection A∈An−1 B∈E(A) {B} is a collection of pairwise disjoint elements in E and as supp dB = B × [0, 1] we have aB dB ||δn ||1 = . A∈An−1 B∈E(A) 1

1

This proves the following local L estimates for T : ||δn ||1 ≤ ||T δn ||1 ≤ A1 ||δn ||1 , A1

(4.1.5)

with A1 = 16. Global L2 -estimates. Let n ∈ N. Define En = E(A) where E(A) = {B ⊂ A : B ∈ An } \ A∗ . A∈An−1

We consider as before, δn =

B∈En

aB dB ,

(4.1.6)


235

for n ∈ N. Clearly the sequence {δn } is biorthogonal in the Hilbert space L2 (Ω × [0, 1]). Observe that the operator T maps {δn } to a biorthogonal sequence in L2 (Ω). Indeed, by Proposition 4.1.2, T δn is Fn measurable and E(T δn |Fn−1 ) = 0, hence {T δn } is a martingale difference sequence, hence biorthogonal. Again by Proposition 4.1.2, the operator T satisfies local L2 estimates, ||δn ||2 ≤ ||T δn ||2 ≤ A2 ||δn ||2 . A2

(4.1.7)

We can take A2 = 16 again. Next we combine the local L2 -estimates (4.1.7) and biorthogonality, to obtain global L2 -estimates as follows. ∞ ∞ 2 δ = ||δn ||22 n n=1

n=1

2

≤ A22

∞ n=1

=

||T δn ||22

A22 T

2 δn . 2

Also the converse holds. With the same proof we obtain, ∞ ∞ δn ≤ A 2 δn . T n=1

n=1

2

2

T acts like a martingale transform. Let g : Ω → R be Fn−1 measurable, and let B ∈ E(A) for A ∈ An−1 . Then by construction of T we have the crucial pointwise identity (4.1.8) g(w)dB (w, t) = T −1 (gT dB )(w, t). An elementary inequality. We continue by provinga nice, elementary inequalm ity. Let {an } be positive real numbers and sm = n=1 an , then the following inequality holds, m 1 −1/2 s an , m ∈ N. (4.1.9) s1/2 m ≥ 2 n=1 n For the proof we put s0 = 0 and estimate using integrals. sm 2s1/2 = t−1/2 dt m 0

= ≥

m n=1 m n=1

sn

t−1/2 dt

sn−1

s−1/2 {sn − sn−1 }. n


236

After the preliminary observations we will show next that T is bounded. Later we prove that T −1 is bounded. Thus we will establish that ||T || · ||T −1 || < ∞. T is bounded. Recall T : X[E] → H 1 [(Fn )] acts on the space of three-valued martingale differences. We let f ∈ X[E] and expand, f= δn , where

δn =

aB dB

B∈En

and where En is given by (4.1.6). We perform now a canonical decomposition of δn by splitting the index set En as follows. For B ∈ En we say that B belongs to Cn iff on the support of dB , the following holds, 4

n−1

δk2 ≤ a2B d2B .

(4.1.10)

k=1

Let αn =

aB dB ,

B∈Cn

and split δn as follows, δn = αn + βn . Next put h=

αn ,

and decompose f = g + h. We continue giving separate estimates for 1T h = T αn and T g = T βn . We start with estimates for the norm of T h in H [(Fn )]. Let hn =

n

αk .

k=1

For hn the associated square function is given as

S(hn ) =

n

1/2 αk2

.

k=1

By (4.1.10) we have the crucial estimates S(hn ) ≥

1 |αn | + S(hn−1 ). 2


237

Summing the telescoping series gives that ∞

|αn | ≤ 2S(h) − 2S(h0 )

(4.1.11)

n=1

≤ 2S(f ). The local L1 estimates for T in combination with (4.1.11) yield upper bounds for T h. Indeed, as T αn is Fn measurable and E(T αn |Fn−1 ) = 0, ||T h||H 1 [(Fn )] = ( (T αn )2 )1/2 , (4.1.12) and

( (T αn )2 )1/2 ≤ |T αn | |αn |. ≤ A1

(4.1.13)

By (4.1.12), (4.1.13) and (4.1.11) we obtain

T h H 1 [(Fn )] ≤ 2A1 f X[E] .

(4.1.14)

Next we will estimate T g where g=

∞

βk .

k=1

Here we will use global L2 estimates for T together with martingale transform techniques. Let k

fk =

δ ,

and S(fk ) =

k

=1

1/2 δ2

.

=1

We begin with the following pointwise estimate for square functions, which uses just that S(fk−1 ) ≤ S(f ), ∞

(T βk )2 ≤ S(f )

k=1

∞

(T βk )2 S(fk−1 )−1 .

k=1

2 1/2 As ||T g||H 1 [(Fn )] = ( ∞ we obtain the following preliminary estik=1 (T βk ) ) mate using the inequality of Cauchy-Schwarz,

||T g||H 1 [(Fn )] ≤

∞

(T βk ) S(fk−1 )

k=1

2

−1

1/2

1/2 S(f )

.


238

We will now show the following bound for the first factor on the right-hand side, ∞ 2 −1 2 (T βk ) S(fk−1 ) ≤ 2A2 S(f ). k=1

By construction S(fk−1 ) is measurable with respect to Fk−1 . By (4.1.8), T acts like a martingale transform and we have the identity (T βk )S(fk−1 )−1/2 = T (βk S(fk−1 )−1/2 ). Global L2 bounds for T give then the estimate ∞ (T βk )2 S(fk−1 )−1 ≤ A22 βk2 S(fk−1 )−1 .

(4.1.15)

k=1

Note that by the negation of (4.1.10) βk was defined to satisfy the pointwise estimate k β2 )1/2 ≤ 3S(fk−1 ). ( =1

Inserting this in the right-hand side of (4.1.15) gives ∞

(T βk )2 S(fk−1 )−1 ≤ 3A22

∞

k=1

βk2 (

k=1

k

β2 )−1/2 .

(4.1.16)

=1

Applying the elementary inequality (4.1.9) gives ∞ k=1

βk2 (

k

β2 )−1/2 ≤ 2(

=1

∞

βk2 )1/2 .

(4.1.17)

k=1

Inserting (4.1.17) into (4.1.16) we finally obtain ∞

(T βk )2 S(fk−1 )−1 ≤ 6A22

k=1

∞ ( βk2 )1/2

≤ 6A22

k=1

S(f ).

We thus have proved the estimate ||T g||H 1 [(Fn )] ≤

√

6A2 ||f ||X[E] .

(4.1.18)

To complete the proof that T is bounded it remains to combine the above estimates (4.1.14) and (4.1.18) obtained for T h and T g, and to use that f = g + h. Thus √ ||T f ||H 1 [(Fn )] ≤ (2A1 + 6A2 )||f ||X[E] .


239

T −1 is bounded. Recall that T −1 : H 1 [(F n )] → X[E] acts on the space of (Fn ) martingales. We fix f ∈ H 1 [(Fn )] with f = 0, and expand it into a sum of martingale differences, so that f= δk , where δk = E(f |Fk ) − E(f |Fk−1 ). For fn = nk=1 δk we denote by S(fn ) the square function of fn with respect to the filtration (Fn ). Thus 1/2

n 2 δk . S(fn ) = k=1

We now define a decomposition procedure with many similarities to the one used in the previous paragraphs. There are differences however, that stem from the fact that δk2 is not, in general, measurable with respect to Fk−1 . We let A be an atom for Fn . Then we put A ∈ Cn iff 2S(fn−1 )(ω) ≤ |δn (ω)|,

for

ω ∈ A.

(4.1.19)

Note that on both sides of the equation (4.1.19) the functions are actually constant when restricted to A. We define Bn = An \ Cn .

(4.1.20)

By the negation of (4.1.19), for B ∈ Bn , and ω ∈ B, the following estimate holds, S(fn )(ω) ≤ 3S(fn−1 )(ω). Let αn =

(4.1.21)

δn 1A − E(δn 1A |Fn−1 ),

A∈Cn

and decompose δn = αn + βn . Notice that αn and βn are martingale difference sequences with respect to the filtration (Fn ). We put h=

∞

αn ,

n=1

and decompose f = g + h. Now we give separate estimates for T −1 h = T −1 αn , and T −1 g = T −1 βn . We begin with T −1 h : By (4.1.19), for A ∈ Cn the pointwise estimate holds, 1 S(fn ) ≥ |δn |1A + S(fn−1 ). 2 Hence, using monotonicity of conditional expectations we obtain the pointwise estimate |αn | ≤ 2S(fn ) − 2S(fn−1 ) + 2E(S(fn ) − S(fn−1 )|Fn−1 ).


240 Summing and integrating gives ∞

|αn | ≤ 4

S(f ).

n=1

By construction the operator T −1 satisfies 1/2 −1 ||T h||X[E] = . (T −1 αn )2

(4.1.22)

Then the obvious estimate 1/2 (T −1 αn )2 ≤ |T −1 αn |, together with local L1 estimates for T −1 , (4.1.5), gives ||T −1 h||X[E] ≤ ||T −1 αn ||1 ≤ A1 |αn | ≤ 4A1 S(f ).

(4.1.23)

Next we prove estimates for the X[E] norm of T −1 g, where g= βn . Notice that by the definition of T −1 , ||T −1 g||X[E] =

(T −1 βn )2

1/2 .

By martingale transform techniques and L2 estimates we give an upper bound for ||T −1 g||X[E] . We begin with the pointwise estimate

(T −1 βn )2 ≤ S(f )

(T −1 βn )2 S(fn−1 )−1 .

The Cauchy–Schwarz inequality gives then ||T −1 g||X[E] ≤

S(f )

1/2

(T −1 βn )2 S(fn−1 )−1

1/2

By the martingale transform property (4.1.8), we have the identity (T −1 βn )S(fn−1 )−1/2 = T −1 (βn S(fn−1 )−1/2 ).

.


241

Hence, with the L2 estimates (4.1.7) we obtain, (T −1 βn )2 S(fn−1 )−1 ≤ A22 βn2 S(fn−1 )−1 . Next we claim that

βn2 S(fn−1 )−1 ≤ 3

(4.1.24)

δn2 S(fn )−1 .

(4.1.25)

We postpone proving the estimate (4.1.25) and continue as follows: Use inequality (4.1.9) with an = δn2 and sn = S 2 (fn ). This gives the pointwise estimate (4.1.26) δn2 S(fn )−1 ≤ 2S(f ). Now combine (4.1.24)–(4.1.26) to obtain −1 2 −1 2 βn2 S(fn−1 )−1 (T βn ) S(fn−1 ) ≤ A2 δn2 S(fn )−1 ≤ 3A22 2 ≤ 6A2 S(f ). We now give the proof for the claim (4.1.25). Recall that An is the collection of atoms in Fn , and that Bn is the subcollection defined by (4.1.20). Next we observe that βn has the representation δn 1B − E(δn 1B |Fn−1 ). (4.1.27) βn = B∈Bn

To see that (4.1.27) holds we write E(δn 1A |Fn−1 ) + E(δn 1B |Fn−1 ) = E(δn |Fn−1 ), A∈Cn

B∈Bn

hence, as E(δn |Fn−1 ) = 0, δn = αn +

δn 1B − E(δn 1B |Fn−1 ).

(4.1.28)

B∈Bn

Comparing (4.1.28) with δn = αn + βn gives (4.1.27). For A ∈ An−1 the L2 estimate holds, ⎡

⎣ A

⎤2 (δn 1B − E(δn 1B |Fn−1 ))⎦ ≤

A

B∈Bn ,B⊆A

=

⎡ ⎣

⎤2

δn 1B ⎦

B∈Bn ,B⊆A

A B∈B ,B⊆A n

δn2 1B .

(4.1.29)


242

We square the identity (4.1.27) and multiply both sides with S(fn−1 )−1 , then integrate. Exploiting that S(fn−1 )−1 is measurable with respect to Fn−1 , we obtain from (4.1.29) that S(fn−1 )−1 δn2 . (4.1.30) βn2 S(fn−1 )−1 ≤ B

B∈Bn

Next recall (4.1.21) asserting that S(fn )(ω) ≤ 3S(fn−1 )(ω) for B ∈ Bn and ω ∈ B. Inserting (4.1.21) into the right-hand side integral of (4.1.30) gives that S(fn−1 )−1 δn2 ≤ 3 S(fn )−1 δn2 B∈Bn

B

B

B∈Bn

≤3

(4.1.31)

S(fn )−1 δn2 .

Combining (4.1.30) and (4.1.31) finishes the proof of the claim (4.1.25). Summing up we showed the following estimates for f ∈ H 1 [(Fn )], ||T −1 (f )||X[E] ≤ ||T −1 (h)||X[E] + ||T −1 (g)||X[E] √ ≤ (4A1 + 6A2 )||f ||H 1 [(Fn )] .

4.2

Isomorphic classification

In this section we prove the classification theorem for martingale H 1 spaces. We determine the isomorphic representatives of the spaces generated by three-valued nested martingale differences which appear through B. Maurey’s isomorphism. Furthermore we classify the family of S ∞ spaces obtained as the weighted intersection between BMO and ∞ . For the spaces H 1 [(Fn )] B. Maurey identified an intrinsic criterion which implies the existence of an isomorphism between the martingale H 1 space and dyadic H 1 : Let An =

{A ∈ Fn : |A| ≤ }

and

A∞ =

∞ +

An .

>0 n=1

It turns out that |A∞ | > 0, precisely when H [(Fn )] and H 1 are isomorphic Banach spaces. See Section 4 of B. Maurey [144]. 1

Classification of martingale H 1 spaces Maurey’s isomorphism reduces the classification for H 1 [(Fn )] spaces to that of the class X[E]. The nested collections E arising through Maurey’s isomorphism satisfy the following condition: If A, B ∈ E, and if A is strictly contained in B, then |A| ≤ |B|/2.

(4.2.1)

4.2. Isomorphic classification

243

Condition (4.2.1) makes E very similar to a subcollection of dyadic intervals. This in turn is the reason that the classification of X[E] is remarkably simple when E satisfies (4.2.1). We will prove that under the hypothesis (4.2.1) the isomorphic types of the class X[E] are given by the spaces H 1 , ( Hn1 )1 or 1 . Each representative is characterized by simple geometric parameters derived from E. These are lim sup E, the set which is contained in infinitely many elements of E, and the Carleson constant, 1 |B|. [[E]] = sup A∈E |A| B⊆A,B∈E

The following theorem presents this in detail. Theorem 4.2.1. Let E be a nested collection of measurable sets such that for A, B ∈ E, A ⊂ B implies |A| ≤ |B|/2. Then X[E] is isomorphic to one of the spaces H 1 , ( Hn1 )1 or 1 . Each of these cases is determined as follows. (a) If | lim sup E| > 0, then X[E] is isomorphic to H 1 . (b) If | lim sup E| = 0 and [[E]] = ∞, then X[E] is isomorphic to (

Hn1 )1 .

(c) If [[E]] < ∞, then X[E] is isomorphic to 1 . This result owes its existence to Pelczy´ nski’s decomposition method. Currently explicit isomorphisms are unknown. The nested collection E consists of measurable sets in the probability space (Ω, F, | . |). Recall that E is canonically decomposed into a sequence of generations. For A ∈ E put G0 (A, E) = {A} and let G1 (A, E) denote the maximal elements of E which are strictly contained in A. By induction for p ≥ 2 we define G1 (B, E). Gp (A, E) = B∈Gp−1 (A,E)

For convenience of notation we write Gp (E) = Gp (Ω, E). Note that lim sup E and the generations Gp (E) are related by the identity lim sup E =

∞ +

G∗p (E),

p=0

#

where we put G∗p (E) = A∈Gp (E) A. It is easy to see that the following version of Proposition 3.1.1 holds: If | lim sup E| = 0, then for every A ∈ E there exists p = p(A) so that 1 |G∗p (A, E)| ≤ |A|. 4


244

Similarly the proof of the condensation Lemma 3.1.4 shows that if [[E]] >

n ,

then there exists B ∈ E so that |G∗n (B, E)| ≥ (1 − )|B|. The first step in the proof of Theorem 4.2.1 determines when X[E] is isomorphic to H 1 . We move to the front an elementary remark that is often helpful in evaluating the dyadic square function and its integral. Let aI h I , f= I∈DN

where DN = {I ∈ D : |I| ≥ 2−N }. Let J ∈ DN , with |J| = 2−N , and let x ∈ J. Then a2I 1I (x) = a2I . I∈DN

I⊇J

Note that the dyadic square function S(f ) is constant over each dyadic interval J satisfying |J| = 2−N . Hence

⎛ S(f )(x)dx = |J| ⎝

J

⎞1/2 a2I ⎠

,

when |J| = 2−N .

I⊇J

Taking the sum over all dyadic intervals J with |J| = 2−N , we get

1

S(f )(x)dx = 0

{J:|J|=2−N }

⎛ |J| ⎝

⎞1/2 a2I ⎠

.

(4.2.2)

I⊇J

It is easy to extend the identity (4.2.2) from the case where f is a finite linear combination of Haar functions to the case where f ∈ H 1 . Indeed, by the monotone convergence theorem we pass to the limit N → ∞ and obtain the H 1 norm of f. Equation (4.2.2) is merely a rewriting of the definitions involved; analogous identities hold of course for the norm in X[E] where dyadic intervals are replaced by nested collections of measurable sets. Theorem 4.2.2. Let E be a nested collection of measurable sets such that for A, B ∈ E, A ⊂ B implies |A| ≤ |B|/2. Then | lim sup E| > 0 implies that X[E] is isomorphic to H 1 .


245

Proof. In Chapter 2 we proved Proposition 2.1.4 to the effect that for any nested collection of measurable sets E the space X[E] is isomorphic to a complemented subspace of H 1 . Hence by the decomposition method it suffices to prove now that if E satisfies (4.2.1) and if the measure of lim sup E is strictly positive, then X[E] contains a complemented copy of H 1 . We claim that E contains pairwise disjoint collections CI , I ∈ D with the following properties: The sets {CI∗ : I ∈ D} form a nested collection with (4.2.3) CI∗ ⊆ CJ∗ iff I ⊆ J; the collections {CI : I ∈ D} satisfy Jones compatibility condition (J), and 1 |I| ≤ |CI∗ | ≤ 2 |I|, 2

(4.2.4)

where = | lim sup E|. We prove the claim by applying the Gamlen–Gaudet construction infinitely often. Begin by recalling that lim sup E can be expressed using generations of E, lim sup E =

∞ +

Gm (E)∗ .

m=1

Hence there is m1 ∈ N such that |G∗m1 (E) \ lim sup E| ≤

, 4

where = | lim sup E|. We define our first collection to be C[0,1] = Gm1 (E). Suppose now that we have already defined the collections CI for all intervals of length ≥ 2−n . We now define the next level of collections using the Gamlen–Gaudet construction. Fix a dyadic interval I such that |I| = 2−n and consider A ∈ CI . Using condition (4.2.1) we obtain a splitting of Gn+2 (A, E) into two collections PA and ∗ | and |M∗A | are almost equal. Precisely their difMA such that the measures |PA ference is small relative to |A| as follows, ∗ | − |M∗A || ≤ |A|2−n−2 . | |PA

Now let I1 be the left half of I and let I2 be the right half of I. Then we define CI1 = MA and CI2 = PA . A∈CI

A∈CI

In this way we have defined now the collections CJ for all dyadic intervals of length ≥ 2−n−1 . This completes the induction step of the construction. We define next block bases {dA : A ∈ CI }, for I ∈ D. bI =


246

The collections {CJ } being obtained through the Gamlen–Gaudet construction satisfy the Jones compatibility condition (J). By Theorem 1.5.9 the orthogonal projection onto span{bI : I ∈ D}, which is bI bI P (f ) = f, , ||bI ||2 ||bI ||2 is a bounded operator on X[E]. It remains to prove that the closed linear span of {bI : I ∈ D} is isomorphic to the to H 1 . We do this by showing that in X[E] the block basis {bI } is equivalent aI bI in X[E], Haar basis in H 1 . To this end we fix a finite linear combination and put CI = CI∗ . Then choose N ∈ N large enough so that each I with aI = 0, satisfies |I| ≥ 2−N . Then using (4.2.3) and (4.2.4) we estimate as follows: aI bI = ( a2I 1CI )1/2 X[E] ≥ ( a2I )1/2 |CJ | (4.2.5) {J:|J|=2−N } I⊇J

1 |C[0,1] | aI h I 1 . 2 H

≥

On the other hand (4.2.3) and (4.2.4) give the upper estimate aI bI ≤2 ( a2I )1/2 |CJ | X[E]

{J:|J|=2−N } I⊇J

aI h I ≤ 4|C[0,1] |

(4.2.6) H

. 1

Combining (4.2.5) and (4.2.6) we showed that the closure of span{bI : I ∈ D} is isomorphic to H 1 . Summing up, the closure of span{bI : I ∈ D} is a complemented subspace of X[E], which is isomorphic to H 1 . We derive now consequences of the fact that | lim sup E| = 0. Proposition 4.2.3. Let E be a nested family of sets for which | lim sup E| = 0. Then E can be decomposed into a sequence of finite collections Ei such that ∞

|Ej∗ ∩ J| ≤

j=i+2

Consequently X[E] ∼

|J| . 4

X[Ei ] ,

and 2 sup[[Ei ]] ≥ [[E]]. i

1

(4.2.7)


247

Proof. The proof starts by defining the decomposition of E. We rename F0 = E, and choose a maximal interval I ∈ F0 . We have as before, lim sup E =

∞ +

Gm (E)∗ .

m=1

By hypothesis | lim sup E| = 0. So the generations Gm (E) are shrinking down to a set of measure zero. Hence we may pick p = p(I) such that |Gp (I, F0 )∗ | ≤

1 |I|. 8

(4.2.8)

Then choose a block of finitely many intervals, call it E(I), satisfying E(I) ⊆ G0 (I, F0 ) ∪ · · · ∪ Gp (I, F0 ),

(4.2.9)

so that the remaining collection defined by F1 (I) = F0 ∩ I \ E(I)

(4.2.10)

is small compared to I and satisfies |F1 (I)∗ | ≤

1 |I|. 4

(4.2.11)

Having defined E(I) and F1 (I) for every maximal element of F0 we form F1 =

{F1 (I) : I is maximal in F0 }.

Note that F0 \ F1 =

{E(I) : I is maximal in F0 }.

Next we consider F1 and apply the above procedure to the maximal elements of F1 , thus forming E(I) for I maximal in F1 . Proceeding by induction we obtain a decreasing chain of collections F0 ⊇ F1 ⊇ · · · ⊇ Fk . . . , where Fk \ Fk+1 =

{E(I) : I is maximal in Fk }.

The blocks E(I) consist of finitely many intervals and satisfy conditions analogous to (4.2.8)–(4.2.11) with F0 replaced by Fk . We relabel the families {E(I)} as {Ei : i ∈ N} such that J ∈ Ej ,

K ∈ Ek and J ⊆ K implies j ≥ k.

(4.2.12)


248

With property (4.2.11) and (4.2.12) it is easy to see that for I ∈ Ei , ∞

|Ej∗ ∩ I| ≤

j=i+2

1 |I|. 4

(4.2.13)

The estimate (4.2.13) in combination with (4.2.12) implies that 2 sup[[Ej ]] ≥ [[E]].

(4.2.14)

Note that in the case when [[E]] = ∞, the estimate 2 sup[[Ej ]] ≥ [[E]] implies that sup[[Ej ]] = ∞. It is in this case that we use the estimate (4.2.14) below. We show now that X[E] is isomorphic to the 1 direct sum of the spaces X[Ei ]. Let fi ∈ X[Ei ]. With the unconditional basis {dA : A ∈ Ei } we expand fi = {aA dA : A ∈ Ei }. It suffices to show that 8

fi X[E] ≥

||fi ||X[Ei ] .

We consider separately the sum over odd and even indices. By unconditionality we get f2i X[E] +

f2i+1 X[E] . 2

fi X[E] ≥

Without loss of generality we treat only the even terms. We let Fi = supp S(f2i ). Hence by (4.2.13), for A ∈ E2i , ∞

|A ∩ Fj | ≤

j=i+1

Now write Di = Fi \

1 |A|. 4

∞

(4.2.15)

Fj .

j=i+1

Since the sets {Di } are disjoint we obtain, using (4.2.15), 1 S(f2i ). S(f2i )1Di ≥ 2 Now we finish as follows,

S(

f2i ) ≥

S(f2i )1Di 1 S(f2i ). ≥ 2

We estimate the odd numbered terms in an analogous way. Thus, X[E] is isomor phic to ( X[Ei ])1 .


249

A combination of Proposition 4.2.3, the Gamlen–Gaudet construction and the condensation lemma leads to the identification of X[E] when | lim sup E| = 0 and [[E]] = ∞. Theorem 4.2.4. Let E be a nested collection of measurable sets such that for A, B ∈ E, 1 A ⊂ B implies |A| ≤ |B|. 2 Then | lim sup E| = 0 and [[E]] = ∞ implies that X[E] ∼ Hn1 . 1

Proof. With Proposition 4.2.3, E can be decomposed into a sequence of finite nested collections Ei so that X[E] ∼ X[Ei ] , 1

and simultaneously sup[[Ei ]] = ∞.

(4.2.16)

i

As Ei is finite, Proposition 2.1.4 implies that for i there exists n = n(i) so that 1 X[Ei ] is well isomorphic to a well complemented subspace of Hn . Consequently 1 the space ( X[Ei ])1 is isomorphic to a complemented subspace of Hn 1 . 1 Next weshow that conversely, Hn 1 is isomorphic to a complemented subspace of ( X[Ei ])1 . Specifically we will show that for n ∈ N there exists i ∈ N such that Id −→ Hn1 Hn1 E Q X[Ei ] where ||E|| ||Q|| ≤ 16 (by the decomposition principle of A. Pelczy´ nski this proves the theorem). Fix n ∈ N, then by (4.2.16) there exists i = i(n) such that [[Ei ]] ≥ 8n . We apply the condensation Lemma 3.1.4 and find A ∈ Ei such that |Gn (A, Ei )∗ | ≥ |A|(1 − 4−n ). Now starting with A ∈ Ei we repeat the basic step in the Gamlen–Gaudet construction n-times. This provides us with collections CI ⊆ Ei where I ∈ Dn , such that C[0,1] = {A}; the sets {CI∗ : I ∈ Dn } form a nested collection, and CI∗ ⊆ CJ∗ iff I ⊆ J;


250

the collections {CI : I ∈ Dn } satisfy Jones compatibility condition (J) and 1 |A| |I| ≤ |CI∗ | ≤ |A| |I|. 2 Now for I ∈ Dn , we form bI =

{dB : B ∈ CI }.

We saw in the proof of Theorem 4.2.2 that the block basis {|A|−1 bI } is equivalent to the Haar basis of Hn1 . More precisely we showed that 1 1 aI h I ≤ aI bI aI h I , ≤ 2 1 1 2 |A| Hn X[Ei ] Hn

(4.2.17)

for any choice of aI ∈ R, with I ∈ Dn . We define the operator E : Hn1 → X[Ei ] to be the linear extension of the map E : hI → |A|−1 bI . The right-hand side of (4.2.17) shows that the operator E has norm ≤ 2. Next we define Q : X[Ei ] → Hn1 by the equation Qf =

f,

I∈Dn

bI hI .

bI 2 hI 2

Clearly Q inverts the action of E so that Id = QE. Furthermore, Jones’s compatibility condition (J) implies that the operator Q has norm ≤ 8. Summing up we proved that X[E] contains a complemented subspace isomorphic to ( Hn1 )1 . For completeness we record again the proposition which identifies the isomorphic type of X[E] when E satisfies a Carleson condition. We stated and proved it already in Chapter 2 (see (2.1.10) and (2.1.11)). Proposition 4.2.5. Let E be a nested family of sets satisfying [[E]] < ∞. Then X[E] ∼ 1 , or X[E] is finite dimensional.


251

Proof of Theorem 4.2.1. We obtain part (a) of Theorem 4.2.1 from Theorem 4.2.2. Part (b) follows from Theorem 4.2.4 and part (c) from Proposition 4.2.5. Next note that for each nested collection E one of the following statements holds true, | lim sup E| > 0, | lim sup E| = 0 and [[E]] = ∞, or [[E]] < ∞. Since these conditions on E are mutually exclusive it follows that the implications in part (a), (b) and (c) are actually equivalences. Thus the following three implications hold 1 true. If X[E] is isomorphic to H , then | lim sup E| > 0, if X[E] is isomorphic to ( Hn1 )1 , then | lim sup E| = 0 and [[E]] = ∞, and finally if X[E] is isomorphic to 1 , then [[E]] < ∞. Next we obtain the classification of the dual spaces to X[E]. Clearly, we are going to apply Theorem 4.2.1. Given the nested collection E the following system of three-valued martingale differences form an unconditional basis for X[E], dA (ω, t) = 1A (ω)rA (t) A ∈ E. Recall that in Section 2.1 we observed that the dualto X[E] can be identified with the space Y [E] consisting of those functions h = A∈E bA dA for which

h Y [E] = sup

A∈E

1/2 1 2 bB |B| < ∞. |A| B∈E∩A

The next theorem classifies the Banach spaces Y [E]. Its interest comes from the fact that there exist just two different spaces in the class Y [E], and that the isomorphic type of Y [E] is determined by the value of the Carleson constant, [[E]] = sup A∈E

1 |B|. |A| B∈E∩A

Theorem 4.2.6. Let E be a nested collection of measurable sets satisfying the following condition. If A, B ∈ E, and if A is strictly contained in B, then|A| ≤ |B|/2. Then the space Y [E] is isomorphic to ∞ or BMO. (a) If [[E]] < ∞, then Y [E] is isomorphic to ∞ . (b) If [[E]] = ∞, then Y [E] is isomorphic to BMO. Proof. The result follows from the classification Theorem 4.2.1 and Wojtaszczyk’s ∼ result that BMO is isomorphic to ( BMOn )∞ . Indeed, if [[E]] < ∞, then X[E] 1 1 . Hence Y [E] ∼ ∞ . On the other hand if [[E]] = ∞, then X[E] ∼ ( H ) or 1 n X[E] ∼ H 1 . Hence Y [E] ∼ ( BMOn )∞ or Y [E] ∼ BMO. By Theorem 2.2.3 the space ( BMOn )∞ is isomorphic to BMO.


252

Let C be a collection of dyadic intervals. Then clearly X[C] is isometric to the closure of span{hI : I ∈ C} in H 1 . Similarly the space Y [C] is isometric to the weak ∗ -closure of span{hI : I ∈ C} in BMO. Thus Theorem 4.2.1 classifies the subspaces in H 1 which are generated by subsequences of the Haar system. The classification for the weak ∗ -closed hull of {hI : I ∈ C} in BMO is given by Theorem 4.2.6.

Classification of weighted intersections Next we turn to the spaces S ∞ defined in Section 2.1. Let (xI ) be a sequence such that 0 ≤ xI ≤ 1. Then the space S ∞ consists of all sequences (aI ) for which

(aI ) S ∞ =

xI aI hI BMO + sup |aI | < ∞. Theorem 2.1.11 asserts that the space S ∞ is isomorphic to a complemented subspace of BMO so that the norm of embedding and projection are independent of the weight (xI ). In the course of proving that S ∞ is isomorphic to a complemented subspace of BMO we exhibited a family of BMO functions {gI } disjointly supported over the Haar system so that g= aI gI , with aI ∈ R, satisfies c g BMO ≤

aI xI hI BMO + sup |aI | ≤ C g BMO .

(4.2.18)

Thus we verified that {gI } is equivalent to the unit vector basis of S ∞ . Moreover in Section 2.1 we showed that the orthogonal projection gI gI f, P (f ) = ||gI ||2 ||gI ||2 I

is bounded in BMO. Now we continue with the analysis of S ∞ and we show that this space is isomorphic to either ∞ or to BMO. Theorem 4.2.7. For every choice of scalars xI ∈ Rwith 0 ≤ xI ≤ 1 the result∞ xI hI ∈ BMO, then S ∞ is ing space S ∞ is isomorphic to BMO or to . If ∞ ∞ xI h I ∈ / BMO, then S is isomorphic to BMO. isomorphic to , and if Proof. Suppose now that f = xI hI ∈ BMO. Let {aI } be a given sequence of scalars. Then we have the upper bound xI hI ||BMO || aI xI hI ||BMO ≤ sup |aI | · || ≤ sup |aI | · ||f ||BMO .


253

Inserting this estimate in the equation defining the norm of S ∞ shows that S ∞ is isomorphic to ∞ when f ∈ BMO. Next we turn to the converse case when f ∈ / BMO. We begin by performing a stopping time argument on the function we perform the stopping time argument on the set of f = xI hI , or rather coefficients defining f = xI hI . For the unit interval [0, 1] we define H[0,1] to be the largest block of dyadic intervals satisfying [0, 1] ∈ H[0,1] and

x2L h2L ≤ 2.

L∈H[0,1]

The maximality condition in the definition of H[0,1] , and the fact that |xI | ≤ 1 imply the following lower estimate. If J ∈ Q([0, 1]) \ H[0,1] , then

x2L h2L ≥ 1 on the interval J.

L∈H[0,1]

Next let K be a maximal interval in the collection Q([0, 1]) \ H[0,1] , Then define HK to be the maximal block of dyadic intervals contained in Q(K) such that

x2L h2L ≤ 2.

L∈HK

Note that if J ∈ Q(K) \ HK , then

x2L h2L ≥ 1 on the interval J.

L∈HK

This process defines a decomposition of the dyadic intervals into a family of blocks {HK : K ∈ E}. In this way we also obtain a decomposition of the function f into = pieces f K L∈HK xL hL . This completes the stopping time decomposition of f= xI hI Now we turn to the last part of the proof examining consequences of the fact that f ∈ / BMO. The proof follows a pattern we are quite familiar with by now. If f ∈ / BMO the index set obtained from the stopping time decomposition does not satisfy the Carleson packing condition. Applying to this index set the condensation Lemma 3.1.4 and then performing the Gamlen–Gaudet construction sufficiently often we find well complemented copies of BMOn in S ∞ . Thus we ∞ find ( BMOn )∞ as a complemented subspace of S . With the decomposition principle we conclude that S ∞ and ( BMOn )∞ are isomorphic spaces. Following are the details. We first observe that if f ∈ / BMO, then the index E satisfies sup I

1 |I|

J∈E,J⊆I

|J| = ∞.


254

The condensation Lemma 3.1.4 implies that for any n ∈ N, there exists a dyadic interval A ∈ E such that |G∗n (A, E)| ≥ |A|(1 − 8−n ). Now let X[0,1] = {A} be the collection containing just the interval A. Let J1 be the left half of [0, 1], and let J2 be the right half of [0, 1]. We also let A1 be the left half of A, and let A2 be the right half of A. Then define XJi to be the collection of maximal intervals in E, which are contained in Ai . (This was the first step of the Gamlen–Gaudet construction.) Next let K be any interval in XJi . Let K1 be the left half of K, and let K2 be the right half of K. Then let X (Kj ) be the collection of maximal intervals in E, which are contained in Kj . We let Ji1 be the left half of Ji , and we let Ji2 be the right half of Ji . We will now take the union and put X (Kj ). XJij = K∈XJi

This was the second step of the Gamlen–Gaudet construction. We repeat the basic step of the Gamlen–Gaudet construction n times. With the assertion of the condensation lemma, that is with |G∗n (A, E)| ≥ |A|(1 − 8−n ), we obtain collections of pairwise disjoint dyadic intervals XJ ⊆ E, for |J| ≥ 2−n , satisfying the following properties. 1. For I ⊆ J let XI be the point-set covered by the collection XI . Then for every K ∈ XJ , (1/2)|K| · |I| ≤ |J| · |XI ∩ K| ≤ 2|K| · |I|. 2. If XI ∩ XJ = ∅, then either XI ⊆ XJ , or XJ ⊆ XI . 3. If XJ ⊆ XI , then J ⊆ I. 4. |A| · |J|/2 ≤ |XJ | ≤ 2|A| · |J|. Let {gI } be the BMO functions satisfying (4.2.18). That is, {gI } is equivalent to the unit vector basis of S ∞ . Their construction is obtained in the course of proving Theorem 2.1.11. Now define gL , GJ = K∈XJ L∈HK

and also HJ =

K∈XJ L∈HK

xL h L .


255

We use {eL } indexed by dyadic intervals to denote the unit vector basis of ∞ . Then put FJ = eL . K∈XJ L∈HK

In the proof of the following statements the systems {HJ }, and {FJ } play the role of auxiliary tools: First, we claim that in BMO the system {GJ : |J| ≥ 2−n } is equivalent to the Haar basis {hJ : |J| ≥ 2−n }. And second, the orthogonal projection Q defined by

Q(y) =

{I:|I|≥2−n }

y,

GI GI ||GI ||2 ||GI ||2

satisfies ||Q(y)||BMO ≤ 4||y||BMO , for y ∈ span{gJ }. We begin by showing the first claim doing the calculations with the system let J2 be the right half of {HJ }. We fix J and let J1 be the left half ofJ and 2 2 J. Note that the square function S 2 (HJ ) = K∈XJ L∈HK xL hL satisfies the 2 pointwise upper estimate S (HJ ) ≤ 2, and also the lower estimates S 2 (HJ ) ≥ 1 on the set XJ1 ∪ XJ1 . From this it follows easily that aJ hJ ||BMO . || aJ HJ ||BMO ∼ || Below we verify that ||

aJ FJ ||∞ = sup |aJ |.

(4.2.19)

(4.2.20)

To see this#fix two dyadic intervals#J = J . Then the corresponding index sets given by IJ = K∈XJ HK and IJ = K∈XJ HK , are disjoint collections of intervals. Thus we proved that (4.2.20) holds true. By (4.2.18) in combination with (4.2.19) and (4.2.20) it follows that || aJ GJ ||BMO ∼ || aJ HJ ||BMO + sup |aI | ∼ || aJ hJ ||BMO . Thus we proved the claim that in BMO the system {GJ : |J| ≥ 2−n } is equivalent to the Haar basis {hJ : |J| ≥ 2−n }. Now we give the BMO estimates showing the boundedness of the orthogonal projection Q defined on span{gI }. We let y= aI gI .


256 Then

y, GJ =

aL x2L |L|,

K∈XJ L∈HK

and

||GJ ||22 =

x2L |L|.

K∈XJ L∈HK

Combining the above expressions with Hölder’s inequality gives y, GJ 2 ≤ 2 ||GJ ||2

a2L x2L |L|.

K∈XJ L∈HK

Recall that XJ consists of pairwise disjoint intervals, hence, |K|. |XJ | = For K ∈ XJ , let fK = 2 2 L∈HK xL hL , satisfies

K∈XJ

L∈HK

xL hL . Then the square function, S 2 (fK ) =

x2L |L| =

L∈HK

S 2 (fK ). K

Next by the stopping time construction and the condensation lemma we obtain, |K| ≤ S 2 (fK ) ≤ 2|K|. 2 K Taking the sum over K ∈ XJ gives

|XJ | ≤ S 2 (fK ) ≤ 2|XJ |. 2 K K∈XJ

Now observe that for J fixed, and for I strictly contained in J , and K ∈ XJ , we have the identities, |XI | ||GI ||22 1 = . G2I = |K| K |XJ | |XJ | With the information collected so far we now derive estimates for the norm of Q(y) in BMO. We fix J and K ∈ XJ , then we compute obtaining global estimates, 1 y, GJ 2 GI 2 1 1 G2I 2 |Q(y) − Q(y)| ≤ + y, |K| K |K| K ||GJ ||42 ||GI ||2 |K| K ||GI ||22 I⊂J 1 2 2 ≤ sup |aI |2 + aL xL |L| |BJ | I⊂J L∈XI aL xL hL ||2BMO ≤ sup |aI |2 + || ≤ || aL gL ||2BMO .

4.3. More on subsystems of the Haar system

257

Finally we point out that by the stopping time process we have the following local estimates. For any interval L ∈ HK with K ∈ XI , 1 1 |GI − GI |2 ≤ C. |L| L |L| L Combining the local estimate and the global estimates above shows that ||Q(y)||BMO ≤ C||y||BMO . So far we showed that for each n ∈ N there is a well complemented copy of BMOn in span{gI }. We obtained these copies using only finite linear combinations of Haar functions. Therefore we actually showed that the infinite direct sum ( BMOn )∞ is isomorphic to a complemented subspace ofthe weak∗ closure of span{gI } in BMO. Hence by Theorem 2.2.3 (asserting that ( BMOn )∞ ∼ BMO) we showed that BMO is isomorphic to a complemented subspace of S ∞ provided / BMO. Since, by Theorem 2.1.11 the space S ∞ is isomorphic that xI hI ∈ to a complemented subspace of BMO, Pelczy´ nski’s decomposition method im∗ closure of span{g } in BMO and BMO are isomorphic when plies that the weak I / BMO. xI hI ∈

4.3

More on subsystems of the Haar system

In this section we study the subspaces in Lp ([0, 1]), 1 < p < ∞, which are spanned by subsequences of the Haar system. We present the theorem of Gamlen and Gaudet to the effect that the closed linear span of every infinite sequence of Haar functions is isomorphic to p or Lp . We also prove a finite-dimensional version of the Gamlen–Gaudet theorem. This is accomplished by replacing certain qualitative considerations with dimension free estimates. The section ends with the discussion of several related open problems.

The theorem of Gamlen and Gaudet We first prove the classical theorem of Gamlen and Gaudet that classifies the spaces generated by a subsequence of the Haar basis in Lp . Later we establish a quantitative refinement of the Gamlen and Gaudet theorem exploiting the properties of Maurey’s isomorphism. Let C be a collection of dyadic intervals. Let 1 < p < ∞, and let X p [C] denote the closure of span{hI : I ∈ C} in Lp . On X p [C] we put the norm induced by the square function characterization of Lp , 1 < p < ∞. Thus for f ∈ X p [C] with Haar expansion f = I∈C aI hI we define

f X p [C] = S(f ) Lp

where S(f )2 (t) =

I∈C

a2I h2I (t).


258

The following theorem of Gamlen and Gaudet asserts that the value of | lim sup C| determines the isomorphic type of X p [C]. Theorem 4.3.1. Let 1 0, then X p [C] is isomorphic to Lp ([0, 1]). Proof. The proof follows from Theorem 4.2.2 and Proposition 4.2.3 by interpolation. We start with the observation that X p [C] is a complemented subspace of p L , 1 0, then by Theorem 4.2.2 there exist sub-collections in C, denoted {CI : I ∈ D} satisfying the compatibility condition (J) together with (4.2.3) and (4.2.4). By Theorem 1.5.9 and interpolation between H 1 and L2 the following orthogonal projection is bounded in X p [C],

P (f ) =

I∈D

f,

bI bI ,

bI 2 bI 2

where bI = {hJ : J ∈ CI }. Furthermore it is#easy to observe that the Lp version of (4.2.5) and (4.2.6) hold true. Let C[0,1] = J∈C[0,1] J. Then 1 aI hI X p [D] ≤ |C[0,1] |−1/p

aI bI X p [C] 2

≤ 2

aI hI X p [D] . (4.3.1)

By Burkholder’s theorem the identity operator is an isomorphism between Lp and X p [D]. In summary if | lim sup C| > 0, then X p [C] contains a complemented copy nski implies that X p [C] of Lp , 1 < p < ∞. The decomposition principle of A. Pelczy´ p is isomorphic to L . Next we turn to the case when | lim sup C| = 0. We will show that then X p [C] is isomorphic to a complemented subspace of p , 1 < p < ∞. If | lim sup C| = 0, then by Proposition 4.2.3 there exists a decomposition of C into finite collections {Ci : i ∈ N} so that for each i ∈ N and J ∈ Ci we have ∞ j=i+2

|J ∩ Cj | ≤

|J| , 4

(4.3.2)

# where Cj = K∈Cj K. Recall that the estimate (4.3.2) appears in the hypothesis of Proposition 3.1.5. Applying it to f= fi , where fi ∈ X p [Ci ], gives that

1/p 1

f X p [C] ≤

fi pX p [Ci ] ≤ 2 f X p [C] . 2


259

Thus X p [C] ∼ ( X p [Ci ])p . For n ∈ N let Dn = {I ∈ D : |I| ≥ 2−n }. Clearly, since Ci is finite there exists n = n(i) so that Ci ⊆ Dn . By Burkholder’s theorem for 1 < p < ∞ the identity operator provides an isomorphism between X p [Dn ] and Lp2n+1 −1 = (span{hI : I ∈ Dn }, · Lp ) .

p . Thus ( X p [Ci ])p is isoNext observe that Lp2n+1 −1 is 2-isomorphic top 2n+1 −1 morphic to a complemented subspace of ( n )p ∼ p . In summary we showed that if | lim sup C| = 0, then X p [C] is isomorphic to a complemented subspace of p . It remains to cite the classic theorem of A. Pelczy´ nski that every infinite-dimensional complemented subspace of p is isomor phic to p , 1 < p < ∞ (see [169] or [130] Volume I, Theorem 2.a.3). The theorem of Gamlen and Gaudet as well as Theorem 4.2.1 are infinite dimensional in nature. They don’t contain specific information when C is a finite collection of intervals. By contrast the following local version of the Gamlen–Gaudet theorem shows that for every finite collection C of dyadic intervals there exists an isomorphism T acting between X p [C] and pN where N = cardinality of C, so that

T · T −1 is bounded independent of N. Specifically we obtain

T · T −1 ≤ Cp . The operator T by which we prove the local Gamlen–Gaudet theorem is just Maurey’s isomorphism. Theorem 4.3.2. Let C be a collection of dyadic intervals and let 1 ≤ p < ∞. Then the closed linear span of {hI : I ∈ C} in Lp is Cp -isomorphic to p , Lp or pn where n is the cardinality of C. Proof. We let Fn be the algebra of sets, generated by the collection {I ∈ C : |I| ≥ 2−n } and let F0 = {[0, 1], ∅}. Let A be an atom for Fn−1 . Then the atoms of Fn contained in A are the dyadic intervals B = {I ∈ C : I ⊆ A, |I| = 2−n } together with A \ B∗ . Now we assume that our collection C was such that A \ B ∗ = ∅, then clearly |A \ B∗ | ≥ 2−n . Note that on the other hand each atom I ∈ B satisfies |I| = 2−n . Therefore the largest atom A∗ ∈ Fn contained in A can be chosen to be A \ B ∗ , provided that A \ B ∗ = ∅. Now we consider the collection E(A), formed by deleting A∗ from the collection of all atoms in Fn that are contained in A, and we let E= E(A), where the union is taken over all atoms in Fn , n ∈ N. Note that by construction E coincides with the collection C we started with. We will use the space X p [C] to mediate between the closure of span{hI : I ∈ C} and Lp (F∞ ). For I ∈ C with |I| = 2−n we form bI = 1I ⊗ rn .


260 By Burkholder’s theorem, p aI bI I∈C

∼

p/2 a2I 1I

.

I∈C

Lp ([0,1]×[0,1])

Hence X p [C] is isomorphic to span{hI : I ∈ C}, and the isomorphism is given by the linear extension of the relation bI → hI . On the other hand Maurey’s operator T : X p [C] → H p [(Fn )] provides the isomorphism between X p [C] and H p [(Fn )], which is isomorphic to Lp ([0, 1], F∞ ) by Burkholder’s theorem. Finally we remark that Lp ([0, 1], F∞ ) is isometric to pn when F∞ is generated by n atoms, and Lp ([0, 1], F∞ ) is isometric to p when F∞ is generated by an infinite sequence of atoms. Finally Lp ([0, 1], F∞ ) is isomorphic to Lp when F∞ is non-atomic. To remove the restrictive assumption A \ B∗ = ∅ from C, we remove from C one interval of length 2−n when B∗ = A for an atom A in Fn−1 . In this way we split C into C1 and C2 , each of which satisfies the additional assumption, and the theorem follows from the unconditionality of the Haar basis.

Related open problems The space S ∞ , consisting of sequences (aI ) for which

(aI ) S ∞ =

xI aI hI BMO + sup |aI | < ∞, appears as a natural endpoint of the scale S p , 2 ≤ p < ∞ for which the norm is given by 1/p hI 1−2/p p

(aI ) S p = xI aI 1/p + | < ∞. (4.3.3) |a I |I| p L

The result of D. Kleper and G. Schechtman in [117] asserts that S p is isomorphic to nski a complemented subspace of Lp . By the decomposition principle of A. Pelczy´ this is the first and crucial step towards the isomorphic classification of S p in [157]: The space S p is isomorphic to Lp or to p . Open, however, is the corresponding finite-dimensional problem of classifying the family Snp , 2 < p < ∞, n ∈ N, where Snp = {(aI )I∈Dn : (aI ) S p < ∞} , and where the norm on Snp is given by (4.3.3).


261

Problem 4.3.3. Determine an isomorphism T : Snp → p2n+1 −1 so that

T · T −1 < Cp . In the special case where xI ∈ {0, 1}, the desired operator T is given by Maurey’s isomorphism (this is the content of Theorem 4.3.2 and its proof ). Next we discuss the product version of the Gamlen–Gaudet problem. Let 1 < p, r < ∞. Then Lp (Lr ) is defined to be the space of measurable functions on the product [0, 1] × [0, 1] so that f ∈ Lp (Lr ) if

f Lp (Lr ) =

1

|f (s, t)| dt r

0

1/p

p/r

1

< ∞.

ds

0

The product form of the Haar system hI×J (s, t) = hI (s)hJ (t),

I, J ∈ D,

forms an unconditional basic sequence in Lp (Lr ). For f ∈ Lp (Lr ), with Haar expansion aI×J hI×J , f= I×J∈D×D p

r

the norm in L (L ) is equivalent to the square-function integral ⎛ ⎝ 0

1

1

( 0

⎞1/p

p/r a2I×J h2I×J )r/2 dt

ds⎠

.

I×J∈D×D

Hence for any collection of dyadic rectangles R ⊆ D × D the closure in Lp (Lr ) of span {hI×J : I × J ∈ R} is complemented in Lp (Lr ). M. Capon in [42] established a subtle, combinatorial criterion for {hI×J : I × J ∈ R} to generate a subspace of Lp (Lr ) which is isomorphic to Lp (Lr ). Following is the product version of the Gamlen–Gaudet theorem due to M. Capon [42]. Theorem 4.3.4. Let R ⊆ D × D be a collection of dyadic rectangles. For I ∈ D and t ∈ [0, 1] define the family of dyadic intervals RI = {J ∈ D : I × J ∈ R}, and put Rt = lim sup{I ∈ D : t ∈ lim sup RI }.


262 If |{t ∈ [0, 1] : |Rt | >

1 }| > 0, 2

(4.3.4)

then the closure in Lp (Lr ) of span {hI×J : I × J ∈ R} is isomorphic to Lp (Lr ). Observe that for R ⊆ D × D or for R = R \ D × D the criterion (4.3.4) is satisfied. The insight into the structure of the product Haar system obtained by M. Capon in [42] should be helpful in solving the following problem. Problem 4.3.5. Determine the isomorphic type of the closure of span {hI×J : I × J ∈ R} in Lp (Lr ). We close this section with the discussion of a problem posed by P. Wojtaszczyk: Decide which subsequences of the Haar system are permutatively equivalent to the whole system. That is, given a collection of dyadic intervals B, when is it possible to find a bijective map τ : D → B such that the induced rearrangement operator T1 : H 1 → H 1 , is bounded and invertible on its range ? This problem links the results on rearrangement operators in Chapter 2 to the classification theorems of the present chapter. By definition, for p < 1 the space H p consists of the sequences {aI } for which

{aI } pH p = ( a2I h2I )p/2 < ∞. Defined that way H p , p < 1, is a quasi-normed space. The importance of Wojtaszczyk’s problem becomes apparent only in the context of H p when p < 1. Indeed for H p , p < 1, P. Wojtaszczyk [215] proved the following results which show fundamental and unexpected differences between H p , p < 1 and H 1 . 1. Any unconditional, normalized basis in a complemented subspace of H p , p < 1 is equivalent to a sub-collection of the Haar basis in H p , p < 1. 2. If a subsystem {hI : I ∈ B} of the Haar basis spans a subspace isomorphic to H p , p < 1, then there exists a bijective mapping τ : D → B, such that the associated permutation operator is bounded and invertible on its range. In other words the subsystem {hI : I ∈ B} is permutatively equivalent to the whole dyadic system. 3. Let two subsystems of the Haar basis {hI : I ∈ B1 } and {hI : I ∈ B2 } be spanning subspaces of H p , p < 1, that are isomorphic. Assume that these spaces are isomorphic to their own square. Then there exists a bijective mapping τ : B1 → B2 , which extends to an isomorphism from the closure of span{hI : I ∈ B1 } onto the closure of span{hI : I ∈ B2 }.


263

4. There exist uncountably many collections Bα , such that the subsystems of the Haar basis {hI : I ∈ Bα } are pairwise non-isomorphic subspaces of H p , p < 1. Note that in particular the last statement is in strong distinction to the H 1 result of Theorem 4.2.1. Indeed, there we classified the subspaces generated by subcollections of the Haar basis, and we showed that in H 1 there are exactly three isomorphic types. In H p with p > 1, there are exactly two isomorphic types by the Gamlen–Gaudet Theorem 4.3.1. Nevertheless the answer to Wojtaszczyk’s conjecture should be the same in all Hardy spaces, hence we discuss it in the context of H 1 . Let B be a collection of dyadic intervals, and let Bk = B ∩ {I ∈ D : |I| = 2−k }. We say that B satisfies Property W if there exists δ > 0 and s ∈ N so that for any k ∈ N, there exists n ∈ [k, k + s] such that |Bn∗ | > δ. The conjecture of P. Wojtaszczyk states that B satisfies Property W if ( and only if ) there exists a bijective map τ : D → B such that the induced rearrangement operator T1 : H 1 → H 1 , is bounded and invertible on its range. Support for this conjecture comes from two results of Wojtaszczyk. First, the conjecture is true for special cases, namely when B is defined by the rule I ∈ B if |I| = 2−kn , where kn is a given increasing sequence of natural numbers. For these examples it is easy to see that B satisfies Property W if sup(kn+1 − kn ) < ∞. n

Second, one implication of Wojtaszczyk ’s conjecture holds in general. There we have the following theorem [215]. Theorem 4.3.6. Let B be a collection of dyadic intervals. Suppose that there exists a bijective map τ : D → B such that the induced rearrangement operator T1 : H 1 → H 1 , is bounded and invertible on its range. Then, the collection B satisfies Property W. Proof. Let τ : D → B be a rearrangement and suppose that the induced rearrangement operator extends to an isomorphism onto its range. Now assume that the theorem is false, and that B does not satisfy Property W. Then for δ = 1/n and s = n there exists kn , such that lim kn = ∞, and for all m ∈ [kn , kn + n] we have 1 ∗ |Bm |≤ . n


264

Note that by the injectivity of τ there are at most 2kn +1 dyadic intervals I for which |τ (I)| ≤ 2−kn . Let Kn = [kn + n/3, kn + 2n/3], and let m ∈ Kn . Now put Cm = {I ∈ D : |I| = 2−m and |τ (I)| ≤ 2−kn }. By the injectivity of τ each of the collections Cm covers almost all of the unit ∗ | ≥ (1 − 2−n/3 ). Now put interval, more precisely, for m ∈ Kn , |Cm hI . xm = I∈Cm

Then we have that ||xm ||H 1 ≥ 1 − 2−n/3 , and also am xm ||H 1 ≤ C( a2m )1/2 . || n m∈Kn

n m∈Kn

The boundedness of the rearrangement operator T1 gives that am T1 xm ||H 1 ≤ C||T1 ||( a2m )1/2 . || n m∈Kn

n m∈Kn

Note that the Haar support of the function T1 xm is a pairwise disjoint collection of dyadic intervals. Summing up the first set of observations, we showed that there exists η > 0, (not depending on n) such that for m ∈ Kn , |τ (I)| ≥ η. I∈Cm

Hence summing over m ∈ Kn gives

|τ (I)| ≥

m∈Kn I∈Cm

n η. 3

Now we prove that this lower estimate contradicts the assumption that Property W does not hold. To see this we consider the index set of the above sum, and split it as follows. First we form A0 = {τ (I) : I ∈ Cm , m ∈ Kn } ∩ {J : |J| > 2−kn }. Note that by definition of Cm this collection is actually empty. Then for k ∈ [kn , kn + n], we form Ak = {τ (I) : I ∈ Cm , m ∈ Kn } ∩ {J : |J| = 2−k }. Note that Ak ⊆ Bk , and that by assumption for each k ∈ [kn , kn + n], |Bk∗ | ≤

1 . n

4.4. Notes

265

And finally recall that for m ∈ Kn the cardinality of Cm is bounded by 2kn +2n/3 . Combining these observations into a single estimate we have that m∈Kn I∈Cm

|τ (I)| ≤ n ×

1 + 2−kn −n n m∈Kn I∈Cm

≤ 1 + n × 2−kn −n × 2kn +2n/3+1 . Note that for n large enough the last upper bound is ≤ 2, and this contradicts the previous lower estimate .

4.4

Notes

Maurey’s isomorphism represents one of several fundamental techniques introduced by B. Maurey in [140], [142], [143], and [144]. The classification of martingale H 1 spaces was obtained as a consequence of Maurey’s isomorphism in [150]. In the text we follow the presentation of [159] and [151]. The subspaces of H 1 that are spanned by subsequences of the Haar basis are classified in [149]. The family S ∞ is classified in [157]. Very interesting applications of threevalued martingale difference sequences to related classification problems are contained in the books of W. Johnson, B. Maurey, G. Schechtman and L. Tzafriri [93], in J. Bourgain [16] and in J. Bourgain, H. Rosenthal and G. Schechtman [29]. Theorem 4.3.1 is a classic result of J. Gamlen and R. Gaudet [71]. Theorem 4.3.2, the finite-dimensional version of the Gamlen–Gaudet theorem, is in [152]; the proof in the text follows that in [159]. Problem 4.3.3 emerged with the analysis of the peculiar rearrangement considered in [160]. A systematic study of the Haar system in vector-valued Lp spaces is contained in the important series of papers by M. Capon [41], [40],[42] and [43]. The conjecture of P. Wojtaszczyk is formulated in [215]. Its discussion in the text is taken from P. Wojtaszczyk’s paper [215]. For its motivation see also [109] and [107].

Chapter 5

Isomorphic Invariants for H 1 In this chapter we give a fairly complete discussion of the known isomorphic invariants of H 1 . We study copies of 2 and prove two dichotomies for complemented subspaces: First, if a complemented subspace of H 1 does not contain a copy of 2 , then it is isomorphic to a complemented subspace of ( Hn1 )1 . Second, if BMO ∼ X ⊕ Y , then one of the spaces X or Y is isomorphic to BMO. We prove that the vector-valued space H 1 (2 ), is not isomorphic to H 1 . The chapter concludes with the proof that BMO satisfies the uniform approximation property.

5.1

Complemented copies of Hilbert spaces

In this section we study subspaces of H 1 containing copies of Hilbert spaces. We prove the theorem of S. Kwapien and A. Pelczy´ nski, establish Johnson’s factorization for H 1 and J. Bourgain’s description of uniformly complemented copies of 2n in H 1 .

Existence, abundance Subspaces of a Banach space which are isomorphic to 2 are the so-called Hilbertian subspaces. We start with the basic result of S. Kwapien and A. Pelczy´ nski, that every Hilbertian subspace of H 1 contains a complemented copy of 2 (where the complementation is in H 1 ). This holds in spite of the fact that there are plenty of Hilbertian subspaces of H 1 which are not complemented. For example the Walsh functions of multiplicity 2 span a non-complemented copy of 2 . We prepare the theorem of S. Kwapien and A. Pelczy´ nski, by showing first a general upper 2 estimate in BMO and by duality a corresponding lower estimate for H 1 . Following is a simple but useful observation. Proposition 5.1.1. Let {x } be a sequence in H 1 . Let {y } be a sequence in BMO. Assume that the sequences {x } and {y } are disjointly supported over the Haar

Chapter 5. Isomorphic Invariants for H 1

268

system. Let {a } and {b } be sequences in R. Then the following estimates hold, 1/2

∞ ∞ 2 2

b y BMO ≤ b y BMO (5.1.1) =1

and

∞

=1

1/2 a2 x 2H 1

∞ √ ≤ 2 2

a x H 1 .

=1

(5.1.2)

=1

Proof. We recall the definition of the localized square function. Fix g ∈ BMO with Haar expansion g = zI hI . Fix a dyadic interval I, then write Q(I) = {J ∈ D : J ⊆ I} and zJ2 h2J . S 2 (g|Q(I)) = J∈Q(I)

Now fix a sequence {y } in BMO that is disjointly supported over the Haar system. Let {b } be a sequence in R, and assume that only finitely many of its entries are non-vanishing. Then define ∞ b y . f= =1

Fix a dyadic interval I. Since the sequence {y } is disjointly supported over the Haar system we obtain the identity S 2 (f |Q(I)) =

∞

b2 S 2 (y |Q(I)).

=1

The definition of the BMO norm gives S 2 (y |Q(I)) ≤ |I| · y 2BMO . Thus ∞ b2 y 2BMO . S 2 (f |Q(I)) ≤ |I| =1

Dividing by |I| and taking the supremum over all dyadic intervals I gives the estimate (5.1.1). We derive the lower H 1 estimate (5.1.2) from (5.1.1) and H 1 − BMO duality. Let {x } be a sequence in H 1 , disjointly supported over the Haar system. By H 1 − BMO duality, Theorem 1.2.3, there exists w ∈ BMO, such that w BMO = 1, and x , w = x H 1 . Clearly we may select w in such a way that its Haar support coincides with that of x . Thus the sequence {w } is disjointly supported over the Haar system. Next we select a sequence of scalars {b } such that b2 = 1 and 1/2

∞ ∞ 2 2 a x H 1 = a x , w b =1

=1 ∞

6 =

=1

a x ,

∞ =1

7 b w

(5.1.3) .

5.1. Complemented copies of Hilbert spaces

269

Applying Fefferman’s inequality to the scalar product appearing in the last line of (5.1.3) we obtain that

∞

1/2 a2 x 2H 1

=1

∞ √ ≤ 2 2 a x =1

H1

∞ b w =1

. BMO

Finally we apply the upper2 estimate (5.1.1) to {w }. In BMO the sequence satisfies w BMO = 1 and b2 = 1, this gives (5.1.2). Let {uj } be a sequence in a Banach space X with Schauder basis {yn }. Assume that the basis constant of {yn } equals K and let {yn∗ } denote the associated biorthogonal functionals defined by the equations ∗ = δn,m . yn , ym

Recall that {uj } is said to be equivalent to the unit vector basis of 2 if there exists c > 0, and C > 0 so that c

a2j

1/2

≤

aj uj X ≤ C

a2j

1/2 ,

for every sequence {aj } ∈ 2 . Below we use that for > 0 there exists a sequence of increasing integers {Nj } and a subsequence {unj } such that

Nj+1

xj =

unj , yn∗ yn

n=Nj +1

satisfies

unj − xj X ≤ 2−j .

Moreover, if the closure of span{xj } is complemented in X by a projection P and if = (K, P , C, c) is sufficiently small, then also the closure of span{unj } is complemented in X. The same conclusion can be derived for sequences {uj } satisfying c ≤ uj X ≤ C and f, uj → 0 ∀f ∈ X ∗ . This result is proved using gliding hump arguments and perturbation results. It is contained in standard texts of functional analysis (for instance in [130] page 7 or [213] page 42). Theorem 5.1.2. Let {uj } be a sequence in H 1 which is equivalent to the unit vector basis on 2 . Then there exists a subsequence {xj } of {uj } such that the closed linear span of {xj } is complemented in H 1 . Proof. The sequence {uj } is equivalent to the unit vector basis on 2 , hence uj → 0 weakly in H 1 . By a standard perturbation argument we may therefore assume that


270

there exists a subsequence {xj }, and pairwise disjoint collections {Bj } consisting of finitely many dyadic intervals, such that xj ∈ span{hJ : J ∈ Bj }. Next by H 1 − BMO duality, that is by Theorem 1.2.3, we may choose zj ∈ BMO, with ||zj ||BMO ≤ 1, so that xj , zj = ||xj ||H 1 , and zj ∈ span{hJ : J ∈ Bj }. Now we define the projection Pf =

j

f, zj

xj . xj , zj

Clearly the range of P equals span{xj }. It remains to verify that P is a bounded operator in H 1 . To this end we fix h ∈ BMO, and we evaluate h, xj P f, h = f, zj xj , zj = f, P ∗ h. We continue by obtaining an upper bound for ||P ∗ h||BMO . Recall that we defined the sequence {zj } to be disjointly supported over the Haar system. Applying the upper 2 estimate of Proposition 5.1.1 gives that 2 h, xj 2 h, xj z ≤

zj 2BMO . j xj , zj BMO xj , zj 2 Inserting this gives

h, xj 2 = C sup{ h, xj aj : a2j = 1} = C sup{h, xj aj : a2j = 1}.

||P ∗ h||2BMO ≤ C

(5.1.4)

We continue giving an upper estimate for the scalar product |h, xj aj | under the hypothesis that a2j = 1. Now recall that the sequence xj is equivalent to the 2 unit vector basis of . Therefore the following upper bound holds true, || xj aj ||H 1 ≤ C( a2j )1/2 . Finally we apply Fefferman’s inequality to the scalar product appearing in the last line of (5.1.4). Thus we obtain the norm estimate of P ∗ , xj aj ||H 1 ||P ∗ h||BMO ≤ C||h||BMO || ≤ C||h||BMO .


271

Johnson’s factorization on H 1 In this subsection we aim at proving dichotomies for complemented subspaces of H 1 . We determine when a projection on H 1 factors through ( Hn1 )1 and when not. To this end we introduce a combinatorial method called Johnson’s factorization. Before we continue we should recall some standard language from Banach space theory. Let T : X → Y be a linear operator acting between two given Banach spaces, X and Y. Let Z be a third Banach space given to us. If there exist bounded linear operators R : X → Z and S : Z → Y such that the following diagram commutes, T X −→ Y R S Z then we say that the operator T factors through Z. In Theorem 4.2.2 we showed that Id : H 1 → H 1 factors through X[E] provided that | lim sup E| > 0. Let {Xn } be a given sequence of finite-dimensional Banach spaces. We say that the Banach space X contains uniformly complemented copies of {Xn } if the following condition holds true. There exist two sequences of linear operators Rn : Xn → X and Sn : X → Xn , such that Id −→ Xn Rn X

Xn Sn

and

sup Rn · Sn < ∞.

n∈N

By Theorem 4.2.4, the space X[E] contains uniformly complemented copies of {Hn1 } if [[E]] = ∞. We prove next a dichotomy for complemented subspaces in H 1 . 2 embeds Theorem 5.1.3. Let X be a complemented subspace of H 1 . Then either 1 ) . into X, or X is isomorphic to a complemented subspace of ( Hn 1 The proof of Theorem 5.1.3 requires a sufficiently good understanding of operators which factor through the space ( Hn1 )1 . Below we will establish the following criterion for a given operator P : H 1 → X to admit such a factorization: If

Id P L2 −→ H 1 −→ X

is a compact operator,

then P

−→

H1 R (

Hn1 )1

X S

and

R · S < ∞.

The proof of this criterion is based on a remarkable combinatorial construction, due to W. B. Johnson. Before we proceed we mention several open problems related


272

to Theorem 5.1.3. Their aim is to obtain richer, and more complete, dichotomies for complemented subspaces of H 1 . Fix a subspace X ⊆ H 1 and assume that there exists a bounded projection P : H 1 → X whose range is X. 1 Problem 5.1.4. Give 2 an intrinsic characterization of operators P : H 1→ X which factor through ( n )1 . Show, specifically, that if the projection P : H → X does not factor through ( 2n )1 , then X contains uniformly complemented copies of Hn1 .

Problem 5.1.5. Give an intrinsic characterization of operators which factor through 1 . Show, specifically, if a projection P : H 1 → X does not factor through 1 , then X contains uniformly complemented copies of 2n . Good solutions to the above problems should allow one to settle the following conjectures concerning complemented subspaces of H 1 : Conjecture 5.1.6. Let X be a complemented subspace of H 1 . Then the following two dichotomies hold true. 1 1. Either X contains a complemented 2 copy of ( Hn )1 , or X is isomorphic to a complemented subspace of ( n )1 . 2. Either X contains a complemented copy of ( 2n )1 or X is isomorphic to 1 . Now we turn to the proof of Theorem 5.1.3. First we describe the isomorphic invariant which controls the dichotomy expressed there. We let F be the orthogonal projection onto the span of the Haar functions {hI : |I| ≤ 2− }. It is given by hI f, hI . F f = |I| − {I:|I|≤2

}

Now suppose that the composition Id P L2 −→ H 1 −→ X is a compact operator. Then of course the sequence of operators F

P

L2 → H 1 → X, converges in norm to zero, it satisfies lim ||P F : L2 → H 1 = 0.

→∞

The main step in the proof of Theorem 5.1.3 is the following result relating compactness and factorization.


273

Proposition 5.1.7. Let X be a complemented subspace of H 1 , and let P : H 1 → X be the projection onto X. Assume that lim ||P F : L2 → H 1 = 0.

→∞

1 Then P factors through 1( Hn )1 . Consequently X is isomorphic to a complemented subspace of ( Hn )1 . We break up the proof of Proposition 5.1.7 into two separate but combinatorially intertwined steps. Combined they constitute Johnson’s factorization. The first step gives the embedding into ( Hn1 )1 , the second one gives the projection 1 from ( Hn )1 onto a subspace isomorphic to X. Below, the Rosenthal space plays an important role. It enters, disguised, with the following spaces that we introduced in Chapter 2, Proposition 2.1.10. Fix natural numbers n < n+1 and a non-negative M , which plays the role of a weight below. Then define the space T (n , n+1 , M ) to consist of span{hJ : 2−n+1 ≤ |J| ≤ 2−n }, equipped with the norm |x|T = inf{||y||H 1 + ||z||L2 M−1 : x = y + z}.

(5.1.5)

For notational simplicity we will write T = T (n , n+1 , M ), when there is no confusion about the choice of n < n+1 and M . The following lemma is the first step in the proof of Proposition 5.1.7. In the course of its proof we make several choices for the sequences {n } and {M }. Note that these choices bear fruit only when we prove Lemma 5.1.9. Lemma 5.1.8. Assume that lim ||P F : L2 → H 1 = 0.

→∞

Then there exist increasing sequences of natural numbers, {n } and {M } such that P x ≤ ||P || |x |T , 1 H

whenever x ∈ T . Proof. First we give an inductive definition of the sequences M+1 and n . This determines the spaces T = T (n , n+1 , M ). Put M1 = 1, n1 = 1, M2 = 2, and n2 = 2. Suppose M1 < · · · < M and n1 < · · · < n have been selected. Then we define M+1 = 2+1 2n .

(5.1.6)


274

Next we pick n+1 ≥ n so large that the following two conditions are satisfied. −1 , ||P Fn+1 ||L2 ,H 1 ≤ ||P ||M+1 2 M+1 ≤ 2n+1 .

(5.1.7)

Having determined the sequences {M+1 } and {n }, the spaces T (n , n+1 , M ) are now defined. We abbreviate T = T (n , n+1 , M ). Fix ∈ N and x ∈ T . Evaluating the norm in the space T , by (5.1.5) we may choose y , z ∈ T , such that |x |T = ||y ||H 1 + ||z ||2 M−1 . Note that we have P z = P Fn z . Hence by our choice of M we obtain from (5.1.7) that ||P z ||H 1 ≤ M−1 ||P || · ||z ||2 . (5.1.8) By the triangle inequality the estimate (5.1.8) gives the upper bound ||P x ||H 1 ≤ ||P y ||H 1 + ||P z ||H 1 ≤ ||P ||( ||y ||H 1 + M−1 ||z ||2 ) = ||P || · |x |T . Therefore we just apply the triangle inequality to the left-hand side expression below to obtain ∞ ∞ P x ≤ ||P || |x |T . =1

=1

H1

Next we are required to find upper bounds for the norm of x in T . An optimal estimate for |x |T is the result of a decomposition x = y + z that minimizes the weighted sum ||y ||H 1 + M−1 ||z ||2 . With stopping time arguments it is quite easy to obtain close to optimal decompositions. In the previous chapters we applied this method for the first time in the course of proving the atomic decomposition and later again in the section on interpolation. Lemma 5.1.9. Let x ∈ T (n , n+1 , M ), be any sequence such that || x ||H 1 < ∞. Then, ∞ ∞ |x |T ≤ 8|| x ||H 1 . =1

=1

Proof. We start by giving good upper bounds for the norm of x in T . We intend to split x in T as x = y + z so that the weighted sum ||y ||H 1 + M−1 ||z ||2 becomes sufficiently small. Below we determine the splitting so that

||y ||H 1 + M−1 ||z ||2 ≤ 8||

x ||H 1 .


275

We use a stopping time argument on the square function of S(x ). Thus we first decompose the Haar support of x and then split x into y + z . Let N be the smallest integer so that the set Ω = {t ∈ [0, 1] : S(x )(t) ≥ N }

(5.1.9)

satisfies the measure estimate |Ω| ≤ 2−n−1 −2 .

(5.1.10)

Let F be the collection of dyadic intervals which are contained in Ω. Then define C = Q([0, 1]) \ F and z =

x ,

J∈C

hJ hJ . |J|

Then put y = x − z . By (5.1.9) z satisfies the pointwise estimate, S(z ) ≤ N. Hence S(z )2 ≤ N ||x ||H 1 . (5.1.11) Next we recall that N was selected by a stopping time rule. Its definition implies that the measure of the set {S(x ) > N − 1} is larger than 2−n−1 /4. Hence, N − 1 ≤ 4 · 2n−1 ||x ||H 1 .

(5.1.12)

Inserting (5.1.12) in the estimate (5.1.11) gives an upper bound for the L2 norm of z . S 2 (z ) ≤ 8 · 2n−1 ||x ||2H 1 (5.1.13) n−1 2 ≤8·2 || xj ||H 1 . By (5.1.6), M satisfies M = 2 2n−1 . We exploit now the seemingly unmotivated factor 2 appearing in the definition of M . We have ∞

2n−1 M−1 ≤

=1

∞

2− ≤ 1.

(5.1.14)

=1

The calculation done in (5.1.14) and the estimate of (5.1.13) combined give that √ ||z ||2 M−1 ≤ 2 2|| x ||H 1 . (5.1.15) Next we give estimates for the norm in H 1 of y . We will exploit that the sequence {y } is essentially disjointly supported. Let K = supp S(y ). For any interval J in the Haar support of y it follows from (5.1.10) that ∞ j=+2

|J ∩ Kj | ≤

1 |J|. 2


276

By Proposition 3.1.5 and the remark thereafter we obtain that y ||H 1 . ||y ||H 1 ≤ 4||

(5.1.16)

Recall that the decomposition x = y + z is obtained by splitting the Haar support of x , hence || y ||H 1 ≤ || x ||H 1 . (5.1.17) Combining (5.1.16) and (5.1.17) with (5.1.15) we obtain finally, x ||H 1 . |x |T ≤ 4||

Proof of Proposition 5.1.7. We let {n } and {M } be the sequencesdefined by (5.1.6) and (5.1.7). We claim that P : H 1 → X factors through ( T )1 . For x ∈ H 1 , we put x = (Fn − Fn−1 )x, and define

Ex = (x )∞ =1 . By Lemma 5.1.9 the operator E : H 1 → ( T )1 is bounded with norm ≤ 8. Next on ( T )1 define Q by Q ((x )) = P x . By Lemma 5.1.8, Q : ( T )1 → X is a bounded operator with norm ≤ ||P ||. Clearly the mappings E and Q are defined such that P = QE. Hence P factors through ( T )1 . Let us combine this with Proposition 2.1.10 asserting that the identity on ( T )1 factors through ( Hn1 )1 . Expressed in diagrams we have that P

−→

H1 E (

X Q

( and

Id −→

T )1 R

T )1

(

Hn1 )1

(

T ) 1 S .

1 Combining the diagrams gives that P : H 1 → X factors through 1 ( Hn )1 . Con sequently X is isomorphic to a complemented subspace of ( Hn )1 . Proof of Theorem 5.1.3. Let us assume that there exists η > 0 such that lim ||P F ||L2 ,H 1 > η.

→∞

Then it is easy to find an increasing sequence of integers {n } and a sequence {y } in L2 such that ||P (Fn − Fn+1 )y ||H 1 ≥ η

and ||(Fn − Fn+1 )y ||L2 = 1.

(5.1.18)


277

Next we define x = P (Fn − Fn+1 )y . We will show now that in H 1 the sequence {x } is equivalent to the unit vector basis of 2 . Fix real numbers {a }, such that a2 < ∞. Recall the lower bound ||x ||H 1 > η. By passing to a subsequence we may assume that x is disjointly supported over the Haar system. By Proposition 5.1.1 we find the lower 2 estimate in H 1 , 1/2 √ a x 1 ≥ 2 2 a2 ||x ||2H 1 H (5.1.19) 1/2 ≥η . a2 Next we observe that the converse estimate is also true. By (5.1.18), a x ≤ ||P || a (F − F )y 1 n n +1 H1 H (5.1.20) 1/2 2 ≤ ||P || a . Combining (5.1.19) and (5.1.20) shows that the sequence {x } is equivalent to the unit vector basis of 2 . In summary we showed that if lim→∞ ||P F ||2,1 > 0, then X contains a subspace isomorphic to 2 . And conversely by Proposition 5.1.7, iflim→∞ ||P F ||2,1 = 0, then X is isomorphic to a complemented subspace of ( Hn1 )1 . Thus holds the dichotomy announced in Theorem 5.1.3.

Intrinsic description By Theorem 1.1.4 the Rademacher functions span complemented copies of 2 in the reflexive Lp spaces. Indeed, by Khintchine’s inequality the unit vectors in 2 are equivalent to the Rademacher functions and the orthogonal projection onto their span is bounded in Lp . Also for H 1 is it true that the Rademacher functions span complemented copies of 2 . This fact is a direct consequence of the definitions. Of particular interest are the converse questions which we describe as follows: Start with a complemented subspace X ⊂ H 1 , and assume that X is isomorphic to 2 . Let {fi } in X be equivalent to the unit vector basis of 2 . We ask now whether isomorphic properties (like being equivalent to the unit vectors in 2 ) are reflected in pointwise estimates of {fi }. We turn to important work of J. Bourgain describing intrinsically the uniformly complemented copies of 2n in H 1 . Loosely speaking we will prove that their unit vectors have to look (more or less) like Rademacher functions. A result like this is not only pleasing aesthetically, but also it leads us to an important isomorphic invariant for H 1 . Below we will test the invariant on the space H 1 (2 ) and obtain that H 1 and H 1 (2 ) are not isomorphic Banach spaces. This is how J. Bourgain solved the important dimension conjecture formulated by B. Maurey in [144].


278

Fix a complemented copy of 2n in H 1 . This is done by specifying linear operators E, P , such that Id 2n −→ 2n E P . H1 We denote the unit vectors in 2n by {ej : j ≤ n}. Consulting the above diagram shows that the H 1 functions Eej , j ∈ {1, . . . , n} satisfy the following upper and lower 2 estimates. ⎛ ⎞1/2 ⎞1/2 ⎛ n n n 1 ⎝ 2 ⎠ ⎝ α ≤ αj Eej αj2 ⎠ . (5.1.21) ≤ ||E|| ||P || j=1 j j=1 1 j=1 H

We restate the above estimates in a more symmetric form. Notice (5.1.21) implies that the map 1/2 T : ej → (||P || · ||E||) Eej extends linearly to an isomorphism from 2n onto its range in H 1 so that for x ∈ 2n ,

||E|| ||P ||

1/2

x 2n ≤ T x H 1 ≤

||P || ||E||

1/2

x 2n ,

and EP is an idempotent map onto T (2n ). Dualizing the operators E, P gives a complemented copy of 2n in BMO. Let ∗ P denote the adjoint operator of P. The BMO functions P ∗ ej , j ∈ {1, . . . , n} are equivalent to the unit vectors of 2n , ⎞1/2 ⎛ n n 1 ⎝ 2 ⎠ ∗ αj ≤ αj P ej ∗ ||E || j=1 j=1

BMO

⎛ ⎞1/2 n ≤ ||P || ⎝ αj2 ⎠ . j=1

As Id = P E the duality between H 1 and BMO preserves biorthogonality as follows, Eej , P ∗ ei = ej , ei (5.1.22) = δi,j . The next proposition clearly indicates that the 2n unit vectors fj = Eej and hj = P ∗ ej display the same pointwise behavior as (rescaled copies of) Rademacher functions. We will apply Proposition 5.1.10 repeatedly below. To ease the reference we rename the operators as M = E and N = P ∗ . n n n(n − 1) S(fj )S(hi ) ≥ 2 . Proposition 5.1.10. 64 ||M ||4 ||N ||4 i=1 j=i+1


279

The analytic backbone of this estimate is the following variant of Fefferman’s inequality. Lemma 5.1.11. Let h ∈ BMO, g ∈ L1 with g ≥ 0, and let f ∈ H 1 , then

1

gS(h) dx ≥ 0

1 2

1

0

f h dx − 2||S(f ) − g||L1 ||h||BMO .

Proof. Following is a repetition of the proof of Fefferman’s inequality. First expand h ∈ BMO into its Haar series h = bJ hJ . Let I be a fixed dyadic interval. The square function localized to Q(I) = {J ⊆ I} is given by b2J h2J (x). S 2 (h|Q(I))(x) = J∈Q(I)

If x ∈ [0, 1] satisfies S(h)(x) < ∞, then there exists a maximal dyadic interval I(x) so that for S(h | Q(I(x)))(x) the following pointwise estimate holds, √ S(h | Q(I(x)))(x) ≤ 2||h||BMO , Moreover it is clear that S(h | Q(I(x)))(x) ≤ S(h)(x). Next recall (1.2.3) asserting that f h dx ≤ 2

S(f )(x) · S(h | Q(I(x)))(x) dx.

Inserting g gives f h ≤ 2 (S(f ) − g)(x)S(h | Q(I(x)))(x) dx + 2 g(x)S(h | Q(I(x)))(x) dx ≤ 4||S(f ) − g||1 ||h||BMO + 2 g(x)S(h) dx.

Proof of Proposition 5.1.10. Fix r ≤ n and i ∈ {1, . . . , r}. Recall that we put fi = M ei and hi = N ei . Let ⎧ ⎛ ⎞1/2 ⎫ ⎪ ⎪ r ⎨ ⎬ 2 ⎝ ⎠ S (fj ) gi = min S(fi ), , ⎪ ⎪ ⎩ ⎭ j=i+1 and apply Lemma 5.1.11 with this choice of gi . By the biorthogonality relation (5.1.22) we have fi , hi = 1. Thus we find that 1 gi S(hi ) ≥ fi , hi − 2||S(fi ) − gi ||1 ||hi ||BMO 2 1 ≥ − 2i ||N ||, 2


280

where we put i = ||S(fi ) − gi ||1 , and used that ||hi ||BMO ≤ ||N ||. We continue with the pointwise minorization r

⎛ S(fj ) ≥ ⎝

j=i+1

⎞1/2

r

S 2 (fj )⎠

j=i+1

≥ gi . Integrating against S(hi ) gives ⎧ ⎨ r ⎩

⎫ ⎬ S(fj )

j=i+1

⎭

S(hi ) ≥

gi S(hi ) (5.1.23)

≥

1 − 2i ||N ||. 2

1 is dominating while 2 i ||N ||BMO is a negligible error term because i is small. Unfortunately this need not be true for each individual i , however in the average this statement is true. Indeed in Lemma 5.1.12 we verify that the following inequality holds, We would like to say now that in the above estimate,

r

i ≤

√

2||M ||r 1/2 .

i=1

Now continue the proof by specifying r = 64||M ||2 ||N ||2 (where we tacitly assume that r is an integer). Take the sum of the estimate (5.1.23) over i ≤ r. This gives r r

r − 2i ||N || 2 i=1 √ r ≥ − 2 2r||M || ||N || 2 ≥ 1. r

S(fj )S(hi ) ≥

i=1 j=i+1

(5.1.24)

The last inequality in (5.1.24) follows from our choice of r. Going over this argument again one notices that there is nothing special about the set {1, . . . , r}, all we used was a lower bound for its cardinality. Hence we showed that for every subset A ⊆ {1, . . . , n} with cardinality larger than r we have S(fj )S(hi ) ≥ 1. (5.1.25) i∈A {j≥i+1}∩A

We continue with a simple double counting argument. For r ≤ n we let Pr be the set of all subsets A ⊆ {1, . . . , n} of cardinality r. Note that Pr is a set of

5.1. Complemented copies of Hilbert spaces cardinality

281

n r

. Given A ∈ Pr we define the indicator function 1 i≥j+1 and i, j ∈ A, bA (i, j) = 0 otherwise.

Now for i, j, ≤ n we consider

ai,j =

S(fj )S(hi ).

Note that for given i, j, ∈ {1, . . . , n} the cardinality of the set {A ∈ Pr : i, j, ∈ any A} equals n−2 r−2 . Changing the order of summation twice gives n n

aij bA (i, j) =

A∈Pr i=1 j=1

=

=

n n

aij bA (i, j)

i=1 j=1 A∈Pr n n

aij |{A ∈ Pr : i, j ∈ A}|

i=1 j=i+1 n n i=1 j=i+1

aij

(5.1.26)

n−2 . r−2

Next we recall (5.1.25) asserting that for any set A of cardinality greater than r we have n n aij bA (i, j) ≥ 1. n

i

j

We also recall that |Pr | = r . Inserting this in identity (5.1.26) gives n n n n−2 ≤ aij , r r − 2 i=1 j=i+1 or simply,

n n

aij ≥

i=1 j=i+1

n(n − 1) . r(r − 1)

Recalling that r is the integer part of 64||M ||2 ||N ||2 , and aij = the desired estimate.

S(fj )S(hi ) gives

We still need to prove that most of the numbers i = ||S(fi ) − gi ||1 are small. This is expressed in the following auxiliary estimate which finishes the proof of Proposition 5.1.10. Lemma 5.1.12. The error terms i = ||S(fi ) − gi ||1 satisfy the upper bound r i=1

i ≤

√

2||M ||r 1/2 .


282

1/2 r 2 Proof. Let 1 ≤ i ≤ r − 1 put Ai = , and Ar = 0. We start j=i+1 S (fj ) with the pointwise estimate ⎧ ⎛ ⎞1/2 ⎫ ⎪ ⎪ r ⎨ ⎬ ≤ Ai−1 − Ai . S 2 (fj )⎠ 0 ≤ S(fi ) − min S(fi ), ⎝ ⎪ ⎪ ⎩ ⎭ j=i+1 Evaluating the telescoping sum from i = 1 to i = r and integrating gives ⎞1/2 ⎛ 1 r r ⎝ i ≤ S 2 (fj )⎠ . (5.1.27) i=1

0

j=1

Theorem 1.1.3 (the Khintchine inequality) implies that ⎛ ⎞1/2 ⎛ ⎞ 1 r r √ ⎝ S 2 (fj )⎠ ≤ 2 S⎝ rj (s)fj ⎠ ds, 0

j=1

(5.1.28)

j=1

where {rj } denotes the Rademacher system. Taking the integral of the pointwise estimate (5.1.28) we obtain that ⎞1/2 ⎛ 1 1 r r √ 2 ⎠ ⎝ S (fj ) ≤ 2 rj (s)fj (5.1.29) ds. 0 0 j=1 1 j=1 H

Now we arrive at a position that allows us to exploit the hypothesis that {fj } is equivalent to the unit vector basis of 2 . Indeed, r ≤ ||M ||r 1/2 , r (s)f j j 1 j=1 H

for s ∈ [0, 1]. It remains to insert this estimate into (5.1.29). Then Lemma 5.1.12 follows from (5.1.27).

5.2

Complemented copies of Hn1

We prove that BMO is primary thereby showing a further dichotomy for complemented subspaces of BMO and H 1 . We obtain J. Bourgain’s pointwise estimates for basis functions of uniformly complemented copies of Hn1 in H 1 . A Banach space X is primary if for every projection P : X → X one of the spaces P (X) or ( Id − P )(X) is isomorphic to X. We are concerned with Banach spaces X for which (5.2.1) X∼ X , p = 1 or p = ∞. p

5.2. Complemented copies of Hn1

283

In view of A. Pelczy´ nski’s decomposition principle (Theorem 2.2.1) a Banach space X satisfying (5.2.1) is primary, if for every projection P : X → X the range P (X) or its complement ( Id − P )(X) contains a complemented subspace isomorphic to X. Equivalently X is primary, if one of the operators H = P or H = Id − P satisfies Id X −→ X M↓ ↑N H

−→

X

X

where M and N are bounded and linear.

Dichotomies We now review Theorem 4.2.6 (on subsequences of the Haar basis) with the aim of seeing it as a result on factorization of operators. Let C be a collection of dyadic intervals, and let TC be the orthogonal projection given by TC (f ) =

I∈C

f, hI

hI . |I|

Clearly one of the collections C or D \ C has unbounded Carleson constant. Hence, by Theorem 4.2.6 the identity on BMO can be factored through H = TC or H = Id − TC , so that Id BMO → BMO E↓ ↑P BMO BMO → H with bounded operators E and P. In this section we will extend this result from the special case of orthogonal projections to arbitrary operators on BMO. The following theorem shows that BMO is a primary space. Theorem 5.2.1. For any operator T : BMO → BMO, either H = T or H = Id−T satisfies Id BMO −→ BMO E↓ ↑P BMO

H

−→ BMO

where E and P are bounded operators . The proof of Theorem 5.2.1 consists of two independentcomponents. First, by P. Wojtaszczyk’s Theorem 2.2.3, we replace BMO with ( BMOn )∞ and we show that it suffices to prove Theorem 5.2.1 for diagonal operators of the form BMOn )∞ D:( BMOn )∞ → ( (xn ) → (Dn xn ).


284

In the second step we prove a localized version of Theorem 5.2.1. (A localized version of a given theorem is characterized by the following two properties: First the spaces involved are replaced by their finite-dimensional analogs. Second the norm estimates appearing in the conclusion of the theorem are independent of the dimension.) Following is a localized version of the assertion that BMO is primary. Theorem 5.2.2. For any n ∈ N there exists N = N (n) such that for any operator T : BMON → BMON , the identity on BMOn factors through H = T or H = Id − T. That is, Id −→

BMOn E↓ BMON

BMOn ↑P

H

−→ BMON

where E, P are linear operators satisfying ||E|| ||P || ≤ 16. The proof of Theorem 5.2.2 involves combinatorial difficulties which we isolate and address in Theorem 5.2.3 below. Its effect is that we find a well complemented copy of BMOn on which T and Id − T are acting similar to a multiplier operator on the Haar basis. Thus with Theorem 5.2.3 we find a well complemented copy of BMOn which has a basis of approximate eigenvectors for T. Moreover, the basis is well equivalent to the Haar system in BMOn . Recall that D denotes the collection of all dyadic intervals. For N ∈ N we defined also that DN = {J ∈ D : |J| ≥ 2−N }. Theorem 5.2.3. Let n ∈ N. Let log N = 8n . Suppose that H : BMON → BMON is a linear operator with H = 1. Then in DN there exist collections of pairwise disjoint dyadic intervals E1 , . . . , E2n+1 −1 so that {hI : I ∈ Ei }, i ≤ 2n+1 − 1, bi = satisfy the following condition: 1. The orthogonal projection Q(f ) =

2n+1 −1

f,

i=1

bi bi ||bi ||2 ||bi ||2

is a bounded operator on BMO with norm ≤ 4. 2. The system b1 , . . . , b2n+1 −1 is equivalent to the Haar basis of BMOn with constants independent of n. 3. For each bi ,

i=j

|Hbj , bi | ≤ 4−i ||bi ||22 .

(5.2.2)


285

As we saw before the Gamlen–Gaudet construction is a very flexible tool allowing us to inductively produce a variety of complemented copies of BMOn . In the proof below we will set up the inductive procedure so that after the completion of the last step there holds the orthogonality condition (5.2.2) We will see later that it implies the eigenvector relation Hbi = mi bi + a small error , and that H, almost leaves invariant the linear span of {bi : i ≤ 2n+1 − 1}. Proof of Theorem 5.2.3. In the first step of the construction we simply put b1 = h[0,1] and E1 = {[0, 1]}. Formulating the induction hypothesis we demand that at stage i of the construction we are given E1 , . . . , Ei and b1 , . . . , bi such that |Hbj , bi | + |bi , H ∗ bj | ≤ 4−i ||bi ||22 . (5.2.3) j≤i−1

Let Q(Ei ) = {J ∈ D : there exists I ∈ Ei and J ⊆ I}. Now consider ⎧ ⎫ i ⎨ ⎬ G = J ∈ Q(Ei ) : |Hbj , hJ | + |hJ , H ∗ bj | ≤ 4−i−1 |J| . ⎩ ⎭

(5.2.4)

j=1

We apply now Lemma 5.2.4 with the following specification of the parameters, k = 4i+1 and l = 8i . The conclusion of Lemma 5.2.4 gives j ≤ i210(i+1) such that the collection of pairwise disjoint dyadic intervals {J ∈ G : J ⊆ I, |J| = 2−j |I|} C= I∈Ei

satisfies

|C ∗ | ≥ (1 − 8−i )|Ei∗ |.

(5.2.5)

Now we are ready to define the collection Ei+1 . We use the Gamlen–Gaudet construction. Hence we distinguish between the cases where i is odd or even, or equivalently between right and left. If i is even, then we define ! " Ei+1 = I ∈ C : I ⊆ {t : bi/2 (t) = −1} . If i is odd, then we define ! " Ei+1 = I ∈ C : I ⊆ {t : b(i−1)/2 (t) = +1} . In either case we put bi+1 =

{hI : I ∈ Ei+1 }.


286

The following estimate is an immediate consequence of the fact that Ei+1 ⊆ G. In fact in trying to establish the estimate (5.2.6) we are forced to define the collection G as we did above. i

|Hbi+1 , bj | + |bj , H ∗ bi+1 | ≤ 4−(i+1) ||bi+1 ||22 .

(5.2.6)

j=1

As we used the Gamlen–Gaudet construction to define {Ej : j ≤ 2n+1 − 1} it is clear that they satisfy Jones’s compatibility condition (J). Applying Theorem 1.5.9 gives the upper bound for the norm of the projection Q. By (5.2.5), the system {bj : j ≤ 2n+1 − 1} is equivalent to the Haar basis of BMOn . Now we verify that the biorthogonality condition (5.2.2) is satisfied, that is we show that |Hbj , bi | ≤ 4−i ||bi ||22 , for every bi . (5.2.7) {j:i=j}

Recall that by (5.2.3) we obtain at once that |Hbj , bi | ≤ 4−i ||bi ||22 . j≤i−1

Next for j ≥ i + 1 the definitions of G in the j-th step of the construction and (5.2.3) imply the estimates |Hbj , bi | = |bj , H ∗ bi | ≤

j−1 k=1 −j

≤4

|bj , H ∗ bk |

(5.2.8)

||bj ||22

≤ 4−j ||bi ||22 . In the last inequality we used the obvious estimate that ||bj ||22 ≤ ||bi ||22 for i ≤ j. Finally we take the sum of the estimates (5.2.8) where j ranges from i + 1 to 2n+1 − 1. This gives |Hbj , bi | ≤ 4−i ||bi ||22 . (5.2.9) j≥i+1 Now we formulate and prove Lemma 5.2.4. Let I be a dyadic interval. Let x ∈ BMO and y ∈ H 1 such that, ||x||BMO ≤ 1

and

||y||H1 ≤ |I|.

Fix k ∈ N and consider the collection of intervals 1 B = J ⊆ I : |x, hJ | + |y, hJ | ≥ |J| . k We claim that the collection B is thin in the following quantitative sense:


287

Lemma 5.2.4. Let l ∈ N and let A = 64k2 l2 + 1. Then there exists j ∈ {1, . . . , A} such that Bj = {J ∈ B : |J| = 2−j |I|} satisfies

|J| ≤

J∈Bj

1 |I|. l

Proof. We assume that Lemma 5.2.4 is false and will derive a contradiction. So we are assuming that each of the collections Bj covers a subset of measure ≥ |I|/l. Recall that for J ∈ Bj , the coefficients satisfy the estimate 1 |J|. (5.2.10) k Summing the estimates (5.2.10) over all J ∈ Bj , and all j ≤ A, gives the lower bound A A 1 |x, hJ | + |y, hJ | ≥ |J| k j=1 j=1 J∈Bj J∈Bj (5.2.11) A|I| . ≥ kl On the other hand we obtain an upper bound for the right-hand side of (5.2.11) by applying Fefferman’s inequality. We start by observing that A √ ±hJ ≤ A|I|, j=1 J∈Bj 1 |x, hJ | + |y, hJ | ≥

H

and also that

A ±hJ j=1 J∈Bj

≤

√ A.

BMO

Now we rewrite, and using Fefferman’s inequality we obtain the estimates A j=1 J∈Bj

|x, hJ | + |y, hJ | =

A

±x, hJ ± y, hJ

j=1 J∈Bj

√ √ ≤ 4 A|I| ||x||BMO + 4 A||y||H 1 √ ≤ 8 A|I|.

(5.2.12)

(Recall that ||x||BMO ≤ 1 and ||y||H 1 ≤ |I|.) Comparing the lower bound (5.2.11) and the upper bound (5.2.12) gives an estimate for A. Indeed, we obtain that A ≤ 64k2 l2 . Note however, that this estimate contradicts our previous choice of A.

By a straightforward application of Theorem 5.2.3 we show now that BM On satisfies the localized version of being primary.


288

Proof of Theorem 5.2.2. Fix n ∈ N. Then we choose N = N (n) such that log log N = 4n2 . Let T : BMON → BMON be a given operator of norm ≤ 1. Let 2 I(n) = {1, . . . , 2n +1 − 1}. Apply Theorem 5.2.3 to T. This gives the system {bi : i ∈ I(n)} disjointly supported over the Haar system and equivalent to the Haar functions in BMOn2 . Moreover the orthogonal projection

Q(f ) =

i∈I(n)

f,

bi bi ||bi ||2 ||bi ||2

(5.2.13)

is a bounded operator on BMO with norm ≤ 4, and for each bi ,

|T bj , bi | ≤ 4−i ||bi ||22 .

i=j

Let Bi = supp bi . We obtained the functions bi by the Gamlen–Gaudet construction. Hence the collection {Bi } forms a nested family of sets. Now consider L = {Bi : |T bi , bi | ≥ ||bi ||22 /2}, and R = {Bi : |( Id − T )bi , bi | ≥ ||bi ||22 /2}. By the triangle inequality, we have {Bi : i ∈ I(n)} = R ∪ L. Hence either R or L has Carleson constant ≥ n2 /2. Assume that this holds for L. Thus n2 1 . (5.2.14) |B| ≥ sup 2 A∈L |A| B∈L, B⊆A

Now we apply the condensation Lemma 3.1.4 and Gamlen–Gaudet construction to the nested collection L. This gives subcollections of L, say {CI : |I| ≥ 2−n }, indexed by dyadic intervals and satisfying the following two properties. First the system {bi : Bi ∈ CI }, fI = is equivalent to the Haar basis in BMOn . Second, {CI : |I| ≥ 2−n } satisfies Jones’s compatibility condition (J). We let L = {i : Bi ∈ L}, and define Y = span{bi : i ∈ L},


289

equipped with the norm of BMO. Summarizing the proof up to this point, we state that there exist linear operators I and R, with norm ≤ 4, so that the following diagram commutes. Id BMOn → BMOn I↓ ↑R (5.2.15) Id Y → Y Next we define the following operator on BMO, f, bi bi T bi , bi −1 . Pf = i∈L

By our definition of the index set L, and by the definition of the nested collection L the following estimate holds, ||P f ||BMO ≤ 2||Qf ||BMO ≤ 8||f ||BMO , where Q is the orthogonal projection defined in (5.2.13). Crucial about P is this: On the subspace Y = span{bi : i ∈ L}, P almost inverts the action of T. Precisely we claim that for any g ∈ Y the following estimate holds, 1 (5.2.16) ||P T g − g||BMO ≤ ||g||BMO . 2 To prove the claim we fix g = i∈L bi ci , and expand P T g as PTg =

ci T bi , bi bi T bi , bi −1 +

i∈L

⎧ ⎨ i∈L

⎩

cj T bj , bi

j∈L,j=i

⎫ ⎬ ⎭

bi T bi , bi −1

= g + e. (5.2.17) With Proposition 5.2.3 we have the estimates |cj T bj , bi | ≤ 4−i ||bi ||22 ||g||BMO .

(5.2.18)

j∈L,j=i

Recall that by our definition of the set L, the following lower bound holds, |T bi , bi | ≥

||bi ||22 . 2

(5.2.19)

Inserting (5.2.18) and (5.2.19) into (5.2.17) gives the following bound on the error term e, 1 ||e||BMO ≤ ||g||BMO . 2


290

This proves our claim in (5.2.16). Hence, by a standard perturbation argument there exists a linear operator E with norm ≤ 4 such that Id →

Y E↓ BMON

T

→

Y ↑P

.

(5.2.20)

BMON

We complete the proof by merging the diagrams (5.2.15) and (5.2.20).

Next we turn to proving that it suffices to show Theorem 5.2.1 for diagonal operators acting on the ∞ sum of the spaces BMOn . By the next proposition for N >> n we can always find in BMON a copy of BMOn and a projection onto it that (almost) annihilates a given n-dimensional subspace in BMON . We prove this assertion by adapting the idea of Lemma 5.2.4. Proposition 5.2.5. Let N ∈ N, and let n ∈ N such that 4n ≤ log log N. Let E be an n-dimensional subspace of BMON . Then in DN there exist pairwise disjoint collections {BJ : J ∈ Dn }, so that the system bJ =

{hI : I ∈ BJ }

is equivalent to the unit vector basis of BMOn , the orthogonal projection Qn (x) =

x,

J∈Dn

bJ bJ ||bJ ||2 ||bJ ||2

is bounded with ||Qn ||BMO ≤ 4 and so that for any x ∈ E, ||Qn (x)||BMO ≤ 4

log n ||x||BMO . n

Proof. Fix n ∈ N. And fix an n-dimensional subspace E ⊂ BMON . In {x ∈ E : ||x|| = 1}, we choose a net {x1 , . . . , xM } of width 1/n. As dim E = n this can be done with M = (2n)n . Let j ≤ M, then we claim that the collection Lj = {J ∈ D : n|xj , hJ | ≥ |J|} satisfies the n2 -Carleson condition. Indeed, for I ∈ Lj , we have the upper bound xj ,

hJ 2 | ≥ n−2 |J|. |J|

5.2. Complemented copies of Hn1 Hence,

291

|J| ≤

J∈Lj ∩I

n2 |xj , hJ |2 |J|−2 |J|

J∈Lj ∩I

≤ n2 ||xj ||2BMO |I| ≤ n2 |I|. Consequently the Carleson constant of the collection G = DN \ {L1 ∪ · · · ∪ LM } is at least as large as log N − M n2 . We apply now the condensation Lemma 3.1.4 to G. Then the Gamlen–Gaudet construction allows us to select in G collections {BJ : |J| ≥ 2−n } which satisfy the Jones compatibility condition (J). Moreover the system {bJ : |J| ≥ 2−n } defined by bJ =

{hI : I ∈ BJ }

is equivalent to the Haar basis in BMOn . Note that we selected BJ ⊆ G, hence we ensured that for I ∈ BJ we have |xj , hI | ≤ n−1 |I|. Consequently the following estimate holds for any xj , |xj , bJ | ≤

{|xj , hI | : I ∈ BJ } ≤ n−1 {|I| : I ∈ BJ }

(5.2.21)

= n−1 ||bJ ||22 . Let x ∈ E be arbitrary satisfying ||x||BMO = 1. Then there exists xj , j ≤ M such that ||x − xj ||BMO ≤ 1/n. With Fefferman’s inequality and with (5.2.21) we obtain that for any bJ , J ∈ Dn , the following estimate holds, |x, bJ | ≤ |x − xj , bJ | + |xj , bJ | √ ≤ 2||bJ ||22 /n + ||bJ ||22 /n.

(5.2.22)

Finally we insert the above upper bound of (5.2.22) into the definition of Qn . This shows that Qn almost annihilates x ∈ E, with ||x||BMO = 1. {x, bJ ||bJ ||−2 2 bJ : J ∈ Dn }||BMO 4 {bJ : J ∈ Dn }||BMO ≤ || n 4 ≤ log n. n

||Qn (x)||BMO = ||

By linearity the conclusion of Proposition 5.2.5 follows.


292

Let {bJ : |J| ≥ 2−n } be the system obtained by the Gamlen–Gaudet construction in the course of proving Proposition 5.2.5. By construction the linear extension of the map in : hJ → bJ is invertible on the range of Qn . It satisfies

and

IdBMOn = i−1 n Qn in ,

(5.2.23)

in BMO · i−1 n BMO ≤ 10.

(5.2.24)

Below we use the following canonical projection operators. Let K be a subset of the natural numbers. We define the projection BMOn )∞ → ( BMOn )∞ PK : ( as follows,

[PK (xn )]k =

xk 0

if k ∈ K, else.

Thus the projection PK acts as a 0, 1 multiplier on the entries of the vector (xn ). We say that D acting on ( BMOn )∞ is a diagonal operator if P{l} DP{k} = 0,

for

l = k.

It is convenient to write Pk = P{k} and to identify Pk with the projection onto asserts the k-th coordinate of the vectors (xn ) ∈ ( BMOn )∞ . The next theorem that it suffices to prove Theorem 5.2.1 for diagonal operators on ( BMOn )∞ . Theorem 5.2.6. For any operator T : ( BMOn )∞ → ( BMOn )∞ there exists a diagonal operator BMOn )∞ D:( BMOn )∞ → ( such that Id − D = R( Id − T )E, where R, E are bounded linear operators on ( BMOn )∞ . D = RT E

and

Proof. By induction we will determine sequences of operators {Qn } and {in } such that Id −→ BMOn BMOn with Qn · in ≤ 40, in i−1 n Qn BMON (n) and so that with Q and I given by (5.2.27) and (5.2.28) below, the operator H = QT I is an almost diagonal operator (satisfying (5.2.31) and (5.2.32) ) on the ∞ sum of BMOn .


293

Let k1 = 1. Then let N (1) be large enough so that the conclusion of Proposition 5.2.5 holds for any 1-dimensional subspace of BMON (1) . By a simple compactness argument we select k2 > k1 such that ||PN (1) T P[k2 ,∞[ || ≤ 4−2 ||T ||−1 . At stage n − 1 we have determined integers k1 < · · · < kn−1 , and N (1) < · · · < N (n − 1), with N (i) ∈ [ki , ∞[. By compactness there exists kn > kn−1 , so that ||PN (n−1) T P[kn ,∞[ || ≤ 4−(n−1) ||T ||−1 .

(5.2.25)

Let Vn−1 = BMO1 ⊕ · · · ⊕ BMON (n−1) . We view Vn−1 as a subspace of the the ∞ sum of BMOn . Hence T acts on Vn−1 . Put Wn−1 = T (Vn−1 ) and let wn−1 denote the algebraic dimension of Wn−1 . Next choose N (n) so large that 4wn−1 ≤ log log N (n), and also N (n) ∈ [k(n), ∞). Applying Proposition 5.2.5 yields an orthogonal projection Qn on BMON (n) so that its range is well isomorphic to BMOn , and so that Qn almost annihilates Zn−1 = PN (n) (Wn−1 ). Specifically, for x ∈ Zn−1 , ||Qn (x)||BMON (n) ≤ 4−n ||T ||−1 ||x||BMO . (5.2.26) Define now the operators Q and I acting on ( BMOn )∞ . First fix a sequence (xn ) in ( BMOn )∞ and determine the sequence I((xn )) by [I(xn )]k = in (xn ) if

k = N (n),

(5.2.27)

(with [I(xn )]k = 0 for k = N (n)). The projection Q is given as [Q((xn ))]k = i−1 k Qk (xN (k) ).

(5.2.28)

To analyze Q and I recall that the operators in , Qn defined by Proposition 5.2.5 satisfy the identity IdBMOn = i−1 n Qn in . Hence Q inverts the action I such that Id = QI, and the operator H = QT I

(5.2.29)

satisfies Id − H = Q( Id − T )I. (5.2.30) Recall that we defined the operator Pk on ( BMOn )∞ to be the orthogonal projection onto the k-th coordinate. The estimates (5.2.25) and (5.2.26) show that H and Id − H are almost orthogonal operators in the following sense. ||Pl HPk || ≤ 4−k ||T ||−1 , (5.2.31) {l:l=k}


294 and

||Pl ( Id − H)Pk || ≤ 4−k ||T ||−1 .

(5.2.32)

{l:l=k}

Next put Tn = Pn HIn : BMOn → BMOn , where In is the canonical embedding on BMOn into ( BMOn )∞ . The operator Tn satisfies sup ||Tn || ≤ ||T ||. n

Finally define the diagonal operator D on (

BMOn )∞ by

D((xn )) = (Tn xn ). By a standard perturbation argument, the estimate (5.2.31) implies that there exist operators E, R on ( BMOn )∞ satisfying Id = RE and D = RT E

and

Id − D = R( Id − T )E.

Summing up, we factored the diagonal operator D through T, and simultaneously we factored Id − D through Id − T. With the above Theorem 5.2.6 we showed that itsuffices to prove the factorization Theorem 5.2.1 for diagonal operators on ( BMOn )∞ . Furthermore we proved the localized factorization Theorem 5.2.2. Combining both gives Theorem 5.2.1. Proof of Theorem 5.2.1. Let T be a bounded operator on ( BMOn )∞ . By Theorem 5.2.6 there exist bounded operators E, R on ( BMOn )∞ and a diagonal operator D such that D = RT E,

and

Id − D = R( Id − T )E.

Let Tn : BMOn → BMOn be the linear maps defining the diagonal operator D, that is D((xn )) = (Tn xn ). Next we apply the local factorization Theorem 5.2.2 to each of the operators Tn . This gives En : BMOn → BMON (n) and Rn : BMON (n) → BMOn with ||En ||, ||Rn || ≤ 2, such that for Hn = Tn or Hn = Id − Tn the identity on BMOn factors through Hn as follows, IdBMOn = Rn Hn En .


295

We may assume that there is an infinite sequence k(n), such that Hk(n) = Tk(n) . (Otherwise we continue with Hk(n) = Id − Tk(n) . ) Now define operators I, P on ( BMOn )∞ by Ek(n) xn if m = k(n), [I(xn )]m = 0 else, [P ((xn ))]m = Rk(m) Tk(m) xk(m) . They satisfy

Id( BMOn )∞ = P DI = P RT EI. Summing up, the identity on ( BMOn )∞ factors boundedly through T. (

remark in closing that the proof of Theorem 5.2.1 shows also that We Hn1 )1 is primary.

Intrinsic description (continued) Next we study pointwise estimates for basis functions in uniformly complemented copies of Hn1 in H 1 . Fix a complemented copy of Hn1 in H 1 , by specifying operators M and P for which Id −→ Hn1 Hn1 M P . H1 Dualizing the operators M and P we obtain complemented copies of BMOn in BMO, Id BMOn −→ BMOn ∗ P M∗ . BMO In H 1 the functions {M hJ } are ||M || ||P || equivalent to the Haar basis of Hn1 , and in BMO the functions {P ∗ hJ } are ||M || ||P || equivalent to the Haar basis in BMOn . Moreover H 1 , BMO duality preserves the biorthogonality of Haar functions. For each I, J ∈ Dn , we have that M hJ , P ∗ hI = hJ , hI .

(5.2.33)

For notational simplicity we write N = P ∗. The next theorem describes the pointwise behavior of the square functions for {M hJ : J ∈ Dn }, and also for {N hI : I ∈ Dn }. We find remarkable similarities to the original system, {hJ : J ∈ Dn }. To formulate this precisely we use the following convention. Let E0 , . . . , EA be disjoint collections of dyadic intervals, each of which


296

consists of pairwise disjoint intervals. We call E0 , . . . , EA linearly ordered if the following two conditions hold, ∗ E0∗ ⊇ · · · ⊇ EA

and for i ∈ {0, . . . , A}, we have that I ∈ Ei , J ∈ Ei+1 implies |J| ≤

1 |I|. 2

The collections E0 , . . . , EA are inducing functions in BMO and H 1 , {N hJ : J ∈ Ei } and gi = {M hJ : J ∈ Ei }|Ei∗ |−1 . ki = The pointwise behavior of their square functions is the content of the next theorem. Its inductive proof is an iteration based on Proposition 5.1.10 that describes the unit vectors of complemented Hilbertian subspaces of H 1 . Theorem 5.2.7. In Dn there exist linearly ordered collections E0 , . . . , EA such that their union E0 ∪ · · · ∪ EA satisfies the Carleson condition with constant 2, and $ A−1 A + (ln n)1/2 , S(ki ) S(gA ) ≥ 32R i=0 where we put R = 642 ||M ||4 ||N ||4 . Remarks. 1. Note that gA ∈ H 1 . Hence the non-negative function S(gA ) is integrable. In the conclusion of Theorem 5.2.7 the role of S(gA ) is that of a weight. It compensates for possible dilations on the BMO side due to the linear map N. 2. Note that if E0 ∪ · · · ∪ EA satisfy the 2-Carleson condition, then the BMO functions k1 , . . . , kA are well equivalent to the unit vectors of ∞ A. 3. Below we repeatedly use the following observation. Let (Ω, µ) be a probability space and let F : Ω → R+ be non-negative, bounded and measurable, satisfying F dµ > α, Ω

for some α > 0. Then µ{F > α/4} ≥ 3α(4 F ∞ )−1 .

(5.2.34)

Indeed, decompose Ω = {F > α/4} ∪ {F ≤ α/4} and estimate the integral accordingly. This gives F dµ ≤ F ∞ µ{F > α/4} + α/4. Recall that

Ω

Ω

F dµ > α and subtract α/4 to obtain 3α ≤ 4 F ∞ µ{F > α/4}.


297

Proof of Theorem 5.2.7. The following proof proceeds by induction where the number of steps is not specified at the beginning. Instead, the induction argument contains a stopping criterion which (if satisfied) implies the validity of the Theorem 5.2.7. If on the other hand the stopping criterion is not satisfied during C(ln n)1/2 iterations of the induction argument, then again we will be able to conclude that Theorem 5.2.7 holds true. Step 1. We cover the unit interval using the collections Gi = {I : |I| = 2−i }, for i ∈ {1, . . . , n}. These collections are contained in Dn , and generate unit vectors of 2n in H 1 respectively BMO. Now put {M hJ : J ∈ Gi }, hi = {N hJ : J ∈ Gi }. fi = In H 1 the functions {fi } are well equivalent to the unit vector basis of 2n . The same holds in BMO for the functions {hi }. Moreover by (5.2.33) they are linked by fi , hj = δi,j . Apply now Theorem 5.1.10 to {fi } and {hi }. This gives n n

S(hi )S(fj ) ≥

i=1 j=i+1

n(n − 1) , R

(5.2.35)

where R = 642 ||M ||4 ||N ||4 . We continue with the proof if for each i ≤ n and j ≥ i + 1, (ln n)1/2 . (5.2.36) S(hi )S(fj ) ≤ R (If condition (5.2.36) does not hold, then we stop, since then there is nothing more to prove, and we put A = 1.) Define Ω = {(i, j) : i + 1 ≤ j ≤ n, i ≤ n} and let µ be the normalized counting measure on Ω. Next define F (i, j) = S(hi )S(fj ) where (i, j) ∈ Ω. By (5.2.35) and (5.2.36) the observation (5.2.34) shows that there exists a constant K(1), with K(1) > n(4(ln n)1/2 )−1 , together with i0 ≤ n and a set of indices D ⊆ {i0 +1, . . . , n}, so that the cardinality of D equals K(1), and so that the following set of K(1) inequalities hold true, 1 , ∀j ∈ D. S(hi0 )S(fj ) ≥ 4R We put E 0 = G i0

and

k0 = hi0 .


298

Then we relabel the collections {Gj : j ∈ D} as G1,1 , . . . , G1,K(1) , and accordingly we rename the H 1 functions {fj : j ∈ D} as f1,1 , . . . , f1,K(1) . This completes the first step of the construction. After B steps. Next we claim that iterating the induction argument B times gives a constant K(B) satisfying K(B) ≥ n(4(log n)1/2 )−B and collections of dyadic intervals GB,j , and Ei with the property that the entire string of collections, E0 , . . . , EB−1 ,

GB,1 , . . . , GB,K(B) ,

is linearly ordered, and satisfies ∗ |I ∩ Ei+1 |=

and

1 |I|, 2

for I ∈ Ei

∗ ∗ GB,j = EB−1 ,

and i ∈ {0, . . . , B − 2},

for j ∈ {0, . . . , K(B)}.

Moreover for the functions induced by the collections GB,j and Ei , that is, for fB,j = 2B−1

{M hJ : J ∈ GB,j } and ki =

the following set of K(B) inequalities hold true, $ B−1 B , S(ki ) S(fB,j ) ≥ 4R i=0

{N hJ : J ∈ Ei },

∀j ≤ K(B).

(5.2.37)

The induction argument. We assume the iteration has been carried out B times and we describe the next step. Start by splitting GB,1 into L and R such that |I ∩ L∗ | = |I ∩ R∗ | =

1 |I|, 2

∀I ∈ EB−1 .

Then for j ≤ K(B) split GB,j accordingly, Aj = {J ∈ GB,j : J ⊆ L∗ } and

Bj = {J ∈ GB,j : J ⊆ R∗ }.

(5.2.38)


299

With Aj , Bj form the H 1 functions aj = 2B

{M hJ : J ∈ Aj } and

bj = 2B

{M hJ : J ∈ Bj }.

This choice gives S(fB,j ) ≤

1 (S(aj ) + S(bj )). 2

(5.2.39)

Combining the estimate (5.2.39) and the induction hypothesis (5.2.37) gives a set of indices D ⊆ {1, . . . , K(B)} of cardinality ≥ K(B)/2, such that one of the following statements is true, $ B−1 B , ∀j ∈ D, S(ki ) S(aj ) ≥ 4R i=1 (5.2.40) $ B−1 B , ∀j ∈ D. S(ki ) S(bj ) ≥ 4R i=1 We assume without loss of generality that the first set of estimates holds true. Note that, by construction, the H 1 functions {aj : j ∈ D} are equivalent to the unit vector basis of 2n (with constant ||M || ||P ||). Also the BMO functions hi =

{N hJ : J ∈ Ai },

i∈D

are equivalent to the unit vector basis of 2n in BMO. The duality between H 1 and BMO preserves biorthogonality, i.e., aj , hi = δij ,

i, j ∈ D.

Hence by Theorem 5.1.10,

i∈D {j≥i+1}∩D

S(hi )S(aj ) ≥

|D|(|D| − 1) , R

(5.2.41)

where |D| denotes the cardinality of D. We assume that for each i ∈ D and j ∈ D with j ≥ i + 1, (ln n)1/2 . (5.2.42) S(hi )S(aj ) ≤ R If the condition (5.2.42) does not hold, then we may stop, and the proof is complete. Otherwise we continue as follows. By (5.2.41) and (5.2.42) the observation (5.2.34) implies that there exists i0 ∈ D and a subset E ⊆ D ∩ {i0 + 1, . . . , K} such that 1 , j ∈ E, (5.2.43) S(hi0 )S(aj ) ≥ 4R


300

and the cardinality of E, denoted by K(B + 1), satisfies the lower estimate K(B + 1) ≥ n(16(ln n)1/2 )−B−1 . We define EB = Ai0 and kB = hi0 , and relabel {Aj : j ∈ E} as GB+1,1 , . . . , GB+1,K(B+1) . Accordingly we relabel the H 1 functions {aj : j ∈ E} as fB+1,1 , . . . , fB+1,K(B+1) . A moment’s reflection shows that by (5.2.38) the collections of dyadic intervals E0 , . . . , EB ,

GB+1,1 , . . . , GB+1,K(B+1)

are linearly ordered. The measures covered by E0 , . . . , EB decrease uniformly as follows, 1 |I|, for I ∈ Ei and i ∈ {0, . . . , B − 1}. 2 cover the same set as EB+1 , that is

∗ |= |I ∩ Ei+1

The collections GB+1,j

∗ ∗ EB+1 = GB+1,j

j ∈ {1, . . . , K(B)}.

Finally we add the inequalities (5.2.40) and (5.2.43) and with the notation just introduced we obtain that $ B B+1 , ∀j ≤ K(B + 1). S(ki ) S(fB+1,j ) ≥ 4R i=0 This completes the induction step. Conclusion. There are two possibilities how this process will terminate. First it will terminate when at some stage of the construction we select a set of indices D for which there exist i ∈ D and j ∈ D with j ≥ i + 1 such that (ln n)1/2 S(hi )S(fj ) ≥ . R In that case the proof is complete. The second possibility is when the index set E becomes empty (that is when the estimate for the lower bound on the cardinality of E becomes < 1). This can happen only when

B 1 n < 1, 16(ln n)1/2 hence B ≥ 16−1 (ln n)1/2 , so that by (5.2.37) the theorem is proven.


301

H 1 with values in 2 Next we prove that H 1 is not isomorphic to the space of H 1 functions taking values in the Hilbert space 2 . This space denoted by H 1 (2 ) consists of those measurable 2 -valued functions f (t) = hI (t)xI , xI ∈ 2 , for which the square function Sf (t) =

h2I (t)||xI ||2

1/2

is integrable. The non-isomorphism of H 1 and H 1 (2 ) is a striking result, especially in view of the obvious analogy in the definitions for H 1 and H 1 (2 ), and also in view of the multitude of analogous theorems that hold in H 1 and H 1 (2 ). The assumption of the next theorem states that there exits a complemented subspace of H 1 which is isomorphic to H 1 (2 ). From this hypothesis we will derive a condition which is in contradiction to Theorem 5.2.7. Hence H 1 does not contain any complemented copy of H 1 (2 ). In particular the spaces H 1 and H 1 (2 ) are not isomorphic. Theorem 5.2.8. Assume that there exists a complemented subspace of H 1 which is isomorphic to H 1 (2 ). Then the following statement is true: For any n ∈ N there exist operators M : Hn1 → H 1 and Q : H 1 → Hn1 satisfying the following conditions: 1. Id = QM, and ||Q|| ||M || ≤ C, with C independent of n. 2. For every choice of linearly ordered collections of dyadic intervals E0 , . . . , EA the functions {Q∗ hJ : J ∈ Ei }, ki = and f=

∗ −1 {M hJ : J ∈ EA }|EA | ,

satisfy the estimate A−1 i=0

A−1 √ S(ki ) S(f ) ≤ A||M || ki $

i=0

. BMO

Comment. Before we prove this theorem we will point out why its conclusion contradicts the assertions of Theorem 5.2.7. √ Note that the right-hand side of the above estimate contains the crucial factor A. Next consider the collections E0 , . . . , EA obtained by Theorem 5.2.7. They are linearly ordered and the union E0 ∪ · · · ∪ EA


302

√ satisfies the 2-Carleson condition. Hence A−1 k ≤ 2||Q∗ ||. Now we i i=0 BMO compare the upper estimate of our present theorem, $ A−1 √ S(ki ) S(f ) ≤ 2A M · Q , i=1

with the lower estimate of Theorem 5.2.7, $ A−1 A + (log n)1/2 , S(ki ) S(f ) ≥ 32R i=1 where R = 642 M 4 · N 4 . Clearly for n large enough we have two conflicting assertions. Thus the hypothesis of Theorem 5.2.8 led to a contradiction, so that H 1 and H 1 (2 ) are not isomorphic Banach spaces. Proof of Theorem 5.2.8. We assume that H 1 contains a complemented subspace isomorphic to H 1 (2 ). Let E, P be bounded linear operators such that Id

H 1 (2 ) −→ H 1 (2 ) E P . H1 Let {ek : k ∈ N} be the orthonormal unit vector basis of 2 . Note that the Hilbert space-valued functions {hI ek : I ∈ D, k ∈ N} are an unconditional basis in H 1 (2 ). Now fix I ∈ D. Then hI ek → 0 weakly in H 1 (2 ) as k → ∞. Consequently, E(hI ek ) → 0 weakly in H 1 ,

(5.2.44)

as k → ∞. Observe that (5.2.44) implies that for > 0 and δ > 0 fixed there exists m = m(, δ) and a finite collection S of dyadic intervals so that for z = E(hI em ) the following conditions hold. If K ∈ S, and

z −

K∈S

|K| ≤ δ,

(5.2.45)

hK

H 1 ≤ . |K|

(5.2.46)

then

z, hK

Next we chose n ∈ N, and we start the construction of the operators M and Q which will eventually satisfy the conclusions of Theorem 5.2.8. Let Dn denote


303

the collection of dyadic intervals of length≥ 2−n . There are 2n+1 − 1 intervals in Dn , we enumerate them as I1 , . . . , I2n+1 −1 , using lexicographic order on the intervals. Next we fix k > 0 such that 2n+1 −1

k < (||E|| ||P ||8)−1 .

(5.2.47)

k=1

Now we start a gliding hump process at the smallest, right-most interval I2n+1 −1 and work backwards toward I1 = [0, 1]. We apply the remark following (5.2.44) with δ = 1, = 2n+1 −1 and I = I2n+1 −1 . Let S2n+1 −1 = S be the finite collection of dyadic intervals so that the conditions (5.2.45) and (5.2.46) hold true. Then put δ=

1 inf{|L| : L ∈ S}. 2

Apply the remark again with this choice of δ with = 2n+1 −2 and I = I2n+1 −2 . We continue inductively until we reach I1 . Thus we obtain a sequence (nk ) and pairwise disjoint collections of dyadic intervals {Sk : k ≤ 2n+1 − 1} so that ||zk − Qk (zk )||H 1 ≤ k ||zk ||,

(5.2.48)

where zk = E(hIk enk ), and where Qk is the orthogonal projection onto span{hJ : J ∈ Sk } given by Qk f =

f, hJ

J∈Sk

hJ . |J|

Furthermore the collections {Sk : k ≤ 2n+1 − 1} are inversely ordered in the following sense. K ∈ Sk and L ∈ Sk+1 implies |K| ≤

1 |L|. 2

Note that for any sequence of scalars {αk }, 2n+1 −1 2n+1 −1 αk hIk enk = αk hIk k=1 k=1 1 2 H ( )

Hence, equipped with the norm of H 1 (2 ) the space Xn = span{hIk enk : k ≤ 2n+1 − 1}

H1

(5.2.49)

.


304

is isometric to Hn1 , and the isometry is given by the linear extension of the map E1 : Xn → Hn1 ,

hIk enk → hIk .

We let R1 : H 1 (2 ) → Xn be the natural restriction operator. R1 is given by a {0, 1} multiplier of the unconditional basis {hI ek : I ∈ D, k ∈ N}, hence ||R1 || ≤ 1. Note that the composition Q0 = E1 R1 P inverts the action of the embedding M0 : Hn1 → H 1 , hIk → zk . Thus Id

−→ Hn1 Hn1 M0 Q0 H1 and ||M0 || ||Q0 || ≤ ||E|| ||P ||. This was the main step of the construction. We exploited the hypothesis to find order inverting embeddings and projections. Now we add two minor modifications to the definition of M0 and Q0 , respectively. We start by modifying M0 . Let wk = Qk zk . By (5.2.48) and (5.2.47) the embedding M : Hn1 → H 1 , hIk → wk , satisfies ||M || ≤ 2||M0 ||, and can be conjugated by a linear operator R : H 1 → Hn1 such that ||R|| ≤ 2||Q0 || and Id

−→ Hn1 Hn1 M R . H1 Now we diagonalize R as follows. Put Q=

2n+1 −1

Tk RQk ,

k=1

where Tk is the orthogonal projection onto the one-dimensional subspace spanned by the function hIk . Then, by inspection, Id = QM, and by unconditionality of the Haar basis we have the norm estimate ||Q|| ≤ √ 2||R||. Summarizing the effect of these modifications we observe that M hIk ∈ span{hJ : J ∈ Sk },

5.2. Complemented copies of Hn1 and also

305

Q∗ hIk ∈ span{hJ : J ∈ Sk }.

We will now verify that the maps M, Q satisfy the conclusion of Theorem 5.2.8. We fix linearly ordered collections E0 , . . . , EA . Then we form ∗ −1 ki = {Q∗ hJ : J ∈ Ei } and f = {M hJ : J ∈ EA }|EA | . Below we prove that g =

A−1

i=0

ki satisfies the surprising estimate,

S(f )S(g) ≤ ||f ||H 1 g BMO .

(5.2.50)

Before that we use (5.2.50) to finish the proof as follows: A−1 i=0

√ S(f )S(ki ) ≤ A S(f )

A−1

1/2 2

S (ki )

i=0

√ = A S(f )S (g) √ ≤ A||f ||H 1 g BMO .

So we are left with proving the estimate (5.2.50). Now we will exploit that the collections Sk are inversely ordered as expressed in (5.2.49). We let F be the σ algebra generated by f . This σ-algebra F is purely atomic. We put g = A−1 i=0 ki , and let φ = S (g) . Now we claim that ||E(φ|F)||∞ ≤ ||g||BMO . To prove this claim we expand g in its Haar series, thus g= aJ h J . Next we fix an atom B of F, and using that {Sk } is an inversely ordered collection of dyadic intervals, we observe the pointwise equality a2J h2J (t), for t ∈ B. (5.2.51) φ2 (t) = J⊆B

The crucial point of equation (5.2.51) is that on the right-hand side the sum is taken over the dyadic intervals that are contained in that atom B (without the assumption of order inversion this does not hold). Averaging the identity (5.2.51) we obtain that 1 |J|a2J , for t ∈ B. (5.2.52) E φ2 |F (t) = |B| J⊆B


306

By Hölder’s inequality we estimate using (5.2.52).

E(φ|F) ∞ ≤ E(φ2 |F)1/2

∞

≤ ||g||BMO . Finally recall that φ = S(

A−1

ki ), and we obtain that S(f )φ = S(f )E(φ|F) ≤ S(f )||g||BMO .

i=0

1

2

Next we show that H ( ) is isomorphic to a space of real-valued functions, the so-called tent space. For 0 < α < 1, and x ∈ [0, 1], we define the cone Γα (x) = {(y, t) ∈ [−1, 2] × [0, 1] : |x − y| ≤ αt}. Then we define the tent space Tα to be the space of all measurable functions f : [0, 1] × [0, 1] → R such that 1 ( f (y, t)2 t−2 dydt)1/2 dx < ∞. ||f ||Tα = 0

Γα (x)

The tent spaces are independent of the cone parameter α, indeed, for α ≤ β there exists C(α, β) > 0 such that ||f ||Tα ≤ C(α, β)||f ||Tβ , for f ∈ Tβ . We will show now that H 1 (2 ) is isomorphic to Tα . To begin with we discuss first convenient representations of the spaces Tα , respectively H 1 (2 ). For I ∈ D we let I0 , I1 , I2 , be three subintervals of I with length |Ij | = |I|/3. We let T (Ij ) = Ij × [|I|, 2|I|[, and we denote by Sαj the subspace of Tα which consists of functions supported in {T (Ij ) : I ∈ D}. It is easy to see that each of the subspaces Sαj is isomorphic to the entire tent space Tα . It thus suffices to show that H 1 (2 ) is isomorphic to Sα1 . We continue with a representation of H 1 (2 ). For I ∈ D we say that a function fI belongs to the Hilbert space L2I iff supp fI is contained in T (I1 ) and dydt 1/2 ) f (y, t)2 < ∞. ||fI ||2 = ( |I| × |I| T (I1 )


307

We let EI : L2I → 2 denote the isomorphism between L2I and the sequence space 2 . Let Z be the space of vector-valued functions F (x) = hI (x)fI , fI ∈ L2I , equipped with the norm ||F || =

( hI (x)2 ||fI ||22 )1/2 .

Z is isomorphic to H 1 (2 ), the isomorphism is given by

T (F ) =

hI (x)EI (fI ).

Theorem 5.2.9. a) For f ∈ Sα1 , let fI = 1T (I1 ) f and let aI = EI (fI ). Then the operator T : Sα1 → H 1 (2 ) f→ h I aI is bounded and invertible. b) For aI ∈ 2 let fI = EI−1 (aI ). Then the operator S : H 1 (2 ) → Sα1 fI h I aI → is bounded, and it is the inverse of T. Proof. As Sα1 is independent of α we make the choice α = 2/3. Note that for x ∈ I we have the inclusion Γα (x) ⊇ T (I1 ). This gives the pointwise estimates,

1I (x)||aI ||22 ≤ 4 ≤4

1I (x)

fI (y, t)2 t−2 dydt

fI (y, t)2 t−2 dydt.

Γα (x)

Taking square roots, interchanging the order of the sum and the integral and finally integrating the pointwise estimate over x ∈ [0, 1], gives ||

hI (x)aI ||H 1 (2 ) ≤ 2

1

( 0

Γα (x)

≤ 2||f ||Sα1 .

fI (y, t)2 t−2 dydt)1/2 dx


308

This proves the first part of the theorem. We start the second part of the theorem by making another choice for α. We let α = 1/6. With this choice of α we observe that for x ∈ / I we have Γα (x) ∩ T (I1 ) = ∅. This gives the pointwise estimate fI (y, t)2 t−2 dydt ≤ 1I (x) fI (y, t)2 t−2 dydt Γα (x)

≤4

1I (x)||aI ||22 .

As above we obtain the boundedness of S from this pointwise estimate by taking square roots and integrating.

5.3

The uniform approximation property of BMO

We review three of the most classical approximation properties in Banach spaces. Recall that a Banach space satisfies the approximation property if for every > 0 and every compact set K ⊆ X there exists a finite rank operator such that ||T x − x||X ≤ ||x||X , for x ∈ K. Note that the approximation property involves qualitative conditions. In that way it asks for very little, in particular the rank of the operator T and its norm are allowed to vary with K, the compact set over which T is supposed to approximate the identity operator. The bounded approximation property puts an upper bound on ||T ||, not depending on K. A Banach space X satisfies the bounded approximation property if there exists λ > 0 so that for every > 0 and every compact set K ⊆ X there exists a finite rank operator satisfying ||T || ≤ λ, and ||T xi − xi ||X ≤ ||xi ||X ,

for i ≤ n.

Let K be a compact subset of X and let > 0. Then, by compactness, there exist x1 , . . . , xn ∈ K such that K⊆ B(xi , ). Assume now that T is a finite rank operator satisfying ||T || ≤ λ, and ||T xi − xi ||X ≤ ||xi ||X ,

for i ≤ n.

For x ∈ K choose k ≤ n such that ||x − xk ||X < , and write T x − x = (T xk − xk ) + (T xk − T x) + (x − xk ). Then the triangle inequality and the norm bound of T give ||T x − x||X ≤ 4λ||x||.

5.3. The uniform approximation property of BMO

309

Hence in the definition of the bounded approximation property we may replace compact sets by finite sets as follows. A Banach space X satisfies the bounded approximation property if there exists λ > 0 so that for every > 0 and x1 , . . . , xn ∈ X there exists a finite rank operator satisfying ||T || ≤ λ, ||T xi − xi ||X ≤ ||xi ||X ,

for i ≤ n.

The bounded approximation property allows the rank of T to vary with the choice of x1 , . . . , xn ∈ X even if > 0 and n ∈ N remain fixed. Imposing an upper bound on the rank of T which depends on and n but not on the specific choice of x1 , . . . , xn ∈ X leads to the uniform approximation property. The importance of this concept was recognized when the Local Theory of Banach spaces emerged. A Banach space X satisfies the uniform approximation property if there exists λ > 0 so that for every > 0 and n ∈ N there exists f (, n) ∈ N so that for every x1 , . . . , xn ∈ X there exists a linear operator T : X → X satisfying ||T || ≤ λ, ||T xi − xi || < ||xi ||, rankT ≤ f (, n). We call x1 , . . . , xn ∈ X the UAP data, T is called a resolving operator, and f (, N ) is called a uniformity function. Our first result establishes the uniform approximation property for the class of Lp spaces. Here the proof for L∞ turns out to be simplest case. We use below a technical device called the Auerbach basis. Let X be a finitedimensional Banach space. Let n be the algebraic dimension of X. Then there exist bi ∈ X (1 ≤ i ≤ n) with bi X = 1 so that for every g ∈ X there exist uniquely determined coefficients ci ∈ R so that g=

n

ci bi

and

|ci | ≤ g X .

i=i

For a proof showing the existence of such a basis see for instance [213] Lemma II.E.11. Theorem 5.3.1. The Banach spaces Lp , 1 ≤ p ≤ ∞, satisfy the uniform approximation property. Proof. Let > 0. Let x1 , . . . , xn ∈ Lp be linearly independent. Then choose an Auerbach basis for span{x1 , . . . , xn }, call it bi , i ≤ n, and define f (x) =

n i=1

|bi (x)|.


310

Then clearly |bi (x)| ≤ f (x), and ||f ||p ≤ n. For i ≤ n fixed there exists m ≤ (2/), pairwise disjoint subsets of the unit interval, G(i,1) , . . . , G(i,m) , and coefficients aj , j ≤ m, such that the following pointwise estimate holds for every x ∈ [0, 1], m bi (x) − f (x) aj 1G(i,j) (x) ≤ f (x). j=1 Let F be the σ-algebra generated by the sets {G(i,j) : i ≤ n and j ≤ m(i, )}. Let A denote the atoms of F. Note that the cardinality of A is ≤ (2/)n . For i ≤ n (i) and A ∈ A there exist coefficients aA , such that (i) aA 1A (x)f (x) ≤ f (x). bi (x) − A∈A

Raising this estimate to the power p and integrating (respectively forming the essential supremum) gives the norm approximation (i) aA 1A f ≤ n. bi − A∈A

p

(Recall that ||f ||p ≤ n.) Now fix g ∈ span{x1 , . . . , xn }. Expanding g using the Auerbach basis gives n g= ci bi , i=1

where |ci | ≤ ||g||p . Put αA =

n

(i)

ci a A .

i=1

Then estimate the degree of approximation, n (i) αA 1A f ≤ g − ci aA 1A f g − i=1 A∈A A∈A p p n (i) ≤ |ci | bi − aA 1A f i=1

≤ 2n

A∈A

n i=1 2

|ci |

≤ 2n ||g||p .

p


311

We use the collection of atoms A to define the resolving operator

1A f yf p−1 p . P (y) = f A A A∈A Let older conjugate index satisfying (p − 1)q = p or p/q = p − 1. Then q be the H¨ ( A f (p−1)q )p/q = ( A f p )p−1 . Hence yf p−1 ≤ A

|y|p A

p−1 fp

.

A

As A is a collection of pairwise disjoint sets it follows that ||P ||p ≤ 1. The dimension of the range of P is the cardinality of A, hence rankP ≤ (2/)n . The following identity holds for every A ∈ A, P (1A f ) = 1A f. Let g ∈ span{x1 , . . . , xn }, then its distance in Lp to span{1A f : A ∈ A} is bounded by 2n2 . Hence ||P g − g||p ≤ 2n2 ||g||p . Thus we verified that P is a resolving operator for the UAP data x1 , . . . , xn .

The following subsections are exclusively devoted to proving this theorem of P. W. Jones [102]. Theorem 5.3.2. BM O satisfies the uniform approximation property. Before we turn to a detailed exposition we summarize the basic pattern of the argument. In Chapter 3 we showed that BMO satisfies the following special case of the uniform approximation property. Let {Bi : i ≤ N } be disjoint collections of dyadic intervals. Assume that the union B = B1 ∪ · · · ∪ BN satisfies the Carleson packing condition. Define hJ , ϕi = J∈Bi

and let M = ϕ1 + · · · + ϕN BMO . Then there exists an orthogonal projection P : BMO → BMO such that ||P ||BMO ≤ 4, P ϕi = ϕi , Rank P ≤ f ( N, M ).

(5.3.1)

The orthogonal projection satisfying (5.3.1) was constructed in Theorem 3.2.3 of Chapter 3.


312

We show below that BMO satisfies the following property which is clearly weaker than the uniform approximation property: Let > 0 and N ∈ N, then there exists f (N, ) so that the following condition holds. Given ϕ1 , . . . , ϕN ∈ BMO there exists an operator R : BMO → BMO so that ||R|| ≤ 10, ||Rϕj − ϕj || ≤ ||ϕj ||, and so that R : BMO → BMO factors through ∞ as follows, R

BMO −→ BMO E F ∞

with

||E|| · ||F || ≤ f (N, ).

(5.3.2)

The factorization (5.3.2) will be obtained in the next subsections. In its proof we will encounter new ideas resulting in a powerful combinatorial technique. Naturally the methods by which we obtain (5.3.1) and (5.3.2) will be important components in the proof of the uniform approximation property of BMO. We will see below that in order to obtain the complete proof that BMO has UAP we only have to merge the ideas that gave (5.3.1) and (5.3.2). It seems to be quite remarkable that proving a special case of UAP and establishing a property that is weaker than UAP suffices to show that indeed BMO satisfies the uniform approximation property.

Splitting the Haar support Here we show that without loss of generality we may always assume that the UAP data are disjointly supported over the Haar system. Specifically we show that if one can find a resolving operator for UAP data with pairwise disjoint Haar support, then one can find a resolving operator for any UAP data. The reduction is based on the disjointification procedure introduced to show that Lp has the uniform approximation property. We apply it here to obtain functions with disjoint Haar support. Proposition 5.3.3. Let > 0 and let x1 , . . . , xn ∈ BMO. Then there exists N ≤ (2n3 /)n , and ϕ1 , . . . , ϕN ∈ BMO disjointly supported over the Haar system so that for g ∈ span{x1 , . . . , xn }, N g − α ϕ ≤ ||g||BMO , j j j=1 BMO

where αj ∈ R are suitably chosen coefficients. Proof. Let bi , i ≤ n be an Auerbach basis for span{x1 , . . . , xn }. Expand bi in its Haar series, bi = bi (I)hI .


313

For every dyadic interval I we form the coefficients f (I) =

n

|bi (I)|,

i=1

inducing the BMO function f (x) = f (I)hI (x). Note that f (I) ≤ n and ||f ||BMO ≤ n. For each i ≤ n, there exist m ≤ (2/), pairwise disjoint collections of dyadic intervals G(i,j) , j ≤ m and constants a1 , . . . , am such that for I ∈ D, m bi (I) − f (I) aj G(i,j) (I) ≤ f (I), j=1 where we denoted by G(i,j) the indicator function of the collection G(i,j) . 1 for I ∈ G(i,j) , G(i,j) (I) = 0 for I ∈ / G(i,j) . Defined that way G(i,j) is a function on D, taking the values {0, 1}. Let A denote the atoms of the σ-algebra generated by the functions {G(i,j) : i ≤ n, j ≤ m(i, )}. Observe that each A ∈ A is a collection of dyadic intervals. It is obtained by intersecting some of the collections {G(i,j) : i ≤ n, j ≤ m(i, )}. Hence for each bi (i) there exist coefficients {aA : A ∈ A}, such that (i) aA f (I)1A (I) ≤ f (I). bi (I) − A∈A

Next we pass from coefficients to functions. For A ∈ A we let ϕA = 1A (I)f (I)hI . I∈D

Then we have that (i) bi (I)hI − aA ϕ A I∈D

A∈A

BMO

≤ 2 f (I)hI I∈D

. BMO

Take g ∈ span{x1 , . . . , xn }. Expand g using the Auerbach basis, then g=

n

ci bi ,

i=1

with |ci | ≤ ||g||BMO . Next fix A ∈ A and form the coefficient αA =

n i=1

(i)

ci a A .


314 This gives the approximation αA ϕA g − A∈A

BMO

n (i) ≤ g − ci aA ϕ A i=1 A∈A BMO n (i) ≤ |ci | bi − aA ϕ A i=1

≤ 2n

A∈A

n

BMO

|ci |

i=1 2

≤ 2n ||g||BMO . Note that the cardinality of A is ≤ (2/)n . It remains to replace by n−2 , and to relabel the functions ϕA , A ∈ A as ϕ 1 , . . . , ϕN ,

with N ≤ (2n3 /)n .

UAP data with large Haar coefficients We have reduced the problem to the case when the data ϕ1 , . . . , ϕN ∈ BMO are disjointly supported over the Haar system. We will first prove the uniform approximation property under the additional hypothesis that if a Haar coefficient does not vanish, then it satisfies the lower bound |ϕj , hJ | > γ|J| · ϕj BMO . Let K denote the Haar support of ϕ1 +· · ·+ϕN . The lower bound on the Haar coefficients implies an upper bound for the Carleson constant of K, namely [[K]] ≤ N γ −2 . Now translate the intrinsic condition on the Haar coefficients into a statement about factorization of operators through ∞ . Let R be the orthogonal projection onto span{hJ : J ∈ K}. Then ||R|| ≤ 1 and Rϕj = ϕj and R factors through ∞ as follows, R BMO −→ BMO E F ∞ where the operators E, F are defined by {aI hI : I ∈ K}. E aI hI = (aI ), and F ((aI )) = They satisfy the norm estimates, ||E|| ≤ 1, and ||F || ≤ [[K]]1/2 . Below the existence of a factorization through ∞ will indicate that the combinatorial methods of Theorem 3.2.1 are applicable. This is the value of re-expressing our hypothesis.


315

Theorem 5.3.4. Let N ∈ N, > 0 and 1 > γ > 0 be given. Let ϕ1 , . . . , ϕN ∈ BMO be disjointly supported over the Haar system. Assume that |ϕi , hI | > γ||ϕi ||BMO |I|, for each I in the Haar support of ϕi , and each i ≤ N. Then there exists an orthogonal projection P such that, ||P ||BMO ≤ 4, ||P (ϕi ) − ϕi ||BMO ≤ ||ϕi ||BMO for i ≤ N, rank(P ) ≤ f (N, , γ). Proof. The proof consists of a straightforward reduction to the deep combinatorial result of Theorem 3.2.3 in Chapter 3. Fix ϕi ∈ BMO. Assume without loss of generality that the Haar coefficients of ϕi are non-negative and that ||ϕi ||BMO = 1. Hence for I in the Haar support of ϕi the following estimate holds, γ|I| < ϕi , hI ≤ |I|. Hence there exists an integer 1 ≤ q ≤ − log γ/ log(1 + ) such that (1 + )−q |I| < ϕi , hI ≤ (1 + )−q+1 |I|. Now we change our point of view. Let M = − log γ/ log(1 + ), and fix 1 ≤ q ≤ M. Define B(i,q) to be the collection of dyadic intervals for which (1 + )−q |I| < ϕi , hI ≤ (1 + )−q+1 |I|.

(5.3.3)

By definition, the collections B(i,q) are pairwise disjoint and for i fixed, B(i,q) is contained in the Haar support of ϕi . Let, b(i,q) = {hJ : J ∈ B(i,q) }, aq = (1 + )−q . By (5.3.3) for I ∈ B(i,q) we obtain that |aq |I| − ϕi , hI | ≤ aq |I|, and also aq |I| ≤ ϕi , hI . Hence we get hI hI |aq − ϕi , | ≤ ϕi , . |I| |I| As ϕi BMO ≤ 1, the following approximation holds, ||ϕi −

M

aq b(i,q) ||BMO ≤ .

q=1

Recall that B(i,q) is contained in the Haar support of ϕi . Hence it satisfies the γ −2 -Carleson condition. Now we apply Theorem 3.2.3 of Chapter 3 to the disjoint


316

collections of dyadic intervals {B(i,q) : i ≤ N, exists an orthogonal projection P such that

and q ≤ M }. It asserts that there

||P ||BMO ≤ 4, ||P (ϕi ) − ϕi ||BMO ≤ 2||ϕi ||BMO for i ≤ N, rank (P ) ≤ f (N, , γ).

UAP data with small Haar coefficients We will now present the proof of Theorem 5.3.2 under the additional hypothesis that the functions ϕj have disjoint Haar support, and that the Haar coefficients of ϕj are sufficiently small. For N ∈ N and > 0 we define the rather small number γ(N, ) =

3 N3

−N N2 ) log( .

Theorem 5.3.5. Let ϕ1 , . . . ϕN ∈ BMO, let > 0. Assume that ϕ1 , . . . ϕN are disjointly supported over the Haar system, and that each Haar coefficient satisfies |ϕj , hI | ≤ γ|I|||ϕj ||BMO , where γ < γ(N, ). Then there exists an operator T on BMO such that ||T ||BMO ≤ 10, ||T ϕj − ϕj ||BMO ≤ ||ϕj ||BMO , rank(T ) ≤ f (N, ). There are two independent components in the proof of Theorem 5.3.5. The first component is the construction of a remarkable operator R that satisfies the following three conditions, ||R|| ≤ 10, ||Rϕj − ϕj || ≤ ||ϕj ||, and R can be factored through ∞ , R


with

||E|| · ||F || ≤ f (N, ).

(5.3.4)

In the second part we are guided by the proof of Theorem 5.3.4. There we observed that the deep combinatorial methods developed for the proof of Theorem 3.2.3 enable us to transform a factorization diagram like (5.3.4) into a finite rank condition with good upper bounds.


317

We recall the definitions for the localized square function and for blocks of dyadic intervals in D. Let f ∈ L2 , with Haar expansion f = aI hI . Given a collection of dyadic intervals H, then the localized square function S(f |H) is defined by the equation a2I h2I . S 2 (f |H) = I∈H

For a given dyadic interval I we define I ∩H to be the collection of dyadic intervals I ∩ H = {J ∈ H : J ⊆ I}. Recall that a collection L(I) ⊆ D is a block of dyadic intervals if I is the only maximal interval of L(I) and if the following connectedness property holds true. Let J ∈ L(I) and let K ∈ D, then J ⊆ K ⊆ I implies K ∈ L(I). Repeatedly in the course of proving Theorem 5.3.5 we use the following simple and general observation relating blocks of dyadic intervals, the Carleson packing condition, the localized square function and the norm in BMO through a simple inequality. Let {L(I) : I ∈ Ω} be a decomposition of D into disjoint blocks of dyadic intervals so that the index set Ω satisfies the Carleson packing condition. Let f ∈ L2 . Assume that for the localized square functions S(f |L(I)) there hold bounds as follows,

S 2 (f |L(I))||∞ ≤ a for I ∈ Ω, and

S 2 (f |L(I)) ≤ b|I|

for I ∈ Ω.

I

Then f ∈ BMO and the norm of f in BMO satisfies an upper bound in terms of a, b and [[Ω]], (5.3.5)

f 2BMO ≤ a + b[[Ω]]. This estimate is useful when b 0 and N ∈ N we choose η, L, A, M, by the formulas η=

, 16N 2

N , η

M = AL.

(5.3.7)

All the above determine γ = γ(, N ) by the relation γ≤

. 2 + N η + 2M

(5.3.8)

We are given the functions ϕ1 , . . . , ϕN , disjointly supported over the Haar system. We assume that they are normalized by ||ϕj ||BMO = 1, and satisfy |ϕj , hI | ≤ γ|I|, for every dyadic interval I. We begin by analyzing their sum ϕ=

N

ϕi .

i=1

Here we use again a standard stopping time procedure introduced in Chapter 1.


319

Proposition 5.3.6. The dyadic intervals D can be grouped into disjoint blocks of dyadic intervals {L(I) : I ∈ Ω} so that the following conditions hold: 1. The index set Ω satisfies the N γ −1 + 1 Carleson condition. 2. The square function S 2 (ϕ|L(I)) satisfies the pointwise estimate ||S 2 (ϕ|L(I))||∞ ≤ γ + γ 2 .

(5.3.9)

3. If B(I) is any block of dyadic intervals, strictly larger than L(I), then the square function S 2 (ϕ|B(I)) satisfies ||S 2 (ϕ|B(I)||∞ ≥ γ + γ 2 . Proof. We use stopping time arguments on the square function of ϕ. We let L0 = L([0, 1]) be the largest block of dyadic intervals such that for any x ∈ [0, 1], S 2 (ϕ|L0 )(x) ≤ γ + γ 2 . Now let K1 = D \ L0 and let {I11 , . . . , Ij1 , . . . } be the maximal intervals of K1 . For each Ij1 we define L(Ij1 ), to be the largest block of dyadic intervals satisfying S 2 (ϕ|L(Ij1 ))(x) ≤ γ + γ 2 .

# 1 2 2 Then we put K2 = K1 \ ∞ j=1 L(Ij ) and let {I1 , . . . , Ij , . . . } be an enumeration of 2 the maximal intervals of K . For each of the intervals Ij2 we define L(Ij2 ) to be the largest block of dyadic intervals such that S 2 (ϕ|L(Ij2 ))(x) ≤ γ + γ 2 . Continuing in this fashion we obtain the index set Ω as Ω = [0, 1] ∪

∞ ∞

Ijn .

n=1 j=1

The decomposition of D is consequently given by D = {L(I) : I ∈ Ω}, where L(I) is the largest block of dyadic intervals satisfying the conditions S 2 (ϕ | L(I))(x) ≤ γ + γ 2 . By the stopping time definition of the blocks L(I) it is evident that for every block B(I) strictly larger than L(I) we obtain

S 2 (ϕ | B(I)) ∞ ≥ γ + γ 2 .


320

Hence condition 3) of Proposition 5.3.6 holds true. Next we verify that Ω satisfies the Carleson packing condition and [[Ω]] ≤ N γ −1 + 1. To this end we fix I ∈ Ω and consider the collection B(I) = L(I) ∪ G1 (I, Ω). A moment’s reflection yields that B(I) is a block of dyadic intervals, and clearly B(I) is strictly larger than L(I). Hence S 2 (ϕ | B(I)) satisfies a lower bound on the set G∗1 (I, Ω). We have, S 2 (ϕ | B(I))(x) ≥ γ + γ 2

for x ∈ G∗1 (I, Ω).

(5.3.10)

Observe that G1 (I, Ω) is a collection of pairwise disjoint dyadic intervals, and by hypothesis the Haar coefficients of ϕ are so small that ϕ,

hJ 2 ≤ γ2 |J|

for J ∈ G1 (I, Ω).

(5.3.11)

Combining (5.3.10) and (5.3.11) yields a crucial lower bound for S 2 (ϕ | L(I)) on the set G∗1 (I, Ω), for x ∈ G∗1 (I, Ω).

S 2 (ϕ | L(I))(x) ≥ γ

By integrating (5.3.12) over I we find that S 2 (ϕ | L(I)) ≥ γ I

|K|.

(5.3.12)

(5.3.13)

K∈G1 (I,Ω)

Recall that ϕ = ϕ1 + · · · + ϕN where the functions ϕi are disjointly supported over the Haar √ system and satisfy ϕi BMO ≤ 1. Hence, by Proposition 5.1.1,

ϕ BMO ≤ N . It follows that for J ∈ Ω, S 2 (ϕ | Q(J)) ≤ N |J|, (5.3.14) J

where Q(J) = {L ∈ D : L ⊆ J}. By exploiting (5.3.13) and (5.3.14) we find the upper estimate for the Carleson constant of Ω as follows. γ |I| = γ|J| + γ |K| I∈Ω∩J

I∈Ω∩J

K∈G1 (I,Ω)

I∈Ω∩J

I

≤ γ|J| +

S 2 (ϕ | L(I))

S 2 (ϕ | Q(J))

≤ γ|J| + J

≤ γ|J| + N |J|.


321

Our next aim is to display more of the structure of the index set Ω. We extract a sub-collection E0 of Ω by defining that I ∈ Ω belongs to E0 if γ (5.3.15) S 2 (ϕ|L(I)) ≤ |I|. 2 We fix j ∈ {1, . . . , N } and define Ej to be the collection of dyadic intervals I ∈ Ω\E0 such that S 2 (ϕj |L(I)) ≤ ηγ|I|.

(5.3.16)

Finally we define a perturbation ψj of ϕj by subtracting the contribution coming from E0 ∪ Ej , hJ (5.3.17) ψj = ϕj − ϕj , hJ . |J| I∈E0 ∪Ej J∈L(I)

In the next proposition we show several things. First we prove that ψj is close to ϕj . Second we decompose the intervals Ω \ E0 into groups {Ci } satisfying the following homogeneity conditions: Up to small relative discretisation errors, the averages 1 S 2 (ψj |L(I)), I ∈ Ci |I| depend only on the indices i and j, and not on the particular choice of interval I ∈ Ci . Recall that the constants L = L(N, ), η = η(N, ) and γ = γ(N, ) appearing in the proposition below have been defined earlier in (5.3.7). Proposition 5.3.7. E0 satisfies the 2-Carleson condition. For 1 ≤ j ≤ N , : (5.3.18) ||ϕj − ψj ||BMO ≤ 4γ + 2ηN . The index set Ω \ E0 can be grouped as Ω \ E0 = C1 ∪ · · · ∪ CL so that the following holds. 1. For i ≤ L and j ≤ N there exist coefficients aij ≥ 0 such that 1 aij ≤ S 2 (ψj |L(I)) ≤ aij , for I ∈ Ci . 2 |I|

(5.3.19)

2. If a coefficient aij is non-zero, then it satisfies the inequalities ηγ ≤ aij ≤ 2γ.

(5.3.20)


322

Proof. We start by showing that the collection E0 satisfies the 2-Carleson condition. Let I0 ∈ E0 and let {Ik } be an enumeration of the intervals in the first generation G1 (I0 , E0 ). So {Ik } is the list of the maximal intervals of E0 which are contained in I0 . In the course of proving Proposition 5.3.6 we established (5.3.13) asserting that the following lower bound holds, S 2 (ϕ|L(I0 )) ≥ γ I0

∞

|Ik |.

k=1

On the other hand as I0 ∈ E0 we have γ S 2 (ϕ|L(I0 )) ≤ |I0 |. 2 The lower estimate and the upper estimate are matching so that γ cancels and

|Ik | ≤

1 |I0 |. 2

1 n By summing a geometric series ∞ n=0 q with q = 2 we obtain the upper bound 2 for the Carleson constant of E0 . Hence, by the square function estimate (5.3.9), the norm in BMO of hJ (5.3.21) ϕj , hJ |J| I∈E0 J∈L(I)

: is bounded by ≤ 2(γ + γ 2 ). Let j ≤ N , and fix I ∈ Ej . Let J ∈ Ej ∩ I. Recall that Ej ⊆ Ω \ E0 . Thus J∈ / E0 . Consequently the following two inequalities hold for L(J), γ S 2 (ϕj |L(J)) ≤ ηγ|J|. S 2 (ϕ|L(J)) ≥ |J| and 2 We combine the two inequalities into one single estimate, namely 2η S 2 (ϕ|L(J)) ≥ S 2 (ϕj |L(J)). Next sum over √ J ∈ Ej ∩ I and take into account that the BMO norm of ϕ is bounded by N . 2 S (ϕj |L(J)) ≤ 2η S 2 (ϕ|L(J)) J∈Ej ∩I

J∈Ej ∩I

≤ 2η

S 2 (ϕ|Q(I))

≤ 2ηN |I|.

(5.3.22)


323

Combining the global estimates (5.3.22) with the local square function estimate (5.3.9) proves that the norm in BMO of the term

ϕj , hJ

I∈Ej J∈L(I)

hJ |J|

(5.3.23)

: is bounded by γ + γ 2 + 2ηN . The estimates for (5.3.23) and (5.3.21) imply (5.3.18). In the next step of the proof we group the intervals in Ω \ E0 . Fix j ∈ {1, . . . , N }. Then recall that we produced ψj from ϕj by removing from the Haar support of ϕj the following blocks of intervals {L(I) : I ∈ E0 ∪ Ej }. Recall that I ∈ Ej , if the condition (5.3.16) holds. The following dichotomy is the result of removing the blocks {L(I) : I ∈ Ej }. Either S 2 (ψj |L(I)) > ηγ|I|, or

S 2 (ψj |L(I)) = 0.

Hence for every I ∈ Ω \ E0 there exist nj ∈ {−1, 0, 1, 2, . . . , | log2 η|, ∞} such that 1 2−nj γ< 2 |I|

S 2 (ψj |L(I)) ≤ 2−nj γ.

Now we change our point of view and fix a vector (n1 , . . . , nN ), with nj ∈ {−1, 0, 1, 2, 1, . . . , − log2 η, ∞}. Then we define the collection C(n1 ,...,nN ) to consist of all dyadic intervals I ∈ Ω \ E0 which satisfy the set of conditions 1 2−nj γ< S 2 (ψj |L(I)) ≤ 2−nj γ, 2 |I| for any j ∈ {1, . . . , N }. Hence for any given interval I ∈ C(n1 ,...,nN ) the index (n1 , . . . , nN ) of the collection stores the discretized values of the integrals 1 S 2 (ψj |L(I)), |I| j ∈ {1, . . . , N }. The relative discretization error is bounded by a factor of 2. By counting all possible choices of indices (n1 , . . . , nN ), we see that there are at


324

most L = L(N, η) different collections C(n1 ,...,nN ) . For simplicity of notation we enumerate them as C1 , . . . , CL . By construction these collections satisfy the conclusions (5.3.19) and (5.3.20) of the Proposition 5.3.7. Finally we remark that no interval in Ω \ E0 has been attributed to more than one of the groups Ci , and that none of the intervals has been overlooked by our construction. Below, we ignore the contribution to ϕj coming from the blocks indexed by intervals in E0 or Ej . This strategy is justified by (5.3.18) showing that the error terms resulting from discarding the index sets E0 and Ej , are sufficiently small in BMO. Now we continue with a more detailed study of the index sets Ci and with the analysis of the functions

ψj , hJ

J∈L(I)

hJ , |J|

I ∈ Ci .

Fix the constant A and one of the index sets Ci . Define Mi to be the collection of all intervals I ∈ Ci , which satisfy,

|J| ≥

J∈GA (I,Ci )

1 |I|. 4

(5.3.24)

Proposition 5.3.8. The collection of dyadic intervals Ci \ Mi satisfies the 2ACarleson packing condition. For each I ∈ Mi , the collection R(I) =

A

L(J)

(5.3.25)

n=0 J∈Gn (I,Ci )

gives the estimate A 2 S 2 (ψj |L(I)) S (ψj |R(I)) ≥ 16

for

Proof. The inclusion {Ci \ Mi } ⊆ Ci implies that |J| ≤ J∈GA (I,Ci \Mi )

1 ≤ j ≤ N.

|J|,

J∈GA (I,Ci )

for every dyadic interval I. Hence by definition of Mi the following estimate holds for I ∈ Ci \ Mi , 1 (5.3.26) |J| ≤ |I|. 4 J∈GA (I,Ci \Mi )

An immediate consequence of (5.3.26) is that each of the collections Ci \Mi satisfies the 2A-Carleson packing condition.


325

Now we fix I ∈ Mi and turn to the relation between the collections R(I) and L(I). Let j ≤ N. The left-hand side estimate of the homogeneity conditions (5.3.19) in Proposition 5.3.7, and the defining condition (5.3.24) give the lower bound A S 2 (ψj |L(J)) S 2 (ψj |R(I)) = n=0 J∈Gn (I,Ci )

≥

A 1 2 n=0

ai,j |J|

(5.3.27)

J∈Gn (I,Ci )

Aai,j |I|. ≥ 8 Conversely the right-hand side of the homogeneity conditions (5.3.19) in Proposition 5.3.7 is just the estimate (5.3.28) S 2 (ψj |L(I)) ≤ ai,j |I|. Combining the inequalities (5.3.27) and (5.3.28) gives the desired estimate between R(I) and L(I). Namely we obtain that A 2 S 2 (ψj |L(I)). S (ψj |R(I)) ≥ 16 For i ≤ L the collection Mi ⊆ Ci was introduced by (5.3.24). Put K = M1 ∪ · · · ∪ ML .

(5.3.29)

Recall that Mi ⊆ Ω. Hence by Proposition 5.3.6, K satisfies the Carleson packing condition with [[K]] ≤ N γ −1 + 1. We define the error function Ej by the equation ϕj =

I∈K J∈L(I)

ψj , hJ

hJ + Ej . |J|

(5.3.30)

Next we estimate the norm in BMO of the error term Ej (this clears the road for part two of the proof of Theorem 5.3.5). Proposition 5.3.6 and the first assertion of Proposition 5.3.8 are the main ingredients in showing that the error term Ej is small. Proposition 5.3.9. Let j ≤ N , then Ej satisfies the estimate ||Ej ||BMO ≤

: 6γ + 2ηN + γ2AL.

(5.3.31)


326

Proof. First we identify the components which constitute the error term Ej . Let E = C1 \ M1 ∪ · · · ∪ CL \ ML , and define zj =

ψj , hJ

I∈E J∈L(I)

(5.3.32)

hJ . |J|

Then the following identity holds, Ej = ϕj − ψj + zj .

(5.3.33)

Note that ϕj − ψj and zj , are disjointly supported over the Haar system. Hence by Proposition 5.1.1,

Ej 2BMO ≤ ϕj − ψj 2BMO + zj 2BMO . Proposition 5.3.7 gives that ϕj − ψj 2BMO ≤ 4γ + 2ηN. We turn to estimating zj . Each of the groups Ci \ Mi satisfies the 2A-Carleson condition. Hence by definition (5.3.32) the collection E satisfies a 2AL-Carleson condition. Recall that by Proposition 5.3.6 for each J ∈ E, we have the estimates 2 S (zj |L(J)) ≤ S 2 (ϕj |L(J)) ≤ 2γ|J|. Let I ∈ E be given and let {Jk } be an enumeration of I ∩ E. Then we estimate as follows, ∞ 2 S 2 (zj |L(Jk )) S (zj |Q(I)) = ≤

k=1 ∞

2γ|Jk |

(5.3.34)

k=1

≤ 4γAL|I|. Combine (5.3.34) with (5.3.9) to obtain that ||zj ||2BMO ≤ γ+γ 2 +4γAL, as claimed. A Provisional Appraisal. Let us pause here to better see where we stand on our way to proving the uniform approximation property. We form now an operator R which almost reproduces the functions ϕj and whose norm is bounded independent of N . Unfortunately R is missing a good estimate on the algebraic dimension of its range. However the


327

range of R satisfies a strong geometric bound expressed by the Carleson packing condition of the index set K. Given ψj and J ∈ K we form the functions xj,J =

ψj , hI

I∈L(J)

hI |I|

and

yj,J =

ψj , hI

I∈R(J)

hI . |I|

Note that we used L(J) to form xj,J and R(J) (defined in Lemma 5.3.8) for yj,J . Define the operator R by the equation Rf =

N

f, yj,J

j=1 J∈K

xj,J . ||yj,J ||22

(5.3.35)

A direct calculation yields that Rψj = ψj −

ψj , hJ

I∈E J∈L(I)

hJ . |J|

Next observe that Rψj = Rϕj , hence Rϕj = ϕj − Ej , where Ej is the error function defined and estimated in Proposition 5.3.9. Thus we know already that ||Rϕj − ϕj ||BMO ≤ . We show now that ||R|| is independent of N, and that R can be factored through the space ∞ . Proposition 5.3.10. ||R||BMO ≤ 100,

(5.3.36)

and R : BMO → BMO admits the following factorization through ∞ , R


with

||E|| · ||F || ≤ f (N, ).

(5.3.37)

Proof. We show first the norm estimate for R. Later we define the operators E and F that provide the factorization through ∞ and prove that the product of their norm can be estimated by a function of and N. The estimate for R . Let f ∈ BMO. Let I be a dyadic interval in the Haar support of Rf. There exists J0 ∈ K such that I ∈ L(J0 ). Let F = {K ∈ L(J0 ) : K ⊆ I}.

(5.3.38)

Let {Ip } be an enumeration of the maximal intervals of K which are strictly contained in I. The following identity holds then for the square function, S 2 (Rf |Q(I)) = S 2 (Rf |F) +

∞ p=1

S 2 (Rf |Q(Ip )).

(5.3.39)


328

Estimates for the sum over p result from the following three simple observations. First note that the square function taken over Q(Ip ) has a very simple form. It is S 2 (Rf |Q(Ip )) =

N

f, yj,J 2

j=1 J∈Ip ∩K

||xj,J ||22 . ||yj,J ||42

(5.3.40)

Second note that the collections {R(J) : J ∈ K} are not pairwise disjoint. However each dyadic interval is contained in at most A of the collections {R(J) : J ∈ K}. Hence applying Bessel’s inequality gives N

2 f, yj,J 2 ||yj,J ||−2 2 ≤ A|Ip | · ||f ||BMO .

(5.3.41)

j=1 J∈Ip ∩K

Third and last, we observe that the following identities hold, ||xj,J ||22 = S 2 (ψj |L(J)), ||yj,J ||22 = S 2 (ψj |R(J)).

(5.3.42)

The integrals appearing on the right in the above equations (5.3.42) are related by the estimate in Proposition 5.3.8. So we find that ||xj,J ||22 ≤

16 ||yj,J ||22 . A

(5.3.43)

Now we insert the estimate (5.3.43) into the identity (5.3.40). After that we use (5.3.41). This gives (5.3.44) S 2 (Rf |Q(Ip )) ≤ 32|Ip | · ||f ||2BMO . We recall that the intervals {Ip } are an enumeration of pairwise disjoint intervals. Indeed the collection of intervals {Ip } is just G1 (I, K). Hence we may take the sum over p in the estimates (5.3.44) and insert the result into (5.3.39). This gives the following estimate summing up the first half of the proof. 2 (5.3.45) S (Rf |Q(I)) ≤ S 2 (Rf |F) + 32|I| · f 2BMO . Now we turn to the estimate for S 2 (Rf |F). This requires more subtle considerations. First recall that F ⊆ L(J0 ). Thus we may rewrite S 2 (Rf |F) as S 2 (Rf |F) =

N j=1

f, yj,J0 2

S 2 (xj,J0 |F) . ||yj,J0 ||42

(5.3.46)


329

Note that we have a pointwise upper estimate for S 2 (xj,J0 |F). Indeed, the second assertion of Proposition 5.3.6 shows that for every j ≤ N , S 2 (xj,J0 |F) ≤ 2γ. Conversely by (5.3.20) the following lower integral estimate holds true, S 2 (xj,J0 |L(J0 )) ≥ ηγ|J0 |.

(5.3.47)

(5.3.48)

We combine next the inequalities (5.3.47) and (5.3.48). After that we use the relation (5.3.43). This gives 2|I| ||xj,J0 ||22 S 2 (xj,J0 |F) ≤ η|J0 | (5.3.49) 32|I| 2 ||yj,J0 ||2 . ≤ Aη|J0 | Note that we found a highly non-trivial estimate for S 2 (xj,J0 |F). By contrast, the estimates for the coefficients f, yj,J0 2 appearing in (5.3.46) are relatively direct. We simply apply Bessel’s inequality and find that f, yj,J0 2 ≤ |J0 | ||f ||2BMO ||yj,J0 ||22 .

(5.3.50)

Insert (5.3.49) and (5.3.50) into the identity (5.3.46). This gives

32|I| f, yj,J0 2 ||yj,J0 ||−2 2 Aη|J0 | j=1 N

S 2 (Rf |F) ≤

(5.3.51)

32N |I| · ||f ||2BMO . ≤ Aη With (5.3.45) we found that 32N S 2 (Rf |Q(I)) ≤ |I| 32 + ||f ||2BMO . Aη The choice of A and η gives that ||R||BMO ≤ 100. The factorization through ∞ . Now we turn to the proof of the factorization (5.3.37). Recall that K ⊆ Ω where Ω is defined in Proposition 5.3.6. Hence K satisfies the Carleson packing condition with [[K]] ≤ N γ −1 + 1.


330

This puts a strong restriction on the range of R which leads to the claimed factorization. Define the operators E, F as follows. Let g ∈ BMO. Then yj,J E(g) = g, : j ≤ N, J ∈ K . ||yj,J ||22 Let (aj,J ), j ≤ N, J ∈ K be a bounded sequence of scalars. Then F ((aj,J )) =

{aj,J xj,J : j ≤ N, J ∈ K}.

Next we show that operator E is bounded. First by the Cauchy–Schwarz inequality we have that 1/2 S 2 (yj,J ) . (5.3.52) S(yj,J ) ≤ |J|1/2 Observe next that S 2 (yj,J ) = S 2 (ψj |R(J)), Combining the lower bounds of Proposition 5.3.8 with (5.3.19) and (5.3.20) we obtain Aγη . (5.3.53) S 2 (yj,J ) ≥ |J| 32 View (5.3.53) as an upper estimate for |J|, and insert it into (5.3.52). This gives : −1 −1 −1 S(yj,J ) ≤ 32A γ η S 2 (yj,J ). Hence by Fefferman’s inequality we obtain |g,

√ yj,J

yj,J H 1 | ≤ 2 2 g BMO 2 ||yj,J ||2 ||yj,J ||22 : ≤ 16 A−1 γ −1 η −1 g BMO .

(5.3.54)

: From (5.3.54) we get ||E : BMO −→ ∞ || ≤ 16 A−1 γ −1 η −1 . Upper bounds for the operator norm of F follow from the upper estimates for the Carleson constant [[K]] and ||xj,J ||BMO . We have ||F : ∞ −→ BMO || ≤ [[K]]1/2 supj,J ||xj,J ||BMO . In summary we obtain the following estimate for the factoring operators, ||E : BMO −→ ∞ || · ||F : ∞ −→ BMO || ≤ f (N, ).

The rank of R is of course infinite, in general, since we cannot guarantee that the cardinality of K is finite. For the uniform approximation property we have to replace K in the above operator by a finite set. The Carleson packing condition [[K]] ≤ N γ −1 + 1, and the factorization of R suggest that we do this by applying the combinatorial methods developed in Chapter 3.


331

Proof of Theorem 5.3.5. Part 2. The reader is advised to review in Section 1.5 (page 109) the glueing process and the pigeon hole principle, and in Section 3.2 the statement of the combinatorial Theorem 3.2.2 on colored intervals and orthogonal projections (page 191). We attach colors to the intervals of K = M1 ∪ · · · ∪ ML . Let 0 ≤ k ≤ A − 1, and put ∞ GmA+k (Mi ). Mi,k = m=0

Note that the collections {Mi,k : 1 ≤ i ≤ L, and 0 ≤ k ≤ A − 1} form a decomposition of K. We use it to attach colors to the intervals of K. For I ∈ K we define its color as follows: The color of I is (i, k) iff I ∈ Mi,k .

(5.3.55)

There are at most M = LA different colors. Thus we turned K into a collection of colored dyadic intervals. There are at most M = M (, N ) different colors. Recall that K satisfies the Carleson packing condition and its Carleson constant admits an upper bound that depends only on and N. Theorem 3.2.2 applied to the colored collection K gives its decomposition into blocks as B(J), (5.3.56) K= J∈J

and for J ∈ J it gives collections F1 (J), . . . , FL(J) (J), so that the following conditions hold. For m ≤ L(J), Fm (J) is monochromatic with respect to (5.3.55) .

(5.3.57)

F1 (J), . . . , FL(J) (J) satisfies Jones’s compatibility condition,

(5.3.58)

B(J) = F1 (J) ∪ · · · ∪ FL(J) (J), and For m ≤ L(J),

|I| > δ|J|.

(5.3.59) (5.3.60)

I∈Fm (J)

The index set J satisfies the Carleson packing condition with constant ≤ 2. This is a relevant piece of information since J ∈ J is the unique maximal interval in B(J). Theorem 3.2.2 puts an upper bound on L(I) and a lower bound on δ. We have L(I) ≤ L(, N ) and δ ≥ δ(, N ). (5.3.61) For each of the collections Fm (J) we use the abbreviation I. Fm (J) = I∈Fm (J)


332

Thus Fm (J) is the point-set covered by the collection Fm (J). Recall that for each J fixed the families {Fm (J) : m ≤ L(J)} form a nested collection of sets. By (5.3.61) the intersection pattern of the nested families {Fm (J) : m ≤ L(J)} is rather limited as J ranges in J . Hence applying the pigeon hole principle discussed after Theorem 1.5.11 in Section 1.5 gives the following: There exist K = K(, N ) and disjoint subcollections Al ⊆ J , so that J = A 1 ∪ · · · ∪ AK ,

(5.3.62)

and so that for each fixed collection Al the families {Fm (J) : m ≤ L(J)}

where J ∈ Al ,

satisfy the so-called rescaling property. It consists of the three conditions stated below. First, there exists L(l) such that for J ∈ Al , L(J) = L(l).

(5.3.63)

Second if I, J ∈ Al and m, n ≤ L(l), then Fm (I) ⊆ Fn (I)

Fm (J) ⊆ Fn (J),

(5.3.64)

|Fm (J)| / |Fn (J)| ≤ 2|Fm (I)| / |Fn (I)|.

(5.3.65)

iff

and The third rescaling condition states that if I, J ∈ Al and m ≤ L(l) are fixed, then the monochromatic collections Fm (I) and Fm (J) satisfy color (Fm (I)) = color (Fm (J)) ,

(5.3.66)

where the coloring is given by (5.3.55). Next we fix l ≤ K and m ≤ L(l). Then we define the collections of dyadic intervals Fl,m = {Fm (J) : J ∈ Al }. In this way we define disjoint and monochromatic families of intervals which form a decomposition of K. The coloring of K is defined in (5.3.55). Hence the fact that the family Fl,m is a monochromatic collection of dyadic intervals implies that there exists i ≤ L, and k ≤ A such that Fl,m ⊆ Mi,k . Consequently the decomposition of K given by {Fl,m : 1 ≤ l ≤ K and 1 ≤ m ≤ L(l)},

(5.3.67)

is a refinement of the initial decomposition {Mi,k : 1 ≤ i ≤ L, and 0 ≤ k ≤ A − 1} with which we started.

(5.3.68)


333

Defining the resolving Operator In Chapter 1 we introduced a glueing procedure to generate new operators from old ones, with the aim of reducing the rank. We determined a condition ensuring that the new operator is bounded on BMO. The examples we considered in Section 1.5 were somewhat simpler than the ones we are facing right now. Nevertheless the basic glueing mechanism remains the same. In particular we exploit the rescaling property and the pigeon hole principle. Recall that in (5.3.29) we defined the collection K of dyadic intervals. We obtained a decomposition of K in (5.3.67). Now we fix 1 ≤ j ≤ N and J ∈ K. As before we form the functions xj,J =

ψj , hI

I∈L(J)

hI , |I|

and

yj,J =

ψj , hI

I∈R(J)

hI . |I|

Let 1 ≤ l ≤ K, and 1 ≤ m ≤ L(l). Let {Jp } be an enumeration of the intervals in Al . Recall that ∞ Fl,m = Fm (Jp ). (5.3.69) p=1

We use the collection Fl,m to define the glueing procedure as follows. xj,l,m = xj,J and yj,l,m = yj,J . J∈Fl,m

(5.3.70)

J∈Fl,m

The rescaling property expressed in (5.3.63)–(5.3.65) and the Jones compatibility condition (5.3.58) allow us to show that the operator T defined below is bounded on BMO. With the rescaling property we are able to modify the proof of Proposition 5.3.10 so that it gives a good norm estimate for the operator T. Proposition 5.3.11. The operator T defined by Tf =

L(l) N K

f, yj,l,m

j=1 l=1 m=1

xj,l,m , ||yj,l,m ||22

(5.3.71)

satisfies the conditions ||T ||BMO ≤ 100, ||T ϕj − ϕj || ≤ , for j ≤ N. Rank (T ) ≤ f (N, ). Proof. Note that by construction the operator T almost leaves invariant the functions ϕj . More precisely, for j ∈ {1, . . . , N }, T ϕj = ϕj − Ej .


334

With the error estimates in Proposition 5.3.9 we obtain ||T ϕj − ϕj || ≤ . The rank of T satisfies the estimate Rank T ≤ N K max L(l). l

By (5.3.62) and (5.3.63) the factors K and maxl L(l) have upper bounds depending on N and . Thus Rank T ≤ f (N, ) as claimed. Now we turn to proving the norm estimate of T. Let I be a dyadic interval in the Haar support of T f. Then there exists J ∈ K, such that I ∈ L(J). We let {Ip : p ∈ N} be the sequence of maximal intervals in K, which are strictly contained in I. This is clearly a sequence of pairwise disjoint dyadic intervals and |Ip | ≤ |I|. Next we define the collection F = I ∩ L(J). Then we have the following identity for the square function of T f, S 2 (T f |Q(Ip )). S 2 (T f |Q(I)) = S 2 (T f |F) +

(5.3.72)

p∈N

At this point we need to apply two auxiliary estimates which we prove later. First we use Lemma 5.3.12 to estimate the term S 2 (T f |F) appearing on the right-hand side of (5.3.72). It states that for I ∈ K and F ⊆ L(I) we have 29 N ||f ||2BMO . (5.3.73) S 2 (T f |F) ≤ |F ∗ | Aη Second, Lemma 5.3.13 is used to obtain a good upper estimate in the sum over p in (5.3.72). It gives (5.3.74) S 2 (T f |Q(I)) ≤ 3 · 210 |I| ||f ||2BMO , for I ∈ K. Integrating the identity (5.3.72) and invoking (5.3.73) and (5.3.74) we find that 2 2 S 2 (T f |Q(Ip )) S (T f |Q(I)) = S (T f |F) + p∈N

⎛ ≤ ||f ||2BMO ⎝

9

2 N |I| + 3 · 210 Aη

⎞ |Ip |⎠

p∈N

≤ C||f ||2BMO |I|, where C = (3 · 210 +

29 N Aη ).

This completes the proof of Proposition 5.3.11.


335

It remains to verify the two crucial lemmata used in the above proof showing the boundedness of the operator T. We first establish that (5.3.73) holds true. Consider the diagram I∈L(J)

F =I∩L(J)

F ⊆F(

F(

,m )

,m ) ⊆Ci0

0 0 0 −−0→ α = (l0 , m0 ) −−−− −− −−−→ i0 . I −−−−−→ J −−−−−−−→ F −−−−−−

It displays the logical dependence of the intervals appearing in the proof above and their relation to the indices chosen in the course of Lemma 5.3.12. Lemma 5.3.12. Let J ∈ K and let F be a sub-collection of L(J), then 29 N ||f ||2BMO . S 2 (T f |F) ≤ |F ∗ | Aη Proof. Fix J ∈ K and F ⊆ L(J). There exists at most one pair α = (l0 , m0 )

(5.3.75)

with l0 ≤ K and m0 ≤ L(l) so that for some j ≤ N , S 2 (xj,l0 ,m0 |F) = 0.

(5.3.76)

Note however that for a given choice of α = (l0 , m0 ) there is no restriction on the set of j ∈ {1, . . . , N } such that (5.3.76) holds. Consequently the square function S 2 (T f |F) assumes the simplified form S 2 (T f |F) =

N j=1

f, yj,α 2

S 2 (xj,α |F) . ||yj,α ||42

Recall the pointwise estimates of Proposition 5.3.6 which imply that S 2 (xj,α |F) ≤ 2γ|F ∗ |.

(5.3.77)

(5.3.78)

Proposition 5.3.7 gives a matching lower bound for the above integral. We have S 2 (xj,α |L(J)) ≥ γη|J|. (5.3.79) Combining the estimates (5.3.79) and (5.3.78) we find that 2|F ∗ | 2 S 2 (xj,α |L(J)). S (xj,α |F) ≤ η|J|

(5.3.80)

Next we relate the L2 norm of the local square function S(xj,α |L(J)) appearing in (5.3.80) to the global L2 norm of yj,α . Specifically we claim that the following estimate holds, .∞ /−1 32||yj,α ||22 1 2 S (xj,α |L(J)) ≤ |Jk | , (5.3.81) |J| A k=1

where {Jk } is an enumeration of the intervals in Fα .


336

Verification of (5.3.81). Here we exploit the rescaling property formulated in (5.3.65). Recall that we defined α = (l0 , m0 ). We let {Jk } be an enumeration of the intervals in Fl0 ,m0 . By construction, for Jk ∈ Fl0 ,m0 we have S 2 (xj,α |L(Jk )) = S 2 (ψj |L(Jk )).

(5.3.82)

Recall that the pigeon hole principle defines Fl0 ,m0 to be monochromatic with respect to the coloring given by the decomposition (5.3.68). Hence there exists i0 so that the following inclusion holds, Fl0 ,m0 ⊆ Mi0 ,k0 ⊆ Ci0 .

(5.3.83)

Combining the observation (5.3.82) with (5.3.83) and applying Proposition 5.3.7 gives 1 ai0 ,j ≤ S 2 (xj,α |L(Jk )) ≤ ai0 ,j . (5.3.84) 2 |Jk | By definition of α = (l0 , m0 ) in (5.3.75) it follows that J ∈ Fl0 ,m0 . Hence 1 ai0 ,j ≤ S 2 (xj,α |L(J)) ≤ ai0 ,j . (5.3.85) 2 |J| Next we merge the estimates (5.3.84) and (5.3.85), and apply Proposition 5.3.8. Thus we obtain, 2 1 2 S (xj,α |L(J)) ≤ S 2 (xj,α |L(Jk )) |J| |Jk | (5.3.86) 32 1 S 2 (yj,α |R(Jk )). ≤ A |Jk | Multiply ∞ the inequality (5.3.86) by |Jk |, and take the sum over k. Finally divide by k=1 |Jk |. This gives ∞ 32 k=1 S 2 (yj,α |R(Jk )) 1 2 ∞ S (xj,α |L(J)) ≤ |J| A k=1 |Jk | (5.3.87) 2 32 ||yj,α ||2 . = A ∞ k=1 |Jk | An L2 bound for ||yj,α ||H 1 . Next we compare the H 1 norm of yj,α with its L2 norm. Simply applying the Cauchy–Schwarz inequality gives the estimate ∞ ∞ S 2 (yj,α |R(Jk )) |Jk | ||yj,α ||2H 1 ≤ ≤ Note that the factor line in (5.3.87).

∞

k=1 |Jk |

k=1 ∞

k =1

(5.3.88)

|Jk | ||yj,α ||22 .

k=1

appeared before in the denominator of the second


337

Conclusion. We finish now the proof as follows. First we integrate over J the identity (5.3.77). Next fix j ∈ {1, . . . , N } and consider separately each of the resulting summands 2 2 S (xj,α |F) . f, yj,α ||yj,α ||42 To the scalar product we apply Fefferman’s inequality, √ |f, yj,α | ≤ 2 2||yj,α ||H 1 ||f ||BMO . To the integral of the square function S 2 (xj,α |F) we apply (5.3.80). Then using the estimates of (5.3.87) we replace the integral of the local square function by the global L2 norm of yj,α . Finally we use (5.3.88) to replace the H 1 norm of yj,α by its L2 norm. Doing this gives that 2 |F ∗ | 2 S (xj,α |F) . ≤ 29 ||f ||2BMO f, yj,α 4 ||yj,α ||2 ηA Summing this estimate over j ∈ {1, . . . , N } we obtain the claimed estimate of Lemma 5.3.12. Lemma 5.3.13. For I0 ∈ K, S 2 (T f |Q(I0 )) ≤ 3 · 210 |I0 | ||f ||2BMO . Proof. We start with an outline of the proof, and show that it suffices to establish the crucial estimate (5.3.90). Its verification requires a detailed analysis of the glueing process and exploits the rescaling properties obtained by applying the pigeon hole principle. The outline of the proof. Let I0 ∈ K. There exists J0 ∈ J such that I0 ∈ B(J0 ). We rewrite the collection K ∩ I0 as B(J). I0 ∩ K = {I0 ∩ B(J0 )} ∪ J∈J ∩J0

For convenience we enumerate the intervals {J ∈ J : J ⊆ J0 } as {Jk : k ∈ N}. Next we introduce the following blocks of dyadic intervals, (5.3.89) B0 = {L(I) : I ∈ I0 ∩ B(J0 )}, and Bk =

{L(I) : I ∈ B(Jk )}.

They induce a corresponding splitting of the square function of T f. We obtain that ∞ S 2 (T f |Bk ). S 2 (T f |Q(I0 )) = S 2 (T f |B0 ) + k=1


338

Below we give estimates for each of the summands appearing in the above sum. First we show that S 2 (T f |B0 ) ≤ C||f ||2BMO |I0 |. (5.3.90) After that we remark that the proof of (5.3.90) actually shows for any Jk ∈ J0 ∩ J the bound S 2 (T f |Bk ) ≤ C||f ||2BMO |Jk |.

(5.3.91)

Recall that J satisfies the Carleson packing condition with constant ≤ 2. Then taking the sum over k ≥ 1 in (5.3.91) and adding (5.3.90) proves the global estimate of Lemma 5.3.13. In the argument below the logical dependence of the intervals and indices is l0 8 ⏐J ∈A ⏐ 0 l0 I0 ∈Fm (J0 )

Fm (J0 )⊆Fm (J0 )

0 J0 −−−−−− −−→ m0 −−−−−−−−−0−−→ m 8 ⏐I ∈B(J ) 0 ⏐0

and

F(l

,m) ⊆Ci0

(l0 , m) −−−0−−−−−→ i0 .

I0 Proof of (5.3.90). Step 1. The first step of the proof consists in verifying the identity (5.3.94) below, which gives a convenient representation for the square function S 2 (T (f )|B0 ). Recall that I0 ∈ B(J0 ), where J0 ∈ K. By (5.3.59) the block B(J0 ) is decomposed as

L(J0 )

B(J0 ) =

Fm (J0 ),

m=1

where the collections {Fm (J0 ) : m ≤ L(J0 )}, satisfy Jones’s compatibility condition. As I0 ∈ B(J0 ), there exists a unique m0 ≤ L(J0 ) such that I0 ∈ Fm0 (J0 ).

(5.3.92)

Um0 = {m : Fm (J0 ) ⊆ Fm0 (J0 )}.

(5.3.93)

Next we define the index set

Recall that the collections {Fm (J0 ) : m ≤ L(J0 )} satisfy the compatibility condition (5.3.58). Hence we may rewrite B0 , using the index set Um0 as follows, B0 =

m∈Um0

{L(I) : I ∈ I0 ∩ Fm (J0 )}.


339

Let l0 ∈ {1, . . . , K} be determined by the relation J0 ∈ Al0 , where the collection Al0 appears in the decomposition (5.3.56) of J . Note that only for the index l0 do there exist j and m such that S 2 (xj,l0 ,m |B0 ) = 0. We have thus observed that the square function of T f simplifies considerably in the presence of B0 . Indeed the following identity holds, S 2 (T f |B0 ) =

N

2 f, yj,l0 ,m 2 ||yj,l0 ,m ||−4 2 S (xj,l0 ,m |B0 ).

(5.3.94)

j=1 m∈Um0

Proof of (5.3.90). Step 2. The second step in the proof of (5.3.90) consists in establishing the two estimates ⎤ ⎡ ⎣ |Fm0 (J)|⎦ S 2 (xj,l0 ,m |B0 ) ≤ 8|I0 | · ||xj,l0 ,m ||22 (5.3.95) J∈Al0

and

16 ||yj,l0 ,m ||22 . (5.3.96) A Once (5.3.95) and (5.3.96) are established the proof of (5.3.90) can be completed as follows. First we remark that the Jones compatibility condition implies that, for J ∈ Al0 , ||xj,l0 ,m ||22 ≤

N

2 f, yj,I 2 ||yj,l0 ,m ||−2 2 ≤ 8A||f ||BMO |Fm0 (J)|.

m∈Um0 j=1 I∈Fm (J)

Hence taking the sum over J ∈ Al0 and applying Bessel’s inequality we obtain that N

2 f, yj,l0 ,m 2 ||yj,l0 ,m ||−2 2 ≤ 8A||f ||BMO

j=1 m∈Um0

|Fm0 (J)|.

(5.3.97)

J∈Al0

Next we integrate the equation (5.3.94) and insert (5.3.95) and (5.3.96). Then use (5.3.97). This gives N 1 1 S 2 (T f |B0 ) = S 2 (xj,l0 ,m |B0 ) f, yj,l0 ,m 2 ||yj,l0 ,m ||−4 2 |I0 | |I0 | j=1 m∈Um0

N 27 A−1 f, yj,l0 ,m 2 ||yj,l0 ,m ||−2 2 |F (J)| m 0 J∈Al j=1

≤

0

≤2

10

m∈Um0

||f ||2BMO .

Thus using (5.3.95) and (5.3.96) we established (5.3.90).


340

Proof of (5.3.95). We turn now to the proof of the first central estimate, which is (5.3.95). We begin by expanding the square function as S 2 (xj,l0 ,m |B0 ) = S 2 (xj,l0 ,m |L(I)). (5.3.98) I∈I0 ∩Fm (J0 )

Next we estimate the integrals appearing on the right-hand side of (5.3.98). By the assertions of the pigeon hole principle the collection Fl0 ,m is monochromatic with respect to the coloring given by (5.3.68). Hence there exists i0 and k0 so that Fl0 ,m ⊆ Mi0 ,k0 ⊆ Ci0 .

(5.3.99)

Observe that for I ∈ Fl0 ,m , we have the identity S 2 (xj,l0 ,m |L(I)) = S 2 (ψj |L(I)). Hence Proposition 5.3.7 gives 1 ai0 ,j ≤ 2 |I|

S 2 (xj,l0 ,m |L(I)) ≤ ai0 ,j ,

(5.3.100)

for every I ∈ Fl0 ,m . Taking the sum of (5.3.100) over the intervals I ∈ I0 ∩ Fm (J0 ) we obtain that |I|. (5.3.101) S 2 (xj,l0 ,m |B0 ) ≤ ai0 ,j I∈I0 ∩Fm (J0 )

The right-hand side of (5.3.101) is just al0 ,j |I0 ∩Fm (J0 )|. The collections {Fm (J0 ) : m ≤ L(l0 )}, satisfy the compatibility condition (J), so that the following uniform estimate holds true, |Fm (J0 )| |I0 ∩ Fm (J0 )| ≤2 . (5.3.102) |I0 | |Fm0 (J0 )| Now we eliminate the dependence on J0 , from the right-hand side of (5.3.102). The rescaling condition (5.3.65) asserts that the ratio |Fm (J)|/|Fm0 (J)| is independent of J ∈ Al0 . By averaging we obtain the inequality |Fm (J0 )| J∈Al0 |Fm (J)| ≤ 2 . (5.3.103) |Fm0 (J0 )| J∈Al |Fm0 (J)| 0

Now combine (5.3.101) with the estimates (5.3.102) and (5.3.103). This proves the inequality 1 J∈Al0 |Fm (J)| 2 S (xj,l0 ,m |B0 ) ≤ 4ai0 ,j . (5.3.104) |I0 | J∈Al |Fm0 (J)| 0


341

Next we give an upper estimate for the constant ai0 ,j appearing on the right-hand side of (5.3.104). Here we exploit the left-hand side of estimate (5.3.100). For J ∈ Al0 the lower bound of the homogeneity relation (5.3.100) implies that ai0 ,j |Fm (J)| ≤ S 2 (xj,l0 ,m |L(I)). 2 I∈Fm (J)

Summing over J ∈ Al0 we obtain that ai0 ,j |Fm (J)| ≤ 2 J∈Al0

S 2 (xj,l0 ,m |L(I))

J∈Al0 I∈Fm (J)

=

(5.3.105)

||xj,l0 ,m ||22 .

View (5.3.105) as an upper bound for ai0 ,j . Inserting it into (5.3.104) gives (5.3.95) by arithmetic. Proof of (5.3.96). It remains to prove (5.3.96) relating the L2 norms of xj,l0 ,m and yj,l0 ,m by the inequality ||xj,l0 ,m ||22 ≤ (16/A)||yj,l0 ,m ||22 . Start by recalling Proposition 5.3.8. It asserts that for every J ∈ Al0 and I ∈ F(J), 16 S 2 (ψj |R(I)). S 2 (ψj |L(I)) ≤ (5.3.106) A Invoking the definitions of xj,l0 ,m and yj,l0 ,m the estimate (5.3.106) translates into 16 2 S 2 (yj,l0 ,m |R(I)). S (xj,l0 ,m |L(I)) ≤ (5.3.107) A Next we sum the estimate (5.3.107) over J ∈ Al0 and I ∈ F(J). This gives that 16 S 2 (xj,l0 ,m |L(I)) ≤ S 2 (yj,l0 ,m |R(I)). (5.3.108) A J∈Al0 I∈F (J)

J∈Al0 I∈F (J)

Observe that the left-hand side of (5.3.108) coincides with ||xj,l0 ,m ||22 and that the right-hand side (5.3.108) equals ||yj,l0 ,m ||22 . Thus (5.3.96) follows from (5.3.108). Conclusion.

We have presented a complete proof of (5.3.90), that is the estimate S 2 (T (f )|B0 ) ≤ 210 ||f ||2BMO |I0 |.

Next we specialize it to the case when J0 = I0 . This gives that for any k ∈ N0 the following holds, S 2 (T (f )|Bk ) ≤ 210 ||f ||2BMO |Jk |.


342

Recall that J satisfies the 2-Carleson constant. Hence 2

S (T (f )|Q(I0 )) =

∞

S 2 (T (f )|Bk )

k=0

≤ 210 ||f ||2BMO |I0 | + 210 ||f ||2BMO

∞

|Jk |

k=1

≤ 3 · 210 ||f ||2BMO |I0 |.

General UAP data To complete the proof that BMO satisfies the uniform approximation property we combine Proposition 5.3.3 with Theorem 5.3.4 and Theorem 5.3.5. Proof of Theorem 5.3.2. Let n ∈ N and > 0. Let x1 , . . . , xn ∈ BMO. Apply Proposition 5.3.3 to x1 , . . . , xn to obtain ϕ1 , . . . , ϕN so that the following conditions hold: (a) For i ≤ n distBMO (xi , span{ϕ1 , . . . , ϕN }) ≤ xi BMO . (b) ϕ1 , . . . , ϕN are disjointly supported over the Haar system. (c)

N≤

2n3

n .

−N , Fix j ≤ N and let Fj be the Haar support of ϕj . For γ = 3 N −3 log(N 2 −1 ) define Bj = {I ∈ Fj : |ϕj , hI | > γ ϕj BMO }. Then put ϕ1j =

I∈Bj

ϕj ,

hI hI |I|

and ϕ2j = ϕj − ϕ1j . Thus {ϕ11 , . . . , ϕ1N } satisfy the hypothesis of Theorem 5.3.4 and {ϕ21 , . . . , ϕ2N } satisfy the hypothesis of Theorem 5.3.5. Moreover the Haar support of {ϕ11 , . . . , ϕ1N } and {ϕ21 , . . . , ϕ2N } are disjoint collections of dyadic intervals. Apply Theorem 5.3.4 to {ϕ11 , . . . , ϕ1N }. This gives a resolving operator T1 so that ||T1 ||BMO ≤ 4, ||T1 (ϕ1i ) − ϕ1i ||BMO ≤ ||ϕ1i ||BMO for i ≤ N, rank(T1 ) ≤ f1 (N, , γ).


343

If J is a dyadic interval not contained in the Haar support of {ϕ11 , . . . , ϕ1N }, then T1 (hJ ) = 0. Next apply Theorem 5.3.5 to {ϕ21 , . . . , ϕ2N }. This gives a resolving operator T2 so that ||T2 ||BMO ≤ 10, ||T2 (ϕ2i )

− ϕ2i ||BMO ≤ ||ϕ2i ||BMO for i ≤ N, rank(T2 ) ≤ f2 (N, ).

If J is not contained in the Haar support of {ϕ21 , . . . , ϕ2N } then T2 (hJ ) = 0. Hence the operator T = T1 + T2 satisfies : ||T ||BMO ≤ 102 + 42 , ||T (xi ) − xi ||BMO ≤ 4||xi ||BMO for i ≤ N, rank(T ) ≤ f1 (N, , γ) + f2 (N, ) = f (, n).

We have now presented a complete proof of P. W. Jones’s theorem that BMO satisfies the uniform approximation property. By a result of S. Heinrich [83] the uniform approximation property is a self-dual isomorphic invariant. Hence H 1 satisfies the uniform approximation property, since BMO does. In addition to the spaces Lp (1 ≤ p ≤ ∞), H 1 and BMO the uniform approximation property has been investigated for the following classical Banach spaces. (a) The reflexive Orlicz spaces satisfy the UAP (theorem of J. Lindenstrauss and L. Tzafriri [129]). (b) L(2 ), the space of bounded linear operators on Hilbert space does not satisfy the approximation property, hence it fails to satisfy the UAP (theorem of A. Szankowski [198]). (c) The Schatten class S r consists of compact operators on Hilbert space for which the singular values are in r . (Singular values of a compact operator T are the eigenvalues of (T ∗ T )1/2 ). The Schatten class S r fails the UAP for r > 80 and 1 ≤ r < 80/79 (theorem of A. Szankowski [199]). Many years ago A. Pelczy´ nski [171] posed the problem whether H ∞ (D), the space of bounded analytic functions on the unit disk, satisfies the UAP. Now, there are available two extremely deep results pertaining to this problem: On the one hand there is the theorem of P. W. Jones asserting that BMO satisfies the UAP, and there are many connections between H ∞ (D) and BMO, as discussed by J. Garnett [72]. This seems to support the conjecture that H ∞ (D) satisfies the uniform approximation property. On the other hand we have a representation of H ∞ (D) as the sub-space of L(2 ), that is formed by the Toeplitz operators (see [120]) and the theorem of A. Szankowski [198] to the effect that L(2 ) does not satisfy


344

the approximation property. This suggests that H ∞ (D) is likely to fail the uniform approximation property. However, for special cases of UAP data in H ∞ (D), resolving operators were found in [28] and [23]. We conclude this section with a list of Banach spaces for which the verification of the uniform approximation property is outstanding. The following spaces are conjectured to satisfy the uniform approximation property. 1. W k,∞ (Rn ), the Sobolev space of functions on Rn (n ≥ 2) with bounded weak derivatives of order ≤ k (k ≥ 1), (conjecture of P. W. Jones). 2. Lp,q (R)(p = q), the Lorentz space consisting of measurable functions on R for which 1/q ∞ 1/p ∗ q dt [t f (t)] < ∞,

f Lp,q (R) = t 0 where f ∗ denotes the non-decreasing rearrangement of f (conjecture of W. B. Johnson). 3. S r , the Schatten class of operators, provided that |1/2 − 1/r| is small enough (conjecture of A. Szankowski).

5.4

Notes

In 1976 S. Kwapien and A. Pelczy´ nski obtained Theorem 5.1.2 for H 1 (T) (the space of integrable functions on the unit circle for which the harmonic extension to the disk is analytic). It appeared in [120]. Historically the work of S. Kwapien and A. Pelczy´ nski establishes the first isomorphic invariant for the space H 1 (T) which strongly suggests that H 1 (T) is a space with an unconditional basis. The proof given in the text is based on ideas of P. Wojtaszczyk [211]. The dichotomy of Theorem 5.1.3 is from [158]. In the context of Lp , W. B. Johnson [91] established the factorization method used in the proof of Theorem 5.1.3. The H 1 version of Johnson’s factorization is in [158]. The dichotomyanalogous to Problem 5.1.5 is established for the complemented subspaces of ( L2 )1 . This is the content of Proposition 8.1 in the memoir [27] by J. Bourgain, P. Cassaza, J. Lindenstrauss and L. Tzafriri. Proposition 5.1.10, the estimate for the square function of 2n unit vectors in 1 H , is due to J. Bourgain [17]. In the famous paper [169] A. Pelczi´ nski establishes the decomposition method and proves that the spaces p (1 ≤ p < ∞) are primary by showing that every infinite-dimensional, complemented subspace of p is isomorphic to p . The complemented subspaces of the non-separable space ∞ are determined by J. Lindenstrauss [127]. The result is analogous to the separable case: Every infinitedimensional, complemented subspace of ∞ is isomorphic to ∞ . In [20] J. Bourgain proved that H ∞ (D), the space of bounded analytic functions in the unit disk is a primary space. In [20], J. Bourgain not only solves a particular problem about

5.4. Notes

345

H ∞ (D), but also introduces a general method capable of treating several classes of non-separable Banach spaces. It is applied in [153] to BMO. Theorem 5.2.1 and its proof is from [153]. Using Bourgain’s method G. Blower [13] proved that the (non-separable) Banach space of bounded linear operators on Hilbert space is primary. Applications of Bourgain’s method to projective tensor products of p spaces were given by A. Arias and J. Farmer [4] who investigate their isomorphic invariants, and use combinatorial methods (Ramsey’s theorem) to study the problem of primarity for these spaces. Consult J. Lindenstrauss and L. Tzafriri [130] for a detailed presentation of the primary spaces among the classical Banach spaces. In particular [130] contains P. Enflo’s result that the Lp spaces are primary for 1 < p < ∞. P. Enflo’s original proof is recorded by B. Maurey in [138]. P. Enflo’s method extends to a large class of separable function spaces with an unconditional basis (see [93] by W. Johnson, B. Maurey, G. Schechtman, L. Tzafriri.) A second proof of P. Enflo’s result is due to D. Alspach, P. Enflo and E. Odell [2], who base their argument on the fact that Lp and Lp (2 ) are isomorphic Banach spaces for 1 < p < ∞. By Bourgain’s Theorem 5.2.8 the spaces H 1 and H 1 (2 ) are not isomorphic Banach spaces. Hence the approach of D. Alspach, P. Enflo and E. Odell [2] does not extend to the H 1 case. Nevertheless H 1 is primary, see [153]. P. Enflo and T. Starbird [63] prove that the space L1 is primary. The space of bounded continuous functions on the unit interval is shown by J. Lindenstrauss and A. Pelczy´ nski [128] to be a primary space. For spaces of continuous functions over countable compact spaces the analogous result is due to P. Billard [11]. Theorem 5.2.7 presents J. Bourgain’s description of complemented copies of Hn1 . Its main application is Theorem 5.2.8, hence the fact that H 1 is not isomorphic to H 1 (2 ). It should have many more applications, in particular it should be useful in connection with the conjecture of P. Wojtaszczyk discussed in Section 4.3. Tent spaces are introduced by R.R Coifman, Y. Meyer and E.M. Stein in [54] who show that H 1 is isomorphic to a complemented subspace of the tent space Tα . The isomorphic identification of tent spaces with H 1 (2 ) is presented in Theorem 5.2.9. It is due to N. Kalton and P. Wojtaszczyk [109]. The concept of the uniform approximation property was introduced by A. Pelczy´ nski and H. Rosenthal in [172]. They show in Theorem 5.3.1 that the Lp spaces satisfy the uniform approximation property. The proof given in the text is due to S. Kwapien. Lower estimates for the uniformity function on Lp are obtained by T. Figiel, W. Johnson and G. Schechtman [67] for the values p = 1 and p = ∞, and by J. Bourgain [25] for the remaining values 1 < p < ∞. For UAP data consisting of independent random variables there exist upper estimates for the uniformity function due to W. Johnson and G. Schechtman [95], which are much lower than the estimates for general UAP data in [25]. J. Lindenstrauss and L. Tzafriri [129] show that a large class of Orlicz spaces satisfies the uniform approximation property. For an overview on approximation properties see [130] and [47]. Solving a problem of A. Pelczy´ nski [171], P. W. Jones established the uniform approximation property for H 1 and BMO in [102].

346


S. Heinrich [83] proved that a Banach space satisfies the uniform approximation property if and only if its dual space does. V. Mascioni [137] gives a constructive proof of S. Heinrich’s result, that could in principle be useful in simplifying the proof that BMO satisfies the uniform approximation property. In particular V. Mascioni’s ideas in [137] should be helpful in passing from operators factoring through ∞ (Proposition 5.3.10) to the resolving operator of finite rank. Executing the original argument of P. W. Jones in the context of tent spaces should result in some simplifications in the proof of the UAP for BMO. The work of A. Szankowski [198] and [199] introduces extremely refined methods to establish that a given Banach space does not satisfy the uniform approximation property. Thereby the space of linear operators on Hilbert space and the Schatten classes S r (for r > 80 or r < 80/79) are shown to fail the UAP.

Chapter 6

Atomic H 1 Spaces This chapter contains the presentation of atomic H 1 spaces and their relation to 1 dyadic H 1 . We treat the example of Hat in considerable detail, obtaining equivalent norms using the Hilbert transform, the Lusin area function and the non-tangential maximal function. We investigate L. Carleson’s biorthogonal system and show 1 . Thereby we prove B. Maurey’s theorem that it is an unconditional basis for Hat 1 1 that Hat and H are isomorphic Banach spaces. We close the chapter with the classification theorem for the Banach spaces H 1 (X, d, µ).

6.1

1 Basic similarities between H 1 and Hat

1 . We prove twoIn this section we give three important characterizations of Hat 1 sided estimates relating the norm of u ∈ Hat to the Lusin function, the nontangential maximal function and the Hilbert transform, respectively. We will now define the Lusin function and the non-tangential maximal function. Let t ∈ R, then define Γt to be the convex hull of eit , and the disk {z ∈ D : |z| ≤ √12 }. Thus defined Γt is a sub domain of the unit disk, it is called the Stolz domain with vertex at eit . In a neighborhood of its vertex the Stolz domain has the shape of a triangle with a right angle at eit . Let F : D → C have continuous partial derivatives. The Lusin function SL (F ) is defined by the equation

1/2 |grad F (z)| dA(z) 2

SL (F )(t) =

,

(6.1.1)

Γt

where t ∈ R, and where dA(z) denotes the area measure in the complex plane. Note that SL (F ) is a periodic function satisfying SL (F )(t+2π) = SL (F )(t). Hence we may regard it as a well-defined function on the boundary of the unit disk. Next we define the non-tangential maximal function of F : D → C. As before

Chapter 6. Atomic H 1 Spaces

348 we let t ∈ R. Then we put

F ∗ (t) = sup |F (z)|.

(6.1.2)

z∈Γt

The non-tangential maximal function of F satisfies F ∗ (t + 2π) = F ∗ (t), it is therefore well defined on the boundary of the unit disk. Next we fix an integrable function u : T → C defined initially on the boundary of the unit disk. To form its Lusin function and its non-tangential maximal function we need to extend u to the unit disk. For that purpose we use the harmonic extension. Recall the Poisson kernel for the unit disk, Pt (z) =

1 − |z|2 , |eit − z|2

where z ∈ D, and t ∈ R. The harmonic extension of u : T → C is obtained by integrating u against the Poisson kernel, thus π 1 U (z) = u(eit )Pt (z)dt. (6.1.3) 2π −π For U : D → C the Lusin function SL (U )(t) and the non-tangential maximal function U ∗ (t) are defined by (6.1.1), respectively (6.1.2). Now we define SL (u) and u∗ by putting SL (u)(t) = SL (U )(t) and

u∗ (t) = U ∗ (t).

We emphasize that for a function u defined on the boundary of the unit disk the Lusin function SL (u) and the non-tangential maximal function u∗ are given only through the harmonic extension of u. Next we recall a basic theorem of Hardy and Littlewood concerning the nontangential maximal function F ∗ when F is analytic in the unit disk. Precisely let F : T → C be integrable. Assume that the harmonic extension of F to the unit disk, is analytic in D. Then, by the theorem of Hardy and Littlewood [82] for every p > 0 the following estimate holds, π π |F ∗ (eit )|p dt ≤ C |F (eit )|p dt. (6.1.4) −π

−π

The analyticity of F is responsible for (6.1.4) to hold for every p > 0. Without analyticity the validity of (6.1.4) is limited to the range of p strictly larger than 1. In fact if u : T → C is integrable, and 1 < p < ∞, then π π ∗ it p p |u (e )| dt ≤ Cp |u(eit )|p dt, −π

−π

where Cp → ∞ as p → 1. By the maximum principle for harmonic functions the above estimate remains true in the limiting case where p → ∞. In particular Cp

1 6.1. Basic similarities between H 1 and Hat

349

stays uniformly bounded as p → ∞. A proof of the Hardy–Littlewood theorem is presented in [72], Theorem II.3.1. The Lusin area function and the non-tangential maximal function show similar boundedness properties in the reflexive Lp spaces. Indeed, a result of J. Marcinkiewicz and A. Zygmund [134] asserts that for 1 < p < ∞, π π it p p |SL (u)(e )| dt ≤ Ap |u(eit )|p dt. (6.1.5) −π

−π

As p → 1 we have Ap → ∞. For large values of p the constants in the theorem of J. √ Marcinkiewicz and A. Zygmund are not uniformly bounded. They satisfy Ap ∼ p, as p → ∞. From a modern point of view, originating with L. H¨ ormander [86], the Lusin area function is regarded as an average of singular integrals. Thus (6.1.5) can be obtained from the Lp boundedness of singular integral operators, see [192]. Finally we recall the definition of the Hilbert transform. Let u be an integrable function defined on the boundary of the unit disk. Equivalently we may regard u as a 2π periodic function on the real line. Then for t ∈] − π, π[ we define t− π y−t 1 dy . (6.1.6) + u(y) cot H(u)(t) = lim →0 2π 2 −π t+ N. Lusin showed that Hu(t) exists for almost every t ∈] − π, π[ provided that u ∈ L2 . In the case where u is merely integrable the existence (almost everywhere) of the limit (6.1.6) is a result of I. Privalov (the book of P. Koosis [118] contains detailed proofs and an interesting historic discussion). Extended to the real line 2π periodically, the Hilbert transform gives a function on the boundary of the unit disk. Let u : T → R be integrable and assume that also the Hilbert transform H(u) : T → R is integrable. For z ∈ D the equation π 1 {u(eit ) + iH(u)(eit )}Pt (z)dt F (z) = 2π π defines an analytic function F (z) in D. Thus the harmonic extension of u + iH(u) is analytic in the unit disk. We equip the unit circle T = {z ∈ D : |z| = 1} with the metric given by the Euclidean distance, and with linear Lebesgue measure. Thus defined the triple (T, dt, | · |) is a space of homogeneous type (for definitions see Section 2.1 in Chapter 2). An atom for (T, dt, | · |) is either a constant function on T or a function a : T → R such that adt = 0 and a2 dt ≤ |I|−1 , T

T

where I is an interval in T such that supp a ⊆ I.


350

In Section 2.1 we defined the associated atomic H 1 space to be the space of functions f : T → R which admit a decomposition into atoms, f= c i ai , where ai are atoms for (T, dt, | · |) and where |ci | < ∞. 1 1 We denote this space by Hat . The norm of f ∈ Hat is 1 = inf{ ||f ||Hat |ci |},

where the infimum is taken over all decompositions of f into atoms. In this section we prove three different characterizations for a function u : T → R to be in the 1 1 . We show that u ∈ Hat is equivalent to (u + iHu) ∈ L1 (T), as well as to space Hat ∗ 1 1 u ∈ L (T), and to SL (u) ∈ L (T). We summarize now the inequalities by which we obtain these characterizations. 1 , is ex1. The equivalence between the conditions u∗ ∈ L1 (T), and u ∈ Hat pressed by the two-sided estimates ∗ 1 ≤ u L1 (T) ≤ C u H 1 . c u Hat at

(6.1.7)

1 The proof of (6.1.7) uses the following three results. First, a function in Hat has an integrable Hilbert transform. Second, a function with an integrable 1 1 . These properties of Hat are non-tangential maximal function belongs to Hat established in Proposition 6.1.3 and Proposition 6.1.2. They give ∗ 1 ≤ C u L1 (T) . c Hu L1 (T) ≤ u Hat

(6.1.8)

The result of Hardy and Littlewood (6.1.4) provides the third component in 1 . We use it to show the proof of the maximal function characterization of Hat that (6.1.8) implies (6.1.7). 2. The Lusin area function and the Hilbert transform are connected by the following result of A. Calder´ on [39], c SL (u) L1 (T) ≤ (u + iHu) L1 (T) ≤ C SL (u) L1 (T) .

(6.1.9)

We present A. Calder´ on’s original proof which is an impressive example of elegance and power. Its key is a remarkable integral inequality, the so-called square-duality relation. Let F : D → C denote the harmonic extension of u + iH(u). Then the Lusin function of |F |1/3 : D → R satisfies the estimate c SL2 |F |1/3 Gdt T ∗ 2/3 ≤ |F |1/3 SL |F |1/3 SL (G)dt, |F | Gdt + T

T


351

whenever G ∈ L3 (T) is non-negative. We obtain important applications of the square-duality relation by exploiting that the homogeneities on its righthand side and left-hand side are different. Indeed we have S 2 (|F |1/3 ) on the left, where as S(|F |1/3 ) appears on the right-hand side.

Characterization by maximal functions 1 We start with the maximal function characterization of Hat .

Theorem 6.1.1. Let u : T → R be integrable, then the following a priori estimates hold, ∗ 1 ≤ u L1 (T) ≤ C u H 1 . c u Hat at The starting point is the following proposition which shows that the Hilbert transform is well behaved on the atoms of T. Proposition 6.1.2. Let a be an atom for T. Then the Hilbert transform of a is integrable, satisfying π |H(a)|(t)dt ≤ C, −π

where C is a universal constant Proof. Let I ⊆ T be an interval. Let a : T → R satisfy supp a ⊆ I, ||a||2 ≤ |I|−1/2 , and

a = 0. T

Assume that there exists N ∈ N such that |I| = 2−N . For 0 ≤ n ≤ N we define (2n I) ⊆ T to be the interval with the same midpoint as I satisfying |(2n I)| = 2n |I|. Next put Jn = {t ∈ [−π, π[: eit ∈ (2n I)}. Note that while (2n I) is an interval in T by definition, it is not necessarily true that Jn is an interval in [−π, π[. (That happens when eiπ ∈ (2n I) and (2n I) = T.) To estimate the L1 norm of H(a) we treat separately the integrals |H(a)|(t)dt, (6.1.10) J1

and

|H(a)|(t)dt. [−π,π[\J1

(6.1.11)


352

First we estimate (6.1.10). By the Cauchy–Schwarz inequality and the L2 boundedness of the Hilbert transform we have that |H(a)|(t)dt ≤ |J1 |1/2 H(a) 2 J1

≤ C|I|1/2 a 2 ≤ C. Now we turn to the corresponding estimate for (6.1.11). For t ∈ / J1 we have y−t 1 dy, a(y) cot H(a)(t) = 2π J0 2 itI where the integral on the right-hand side is absolutely converging. Let e denote the midpoint of I. Next use that T a = 0, to find the identity tI − t y−t H(a)(t) = − cot dy. a(y) cot 2 2 J0

Let n ≤ N, let t ∈ Jn \ Jn−1 , and let y ∈ J0 , then we have the following pointwise estimate, resulting from the periodicity of the cotangent function cot y − t − cot tI − t ≤ C|I| . 2 2 |(2n I)|2 Invoking this estimate we obtain that −n |H(a)(t)|dt ≤ C2 Jn \Jn−1

π −π

|a(t)|dt.

Summing up we showed the estimate |H(a)|(t)dt = [−π,π[\J1

N n=1

≤C

Jn \Jn−1 π

−π

|H(a)(t)|dt

|a(t)|dt.

The Cauchy–Schwarz inequality implies that T |a(t)|dt ≤ 1. 1 λI aI where the aI are atoms and where λI ∈ R If u ∈ Hat , then u = satisfies 1 . |λI | ≤ C u Hat Hence Proposition 6.1.2 implies that H(u) is integrable and π 1 . |u + iH(u)|dt ≤ C u Hat −π

(6.1.12)


353

Next we prove that functions with integrable non-tangential maximal func1 tion are contained in Hat . We show, if u∗ is integrable, then u can be decomposed into an absolutely convergent series of atoms. Proposition 6.1.3. Let u : T → R be integrable and assume that also u∗ , its non1 , and tangential maximal function, is integrable. Then u ∈ Hat π 1 ≤ C u∗ dt. (6.1.13)

u Hat −π

Proof. Let k ∈ Z and define Ek = {eit ∈ T : u∗ (t) > 2k }. As u∗ is the supremum of continuous functions, Ek is an open set in T. We let Ck be the collection of disjoint open intervals such that Ek = I. I∈Ck

Let I ∈ Ck . Denote by mI (u) the average of u over the interval I. We claim that for I ∈ Ck the mean value of u over I admits the upper bound |mI (u)| ≤ C2k .

(6.1.14)

Skipping over the details, we remark that (6.1.14) can be verified using the following version of the mean value property: For a smooth harmonic function defined on a square, the average over the sides of the boundary is equal to the average over the diagonals. For details consult the book of C. Bennett and R. Sharpley [8] p. 370. Next we define ⎡ ⎤ (u − mJ (u))1J ⎦ . aI = 2−k |I|−1 ⎣(u − mI (u))1I − {J∈Ck+1 :J⊆I}

Observe that supp aI ⊆ I, and

T

aI dt = 0.

The claimed estimate (6.1.14) implies

aI ∞ ≤ C|I|−1 . Putting λI = 2k |I| we obtain u=

∞ k=−∞ I∈Ck

λI aI .


354

Next we claim that the series on the right-hand side represents an atomic decom1 . To verify the claim we recall position of u which gives correct estimates for u Hat the definition of the sequence λI and estimate ∞

λI =

k=−∞ I∈Ck

=

∞ k=−∞ ∞ k=−∞ π

≤C

2k

|I|

I∈Ck

2k |Ek | u∗ dt.

−π

1 Proof of Theorem 6.1.1. By (6.1.12) for every u ∈ Hat we have π 1 . |H(u)| dt ≤ C u Hat −π

Write F = u + iH(u). The harmonic extension of F to the unit disk defines an analytic function. Hence (6.1.4) — the estimate of Hardy and Littlewood — is applicable. It implies that F ∗ is integrable and satisfies π π ∗ F dt ≤ C |F |dt. −π

−π

Taking real parts in the definition of F ∗ and using (6.1.12) we obtain the following 1 estimate for u ∈ Hat , π 1 . u∗ dt ≤ C u Hat (6.1.15) −π

Conversely by Proposition 6.1.3, if u∗ ∈ L1 (T), then u admits an atomic decomposition showing that π 1 ≤ C u∗ dt. (6.1.16)

u Hat −π

Combining (6.1.15) and (6.1.16) gives the maximal function characterization as claimed. Observe that by proving Theorem 6.1.1 we showed simultaneously that 1 ≤ C (u + iH(u) L1 (T) . c (u + iH(u) L1 (T) ≤ u Hat


355

Characterization by square functions Now we study the connection between the Lusin square function and the Hilbert 1 transform. The next result provides the square function characterization for Hat due to A. Calder´ on. Theorem 6.1.4. Let u : T → R be integrable, then the following a priori estimates hold, c SL (u) L1 (T) ≤ (u + iHu) L1 (T) ≤ C SL (u) L1 (T) . Before we start proving Calder´ on’s theorem we should single out some of the ingredients that are used in the proof. We will repeatedly apply Green’s theorem in the following form. Let u be a smooth function defined in a neighborhood of the unit disk, then π 1 1 1 ∆u(z) log dA(z) = u(eit )dt − u(0). (6.1.17) π |z| 2π −π D We also use a very interesting set of identities holding for the modulus of analytic functions: Let F be analytic, p ≥ 1 and q > 0, then (6.1.18) grad|F |p/q = p|F |(p−1)/q grad|F |1/q . This identity relates first order differentiation of various powers of |F |. The next identity is between the Laplacian, thus second order differentiation, and the gradient of |F |. Let q > 0 and let F be analytic. Then 2 ∆(|F |2/q ) = 4 grad|F |1/q .

(6.1.19)

As a first application of the identity (6.1.19) and Green’s Theorem (6.1.17) we prove the following lemma. It will be used several times below. Lemma 6.1.5. Let F : T → C be integrable. Assume that the harmonic extension of F to the unit disk is analytic, and that F (0) = 0. Then ≤ C F L1 (T) , c F L1 (T) ≤ SL2 |F |1/2 L1 (T)

where

SL2 |F |1/2 (eit ) =

2 grad|F (z)|1/2 dA(z).

Γt

Proof. The starting point is Green’s theorem. It expresses the L1 (T) norm as an area integral. Recall that F (0) = 0, hence (6.1.17) gives that π 1 1 1 |F |(eit )dt = ∆(|F |) log dA(z). 2π −π π |z| D


356

Next by (6.1.19) we may replace the Laplacian in the above integral as follows, 2 4 1 1 1 (6.1.20) ∆(|F |) log dA(z) = grad|F |1/2 log dA(z). π |z| π |z| D D 1 Next we replace the factor log |z| by 1−|z|. This is possible since the following three 1 1 simple facts hold. First for |z| ≥ 1/4 we have that log |z| ∼ 1 − |z|. Second log |z| is 2 integrable over the unit disk with respect to the area measure. Third grad|F |1/2 is a subharmonic function. Combined these facts imply that the integral on the right-hand side of (6.1.20) is bounded above and below by (a constant multiple of) the integral 2 (6.1.21) grad|F |1/2 (1 − |z|)dA(z). D

Applying Fubini’s theorem shows that the integral in (6.1.21) is (up to universal constants) the same as π SL2 ( |F |1/2 )(eit )dt. −π

The next proposition contains the first part of the square function charac1 . terization of Hat Proposition 6.1.6. Let F : T → C be integrable. Assume that the harmonic extension of F to the unit disk is analytic. Then ||SL (F )||L1 (T) ≤ C||F ||L1 (T) . Proof. We start with the identity (6.1.18). Putting p = q = 2 we obtain that |grad |F || = 2|F |1/2 grad |F |1/2 . Taking the supremum of |F |1/2 over the Stolz domain Γt , and then integrating 1/2 the square of grad |F | over Γt , gives the following estimate between the Lusin function and the non-tangential maximal function. ∗ SL (|F |) ≤ 2 |F |1/2 SL |F |1/2 . Integrating and applying the Cauchy–Schwarz inequality gives that ∗

SL (F ) L1 (T) ≤ 2 |F |1/2 SL |F |1/2 1 L (T) ∗ 1/2 ≤ |F | 2 SL |F |1/2 2 . L (T)

(6.1.22)

L (T)

Next we show that both factors appearing in the second line of (6.1.22) are 1/2 bounded by C||F ||L1 (T) . The estimate for the factor involving SL |F |1/2 is the content of Lemma 6.1.5.


Observe that

357

∗ 1/2 |F |1/2 = (|F |∗ ) .

Hence the following inequality for the non-tangential maximal function holds true, by (6.1.4), ∗ 1/2 (6.1.23) |F |1/2 2 ≤ C F L1 (T) . L (T)

Thus with (6.1.23) and Lemma 6.1.5 the conclusion of Proposition 6.1.6 follows from (6.1.22). Proposition 6.1.7. Let F : T → C be integrable with F = 0. Assume that the harmonic extension of F to the unit disk is analytic. Then ||F ||L1 (T) ≤ C||SL (F )||L1 (T) . Proof. We evaluate the identities (6.1.18) with p = 2 and q = 2. Thus |grad |F || = 2|F |1/2 grad |F |1/2 . Next we take p = 3/2 and q = 3 and we obtain 3 grad |F |1/2 = |F |1/6 grad |F |1/3 . 2 It follows that 3/4 1/4 (1/2 |grad |F ||) . grad |F |1/2 = 3/2 grad |F |1/3

(6.1.24)

Next we take the square of (6.1.24), then we integrate over the cone Γt and finally we apply H¨ older’s inequality with the parameters 4/3 and 4. This gives the following pointwise estimate for the Lusin function, 3/2 1/2 1 3 1/3 SL |F | SL (|F |) . (6.1.25) SL2 (|F |1/2 ) ≤ 2 2 Next we apply Lemma 6.1.5 followed by the Schwarz inequality. This gives that ||F ||L1 (T) ≤ C SL2 |F |1/2 1 L (T) 3/2 1/3 |F | ≤ C SL 2

estimate in (6.1.25) and the Cauchy–

L (T)

1/2 SL (|F |)

L2 (T)

(6.1.26) . 3/2

Note that the first factor in the second line of (6.1.26) is just SL (|F |1/3 ) L3 (T) . 1/2

Similarly we may rewrite the second factor as SL (|F |) L1 (T) . It is also true that grad |F | = |F |, hence the following identity holds for the Lusin function, SL (|F |) = SL (F ) .


358 In summary we showed that 3/2 ||F ||L1 (T) ≤ C SL |F |1/3 3

L (T)

1/2

||SL (F )||L1 (T) .

(6.1.27)

In Proposition 6.1.8 we will prove that the following crucial inequality holds, 3/2 SL |F |1/3 3

L (T)

1/2

≤ C F L1 (T) .

(6.1.28)

Inserting (6.1.28) into (6.1.27) gives that ||F ||L1 (T) ≤ C||SL (F )||L1 (T) .

To complete the proof of Proposition 6.1.7 we show now that (6.1.28) holds. Proposition 6.1.8.

SL |F |1/3

1/3

L3 (T)

≤ C F L1 (T) .

Proof. Let G ∈ L3 (T) be non-negative. We denote its harmonic extension to the unit disk again by G. We claim that the following integral inequality holds, π π π 2 ∗ |F |1/3 SL |F |1/3 SL (G)dt. SL |F |1/3 G dt ≤ C |F |2/3 G dt+C −π

−π

−π

From this claim we derive the conclusion of Proposition 6.1.8 as follows. We estimate the first integral on the right-hand side using H¨ older’s inequality with the parameters 1/3 + 2/3 = 1. For the second integral use Hölder’s inequality with 1/3 + 1/3 + 1/3 = 1. π SL2 |F |1/3 G ≤ C |F |2/3 3/2 G L3 (T) L (T) −π ∗ 1/3 + C |F | 3 SL |F |1/3 3 SL (G) L3 (T) . L (T)

L (T)

(6.1.29) Let us now comment on estimates for the third and the first factor appearing in the second line of (6.1.29). Recall that by (6.1.5) a harmonic function G in the unit disk satisfies SL (G) L3 (T) ≤ C G L3 (T) . As F is analytic, the estimate of Hardy and Littlewood (6.1.4) implies that taking the non-tangential maximal function is a bounded operation in L1 (T). Rewriting (6.1.4) gives that ∗ |F |1/3 3 ≤ C F 1/3 3 . L (T)

L (T)

Notice also that clearly F 1/3 3 = F 1/3 1 . Hence taking the supremum in (6.1.29) over all non-negative G with G 3 ≤ 1, we find that 2 2/3 1/3 SL |F |1/3 3/2 ≤ F L1 (T) + C F L1 (T) SL |F |1/3 3 . L

(T)

L (T)


359

By simple arithmetic, the homogeneities of this inequality give the estimate 1/3 SL |F |1/3 3 ≤ C F L1 (T) . L (T)

Now we turn to the proof of the square-duality relation, π π π ∗ |F |1/3 SL |F |1/3 SL (G). SL2 |F |1/3 G ≤ C |F |2/3 G + C

−π

−π

−π

(6.1.30) Recalling first the definition of the Lusin function and then that G is non-negative we find with Fubini’s theorem the estimate (1−|w|)π π w 2 1/3 1/3 2 )dtdA(w). Gdt ≤ C SL |F | |grad(|F | )| (w) G(eit |w| −π −(1−|w|)π D Next by pointwise estimates for the Poisson kernel and by (1 − |w|) ≤ 2 log obtain the following estimate for the non-negative function G,

(1−|w|)π

G(eit −(1−|w|)π

w 1 )dt ≤ C log( )G(w). |w| |w|

Inserting the estimate (6.1.31) we obtain the inequality π 2 2 1 SL |F |1/3 Gdt ≤ C 4G grad |F |1/3 log( )dA(z). |z| D −π

1 |w|

we

(6.1.31)

(6.1.32)

We thus replaced the integral over T by an integral over the unit disk D, and thereby we made a first step towards applying Green’s theorem below. The second step consists in finding pointwise upper bounds for the integrand on the righthand side of (6.1.32). That is, we seek estimates for G|grad(|F |1/3 )|2 . We start by calculating ∆(|F |2/3 G) = G∆(|F |2/3 ) + 2grad(|F |2/3 ) · grad(G).

(6.1.33)

We rewrite the identity (6.1.33) using that F is an analytic function. By (6.1.19) with q = 3 we have 2 ∆(|F |2/3 ) = 4 grad(|F |1/3 ) . Equation (6.1.18) with p = 2 and q = 3 gives that grad(|F |2/3 ) = 2|F |1/3 grad(|F |1/3 ) . Inserting back into equation (6.1.33) and rearranging terms gives the inequality 2 4G grad(|F |1/3 ) ≤ ∆(G|F |2/3 ) + 4|F |1/3 grad(|F |1/3 ) · |grad(G)|. (6.1.34)


360

We return to estimating integrals. Multiply the inequality (6.1.34) by log(1/|z|) and integrate over the unit disk. This gives that 2 1 4G grad(|F |1/3 ) log dA(z) |z| D 1 1 ≤ ∆(|F |2/3 G) log dA(z) + 4 |F |1/3 |grad(|F |1/3 )| |grad(G)| log dA(z). |z| |z| D D (6.1.35) Applying Green’s theorem we rewrite the first integral appearing in the second line of (6.1.35). By Theorem 6.1.5 we obtain that π 1 |F |2/3 Gdt. (6.1.36) 2 ∆(|F |2/3 G) log dA(z) = |z| −π D Finally we estimate the second integral using Fubini’s theorem and the Cauchy Schwarz inequality. This gives π 1 1/3 1/3 |F | grad(|F | ) ·|grad(G)| log dA(z) ≤ C (|F |1/3 )∗ SL (|F |1/3 )SL (G). |z| −π D (6.1.37) Summing up we showed that (6.1.32) followed by (6.1.35)– (6.1.37) implies the inequality π π π 2 ∗ |F |1/3 SL |F |1/3 SL (G). SL |F |1/3 G ≤ C |F |2/3 G + C −π

−π

−π

n

For a power series F (z) = an z in the unit disk, R. E. A. C. Paley [168] shows that the sequence of lacunary coefficients is square summable provided that the boundary values are integrable on the unit circle. Paley’s classic inequality asserts that 1/2 1 π 2 ≤ |F (eit )|dt. |a|2n π −π The square function estimate of Proposition 6.1.7 reveals the connection between Paley’s classic inequality and J. Bourgain’s estimate (1.3.32) for Rademacher projections on H 1 . Indeed, by Proposition 6.1.7 and Paley’s inequality, it follows that π π |a|22n ≤ C |F (eit )|dt SL (F )(eit )|dt. −π

6.2

−π

Carleson’s biorthogonal system

1 In this section we prove B. Maurey’s result that H 1 and Hat are isomorphic Ba1 . We present L. nach spaces, and that there exists an unconditional basis in Hat Carleson’s approach to B. Maurey’s theorem since it determines explicitly the un1 and gives a simple characterization of its dual space using conditional basis in Hat biorthogonal coefficients.

6.2. Carleson’s biorthogonal system

361

1 is the atomic H 1 space defined over the space of homogeneous Recall that Hat 1 type (T, | · |, dt). In Chapter 2, on page 124, we showed that the dual space to Hat can be represented as the space of integrable functions h : T → R satisfying dt < ∞, sup |h − mI (h)|2 |I| I I

where the supremum is extended over all intervals I ⊆ T. We find it convenient to identify functions on the unit circle with one-periodic functions on the real line. Intervals on T correspond to sets of the form A + Z where A is an interval on the real line satisfying |A| < 1. Having identified functions on T with one-periodic 1 by BMO([0, 1)). functions on R we denote the dual space of Hat This section is devoted to the analysis of Carleson’s system {1[0,1[ }∪{ϕJ : J ∈ 1 D}, defined below in (6.2.3), and its relation to the spaces Hat and BMO([0, 1)). Carleson’s system consists of continuous and piecewise linear functions which form an unconditional basis in L2 . Moreover it is equivalent to the L2 normalized Haar system. Hence there exists a biorthogonal system {ψI : I ∈ D} defined uniquely by the equations ψI = 0 and ϕJ , ψI = δI,J . In this section we establish the following results of L. Carleson [45]. 1. Let f ∈ L2 , then the biorthogonal coefficients {f, ψI : I ∈ D} determine the norm in L2 , c||f ||22 ≤ f, 1[0,1] 2 + f, ψI 2 ≤ C||f ||22 . I∈D

2. Let f ∈ BMO([0, 1)), then the biorthogonal coefficients satisfy the upper estimate |f, ψI | ≤ C||f ||BMO([0,1)) |I|1/2 . 3. For f ∈ BMO([0, 1)) the coefficients {f, ψI : I ∈ D} determine its norm in the following way, c||f ||2BMO([0,1)) ≤ sup

I∈D

1 f, ψJ 2 ≤ C||f ||2BMO([0,1)) . |I| J⊆I

1 4. The biorthogonal functionals {ψI } form an unconditional basis for Hat . 1 1 Specifically, for g ∈ Hat , with 0 g = 0, 1 ≤ sup{ c||g||Hat

1 . |g, ϕI ψI , f | : ||f ||BMO([0,1)) ≤ 1} ≤ C||g||Hat


362

We present a detailed exposition of Carleson’s original work. Central to this study is the verification of the coefficient estimates |f, ψI | ≤ C||f ||BMO([0,1)) |I|1/2 ,

(6.2.1)

where C is independent of I or f. The starting point is the equation f, hI = f, ψI ϕI , hI + f, ψJ ϕJ , hI . {J:J=I}

We then proceed by distinguishing between two cases. In the case where |f, ψI ϕI , hI | ≤ C|f, hI |, the bound (6.2.1) for |f, ψI | follows quite easily. The converse case is characterized by the estimate (6.2.2) |f, ψI ϕI , hI | |f, hI |. For (6.2.2) to hold the value of f, ϕI ϕI , hI must be matched with opposite sign and with precision of order |I| by the value of f, ψJ ϕJ , hI . {J:J=I}

With beautiful combinatorial reasoning L. Carleson exploits this observation and turns it into a proof of the coefficient estimates (6.2.1). The core of L. Carleson’s proof is formed by the so-called compensation argument. Starting from (6.2.2) its main objective is to establish the compensation inequalities, and to show that they imply (6.2.1).

The definition of Carleson’s biorthogonal system We give now the definition of the Carleson system. Let k ∈ N with k ≥ 10 . Let ϕ : R → [−1, 1] be continuous, and piecewise linear with nodes at the following eight points, listed in increasing order, −2−k+1 , −2−k ,

1 1 1 1 − 2−k+1 , − 2−k , + 2−k , + 2−k+1 , 1 + 2−k , 1 + 2−k+1 . 2 2 2 2

Hence ϕ is determined by its values at these points. We put ϕ(−2−k+1 ) = 0, ϕ(−2−k ) = +1, ϕ( 21 − 2−k ) = 0, ϕ( 21 + 2−k ) = 0, ϕ(1 + 2−k ) = −1 and

ϕ( 21 − 2−k+1 ) = +1, ϕ( 21 + 2−k+1 ) = −1, ϕ(1 + 2−k+1 ) = 0.

The value of the parameter 2−k determines how close ϕ is to the first Haar function. We have 2−k . |ϕ − h[0,1] |2 = 4 · 2−k + 4 3 R


363

1 /2 Ŧ2Ŧk

1

0

/2 +2Ŧk

1

1

1+2Ŧk

1

Ŧ2Ŧk 0

/2

Ŧ1

Figure 6.1: The graph of ϕ. It also determines the Lipschitz constant of ϕ. For x, y ∈ R, |ϕ(x) − ϕ(y)| ≤ 2k |x − y|. We define the entire system by rescaling using the single function ϕ, as follows. Given a dyadic interval I ∈ D, let l(I) be its left endpoint. For |I| ≤ 1/2 put x − l(I) −1/2 . (6.2.3) ϕ ;I (x) = |I| ϕ |I| Define ϕI to be the 1-periodic extension of ϕ ;I . Thus ϕI ∈ BMO([0, 1)). For ϕ[0,1] (and only for ϕ[0,1] ) a minor adjustment is required; we proceed as follows. Let a(x) be the affine function given by the conditions a(0) = −2−k+1

and

a(1) = 1 + 2−k+1 .

Then put ϕ ;[0,1] (x) = ϕ(a(x)) x ∈ R. Define ϕ[0,1] to be the 1-periodic extension of ϕ ;[0,1] . We begin the investigation of the Carleson system by comparing it to the L2 normalized Haar system. Unwinding the definition of ϕI or equivalently glancing at its graph shows that ϕI ,

hI = 1 − 3 · 2−k , |I|1/2

|I| ≤ 1/2.

Next expand ϕI (with |I| ≤ 1/2) using the L2 normalized Haar system, hI hJ ϕI = (1 − 3 · 2−k ) 1/2 + H(I, J) 1/2 , |I| |J|

(6.2.4)

(6.2.5)

{J:J=I}

where we defined H(I, J) = ϕI ,

hJ . |J|1/2

Clearly the size of the off-diagonal coefficients {H(I, J) : I = J} determines how close ϕI is to hI /|I|1/2 . Our next proposition gives a quantitative estimate for the assertion that the first term in the Haar expansion (6.2.5) dominates the sum over {J : J = I}.


364

Proposition 6.2.1. For every I ∈ D, |H(I, J)| + |H(J, I)| ≤ 2−k/2+8 . {J:J=I}

Proof. Let I, J be fixed dyadic intervals such that J = I. Assume without loss of generality that J = [0, 1]. The relation between |I|, |J| and the smoothing parameter 2−k determines the value of H(I, J). Let |I| = 2−i and |J| = 2−j . We will distinguish between the following three cases. 1. j ≥ i + k. 2. i + k > j > i. 3. j ≤ i. Let j ≥ i + k. Let m(J) denote the midpoint of J. Then, using that hJ has mean zero, we find hJ |H(I, J)| ≤ (ϕI − ϕI (m(J))) 1/2 |J| (6.2.6) 3/2 k−2 |J| ≤2 . |I|3/2 Let i + k > j > i. Then by comparing the graph of ϕI with that of hJ , we find |H(I, J)| ≤ 2−k

3 |I| 2 |I|1/2 |J|1/2 |I|1/2 . 2 |J|1/2

−k 3

=2

(6.2.7)

Let j ≤ i. If I = J then we have again the estimate |H(I, J)| ≤ 2−k

3 |I|1/2 . 2 |J|1/2

(6.2.8)

With the above estimates for the coefficients H(I, J) we will now prove that |H(I, J)| ≤ 2−k/2 26 . (6.2.9) {J:J=I}

Let j ≤ i. Then the cardinality of the collection {J : H(I, J) = 0, |J| = 2−j } is bounded by 2. By (6.2.8) we obtain that i

j=0

{J:|J|=2−j }

|H(I, J)| ≤ 3 · 2−k

i j=0

2−i/2+j/2

√ √ ≤ 3 2( 2 + 1) · 2−k .

(6.2.10)


365

Let i + k > j > i. The cardinality of the collection {J : J = I, H(I, J) = 0, |J| = 2−j } is now bounded by 4. Thus by (6.2.7), i+k−1

|H(I, J)| ≤ 6 · 2−k

j=i+1 {J:|J|=2−j }

i+k−1

2−i/2+j/2

j=i+1

√ ≤ 6( 2 + 1) · 2−k/2 .

(6.2.11)

Let j ≥ i + k. The cardinality of the collection {J : H(I, J) = 0, |J| = 2−j } equals 4 · 2j−i−k . Hence by (6.2.6), ∞

j=i+k

{J:|J|=2−j }

|H(I, J)| ≤ 2

∞

(2j−i−k )(2k 23i/2−3j/2 )

j=i+k

≤2

∞

(6.2.12)

2i/2−j/2

j=i+k

√ √ ≤ 2 2( 2 + 1) · 2−k/2 . Taking the sum of the estimates (6.2.10)–(6.2.12) gives (6.2.9). Parallel considerations show that |H(J, I)| ≤ 2−k/2 27 . {J:J=I}

The estimate of Proposition 6.2.1 is used to show that the Carleson system is equivalent to the L2 normalized Haar system in L2 . It allows us to use a perturbation argument based on Neumann series and Riesz–Thorin interpolation. Theorem 6.2.2. Carleson’s system is equivalent to the L2 normalized Haar system. The operator T : L2 → L2 , defined by the relations hI /|I|1/2 → ϕI and 1[0,1] → 1[0,1] , is close to the identity || Id − T ||L2 ,L2 ≤ 2−k/2+8 . For k ≥ 17 the operator T is an isomorphism, and the Carleson system is an unconditional basis in L2 . Proof. Let I be a dyadic interval. For convenience assume that |I| ≤ 1/2. We expand T (hI /|I|1/2 ) = ϕI in its Haar series using the the matrix H = (H(I, J)). ϕI = (1 − 3 · 2−k )

hI + |I|1/2

{J:J=I}

H(I, J)

hJ . |J|1/2


366

Next, for any sequence of scalars {cI }, 2 2 −k 1/2 1/2 (1 − 3 · 2 ) c h /|I| − c ϕ = c H(I, J)h /|J| I I I I I J I∈D {J:J=I} I I 2 2 ⎡ ⎤2 ⎣ = cI H(I, J)⎦ . J∈D

{I:J=I}

By Proposition 6.2.1 we have sup |H(I, J)| + |H(J, I)| ≤ 2−k/2+8 . I

{J:J=I}

Combined with the M. Riesz interpolation theorem this estimate gives the following conclusion, ⎤2 ⎡ ⎣ cI H(I, J)⎦ ≤ 2−k+16 c2I . J∈D

{I:J=I}

I 2

By Theorem 6.2.2, in the space L , the Carleson system is equivalent to the L2 normalized Haar system. In particular {ϕI : I ∈ D}∪{1[0,1] } is an unconditional basis in L2 . It has therefore associated to it a biorthogonal system. We denote it by {ψI : I ∈ D} ∪ {1[0,1] }. The equations defining the biorthogonal system are ψJ = 0 and ϕI , ψJ = δI,J , where δI,J = 0 for I = J, and δI,I = 1. It follows that every f ∈ L2 admits a unique expansion along the system, {ϕI : I ∈ D} ∪ {1[0,1] }, where the coefficients are computed by integration against the biorthogonal functionals, {ψI : I ∈ D} ∪ {1[0,1] }. Let f ∈ L2 , then f, ψI ϕI , f = f, 1[0,1] + and the convergence of the above series is in L2 . The isomorphism T of Theorem 6.2.2 established the equivalence between the L2 normalized Haar system and {ϕI : I ∈ D}. It gives that f, ψI 2 ≤ C f 22 . c f 22 ≤ f, 1[0,1] 2 +

Carleson coefficients versus Haar coefficients In the previous subsection we showed that {ϕJ } has associated to it a system of biorthogonal functionals with a uniform bound on their L2 norm. As a result every f ∈ L2 can be expanded as f = f, 1[0,1] + f, ψI ϕI . (6.2.13)


367

ψ[0,1)

2 0 Ŧ2 0

0.5

5

1

ψ[0,1/2)

0

Ŧ5 0 5

0.5

1

0.5

1

ψ[1/2,1)

0

Ŧ5 0 40

ψ[11/16,3/4)

20 0 Ŧ20 Ŧ40

0

0.25

0.5

0.75

1

Figure 6.2: The graph of ψ[0,1) , ψ[0,1/2) , ψ[1/2,1) and ψ[11/16,3/4) .

We obtained (6.2.13) by comparing {ϕJ } to the L2 normalized Haar functions. Now we continue the comparison of the Carleson system and the Haar system. Eventually we will learn how to exploit the expansion on the right-hand side of the equation f, ψJ ϕJ , hI , f, hI = to obtain specific and non-trivial information about the size of the Haar coefficient f, hI . The first step consists in evaluating precisely the scalar product ϕJ , hI , where I, J ∈ D. This is the purpose of the following discussion which provides a thorough preparation for the most delicate part of Carleson’s proof. The values of the scalar product ϕJ , hI . We fix numbers n, q ∈ N such that n ≥ q + 1. Consider Iq−1 a dyadic interval of length 2−q+1 . Let µ be the midpoint of Iq−1 . For each j ≥ q we define the dyadic intervals Jj = [µ − 2−j , µ),

and

Ij = [µ, µ + 2−j ).

(6.2.14)


368

Observe that for j ≥ q, Jj and Ij are adjacent dyadic intervals and that Jq ∪ Iq = Iq−1 . The interval Iq−1 is then the smallest dyadic interval containing In and Jn . Below we also need to consider the dyadic interval Rn = In−1 \ In . Thus Rn is adjacent to the right of In and satisfies |Rn | = 2−n . The intervals we listed in (6.2.14) and their relation to In will play a central role in this section. Fix now j ∈ {1, . . . , n} and a dyadic interval J with |J| = 2−j . We will evaluate the Haar coefficients ϕJ , hIn . In the discussion below we will see that ϕJ , hIn depends on the size and the relative location of the dyadic intervals J, and In . In particular the value depends on the relation between n, j, q, and k . We proceed therefore very carefully. Below we consider separately the following cases for j. 1. n ≥ j ≥ q. 2. j = q − 1. 3. q − 2 ≥ j ≥ q − k + 2. 4. q − k + 1 ≥ j ≥ 0. Case 1. Let n ≥ j ≥ q. In this case the only intervals J for which ϕJ , hIn = 0 are the intervals Jj defined by (6.2.14), together with In , In−1 and Rn = In−1 \ In . First we note that 3 3√ ϕRn , hIn = − 2−k 2−n/2 , and ϕIn−1 , hIn = 2 · 2−k 2−n/2 . (6.2.15) 2 2 We determine now the Haar coefficients ϕJj , hIn . Aiming at precise values we distinguish between the following three sub-cases. If j ≤ n − k, then ϕJj , hIn = 0. 1 If j = n − k + 1, then ϕJj , hIn = − 2−k 2−j/2 . 2 3 −k −j/2 . If j ≥ n − k + 2, then ϕJj , hIn = − 2 2 2

(6.2.16) (6.2.17) (6.2.18)

Case 2. Let j = q − 1. Define Jq−1 = Iq−1 . In this case Jq−1 is the only interval J with |J| = 2−q+1 for which ϕJ , hIn = 0. We consider again three sub-cases as above. Consequently we find a similar set of values for the Haar coefficients, however in the present case we observe opposite signs. Precisely, the following holds. (6.2.19) If q − 1 ≤ n − k, then ϕJq−1 , hIn = 0.


369

1 If q − 1 = n − k + 1, then ϕJq−1 , hIn = + 2−k 2−(q−1)/2 . 2

(6.2.20)

3 If q − 1 ≥ n − k + 2, then ϕJq−1 , hIn = + 2−k 2−(q−1)/2 . 2

(6.2.21)

Case 3. Let q − 2 ≥ j ≥ q − k + 2. In this case it is easy to see that the geometry of the graph of ϕ causes the Haar coefficient at In to vanish. Thus, for any dyadic interval K satisfying |K| = 2−j , we obtain that ϕK , hIn = 0. Case 4. Finally let q − k + 1 ≥ j ≥ 0. In this case there exists exactly one dyadic interval Kj such that |Kj | = 2−j and ϕKj , hIn = 0. Recall that ϕKj is a piecewise linear function and that its Lipschitz constant is 2k+3j/2 . Moreover, when restricted to the interval In the function ϕKj is just affine with slope ≤ 2k+3j/2 . Using that the mean value of hIn vanishes, we estimate now the Haar coefficient. Let m be the midpoint of In . Then the following estimate holds. |ϕKj , hIn | ≤ |ϕKj − ϕKj (m)| In

= 2+k−2 2+3j/2−2n . We complete the preparatory considerations by collecting information about the dyadic intervals NIn = {J : |J| ≥ |In |, J = In , ϕJ , hIn = 0} \ {Rn , In−1 }. The discussion above allows us to obtain a representation of the collection NIn . Of course it is the relation of n to k and q that determines the structure of NIn . We distinguish between the following two cases, (a) n ≤ q + k − 3. (b) n ≥ q + k − 2. Case a. Let n ≤ q + k − 3. Then n − k + 1 ≤ q − 2 and the interval Jq−1 is the largest interval Jj ∈ {Jn , . . . , Jq−1 } for which ϕJj , hIn = 0. It follows that NIn ⊆ {Jn , . . . , Jq−1 , Kq−k+1 , . . . , K0 },

(6.2.22)

where for 0 ≤ j ≤ q − k + 1, we denote by Kj a dyadic interval of length 2−j .


370

Case b. Let n ≥ q + k − 2. Then n − k + 1 ≤ q − 1 (in the limiting case n = q + k − 2 we have n − k + 1 = q − 1). The interval Jn−k+1 is the largest interval Jj ∈ {Jn , . . . , Jq−1 } for which ϕJj , hIn = 0. This gives the representation NIn ⊆ {Jn , . . . , Jn−k+1 , Kq−k+1 , . . . , K0 }.

(6.2.23)

The information collected about the Haar coefficients ϕJ , hIn will be needed in our analysis of Carleson’s biorthogonal system. Two helpful examples. Before we approach the main results in this section we will work out two examples in detail. We define a very simple function f and we will compare the coefficients coming from the Carleson system with its Haar coefficients. Recall that the Haar functions are normalized in L∞ whereas the functions in the Carleson system are approximately normalized in L2 . Consequently we will be comparing the following two numbers, f, hIn and f, ψIn 2−n/2 . The numerical values appearing in the conclusion of our example are difficult to interpret at first sight. It may be helpful to apply a change of scale and use the following two rules of thumb: First, think of 2−n as being the unit size. Second, consider 2−n−k as being so small that it cannot be distinguished from zero. At the same time 2−n k should be viewed as a very large number. Example 6.2.3. Let n ≥ q + k − 1, and define f=

n

2−j/2 ϕIj + ϕJj .

j=n−k+1

For this function the ratio between the Haar coefficient at In and the coefficient with respect to the Carleson system is of order 2−k . The Haar coefficient at In is given as 3 f, hIn = 2−n−k . (6.2.24) 2 The coefficient with respect to the Carleson system satisfies f, ψIn 2−n/2 = 2−n .

(6.2.25)

Moreover the integrals of f over In and Jn behave differently in the following sense, ≥ 2−n (2k − 6). f − f (6.2.26) In

Jn


371

Remark. We move to the front some comments concerning the estimate (6.2.26). Dividing (6.2.26) by 2−n gives a lower bound for the BMO norm of f. Indeed it follows from (6.2.26) that f BMO ≥ (k − 3). Next observe that the norm of 2−j/2 ϕIj + ϕJj in BMO is bounded by 2. Hence we obtained a lower bound for f BMO that is of the same order as the upper bound given by the triangle inequality. It follows that very little cancellation takes place in the sum defining f. 4 3 2 1 0 Ŧ1 Ŧ2 Ŧ3

µ

Ŧ4

Figure 6.3: f =

n

−j/2 (ϕIj j=n−k+1 2

+ ϕJj ), k = 5, n = 10, µ = 1/2.

Proof of (6.2.24)–(6.2.26). By the biorthogonality of the functions ϕI and ψI we obtain immediately that f, ψIn = 2−n/2 . Hence (6.2.25) follows. Next we evaluate the Haar coefficient of f at In . First observe that we already know the values of the coefficients appearing in the first line of (6.2.27) below. Indeed ϕIn , hIn is determined in (6.2.4). The remaining coefficients correspond to In−1 and Jj satisfying n − k + 1 ≤ j ≤ n. The values of the coefficients are recorded in (6.2.15) and in the equations (6.2.17) and (6.2.18). f, hIn = 2−n/2 ϕIn , hIn + 2−(n−1)/2 ϕIn−1 , hIn + 1 3 = 2−n (1 − 2−k 3) + 2−n−k 3 − 2−n−1 − 2 2

n

2−j/2 ϕJj , hIn

j=n−k+1 n −k−j

2

.

j=n−k+2

(6.2.27) To simplify the above expression we introduce the abbreviation 1 3 X = 2k − 2k−1 − 2 2

n j=n−k+2

2−j+n .


372

Then collecting terms and rewriting the second line of (6.2.27) using X we find that f, hIn = 2−k−n X. Below we will calculate that X = 32 . Hence the above equation implies (6.2.24). The value of X is determined as follows. First we evaluate the geometric sum and obtain n 2−j+n = 2k−1 − 1. j=n−k+2

Then we insert this value into the defining equation for X. This gives 3 1 3 3 X = 2k − 2k−1 − 2k−1 + = . 2 2 2 2 Next we turn to the verification of (6.2.26). For a dyadic interval J we introduce the notation ϕJ − ϕJ . ∆(J) = In

Jn

Then by the definition of f we obtain that

f− In

f= Jn

n

2−j/2 (∆(Jj ) + ∆(Ij )) .

j=n−k+1

Next by drawing the graph of ϕJj and ϕIj and by taking into account the position of the intervals Jj and Ij relative to In we determine the values of ∆(Jj ), ∆(Ij ) for n ≥ j ≥ n − k + 1. First notice that throughout we have ∆(Jj ) = ∆(Ij ). Next consider separately the cases j = n, j = n − 1 and n − 2 ≥ j ≥ n − k + 1. We get 3 ∆(In ) = ∆(Jn ) = − 2−k 2−n/2 , 2

(6.2.28)

√ 2−n/2 ∆(In−1 ) = ∆(Jn−1 ) = √ − 3 2 · 2−k 2−n/2 , 2

(6.2.29)

and 3 ∆(Jj ) = ∆(Ij ) = 2−n 2j/2 − 2−k 2−j/2 , 2

for n − 2 ≥ j ≥ n − k + 1. (6.2.30)

These values will now be inserted into the above equation for the difference. This gives

f− In

Jn

f = 2 · 2−n − 15 · 2−k−n + 2

n−2

3 (2−n − 2−k−j ). 2

j=n−k+1


373

Summing a geometric series we determine the value of the above expression. We have that n−2 2−k−j = 2−n (1 − 4 · 2−k ). j=n−k+1

Thus by inserting we find that f dt − In

Jn

f dt ≥ 2−n (2k − 6).

In the following example we define a function g which appears to be very similar to the function f considered above. In fact the only difference is that the defining sum for g starts with the index j = n − k + 2, as opposed to j = n − k + 1 in the definition of f. Despite the similarities between f and g we will find that the Haar coefficient g, hIn is much larger than f, hIn . It turns out that the ratio between the two Haar coefficients is of order 2k . As we will see below it is just the absence of Jn−k+1 which is responsible for this fact. Example 6.2.4. Let n ≥ q + k − 1, and define g=

n

2−j/2 ϕIj + ϕJj .

j=n−k+2

Then at In the coefficient with respect to the Carleson system and the Haar coefficient are comparable within a factor of 4. Precisely we have g, ψIn 2−n/2 = 2−n , and the Haar coefficient is g, hIn =

1 −n 3 −n−k 2 + 2 . 4 2

(6.2.31)

Proof. The biorthogonality of ϕI and ψI implies that g, ψIn = 2−n/2 . This gives the value for the coefficient with respect to the Carleson system. We evaluate the Haar coefficient of g at In following the pattern of the previous example. The definition of g gives g, hIn = 2−n/2 ϕIn , hIn + 2−(n−1)/2 ϕIn−1 , hIn +

n

2−j/2 ϕJj , hIn

j=n−k+2

= 2−n (1 − 2−k 3) + 2−k−n 3 −

3 2

n

2−k−j .

j=n−k+2

(6.2.32) To simplify the above expression we introduce the abbreviation Y = 2k −

3 2

n j=n−k+2

2−j+n .


374

We use Y to rewrite the second line of (6.2.32). Collecting terms gives g, hIn = +2−k−n Y.

(6.2.33)

Below we will calculate that

1 k 3 2 + . 4 2 Hence the equation (6.2.33) implies (6.2.31). The value of Y is determined quite simply as follows. As in the previous example we write down the value of the geometric sum n 2−j+n = 2k−1 − 1. Y =

j=n−k+2

Then we insert into the defining equation for Y. This gives 3 3 Y = 2k − 2k−1 + 2 2 1 k 3 = 2 + . 4 2

The compensation argument 1 The following theorem provides an upper estimate for the Hat norm of the biorthogonal functionals {ψI : I ∈ D}.

Theorem 6.2.5. The biorthogonal coefficients satisfy the inequality |f, ψI | ≤ A||f ||BMO([0,1)) |I|1/2 ,

(6.2.34)

where A = A(k) is independent of I ∈ D or f ∈ BMO([0, 1)). To help organizing the set of coefficients {f, ψI : I ∈ D} we introduce a simple and useful device. Let m ∈ N, then define Mm = sup{|f, ψI | · |I|−1/2 : |I| ≥ 2−m }. The assertion of Theorem 6.2.5 is equivalent to saying that the sequence {Mm } is bounded by A||f ||BMO([0,1)) . Clearly {Mm } is an increasing sequence. Thus it suffices to show that there exists a subsequence n(j) ↑ ∞, such that Mn(j) ≤ A||f ||BMO([0,1)) . To get started we show the following short proposition. Proposition 6.2.6. There exists a strictly increasing sequence n(j) ↑ ∞, such that for each n(j) the following holds, Mn(j)+p ≤ 2p/2 Mn(j) , ∀p ∈ N.

(6.2.35)


375

Proof. Assume to the contrary that there exists K ∈ N such that for all n ≥ K there exists p ≥ 1 satisfying Mn+p ≥ 2p/2 Mn . Iterating this assumption we find a sequence n(j) ∈ N, such that Mn(j) ≥ Mn(j−1) 2(n(j)−n(j−1))/2 , and n(j) − n(j − 1) ≥ 1. Equivalently we have for every j ≥ 1, that Mn(j) ≥ 2n(j)/2 Mn(1) 2−n(1)/2 . By definition of Mn(j) there exists a dyadic interval Ij such that |Ij | ≥ 2−n(j) and 2 f, ψIj 2 ≥ Mn(j) |Ij |.

Inserting the growth condition of the sequence Mn(j) we find that for every j ∈ N, 2 2−n(1) . f, ψIj 2 ≥ Mn(1)

Passing to an infinite subcollection of the intervals {Ij }, if necessary we obtained 2 f, ψIj 2 ≥ Mn(1) 2−n(1) .

Clearly this contradicts the fact that the Carleson system is an unconditional basis in L2 , and that 2 f, ψIj ≤ 4||f − f ||22 ≤ 4||f ||2BMO([0,1)) . j

With the above proposition the inequality (6.2.34) for the biorthogonal functionals is a result of the next theorem. Theorem 6.2.7. Let n ∈ N such that Mn satisfies (6.2.35). Then Mn ≤ A||f ||BMO([0,1)) , where A = A(k) is independent of n or f ∈ BMO([0, 1)). Proof. We show how Theorem 6.2.7 follows from Theorem 6.2.8 below. Fix n ∈ N. Select a dyadic interval In for which |f, ψIn | · |In |−1/2 = Mn . Without losing any generality we assume that In satisfies also the following conditions. (a) |In | = 2−n .


376 (b) f, ψIn ≥ 0. (c) In is the left half of its dyadic predecessor.

Let us now describe the interval In , its position within the tree of all dyadic intervals and some intervals around In , which play an important role in the proof to come. Let µ denote the left endpoint of In . Then In = [µ, µ + 2−n ). We define Jn to be the dyadic interval of length 2−n such that µ is the right endpoint of Jn . Thus Jn = [µ − 2−n , µ). We denote the dyadic predecessors of In respectively Jn by In−1 ⊃ · · · ⊃ I0 = [0, 1), respectively Jn−1 ⊃ · · · ⊃ J0 = [0, 1). Clearly the chains of dyadic intervals Ij , Jj will have to merge somewhere. We encode this fact as follows. Define q ∈ N by the following two conditions. 1. Iq ∩ Jq = ∅. 2. Iq ∪ Jq = Iq−1 = Jq−1 . Notice that Iq−1 is the smallest dyadic interval containing both In and Jn . Hence the assumption that In is the left half of its dyadic predecessor implies that q ≤ n − 1. At this point we have reduced the proof of Theorem 6.2.7 to the following critical implication. Theorem 6.2.8. Assume that log2 k ≥ 88. If |f, hIn | ≤ 2−k−n Mn , and if also |f, hJm | ≤ 2−k−m Mn , then

In

f− Jn

for

n − k ≤ m ≤ n,

f ≥ c log k · 2−n Mn ,

where c > 0 is a universal constant independent of k, or n. We obtain Theorem 6.2.8 by combining Lemma 6.2.11 and Lemma 6.2.12. Theorem 6.2.8 is the central piece in Carleson’s solution to the unconditional 1 . Its proof introduces a new method, called by Carleson, the basis problem for Hat compensation argument. Quoting from Carleson’s introduction [45], the compensation argument is the delicate part of a proof that is not quite easy. Before we proceed we will make some remarks about the structure of the proof of Theorem 6.2.8. To this end we introduce the following notation. Let 0 ≤ i ≤ n, and i = q − 1. Then select non-negative numbers 0 ≤ αi ≤ 2, and 0 ≤ βi ≤ 2, such that f, ψJi |Ji |−1/2 = Mn (1 − αi ), (6.2.36) f, ψIi |Ii |−1/2 = Mn (1 − βi ).


377

For the remaining case where i = q − 1 we choose 0 ≤ αq−1 ≤ 2, and 0 ≤ βq−1 ≤ 2, such that f, ψJq−1 |Jq−1 |−1/2 = Mn (αq−1 − 1), (6.2.37) f, ψIq−1 |Iq−1 |−1/2 = Mn (βq−1 − 1). We treat the case i = q − 1 and i = q − 1 differently to take into account the plus sign in (6.2.20) and (6.2.21) and the minus sign in the corresponding equations (6.2.17) and (6.2.18). Doing this allows us to write the equations (6.2.61) and (6.2.62) below in closed form. The main lemma in the proof of Theorem 6.2.8 is Lemma 6.2.9 below. It asserts that the size of the numbers αi is intimately related to the hypothesis that at In the Haar coefficient of f satisfies |f, hIn | ≤ 2−k−n Mn . Denote P (In ) =

3 2

n

αi 2n−i .

(6.2.38)

i=n−k+2

Lemma 6.2.9 shows that |f, hIn | ≤ 2−k−n Mn implies that P (In ) ≤ 44.

(6.2.39)

This inequality is called the compensation inequality for the interval In , its proof is called the compensation argument. In Lemma 6.2.10 we apply the compensation argument to the intervals Jm with n − k ≤ m ≤ n. Using that P (In ) ≤ 44 we obtain from Lemma 6.2.10 a set of compensation inequalities showing that n−k+A

βi ≤ 88,

(6.2.40)

i=n−k+2

where A = log(k − 1). In Lemma 6.2.12 which is the final step in the proof of Theorem 6.2.8 we exploit that the non-negative coefficients αi , βi , are small in the sense that (6.2.39) and (6.2.40) hold true. We show there the following lower bound for the integral of f over the adjacent intervals Jn , In , ≥ cMn log k|In |. f − f (6.2.41) In

Jn

The above estimate implies an upper bound for Mn . Indeed since ≤ C f BMO([0,1)) |In |, f − f In

Jn

we obtain from (6.2.41) that −1 Mn ≤ C (log k) f BMO([0,1)) .


378

Now we turn to the main step of the proof of Theorem 6.2.8. It shows that 3 2

P (In ) =

n

αi 2n−i

i=n−k+2

is strongly related to lower estimates for the Haar coefficient f, hIn . Recall that the definition of the non-negative numbers αi involves only the coefficients f, ψJ formed by the Carleson system. Hence their important role in linking the Carleson system to the Haar system and BMO. Lemma 6.2.9. If n ≤ q + k − 3, then |f, hIn | ≥ 2−n Mn /16.

(6.2.42)

|f, hIn | ≥ 2−n−k Mn P (In ) − 2−n−k Mn 43.

(6.2.43)

If n ≥ q + k − 2, then

The essence of Lemma 6.2.9 could be formulated briefly as follows: If |f, hIn | ≤ 2−n−k Mn , then n ≥ q + k − 2, and P (In ) ≤ 44. Observe that the lower estimate (6.2.42) holding for n ≤ q + k − 3, is already quite satisfactory. After dividing by 2−n we see that the first assertion of Lemma 6.2.9 gives Mn ≤ 2n+4 |f, hIn | ≤ C||f ||BMO([0,1)) . In the case when n ≥ q + k − 2, we have identified the value of the parameter P (In ) as the crucial quantity: A lower estimate for P (In ) gives a lower estimate for |f, hIn |, and thus an upper bound for Mn . If however |f, hIn | ≤ 2−n−k Mn , then the conclusion of Lemma 6.2.9 states that 3 2

n

αi 2n−i ≤ 44,

i=n−k+2

which means in a very strong sense that the non-negative coefficients αi satisfy αi ≈ 0. Proof of Lemma 6.2.9. Part 1. The first part of the proof establishes the basic estimate (6.2.52). We simplify notation and put M = Mn . Using that the Carleson system is an unconditional basis for L2 we expand the Haar coefficient f, hIn as follows. f, hIn = f, ψI ϕI , hIn . I∈D


379

Actually contributing to the above expansion are the following dyadic intervals {I ∈ D : ϕI , hIn = 0}. Next split this collection of dyadic intervals into the following two parts (the notation should remind us of “north of In ” and “south of In ”). We put NIn = {I : |I| ≥ |In |, In = I, ϕI , hIn = 0} \ {Rn , In−1 }, and SIn = {I : |I| < |In |, ϕI , hIn = 0} ∪ {Rn , In−1 }. Rewrite the equation for f, hIn using NIn and SIn , f, hIn = f, ψIn ϕIn , hIn + f, ψI ϕI , hIn .

(6.2.44)

I∈NIn ∪ SIn

Clearly our aim is to understand better the sum appearing on the right-hand side of the above equation. We start by showing that the sum over SIn represents a negligible quantity for the problem at hand. At this point of the proof we are using the hypothesis that Mn+p ≤ 2p/2 M, ∀p ∈ N. We claim the following upper bound for the sum over SIn , |f, ψI ϕI , hIn | ≤ 34 · M 2−n−k .

(6.2.45)

I∈SIn

To prove the claim, fix l ∈ N with l ≥ n + 1. Observe that the collection {J ∈ SIn : |J| = 2−l } consists of six intervals. Indeed, if I ∈ {J ∈ SIn : |J| = 2−l }, then either I is adjacent to the midpoint of In , to the right endpoint of In , or to the left endpoint of In . Each of these cases accounts for two intervals in {J ∈ SIn : |J| = 2−l }. Next let I ∈ {J ∈ SIn : |J| = 2−l }. We determine the value of the scalar product ϕI , hIn as follows. If I is adjacent to the midpoint of In , then (6.2.46) |ϕI , hIn | = 3 2−k |I|1/2 . In the remaining cases where I is adjacent to the right or to the left endpoint of In , we have that 3 −k 1/2 2 |I| . (6.2.47) |ϕI , hIn | = 2 Moreover as Ml ≤ 2(l−n)/2 M, we have that |f, ψI | ≤ 2−l/2 Ml ≤ 2−n/2 M.

(6.2.48)


380

To summarize we have four intervals in {J ∈ SIn : |J| = 2−l } for which (6.2.47) holds and the remaining two intervals satisfy (6.2.46). Combining this with (6.2.48) gives satisfactory estimates for the products f, ψI ϕI , hIn and their sum ∞

|f, ψI ϕI , hIn | ≤ 12M 2−k

l=n+1 {I∈SIn :|I|=2−l }

∞

2−(n+l)/2

l=n+1

√ = 12( 2 + 1)M 2−k−n .

(6.2.49)

Next recall that the contribution of the intervals Rn = In−1 \ In and In−1 is given by 3 ϕRn , hIn = − 2−k 2−n/2 , 2

and ϕIn−1 , hIn =

3√ 2 · 2−k 2−n/2 . 2

(6.2.50)

This proves (6.2.45), the claimed estimate for the sum over SIn . Below we insert (6.2.45) into (6.2.44). We also use that f, ψIn ϕIn , hIn = M 2−n (1 − 2−k 3). Thus we obtain the lower bound −n |f, hIn | ≥ M 2 + f, ψI ϕI , hIn − {34 + 3}M 2−n−k . I∈NIn

(6.2.51)

(6.2.52)

Next distinguish between the cases n ≤ q + k − 3 and n ≥ q + k − 2. In both cases the preceding estimate (6.2.52) is the starting point of further analysis. Part 2. Now we prove (6.2.42). Let n ≤ q + k − 3. This case is the easier of the two. We claim that if n ≤ q + k − 3, then I∈NIn

|f, ψI ϕI , hIn | ≤

7 M 2−n . 8

(6.2.53)

What counts here is that 1 − 78 is large compared to 2−k . We start proving (6.2.53) by recalling the following inclusion for the index set NIn . It is valid when n ≤ q + k − 3. (6.2.54) NIn ⊆ {Jn , . . . , Jq−1 , Kq−k+1 , . . . , K0 }. Here, for 0 ≤ j ≤ q − k + 1 the intervals Kj satisfy |Kj | = 2−j . Fix 0 ≤ j ≤ q − k. Then |f, ψKj | ≤ M |Kj |1/2 , and |ϕKj , hIn | = 2+k−2 |In |2 |Kj |−3/2 .


381

Inserting these values and taking the product gives |f, ψKj ϕKj , hIn | ≤ M 2+k−2 2−2n 2j . Summing over the index set 0 ≤ j ≤ q − k + 1, we find that q−k+1

|f, ψKj ϕKj , hIn | ≤ M 2k−2n−2

j=0

q−k+1

2j

j=0

≤ M2

q−2n

(6.2.55)

.

Next fix q − 1 ≤ i ≤ n. Then we have |f, ψJi | ≤ M |Ji |1/2

and

|ϕJi , hIn | = 3(2−k−1 )|Ji |1/2 .

This gives n

n

|f, ψJi ϕJi , hIn | ≤ 3M · 2−k−1

i=q−1

2−i

i=q−1

(6.2.56)

≤ 6M · 2−k−q . Adding the estimates (6.2.55) and (6.2.56) we find that

|f, ψI ϕI , hIn | ≤ M (6 · 2−k−q + 2−2n+q ).

(6.2.57)

I∈NIn

The assumption that n ≤ q + k − 3, and the fact that n ≥ q + 1 gives M (6 · 2−k−q + 2−2n+q ) ≤

7 M 2−n . 8

(6.2.58)

Indeed, verify first that (6.2.58) holds for n = q+k−3 and n = q+1. Then estimate in a straightforward way using the improved hypothesis q+2 ≤ n ≤ q+k−4. Insert the estimate (6.2.57) into the previously established inequality (6.2.52). Using that 1 − 7/8 = 1/8, and that k ≥ 10 this gives the following lower bound for the Haar coefficients f, hIn (it is valid when n ≤ q + k − 3). −n f, ψI ϕI , hIn − {34 + 3} M 2−k−n |f, hIn | ≥ M 2 + I∈NIn 1 M 2−n − {34 + 3} M 2−k−n 8 1 M 2−n . ≥ 16 ≥

(6.2.59)


382

Part 3. Next we prove (6.2.43). Let n ≥ q + k − 2. For n ≥ q + k − 2 we have the following representation for NIn , NIn ⊆ {Jn , · · · , Jn−k+1 , Kq−k+1 , · · · , K0 }. Rewrite the inequality n ≥ q + k − 2, as q − 2n ≤ −k − n + 2. Inserting this into (6.2.55) we obtain q−k+1

|f, ψKj ϕKj , hIn | ≤ M 2q−2n

j=0

≤ 4M 2−n−k . We insert this estimate into (6.2.52), the lower bound for |f, hIn | obtained in the first part of the proof. This gives that n −n f, ψJi ϕJi , hIn − {34 + 7} M 2−n−k . (6.2.60) |f, hIn | ≥ M 2 + i=n−k+1

We continue by recalling what is known about the values of f, ψJi ϕJi , hIn appearing in the above sum. We distinguish between the cases n − k + 2 ≤ i ≤ n, and i = n − k + 1. This distinction is necessary in view of the different factors −3/2 and −1/2 appearing in the formulas below: It follows from 1. If n − k + 2 ≤ i ≤ n, then 3 f, ψJi ϕJi , hIn = − 2−i−k M (1 − αi ). 2

(6.2.61)

The equation (6.2.61) follows from (6.2.18) and (6.2.21) in combination with (6.2.36) and (6.2.37). 2. If i = n − k + 1, then 1 f, ψJi ϕJi , hIn = − 2−i−k M (1 − αi ). 2

(6.2.62)

The equation (6.2.62) follows from (6.2.17) and (6.2.20) in combination with (6.2.36) and (6.2.37). We displayed the above identities with the purpose of inserting them into (6.2.60). Before we do that we introduce, for sake of convenience, the abbreviation 1 3 X = 2+k − 2+k−1 − 2 2

n

2−i+n .

i=n−k+2

Later we will calculate the value of X and find that X = 3/2. However for the time being it is really more instructive to write it in just the form we used above.


383

We insert now into (6.2.60) the identities we obtained in (6.2.61) and (6.2.62). Next we pull out the common factor 2−n−k M. This gives the equations n −n f, ψJi ϕJi , hIn M 2 + i=n−k+1 n 3 +k 1 +k−1 −n−k −i+n =2 M 2 − 2 (1 − αn−k+1 ) − 2 (1 − αi ) 2 2 i=n−k+2 n 1 3 = 2−n−k M X + 2+k−1 αn−k+1 + 2−i+n αi . 2 2 i=n−k+2

Next we determine the value of X. First we re-index the sum on the right of its definition. This gives n

2−i+n =

k−2

2j = 2k−1 − 1.

j=0

i=n−k+2

Then we insert the value for the geometric sum and find that X=

3 . 2

Below it is important that we have a universal upper bound for X. By (6.2.60) the resulting lower bound for |f, hIn | is now / . n 3 −n−k −i+n |f, hIn | ≥ 2 M 2 αi − 2−n−k M {34 + 9} . 2 i=n−k+2

As 34 + 9 = 43 this completes the proof of Lemma 6.2.9.

The proof given above constitutes the compensation argument. We applied it to the interval In . Next we apply it to the intervals Jm , n−k ≤ m ≤ n. We consider the Haar coefficients, f, hJm where n−k ≤ m ≤ n and try to obtain lower bounds for |f, hJm |. We will therefore repeat the proof of the previous lemma. We execute Carleson’s compensation argument and uncover again the associated compensation inequalities. It turns out that now the value of the following parameter controls the lower estimates for the Haar coefficient |f, hJm |, m 1 3 P (Jm ) = −αm 2+k + 2+k βm−k+1 + 2−i+m βi . 4 2 i=m−k+2

In its formulation and in its proof the next lemma is an “affine image” of Lemma 6.2.9. Its presentation here is motivated only by the desire to be definite and complete. The proof below uses once more the compensation argument — shifted and rescaled to the interval Jm . We will apply Lemma 6.2.10 in the course of proving Lemma 6.2.11, where we will consider only those m for which αm ≤ 2−5 .


384 Lemma 6.2.10. If m ≤ q + k − 3, then |f, hJm | ≥ 2−m M

|1 − αm | −

15 16

.

If m ≥ q + k − 2, then |f, hJm | ≥ 2−m−k M P (Jm ) − 43 · 2−m−k M. Proof. We define first the following two collections of dyadic intervals, NJm = {I : |I| ≥ |Jm |, Jm = I, ϕI , hJm = 0} \ {Jm−1 , Jm−1 \ Jm }, and SJm = {I : |I| < |Jm |, ϕI , hJm = 0} ∪ {Jm−1 , Jm−1 \ Jm }. With this notation we expand f, hJm . The expansion is naturally centered around the interval Jm . f, ψI ϕI , hJm . (6.2.63) f, hJm = f, ψJm ϕJm , hJm + I∈NJm ∪SJm

The intervals contained in SJm contribute relatively little to the above sum. Indeed the proof of (6.2.45) shows that the sum over the index set SJm is bounded as follows, |f, ψI ϕI , hJm | ≤ 2−m−k 34M. (6.2.64) I∈SJm

To obtain (6.2.64) we use the additional hypothesis that Mn+p ≤ 2p/2 Mn . Next recall that f, ψJm = M (1 − αm )2−m/2 , and that ϕJm , hJm = 2−m/2 (1 − 3 · 2−k ). Multiplying allows us to rewrite the leading term in (6.2.63) as f, ψJm ϕJm , hJm = M (1 − αm )2−m (1 − 3 · 2−k ). Then with the estimate (6.2.64) for the sum over SJm we obtain the lower bound −m + f, ψI ϕI , hJm − 37M 2−m−k . (6.2.65) |f, hJm | ≥ M (1 − αm )2 I∈NJm Next distinguish between the cases m ≤ q + k − 3 and m ≥ q + k − 2. In both cases the preceding estimate is the place from which we start developing the proof. Let m ≤ q + k − 3. As in the proof of Lemma 6.2.9 the following estimate holds, 7 |f, ψI ϕI , hJm | ≤ M 2−m . (6.2.66) 8 I∈NJm


385

Inserting the estimate (6.2.66) into the lower bound (6.2.65) gives that 7 |f, hJm | ≥ M |1 − αm |2−m − M 2−m − 37M 2−k−m . 8 With k ≥ 10 we obtain therefore that |f, hJm | ≥ 2

−m

M

15 |1 − αm | − 16

.

This completes the proof for the case m ≤ q + k − 3. Next we consider m ≥ q + k − 2. We return to the lower estimate obtained in (6.2.65). Recall that for m ≥ q + k − 2 the following representation holds, NJm ⊆ {Im , . . . , Im−k+1 , Kq−k+1 , . . . , K0 }. We estimate now the contribution to the sum on the right-hand side of (6.2.65) which comes from of the intervals Kq−k+1 , . . . , K0 . Recall (6.2.55), where we proved that q−k+1 |f, ψKj ϕKj , hJm | ≤ M 2q−2m . j=0

As m ≥ q + k − 2, we have q − 2m − 1 ≤ −k − m + 1, hence q−k+1

|f, ψKj ϕKj , hJm | ≤ 4M 2−m−k .

j=0

Inserting this estimate into the lower bound (6.2.65) shows that, m −m |f, hJm | ≥ M (1 − αm )2 + f, ψIi ϕIi , hJm − 41M 2−m−k . i=m−k+1

(6.2.67) The preceding analysis showed that the crucial contribution to the right-hand side of (6.2.65) comes from the intervals Im , . . . , Im−k+1 . Indeed all the rest contributes an amount that is less than 41M 2−m−k . We will now calculate the value of the sum m f, ψIi ϕIi , hJm , i=m−k+1

with as much precision as possible. We have to distinguish between the following two cases. The case where m − k + 2 ≤ i ≤ m, and i = m − k + 1. 1. If m − k + 2 ≤ i ≤ m, then 3 f, ψIi ϕIi , hJm = − M (1 − βi )2−i−k . 2


386 2. If i = m − k + 1, then

1 f, ψIi ϕIi , hJm = − M (1 − βi )2−i−k . 2 We use again the somewhat artificial abbreviation 1 3 X = 2+k − 2+k−1 − 2 2

m

2−i+m .

i=m−k+2

Recall that X = 3/2. Recall further that we put 1 3 P (Jm ) = −αm 2+k + 2+k βm−k+1 + 4 2

m

−i+m

2

i=m−k+2

βi .

With this notation and with the values for f, ψIi ϕIi , hJm we obtain the following identities. m −m + f, ψIi ϕIi , hJm M (1 − αm )2 i=m−k+1 m 1 3 = 2−m−k M (1 − αm )2+k − 2+k−1 (1 − βm−k+1 ) − 2−i+m (1 − βi ) 2 2 i=m−k+2

= 2−m−k M |X − P (Jm )| . Thus we showed that the Haar coefficient f, hJm is bounded from below as follows, |f, hJm | ≥ 2−m−k M P (Jm ) − 43M 2−m−k . The following lemma is a simple bookkeeping device. We use it to merge the conclusions of Lemma 6.2.9 and Lemma 6.2.10. Lemma 6.2.11. Assume that |f, hIn | < 2−k−n M,

(6.2.68)

and that for each m satisfying n − k ≤ m < n, |f, hJm | ≤ 2−k−m M.

(6.2.69)

Then the following compensation inequalities hold true, n i=n−k+2

where A = log2 (k − 1).

αi ≤ 44,

and

n−k+A i=n−k+2

βi ≤ 88,

(6.2.70)


387

Proof. By Lemma 6.2.9 the hypothesis (6.2.68) implies that n ≥ q + k − 2, and that P (In ) ≤ 44. By definition of P (In ) this gives n

αi ≤ 44.

i=n−k+2

Let A = log2 (k−1) and assume that it is a natural number. The compensation inequality P (In ) ≤ 44 implies that 0 ≤ αm ≤ 1/32, for n − k + 1 ≤ m ≤ n − k + A. Hence 31 . (6.2.71) |1 − αm | ≥ 32 With (6.2.71) and the hypothesis (6.2.69) the conclusion of Lemma 6.2.10 gives that m ≥ q + k − 2 and P (Jm ) ≤ 44 for n − k + 1 ≤ m ≤ n − k + A. Writing out the inequality P (Jm ) ≤ 44 and taking into account that 3/2 ≥ 1, we find that m

2−i+m βi ≤ 44 + αm 2k .

(6.2.72)

i=m−k+2

We would like to take advantage of the compensation inequality for In . So we multiply the inequalities (6.2.72) with the factor 2−k−m+n . This gives m

2−i−k+n βi ≤ 44 · 2−k−m+n + αm 2−m+n ,

(6.2.73)

i=m−k+2

where n − k + 2 ≤ m ≤ n − k + A. In this way the coefficient in front of αm has the same form as in the compensation inequality. We take the sum of the estimates (6.2.73) over the range n − k + 2 ≤ m ≤ n − k + A. This gives n−k+A

m

−i−k+n

2

m=n−k+2 i=m−k+2

βi ≤ 44

n−k+A

−k−m+n

2

n−k+A

+

m=n−k+2

αm 2−m+n .

m=n−k+2

(6.2.74) The geometric sum appearing on the right-hand side of (6.2.74) is easily evaluated. Its value is ≤ 1. The second sum is bounded by P (In ). Hence, by (6.2.74) we obtain that m n−k+A 2−i−k+n βi ≤ 88. (6.2.75) m=n−k+2 i=m−k+2

We now change the order of summation on the left-hand side of (6.2.75). Doing this allows us to exploit that the summands on the left do not depend on m. We find thus the identity n−k+A

m

m=n−k+2 i=m−k+2

2−i−k+n βi = (k − 1)

n−k+A i=n−2k+4

2−i−k+n βi .


388

Insert now the above line into (6.2.75). Observe that for n − 2k + 4 ≤ i ≤ n − k + A we obtain that 2−i−k+n ≥ 2−A . Hence, the non-negative numbers βi satisfy the simple estimate n−k+A βi ≤ 88. (6.2.76) 2−A (k − 1) i=n−2k+4

We observe that by our choice for A, the factor 2−A (k−1) equals 1. Hence, dividing (6.2.76) by 2−A (k − 1) shows that (6.2.70) holds. Next we prove that the compensation inequalities (6.2.70) imply an upper bound for M = Mn . The idea of the proof was present already in Example 6.2.3. Note that by Lemma 6.2.11 the hypothesis of the following lemma is satisfied with A = log(k − 1), B = 44 and C = 88. Lemma 6.2.12. Let A, B, C ∈ N, be fixed. Assume that there hold compensation inequalities in the form n−k+A

βi ≤ C

i=n−k+2

Then

αi ≤ B.

(6.2.77)

i=n−k+2

f−

In

n

and

Jn

f ≥ 2−n M (2A − B − C − 12).

Proof. For any dyadic interval J we denote ∆(J) = ϕJ − In

ϕJ .

Jn

We define the collection of dyadic intervals N = {J : |J| ≥ 2−n+2 , ∆(J) = 0} With this notation we have f− In

Jn

f=

and

S = {J : |J| < 2−n+2 , ∆(J) = 0}.

f, ψJ ∆(J).

J∈N ∪S

We easily observe that the proof of (6.2.45) gives |f, ψJ ∆(J)| ≤ 34M 2−n−k .

(6.2.78)

J∈S

Now we show the lower estimate f, ψJ ∆(J) ≥ 2−n M (2A − B − C − 11), J∈N

(6.2.79)


389

where A, B are the constants appearing in the hypothesis of the lemma. Clearly the estimates (6.2.78) and (6.2.79) imply the conclusion of the lemma. We now turn to the verification of (6.2.79). We start the proof by analyzing the index set N . For N we have the representation N ⊆ {In−2 , Jn−2 , . . . , In−k+1 , Jn−k+1 , Kq−k+1 , . . . , K0 }. Correspondingly, we have that

n−2

f, ψJ ∆(J) =

J∈N

f, ψJj ∆(Jj ) + f, ψIj ∆(Ij )

j=n−k+1

+

(6.2.80)

q−k+1

f, ψKi ∆(Ki ).

i=0

Next we show that the sum in the second line of (6.2.80) is bounded by M 2−n+3 . Fix i ≤ q − k + 1. Recall that Lip(ϕKi ) ≤ 2+3i/2+k . As In ∪ Jn is an interval of diameter ≤ 2−n+1 , we obtain that ϕKi − ϕKi ≤ 2−2n+3i/2+k+1 . In

Jn

Next recall that |f, ψKi | ≤ M 2−i/2 by definition. Hence for i ≤ q − k + 1, |f, ψKi ∆(Ki )| ≤ M 2−2n+k+i+1 , and by summing the geometric series we find q−k+1

|f, ψKi ∆(Ki )| ≤ M 2q−2n+3 .

(6.2.81)

i=0

Using that n ≥ q + 1 we obtain from (6.2.80) and (6.2.81) that

−n+2

f, ψJ ∆(J) ≥ −M 2

J∈N

+

n−2

f, ψJj ∆(Jj ) + f, ψIj ∆(Ij ). (6.2.82)

j=n−k+1

Hence to verify (6.2.79) it remains to find a lower bound for the sum appearing in (6.2.82). To this end we let n − k + 1 ≤ j ≤ n − 2 and Ij , Jj ∈ N . The values of the differences ∆(Ij ), ∆(Jj ) are calculated most easily by looking at the graph of ϕJj respectively ϕIj . We obtained in (6.2.30) that 3 ∆(Ij ) = ∆(Jj ) = 2−n 2+j/2 − 2−k 2−j/2 2

for n − k + 1 ≤ j ≤ n − 2.


390

Now we rewrite the sum appearing on the right-hand side (6.2.82) using some of the notation we introduced earlier. We use f, ψJj 2+j/2 = M (1 − αj ) and f, ψIj 2+j/2 = M (1 − βj ). Thus we have that 3 f, ψJj ∆(Jj ) = M (1 − αj ){2−n − 2−k 2−j }, 2 and also

3 f, ψIj ∆(Ij ) = M (1 − βj ){2−n − 2−k 2−j }. 2 Now we sum the above identities over n − k + 1 ≤ j ≤ n − 2. This gives a new formula for the sum (6.2.79). Most interestingly, the right-hand side of (6.2.83) below contains only terms about which we have information available due to the hypothesis of the lemma. n−2

f, ψJj ∆(Jj ) + f, ψIj ∆(Ij )

j=n−k+1 −n+1

≥ −2

M+

n−2

(6.2.83) −n

(2

j=n−k+2

3 − 2−k−j ) (M (1 − αj ) + M (1 − βj )) . 2

Our aim is to find a lower bound for the right-hand side of (6.2.83). To this end we consider separately the summands containing M (1 − αj ), and M (1 − βj ). We will first obtain the lower bound for the sum involving the terms M (1 − αj ). We claim that the following estimate holds true. n−2 j=n−k+2

3 2−n − 2−k−j M (1 − αj ) ≥ 2−n M (k − B − 4). 2

(6.2.84)

To verify (6.2.84) it suffices to combine the following three simple observations. n−2 First counting the number of summands gives j=n−k+2 2−n = 2−n (k−3). Second we sum the geometric series n−2

2−k−j = 2−n−1 (1 − 2−k+3 ).

(6.2.85)

j=n−k+2

Third we invoke the compensation inequality as stated in the hypothesis, that is n−2

αj ≤ B.

j=n−k+2

Combining these three remarks gives the lower bound stated in (6.2.84). Now we turn to the remaining sum in (6.2.83) where the summands contain M (1 − βj ).


391

We claim that the compensation inequalities for βj stated in the hypothesis give the lower bound n−2 3 2−n − 2−k−j M (1 − βj ) ≥ 2−n M (A − k − 1). (6.2.86) 2 j=n−k+2

To verify (6.2.86) we split the range of summation into two groups. We consider separately the cases n − k + 2 ≤ j ≤ n − k + A and n − k + A < j ≤ n − 2. For the first case where n − k + 2 ≤ j ≤ n − k + A, we invoke the hypothesis stating that n−k+A

βj ≤ C.

(6.2.87)

j=n−k+2

Using (6.2.85) and (6.2.87) gives that n−k+A

3 (1 − βj ) 2−n − 2−k−j ≥ 2−n (A − C − 3). 2

(6.2.88)

j=n−k+2

On the other hand when n − k + A + 1 ≤ j ≤ n − 2 we just use the trivial estimate 3 3 (1 − βj )(2−n − 2−k−j ) ≥ −2−n − 2−k−j . 2 2 Summing over j and counting the number of summands gives that trivially n−2

3 (1 − βj ) 2−n − 2−k−j ≥ 2−n (A − k − 1). 2

(6.2.89)

j=n−k+A+1

It remains to take the sum of (6.2.88) and (6.2.89) to see that (6.2.86) holds. Compare now the lower bound (6.2.86) with the right-hand side of (6.2.84). Note that the term −k appearing in (6.2.86) is matched by +k in (6.2.84). Hence the combination of (6.2.83) with (6.2.84) and (6.2.86) gives that n−2

f, ψJj ∆(Jj ) + f, ψIj ∆(Ij ) ≥ 2−n M (2A − B − C − 7).

(6.2.90)

j=n−k+1

By (6.2.82) the estimate (6.2.90) implies (6.2.79).

We have now completed the proof of Theorem 6.2.8. We deduce it by putting together Lemma 6.2.11 and Lemma 6.2.12. 1 Carleson’s unconditional basis for Hat

In this subsection we prove that the Carleson system is an unconditional basis in 1 . We begin by proving first the corresponding dual estimate in BMO([0, 1)). Hat


392

Theorem 6.2.13. For f ∈ BMO([0, 1)) the biorthogonal coefficients {f, ψI : I ∈ D} determine the norm as 1 A−1 ||f ||2BMO([0,1)) ≤ sup f, ψI 2 ≤ A||f ||2BMO([0,1)) . (6.2.91) |J| J∈D I⊆J

The constant A = A(k) depends only on the smoothing parameter k. Proof. Let f ∈ BMO([0, 1)), and normalize such that ||f ||BMO([0,1)) = 1. We start proving the right-hand side estimate. Viewing f ∈ BMO([0, 1)) as an element of L2 and assuming that f = 0, we expand f using Carleson’s biorthogonal system, f= f, ψI ϕI . (6.2.92) I∈D

By Theorem 6.2.2 the convergence in (6.2.92) is unconditional in L2 . Let J be a fixed dyadic interval. Depending on J we split the collection of all dyadic intervals D into the following three families. We let A1 = {I ∈ D : I ⊆ J}, A2 = {I ∈ A1 : |I| ≤ |J|}, A3 = {I ∈ D : |I| > |J|}. Accordingly we split f = f1 + f2 + f3 where for i ∈ {1, 2, 3}, we put fi = f, ψI ϕI . I∈Ai

Our normalizing assumption that ||f ||BMO([0,1)) = 1, the definition of the norm in BMO([0, 1)) and the triangle inequality, give the lower bounds 1/2 2 dt 1≥ |f − mJ (f )| |J| J 1/2 1/2 1/2 (6.2.93) 2 dt 2 dt 2 dt ≥ |f1 | − |f2 | − |f3 − mJ (f )| . |J| |J| |J| J J J Now we analyze the three terms appearing above. We find a lower bound for the first term and upper bounds for the remaining ones. From now on, the coefficient estimates |f, ψI | ≤ C|I|1/2 , established in Theorem 6.2.5 play a decisive role. First we estimate J |f1 |2 from below. Let |J| = 2−j . Let R denote the intervals in A1 which are adjacent to the right endpoint of J. Similarly let L denote the intervals in A1 which are adjacent to the left endpoint of J. Then with E = R ∪ L we define h= f, ψI ϕI , and g = f1 − h. (6.2.94) I∈E


393

Recall that |J| = 2−j . For ≥ 0 let E = {I ∈ E : |I| = 2−j− }. Note that E0 = {J}, and that for ≥ 1 the collection E consists of two intervals. By Theorem 6.2.2 it follows that |h|2 ≤ 4 f, ψI 2 . (6.2.95) J

I∈E

Now estimate the sum on the right-hand side. For I ∈ E , we have |I| = 2−j− . Hence Theorem 6.2.5 gives that f, ψI 2 ≤ C2−j− .

(6.2.96)

Inserting the estimate (6.2.96) into (6.2.95) and summing the geometric series gives that f, ψI 2 ≤ C|J|. (6.2.97) I∈E

Next, observe that for I ∈ A1 \ E the support of ϕI is contained in the interval J. Now Theorem 6.2.2 implies that the L2 norm of g can be expressed through its coefficients as 2 f, ψI ≤ 4 |g|2 J

I∈A1 \E

≤8

(6.2.98)

|f1 | + 8

|h| .

2

J

2

J

We combine (6.2.98) with the upper bounds obtained in (6.2.97) and (6.2.95). This gives 2 f, ψI ≤ C|J| + C |f1 |2 . I∈A1

Next use the upper bound for I∈A1

J

J

|f1 |2 given by (6.2.93). Thus we obtain that

f, ψI ≤ C|J| + C

|f3 − mJ (f )| dt + C

2

|f2 |2 dt.

2

J

J

(6.2.99)

Now we turn to the estimate for the second term appearing on the right-hand side of (6.2.99). We first rewrite mJ (f ) = mJ (f1 ) + mJ (f2 ) + mJ (f3 ). Then we estimate the integrand as |f3 (x) − mJ (f )| ≤ |f3 (x) − mJ (f3 )| + |mJ (f2 )| + |mJ (f1 )|.

(6.2.100)

We continue finding an estimate for |mJ (f1 )| appearing in the last line. Observe with h defined in (6.2.94), mJ (f1 ) = mJ (h).


394

Hence by using again Theorem 6.2.5 and the estimate |mJ (ϕI )| ≤ 2−k (3|I|)/(2|J|) holding for I ∈ E , we find that |f, ψI mJ (ϕI )| ≤ C2− ,

for I ∈ E .

Consequently the following estimate holds for mJ (f1 ). |mJ (f1 )| ≤

∞

|f, ψI mJ (ϕI )|

=0 I∈E

(6.2.101)

≤ C. Next we prove a pointwise upper bound for |f3 (x) − mJ (f3 )|. Recall that the definition of ϕI (x) gives this Lipschitz estimate Lip(ϕI ) ≤ 2+k |I|−3/2 . Hence by Theorem 6.2.5 for I ∈ A3 we find that for x ∈ J, |f, ψI (ϕI (x) − mJ (ϕI ))| ≤ C

2k diam(J) . |I|

Let (2 · I) denote the interval with the same midpoint as I and twice its diameter. The support of ϕI is ( clearly ) contained in the interval (2 · I). In summary for x ∈ J, |f, ψI (ϕI (x) − mJ (ϕI ))| |f3 (x) − mJ (f3 )| ≤ I∈A3

≤C

2k diam(J) 1(2·I) (x) |I|

(6.2.102)

I∈A3

≤ C2k . It remains to find good estimates for mJ (f2 ) and estimate the latter integral since |mJ (f2 )| ≤

dt |f2 | |J| J 2

J

|f2 |2 . Clearly it suffices to

1/2 .

Now estimates for |f2 |2 over the interval J are obtained just like the estimates for |h|2 in (6.2.95) and (6.2.97). We find |f2 |2 dt ≤ C|J|. (6.2.103) J

Summing up we combine the estimates (6.2.101), (6.2.102) and (6.2.103) and insert them first into (6.2.100) and then into (6.2.99). This proves that f, ψI 2 ≤ C|J|. I∈A1


395

Now we prove the left-hand side estimate of Theorem 6.2.13. We start by fixing an interval D ⊆ [0, 1). The interval D is not necessarily dyadic. Note that it suffices to show that there exists m = m(D) so that 1 1 |f (x) − m|2 dx ≤ 2k C sup f, ψI 2 . |D| D J∈D |J| I⊆J

Choose two (adjacent) dyadic intervals J1 , J2 such that D ⊆ J1 ∪ J2 , and such that every y ∈ J1 ∪ J2 satisfies dist(y, D) ≥

|D| . 2

Next let I be a dyadic interval such that I ∩ J1 = ∅, I ∩ J2 = ∅, and |I| ≤ |D|. For this choice of I our definition of J1 , J2 , gives that ϕI (x) = 0, for x ∈ D. Now define f, ψI ϕI (x), and h(x) = f (x) − g(x). g(x) = I⊆J1 ∪J2

Fix x ∈ D and x0 ∈ D. Then using that Lip(ϕI ) ≤ 2k |I|−3/2 we obtain for x ∈ D, the following pointwise estimate, |h(x) − h(x0 )| ≤ 2k C sup |I|−1/2 |f, ψI |.

(6.2.104)

I

Next write f (x) − h(x0 ) = g(x) + h(x) − h(x0 ). Combining Theorem 6.2.2 and (6.2.104) gives that

|f (x) − h(x0 )| dx ≤ C

|g|2 + 2k C|D| sup

2

D

I

D

f, ψI 2 |I|

1 ≤ 2 C|D| sup f, ψI 2 . |J| J∈D k

I⊆J

1 . Next we prove that Carleson’s system is an unconditional basis in Hat

Theorem 6.2.14. The biorthogonal functionals {ψI : I ∈ D} form an unconditional 1 1 . The Banach spaces Hat and H 1 are isomorphic. Specifically, the basis in Hat linear extension of the map T : ψI →

hI , |I|1/2

and

1 defines an isomorphism between Hat and H 1 .

1[0,1] → 1[0,1]


396

Proof. Recall first that by Theorem 6.2.2 the map T defines an isomorphism on 1 1 L2 . Hence, as Hat ∩ L2 is dense in Hat , is suffices to prove that 1 ≤ T g H 1 ≤ C g H 1 , c g Hat at

(6.2.105)

1 1 for every g ∈ Hat ∩ L2 . Theorem 6.2.2 asserts that for g ∈ Hat ∩ L2 and f ∈ BMO([0, 1)), g, f = g, ϕI f, ψI ,

where the sum on the right-hand side is absolutely converging. Hence by duality, % & 1 ∼ sup ||g||Hat g, ϕI f, ψI : ||f ||BMO([0,1)) ≤ 1 . By Theorem 6.2.13 we obtain that 1 ∼ sup ||g||Hat

%

& g, ϕI bI ,

(6.2.106)

where the supremum is taken over all sequences (bI ) satisfying sup J

1 2 bI ≤ 1. |J|

(6.2.107)

I⊆J

Clearly if (bI ) satisfies (6.2.107), then the same holds for (±bI ). Consequently 1 ∼ sup ||g||Hat

%

& |g, ϕI bI | ,

where the supremum is taken over all sequences (bI ) satisfying (6.2.107). In particular 1 ≤ C||g||H 1 .

±g, ϕI ψI Hat at 1 norm of g with the norm of the function in Let us now compare the Hat 1 (dyadic) H , hI Tg = G = g, ϕI 1/2 . |I|

By the duality between the dyadic H 1 and dyadic BMO we obtain that ⎧ ⎫ ⎨ ⎬ 1 2 ||G||H 1 ∼ sup |g, ϕI bI | : sup bI ≤ 1 . (6.2.108) ⎩ ⎭ J |J| I⊆J

A direct comparison between (6.2.106) and (6.2.108) gives that (6.2.105) holds.

6.3. Spaces of homogeneous type

6.3

397

Spaces of homogeneous type

The remaining sections of this chapter are devoted to proving the classification theorem for the Banach spaces H 1 (X, d, µ). The style of the exposition below will be more cursory than in the previous chapters of this book and I refer to [155] for additional details. On a set X × X let d be a quasi-metric, that is, d is symmetric, positive definite and satisfies the following so-called quasi-triangle inequality, d(x, z) ≤ K(d(x, y) + d(y, z)),

(6.3.1)

for x, y, z, ∈ X. Let µ be a probability measure on X. We say that µ satisfies the doubling condition, if there exists A > 0 so that for all x ∈ X, and r > 0, µ(B(x, 2r)) ≤ Aµ(B(x, r)).

(6.3.2)

If for d the quasi-triangle inequality (6.3.1) holds and if the probability measure µ satisfies (6.3.2), then the triple (X, d, µ) is called a space of homogeneous type. We consider quasi-metrics only which satisfy the following Lipschitz condition. There exists 0 < β ≤ 1 and C > 0 so that for each x, y, z ∈ X, |d(x, y) − d(y, z)| ≤ Cd(x, z)β (d(x, y) + d(y, z))

1−β

.

(6.3.3)

Finally we assume that the measure µ and the quasi-metric d are related as follows. If x ∈ X and µ({x}) ≤ r ≤ 1, then the measure of a ball centered at x with radius r is bounded above and below by its radius. Thus, cr ≤ µ(B(x, r)) ≤ Cr,

for µ({x}) ≤ r ≤ 1,

(6.3.4)

where c > 0, C > 0. For the remaining values of r where r < µ({x}) we assume that the ball B(x, r) contains only one point, hence B(x, r) \ {x} = ∅,

for r < µ({x}).

(6.3.5)

Clearly the conditions (6.3.3)–(6.3.5) imply (6.3.1) and (6.3.2). Conversely there exists a theorem of R. Macias and C. Segovia [131] and [132] to the effect that for every space of homogeneous type there exists a metric d and α > 0 so that (6.3.3)–(6.3.5) are satisfied and C1 d (x, y) ≤ dα (x, y) ≤ C2 d (x, y),

for x, y ∈ X.

On a space of homogeneous type we define the following H 1 space of functions. An atom for (X, d, µ) is either a constant or a real-valued function a on X satisfying adµ = 0, X


398 and

a2 dµ ≤ µ(B)−1 ,

X

where B is a ball in X containing the support of a. We define the associated atomic H 1 space to be the space of functions f admitting a decomposition into atoms ai , (6.3.6) f= ci a i , where ci ∈ R and

|ci | < ∞.

The right-hand side of (6.3.6) we call atomic decomposition of f. The space of functions defined that way is denoted H 1 (X, d, µ). The norm of f ∈ H 1 (X, d, µ) is given by ||f ||H 1 (X,d,µ) = inf{ |ci |}, where the infimum is taken over all decompositions (6.3.6). In this chapter we will present the isomorphic classification of these spaces. We will prove the following result. Theorem 6.3.1. Let (X, d, µ) be a space of homogeneous type. The Banach space H 1 (X, d, µ), is isomorphic to one of the following spaces: dyadic H 1 , ( Hn1 )1 , or 1 . We will isolate geometric conditions on (X, d, µ), which determine the isomorphic type of H 1 (X, d, µ). The simplest of these conditions is µ{x ∈ X : µ({x}) = 0} > 0,

(6.3.7)

characterizing the case when H 1 (X, d, µ) is isomorphic to the usual dyadic H 1 . It is easy to see from (6.3.3) and (6.3.5) that in a space of homogeneous type a point x ∈ X is an isolated point precisely when µ({x}) > 0. Hence, the mere absence of isolated points in X implies that H 1 (X, d, µ) is isomorphic to dyadic H 1 . Theremaining possibilities for the isomorphic types of H 1 (X, d, µ) are the spaces ( Hn1 )1 , or 1 . These are possible only when µ{x ∈ X : µ({x}) = 0} = 0, hence only when, up to a set of measure zero, X consists of isolated points. We will use a quantitative measure, modeled after the Carleson constant, to define the isomorphic invariant by which we will distinguish the spaces ( Hn1 )1 and 1 . This requires a bit more preparation: Let q ∈ N be large enough so that q −β ≤

1 , 2

(6.3.8)

where β > 0 is given by (6.3.3). In a space of homogeneous type there exist collections of balls B n which serve as a substitute and generalization of dyadic intervals of length 2−n . Their definition is as follows. First select any maximal set of points x1 , . . . , xm ,

6.3. Spaces of homogeneous type satisfying

399

d(xi , xj ) ≥ q −n

for i = j.

(6.3.9)

Then form the balls Bi = B(xi , 4q −n ), and define B n = {Bi : 1 ≤ i ≤ m }. In Proposition 6.3.2 below we will show that Bn is a covering of X by balls with bounded overlap. The collections {Bn : n ∈ N}, determine an isomorphic invariant for the space H 1 (X, d, µ). Indeed, fix B ∈ B n , and let w(B) =

∞

diam(Q).

m=n {Q∈Bm ,B∩Q=∅}

Then define the Carleson constant of the collection B = [[B]] = sup sup

B∈Bn

n

# n

Bn to be

w(B) . diam(2B)

(6.3.10)

We will now summarize how the underlying space of homogeneous type determines the isomorphic class of H 1 (X, d, µ). (a) H 1 (X, d, µ) is isomorphic to dyadic H 1 , if and only if µ{x ∈ X : µ({x}) = 0} > 0. (b) H 1 (X, d, µ) is isomorphic to (

Hn1 )1 , if and only if

µ{x ∈ X : µ({x}) = 0} = 0,

and

[[B]] = ∞.

(b) H 1 (X, d, µ) is isomorphic to 1 , if and only if [[B]] < ∞. Next we present a detailed construction of the families {Bn : n ∈ N}, referred to above. They will serve as a substitute for dyadic intervals. The proof illustrates very well how the measure and the metric interact on spaces of homogeneous type. From now on we fix a large integer q so that (6.3.8) holds. Proposition 6.3.2. For n ∈ N there exists a collection of balls Bn , so that the following conditions hold. 1. X=

B.

B∈Bn

2. Each ball B ∈ Bn intersects at most A1 = A1 (K, A) different balls in Bn .


400 3. For each B ∈ Bn ,

q −n ≤ diam B ≤ Kq −n .

Proof. Fix n ∈ N. Then select a maximal set of points x1 , . . . , xm , such that

d(xi , xj ) ≥ q −n ,

for i = j.

(6.3.11)

−n

We claim that the union of the sets Bi = B(xi , 4q ) covers X. Indeed #m suppose that there exists w ∈ X which is not contained in the union i=1 Bi . Then d(w, xi ) ≥ 4q −n , for each i ≤ m. This contradicts the maximality condition (6.3.11) used to define the points {xi }. Recall that K > 0 is defined by the quasi-triangle inequality (6.3.1). We claim that the balls Di = B(xi , (4K)−1 q −n ) are pairwise disjoint. If not there exist j = k, and w ∈ Dj ∩ Dk . Hence d(w, xj ) ≤ (4K)−1 q −n and also d(w, xk ) ≤ (4K)−1 q −n . Consequently d(xj , xk ) ≤ K(d(w, xj ) + d(w, xk )) ≤ K(2K)−1 q −n ≤ q −n /2. This contradicts the lower estimate (6.3.11). Now let A1 ∈ N and suppose that there exists a point w ∈ X which intersects more than A1 of the sets Bi . Next we show that there exists an upper bound for A1 . Here we exploit the doubling property (6.3.2) of the measure µ. Without loss of generality we may assume that w intersects B1 , . . . , BA1 . Then by triangle inequality the union B1 ∪ · · · ∪ BA1 is contained in B(w, 8Kq −n ). Hence µ(B1 ∪ · · · ∪ BA1 ) ≤ CKq −n ,

(6.3.12)

for some universal constant C. On the other hand Bj ⊇ Dj . Note that we obtain Bj by doubling Dj approximately log K times. Thus we find that µ(Dj ) ≥ c1 µ(Bj )/(A log K). Recall now that the sets Dj are pairwise disjoint. This allows us to estimate µ(B1 ∪ · · · ∪ BA1 ) ≥

A1

µ(Dj )

j=1

≥ c1

A1

(6.3.13) µ(Bj )/(A log K)

j=1

≥ c2 A1 q −n /(A log K).


401

Merging the estimates (6.3.12) and (6.3.13) and cancelling q −n gives the following upper bound for A1 , A1 ≤ CAK log K. With the following proposition we refine the properties of the collection Bn . Proposition 6.3.3. Let L ∈ N. The covering Bn can be split into A2 = A2 (L, A1 ) disjoint collections n , B1n , . . . , BA 2 such that,

for B, B ∈

dist(B, B ) ≥ Lq −n ,

(6.3.14)

Bin .

Proof. Let B ∈ Bn , and fix xB ∈ B, then choose R = 4LK 2 .

(6.3.15)

Next we define the collection A = {B(xB , Rq −n ) : B ∈ Bn }. This is a finite collection of overlapping balls in X. Note that each ball in A intersects at most N = N (R) different elements of A. Now we split A into N + 1 collections {Li }, in such a way that the balls in Li are pairwise disjoint. We let {U1 , . . . , Um } be an enumeration of the sets in A. The argument proceeds now inductively. Let ≤ m. Suppose that we have been able to obtain a splitting of {U1 , . . . , U } into L1 , . . . , LN +1 such that each Li contains pairwise disjoint balls. Now consider U+1 . Assume for the moment that for each i ≤ N + 1 there exists Li ∈ Li , such that U+1 ∩ Li = ∅. This means that U+1 intersects at least N + 1 elements of A, which can’t happen. Therefore we found i0 ≤ N + 1 so that for each L ∈ Li0 , U+1 ∩ L = ∅. In this way we obtained a proper splitting of A = {U1 , . . . , U+1 }. This completes the proof that A may be split into collections L1 , . . . , LN +1 so that each Li consists of pairwise disjoint sets. Clearly the elements in A are indexed by the elements in Bn . Thus the splitting of A into disjoint collections L1 , . . . , LN +1 induces a corresponding splitting n n n of Bn into subcollections B1n , . . . , BN +1 . We claim now that B1 , . . . , BN +1 satisfies the conclusion of our proposition. Indeed suppose that there exist A, B ∈ Bin , b ∈ B, and a ∈ A such that d(a, b) ≤ Lq −n . Then by the quasi-triangle inequality we obtain that d(xA , xB ) ≤ K(d(b, xB ) + K(d(b, a) + d(a, xA , ))) ≤ (K 2 L + K 2 + K)q −n .

(6.3.16)


402

With our choice of (6.3.15) the estimate (6.3.16) contradicts the fact that the collection Li consists of pairwise disjoint sets. Let M ∈ N be a large integer to be specified below. Now we fix 1 ≤ i ≤ A2 and 1 ≤ k ≤ M. Then for j = j(k, i) we form Mj =

∞

BiM n+k .

(6.3.17)

n=1

In that way we define N = A2 × M

(6.3.18)

collections of balls. Next fix j ≤ N. We discard those balls B ∈ Mj for which µ(B) is much larger than diam(B). Note that if a ball B satisfies µ(B) ≥ LdiamB, then B consists of one isolated point. This is a consequence of the doubling property of the measure µ. We put Nj = {B ∈ Mj : L · diam(B) ≥ µ(B)},

for j ≤ N , and N = N1 ∪ · · · ∪ NN . (6.3.19) Now we construct a nested family of measurable sets associated to Nj . Proposition 6.3.4. For any B ∈ Nj there exists a set T (B), such that B ⊆ T (B)

and

diam(T (B)) ≤ Cdiam(B),

and so that the entire collection Ej = {T (B) : B ∈ Nj }

(6.3.20)

forms a nested family of measurable sets. Proof. Let n ∈ N, and let

Ln = BiM n+k ∩ Nj

be the layers of Nj . Now fix any B ∈ Ln . We define the set T (B) as follows. We put x ∈ T (B), if there exists a chain of sets {Bi : i ∈ N}, so that 1. Bi ∈ Ln ∪ · · · ∪ Ln+k · · · , 2. B1 = B, 3. Bi ∩ Bi+1 = ∅, and x∈

∞ i=1

Bi .


403

This definition makes the collection Ej = {T (B) : B ∈ Nj },

j ≤ N,

(6.3.21)

a nested collection of measurable sets (automatically!). Moreover with the separation property of Proposition 6.3.3 one verifies easily that the chain {Bi : i ≥ 1}, does not extend “horizontally” by an amount larger than Cdiam(B). Thus, diam(T (B)) ≤ Cdiam(B).

The nested collections {Ej : j ≤ N } defined by (6.3.20) reflect the geometry of (X, d, µ) as follows. The condition µ{x ∈ X : µ({x}) = 0} > 0 is equivalent to max µ(lim sup Ej ) > 0. j≤N

(6.3.22)

# Recall that B = n Bn . We defined the Carleson constant of B in (6.3.10). It follows that [[B]] = ∞ if and only if max sup

j≤N A∈Ej

{B∈Ej ∩A}

µ(B) = ∞. µ(A)

(6.3.23)

We use the generations of Ej to define a sequence of increasing σ-algebras on X. We put F0j = {X}, and let An = Gn (Ej ),

n ∈ N.

Then we define inductively j = σ(Fnj , An+1 ) Fn+1

n ∈ N.

(6.3.24)

Observe that by B. Maurey’s isomorphism established in Theorem 4.1.3, the martingale space H 1 [(Fnj )∞ n=1 ] is isomorphic to X[Ej ], the space spanned by threevalued martingale differences associated to the nested family Ej . In the following two sections we show that H 1 [(Fn1 )] ⊕ · · · ⊕ H 1 [(FnN )] is isomorphic to H 1 (X, d, µ). We obtain this result by applying the decomposition principle of A. Pelczy` nski and the two theorems stated next. First in Section 6.4 we prove that Theorem 6.3.5. X[E1 ] ⊕ · · · ⊕ X[EN ] is isomorphic to a complemented subspace of H 1 (X, d, µ). Second, in Section 6.5 we show the converse to Theorem 6.3.5. Theorem 6.3.6. H 1 [(Fn1 )] ⊕ · · · ⊕ H 1 [(FnN )] contains a complemented copy of H 1 (X, d, µ) Thus the problem of identifying the isomorphic type of atomic H 1 spaces is reduced to the case of martingale H 1 spaces. The latter ones are classified by Theorem 4.1.3 and Theorem 4.2.1.


404

Lipschitz partitions of unity The covering Bn defined in Proposition 6.3.3 induces the following Lipschitz partition of unity on (X, d, µ). We let g be a smooth and non-negative function on R such that g is equal to 1 on the interval ] − 1, 1[, and 0 outside of ] − 2, 2[. Then fix B ∈ Bn , and define d(x, B) rB (x) = g . (6.3.25) diamB Note that rB (x) = 1 for x ∈ B, and that rB (x) = 0, when d(x, B) ≥ 2diamB. Hence, rB (x) ≤ K1 1≤ B∈Bn

where K1 = K1 (A, K, A1 ). Then we define a partition of unity by the formula

−1 rA (x) . sB (x) = rB (x) A∈Bn

Note that sB satisfies the Lipschitz estimate |sB (x) − sB (y)| ≤ Cµ(B)−β d(x, y)β ,

(6.3.26)

and its support is centered around B, B ⊆ supp sB ⊆ KB. The partition of unity {sB : B ∈ Bn } induces a positive and symmetric kernel Sn (x, y) = sB (x)sB (y)/||sB ||1 . (6.3.27) B∈Bn

The kernel satisfies a Lipschitz estimate |Sn (x, y) − Sn (u, y)| ≤ Cd(x, u)β q n+βn , and

(6.3.28)

Sn (x, y)dµ(x) = 1.

(6.3.29)

Moreover Sn (x, y) is supported around the diagonal as follows: If d(x, y) ≥ Kq −n , then Sn (x, y) = 0. For β > 0 the Lipschitz class Lipβ consists of functions f : X → R for which there is C > 0 so that |f (x) − f (y)| ≤ Cd(x, y)β ,

for x, y ∈ X.

(6.3.30)

We let Lipβ (f ) denote the infimum over all C > 0 for which (6.3.30) holds true.


405

H 1 (X, d, µ) estimates for molecules In this subsection we establish a general and widely used condition asserting that a given operator is bounded on H 1 (X, d, µ). Suppose that a linear operator T defined initially on L2 (X, µ) maps atoms into atoms. Then, of course, T admits a unique extension to a bounded operator on H 1 (X, d, µ). The same conclusion holds true if the operator T has the property that it maps a single atom into a convex combination of atoms. This observation motivates the search for conditions implying that a given function in L2 (X, µ) can be written as a convex combination of atoms. Most frequently the following criterion is used. We say that a function m : X → R is a molecule centered at x0 if there exists > 0 such that

1/

m2 (x)dµ(x) X

m2 (x)d(x0 , x)1+ dµ(x)

≤ 1,

X

and

mdµ = 0. Next we will show that a molecule can be written as an absolutely convergent series of atoms, so that the sum of the absolute values of the coefficients depends only on > 0 and a universal constant derived from the space of homogeneous type (X, d, µ). It follows that a linear operator mapping atoms into molecules (with fixed > 0) admits an extension to a bounded operator on H 1 (X, d, µ). Theorem 6.3.7. Any molecule admits an absolutely convergent decomposition into atoms, and hence belongs to H 1 (X, d, µ). Proof. Let m be an -molecule centered at x0 . Then put R = ||m||−2 and let 2 A0 = B(x0 , R). Define the following sequence of annuli centered at x0 . Ai = {x : R2i−1 ≤ d(x, x0 ) ≤ R2i }.

Then define coefficients ai =

mdµ, Ai

and the following pieces of m, mi = m1Ai − ai µ(Ai )−1 1Ai . Note that mi is supported on a ball of radius R2i , and ci =

∞ k=i

ak .

mi dµ = 0. Now let


406 By assumption c0 = m=

∞

mdµ = 0, hence partial summation gives the identities mi +

i=0

=

∞ i=0

∞

ai µ(Ai )−1 1Ai

i=0

mi +

∞

ci+1 (µ(Ai+1 )−1 1Ai+1 − µ(Ai )−1 1Ai ).

i=0

Now we will show that this is indeed an atomic decomposition of m. Note that mi is supported on a ball of radius R2i , and mi dµ = 0. Analogously (µ(Ai+1 )−1 1Ai+1 − µ(Ai )−1 1Ai ) is supported on a ball of radius R2i+1 . Next we verify the L2 estimates i 2 m2i ≤ 4(R2i )−1 , (6.3.31) (µ(Ai+1 )−1 1Ai+1 − µ(Ai )−1 1Ai )2 ≤ C(R2i )−1 , and the upper bound for the coefficients ci ≤ 2−i/2 .

(6.3.32)

It is worth pointing out that, in proving the first estimate in (6.3.31) we use that -molecules satisfy the upper bound m2 (x)d(x, x0 )1+ dµ(x) ≤ ||m||−2 2 . Fix i ∈ N and write 2 mi (x)dµ ≤

m2 (x)d(x, x0 )1+ d(x, x0 )−1− dµ

Ai

≤ (R2i−1 )(−1−) ||m||−2 2 ≤ 21+ (R2i )−1 2−i . For the second estimate in (6.3.31) observe that, pointwise, |µ(Ai+1 )−1 1Ai+1 − µ(Ai )−1 1Ai | ≤ C(R2i )−1 . It remains to square this and integrate over its support. The support has measure ≤ CR2i . Finally we prove the estimate (6.3.32) for the coefficients ci . We begin with Cauchy–Schwarz, ∞ ∞ |m| ≤ µ(Ak )1/2 m2k 2 . (6.3.33) |ci | =

k=i

Ak

k=i

The bound for m2k , shows that the summands of the series on the right-hand side of (6.3.33) are dominated by a geometric progression, 1/2 µ(Ak ) ( m2k )1/2 ≤ C(R2k )1/2 (R2k )−1/2 2−k/2 .

6.4. Orthogonal projections in atomic H 1 spaces

407

Summing the resulting geometric series, ∞

2−k/2 ,

k=i

gives the upper estimate |ci | ≤ C 2−i/2 .

6.4

Orthogonal projections in atomic H 1 spaces

The previous section was devoted to the construction of the nested collections {Ej : j ≤ N }. See Proposition 6.3.4. Here we will show that X[Ej ] is isomorphic to a complemented subspace of H 1 (X, d, µ). Following is an outline of how this is done below. We start with a martingale difference sequence {hA : A ∈ Nj } on a probability space (X, µ), generating a copy of X[Ej ]. Then we regularize the martingale difference sequence using the kernels {Sn : n ∈ N}, defined by (6.3.27) in the previous section. We will show that for the resulting family of L2 normalized functions {gA : A ∈ Nj } the Gram matrix G(A, B) = gA gB dµ, is a positive definite and almost diagonal matrix. Hence we may use the square root of its inverse to form a biorthogonal system defined by fA = G−1/2 (A, B)gB . This system gives rise to an orthogonal projection Pf = f, fA fA . We will prove that P is bounded on H 1 (X, d, µ) and that its range is isomorphic to X[Ej ]. In this way we will obtain an explicit form of the embedding and projection showing that X[Ej ] is isomorphic to a complemented subspace of H 1 (X, d, µ). Given Ej = {T (B) : B ∈ Nj }, we start by constructing L2 normalized martingale difference sequences, {hB : B ∈ Nj } satisfying, (a) supp hB ⊆ T (B), (b) hB dµ = 0, (c) supx∈B |hB (x)| ≤ C inf x∈B |hB (x)|, for some universal constant C, (d) hB is constant when restricted to any A ∈ G1 (T (B), Ej ). Now we continue by regularizing the martingale differences defined above. Let k0 ∈ N be a fixed, large integer depending on (X, d, µ). Let n ∈ N and B ∈ B n ∩Nj . For M (n) = n + k0 , define φB (x) = SM (n) (x, y)hB (y)dµ(y), (6.4.1)


408 and renormalize in L2 ,

gB (x) = φB (x)/||φB ||2 .

(6.4.2)

The Gram matrix G = (G(A, B)) of the system {gB : B ∈ Nj } is defined to be the matrix whose entries are gA gB dµ, where A, B ∈ Nj . (6.4.3) G(A, B) = X

With the next proposition we summarize the basic properties of the system {gB : B ∈ Nj }. We show that G is positive definite, and that its entries decay rapidly off the diagonal. Recall that the constant M appearing in the conclusion of Proposition 6.4.1 is defined in (6.3.17). Proposition 6.4.1. For any A ∈ Nj , the function gA , is supported in a constant multiple of the ball A. Let A ∈ Bn ∩ Nj , and let B ∈ Bm ∩ Nj . If m = n, then |G(A, B)| ≤ C(k0 ) min

µ(A) µ(B) , µ(B) µ(A)

1/2+β .

(6.4.4)

If m = n and A = B, then G(A, B) = 0.

(6.4.5)

−M β

so that for any sequence of scalars {cA : A ∈ There exists 0 < (M ) ≤ C(k0 )q Nj }, (1 − (M )) c2A ≤ || cA gA ||22 ≤ (1 + (M )) c2A . Hence, the Gram matrix G is invertible and satisfies

Id − G 2 ≤ (M )

with

0 < (M ) ≤ C(k0 )q −M β .

(6.4.6)

Proof. Let A, B ∈ Bn ∩ Nj , and A = B, then by Proposition 6.3.3, supp gA ∩ supp gB = ∅. Hence G(A, B) = 0, and (6.4.5) holds. Next we turn to the verification of (6.4.4). Let A ∈ Nj ∩ Bm . We first obtain Lipschitz estimates for gA directly from Lipschitz estimates of the kernel SM (n) . |gA (x) − gA (y)| ≤ C |SM (n) (x, u) − SM (n) (y, u) · |hA (u)|dµ(u) ≤ C(k0 )µ(A)−1/2−β d(x, y)β . Fix A, B ∈ Nj with µ(B) ≤ µ(A), then fix a ∈ A. Now estimate using that gB dµ = 0, |G(A, B)| ≤ |gB (x)(gA (x) − gA (a))|dµ (6.4.7) ≤ |gB |dµLipβ (gA )µ(B)β ≤ C(k0 )(µ(B)/µ(A))1/2+β .


409

Thus we verified (6.4.4). Let {cA : A ∈ Nj } be given, then || cA gA ||22 = c2A + cA cB G(A, B). A B=A

Exploiting the estimate (6.4.7) we prove now good estimates for the off diagonal terms appearing on the right-hand side. We claim that these are bounded by a small fraction of c2A . Fix A ∈ Nj and l ≥ 1. Define Gl (A) to be the l-th generation of T (A) with respect to the nested collection Ej . Then put Hl (A) = {B : T (B) ∈ Gl (A)}. By (6.4.5) we have the identity

cA cB G(A, B) =

A B=A

∞

cA

l=1 A

cB G(A, B).

B∈Hl (A)

Note that for l ≥ 1 and B ∈ Hl (A) we have that µ(B) ∼ q −lM µ(A), hence by (6.4.7), |G(A, B)| ≤ C(k0 )(µ(B)/µ(A))1/2+β ≤ C(k0 )q −lM (1/2+β) . Using also the symmetry of the Gram matrix, G(A, B) = G(B, A), we obtain that

|cA cB G(A, B)| ≤ C(k0 )

A B=A

∞

|cA |

l=1 A

|cB |q −lM (1/2+β) .

(6.4.8)

B∈Hl (A)

The cardinality of the index set Hl (A) is of order q lM . Next we use the Cauchy– Schwarz inequality and invoke the estimate for the cardinality of Hl (A). Then the right-hand side of the inequality (6.4.8) can be dominated by C(k0 )

c2A

∞ 1/2

q −lM β

c2B

1/2 .

(6.4.9)

l=1

Note that the geometric series in (6.4.9) starts with l = 1. Evaluating its sum we obtain finally the following bound for the off diagonal terms, |cA cB G(A, B)| ≤ C(k0 )q −M β c2A . A B=A

Putting (M ) = C(k0 )q −M β finishes the proof.

A

Proposition 6.4.1 implies immediately that the Gram matrix G is positive definite. Its inverse G−1 has therefore a well-defined square root denoted by G−1/2 .


410

Its existence is guaranteed by the functional calculus of spectral theory. By (6.4.6) the functional calculus yields an expansion of G−1/2 in an absolutely convergent power series of the form G−1/2 =

∞

ck ( Id − G)k ,

k=0

where the coefficients satisfy ck ∼ k−1/2 . Below, we use the power series expansion of G−1/2 to show that (6.4.4) and (6.4.6) imply that the entries of G−1/2 ( indexed by A, B ∈ Nj ) satisfy the estimate 1/2+β/8 −1−β/8 d(A, B) µ(A) µ(B) −1/2 , 1+ (A, B)| ≤ C min . |G µ(B) µ(A) max{µ(A), µ(B)} See Theorem 6.4.6 and (6.4.45). The off-diagonal decay of G−1/2 allows us to construct an orthonormal system which satisfies good localization and decay properties. This is the content of the next theorem. Theorem 6.4.2. There exists an orthonormal system of functions {fB : B ∈ Nj } on (X, d, µ), so that the following conditions hold. 1. span{fB : B ∈ Nj } = span{gB : B ∈ Nj }. 2. fB dµ = 0. −1−β/(16) dist(x, B) 1+ . µ(B) −1−β/(16) dist(x, B) d(x, y)β/(16) 4. |fB (x) − fB (y)| ≤ C 1+ . µ(B) µ(B)1/2+β/(16)

3. |fB (x)| ≤

C µ(B)1/2

Proof. By Theorem 6.4.1 the Gram matrix G is positive definite hence its inverse and the square root of its inverse are well defined. We obtain the system fB by applying the matrix G−1/2 , to the functions {gA }, that is, we let fB (x) = G−1/2 (A, B)gA (x). Theorem 6.4.6 below describes the decay of the off diagonal coefficients in the matrix G−1/2 . It asserts that 1/2+β/8 −1−β/8 d(A, B) µ(A) µ(B) , 1+ . G−1/2 (A, B) ≤ C min µ(B) µ(A) max{µ(A), µ(B)} For x ∈ X, fixed, the support of gA (x), is contained in a set of diameter ≤ Cµ(A). Hence the pointwise estimate for fB (x), |fB (x)| ≤ C µ(A)−1/2 G−1/2 (A, B). {A : dist(x,A)≤Cµ(A)}


411

Now we split the index set {A ∈ Nj : dist(x, A) ≤ Cµ(A)} in three parts. Define K0 ∈ N such that dist(x, B) K0 ≤1+ ≤ K0 . 2 µ(B) Depending on the value of K0 we split the index set {A : dist(x, A) ≤ Cµ(A)}. Start by “drawing a disk” around x with radius K0 . Then let A be the part of the index set which is not contained in this disk. The remaining part of the index set is divided into balls with large measure and those with small measure. Precisely, we define A = {A ∈ Nj : dist(x, A) ≤ Cµ(A) and µ(A) > K0 µ(B)}, B = {A ∈ Nj : dist(x, A) ≤ Cµ(A) and µ(B) ≤ µ(A) ≤ K0 µ(B)}, C = {A ∈ Nj : dist(x, A) ≤ Cµ(A) and µ(A) < µ(B)}.

(6.4.10)

We record now estimates for each of these cases. In the first case we have −1−β/8 µ(A)−1/2 G−1/2 (A, B) ≤ K0 µ(B)−1/2 . A∈A

For the second case we obtain −1−β/8 µ(A)−1/2 G−1/2 (A, B) ≤ K0 (log K0 )µ(B)−1/2 . A∈B

Finally in the third case we consider two sub-cases. First if dist(x, B) ≤ Cµ(B), then µ(A)−1/2 G−1/2 (A, B) ≤ µ(B)−1/2 . A∈C

Second if dist(x, B) ≥ Cµ(B), then we simply get µ(A)−1/2 G−1/2 (A, B) = 0. A∈C

Combining the estimates of these three cases into one statement we formulate µ(A)−1/2 G−1/2 (A, B) {A: dist(x,A)≤Cµ(A)} −1−β/(16) dist(x, B) −1/2 1+ ≤ µ(B) . µ(B) The H 1 (X, d, µ) bounds for fB appearing in the next theorem are a direct consequence of the pointwise decay established in Theorem 6.4.2. Theorem 6.4.3. The orthonormal functions fB , B ∈ Nj defined by Theorem 6.4.2 satisfy the H 1 (X, d, µ) estimates ||fB ||H 1 (X,d,µ) ≤ Cµ(B)1/2 .


412

−1/2 fB is a Proof. Let B ∈ Nj . By Theorem 6.3.7 it suffices to check that µ(B) multiple of a β/(16)-molecule. Recall first that fB (x)dµ(x) = 0, and fB 2 = 1. Now fix xB ∈ B. We will show that fB2 (x)d(xB , x)1+β/(16) dµ(x) ≤ Cµ(B)1+β/(16) . X

Let A0 = B. For j ≥ 1, we form the annuli Aj = {x ∈ X : 2j−1 µ(B) ≤ d(xB , x) ≤ 2j µ(B)}. The pointwise estimate of Proposition 6.4.2 and the defining equation of the annuli Aj give that for j ≥ 1, fB2 (x)d(xB , x)1+β/(16) dµ(x) Aj

≤ µ(Aj ) sup fB2 (x) sup d(xB , x)1+β/(16) x∈Aj

x∈Aj

54 5 =4 ≤ C 2j µ(B) 2−j(2+β/8) µ(B)−1 (µ(B)2j )1+β/(16) .
µ(I)}.

(6.4.14)

Before we start estimating we will decompose these collections into levels consisting of sets of comparable measure. Recall that Nj ⊆ BiM n+k , (6.4.15) n

for some fixed k ≤ M and i ≤ A2 . Now we select the level n0 which is determined by µ(I). We choose n0 ∈ N by the following rule. If B ∈ A, then there exists n ≤ n0 such that B ∈ BiM n+k ; if B ∈ B ∪ C, then there exists n ≥ n0 + 1 so that B ∈ BiM n+k . Part 2.

Here we estimate ||

a, fB fB ||H 1 (X,d,µ) .

B∈A

With the triangle inequality this sum is dominated by

a, fB fB H 1 (X,d,µ) . B∈A

Recall that by Theorem 6.4.3, we have

and

||fB || ≤ Cµ(B)1/2 ,

(6.4.16)

adµ = 0. The Lipschitz estimate of Proposition 6.4.2 gives that

|a, fB | ≤ diam(I)

β/(16)

µ(B)

−1/2−β/(16)

−1−β/(16) dist(B, I) 1+ . (6.4.17) µ(B)

Now we split the index set A into its levels. For n ≤ n0 we put An = A ∩ BiM n+k . Observe that

−1−β/(16) dist(B, I) 1+ ≤ C, µ(B)

B∈An

(6.4.18)


414

where C > 0 is independent of the level n. Moreover for B ∈ An we have µ(B)−β/(16) ∼ q (M n+k)β/(16) . Hence combining (6.4.16)–(6.4.18) we find that

a, fB fB H 1 (X,d,µ) ≤ Cdiam(I)β/(16)

n0

q (M n+k)β/(16) .

(6.4.19)

n=1

B∈A

It remains to analyze the factor in (6.4.19) given by the geometric series. First note that q (M n0 +k)β/(16) ∼ µ(I)−β/(16) ∼ diam(I)−β/(16) . Inserting this in the estimate (6.4.19) we obtain that

a, fB fB H 1 (X,d,µ) ≤ C.

B∈A

Part 3.

Now we turn to estimating || a, fB fB ||H 1 (X,d,µ) .

(6.4.20)

B∈C

Recall that the index set C contains those B ∈ Nj which are well separated from I and which satisfy µ(B) ≤ µ(I). Again the first move is to apply the triangle inequality. So we will actually find an estimate for the sum ||fB ||H 1 (X,d,µ) |fB (x)a(x)|dµ(x). (6.4.21) I

B∈C

By Theorem 6.4.2 for B ∈ C the following pointwise estimate holds for x ∈ I, |fB (x)| ≤ µ(B)+1/2+β/(16) dist(I, B)−1−β/(16) . We insert this, and recall that ||fB || ≤ Cµ(B)1/2 . Thus we obtain that ||fB ||H 1 (X,d,µ) |fB (x)a(x)|dµ(x) ≤ Cµ(B)1+β/(16) dist(I, B)−1−β/(16) . I

(6.4.22)

Next we split the index set C in its natural layers. For n ≥ n0 + 1, define Cn = C ∩ BiM n+k , where k, i are determined by (6.4.15). We thus arrived at the estimate, ∞

n=n0 +1 B∈Cn

µ(B)1+β/(16) dist(I, B)−1−β/(16) .

(6.4.23)


415

To this end we fix n ≥ n0 + 1. Then for B ∈ Cn we have dist(B, I) ≥ (µ(I) + mµ(B)) for some m ∈ N. Moreover, the sets B ∈ Cn satisfy the measure estimate µ(B) ∼ q −(M n+k) . Now we abbreviate A0 = µ(I)q (M n+k) . Then the inner sum of (6.4.23), has the upper bound

µ(B)

1+β/(16)

dist(I, B)

−1−β/(16)

≤C

∞

(m + A0 )−1−β/(16)

m=1

B∈Cn

≤

(6.4.24)

−β/(16) CA0 .

We have now reduced our task of estimating (6.4.20) to that of evaluating a geometric series. Indeed, as A0 = µ(I)q (M n+k) , we obtained with (6.4.22) and (6.4.24), that ∞ a, fB fB || ≤ Cµ(I)−β/(16) q −(M n+k)β/(16) . (6.4.25) || n=n0 +1

B∈C

Note that by choice of n0 , ∞

q −(M n+k)β/(16) ≤ Cµ(I)β/(16) .

(6.4.26)

n=n0

In (6.4.25) the factor µ(I)−β/(16) is thus compensated by the value of the geometric sum in (6.4.26). Thus we find, || a, fB fB || ≤ C. B∈C

Part 4.

Now we turn to the estimates for a, fB fB . b= B∈B

Here the index set comprises the elements in Nj which are close to I, and for which µ(B) ≤ Cµ(I). Therefore loosely speaking the sum b is strongly concentrated around I. This observation makes it plausible that b is a molecule, centered around ( any point of ) I. We will now prove that this is in fact the case. To begin with, we have by orthonormality of the system fB , b2 dµ ≤ a2 dµ ≤ µ(I)−1 . Hence in order for b to be a molecule, it suffices that b(x)2 d(x, xI )1+β/(16) dµ(x) ≤ µ(I)β/(16) . X


416

Now we split the domain of integration. Consider first the integral over the region 4I. Here we estimate simply by pulling out the factor d(x, xI )1+β/(16) , hence 2 1+β/(16) 1+β/(16) b2 (x)dµ(x) b(x) d(x, xI ) dµ(x) ≤ µ(I) 4I

≤ µ(I)β/(16) . Now we turn to the remaining integral over X \ 4I. We will show that b(x)2 d(x, xI )1+β/(16) dµ(x) ≤ µ(I)β/(16) .

(6.4.27)

X\4I

Here we start with an unusual estimate, namely we observe that we have the upper bound a, fB 2 )( fB2 (x)). (6.4.28) b2 (x) ≤ ( B∈B

B∈B

An estimate like the one above is rarely used, mainly because it is almost always too crude. Here however it works pretty well, because the factor B∈B fB2 (x) is concentrated around I and we are now interested in estimates over the set X \ 4I. Note that by orthonormality ( B∈B a, fB 2 ) ≤ µ(I)−1 . Hence by (6.4.28) 2 1+β/(16) −1 b(x) d(x, xI ) dµ(x) ≤ µ(I) fB2 (x)d(x, xI )1+β/(16) dµ(x). X\4I

X\4I B∈B

(6.4.29) Hence (6.4.27) is implied by the estimate fB2 (x)d(x, xI )1+β/(16) dµ(x) ≤ µ(I)1+β/(16) . B∈B

(6.4.30)

X\4I

Its verification requires again that we display the layers of the index set B. Let k ≤ M, and i ≤ A2 be determined by the inclusion (6.4.15). Recall that n0 ∈ N is chosen in the first part of the proof. For n ≥ n0 + 1 we write Ln = B ∩ BiM n+k .

(6.4.31)

For each of these collections we will find that the following estimate holds, fB2 (x)d(x, xI )1+β/(16) dµ(x) ≤ µ(I)1−β/(16) q −(M n+k)β/8 . (6.4.32) B∈Ln

X\4I

Let us first observe how the estimate (6.4.32) implies (6.4.30). We take the righthand side of (6.4.32) and sum over n ≥ n0 + 1. Evaluating the geometric series we find that µ(I)1−β/(16)

∞ n=n0 +1

q −(M n+k)β/8 ≤ µ(I)1−β/(16) q −(M n0 +k)β/8 ∼ µ(I)1+β/(16) .


417

We now turn to the proof of (6.4.32). We begin with an estimate for the integrand. We claim that for x ∈ X \ 4I, fB2 (x) ≤ Cµ(I)q −(M n+k)β/8 dist(x, I)−2−β/8 . (6.4.33) B∈Ln

To prove the claim (6.4.33) we invoke the pointwise bounds established for the Franklin functions in Theorem 6.4.2, fB2 (x) ≤ µ(B)1+β/8 dist(x, B)−2−β/8 .

(6.4.34)

Note that for any x ∈ X \ 4I and any B ∈ Ln , we have the uniform estimate dist(x, I) ≤ dist(x, B). Hence the summands of (6.4.35) below, can be estimated independent of B ∈ Ln . To find an upper bound of (6.4.35) we take out one of the summands and multiply it with the cardinality of the index set, (denoted here |Ln | ). This gives dist(x, B)−2−β/8 ≤ |Ln | dist(x, I)−2−β/8 . (6.4.35) B∈Ln

By (6.4.31) for B ∈ Ln we have µ(B) ∼ q −(M n+k) . Hence the cardinality of Ln satisfies the upper bound |Ln | ≤ Cµ(I)q (M n+k) .

(6.4.36)

Combining the estimates (6.4.34) – (6.4.36) proves (6.4.33). It follows from (6.4.33) that for x ∈ X \ 4I the following estimate holds, fB2 (x) dist(x, I)1+β/(16) ≤ Cµ(I)q −(M n+k)β/8 dist(x, I)−1−β/(16) . B∈Ln

Integrating this bound and changing the order of summation and integration we obtain fB2 (x) dist(x, I)1+β/(16) dµ(x) B∈Ln

X\4I

≤ µ(I)q −(M n+k)β/8

(6.4.37) dist(x, I)−1−β/(16) dµ(x). X\4I

It remains to find the value of the integral appearing in the second line of (6.4.37). A sufficiently good approximation is obtained as follows. First split X \ 4I into dyadic annuli Ai = 2i+1 I \ 2i I. Then for x ∈ Ai the ratio dist(x, I)/ dist(Ai , I) is bounded above and below by a constant independent of i or I. Finally sum the values µ(Ai ) dist(Ai , I)−1−β/(16) . The resulting estimate for the integral in (6.4.37) is dist(x, I)−1−β/(16) dµ(x) ≤ µ(I)−β/(16) .

X\4I

This completes the proof of (6.4.30).


418

By Theorem 6.4.4 the closed linear span of the orthonormal system {fB : B ∈ Nj } is a complemented subspace of H 1 (X, d, µ). Recall that the nested collection Ej is defined by Proposition 6.3.4. By a reasoning similar to the one appearing in the proof of Theorem 6.4.4 the space X[Ej ] (spanned by three-valued martingale differences) is isomorphic to the closed linear span of the system {fB : B ∈ Nj } in H 1 (X, d, µ). For the details of the proof the reader is referred to [155]. In summary we proved that the martingale space X[Ej ] is isomorphic to a complemented subspace of H 1 (X, d, µ).

The square root of the Gram matrix For A, B ∈ Nj and γ > 0 we define σ(A, B, γ) = min

µA µB , µB µA

1/2+γ 1+

d(A, B) max{µA, µB}

−1−γ .

A matrix R = (R(A, B)) is called an almost diagonal matrix if there exists C > 0 and γ > 0 such that |R(A, B)| ≤ Cσ(A, B, γ),

A, B ∈ Nj .

For instance the Gram matrix of the system {gA , A ∈ Nj } is an almost diagonal matrix. Our first result states that the class of almost diagonal matrices is stable under the formation of products. Proposition 6.4.5. Let R be an infinite matrix with entries R(A, B), A, B ∈ Nj , and assume that |R(A, B)| ≤ M1 σ(A, B, γ), for all A, B ∈ Nj , and some M1 . Let Rk be the k-th power of R and let its entries be Rk (A, B). Then there exists M2 ≥ 0 depending only on (X, d, µ) such that |Rk (A, B)| ≤ (M1 M2 )k σ(A, B, γ/2), for any k ∈ N and A, B ∈ Nj . Proof. The proof is inductive. We let S = (S(A, B)) be an almost diagonal matrix satisfying |S(A, B)| ≤ (M1 M2 )k−1 σ(A, B, γ/2). Explicitly this means that, |S(A, B)| ≤ (M1 M2 )

k−1

min

µA µB , µB µA

1/2+γ/2

d(A, B) 1+ max{µA, µB}

−1−γ/2 .

In this way the matrix S assumes the role of Rk−1 and we are assuming that the theorem holds true for Rk−1 . By hypothesis, R satisfies |R(A, B)| ≤ M1 σ(A, B, γ). Note that the entries of the product SR are given by S(A, P )R(P, B). (6.4.38) w(A, B) = P ∈Nj


419

Now we claim that the following estimate holds, |w(A, B)| ≤ (M1 M2 ) min k

µA µB , µB µA

1/2+γ/2

d(A, B) 1+ max{µA, µB}

−1−γ/2 .

We will only be sketching the proof of the claim, omitting the details. The idea is to split the sum P ∈Nj in (6.4.38) into three pieces depending on the measure of P relative to the measure of A and B. This is done as follows. Let µA ≤ µB, then we have = + + . (6.4.39) P ∈Nj

{P ∈Nj :µP ≤µA}

{P ∈Nj :µA≤µP ≤µB}

{P ∈Nj :µB≤µP }

Each of the sums in (6.4.39) is now estimated using the assumptions we made above. These are the induction hypothesis and the fact that R is an almost diagonal matrix, |S(A, P )| ≤ (M1 M2 )k−1 σ(A, B, γ/2),

and

|R(P, B)| ≤ M1 σ(A, B, γ).

The calculations involved are quite long; we omit them and refer to [155] and [70] for the details. The constant M2 appearing in Proposition 6.4.5 is an absolute constant that can be traced to the quasi-metric constant (6.3.1) and the doubling constant in (6.3.2). Under the additional hypothesis that the almost diagonal matrix G is positive definite and sufficiently close to the identity, we will show now that also the square root of the inverse G−1/2 is an almost diagonal matrix. Theorem 6.4.6. Let G be a symmetric matrix with entries G(A, B), A, B ∈ Nj . Assume that (6.4.40) |G(A, B)| ≤ K1 σ(A, B, γ), and that the induced operator satisfies || Id − G||2 ≤ 1/K2

with

K2 ≥ ((K1 + 1)M2 )(4/γ)−1 .

(6.4.41)

(Here M2 is the constant appearing in Proposition 6.4.5.) Then G−1/2 is diagonally dominant and the entries of G−1/2 satisfy |G−1/2 (A, B)| ≤ K3 σ(A, B, γ/2)1−γ/4 .

(6.4.42)

Proof. Let R = Id − G and let Rk be the k-th power of R. The link between G−1/2 and the powers Rk comes from elementary spectral theory of symmetric operators. With the spectral theorem G−1/2 has a series representation as G−1/2 =

∞ k=1

ck R k ,


420

where the coefficients ck satisfy ck ≤ Ck−1/2 . Let the entries of Rk be Rk (A, B). We will verify that for a fixed choice of A, B ∈ Nj the following estimate holds for the sum of the powers, ∞

|Rk (A, B)| ≤ σ(A, B, γ/2)1−γ/4 .

k=1

By Proposition 6.4.5, we have |Rk (A, B)| ≤ (K1 + 1)k M2k σ(A, B, γ/2)

(6.4.43)

for k ∈ N. Note that we also have the following inequality as a result of the operator estimate (6.4.41), |Rk (A, B)| ≤ K2−k .

(6.4.44)

We will use the estimate (6.4.43) for small values of k, and (6.4.44) for large values of k. The transition from one regime to the next happens at the critical value k0 . Define k0 by the relation (K1 + 1)k0 M2k0 ≤ σ(A, B, γ/2)−γ/4 ≤ (K1 + 1)k0 +1 M2k0 +1 . The following estimate holds for k ≤ k0 , k0

|Rk (A, B)| ≤

k=1

k0

(K1 + 1)k M2k σ(A, B, γ/2)

k=1

≤ 2(K1 + 1)k0 M2k0 σ(A, B, γ/2). The choice of k0 implies that k0

|Rk (A, B)| ≤ 2σ(A, B, γ/2)1−γ/4 .

k=1

Now we turn to the remainder. Estimating the geometric sum gives ∞

|Rk (A, B)| ≤ 2K2−k0 .

k=k0 +1

The assumption that K2 ≥ ((K1 + 1)M2 )(4/γ)−1 together with the choice of k0 implies that K2−k0 ≤ Cσ(A, B, γ/2)1−γ/4 . We thus verified that ∞ k=1

|Rk (A, B)| ≤ Cσ(A, B, γ/2)1−γ/4 .

6.5. Martingale approximation in atomic H 1 spaces

421

Finally,the spectral theorem implies that G−1/2 =

∞

ck R k ,

k=1

where the coefficients satisfy ck ≤ Ck−1/2 . This gives G−1/2 (A, B) ≤ Cσ(A, B, γ/2)1−γ/4 . Remarks.

1. An elementary calculation shows that σ(A, B, γ/2)1−γ/4 ≤ Cσ(A, B, γ/8).

(6.4.45)

2. The hypothesis of Theorem 6.4.6 is matched by the conclusion of Proposition 6.4.1. Hence by choosing the constant M large enough in (6.3.17) it follows that the inverse to the square root of G(A, B), A, B ∈ Nj is an almost diagonal matrix.

6.5

Martingale approximation in atomic H 1 spaces

In this section we will prove that H 1 (X, d, µ) is isomorphic to a complemented subspace of the direct sum H 1 [(Fn1 )] ⊕ · · · ⊕ H 1 [(FnN )]. The theorem of B. Maurey implies then that H 1 (X, d, µ) is isomorphic to a complemented subspace of dyadic H 1 . More can be said by taking into account Theorem 6.3.5 and the classification theorems for martingale H 1 spaces (Theorem 4.1.3 and Theorem 4.2.1): Combining these results with Theorem 6.5.2 below gives the isomorphic classification of atomic H 1 spaces. We use a remarkable family of symmetric kernels {kA : A ∈ N }, constructed by R. Coifman. Its properties are summarized as follows. (a) For f ∈ BMO(X, d, µ), respectively f ∈ L2 (X, µ), and λA (f )(x) = f (y)kA (x, y)dµ(y), the series f1 =

λA (f )

A∈N

converges unconditionally in BMO(X, d, µ), respectively in L2 (X, µ).

(6.5.1)

(6.5.2)


422

(b) The series (6.5.2) provides an approximation of f, ||f − f1 ||BMO(X) ≤

1 ||f ||BMO(X) , 2

(6.5.3)

1 ||f ||L2 (X) . 2

(6.5.4)

and simultaneously ||f − f1 ||L2 (X) ≤ (c) The operator Kj (f ) =

λB (f )

(6.5.5)

B∈Nj

is bounded and linear on BMO(X, d, µ) and on H 1 (X, d, µ). By interpolation Kj is bounded on Lp (X, µ), 1 < p < ∞. We next define the space Z∞ = {(K1 (f ), . . . , KN (f )) : f ∈ BMO(X, d, µ)}, equipped with the norm max ||Kj (f )||BMO(X,d,µ) . j

By (6.5.2), the space Z∞ is naturally isomorphic to BMO(X, d, µ) where the isomorphism is given by the map f → (K1 (f ), . . . , KN (f )). By (6.5.2) we obtain a corresponding renorming of L2 . We define Z2 = {(K1 (f ), . . . , KN (f )) : f ∈ L2 (X)}, and on Z2 we define the norm ⎛ ⎞1/2 ⎝ ||Kj (f )||22 ⎠ . j

As a result the space Z2 is a closed subspace of the N -fold direct sum L2 (X, µ) ⊕ · · · ⊕ L2 (X, µ). Hence the orthogonal projection P defined on L2 (X, µ) ⊕ · · · ⊕ L2 (X, µ) with range in Z2 is bounded. The article by Y. S. Han and G. Weiss [81] presents the unpublished results of R. Coifman referred to above. They are established by isolating and exploiting the singular integral representation for the operators Kj and applying a version of the T (1)-theorem valid in spaces of homogeneous type. The representation of Kj as a singular integral is possible sincethe kernels kA , used in the definition of Kj , are of mean zero, kA (x, y)dµ(x) = kA (x, y)dµ(y) = 0, they satisfy a Lipschitz estimate, |kA (x, y) − kA (x, z)| ≤ Cdiam(A)−1−β d(y, z)β ,


423

and moreover, they are localized around a fixed multiple of A × A. The T (1) theorem asserts that for any subcollection G ⊆ N , the resulting kernel, kA (x, y), k(x, y) = A∈G

defines a bounded singular integral operator an L2 . For details we refer to the book of S. Y. Han and E. T. Sawyer [80] or to A. Nahmod [161]. We will establish below that the operators Kj decompose any f ∈ BMO(X, d, µ) into approximate martingales. The first step in this direction is the next proposition. It asserts that for a fixed j the function Kj (f ) defined by (6.5.5) is well adapted to the increasing sequence of σ-algebras (Fnj )∞ n=1 defined in (6.3.24). Proposition 6.5.1. The following a priori estimates hold for the linear operator Kj , ||Kj (f )||BMO[(Fnj )] ≤ C||f ||BMO(X,d,µ) , and ||Kj (f )||H 1 [(Fnj )] ≤ C||f ||H 1 (X,d,µ) . Proof. We first prove the BMO estimates. Let h = Kj (f ). Let D be an atom in the σ-algebra Fnj . We will show that j |h − E(h|Fn−1 )|2 dµ ≤ µ(D)||f ||2BMO(X,d,µ) . (6.5.6) D

We need to isolate the index set in the sum defining h = Kj (f ), which contributes to the integral on the left-hand side of (6.5.6). The index set consists of three parts as follows. First there is the collection j A = {A ∈ Nj : T (A) ⊇ D and T (A) ∈ Fn−2 },

second B = {A ∈ Nj : T (A) ⊆ D}, j j and third, there is A0 ∈ Nj satisfying T (A0 ) ⊇ D and T (A0 ) ∈ Fn−1 \ Fn−2 . Note that on the atom D, there holds the following identity, j j )= E(λA (f )|Fn−1 ). E(h|Fn−1 A∈A j ) appearing in Thus we obtain a representation for the difference h − E(h|Fn−1 (6.5.6). Indeed on D the following identity holds, j j h − E(h|Fn−1 )= λA (f ) − E(λA (f )|Fn−1 )+ λA (f ). A∈A

A∈B∪{A0 }


424

Next we give proper estimates for each of the above sums. We start with the second term. The integral representation for λA (f ), and the resulting singular integral representation for A∈B∪{A0 } λA (f ) justifies the application of the T (1) theorem. Hence, the following L2 estimate holds, | λA (f )|2 dµ ≤ µ(D)||λA0 (f )||2∞ + ||λA (f )||22 . (6.5.7) D A∈B∪{A } 0

A∈B

We continue with pointwise estimates for the term We start by applying the triangle inequality,

j |λA (f ) − E(λA (f )|Fn−1 )| ≤ diamDβ

A∈A

A∈A

j λA (f )−E(λA (f )|Fn−1 ).

Lipβ (λA (f )).

(6.5.8)

A∈A

To analyze the right-hand side of (6.5.8), observe that for each k ≤ n − 1, there j . Invokexists at most one A ∈ Nj such that T (A) ⊇ D, and T (A) ∈ Fkj \ Fk−1 ing the Lipschitz estimates Lipβ (λA (f )) ≤ CdiamA−β f BMO(X) we obtain from (6.5.7) and (6.5.8) the BMO estimate (6.5.6). Now we prove the H 1 part of our proposition. Let I ⊆ X be a ball and let a : X → R be an atom supported on I such that ||a||22 ≤ µ(I)−1 . Now put h = Kj (a). Next denote by Sj the martingale square function induced by the filtration (Fnj )∞ n=1 . Thus Sj (h)2 =

∞

j (E(h|Fnj ) − E(h|Fn−1 ))2 .

n=1

Now observe that the support of Sj (h)2 is contained in the set S = {T (A) : A ∈ Nj , A ∩ I = ∅, µ(A) ≤ µ(I)}. Note that µ(S) ≤ Cµ(I). Hence we estimate using Cauchy–Schwarz, and ||h||2 ≤ C||a||2 ≤ Cµ(I)−1/2 , ||Sj (h)||1 ≤ µ(S)1/2 ||Sj (h)||2 ≤ Cµ(S)1/2 ||h||2 ≤ C.

Let P be the orthogonal projection of L (X, µ) ⊕ · · · ⊕ L (X, µ) onto the closed subspace 2

Z2 = {(K1 (f ), . . . , KN (f )) : f ∈ L2 (X, µ)},

2

6.5. Martingale approximation in atomic H 1 spaces equipped with the norm (

425

||Kj (f )||22 )1/2 .

j

We write Pj (f1 , . . . , fN ) to denote the j -th component of the vector P (f1 , . . . , fN ). Now we are ready to prove the main theorem of this section which expresses an a priori estimate for the orthogonal projection P. Theorem 6.5.2. For every choice hj ∈ BMO[(Fnj )] the following a priori estimate holds for the components of the orthogonal projection P. max ||Pj (h1 , . . . , hN )||BMO(X,d,µ) ≤ C(N ) max ||hj ||BMO[(Fnj )] . j

j

Hence, P extends to a bounded projection from BMO[(Fn1 )] ⊕ · · · ⊕ BMO[(FnN )] onto Z∞ . With the duality theorem between H 1 (X, d, µ) and BMO(X, d, µ) we obtain the a priori bound of Theorem 6.5.2 from the following proposition. Proposition 6.5.3. Let hj ∈ BMO[(Fnj )]. Then there exists f ∈ BMO(X, d, µ) such that P (h1 , . . . , hN ) = (K1 (f ), . . . , KN (f )), and for any atom aI in (X, d, µ), the following estimate holds, f aI dµ ≤ C(N ) max ||hj ||BMO[(Fnj )] + Cq −M β ||f ||BMO(X,d,µ) , where the constant C is a universal constant, independent of q, or M. Proof. First we rewrite f as f = f − f1 +

N

Ki (f ).

(6.5.9)

i=1

We let I be the ball in X supporting the atom aI . For j we define the collection Nj (I) = {B ∈ Nj : µ(B) ≤ q M β µ(I), I ∩ B = ∅}. This defines KjI (f ) =

λB (f ).

B∈Nj (I)

We let P I be the orthogonal projection of L2 (X, µ)⊕· · ·⊕L2 (X, µ) onto the closed subspace I (f )) : f ∈ L2 (X, µ)}. Z2I = {(K1I (f ), . . . , KN Next put qj = PjI (aI , . . . , aI ). We find b ∈ L2 so that λQ (b), qj = Q∈Nj (I)


426 moreover we may have that

supp b ⊆

b = 0, ||b||2 ≤ 2||aI ||2 , and {(C · Q) ∈ Nj (I) : j ≤ N },

I where C > 0 is an absolute constant. Note that the vector (K1I (f ), . . . , KN (f )) ∈ I I Z2 is contained in the range of the operator P . Hence by the orthogonality of the projection P I , we find N

KjI (f )aI =

N

j=1

KjI (f )qj .

j=1

Thus we may rewrite N

Kj (f )aI =

j=1

N

Kj (f )qj

j=1

+

N

(Kj (f ) − KjI (f ))aI

(6.5.10)

j=1

+

N

(KjI (f ) − Kj (f ))qj .

j=1

Now we let 1 ≤ j ≤ N and put rj = Q∈N / j (I) λQ (b). By this choice of rj , the vector (qj + rj ) is contained in Z2 . Hence the orthogonal projection leaves it invariant and we have P ((qj + rj )) = (qj + rj ). Next recall that P (h1 , . . . , hN ) = (K1 (f ), . . . , KN (f )), so that by orthogonality we obtain N

Kj (f )(qj + rj ) =

j=1

N

hj (qj + rj ).

j=1

Regrouping terms we obtain from the last identity that N j=1

Kj (f )qj =

N j=1

hj (qj + rj ) −

N

Kj (f )rj .

(6.5.11)

j=1

We add the equation (6.5.11) to the decomposition (6.5.10) and arrive thereby at


427

the representation N

Kj (f )aI =

j=1

N

hj (qj + rj )

j=1

+

N

(Kj (f ) − KjI (f ))aI

j=1

+

N

(KjI (f ) − Kj (f ))qj

j=1

−

N

Kj (f )rj .

j=1

The first sum on the right-hand side is the leading term. We will estimate it using duality and Proposition 6.5.1. It will turn out that each of the remaining three sums is bounded by a small fraction of ||f ||BMO(X,d,µ) . For our purposes the meaning of small is “ small relative to 1/N.” We now turn to estimating the integrals | hj (qj + rj )|, using Fefferman’s inequality and Proposition 6.5.1 . We have that, | hj (qj + rj )| ≤ C||hj ||BMO[(Fnj )] · ||qj + rj ||H 1 [(Fnj )] ≤ C||hj ||BMO[(Fnj )] · ||b||H 1 (X,d,µ)

(6.5.12)

≤ C(M, q)||hj ||BMO[(Fnj )] . We insert this estimate in the above representation. Thus we obtain the estimate | f aI | ≤ ||f − f1 ||BMO(X,d,µ) + C(M, N, q) max ||hj ||BMO[(Fnj )] + E1 + E2 + E3 , where we used the abbreviation E1 =

N (Kj (f ) − KjI (f ))aI dµ , j=1

E2 =

N (Kj (f ) − KjI (f ))qj dµ , j=1

E3 = ||f ||BMO(X,d,µ)

N

||rj ||H 1 (X,d,µ) .

j=1

We will now give estimates for the terms E1 , E2 , E3 , showing that they represent


428 negligible error terms. We start estimating E2 . E2 =

N | KjI (f ) − Kj (f ))qj |dµ j=1

≤

N

(6.5.13)

|

λB (f )qj dµ|.

j=1 B ∈N / j (I)

/ Nj (I), and seek estimates for the integrals appearing Now we fix B ∈ Nj with B ∈ in the sum in the second line of (6.5.13). We let U be the collection of maximal sets in Nj (I). Our claim is that the following estimate holds, | λB (f )qj dµ| ≤ sup{diam(P ∩ B)β : P ∈ U}µ(B)−β ||f ||BMO(X,d,µ) . (6.5.14) We prove the claim as follows. Let b ∈ L2 such that qj = KjI (b). Then for R ∈ U we put bR =

λA (b).

A∈Nj ,A⊆R

Using that the collection U consists of pairwise disjoint sets we obtain ||bR ||22 ≤ aI 22 . ||qj ||22 = R∈U

Next we fix R ∈ U. As bR = 0, we use below the Lipschitz continuity of λB (f ), and the fact that bR is supported on R. Thus by H¨ older’s inequality we obtain that (6.5.15) | λB (f )bR dµ| ≤ ||f ||BMO(X) diam(R ∩ B)β µ(B)−β µ(R)1/2 ||bR ||2 . Next we note that, by H¨ older’s inequality we find µ(R)1−1/s ||bR ||Ls ≤ q M β/2 µ(I)1/2 ||aI ||L2 .

(6.5.16)

R∈U

Recall also that for any atom, µ(I)1/2 ||aI ||L2 ≤ 1. Hence by (6.5.15) and (6.5.16) we obtain the following estimate, by summing over R ∈ U and taking out the supremum of the terms diam(R ∩ B)β µ(B)−β . | λB (f )bR | ≤ q M β/2 ||f ||BMO sup{diam(R ∩ B)β : R ∈ U}µ(B)−β . R∈U


429

To complete the proof of the claim (6.5.14) it remains to use the triangle inequality | λB (f )bR |. | λB (f )qj | ≤ R∈U

Now we return to estimating E2 . Recall that N

E2 ≤

|

λB (f )qj dµ|.

j=1 B ∈N / j (I)

With (6.5.14) we find that

E2 ≤ N q M β/2 ||f ||BMO

µ(B)−β sup{diam(R ∩ B)β : R ∈ U}.

B ∈N / j (I)

Now we consider the geometric series appearing on the right-hand side of the last line. Its sum is bounded by Cq −M β , where C > 0 is a universal constant. Thus we obtain that E2 ≤ CN q −M β/2 ||f ||BMO . Note that choosing q large enough makes q −M β/2 of order 2−M . This is sufficiently small since N = A2 M, where A2 is a universal constant. Finally we turn to estimating E3 . We show that the norm in H 1 (X, d, µ) of rj is a small multiple of 1/N. Recall that λQ (bI ). rj = Q∈N / j (I)

By the triangle inequality it suffices to obtain estimates for the norm of λQ (bI ) in H 1 (X, d, µ), when Q ∈ / Nj (I). First we note that the support of λQ (bI ), is contained in a ball of diameter ≤ Cµ(Q), and λQ (bI ) = 0. Hence λQ (bI ) is a multiple of an atom for (X, d, µ). We have that ||λQ (bI )||H 1 (X,d,µ) ≤ µ(Q)1/2 ||λQ (bI )||2 . Next examining the kernel representation of λQ (bI ), we observe the following L2 estimate, ||λQ (bI )||2 ≤ Cµ(Q)−1/2−β (diam supp bI )β . Summing up we find that ||rj ||H 1 (X,d,µ) ≤ C(diam supp bI )β

{µ(Q)−β : Q ∩ supp bI = ∅}

Q∈N / j (I)

≤ Cq

−M β

.

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List of Symbols

[[C]], 171 C ∗ , 170

lim sup C, 170 Lipβ , 404 Lipβ (f ), 404 L(2 ), 343 2n , 142 L1 (2 ), 39 L∞ (2 ), 39 Lp,q (R), 344 Lpn , 119

D, 1 Dn , 1

max F, 42 Mq (f ), 47

BMO, 34 BMO[(Fn )], 120 BMO([0, 1)), 361 1 BMO n , 119 ( BMOn )ind , 134 BMO(X, d, µ), 123

E ∼ F , 118 ∗

F (t), 348 f(I), 148 Γ(x), 32 Γt , 347 Gp (J, C), 170 Gp (C), 170 1 Hat , 122 H 1 (X, d, µ), 122 hI , 2 H ∞ (D), 343 H 1 [(Fn )], 120 H 1 , 34 H 1 (2n ), 142 Hn1 , 119 H p , p < 1, 262 h , 37

I ∩ H, 170, 317

Pa (f ), 89 ∂i , 21 Q(C), 75, 179 Q(I), 42, 75, 179 rn , 6 S r , 343 S(f ), 16 S 2 (f |H), 104 SL∞ , 52 SL (F ), 347 S k−1 , 66 S(p, q, M ), 135 S(hn ), 236 S ∞ , 137 Tα , 306 Tm , 92 T (p, q, M ), 135

450 Um , 92 wA , 7 W k,∞ (Rn ), 344 (X, d, µ), 121 X[E], 124 ( X)p , 140 Y [E], 124

List of Symbols

Index Analytic family of operators, 159–166 Approximation property, 308 Atomic H 1 spaces, 121, 347–429 Atomic decomposition, 41–45, 353 Auerbach basis, 309, 312 Averaging projection, 104–112 Banach space decomposition method, 139–144, 243–261, 421–429 Banach–Mazur distance, 118 Basic sequence, 12 Basis constant, 6 Biorthogonal functionals, 6 Block of dyadic intervals, 45, 107–112 Bonami–Kiener inequality, 26 Bounded approximation property, 308 Burkholder’s inequality, 13–19 Calder´ on product, 146, 167 Calder´ on–Zygmund kernel, 101–104 Carleson constant, 171 Carleson packing condition, 45, 170 Carleson’s biorthogonal system, 360– 396 1 , 395 Carleson’s system in Hat Carleson’s system in L2 , 365 Colored dyadic intervals, 169–228 Compensation argument, 362, 377– 386 Compensation inequality, 362, 377– 386 Complemented subspace, 117 Complex interpolation, 144–166

Condensation lemma, 172, 249–257, 284–295 Copy of a Banach space, 118 Diagonal operator, 292 Dual Banach space, 6 Dual space of H 1 (X, d, µ), 123 Dual space of H 1 [(Fn )], 120 Dyadic atom, 41 Dyadic chain rule, 10 Dyadic derivative, 7 Dyadic gradient, 20 Dyadic interval, 1 Dyadic Poincaré inequality, 20 Dyadic square function, 16 Fefferman’s inequality, 34–45, 279 Figiel’s compatibility condition, 216 Figiel’s expansion, 84, 85, 89 Figiel’s representation of integral operators, 92–104 Gamlen–Gaudet construction, 176, 249– 257, 284–295 Gamma function, 32 Generations in nested collecting, 127 Generations of dyadic intervals, 169 Glueing process, 107–112, 193, 331 Good λ inequality, 56–60 Gram matrix, 407 Green’s theorem, 355 Hölder conjugate exponent, 13 Haar basis, 1 Haar coefficient, 5

452 Haar expansion, 5 Haar multiplier, 88 Haar support, 41 Hardy–Littlewood maximal function, 45–51 Harmonic extension, 348 Hilbert transform, 100, 347 Independent sum of BMOn , 134 Inequalities of Bonami and Kiener, 26 of Bourgain, 62 of Burkholder, 13 of Fefferman, 35 of Hardy and Littlewood, 47 of Kahane, 10 of Khintchine, 7 of Paley, 16 of Pisier, 23 of Stein, 79 Interpolation of operators, 144–166 Isomorphic invariant, 118, 120, 267– 343 Isomorphic to a complemented subspace, 118 Isomorphism, 118 Johnson’s factorization, 271–277 Jones’s compatibility condition, 105– 112, 181–196, 249–257, 284– 295, 331–343 Kahane’s inequality, 10 Khintchine’s inequality, 7–13 Kiener’s integral representation, 29– 32 Large deviation inequalities, 52–56 Linearly ordered collections, 296 Lipschitz class, 404 Lipschitz partition of unity, 404 Localized square function, 104, 317 Lorentz space, 344 Lusin function, 347

Index 1 Lusin function characterization of Hat , 355

M -Carleson condition, 171 Martingale, 74 Martingale H 1 spaces, 120, 229–265 Martingale difference sequence, 74 Martingale square function, 79 Maurey’s isomorphism, 229–242 Maximal function, 45–51 Maximal function characterization of 1 , 351 Hat Molecules, 405 Multiplicity of Walsh functions, 26 Multiplier, 88 Nested collection, 124–130, 229, 242– 252 Non-tangential maximal function, 347 Open problems, 34, 135, 260, 262, 265, 272, 344 Order inversing embedding, 198, 200– 203, 301–306 Orthogonal projection, 104–112, 193– 196 Paley’s identity, 24 Paraproduct, 89, 92–104 Partial sum operators, 6 Pelczy´ nski’s decomposition method, 139–144, 243–261, 421–429 Pigeon hole principle, 112, 193, 331, 332 Pisier’s inequality, 23 Pisier’s renorming of H 1 , 153 Poincaré inequality, 20 Poisson kernel, 348 Positive homogeneity, 171 Primary, 283 Projection, 117 Property P, 197, 207–215 Quasi-metric, 121, 397 Quasi-triangle inequality, 397

Index Rademacher system, 7 Rearrangement operator, 92–104, 146, 154–159, 196–228 Relative distributional estimate, 56– 60 Research problems, 34, 135, 260, 262, 265, 272, 344 Resolving operator, 309 Riesz convexity theorem, 144 Roider’s example, 28 Rosenthal space, 130–136, 271–277 Schatten class, 343, 344 Schauder basis, 6 Schechtman’s sign-embedding, 52, 63 Semenov’s criterion, 197, 216, 223 Sharp function, 37, 45–51 Sign-embedding, 63 Singular values of a compact operator, 343 Sobolev space, 344 Space of homogeneous type, 121, 397– 429 Square function, 16 Square function characterization, 16 1 Square function characterization of Hat , 355 Square-duality relation, 350 Stein’s martingale inequality, 79 Stolz domain, 347 Stopping time decomposition, 41–45, 149–151, 211–213, 253, 275– 276, 319 Tent space, 306–308 Three lines theorem, 159, 163–166 UAP data, 309 UMD property, 18 Unconditional basis, 18 Unconditional basis constant, 18 1 Unconditional basis for Hat , 391 Uniform approximation property, 181, 309–344

453 Uniformity function, 309 Uniformly complemented copies, 136 Uniformly complemented subspaces, 136 Walsh series, 7 Walsh system, 7, 19–34 Walsh–Paley order, 25 Weak type estimate, 48 Well isomorphic, 118, 121, 136

Isomorphisms between H1 Spaces

Isomorphisms Between H1 Spaces

Isomorphisms Between H1 Spaces (Monografie Matematyczne)

The Spaces in Between

The Spaces in Between

Complemented subspaces of A, H1, and H

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p-equivariant maps between lens spaces and spheres

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Between

Between

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Orlicz Spaces and Modular Spaces

H1-estimates of Jacobians by subdeterminants

BMO, H1, and Calderon-Zygmund operators for non doubling measures

Classical Banach Spaces I and II. Sequence Spaces. Function Spaces

Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry

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Isomorphisms between H1 Spaces