Extensions of the theory of tent spaces

and applications to boundary value

problems

Alex Amenta

June 2016

A thesis submitted for the degree of Doctor of Philosophyof the Australian National University

© Copyright by Alexander Joseph Amenta 2016

Declaration

I hereby declare that the material in this thesis is my own original work exceptwhere stated otherwise.

Part I of this thesis consists of work that has either been published or hasbeen submitted for publication. Chapter 1 has been published as ‘Tent spacesover metric measure spaces under doubling and related assumptions’ in the con-ference proceedings Operator theory in harmonic and non-commutative analysis[3]. Chapter 2 is joint work with Mikko Kemppainen, and has been published as‘Non-uniformly local tent spaces’ in Publicacions Matemàtiques [5]. Chapter 3has been submitted for publication under the title ‘Interpolation and embeddingsof weighted tent spaces’, and is available as an arXiv preprint [4].

Part II will be submitted for publication, pending further additions and mod-ifications.

Alex Amenta (Alexander Joseph Amenta)

iii

Acknowledgements

I would like to thank my supervisors, Pierre Portal (without whom this thesiswould never have been started), and Pascal Auscher (without whom it wouldnever have been finished). It has been an absolute pleasure to work with them,and I look forward to continuing our collaboration. I would also like to thank myhonours supervisor, Bryan Wang, for (unintentionally) introducing me to researchin analysis. I hope his influence is visible in this work, even 3.5 years after mytime with him.

I realise as I write these acknowledgements that there is no way to thank allthe friends who have helped me through this thesis—who truly deserve to bethanked individually—without forgetting anybody. Therefore, I thank them allnot by name, but with the knowledge that they know who they are.

I thank my collaborators for their high-level distraction: Lashi Bandara,Mikko Kemppainen (whose work I have the honour of including in my thesis),and Jonas Teuwen (whose thesis I have the honour of my work being includedin). I also thank my mathematical brothers and sisters: Li Chen, Angus Gri�th,Yi Huang, and Sebastian Stahlhut (and also Moritz Egert and Dorothee Frey,although they don’t really fit into this category) for all kinds of mathematicaland personal support.

I also thank Vincent, Lila, and Céleste, who were ideal housemates while themost di�cult parts of this thesis were being completed.

I thank Ping for her support and for suggesting notation for the real inter-polants of weighted tent spaces.

Finally: I thank my family, particularly my parents, for putting up with myextended absence for so long. This thesis is for Dane, even if he doesn’t quitehave the prerequisites to read it yet.

v

...there is a sort of magic in the written word. The idea acquiressubstance by taking on a visible nature, and then stands in the wayof its own clarification.

W. Somerset Maugham, The Summing Up

tout juste un peu de bruitpour combler le silencetout juste un peu de bruitet rien de plus

Brigitte Fontaine, Comme á la radio

Abstract

We extend the theory of tent spaces from Euclidean spaces to various types ofmetric measure spaces. For doubling spaces we show that the usual ‘global’theory remains valid, and for ‘non-uniformly locally doubling’ spaces (includingRn with the Gaussian measure) we establish a satisfactory local theory. In thedoubling context we show that Hardy–Littlewood–Sobolev-type embeddings holdin the scale of weighted tent spaces, and in the special case of unbounded AD-regular metric measure spaces we identify the real interpolants (the ‘Z-spaces’)of weighted tent spaces.

Weighted tent spaces and Z-spaces on Rn are used to construct Hardy–Sobolevand Besov spaces adapted to perturbed Dirac operators. These spaces play a keyrole in the classification of solutions to first-order Cauchy–Riemann systems (orequivalently, the classification of conormal gradients of solutions to second-orderelliptic systems) within weighted tent spaces and Z-spaces. We establish this clas-sification, and as a corollary we obtain a useful characterisation of well-posednessof Regularity and Neumann problems for second-order complex-coe�cient ellip-tic systems with boundary data in Hardy–Sobolev and Besov spaces of orders œ (≠1, 0).

Contents

Declaration iii

Acknowledgements v

Introduction 1

I Extensions of the theory of tent spaces 7

1 Tent spaces over doubling metric measure spaces 91.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Spatial assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 The basic tent space theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 Initial definitions and consequences . . . . . . . . . . . . . . . . . . . . . . 121.3.2 Duality, the vector-valued approach, and complex interpolation . . . . . . 201.3.3 Change of aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.3.4 Relations between A and C . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.4 Appendix: Assorted lemmas and notation . . . . . . . . . . . . . . . . . . . . . . 351.4.1 Tents, cones, and shadows . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.4.2 Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.4.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2 Non-uniformly local tent spaces 412.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.2 Weighted measures and admissible balls . . . . . . . . . . . . . . . . . . . . . . . 432.3 Local tent spaces: the reflexive range . . . . . . . . . . . . . . . . . . . . . . . . . 492.4 Endpoints: t1,q and tŒ,q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.4.1 Atomic decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.4.2 Duality, interpolation and change of aperture . . . . . . . . . . . . . . . . 55

2.5 Appendix 1: Local maximal functions . . . . . . . . . . . . . . . . . . . . . . . . 572.6 Appendix 2: Cone covering lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.6.1 Review of non-negatively curved spaces . . . . . . . . . . . . . . . . . . . 602.6.2 Cone covering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

xi

3 Interpolation and embeddings of weighted tent spaces 653.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.1 Metric measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.2.2 Unweighted tent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2.3 Weighted tent spaces: definitions, duality, and atoms . . . . . . . . . . . . 73

3.3 Interpolation and embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.3.1 Complex interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.3.2 Real interpolation: the reflexive range . . . . . . . . . . . . . . . . . . . . 803.3.3 Real interpolation: the non-reflexive range . . . . . . . . . . . . . . . . . . 843.3.4 Hardy–Littlewood–Sobolev embeddings . . . . . . . . . . . . . . . . . . . 91

3.4 Deferred proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.4.1 T p,Œ–LŒ estimates for cylindrically supported functions . . . . . . . . . . 943.4.2 T p,Œ atomic decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 96

II Abstract Hardy–Sobolev and Besov spaces 1014 Introduction 105

4.1 Introduction and context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.1.1 Formulation of boundary value problems . . . . . . . . . . . . . . . . . . . 1064.1.2 The first-order approach: perturbed Dirac operators and CR systems . . 1124.1.3 Solutions to CR systems and well-posedness . . . . . . . . . . . . . . . . . 118

4.2 Summary of the article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5 Technical preliminaries 1255.1 Function space preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.1.1 Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1.2 Tent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.1.3 Z-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.1.4 Unification: tent spaces, Z-spaces, and slice spaces . . . . . . . . . . . . . 1415.1.5 Homogeneous smoothness spaces . . . . . . . . . . . . . . . . . . . . . . . 148

5.2 Operator-theoretic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.2.1 Bisectorial operators and holomorphic functional calculus . . . . . . . . . 1515.2.2 O�-diagonal estimates and the Standard Assumptions . . . . . . . . . . . 1565.2.3 Integral operators on tent spaces . . . . . . . . . . . . . . . . . . . . . . . 1595.2.4 Extension and contraction operators . . . . . . . . . . . . . . . . . . . . . 164

6 Adapted function spaces 1716.1 Adapted Hardy–Sobolev and Besov spaces . . . . . . . . . . . . . . . . . . . . . . 171

6.1.1 Initial definitions, equivalent norms, and duality . . . . . . . . . . . . . . 1716.1.2 Mapping properties of holomorphic functional calculus . . . . . . . . . . . 1766.1.3 Completions and interpolation . . . . . . . . . . . . . . . . . . . . . . . . 1776.1.4 The Cauchy operator on general adapted spaces . . . . . . . . . . . . . . 182

xii

6.2 Spaces adapted to perturbed Dirac operators . . . . . . . . . . . . . . . . . . . . 1856.2.1 Identification of spaces adapted to D, DB, and BD . . . . . . . . . . . . 1866.2.2 The Cauchy operator on DB-adapted spaces . . . . . . . . . . . . . . . . 193

7 Elliptic equations, CR systems, and boundary value problems 2037.1 Basic properties of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037.2 Decay of solutions at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2057.3 Classification of solutions to Cauchy–Riemann systems . . . . . . . . . . . . . . . 207

7.3.1 Construction of solutions via Cauchy extension . . . . . . . . . . . . . . . 2087.3.2 Initial limiting arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . 2097.3.3 Proof of Theorem 7.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2117.3.4 Proof of Theorem 7.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

7.4 Applications to boundary value problems . . . . . . . . . . . . . . . . . . . . . . 2307.4.1 Characterisation of well-posedness and corollaries . . . . . . . . . . . . . . 2307.4.2 The regularity problem for real coe�cient scalar equations . . . . . . . . . 2367.4.3 Additional boundary behaviour of solutions . . . . . . . . . . . . . . . . . 2397.4.4 Layer potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

xiii

Introduction

This thesis consists of two main parts. In the first part we provide various gen-eralisations and extensions of the theory of tent spaces. In the second part weestablish results concerning the well-posedness of certain elliptic boundary valueproblems, using some of our extended tent space theory in the process.

Part I: Extensions of the theory of tent spacesTent spaces were first introduced by Coifman, Meyer, and Stein [32, 33] as aunification of fundamental ideas in modern harmonic analysis. Each of the threechapters of this part provides a di�erent extension of their theory.

Chapter 1: Tent spaces over metric measure spaces under doublingand related assumptions.

The main focus here is on doubling metric measure spaces (X, d, µ): (X, d) is ametric space, µ is a Borel measure on (X, d), and the doubling condition

µ(B(x, 2r)) . µ(B(x, r)) (x œ X, r > 0)

is satisfied. We define tent spaces T p,q,–(X) associated with such a doublingmetric measure space, and establish properties of T p,q,–(X) analogous to thoseestablished by Coifman, Meyer, and Stein in the case where X is Rn, d is theEuclidean distance, and µ is the Lebesgue measure.

In particular, we show that these tent spaces are complete (Proposition 1.3.5),that the tent space scale is closed under duality (Propositions 1.3.10 and 1.3.15)and forms a complex interpolation scale (Propositions 1.3.12 and 1.3.18), andthat the space T p,q,–(X) is independent of the ‘aperture’ parameter – (Proposi-tion 1.3.21). The proofs of these results are generally more technical than thecorresponding Euclidean proofs, and we also point out that our proof of the com-plex interpolation result avoids an error in the original Coifman–Meyer–Steinargument.

1

The prototypical example of a doubling metric measure space is the Euclideanspace Rn with the Euclidean distance and Lebesgue measure. More generally, onecan consider a Riemannian manifold of non-negative Ricci curvature, equippedwith the geodesic distance and Riemannian volume (the curvature assumptionensures that the doubling condition is satisfied, by the Bishop–Gromov compar-ison theorem). Tent spaces associated with doubling Riemannian manifolds arethe foundation for the Hardy spaces of di�erential forms developed by Auscher,McIntosh, and Russ [13] (see also the more recent work on this topic by Auscher,McIntosh, and Morris [11]). However, full details of this tent space theory hadnot appeared in the literature (with the exception of the atomic decompositiontheorem, which was proven explicitly by Russ [81]). Therefore the material ofthis chapter fills a gap which was perhaps neglected in the past.

Chapter 2: Non-uniformly local tent spaces.

In this chapter we consider metric measure spaces (X, d, “) which are not dou-bling, but which are—in a certain quantified and non-uniform sense—locally dou-bling (for the precise definition see Section 2.2). Given such a space, we constructnon-uniformly local tent spaces tp,q

– (“).1 The main di�erence between these spacesand those constructed in Chapter 1 is that instead of the full ‘upper half-space’X ◊ R+, we use an admissible region D µ X ◊ R+ defined in terms of the‘non-uniform local doubling’ data (see Definition 2.3.1).

Our theory of non-uniformly local tent spaces runs parallel to the theory con-structed in Chapter 1. We also prove an atomic decomposition theorem (Theorem2.4.5). Technicalities imposed on us by the non-uniform local doubling assump-tion force us to require that the metric space (X, d) is complete in the proof ofthis theorem.

The model non-uniformly locally doubling metric measure space is the Eu-clidean space Rn equipped with the Euclidean distance and, in place of theLebesgue measure, the Gaussian measure

d“(x) = 1(2fi)n/2 e≠|x|2/2 dx.

Non-uniformly local tent spaces associated with this space correspond to theGaussian tent spaces defined by Maas, van Neerven, and Portal [63]. These areused in the construction of Gaussian Hardy spaces by Portal [78]. In Examples

1Note that the notation has changed from the first article: such notation changes will occurin each article.

2

2.2.2 and 2.2.4 we provide many other examples of non-uniformly locally doublingspaces, given by weighted measures analogous to the Gaussian measure.

Chapter 3: Interpolation and embeddings of weighted tent spaces.

Here we return to the setting of doubling metric measure spaces (X, d, µ) as inChapter 1. The tent space scale T p,q(X) introduced there (we need not makereference to the aperture parameter –, as we have already shown the tent spacesdo not depend on it) is expanded: we define weighted tent spaces T p,q

s (X) anal-ogously to the spaces T p,q(X) = T p,q

0 (X), the di�erence being the presence of aweight µ(B(x, t))≠s in the norm. This is motivated by applications to boundaryvalue problems (which appear in Part II), where it is often natural to measurethe function (t, x) ‘æ t≠sÒu(t, x) in T p,2(Rn) when u is the solution to an ellipticPDE.

The weighted tent space scale satisfies the following embedding property:when the parameters p0, p1, s0, s1 satisfy the relation2

s1 ≠ s0 = 1p1

≠ 1p0

,

we have a continuous embedding

T p0,qs0 (X) Òæ T p1,q

s1 (X)

(Theorem 3.3.19). These embeddings are actually quite counterintuitive. Forhomogeneous Sobolev spaces a similar embedding property (related to the Hardy–Littlewood–Sobolev lemma) holds, but this is interpreted as an interchange ofregularity for integrability. In the context of weighted tent spaces, the parameters does not actually reflect any kind of regularity.

When X is unbounded and AD-regular, so that in particular we have

µ(B(x, r)) ƒ rn (x œ X, r > 0)

for some n > 0, we identify the real interpolation spaces

(T p0,qs0 (X), T p1,q

s1 (X))◊,p◊

= Zp◊

,qs

◊

(X) (1)

when p0, p1, q > 1 (Theorem 3.3.4; see Definition 3.3.3 for the definition of thespaces Zp,q

s (X)). When X = Rn we extend this result to p0, p1 > 0 (Theorem2Normally a factor of some ‘dimension’ n should appear on the right hand side, but this

does not appear here because of our convention of using ball volumes as weights.

3

3.3.9). The ‘Z-spaces’ Zp,qs (X) are defined in terms of weighted Lp(X◊R+)-norms

of Lq Whitney averages. They have appeared in the work of Barton and May-boroda on elliptic boundary value problems with data in Besov spaces [21], butthis connection with weighted tent spaces is new. Furthermore, this shows thatWhitney averages arise naturally from the consideration of tent spaces, whereasin the past their use had always been justified by applications to PDE.

Part II: Abstract Hardy–Sobolev and Besov spaces for el-liptic boundary value problems with complex LŒ coe�-cients.

This part of the thesis, unlike the previous part, consists of one single (long)article. Broadly speaking, in this article we construct abstract Hardy–Sobolevand Besov spaces associated with perturbed Dirac operators, and we apply thesespaces to the classification of solutions to Cauchy–Riemann systems. The foun-dation for our abstract Hardy–Sobolev and Besov spaces is the theory of weightedtent spaces (and their real interpolants, the Z-spaces) introduced in Chapter 3.

The main trajectory of this article follows the recent works of Auscher andStahlhut [16] and Auscher and Mourgoglou [14]. However, we introduce manynew techniques and shed some additional light on their results. For example,we introduce a new ‘exponent notation’, where boldface letters p are used todenote pairs (p, s) or triples (Œ, s; –). The purpose of this notation is to combineintegrability and regularity, and in turn to make the exponent calculations used inembeddings and interpolation more intuitive. We also refer to tent spaces T p

s andZ-spaces Zp

s simply as Xp, in order to emphasise the fact that these spaces behavein essentially identical ways. This allows us to streamline our proofs, to handlespaces T p

s and T Œs;– on an equal footing, and to prove results for Hardy–Sobolev

and Besov spaces simultaneously.A much more detailed overview of the article is contained in the introduction

given there (Chapter 4).

The structure of the thesis

As we have already pointed out, this thesis consists of four distinct articles, andeach article uses di�erent notational conventions. They may be read indepen-dently, although the later articles do refer to the earlier ones. Their bibliogra-phies have been consolidated into one single bibliography. With the exception of

4

cosmetic changes and the correction of a few minor errors, the first two articles(Chapters 1 and 2) are identical to the publications [3] and [5], and the thirdarticle (Chapter 3) is identical to the preprint [4].

5

6

Part I

Extensions of the theory of tentspaces

7

Chapter 1

Tent spaces over metric measurespaces under doubling andrelated assumptions

Abstract

In this article, we define the Coifman–Meyer–Stein tent spaces T p,q,–(X) asso-ciated with an arbitrary metric measure space (X, d, µ) under minimal geomet-ric assumptions. While gradually strengthening our geometric assumptions, weprove duality, interpolation, and change of aperture theorems for the tent spaces.Because of the inherent technicalities in dealing with abstract metric measurespaces, most proofs are presented in full detail.

1.1 Introduction

The purpose of this article is to indicate how the theory of tent spaces, as devel-oped by Coifman, Meyer, and Stein for Euclidean space in [33], can be extendedto more general metric measure spaces. Let X denote the metric measure spaceunder consideration. If X is doubling, then the methods of [33] seem at first tocarry over without much modification. However, there are some technicalities tobe considered, even in this context. This is already apparent in the proof of theatomic decomposition given in [81].

Further still, there is an issue with the proof of the main interpolation resultof [33] (see Remark 1.3.20 below). Alternate proofs of the interpolation resulthave since appeared in the literature — see for example [44], [23], [31], and [59]

9

— but these proofs are given in the Euclidean context, and no indication is givenof their general applicability. In fact, the methods of [44] and [23] can be usedto obtain a partial interpolation result under weaker assumptions than doubling.This result relies on some tent space duality; we show in Section 1.3.2 that thisholds once we assume that the uncentred Hardy–Littlewood maximal operator isof strong type (r, r) for all r > 1.1

Finally, we consider the problem of proving the change of aperture result whenX is doubling. The proof in [33] implicitly uses a geometric property of X whichwe term (NI), or ‘nice intersections’. This property is independent of doubling,but holds for many doubling spaces which appear in applications — in particular,all complete Riemannian manifolds have ‘nice intersections’. We provide a proofwhich does not require this assumption.

Acknowledgements

We thank Pierre Portal and Pascal Auscher for their comments and suggestions,particularly regarding the proofs of Lemmas 1.3.3 and 1.4.6. We further thankLashi Bandara, Li Chen, Mikko Kemppainen and Yi Huang for discussions onthis work, as well as the participants of the Workshop in Harmonic Analysis andGeometry at the Australian National University for their interest and suggestions.Finally, we thank the referee for their detailed comments.

1.2 Spatial assumptions

Throughout this article, we implicitly assume that (X, d, µ) is a metric measurespace; that is, (X, d) is a metric space and µ is a Borel measure on X. The ballcentred at x œ X of radius r > 0 is the set

B(x, r) := {y œ X : d(x, y) < r},

and we write V (x, r) := µ(B(x, r)) for the volume of this set. We assume thatthe volume function V (x, r) is finite2 and positive; one can show that V is auto-matically measurable on X ◊ R+.

There are four geometric assumptions which we isolate for future reference:

1This fact is already implicit in [33].2Since X is a metric space, this implies that µ is ‡-finite.

10

(Proper) a subset S µ X is compact if and only if it is both closed and bounded,and the volume function V (x, r) is lower semicontinuous as a function of(x, r);3

(HL) the uncentred Hardy–Littlewood maximal operator M, defined for mea-surable functions f on X by

M(f)(x) := supB–x

1µ(B)

ˆB

|f(y)| dµ(y) (1.1)

where the supremum is taken over all balls B containing x, is of strong type(r, r) for all r > 1;

(Doubling) there exists a constant C > 0 such that for all x œ X and r > 0,

V (x, 2r) Æ CV (x, r);

(NI) for all –, — > 0 there exists a positive constant c–,— > 0 such that for allr > 0 and for all x, y œ X with d(x, y) < –r,

µ(B(x, –r) fl B(y, —r))V (x, –r) Ø c–,—.

We do not assume that X satisfies any of these assumptions unless mentionedotherwise. However, readers are advised to take (X, d, µ) to be a complete Rie-mannian manifold with its geodesic distance and Riemannian volume if they arenot interested in such technicalities.

It is well-known that doubling implies (HL). However, the converse is nottrue. See for example [37] and [82], where it is shown that (HL) is true for R2

with the Gaussian measure. We will only consider (NI) along with doubling,so we remark that doubling does not imply (NI): one can see this by takingR2 (now with Lebesgue measure) and removing an open strip.4 One can showthat all complete doubling length spaces—in particular, all complete doublingRiemannian manifolds—satisfy (NI).

3Note that this is a strengthening of the usual definition of a proper metric space, as theusual definition does not involve a measure. We have abused notation by using the word ‘proper’in this way, as it is convenient in this context.

4One could instead remove an open bounded region with su�ciently regular boundary, forexample an open square. This yields a connected example.

11

1.3 The basic tent space theory

1.3.1 Initial definitions and consequencesLet X+ denote the ‘upper half-space’ X◊R+, equipped with the product measuredµ(y) dt/t and the product topology. Since X and R+ are metric spaces, withR+ separable, the Borel ‡-algebra on X+ is equal to the product of the Borel‡-algebras on X and R+, and so the product measure on X+ is Borel (see [26,Lemma 6.4.2(i)]).

We say that a subset C µ X+ is cylindrical if it is contained in a cylinder:that is, if there exists x œ X and a, b, r > 0 such that C µ B(x, r) ◊ (a, b).Note that cylindricity is equivalent to boundedness when X+ is equipped withan appropriate metric, and that compact subsets of X+ are cylindrical.

Cones and tents are defined as usual: for each x œ X and – > 0, the cone ofaperture – with vertex x is the set

�–(x) := {(y, t) œ X+ : y œ B(x, –t)}.

For any subset F µ X we write

�–(F ) :=€

xœF

�–(x).

For any subset O µ X, the tent of aperture – over O is defined to be the set

T –(O) := (�–(Oc))c.

WritingFO(y, t) := dist(y, Oc)

t= t≠1 inf

xœOc

d(y, x),

one can check that T –(O) = F ≠1O ([–, Œ)). Since FO is continuous (due to the

continuity of dist(·, Oc)), we find that tents are measurable, and so it follows thatcones are also measurable.

Let F µ X be such that O := F c has finite measure. Given “ œ (0, 1), wesay that a point x œ X has global “-density with respect to F if for all balls B

containing x,µ(B fl F )

µ(B) Ø “.

We denote the set of all such points by F ú“ , and define Oú

“ := (F ú“ )c. An important

fact here is the equality

Oú“ = {x œ X : M(1O)(x) > 1 ≠ “},

12

where 1O is the indicator function of O. We emphasise that M denotes theuncentred maximal operator. When O is open (i.e. when F is closed), this showsthat O µ Oú

“ and hence that F ú“ µ F . Furthermore, the function M(1O) is

lower semicontinuous whenever 1O is locally integrable (which is always true,since we assumed O has finite measure), which implies that F ú

“ is closed (hencemeasurable) and that Oú

“ is open (hence also measurable). Note that if X isdoubling, then since M is of weak-type (1, 1), we have that

µ(Oú“) .“,X µ(O).

Remark 1.3.1. In our definition of points of “-density, we used balls containingx rather than balls centred at x (as is usually done). This is done in order toavoid using the centred maximal function, which may not be measurable withoutassuming continuity of the volume function V (x, r).

Here we find it convenient to introduce the notion of the –-shadow of a subsetof X+. For a subset C µ X+, we define the –-shadow of C to be the set

S–(C) := {x œ X : �–(x) fl C ”= ?}.

Shadows are always open, for if A µ X+ is any subset, and if x œ S–(A),then there exists a point (z, tz) œ �–(x) fl A, and one can easily show thatB(x, –tz ≠ d(x, z)) is contained in S–(A).

The starting point of the tent space theory is the definition of the operatorsA–

q and C–q . For q œ (0, Œ), the former is usually defined for measurable functions

f on Rn+1+ (with values in R or C, depending on context) by

A–q (f)(x)q :=

¨�–(x)

|f(y, t)|q d⁄(y) dt

tn+1

where x œ Rn and ⁄ is Lebesgue measure. There are four reasonable ways togeneralise this definition to our possibly non-doubling metric measure space X:5

these take the form

A–q (f)(x)q :=

¨�–(x)

|f(y, t)|q dµ(y)V (a, bt)

dt

t

where a œ {x, y} and b œ {1, –}. In all of these definitions, if a function f on X+

is supported on a subset C µ X+, then A–q (f) is supported on S–(C); we will

use this fact repeatedly in what follows. Measurability of A–q (f)(x) in x when

5We do not claim that these are the only reasonable generalisations.

13

a = y follows from Lemma 1.4.6 in the Appendix; the choice a = x can be takencare of with a straightforward modification of this lemma. The choice a = x,b = 1 appears in [13, 81], and the choice a = y, b = 1 appears in [63, §3]. Thesedefinitions all lead to equivalent tent spaces when X is doubling. We will takea = y, b = – in our definition, as it leads to the following fundamental technique,which works with no geometric assumptions on X.

Lemma 1.3.2 (Averaging trick). Let – > 0, and suppose � is a nonnegativemeasurable function on X+. Then

ˆX

¨�–(x)

�(y, t) dµ(y)V (y, –t)

dt

tdµ(x) =

¨X+

�(y, t) dµ(y) dt

t.

Proof. This is a straightforward application of Fubini–Tonelli’s theorem, whichwe present explicitly due to its importance in what follows:

ˆX

¨�–(x)

�(y, t) dµ(y)V (y, –t)

dt

tdµ(x)

=ˆ

X

ˆ Œ

0

ˆX

1B(x,–t)(y)�(y, t) dµ(y)V (y, –t)

dt

tdµ(x)

=ˆ Œ

0

ˆX

ˆX

1B(y,–t)(x) dµ(x) �(y, t) dµ(y)V (y, –t)

dt

t

=ˆ Œ

0

ˆX

V (y, –t)V (y, –t)�(y, t) dµ(y) dt

t

=¨

X+�(y, t) dµ(y) dt

t.

We will also need the following lemma in order to prove that our tent spacesare complete. Here we need to make some geometric assumptions.

Lemma 1.3.3. Let X be proper or doubling. Let p, q, – > 0, let K µ X+ becylindrical, and suppose f is a measurable function on X+. Then

------A–

q (1Kf)------Lp(X)

. ||f ||Lq(K) .------A–

q (f)------Lp(X)

, (1.2)

with implicit constants depending on p, q, –, and K.

Proof. WriteK µ B(x, r) ◊ (a, b) =: C

14

for some x œ X and a, b, r > 0. We claim that there exist constants c0, c1 > 0such that for all (y, t) œ C,

c0 Æ V (y, –t) Æ c1.

If X is proper, this is an immediate consequence of the lower semicontinuity ofthe ball volume function (recall that we are assuming this whenever we assumeX is proper) and the compactness of the closed cylinder B(x, r) ◊ [a, b]. If X isdoubling, then we argue as follows. Since V (y, –t) is increasing in t, we have that

min(y,t)œC

V (y, –t) Ø minyœB(x,r)

V (y, –a)

andmax

(y,t)œCV (y, –t) Æ max

yœB(x,r)V (y, –b).

By the argument in the proof of Lemma 1.4.4 (in particular, by (1.16)), thereexists c0 > 0 such that

minyœB(x,r)

V (y, –a) Ø c0.

Furthermore, sinceV (y, –b) Æ V (x, –b + r)

for all y œ B(x, r), we have that

maxyœB(x,r)

V (y, –b) Æ V (x, –b + r) =: c1,

proving the claim.To prove the first estimate of (1.2), write

------A–

q (1Kf)------Lp(X)

=Q

aˆ

S–(K)

A¨�–(x)

1K(y, t)|f(y, t)|q dµ(y)V (y, –t)

dt

t

B p

q

dµ(x)R

b

1p

.c0,q

Q

aˆ

S–(K)

A¨K

|f(y, t)|q dµ(y) dt

t

B p

q

dµ(x)R

b

1p

.K ||f ||Lq(K) .

To prove the second estimate, first choose finitely many points (xn)Nn=1 such

that

B(x, r) µN€

n=1B(xn, –a/2)

15

using either compactness of B(x, r) (in the proper case) or doubling.6 WriteBn := B(xn, –a/2). We then have

A¨K

|f(y, t)|q dµ(y) dt

t

B 1q

.c1

A¨K

Nÿ

n=11B

n

(y)|f(y, t)|q dµ(y)V (y, –t)

dt

t

B 1q

.X,q

Nÿ

n=1

A¨K

1Bn

(y)|f(y, t)|q dµ(y)V (y, –t)

dt

t

B 1q

.

If x, y œ Bn, then d(x, y) < –a < –t (since t > a), and so¨K

1Bn

(y)|f(y, t)|q dµ(y)V (y, –t)

dt

tƨ

�–(x)|f(y, t)|q dµ(y)

V (y, –t)dt

t. (1.3)

We then haveNÿ

n=1

A¨K

1Bn

(y)|f(y, t)|q dµ(y)V (y, –t)

dt

t

B1/q

=Nÿ

n=1

Q

aˆ

Bn

A¨K

1Bn

(y)|f(y, t)|q dµ(y)V (y, –t)

dt

t

Bp/q

dµ(x)R

b1/p

ÆNÿ

n=1

Q

aˆ

Bn

A¨�–(x)

|f(y, t)|q dµ(y)V (y, –t)

dt

t

Bp/q

dµ(x)R

b1/p

Æ N3

maxn

µ(Bn)≠1/p4 Q

aˆ

X

A¨�–(x)

|f(y, t)|q dµ(y)V (y, –t)

dt

t

Bp/q

dµ(x)R

b1/p

.K,p,–

------A–

q (f)------Lp(X)

,

which completes the proof.

As usual, with – > 0 and p, q œ (0, Œ), we define the tent space (quasi-)normof a measurable function f on X+ by

||f ||T p,q,–(X) :=------A–

q (f)------Lp(X)

,

and the tent space T p,q,–(X) to be the (quasi-)normed vector space consisting ofall such f (defined almost everywhere) for which this quantity is finite.Remark 1.3.4. One can define the tent space as either a real or complex vectorspace, according to one’s own preference. We will implicitly work in the complexsetting (so our functions will always be C-valued). Apart from complex interpo-lation, which demands that we consider complex Banach spaces, the di�erence isimmaterial.

6In the doubling case, this is a consequence of what is usually called ‘geometric doubling’.A proof that this follows from the doubling condition can be found in [34, §III.1].

16

Proposition 1.3.5. Let X be proper or doubling. For all p, q, – œ (0, Œ), thetent space T p,q,–(X) is complete and contains Lq

c(X+) (the space of functionsf œ Lq(X+) with cylindrical support) as a dense subspace.

Proof. Let (fn)nœN be a Cauchy sequence in T p,q,–(X). Then by Lemma 1.3.3,for every cylindrical subset K µ X+ the sequence (1Kfn)nœN is Cauchy in Lq(K).We thus obtain a limit

fK := limnæŒ

1Kfn œ Lq(K)

for each K. If K1 and K2 are two cylindrical subsets of X+, then fK1 |K1flK2 =fK2 |K1flK2 , so by making use of an increasing sequence {Km}mœN of cylindricalsubsets of X+ whose union is X+ (for example, we could take Km := B(x, m) ◊(1/m, m) for some x œ X) we obtain a function f œ Lq

loc(X+) with f |Km

= fKm

for each m œ N.7 This is our candidate limit for the sequence (fn)nœN.To see that f lies in T p,q,–(X), write for any m, n œ N

||1Km

f ||T p,q,–(X) .p,q ||1Km

(f ≠ fn)||T p,q,–(X) + ||1Km

fn||T p,q,–(X)

Æ Cp,q,–,X,m ||f ≠ fn||Lq(Km

) + ||fn||T p,q,–(X) ,

the (p, q)-dependence in the first estimate being relevant only for p < 1 or q < 1,and the second estimate coming from Lemma 1.3.3. Since the sequence (fn)nœN

converges to 1Km

f in Lq(Km) and is Cauchy in T p,q,–(X), we have that

||1Km

f ||T p,q,–(X) . supnœN

||fn||T p,q,–(X)

uniformly in m. Hence ||f ||T p,q,–(X) is finite.We now claim that for all Á > 0 there exists m œ N such that for all su�ciently

large n œ N, we have ------1Kc

m

(fn ≠ f)------T p,q,–(X)

Æ Á.

Indeed, since the sequence (fn)nœN is Cauchy in T p,q,–(X), there exists N œ Nsuch that for all n, nÕ Ø N we have ||fn ≠ fnÕ||T p,q,–(X) < Á/2. Furthermore, since

limmæŒ

------1Kc

m

(fN ≠ f)------T p,q,–(X)

= 0

by the Dominated Convergence Theorem, we can choose m such that------1Kc

m

(fN ≠ f)------T p,q,–(X)

< Á/2.

7We interpret ‘locally integrable on X+’ as meaning ‘integrable on all cylinders’, rather than‘integrable on all compact sets’.

17

Then for all n Ø N ,------1Kc

m

(fn ≠ f)------T p,q,–(X)

.p,q

------1Kc

m

(fn ≠ fN)------T p,q,–(X)

+------1Kc

m

(fN ≠ f)------T p,q,–(X)

Æ ||fn ≠ fN ||T p,q,–(X) +------1Kc

m

(fN ≠ f)------T p,q,–(X)

< Á,

proving the claim.Finally, by the previous remark, for all Á > 0 we can find m such that for all

su�ciently large n œ N we have

||fn ≠ f ||T p,q,–(X) .p,q ||1Km

(fn ≠ f)||T p,q,–(X) +------1Kc

m

(fn ≠ f)------T p,q,–(X)

< ||1Km

(fn ≠ f)||T p,q,–(X) + Á

Æ C(p, q, –, X, m) ||fn ≠ f ||Lq(Km

) + Á.

Taking the limit of both sides as n æ Œ, we find that limnæŒ fn = f in T p,q,–(X),and therefore T p,q,–(X) is complete.

To see that Lqc(X+) is dense in T p,q,–(X), simply write f œ T p,q,–(X) as the

pointwise limitf = lim

næŒ1K

n

f.

By the Dominated Convergence Theorem, this convergence holds in T p,q,–(X).

We note that Lemma 1.3.2 implies that in the case where p = q, we haveT p,p,–(X) = Lp(X+) for all – > 0.

In the same way as Lemma 1.3.2, we can prove the analogue of [33, Lemma1].

Lemma 1.3.6 (First integration lemma). For any nonnegative measurable func-tion � on X+, with F a measurable subset of X and – > 0,ˆ

F

¨�–(x)

�(y, t) dµ(y) dt dµ(x) ƨ

�–(F )�(y, t)V (y, –t) dµ(y) dt.

Remark 1.3.7. There is one clear disadvantage of our choice of tent space norm:it is no longer clear that

||·||T p,q,–(X) Æ ||·||T p,q,—(X) (1.4)

when – < —. In fact, this may not even be true for general non-doubling spaces.This is no great loss, since for doubling spaces we can revert to the ‘original’ tentspace norm (with a = x and b = 1) at the cost of a constant depending only onX, and for this choice of norm (1.4) is immediate.

18

In order to define the tent spaces T Œ,q,–(X), we need to introduce the operatorC–

q . For measurable functions f on X+, we define

C–q (f)(x) := sup

B–x

A1

µ(B)

¨T –(B)

|f(y, t)|q dµ(y) dt

t

B 1q

,

where the supremum is taken over all balls containing x. Since C–q (f) is lower

semicontinuous (see Lemma 1.4.7), C–q (f) is measurable. For functions f on X+

we define the (quasi-)norm ||·||T Œ,q,–(X) by

||f ||T Œ,q,–(X) :=------C–

q (f)------LŒ(X)

,

and the tent space T Œ,q,–(X) as the (quasi-)normed vector space of measurablefunctions f on X+, defined almost everywhere, for which ||f ||T Œ,q,–(X) is finite.The proof that T Œ,q,–(X) is a (quasi-)Banach space is similar to that of Propo-sition 1.3.5 once we have established the following analogue of Lemma 1.3.3.

Lemma 1.3.8. Let q, – > 0, let K µ X+ be cylindrical, and suppose f is ameasurable function on X+. Then

||f ||Lq(K) . ||f ||T Œ,q,–(X) , (1.5)

with implicit constant depending only on –, q, and K (but not otherwise on X).Furthermore, if X is proper or doubling, then we also have

||1Kf ||T Œ,q,–(X) . ||f ||Lq(K) ,

again with implicit constant depending only on –, q, and K.

Proof. We use Lemma 1.4.4. To prove the first estimate, for each Á > 0 we canchoose a ball BÁ such that T –(BÁ) ∏ K and µ(BÁ) < —1(K) + Á. Then

||f ||Lq(K) Æ------1T –(B

Á

)f------Lq(X+)

= µ(BÁ)1q µ(BÁ)≠ 1

q

------1T –(B

Á

)f------Lq(X+)

Æ (—1(K) + Á)!q ||f ||T Œ,q,–(X) .

In the final line we used that µ(BÁ) > 0 to conclude that

µ(BÁ)≠1/q------1T –(B

Á

)f------Lq(X+)

is less than the essential supremum of C–q (f). Since Á > 0 was arbitrary, we have

the first estimate.

19

For the second estimate, assuming that X is proper or doubling, observe that

||1Kf ||T Œ,q,–(X) Æ supBµX

A1

µ(B)

¨T –(B)flK

|f(y, t)|q dµ(y) dt

t

B 1q

ÆA

1—0(K)

¨K

|f(y, t)|q dµ(y)dt

t

B 1q

= —0(K)≠ 1q ||f ||Lq(K) ,

completing the proof.

Remark 1.3.9. In this section we did not impose any geometric conditions on ourspace X besides our standing assumptions on the measure µ and the proper-ness assumption (in the absence of doubling). Thus we have defined the tentspace T p,q,–(X) in considerable generality. However, what we have defined is aglobal tent space, and so this concept may not be inherently useful when X isnon-doubling. Instead, our interest is to determine precisely where geometricassumptions are needed in the tent space theory.

1.3.2 Duality, the vector-valued approach, and complexinterpolation

Midpoint results

The geometric assumption (HL) from Section 1.2 now comes into play. For r Ø 0,we denote the Hölder conjugate of r by rÕ := r/(r ≠ 1) with rÕ = Œ when r = 1.

Proposition 1.3.10. Suppose that X is either proper or doubling, and satisfiesassumption (HL). Then for p, q œ (1, Œ) and – > 0, the pairing

Èf, gÍ :=¨

X+f(y, t)g(y, t) dµ(y) dt

t(f œ T p,q,–(X), g œ T pÕ,qÕ,–(X))

realises T pÕ,qÕ,–(X) as the Banach space dual of T p,q,–(X), up to equivalence ofnorms.

This is proved in the same way as in [33]. We provide the details in the interestof self-containment.

Proof. We first remark that if p = q, the duality statement is a trivial consequenceof the equality T p,p,–(X) = Lp(X+).

20

In general, suppose f œ T p,q,–(X) and g œ T pÕ,qÕ,–(X). Then by the averagingtrick and Hölder’s inequality, we have

|Èf, gÍ| ƈ

X

¨�–(x)

|f(y, t)g(y, t)| dµ(y)V (y, –t)

dt

tdµ(x)

ƈ

X

A–q (f)(x)A–

qÕ(g)(x) dµ(x)

Æ ||f ||T p,q,–(X) ||g||T p

Õ,q

Õ,–(X) . (1.6)

Thus every g œ T pÕ,qÕ,–(X) induces a bounded linear functional on T p,q,–(X) viathe pairing È·, ·Í, and so T pÕ,qÕ,–(X) µ (T p,q,–(X))ú.

Conversely, suppose ¸ œ (T p,q,–(X))ú. If K µ X+ is cylindrical, then bythe properness or doubling assumption, we can invoke Lemma 1.3.3 to showthat ¸ induces a bounded linear functional ¸K œ (Lq(K))ú, which can in turnbe identified with a function gK œ LqÕ(K). By covering X+ with an increasingsequence of cylindrical subsets, we thus obtain a function g œ LqÕ

loc(X+) such thatg|K = gK for all cylindrical K µ X+.

If f œ Lq(X+) is cylindrically supported, then we have

¨X+

f(y, t)g(y, t) dµ(y) dt

t=¨

supp f

f(y, t)gsupp f (y, t) dµ(y) dt

t

= ¸supp f (f)= ¸(f), (1.7)

recalling that f œ T p,q,–(X) by Lemma 1.3.3. Since the cylindrically supportedLq(X+) functions are dense in T p,q,–(X), the representation (1.7) of ¸(f) in termsof g is valid for all f œ T p,q,–(X) by dominated convergence and the inequality(1.6), provided we show that g is in T pÕ,qÕ,–(X).

Now suppose p < q. We will show that g lies in T pÕ,qÕ,–(X), thus showingdirectly that (T p,q,–(X))ú is contained in T pÕ,qÕ,–(X). It su�ces to show this forgK , where K µ X+ is an arbitrary cylindrical subset, provided we obtain anestimate which is uniform in K. We estimate

||gK ||qÕ

T p

Õ,q

Õ,–(X) =

------A–

qÕ(gK)qÕ------Lp

Õ/q

Õ (X)

by duality. Let  œ L(pÕ/qÕ)Õ(X) be nonnegative, with ||Â||L(p

Õ/q

Õ)Õ (X) Æ 1. Then by

21

Fubini–Tonelli’s theorem,ˆX

A–qÕ(gK)(x)qÕ

Â(x) dµ(x)

=ˆ

X

¨X+

1B(y,–t)(x)|gK(y, t)|qÕ dµ(y)V (y, –t)

dt

tÂ(x) dµ(x)

=ˆ Œ

0

ˆX

1V (y, –t)

ˆB(y,–t)

Â(x) dµ(x) |gK(y, t)|qÕdµ(y) dt

t

=¨

X+M–tÂ(y)|gK(y, t)|qÕ

dµ(y) dt

t,

where Ms is the averaging operator defined for y œ X and s > 0 by

MsÂ(y) := 1V (y, s)

ˆB(y,s)

Â(x) dµ(x).

Thus we can write formallyˆX

A–qÕ(gK)(x)qÕ

Â(x) dµ(x) = ÈfÂ, gKÍ, (1.8)

where we define

fÂ(y, t) :=Y]

[M–tÂ(y)gK(y, t)qÕ/2

gK(y, t)(qÕ/2)≠1 when gK(y, t) ”= 0,0 when gK(y, t) = 0,

noting that gK(y, t)(qÕ/2)≠1 is not defined when gK(y, t) = 0 and qÕ < 2. However,the equality (1.8) is not valid until we show that f lies in T p,q,–(X). To this end,estimate

A–q (fÂ) Æ

A¨�–(x)

M–tÂ(y)q|gK(y, t)|q(qÕ≠1) dµ(y)V (y, –t)

dt

t

B 1q

ÆA¨

�–(x)MÂ(x)q|gK(y, t)|qÕ dµ(y)

V (y, –t)dt

t

B 1q

= MÂ(x)A–qÕ(gK)(x)qÕ/q.

Taking r such that 1/p = 1/r + 1/(pÕ/qÕ)Õ and using (HL), we then have------A–

q (fÂ)------Lp(X)

Æ------(MÂ)A–

qÕ(gK)qÕ/q------Lp(X)

Æ ||MÂ||L(p

Õ/q

Õ)Õ (X)

------A–

qÕ(gK)qÕ/q------Lr(X)

.X ||Â||L(p

Õ/q

Õ)Õ (X)

------A–

qÕ(gK)------qÕ/q

Lrq

Õ/q(X)

Æ------A–

qÕ(gK)------qÕ/q

Lrq

Õ/q(X)

.

22

One can show that rqÕ/q = pÕ, and so f is in T p,q,–(X) by Lemma 1.3.3. By(1.8), taking the supremum over all  under consideration, we can write

||gK ||qÕ

T p

Õ,q

Õ,–(X) Æ ||¸|| ||fÂ||T p,q,–(X)

.X ||¸|| ||gK ||qÕ/q

T p

Õ,q

Õ,–(X) ,

and consequently, using that ||gK ||T p

Õ,q

Õ,–(X) < Œ,

||gK ||T p

Õ,q

Õ,–(X) .X ||¸|| .

Since this estimate is independent of K, we have shown that g œ T pÕ,qÕ,–(X), andtherefore that (T p,q,–(X))ú is contained in T pÕ,qÕ,–(X). This completes the proofwhen p < q.

To prove the statement for p > q, it su�ces to show that the tent spaceT pÕ,qÕ,–(X) is reflexive. Thanks to the Eberlein–Smulian theorem (see [1, Corollary1.6.4]), this is equivalent to showing that every bounded sequence in T pÕ,qÕ,–(X)has a weakly convergent subsequence.

Let {fn}nœN be a sequence in T pÕ,qÕ,–(X) with ||fn||T p

Õ,q

Õ,–(X) Æ 1 for all n œ

N. Then by Lemma 1.3.3, for all cylindrical K µ X+ the sequence {fn}nœN isbounded in LqÕ(K), and so by reflexivity of LqÕ(K) we can find a subsequence{fn

j

}jœN which converges weakly in LqÕ(K). We will show that this subsequencealso converges weakly in T pÕ,qÕ,–(X).

Let ¸ œ (T pÕ,qÕ,–(X))ú. Since pÕ < qÕ, we have already shown that there existsa function g œ T p,q,–(X) such that ¸(f) = Èf, gÍ. For every Á > 0, we can find acylindrical set KÁ µ X+ such that

||g ≠ 1KÁ

g||T p,q,–(X) Æ Á.

Thus for all i, j œ N and for all Á > 0 we have

¸(fni

) ≠ ¸(fnj

) = Èfni

≠ fnj

, 1KÁ

gÍ + Èfni

≠ fnj

, g ≠ 1KÁ

gÍÆ Èfn

i

≠ fnj

, 1KÁ

gÍ

+3

||fni

||T p

Õ,q

Õ,–(X) +

------fn

j

------T p

Õ,q

Õ,–(X)

4||g ≠ 1K

Á

g||T p,q,–

Æ Èfni

≠ fnj

, 1KÁ

gÍ + 2Á.

As i, j æ Œ, the first term on the right hand side above tends to 0, and sowe conclude that {fn

j

}nœN converges weakly in T pÕ,qÕ,–(X). This completes theproof.

23

Remark 1.3.11. As mentioned earlier, property (HL) is weaker than doubling, butthis is still a strong assumption. We note that for Proposition 1.3.10 to hold fora given pair (p, q), the uncentred Hardy–Littlewood maximal operator need onlybe of strong type ((pÕ/qÕ)Õ, (pÕ/qÕ)Õ). Since (pÕ/qÕ)Õ is increasing in p and decreasingin q, the condition required on X is stronger as p æ 1 and q æ Œ.

Given Proposition 1.3.10, we can set up the vector-valued approach to tentspaces (first considered in [44]) using the method of [23]. Fix p œ (0, Œ), q œ(1, Œ), and – > 0. For simplicity of notation, write

Lq–(X+) := Lq

A

X+; dµ(y)V (y, –t)

dt

t

B

.

We define an operator T– : T p,q,–(X) æ Lp(X; Lq–(X+)) from the tent space into

the Lq–(X+)-valued Lp space on X (see [35, §2] for vector-valued Lebesgue spaces)

by settingT–f(x)(y, t) := f(y, t)1�–(x)(y, t).

One can easily check that

||T–f ||Lp(X;Lq

–

(X+)) = ||f ||T p,q,–(X) ,

and so the tent space T p,q,–(X) can be identified with its image under T– inthe space Lp(X; Lq

–(X+)), provided that T–f is indeed a strongly measurablefunction of x œ X. This can be shown for q œ (1, Œ) by recourse to Pettis’measurability theorem [35, §2.1, Theorem 2], which reduces the question to thatof weak measurability of T–f . To prove weak measurability, suppose g œ LqÕ

– (X);then

ÈT–f(x), gÍ =¨

�–(x)f(y, t)g(y, t) dµ(y)

V (y, –t)dt

t,

which is measurable in x by Lemma 1.4.6. Thus T–f is weakly measurable, andtherefore T–f is strongly measurable as claimed.

Now assume p, q œ (1, Œ) and consider the operator �–, sending X+-valuedfunctions on X to C-valued functions on X+, given by

(�–F )(y, t) := 1V (y, –t)

ˆB(y,–t)

F (x)(y, t) dµ(x)

whenever this expression is defined. Using the duality pairing from Proposition1.3.10 and the duality pairing ÈÈ·, ·ÍÍ for vector-valued Lp spaces, for f œ T p,q,–(X)

24

and G œ LpÕ(X; LqÕ– (X+)) we have

ÈÈT–f, GÍÍ =ˆ

X

¨X+

T–f(x)(y, t)G(x)(y, t) dµ(y)V (y, –t)

dt

tdµ(x)

=¨

X+

f(y, t)V (y, –t)

ˆX

1B(y,–t)(x)G(x)(y, t) dµ(x) dµ(y) dt

t

=¨

X+f(y, t)(�–G)(y, t) dµ(y) dt

t

= ÈÈf, �–GÍÍ.

Thus �– maps LpÕ(X; LqÕ– (X+)) to T pÕ,qÕ,–(X), by virtue of being the adjoint of

T–. Consequently, the operator P– := T–�– is bounded from Lp(X; Lq–(X+)) to

itself for p, q œ (1, Œ). A quick computation shows that �–T– = I, so that P–

projects Lp(X; Lq–(X+)) onto T–(T p,q,–(X)). This shows that T–(T p,q,–(X)) is a

complemented subspace of Lp(X; Lq–(X+)). This observation leads to the basic

interpolation result for tent spaces. Here [·, ·]◊ denotes the complex interpolationfunctor (see [22, Chapter 4]).

Proposition 1.3.12. Suppose that X is either proper or doubling, and satisfiesassumption (HL). Then for p0, p1, q0, and q1 in (1, Œ), ◊ œ [0, 1], and – > 0, wehave (up to equivalence of norms)

[T p0,q0,–(X), T p1,q1,–(X)]◊ = T p,q,–(X),

where 1/p = (1 ≠ ◊)/p0 + ◊/p1 and 1/q = (1 ≠ ◊)/q0 + ◊/q1.

Proof. Recall the identification

T r,s,–(X) ≥= T–T r,s,–(X) µ Lr(X; Ls–(X+))

for all r œ (0, Œ) and s œ (1, Œ). Since

[Lp0(X; Lq0– (X+)), Lp1(X; Lq1

– (X+))]◊ = Lp(X; [Lq0– (X+), Lq1

– (X+)]◊)= Lp(X; Lq

–(X+))

applying the standard result on interpolation of complemented subspaces withcommon projections (see [89, Theorem 1.17.1.1]) yields

[T p0,q0,–(X), T p1,q1,–(X)]◊ = Lp(X; Lq–(X+)) fl (T p0,q0,–(X) + T p1,q1,–(X))

= T p,q,–(X).

25

Remark 1.3.13. Since [89, Theorem 1.17.1.1] is true for any interpolation functor(not just complex interpolation), analogues of Proposition 1.3.12 hold for anyinterpolation functor F for which the spaces Lp(X; Lq

–(X+)) form an appropriateinterpolation scale. In particular, Proposition 1.3.12 (appropriately modified)holds for real interpolation.

Remark 1.3.14. Following the first submission of this article, the anonymous ref-eree suggested a more direct proof of Proposition 1.3.12, which avoids inter-polation of complemented subspaces. Since T– acts as an isometry both fromT p0,q0,–(X) to Lp0(X; Lq0

– (X+)) and from T p1,q1,–(X) to Lp1(X; Lq1– (X+)), if f is

in the interpolation space [T p0,q0,–(X), T p1,q1,–(X)]◊, then

||f ||T p,q,–(X) = ||T–f ||Lp(X;Lq

–

(X+)) Æ ||f ||[T p0,q0,–(X),T p1,q1,–(X)]◊

due to the exactness of the complex interpolation functor (and similarly for thereal interpolation functor). Hence [T p0,q0,–(X), T p1,q1,–(X)]◊ µ T p,q,–(X), and thereverse containment follows by duality. We have chosen to include both proofsfor their own intrinsic interest.

Endpoint results

We now consider the tent spaces T 1,q,–(X) and T Œ,q,–(X), and their relation tothe rest of the tent space scale. In this section, we prove the following dualityresult using the method of [33].

Proposition 1.3.15. Suppose X is doubling, and let – > 0 and q œ (1, Œ). Thenthe pairing È·, ·Í of Proposition 1.3.10 realises T Œ,q,–(X) as the Banach space dualof T 1,q,–(X), up to equivalence of norms.

As in [33], we require a small series of definitions and lemmas to prove thisresult. We define truncated cones for x œ X, –, h > 0 by

�–h(x) := �–(x) fl {(y, t) œ X+ : t < h},

and corresponding Lusin operators for q > 0 by

A–q (f |h)(x) :=

Q

a¨

�–

h

(x)|f(y, t)|q dµ(y)

V (y, –t)dt

t

R

b

1q

.

One can show that A–q (f |h) is measurable in the same way as for A–

q (f).

26

Lemma 1.3.16. For each measurable function g on X+, each q œ [1, Œ), andeach M > 0, define

h–g,q,M(x) := sup{h > 0 : A–

q (g|h)(x) Æ MC–q (g)(x)}

for x œ X. If X is doubling, then for su�ciently large M (depending on X, q,and –), whenever B µ X is a ball of radius r,

µ{x œ B : h–g,q,M(x) Ø r} &X,– µ(B).

Proof. Let B µ X be a ball of radius r. Applying Lemmas 1.4.5 and 1.3.6, thedefinition of C–

q , and doubling, we haveˆ

B

A–q (g|r)(x)q dµ(x) =

ˆB

¨�–

r

(x)1T –((2–+1)B)(y, t)|g(y, t)|q dµ(y)

V (y, –t)dt

tdµ(x)

ƈ

B

¨�–(x)

1T –((2–+1)B)(y, t)|g(y, t)|q dµ(y)V (y, –t)

dt

tdµ(x)

ƨ

T –((2–+1)B)|g(y, t)|q dµ(y) dt

t

Æ µ((2– + 1)B) infxœB

C–q (g)(x)q

.X,– µ(B) infxœB

C–q (g)(x)q.

We can estimateˆB

A–q (g|r)(x)q dµ(x) Ø

3M inf

xœBC–

q (g)(x)4q

·

· µ;

x œ B : A–q (g|r)(x) > M inf

xœBC–

q (g)(x)<

,

and after rearranging and combining with the previous estimate we get

M q3

µ(B) ≠ µ{x œ B : A–q (g|r)(x) Æ M inf

xœBC–

q (g)(x)}4.X,– µ(B).

More rearranging and straightforward estimating yields

µ{x œ B : A–q (g|r)(x) Æ MC–

q (g)(x)} Ø (1 ≠ M≠qCX,–)µ(B).

Since h–g,q,M(x) Ø r if and only if A–

q (g|r)(x) Æ MC–q (g)(x) as A–

q (g|h) is increas-ing in h, we can rewrite this as

µ{x œ B : h–g,q,M(x) Ø r} Ø (1 ≠ M≠qCX,–)µ(B).

Choosing M > C1/qX,– completes the proof.

27

Corollary 1.3.17. With X, g, q, and – as in the statement of the previouslemma, there exists M = M(X, q, –) such that whenever � is a nonnegativemeasurable function on X+, we have¨

X+�(y, t)V (y, –t) dµ(y) dt .X,–

ˆX

¨�–

h

–

g,q,M

(x)/–

(x)�(y, t) dµ(y) dt dµ(x).

Proof. This is a straightforward application of Fubini–Tonelli’s theorem alongwith the previous lemma. Taking M su�ciently large, Lemma 1.3.16 gives

¨X+

�(y, t)V (y, –t) dµ(y) dt

.X,–

¨X+

�(y, t)ˆ

{xœB(y,–t):h–

g,q,M(x)Ø–t}dµ(x) dµ(y) dt

=ˆ

X

ˆ h–

g,q,M

(x)/–

0

ˆB(x,–t)

�(y, t) dµ(y) dt dµ(x)

=ˆ

X

¨�–

h

–

g,q,M

(x)/–

(x)�(y, t) dµ(y) dt dµ(x)

as required.

We are now ready for the proof of the main duality result.

Proof of Proposition 1.3.15. First suppose f œ T 1,q,–(X) and g œ T Œ,qÕ,–(X). ByCorollary 1.3.17, there exists M = M(X, q, –) > 0 such that

¨X+

|f(y, t)||g(y, t)| dµ(y) dt

t

.X,–

ˆX

¨�–

h(x)(x)|f(y, t)||g(y, t)| dµ(y)

V (y, –t)dt

tdµ(x),

where h(x) := h–g,qÕ,M(x)/–. Using Hölder’s inequality and the definition of h(x),

we find thatˆ

X

Q

a¨

�–

h(x)(x)|f(y, t)||g(y, t)| dµ(y)

V (y, –t)dt

t

R

b dµ(x)

ƈ

X

A–q (f |h(x))(x)A–

qÕ(g|h(x))(x) dµ(x)

Æ M

ˆX

A–q (f)(x)C–

qÕ(g)(x) dµ(x)

.X,q,– ||f ||T 1,q,–(X) ||g||T Œ,q,–(X) .

28

Hence every g œ T Œ,qÕ,–(X) induces a bounded linear functional on T 1,q,–(X) viathe pairing Èf, gÍ above, and so T Œ,qÕ,–(X) µ (T 1,q,–(X))ú.

Conversely, suppose ¸ œ (T 1,q,–(X))ú. Then as in the proof of Proposition1.3.10, from ¸ we construct a function g œ LqÕ

loc(X+) such that¨X+

f(y, t)g(y, t) dµ(y) dt

t= ¸(f)

for all f œ T 1,q,–(X) with cylindrical support. We just need to show that g is inT Œ,qÕ,–(X). By the definition of the T Œ,qÕ,–(X) norm, it su�ces to estimate

A1

µ(B)

¨T –(B)

|g(y, t)|qÕdµ(y) dt

t

B 1q

Õ

,

where B µ X is an arbitrary ball.For all nonnegative  œ Lq(T –(B)) with ||Â||Lq(T –(B)) Æ 1, using that

S–(T –(B)) = B

we have that

||Â||T 1,q,–(X) =ˆ

B

A–q (Â)(x) dµ(x)

Æ µ(B)1/qÕ ||Â||T q,q,–(X)

= µ(B)1/qÕ ||Â||Lq(X+)

Æ µ(B)1/qÕ.

In particular, Â is in T 1,q,–(X), so we can write¨T –(B)

g dµdt

t= ¸(Â).

Arguing by duality and using the above computation, we then haveA

1µ(B)

¨T –(B)

|g(y, t)|qÕdµ(y) dt

t

B1/qÕ

= µ(B)≠1/qÕ supÂ

¨T –(B)

g dµdt

t

= µ(B)≠1/qÕ supÂ

¸(Â)

Æ µ(B)≠1/qÕ ||¸|| ||Â||T 1,q,–(X)

Æ ||¸|| ,

where the supremum is taken over all  described above. Now taking the supre-mum over all balls B µ X, we find that

||g||T Œ,q

Õ,–(X) Æ ||¸|| ,

which completes the proof that (T 1,q,–(X))ú µ T Œ,qÕ,–(X).

29

Once Proposition 1.3.15 is established, we can obtain the full scale of inter-polation using the ‘convex reduction’ argument of [23, Theorem 3] and Wol�’sreiteration theorem (see [92] and [54]).

Proposition 1.3.18. Suppose that X is doubling. Then for p0, p1 œ [1, Œ] (notboth equal to Œ), q0 and q1 in (1, Œ), ◊ œ [0, 1], and – > 0, we have (up toequivalence of norms)

[T p0,q0,–(X), T p1,q1,–(X)]◊ = T p,q,–(X),

where 1/p = (1 ≠ ◊)/p0 + ◊/p1 and 1/q = (1 ≠ ◊)/q0 + ◊/q1.

Proof. First we will show that

[T 1,q0,–(X), T p1,q1,–(X)]◊ ∏ T p,q,–(X). (1.9)

Suppose f œ T p,q,–(X) is a cylindrically supported simple function. Then thereexists another cylindrically supported simple function g such that f = g2. Then

||f ||T p,q,–(X) = ||g||2T 2p,2q,–(X) ,

and so g is in T 2p,2q,–(X). By Proposition 1.3.12 we have the identification

T 2p,2q,–(X) = [T 2,2q0,–(X), T 2p1,2q1,–(X)]◊ (1.10)

up to equivalence of norms, and so by the definition of the complex interpolationfunctor (see Section 1.4.3), there exists for each Á > 0 a function

GÁ œ F(T 2,2q0,–(X), T 2p1,2q1,–(X))

such that GÁ(◊) = g and

||GÁ||F(T 2,2q0,–(X),T 2p1,2q1,–(X) Æ (1 + Á) ||g||[T 2,2q0,–(X),T 2p1,2q1,–(X)]◊

ƒ (1 + Á) ||g||T 2p,2q,–(X) ,

the implicit constant coming from the norm equivalence (1.10). Define FÁ := G2Á.

Then we haveFÁ œ F(T 1,q0,–(X), T p1,q1,–(X)),

with

||FÁ||F(T 1,q0,–(X),T p1,q1,–(X)) = ||GÁ||2F(T 2,2q0,–(X),T 2p1,2q1,–(X))

. (1 + Á)2 ||g||2T 2p,2q,–(X)

= (1 + Á)2 ||f ||T p,q,–(X) .

30

Therefore||f ||[T 1,q0,–(X),T p1,q1,–(X)]

◊

. ||f ||T p,q,–(X) ,

and so the inclusion (1.9) follows from the fact that cylindrically supported simplefunctions are dense in T p,q,–(X).

By the duality theorem [22, Corollary 4.5.2] for interpolation (using reflexivityof T p1,q1,–(X)), the inclusion (1.9), and Propositions 1.3.10 and 1.3.15, we have

[T pÕ1,qÕ

1,–(X), T Œ,qÕ0,–(X)]1≠◊ µ T pÕ,qÕ,–(X).

Therefore we have the containment

[T p0,q0,–(X), T Œ,q1,–(X)]◊ µ T p,q,–(X). (1.11)

The reverse containment can be obtained from

[T 1,q0,–(X), T p1,q1,–(X)]◊ µ T p,q,–(X) (1.12)

(for p1, q0, q1 œ (1, Œ)) by duality. The containment (1.12) can be obtained asin Remark 1.3.14, with p0 = 1 not changing the validity of this method.8

Finally, it remains to consider the case when p0 = 1 and p1 = Œ. Thisis covered by Wol� reiteration. Set A1 = T 1,q0,–(X), A2 = T p,q,–(X), A3 =T p+1,q3,–(X), and A4 = T Œ,q1,–(X) for an approprate choice of q3.9 Then foran appropriate index ÷, we have [A1, A3]◊/÷ = A2 and [A2, A4](÷≠◊)/(1≠◊) = A3.Therefore by Wol� reiteration, we have [A1, A4]◊ = A2; that is,

[T 1,q0,–(X), T Œ,q1,–(X)]◊ = T p,q,–(X).

This completes the proof.

Remark 1.3.19. Note that doubling is not explicitly used in the above proof; itis only required to the extent that it is needed to prove Propositions 1.3.10 and1.3.15 (as Proposition 1.3.12 follows from 1.3.10). If these propositions could beproven under some assumptions other than doubling, then it would follow thatProposition 1.3.18 holds under these assumptions.

Remark 1.3.20. The proof of [33, Lemma 5], which amounts to proving the con-tainment (1.9), contains a mistake which is seemingly irrepairable without re-sorting to more advanced techniques. This mistake appears on page 323, line

8We thank the anonymous referee once more for this suggestion.9More precisely, we need to take 1/q

3

= (1 ≠ 1/pÕ)/q0

+ (1/pÕ)/q1

.

31

-3, when it is stated that “A(fk) is supported in Oúk ≠ Ok+1” (and in particular,

that A(fk) is supported in Ock+1). However (reverting to our notation), since

fk := 1T ((Ok

)ú“

)\T ((Ok+1)ú

“

)f , A12(fk) is supported on

S1(T ((Ok)ú“) \ T ((Ok+1)ú

“)) = (Ok)ú“

and we cannot conclude that A12(fk) is supported away from Ok+1. Simple 1-

dimensional examples can be constructed which show that this is false in general.Hence the containment (1.9) is not fully proven in [33]; the first valid proof inthe Euclidean case that we know of is in [23] (the full range of interpolation isnot obtained in [44].)

1.3.3 Change of apertureUnder the doubling assumption, the change of aperture result can be provenwithout assuming (NI) by means of the vector-valued method. The proof is acombination of the techniques of [44] and [23].

Proposition 1.3.21. Suppose X is doubling. For –, — œ (0, Œ) and p, q œ(0, Œ), the tent space (quasi-)norms ||·||T p,q,–(X) and ||·||T p,q,—(X) are equivalent.

Proof. First suppose p, q œ (1, Œ). Since X is doubling, we can replace ourdefinition of A–

q with the definition

A–q (f)(x)q :=

¨�–(x)

|f(y, t)|q dµ(y)V (y, t)

dt

t;

using the notation of Section 1.3.1, this is the definition with a = y and b = 1.Having made this change, the vector-valued approach to tent spaces (see Section1.3.2) transforms as follows. The tent space T p,q,–(X) now embeds isometricallyinto Lp(X; Lq

1(X+)) via the operator T– defined, as before, by

T–f(x)(y, t) := f(y, t)1�–(x)(y, t)

for f œ T p,q,–(X). The adjoint of T– is the operator �–, now defined by

(�–G)(y, t) := 1V (y, t)

ˆB(y,–t)

G(z)(y, t) dµ(z)

for G œ Lp(X; Lq1(X+)). The composition P– := T–�– is then a bounded projec-

tion from Lp(X; Lq1(X+)) onto T–T p,q,–(X), and can be written in the form

P–G(x)(y, t) = 1�–(x)(y, t)V (y, t)

ˆB(y,–t)

G(z)(y, t) dµ(z).

32

For f œ T p,q,–(X), we can easily compute

P—T–f(x)(y, t) = T—f(x)(y, t)V (y, min(–, —)t)V (y, t) . (1.13)

Without loss of generality, suppose — > –. Then we obviously have

||·||T p,q,–(X) .q,–,—,X ||·||T p,q,—(X)

by Remark 1.3.7. It remains to show that

||·||T p,q,—(X) .p,q,–,—,X ||·||T p,q,–(X) . (1.14)

From (1.13) and doubling, for f œ T p,q,–(X) we have that

T—f(x)(y, t) .X,– P—T–f(x)(y, t),

and so we can write

||f ||T p,q,—(X) = ||T—f ||Lp(X;Lq

1(X+))

.X,– ||P—T–f ||Lp(X;Lq

1(X+))

Æ ||P—||L(Lp(X;Lq

1(X+))) ||T–f ||Lp(X;Lq

1(X+))

.p,q,—,X ||f ||T p,q,–(X)

since P— is a bounded operator on Lp(X; Lq1(X+)). This shows (1.14), and com-

pletes the proof for p, q œ (1, Œ).Now suppose that at least one of p and q is not in (1, Œ), and suppose f œ

T p,q,–(X) is a cylindrically supported simple function. Choose an integer M suchthat both Mp and Mq are in (1, Œ). Then there exists a cylindrically supportedsimple function g with gM = f . We then have

||f ||1/MT p,q,–(X) =

------gM

------1/M

T p,q,–(X)

= ||g||T Mp,Mq,–(X)

ƒp,q,–,—,X ||g||T Mp,Mq,—(X)

= ||f ||1/MT p,q,—(X) ,

and so the result is true for cylindrically supported simple functions, with animplicit constant which does not depend on the support of such a function. Sincethe cylindrically supported simple functions are dense in T p,q,–(X), the proof iscomplete.

33

Remark 1.3.22. Written more precisely, with p, q œ (0, Œ) and — < 1, the in-equality (1.14) is of the form

||·||T p,q,1(X) .p,q,X sup(y,t)œX+

AV (y, t)

V (y, —t)

BM

||·||T p,q,—(X) .

where M is such that Mp, Mq œ (1, Œ).

1.3.4 Relations between A and C

Again, this proposition follows from the methods of [33].

Proposition 1.3.23. Suppose X satisfies (HL), and suppose 0 < q < p < Œand – > 0. Then ---

---C–q (f)

------Lp(X)

.p,q,X

------A–

q (f)------Lp(X)

.

Proof. Let B µ X be a ball. Then by Fubini–Tonelli’s theorem, and using thatS–(T –(B)) = B,

1µ(B)

¨T –(B)

|f(y, t)|q dµ(y) dt

t

= 1µ(B)

¨T –(B)

|f(y, t)|qV (y, –t)

ˆB(y,–t)

dµ(x) dµ(y) dt

t

= 1µ(B)

ˆX

¨T –(B)

1B(y,–t)(x)|f(y, t)|q dµ(y)V (y, –t)

dt

tdµ(x)

= 1µ(B)

ˆB

¨T –(B)

1B(x,–t)(y)|f(y, t)|q dµ(y)V (y, –t)

dt

tdµ(x)

Æ 1µ(B)

ˆB

¨X+

1B(x,–t)(y)|f(y, t)|q dµ(y)V (y, –t)

dt

tdµ(x)

= 1µ(B)

ˆB

A–q (f)(x)q dµ(x).

Now fix x œ X and take the supremum of both sides of this inequality over allballs B containing x. We find that

C–q (f)(x)q Æ M(A–

q (f)q)(x).

Since p/q > 1, we can apply (HL) to get------C–

q (f)------Lp(X)

Æ------M(A–

q (f)q)1/q------Lp(X)

=------M(A–

q (f)q)------1/q

Lp/q(X)

.p,q,X

------A–

q (f)q------1/q

Lp/q(X)

=------A–

q (f)------Lp(X)

34

as desired.

Remark 1.3.24. If X is doubling, and if p, q œ (0, Œ), then for – > 0 we also havethat ---

---A–q (f)

------Lp(X)

.p,q,X

------C–

q (f)------Lp(X)

.

This can be proven as in [33, §6], completely analogously to the proofs above.

1.4 Appendix: Assorted lemmas and notation

1.4.1 Tents, cones, and shadows

Lemma 1.4.1. Suppose A and B are subsets of X, with A open, and supposeT –(A) µ T –(B). Then A µ B.

Proof. Suppose x œ A. Then dist(x, Ac) > 0 since A is open, and so dist(x, Ac) >

–t for some t > 0. Hence (x, t) œ T –(A) µ T –(B), so that dist(x, Bc) > –t > 0.Therefore x œ B.

Lemma 1.4.2. Let C µ X+ be cylindrical, and suppose – > 0. Then S–(C) isbounded.

Proof. Write C µ B(x, r) ◊ (a, b) for some x œ X and r, a, b > 0. Then S–(C) µS–(B(x, r) ◊ (a, b)), and one can easily show that

S–(B(x, r) ◊ (a, b)) µ B(x, r + –b),

showing the boundedness of S–(C).

Lemma 1.4.3. Let C µ X+, and suppose – > 0. Then T –(S–(C)) is the minimal–-tent containing C, in the sense that T –(S) ∏ C for some S µ X implies thatT –(S–(C)) µ T –(S).

Proof. A straightforward set-theoretic manipulation shows that C is contained inT –(S–(C)). We need to show that S–(C) is minimal with respect to this property.

Suppose that S µ X is such that C µ T –(S), and suppose (w, tw) is inT –(S–(C)). With the aim of showing that dist(w, Sc) > –tw, suppose that y œ Sc.Then �–(y) fl T –(S) = ?, and so �–(y) fl C = ? since T –(S) contains C. Thusy œ S–(C)c, and so

d(w, y) Ø dist(w, S–(C)c) > –tw

35

since (w, tw) œ T –(S–(C)). Taking an infimum over y œ Sc, we get that

dist(w, Sc) > –tw,

which says precisely that (w, tw) is in T –(S). Therefore T –(S–(C)) µ T –(S) asdesired.

Lemma 1.4.4. For a cylindrical subset K µ X+, define

—0(K) := infBµX

{µ(B) : T –(B) fl K ”= ?} and

—1(K) := infBµX

{µ(B) : T –(B) ∏ K},

with both infima taken over the set of balls B in X. Then —1(K) is positive, andif X is proper or doubling, then —0(K) is also positive.

Proof. We first prove that —0 := —0(K) is positive, assuming that X is proper ordoubling. Write

K µ C := B(x0, r0) ◊ [a0, b0]

for some x0 œ X and a0, b0, r0 > 0. If B is a ball such that T –(B) fl K ”= ?, thenwe must have T –(B) fl C ”= ?, and so we can estimate

—0 Ø infBµX

{µ(B) : T –(B) fl C ”= ?}.

Note that if B = B(c(B), r(B)) is a ball with c(B) œ B(x0, r0), then T –(B) flC ”= ? if and only if r(B) Ø –a0. Defining

I(x) := inf{V (x, r) : r > 0, T –(B(x, r)) fl C ”= ?}

for x œ X, we thus see that I(x) = V (x, –a0) when x œ B(x0, r0), and so I|B(x0,r0)is lower semicontinuous as long as the volume function is lower semicontinuous.

Now suppose B = B(y, fl) is any ball with T –(B) fl C ”= ?. Let (z, tz) be apoint in T –(B) fl C. We claim that the ball

ÂB := B3

z,12 (fl ≠ d(z, y) + –tz)

4

is contained in B, centred in B(x0, r0), and is such that T –( ÂB) fl C ”= ?. Thesecond fact is obvious: (z, tz) œ C implies z œ B(x0, r0). For the first fact, observethat

ÂB µ B(y, d(z, y) + (fl ≠ d(z, y) + –tz)/2)= B(y, (fl + d(z, y) + –tz)/2)µ B(y, (fl + (fl ≠ –tz) + –tz)/2)= B(y, fl),

36

since (z, tz) œ T –(B) implies that d(z, y) < fl ≠ –tz. Finally, we have (z, tz) œT –( ÂB): since c( ÂB) = z, we just need to show that tz < r( ÂB)/–. Indeed, we have

r( ÂB)–

= 12

Afl ≠ d(z, y)

–+ tz

B

,

and tz < (fl ≠ d(z, y))/– as above.The previous paragraph shows that

infxœX

I(x) Ø infxœB(x0,r0)

I(x),

and so we are reduced to showing that the right hand side of this inequality ispositive, since —0 Ø infxœX I(x).

If X is proper: Since B(x0, r0) is compact and I|B(x0,r0) is lower semicontinu-ous, I|B(x0,r0) attains its infimum on B(x0, r0). That is,

infxœB(x0,r0)

I(x) = minxœB(x0,r0)

Ix > 0, (1.15)

by positivity of the ball volume function.

If X is doubling: Since I(x) = V (x, –a0) when x œ B(x0, r0), we can write

infxœB(x0,r0)

I(x) Ø infxœB(x0,r0)

V (x, Á),

where Á = min(–a0, 3r0). If x œ B(x0, r0), then B(x0, r0) µ B(x, 2r0) µB(x, 3r0), and so since 3r0/Á Ø 1,

V (x0, r0) Æ V (x, 3r0)= V (x, Á(3r0/Á)).X V (x, Á).

Hence V (x, Á) &X V (x0, r0), and therefore

infxœB(x0,r0)

V (x, Á) & V (x0, r0) > 0 (1.16)

as desired.

We now prove that —1 = —1(K) is positive. Recall from Lemma 1.4.3 that ifT –(B) ∏ K, then T –(B) ∏ T –(S–(K)). Since shadows are open, Lemma 1.4.1tells us that B ∏ S–(K). Hence µ(B) Ø µ(S–(K)), and so

—1 Ø µ(S–(K)) > 0

by positivity of the ball volume function.10

10If S–(K) is a ball, then —1

(K) = µ(S–(K)).

37

Lemma 1.4.5. Let B be an open ball in X of radius r. Then for all x œ B, thetruncated cone �–

r (x) is contained in T –((2– + 1)B).

Proof. Suppose (y, t) œ �–r (x) and z œ ((2– + 1)B)c, so that d(y, x) < –t < –r

and d(c(B), z) Ø (2– + 1)r. Then by the triangle inequality

d(y, z) Ø d(c(B), z) ≠ d(c(B), x) ≠ d(x, y)> (2– + 1)r ≠ r ≠ –r

= –r

> –t,

so that dist(y, ((2– + 1)B)c) > –t, which yields (y, t) œ T –((2– + 1)B).

1.4.2 Measurability

We assume (X, d, µ) has the implicit assumptions from Section 1.2.

Lemma 1.4.6. Let – > 0, and suppose � is a non-negative measurable functionon X+. Then the function

g : x ‘æ¨

�–(x)�(y, t) dµ(y) dt

t

is µ-measurable.

We present two proofs of this lemma: one uses an abstract measurabilityresult, while the other is elementary (and in fact stronger, proving that g is notonly measurable but lower semicontinuous).

First proof. By [67, Theorem 3.1], it su�ces to show that the function

F (x, (y, t)) := 1B(y,–t)(x)�(y, t)

is measurable on X ◊ X+. For Á > 0, define

fÁ(x, (y, t)) := dist(x, B(y, –t))dist(x, B(y, –t)) + dist(x, B(y, –t + Á)c)

.

Then fÁ(x, (y, t)) is continuous in x, and converges pointwise to 1B(y,–t)(x) asÁ æ 0. Hence

F (x, (y, t)) = limÁæ0

fÁ(x, (y, t))�(y, t) =: limÁæ0

FÁ(x, (y, t)),

38

and therefore it su�ces to show that each FÁ(x, (y, t)) is measurable on X ◊ X+.Since FÁ is continuous in x and measurable in (y, t), FÁ is measurable on X◊X+,11

and the proof is complete.

Second proof. For all x œ X and Á > 0, define the vertically translated cone

�–Á (x) := {(y, t) œ X+ : (y, t ≠ Á) œ �–(x)} µ �–(x).

If y œ B(x, –Á), then is it easy to show that �–Á (x) µ �–(y): indeed, if (z, t) œ

�–Á (x), then d(z, x) < –(t ≠ Á), and so

d(z, y) Æ d(z, x) + d(x, y) < –(t ≠ Á) + –Á = –t.

For all x œ X and Á > 0, define

gÁ(x) :=¨

�–

Á

(x)�(y, t) dµ(y) dt

t.

For each x œ X, as Á √ 0, we have gÁ(x) ¬ g(x) by monotone convergence. Fix⁄ > 0, and suppose that g(x) > ⁄. Then there exists Á(x) such that gÁ(x)(x) > ⁄.If y œ B(x, –Á(x)), then by the previous paragraph we have

g(y) Ø gÁ(x)(x) > ⁄.

Therefore g is lower semicontinuous, and thus measurable.

Lemma 1.4.7. Let f be a measurable function on X+, q œ (0, Œ), and – > 0.Then C–

q (f) is lower semicontinuous.

Proof. Let ⁄ > 0, and suppose x œ X is such that C–q (f)(x) > ⁄. Then there

exists a ball B – x such that

1µ(B)

¨T –(B)

|f(y, t)|q dµ(y) dt

t> ⁄q.

Hence for any z œ B, we have C–q (f)(z) > ⁄, and so the set {x œ X : C–

q (f)(x) >

⁄} is open.11See [41, Theorem 1], which tells us that FÁ is Lusin measurable; this implies Borel measur-

ability on X ◊ X+.

39

1.4.3 InterpolationHere we fix some notation involving complex interpolation.

An interpolation pair is a pair (B0, B1) of complex Banach spaces which admitembeddings into a single complex Hausdor� topological vector space. To such apair we can associate the Banach space B0 + B1, endowed with the norm

||x||B0+B1:= inf{||x0||B0

+ ||x1||B1: x0 œ B0, x1 œ B1, x = x0 + x1}.

We can then consider the space F(B0, B1) of functions f from the closed strip

S = {z œ C : 0 Æ Re(z) Æ 1}

into the Banach space B0 + B1, such that

• f is analytic on the interior of S and continuous on S,

• f(z) œ Bj whenever Re(z) = j (j œ {0, 1}), and

• the traces fj := f |Re z=j (j œ {0, 1}) are continuous maps into Bj whichvanish at infinity.

The space F(B0, B1) is a Banach space when endowed with the norm

||f ||F(B0,B1) := maxA

supRe z=0

||f(z)||B0, sup

Re z=1||f(z)||B1

B

.

We define the complex interpolation space [B0, B1]◊ for ◊ œ [0, 1] to be the sub-space of B0 + B1 defined by

[B0, B1]◊ := {f(◊) : f œ F(B0, B1)}

endowed with the norm

||x||[B0,B1]◊

:= inff(◊)=x

||f ||F(B0,B1) .

40

Chapter 2

Non-uniformly local tent spaces

This article is joint work with Mikko Kemppainen.

Abstract

We develop a theory of ‘non-uniformly local’ tent spaces on metric measure spaces.As our main result, we give a remarkably simple proof of the atomic decomposi-tion.

2.1 IntroductionThe theory of global tent spaces on Euclidean space was first considered by Coif-man, Meyer, and Stein [33], and has since become a central framework for un-derstanding Hardy spaces defined by square functions. Upon replacing Euclideanspace with a doubling metric measure space, the theory is largely unchanged.Details of this generalisation can be found in [3], although this was known toharmonic analysts for some time.

Tent spaces on Riemannian manifolds with doubling volume measure wereused by Auscher, McIntosh, and Russ in [13], where a ‘first order approach’ toHardy spaces associated with the Laplacian ≠� (or more accurately, the corre-sponding Hodge–Dirac operator) was investigated. A corresponding local tentspace theory, now on manifolds with exponentially locally doubling volume mea-sure, was considered by Carbonaro, McIntosh, and Morris [30], with applicationsto operators such as ≠� + a for a > 0. The locality arises from the ‘spectralgap’ between 0 and ‡(≠�+a) µ [a, Œ) and means that the relevant informationof a function can be captured from small time di�usion, which in turn allowsone to exploit the locally doubling nature of the manifold under investigation.

41

Hence the related tent spaces consist of functions of space-time variables (y, t)with 0 < t < 1 instead of 0 < t < Œ.

The motivation for non-uniformly local tent spaces comes from the settingof Gaussian harmonic analysis, in which one considers the Ornstein–Uhlenbeckoperator L = ≠� + x · Ò on Rn equipped with the usual Euclidean distance andthe Gaussian measure d“(x) = (2fi)≠n/2e≠|x|2/2 dx. Here ‡(L) = {0, 1, 2, . . .}, butdespite the evident spectral gap, one cannot make use of a uniformly local tentspace because the rapidly decaying measure “ is non-doubling. This was remediedby Maas, van Neerven, and Portal [64], who defined the ‘Gaussian tent spaces’tp(“) to consist of functions on the region D = {(y, t) œ Rn ◊ (0, Œ) : t < m(y)}.Here m(y) = min(1, |y|≠1) is the admissibility function of Mauceri and Meda [68],who showed that “ is doubling on the family of ‘admissible balls’ B(x, t) witht Æ m(x). In [78], Portal then defined the ‘Gaussian Hardy space’ h1(“) usingthe conical square function

Su(x) =Aˆ 2m(x)

0

ˆB(x,t)

|tÒe≠t2Lu(y)|2 d“(y)dt

t

B1/2

,

and showed that the Riesz transform ÒL≠1/2 is bounded from h1(“) to L1(“).This relied on the atomic decomposition on t1(“), which was established in [64],along with a square function estimate from [63]. The Gaussian Hardy space isalso known to interpolate with L2(“), in the sense that [h1(“), L2(“)]◊ = Lp(“)for 1/p = 1 ≠ ◊/2 [77]. Note that dimension-independent boundedness of ÒL≠1/2

on Lp(“) for 1 < p < Œ is a classical result of Meyer [74].Our long-term aim is to generalise this theory to the setting where, given an

appropriate ‘potential function’ „ on a Riemannian manifold X (or some moregeneral space) with volume measure µ, one considers the Witten Laplacian L =≠� + Ò„ · Ò equipped with the geodesic distance and the measure d“ = e≠„dµ.An admissibility function can then be defined by m(x) = min(1, |Ò„(x)|≠1), witha suitable interpretation of Ò if „ is not di�erentiable, and the setting of Gaussianharmonic analysis is recovered by taking X = Rn and „(x) = n

2 log(2fi) + |x|22 .

The Riesz transform associated with the Witten Laplacian has been studied forinstance by Bakry in [20], where Lp(“) boundedness for 1 < p < Œ is provenunder a „-related curvature assumption.

In this article we define and study the corresponding local tent spaces tp,q(“).Our main result is the atomic decomposition Theorem 2.4.5. This allows us toidentify the dual of t1,q(“) with the local tent space tŒ,qÕ(“), and to show that thelocal tent spaces form a complex interpolation scale. In Appendix 2.6 we prove a

42

‘cone covering lemma’ for non-negatively curved Riemannian manifolds. It givesa stronger version of Lemma 2.4.4 that is applicable also in the vector-valuedtheory of tent spaces (see [59, 60]).

A di�erent approach to Gaussian Hardy spaces was introduced in [68], wherethe atomic Hardy space H1(“) was introduced. This theory has also been ex-tended to certain metric measure spaces (see [28, 29]). While many interestingsingular integral operators, such as imaginary powers of the Ornstein–Uhlenbeckoperator, have been shown to act boundedly from H1(“) to L1(“) (see [68]), itshould be noted that this is not the case for the Riesz transform (see [69]). Thismarks the crucial di�erence between the atomic Hardy space H1(“) and h1(“).

Acknowledgements

The first author acknowledges the financial support from the Australian ResearchCouncil Discovery grant DP120103692. The second author acknowledges the fi-nancial support from the Väisälä Foundation and from the Academy of Finlandthrough the project Stochastic and harmonic analysis: interactions and applica-tions (133264). He is grateful for the hospitality of the Mathematical SciencesInstitute at the Australian National University during his stay.

2.2 Weighted measures and admissible ballsWe begin by formulating the abstract framework in which we develop our theory.Let (X, d, µ) be a metric measure space: that is, a metric space (X, d) equippedwith a Borel measure µ. We assume that every ball B µ X comes with a givencenter cB and a radius rB > 0, and that the volume µ(B) is finite and nonzero.Furthermore, we assume that the metric space (X, d) is geometrically doubling:that is, we assume that there exists a natural number N Ø 1 such that for everyball B µ X of radius rB, there exist at most N mutually disjoint balls of radiusrB/2 contained in B.

Given a measurable real-valued function „ on X, we consider the weightedmeasure

d“(x) := e≠„(x) dµ(x).

Furthermore, we fix a function m : X æ (0, Œ), which we call an admissibilityfunction. For every – > 0, this defines the family of admissible balls

B– := {B µ X : 0 < rB Æ –m(cB)}.

43

These objects are required to satisfy the following doubling condition:

(A) For every – > 0, “ is doubling on B–, in the sense that there exists a constantC– Ø 1 such that for all –-admissible balls B œ B–,

“(2B) Æ C–“(B).

Here and in what follows, we write ⁄B = B(cB, ⁄rB) for the expansion of a ballB by ⁄ Ø 1.

Remark 2.2.1. Condition (A) implies that for every – > 0 and every ⁄ Ø 1, thereexists a constant C–,⁄ Ø 1 such that for all –-admissible balls B œ B–,

“(⁄B) Æ C–,⁄“(B). (2.1)

We now describe two classes of examples of „ and m.

Example 2.2.2 (Distance functions). Assume that the underlying measure µ isdoubling, let � µ X be a measurable set of ‘origins’, and let a, aÕ > 0. Define „

by„(x) := a + aÕ dist(x, �)2.

An admissibility function can then be defined by

m(x) = minA

1,1

dist(x, �)

B

.

Taking X to be Rn (equipped with the usual Euclidean distance and Lebesguemeasure), � = {0}, and (a, aÕ) = (n log(2fi)/2, 1/2), we recover the setting ofGaussian harmonic analysis.

Claim 2.2.3. Condition (A) is satisfied with C– = DµeaÕ–(5–+6), where Dµ is thedoubling constant of the underlying measure µ.

Proof. Since µ is doubling, it su�ces to show that for every –-admissible ballB œ B– we have

Y_]

_[

e≠„(x) Æ C Õ–e≠„(c

B

) when x œ 2B, ande≠„(x) Ø C ÕÕ

–e≠„(cB

) when x œ B.(2.2)

Indeed, this would imply that

“(2B) =ˆ

2B

e≠„(x) dµ(x) Æ C Õ–µ(2B)e≠„(c

B

)

44

and“(B) =

ˆB

e≠„(x) dµ(x) Ø C ÕÕ–µ(B)e≠„(c

B

),

so that“(2B)“(B) Æ C Õ

–

C ÕÕ–

µ(2B)µ(B) Æ C– := Dµ

C Õ–

C ÕÕ–

.

To see that the first inequality in (2.2) holds with C Õ– = e4aÕ–(–+1), observe

that if x œ 2B, then

dist(cB, �) Æ 2–m(x) + dist(x, �).

Indeed, if dist(cB, �) Ø dist(x, �), then m(cB) Æ m(x), and so

dist(cB, �) Æ d(cB, x)+dist(x, �) Æ 2–m(cB)+dist(x, �) Æ 2–m(x)+dist(x, �).

Consequently we have

dist(cB, �)2 Æ 4–m(x)2 +4–m(x) dist(x, �)+dist(x, �)2 Æ 4–2 +4–+dist(x, �)2,

and soe≠aÕ dist(x,�)2 Æ e4aÕ–(–+1)e≠aÕ dist(c

B

,�)2.

Similarly, the second inequality in (2.2) with C ÕÕ– = e≠aÕ–(–+2) follows after

noting that if x œ B, then

dist(x, �) Æ d(x, cB) + dist(cB, �) Æ –m(cB) + dist(cB, �).

Thusdist(x, �)2 Æ –2 + 2– + dist(cB, �)2

ande≠aÕ dist(x,�)2 Ø e≠aÕ–(–+2)e≠aÕ dist(c

B

,�)2.

Putting these estimates together, we have

C– = Dµe4aÕ–(–+1)eaÕ–(–+2) = DµeaÕ–(5–+6)

as claimed.

Example 2.2.4 (C2 potentials). In this example, let (X, g) be a connectedRiemannian manifold (C2 is su�cient) with doubling volume measure, let „ œC2(X), and assume that the following condition is satisfied:

45

(B) there exists a constant M > 0 such that for every unit speed geodesicfl : [0, ¸] æ X, we have

|(„ ¶ fl)ÕÕ(t)| Æ M |(„ ¶ fl)Õ(t)| (2.3)

for all t œ (0, ¸) such that |(„ ¶ fl)Õ(t)| > 1.

Alternatively, we can assume the following inequivalent condition, which is neaterbut generally harder to verify:

(H) there exists a constant M > 0 such that

||Hess „(x)|| Æ M |Ò„(x)| (2.4)

for all x œ X such that |Ò„(x)| > 1.

Note that (B) can be interpreted as a one-dimensional version of (H); indeed,when X is one-dimensional, both conditions are equivalent.

If either of the above conditions are satisfied, we define an admissibility func-tion by

m(x) := minA

1,1

|Ò„(x)|

B

for x œ X, with m(x) := 1 when |Ò„(x)| = 0.

Claim 2.2.5. If d(x, y) Æ – then m(x) Æ eM–m(y).

Proof. Here we assume condition (H); the proof under assumption (B) requiresonly a simple modification.

Given Á > 0, we first take a continuous arclength-parametrised path

fl : [0, d(x, y) + Á] æ X

connecting x to y (we may take Á = 0 when X is complete, and the argumentis slightly simpler in this case). Since „ is twice continuously di�erentiable, thefunction mfl := m ¶ fl is absolutely continuous on [0, d(x, y) + Á], and hencedi�erentiable almost everywhere on this interval. We compute the derivativeof mfl(t) whenever mfl is di�erentiable. If t is such that |Ò„(fl(t))| Æ 1 in aneighbourhood of t, then ˆtmfl(t) = 0. If t is such that |Ò„(fl(t))| > 1 in aneighbourhood of t, then

ˆtmfl(t) = ˆt(|Ò„(fl(t))|≠1) = ≠ˆt|Ò„(fl(t))||Ò„(fl(t))|2 .

46

Using the estimate|ˆt|Ò„(fl(t))|| Æ ||Hess „(fl(t))||

along with assumption (H), we find that

|ˆtmfl(t)| Æ ||Hess „(fl(t))|||Ò„(fl(t))|2 Æ M

|Ò„(fl(t))|

for all t such that mfl(t) is di�erentiable.Since mfl(t) is di�erentiable almost everywhere, we have

| log mfl(d(x, y)) ≠ log mfl(0)| Æ sup0<t<d(x,y)+Á

|ˆt log mfl(t)|d(x, y)

Æ sup0<t<d(x,y)+Á

|ˆt log mfl(t)|–,

where the supremum is taken over all t œ (0, d(x, y) + Á) such that mfl(t) isdi�erentiable. Note that

|ˆt log mfl(t)| = |ˆtmfl(t)||mfl(t)| ,

and so by the estimate above we have that

|ˆt log mfl(t)| Æ M

|Ò„(fl(t))| |Ò„(fl(t))| = M.

Therefore| log mfl(d(x, y) + Á) ≠ log mfl(0)| Æ M–,

and soe| log(m(y)/m(x))| Æ eM(–+Á) =: cÕ

–eMÁ.

This holds for every Á > 0, so by taking the limit of both sides as Á æ 0 we obtain

e| log(m(y)/m(x))| Æ cÕ–. (2.5)

Without loss of generality, we can suppose that m(x) Ø m(y). Then

| log(m(y)/m(x))| = log(m(x)/m(y)),

and (2.5) implies thatm(x)m(y) Æ cÕ

–,

which completes the proof.

Claim 2.2.6. Condition (A) is satisfied, with C– = Dµe3–eM–.

47

Proof. As in the previous example, it su�ces to show that for every B œ B– wehave Y

_]

_[

e≠„(x) Æ C Õ–e≠„(c

B

), when x œ 2B,e≠„(x) Ø C ÕÕ

–e≠„(cB

), when x œ B.(2.6)

This is implied (with C Õ– = e–cÕ

– and C ÕÕ– = e≠2–cÕ

–) by the estimate

|„(x) ≠ „(cB)| Æ ⁄–cÕ– ’x œ ⁄B,

for all ⁄ Ø 1 and x œ ⁄B, which we now show. If x œ ⁄B, then we have

|„(x) ≠ „(cB)| Æ supyœ⁄B

|Ò„(y)|d(x, cB).

Since B is –-admissible, for all x, y œ ⁄B Claim 2.2.5 yields

d(x, cB) Æ ⁄rB Æ ⁄–m(cB) Æ ⁄–cÕ–m(y) Æ ⁄–cÕ

–|Ò„(y)|≠1,

and so |„(x) ≠ „(cB)| Æ ⁄–cÕ–. As in the previous example, we then have

C– = DµC Õ

–

C ÕÕ–

= Dµe3–cÕ– .

Using cÕ– = eM– (from Claim 2.2.5) yields the result.

For a concrete subexample, let (X, d, µ) be the Euclidean space Rn with theusual Euclidean distance and Lebesgue measure, and let „ œ R[x1, . . . , xn] bea polynomial. Condition (B) is easily verified, although condition (H) may nothold when n Ø 2. Taking „(x) = n log(2fi)

2 + 12

qni=1 x2

i , we again recover thesetting of Gaussian harmonic analysis. However, in this case the constants cÕ

–

and C– have significantly worse –-dependence than the constants we found inthe previous example. This is because conditions (B) and (H) are less restrictivethan assuming „ is given in terms of a distance function.

Remark 2.2.7. The utility of an admissibility function is eventually judged by itsapplicability to the local Hardy space theory. More precisely, one needs to obtainsuitable ‘error estimates’ in the spirit of [78, Section 5]. The only known exampleof such at the time of writing is the setting of Rn with „(x) = n

2 log fi + |x|2 andm(x) = min(1, |x|≠1).

48

2.3 Local tent spaces: the reflexive rangeWe now introduce the main topic of the paper — the non-uniformly local tentspaces. Let „ and m be given and satisfy (A) from Section 2.2. Denote theresulting weighted measure by “.

Definition 2.3.1. Let 0 < p, q < Œ and – > 0. The local tent space tp,q– (“) is

the set of all measurable functions f defined on the admissible region

D = {(y, t) œ X ◊ (0, Œ) : t < m(y)}

such that the functional

A–q f(x) =

A¨�

–

(x)|f(y, t)|q d“(y)

“(B(y, t))dt

t

B1/q

satisfiesÎfÎtp,q

–

(“) := ÎA–q fÎLp(“) < Œ.

Here �–(x) = {(y, t) œ D : d(x, y) < –t} is the admissible cone of aperture – atx œ X.

It is clear that ||·||tp,q

–

(“) is a norm on tp,q(“) when p, q œ [1, Œ), and a quasi-norm when p < 1 or q < 1. Following the argument of [3, Proposition 3.4] withdoubling replaced by local doubling, we can show that tp,q

– (“) is complete in this(quasi-)norm.

Remark 2.3.2. The choice „ = 0 and m = Œ recovers the setting of global tentspaces [3], whereas „ = 0 and m = 1 gives the setting of uniformly local tentspaces by Carbonaro, McIntosh and Morris [30].

For 1 < p, q < Œ, the properties of tp,q– (“) can be studied, as in [44], by

embedding the space into an Lp-space of Lq-valued functions. More precisely, letus write Lq(D) for the space of q-integrable functions on D with respect to themeasure d“(y) dt

t“(B(y,t)) , so that

J– : tp,q– (“) Òæ Lp(“; Lq(D)), J–f(x) = 1�

–

(x)f

defines an isometry. We will show that J– embeds tp,q– (“) as a complemented

subspace of Lp(“; Lq(D)), with

N–U(x; y, t) = 1B(y,–t)(x)ˆ

B(y,–t)U(z; y, t) d“(z)

49

defining a bounded projection of Lp(“; Lq(D)) onto the image of tp,q– (“), where

U œ Lp(“; Lq(D)), x œ X, and (y, t) œ D.To see that N– is bounded, we first observe that

|N–U(x; y, t)| Æ 1B(y,–t)(x)ˆ

B(y,–t)|U(z; y, t)| d“(z)

Æ supB–x

BœB–

ˆB

|U(z; y, t)| d“(z)

= M–U(x; y, t),

where M– is the Lq(�)-valued –-local maximal function from Appendix 2.5, with� = (D, d“(y) dt

t“(B(y,t))). Consequently,

ÎN–UÎLp(“;Lq(D)) Æ ÎM–UÎLp(“;Lq(D)) .p,q cXC–,cX

ÎUÎLp(“;L2(D)),

(see Appendix 2.5).An immediate consequence of this vector-valued approach is the following

theorem, detailing the behaviour of the local tent spaces in the reflexive range.

Theorem 2.3.3. Let 1 < p, q < Œ. We have

• (change of aperture) ÎfÎtp,q

–

(“) hp,q,–,— ÎfÎtp,q

—

(“) for 0 < — < – < Œ,

• (duality) tp,q– (“)ú = tp

Õ,qÕ– (“), realised by the duality pairing

Èf, gÍ =¨

D

f(y, t)g(y, t) d“(y)dt

t,

• (complex interpolation) [tp0,q0– (“), tp1,q1

– (“)]◊ = tp,q– (“) when 1 < p0 Æ p1 < Œ

and 1 < q0 Æ q1 < Œ, with 1/p = (1≠◊)/p0 +◊/p1, 1/q = (1≠◊)/q0 +◊/q1.

Proof. For our claim on change of aperture, we follow [44] and begin by notingthat for suitable f we have

N–J—f(x; y, t) = “(B(y, —t))“(B(y, –t))J–f(x; y, t).

Then

ÎfÎtp,q

–

(“) = ÎJ–fÎLp(“;Lq(D)) = “(B(y, –t))“(B(y, —t))ÎN–J—fÎLp(“;Lq(D))

.p,q C—,–/—C–,cX

ÎJ—fÎLp(“;Lq(D))

= C—,–/—C–,cX

ÎfÎtp,q

—

(“),

50

where the constants are from Remark 2.2.1.Now tp,q

– (“) is embedded in Lp(“; Lq(D)) as the range of the projection N–,whose dual is isomorphic to the range of Nú

– on Lp(“; Lq(D))ú = LpÕ(“; LqÕ(D)),which, in turn, is isometrically isomorphic to tp

Õ,qÕ– (“) (because Nú

– = N–). Theduality is realised as

Èf, gÍ = ÈJ–f, J–gÍ

=ˆ

X

È1�–

(x)f, 1�–

(x)gÍ d“(x)

=ˆ

X

¨�

–

(x)f(y, t)g(y, t) d“(y)

“(B(y, t))dt

td“(x)

=¨

D

f(y, t)g(y, t) d“(y)dt

t.

For 1 < p0 Æ p1 < Œ and 1 < q0 Æ q1 < Œ the interpolation of tent spacesfollows, by the standard result on interpolation of complemented subspaces [89,Section 1.17], from the fact that

[Lp0(“; Lq0(D)), Lp1(“; Lq1(D))]◊ = Lp(“; Lq(D)).

Remark 2.3.4. The dependence on – in the aperture change constant C1,–C–,cX

(between tp,q– (“) and tp,q

1 (“)) is not optimal in general. For instance, on (Rn, dx),the optimal dependence is –n/ min(p,2) (see [6]), while C1,–C–,c

X

h –n. Note how-ever, that on (Rn, “) we have C1,–C–,c

X

. ec–2 for some constant c. We return tothis in Section 2.4.

The change of aperture and interpolation results extend to 1 Æ p, q < Œ bya convex reduction due to Bernal ([23], see also [3]).

Corollary 2.3.5. Let 1 Æ q < Œ. We have

• (change of aperture) ÎfÎt1,q

–

(“) hq,–,— ÎfÎt1,q

—

(“) for 0 < — < – < Œ,

• (complex interpolation) [tp0,q0– (“), tp1,q1

– (“)]◊ = tp,q– (“) when 1 Æ p0 Æ p1 < Œ

and 1 < q0 Æ q1 < Œ, with 1/p = (1≠◊)/p0 +◊/p1, 1/q = (1≠◊)/q0 +◊/q1.

2.4 Endpoints: t1,q and tŒ,q

In this section, under the assumption that the space X is complete, we studythe endpoints of the local tent space scale: the spaces t1,q

– (“) and tŒ,q– (“) (with

51

1 Æ q < Œ). In particular, employing Corollary 2.3.5 we prove, following theargument in [59], that elements of t1,q

– (“) can be decomposed into ‘atoms’. Fromthis we deduce duality, interpolation, and (quantified) change of aperture resultsfor the full local tent space scale tp,q

– (“) (1 Æ p Æ Œ, 1 Æ q < Œ). We writet1,q := t1,q

1 for notational simplicity. We do not consider q = Œ. As in [33], thisrequires additional continuity and convergence assumptions.

2.4.1 Atomic decompositionFix (X, d, µ), „, and m as in the previous section. The admissible tent T (O) overan open set O µ X is given by

T (O) := D \ �(Oc),

where �(Oc) := fixœOc�(x).

Definition 2.4.1. Fix – > 0 and q Ø 1. A function a on D is called an –-t1,q-atom (or more succinctly, a –-atom) if there exists an –-admissible ball B œ B–

such that supp a µ T (B) and¨

T (B)|a(y, t)|q d“(y)dt

tÆ 1

“(B)q≠1 .

Observe that for such a function a,

ÎaÎt1,q(“) =ˆ

B

Aqa(x)q d“(x) Æ “(B)q≠1

q

AˆB

Aqa(x)q d“(x)B1/q

. 1.

Furthermore, if (ak)kœN is a sequence of –-t1,q-atoms for some – > 0, then theseries f = q

k ⁄kak converges in t1,q(“) when qk |⁄k| < Œ. The atomic tent space

t1,qat (“) consisting of such functions f becomes a Banach space when normed by

ÎfÎt1,q

at (“) = inf; ÿ

k

|⁄k| : f =ÿ

k

⁄kak

<.

Lemma 2.4.2. Suppose that E µ X is a bounded open set. Then there exists acountable sequence of disjoint admissible balls Bj µ E such that

T (E) µ€

jØ1T (5Bj).

Proof. Let ”1 = sup{rB : B µ E admissible} and begin by choosing an admissibleball B1 µ E with radius r1 > ”1/2. Proceeding inductively we put

”k+1 = sup{rB : B µ E admissible, B fl Bj = ?, j = 1, . . . , k}

52

and choose (if possible) an admissible ball Bk+1 µ E with radius rk+1 > ”k+1/2disjoint from B1, . . . , Bk. Given a (y, t) œ T (E) we show that B(y, t) µ 5Bj forsome j. It is possible to pick the first index j for which B(y, t) fl Bj is nonempty.Indeed, if on the contrary B(y, t) was disjoint from every Bj, then, B(y, t) beingadmissible and contained in E, we would have t Ø ”j for all j which under theassumption that (X, d) is geometrically doubling contradicts the boundedness ofE. By construction, we have t Æ ”j Æ 2rj and so B(y, t) µ 5Bj, as required.

Remark 2.4.3. The above lemma is a stronger version of a ‘local Vitali coveringlemma’, which is otherwise identical but claims only that E µ t

jØ1 5Bj withoutreference to tents (see also Remark 2.5.2).

The following lemma regarding pointwise estimates for A-functionals, whichappears implicitly in [33, Theorem 4’], lies at the heart of our proof of the atomicdecomposition. This is the only point at which we seem to need completeness;we suspect that this assumption can be removed or at least weakened.

Lemma 2.4.4. Suppose X is complete, let q Ø 1 and let f be a measurablefunction in D. Let ⁄ > 0 and write E = {x œ X : A3

qf(x) > ⁄}. ThenAq(f1D\T (E))(x) Æ ⁄ for all x œ X.

Proof. If x ”œ E, then Aq(f1D\T (E))(x) Æ A3qf(x) Æ ⁄.

If x œ E, then by completeness of X we can choose a point x0 œ X\E such thatd(x, x0) = d(x, X\E). We show that �(x)\T (E) µ �3(x0): let (y, t) œ �(x)\T (E)so that d(x, y) < t and B(y, t) ”µ E. Now B(y, t) µ B(x, 2t), which meansthat B(x, 2t) ”µ E and so x0 œ B(x, 2t). Moreover B(x, 2t) µ B(y, 3t) so that(y, t) œ �3(x0). Therefore Aq(f1D\T (E))(x) Æ A3

qf(x0) Æ ⁄.

Theorem 2.4.5. Suppose X is complete, and let q Ø 1. For every f œ t1,q(“),there exist 5-t1,q-atoms ak and scalars ⁄k such that

f =ÿ

k

⁄kak, (2.7)

withÿ

k

|⁄k| ƒ ||f ||t1,q(“) .

We call the series (2.7) an atomic decomposition of f .

Proof. We first derive atomic decompositions for the dense class of boundedly-supported functions in t1,q(“), and then argue by completeness of t1,q

at (“). Givena function f in t1,q(“) with bounded support, we consider the bounded open sets

Ek = {x œ X : A3qf(x) > 2k}

53

for each integer k. Applying Lemma 2.4.2 to these sets provides us with disjointballs Bj

k µ Ek such thatT (Ek) µ

€

jØ1T (5Bj

k).

In addition, we take a collection of functions ‰jk (cf. [59, Theorem 11]) satisfying

0 Æ ‰jk Æ 1,

ÿ

jØ1‰j

k = 1 on T (Ek), and supp ‰jk µ T (5Bj

k).

Writing Ak := T (Ek) \ T (Ek+1), we can decompose f as

f =ÿ

kœZ1A

k

f =ÿ

kœZ

ÿ

jØ1‰j

k1Ak

f =ÿ

kœZ

ÿ

jØ1⁄j

kajk,

where

⁄jk = “(5Bj

k)1/qÕ

Q

aˆ

5Bj

k

Aq(f1Ak

)(x)q d“(x)R

b1/q

.

Observe that ajk = ‰j

k1Ak

f/⁄jk is a 5-atom supported in T (5Bj

k).What remains is to control the sum of the scalars ⁄j

k. By Lemma 2.4.4, wehave

Aq (f1Ak

) (x) Æ Aq

1f1D\T (E

k+1)2

(x) Æ 2k+1

for all x œ X, and so⁄j

k Æ “(5Bjk)2k+1.

Consequently,

ÿ

kœZ

ÿ

jØ1⁄j

k Æÿ

kœZ2k+1 ÿ

jØ1“(5Bj

k) .ÿ

kœZ2k+1“(Ek) . ÎA3

qfÎL1(“) . ÎfÎt1,q(“),

where the last step follows by Corollary 2.3.5.We have thus shown that ÎfÎt1,q

at (“) h ÎfÎt1,q(“) for boundedly supported f int1,q(“). Since the class of such functions is dense in t1,q(“), the completeness oft1,qat (“) guarantees that every f œ t1,q(“) has an atomic decomposition.

Remark 2.4.6. Maas, van Neerven and Portal established the above result in thesetting of Gaussian Rn by a di�erent method, which relies on Gaussian Whitneydecompositions [64, Theorem 3.4]. In addition, they showed that decompositionsinto –-atoms exist for every – > 1 [64, Lemma 3.6]. Such a result may not holdin this level of generality due to the lack of geometric information.

54

2.4.2 Duality, interpolation and change of aperture

We present three corollaries of the atomic decomposition theorem, which holdswhen X is complete.

The dual of t1,q(“) can be identified with the space tŒ,qÕ(“), consisting of thosefunctions g on D for which

ÎgÎtŒ,q

Õ (“) = supBœB5

A1

“(B)

¨T (B)

|g(y, t)|qÕd“(y)dt

t

B1/qÕ

is finite. Note that we take a supremum over 5-admissible balls, reflecting thefact that we have atomic decompositions of elements of t1,q(“) into 5-atoms. Forthe reader’s convenience, we present the standard proof, following [33, Theorem1 (b)].

Corollary 2.4.7. Suppose X is complete, and let q Ø 1. Then the pairing

Èf, gÍ =¨

D

f(y, t)g(y, t) d“(y)dt

t, f œ t1,q(“), g œ tŒ,qÕ(“), (2.8)

realises tŒ,qÕ(“) as the dual of t1,q(“).

Proof. To see that (2.8) defines a bounded linear functional on t1,q(“) for everyg œ tŒ,qÕ(“), it su�ces (by Theorem 2.4.5) to test the pairing on atoms. For anyatom a associated with a ball B œ B5 we have

|Èa, gÍ| ƨ

T (B)|ag| d“

dt

t

ÆA¨

T (B)|a|q d“

dt

t

B1/q A¨T (B)

|g|qÕd“

dt

t

B1/qÕ

Æ ÎgÎtŒ,q

Õ (“).

To show that every functional � œ t1,qÕ(“)ú arises in this way, we first notethat each f œ Lq(T (B)), with B œ B5, satisfies

ÎfÎt1,q(“) Æ “(B)1/qÕÎfÎLq(T (B))

(we equip the space T (B) with the product measure d“(y)dt/t). Hence � restrictsto a bounded linear functional on Lq(T (B)), and is thus given by

�f =¨

T (B)fgB d“

dt

t, f œ Lq(T (B)),

55

for some gB œ LqÕ(T (B)), with the estimate

ÎgBÎLq

Õ (T (B)) Æ “(B)1/qÕÎ�Ît1,q

Õ (“)ú .

A single function g on D can then be obtained from the family (gB)BœB5 in awell-defined manner, since for any two balls B, BÕ œ B5, the functions gB and gBÕ

agree on T (B) fl T (BÕ). It remains to be checked that ÎgÎtŒ,q

Õ (“) = Î�Ît1(“)ú . Onthe one hand, for any B œ B5 we have

A¨T (B)

|g|qÕd“

dt

t

B1/qÕ

= ÎgBÎLq

Õ (T (B)) Æ “(B)1/qÕÎ�Ît1,q

Õ (“)ú .

On the other hand, due to Theorem 2.4.5, Î�Ît1,q(“)ú is achieved (up to a constant)by testing against all atoms, and so the proof is completed after checking that

|�a| ƨ

T (B)|ga| d“

dt

t

ÆA¨

T (B)|g|qÕ

d“dt

t

B1/qÕ A¨T (B)

|a|q d“dt

t

B1/q

Æ “(B)1/qÕ ||g||tŒ,q

Õ (X,“) “(B)≠1/qÕ

= ||g||tŒ,q

Õ (“) ,

Corollary 2.4.8. Suppose X is complete. For 1 Æ p0 Æ p1 Æ Œ (excluding thecase p0 = p1 = Œ) and 1 Æ q0 Æ q1 < Œ, we have [tp0,q0(“), tp1,q1(“)]◊ = tp,q(“),when 1/p = (1 ≠ ◊)/p0 + ◊/p1, 1/q = (1 ≠ ◊)/q0 + ◊/q1, and 0 Æ ◊ Æ 1.

Proof. This follows directly from Theorem 2.3.3 and Corollary 2.4.7, by convexreduction and reiteration (see Remark 2.3.4).

Corollary 2.4.9. Let q Ø 1. For all 1 Æ p Æ q and – Ø 1 we have

ÎfÎtp,q

–

(“) . C1/q1,– C1/p≠1/q

5,– ÎfÎtp,q(“).

Proof. In order to argue by interpolation, consider first the case p = q:

ÎfÎqtq,q

–

(“) =ˆ

X

¨�–(x)

|f(y, t)|q d“(y)“(B(y, t))

dt

td“(x)

=ˆ

X

ˆ Œ

0|f(y, t)|q1(0,m(y))(t)

“(B(y, –t))“(B(y, t))

dt

td“(y)

Æ C1,–

¨D

|f(y, t)|q d“(y)dt

t

= C1,–ÎfÎqtq,q(“).

56

For p = 1 we make use of the atomic decomposition. If a is a 5-atom associatedwith B œ B5, then, since �–(x) fl T (B) is non-empty exactly when x œ –B, wehave

ÎaÎt1,q

–

(“) Æ “(–B)1/qÕÎaÎtq,q

–

(“)

Æ C1/q1,– “(–B)1/qÕÎaÎtq,q(“)

Æ C1/q1,–

A“(–B)“(B)

B1/qÕ

Æ C1/q1,– C1≠1/q

5,– .

Thus ÎfÎt1,q

–

(“) Æ C1/q1,– C1≠1/q

5,– ÎfÎt1,q(“) for all f œ t1,q(“), and the result thenfollows by interpolation.

Remark 2.4.10. Note that on (Rn, dx) this gives the optimal dependence on –

for 1 Æ p Æ 2, which we could not obtain from the vector-valued approach,since C1/2

1,– C1/p≠1/25,– = –n/p (see Remark 2.3.4). On Gaussian Rn this merely

extends the aperture change to t1(“) with the constant ec–2 , the improvementfrom interpolation being immaterial.

2.5 Appendix 1: Local maximal functions

Here we present a brief justification of the boundedness of the maximal functionsused above and in Appendix 2.6. We use dyadic methods, particularly the exis-tence of finitely many ‘adjacent’ dyadic systems, combined with some methodsfrom Martingale theory. At the end of this section we indicate another approach,which is more elementary but does not adapt well to vector-valued contexts.

By a dyadic system on a measure space (X, “) we mean a countable collectionD = {Dk}kœZ, where each Dk is a partition of X into measurable sets of finitenonzero measure, such that the containment relations

Q œ Dk, R œ Dl, l Ø k =∆ R µ Q or Q fl R = ÿ

hold. The elements of Dk are called dyadic cubes (of generation k).Associated to each dyadic system D is a dyadic maximal function, defined by

MDu(x) = supQ–xQœD

ˆQ

|u| d“

57

for all u œ L1loc(“). Since MD coincides with the martingale maximal function

for the (increasing) filtration (Fk)kœZ when each Fk is the ‡-algebra generated byDk, it follows that MD satisfies a weak type (1,1) inequality

“({x œ X : MDu(x) > ⁄}) Æ 1⁄

ÎuÎL1(“) (2.9)

for all ⁄ > 0 (see for instance [91, Theorem 14.6] or [86, Chapter IV, Section 1]).Now suppose that (X, d) is a geometrically doubling metric space. Hytönen

and Kairema showed in [52] (see also [72]) the existence of a finite collection ofadjacent dyadic systems.

Theorem 2.5.1. There exists a finite collection {Di}Ni=1 of dyadic systems on X,

with N bounded by a constant depending only on the geometric doubling constantof (X, d), such that every open ball B µ X is contained in a dyadic cube QB fromone of the dyadic systems, with diam(QB) Æ cX diam(B).

Now let (X, d, µ), “, and m be as in Section 2.2, and let – > 0. Combiningthe theorem above with the weak type (1,1) estimate for the dyadic maximalfunction yields a corresponding weak type (1,1) estimate for the –-local maximaloperator M–.

Indeed, for each –-admissible ball B œ B– we have that B µ QB for somedyadic cube QB that satisfies QB µ cXB, and so

1B(x)ˆ

B

|u| d“ Æ 1QB

(x)“(QB)“(B)

ˆQ

B

|u| d“

Æ 1QB

(x)“(cXB)“(B)

ˆQ

B

|u| d“

Æ 1QB

(x)C–,cX

ˆQ

B

|u| d“.

Here C–,cX

is the doubling constant from Remark 2.2.1. Summing over finitelymany dyadic systems, we find that

M–u(x) Æ C–,cX

ÿ

DMDu(x),

and using the estimate (2.9) yields

“({x œ X : M–u(x) > ⁄}) . C–,cX

||u||L1(“)

for all ⁄ > 0.

58

Similarly, given a ‡-finite measure space �, we can consider the –-local max-imal function M–, given by

M–U(x, s) = supBœB

–

B–x

ˆB

|U(z, s)| d“(z)

for U œ L1loc(“; Lq(�)) with q œ (1, Œ) (see [80] for a general overview). Again,

this is controlled pointwise by a finite sum of its dyadic counterparts, that is,

M–U(x, s) Æ C–,cX

ÿ

DMDU(x, s) (2.10)

for some finite collection of dyadic systems D. The dyadic lattice maximal op-erators MD are again amenable to Martingale theory. Indeed, according to themartingale version of Fe�erman–Stein inequality (see [66, Subsection 3.1]) wehave for 1 < p < Œ that

ÎMDUÎLp(“;Lq(�)) .p,q ÎUÎLp(“;Lq(�)),

and consequently

ÎM–UÎLp(“;Lq(�)) .p,q cXC–,cX

ÎUÎLp(“;Lq(�)).

Although the explicit statement in [66] concerns the case of sequences, i.e. thecase � = N, it immediately extends to more general measure spaces � by meansof lattice finite representability: Lq(�) is lattice finitely representable in ¸q inthe sense that for every finite dimensional sublattice E of Lq(�) and every Á >

0 there exists a sublattice F of ¸q and a lattice isomorphism � : E æ F forwhich ||�|| ||�≠1|| Æ 1 + Á (see for instance [40] and the references therein). Forboundedness of MD it su�ces to consider simple functions U : X æ Lq(�) andthe boundedness is therefore transferable in lattice finite representability.

Remark 2.5.2. Martingale theory can be avoided by analysing M– by means ofa ‘local Vitali covering lemma’, analogous to the usual analysis of the (global)maximal operator through the usual Vitali covering lemma. One can then provethe duality of tp,q

– and tpÕ,qÕ

– for 1 < p, q < Œ, and recover the boundedness ofthe projections N– by realising them as the adjoints of the (bounded) inclusionsfrom tp,q

– into the appropriate Lq-valued Lp-space. This is the method of Bernal[23], used by the first author for global tent spaces in [3]. In this way we alsoavoid the use of the Lq(�)-valued maximal function M–, but we do not achievethe potential generality of the above method.

59

2.6 Appendix 2: Cone covering lemma for non-negatively curved Riemannian manifolds

In this section we prove a stronger version of Lemma 2.4.4 that will be useful forthe theory of vector-valued tent spaces. This is based on a ‘cone covering lemma’,the Euclidean version of which appears in [59, Lemma 10].

2.6.1 Review of non-negatively curved spacesRecall that a complete length space (X, d) has non-negative curvature if and onlyif for every point x œ X and for every pair of geodesics fl1, fl2 with fl1(0) = fl2(0) =x, the comparison angle

\fl1(t)xfl2(t) := cos≠1A

d(x, fl1(t))2 + d(x, fl2(t))2 ≠ d(fl1(t), fl2(t))2d(x, fl1(t))d(x, fl2(t))

B

is nonincreasing in t (this is the corresponding angle of a Euclidean triangle withsidelengths d(x, fl1(t)), d(x, fl2(t)), and d(fl1(t), fl2(t))). Actually, this monotonic-ity is a combination of the usual (local) definition of non-negative curvature andthe conclusion of Topogonov’s theorem: see [27, Definition 4.3.1 and Theorem10.3.1] for details.

We have the following simple corollary of this characterisation of non-negativecurvature.

Corollary 2.6.1. Suppose (X, d) is a complete length space with non-negativecurvature. Let x, y, z œ X, let flxy and flxz be two unit speed minimising geodesicsfrom x to y and z respectively, and denote the angle \(flÕ

xy(0), flÕxz(0)) by ◊. Then

d(y, z) Æ d(x, z) tan ◊.

Proof. We have◊ = lim

tæ0\(flÕ

xy(t), flÕxz(t)) Ø ◊Õ

by Topogonov’s theorem (as stated above), where ◊Õ is the comparison angleÂ\yxz. By basic trigonometry,

tan ◊Õ = d(y, z)d(x, z) ,

and so we havetan ◊ Ø d(y, z)

d(x, z) .

This yields the result.

60

In particular, if fl1 and fl2 are two unit speed geodesics emanating from a pointx œ X with \(flÕ

1(0), flÕ2(0)) Æ tan≠1(1/4), then

d(fl1(t), fl2(t)) Æ t/4

for all t > 0, since d(fl2(0), fl2(t)) Æ t.

2.6.2 Cone covering

In this section, we assume that X is a complete geometrically doubling Rieman-nian manifold, so that (X, d) is a complete length space. We also fix „ and m

satisfying condition (A) as in Section 2.2 and assume in addition the followingcomparability condition:

(C) For every – > 0, there exists a constant c– such that for all pairs of pointsx, y œ X,

d(x, y) Æ –m(x) =∆ m(x) Æ c–m(y).

Remark 2.6.2. We could work in the context of complete geometrically doublingnon-negatively curved length spaces; we have imposed smooth structure in orderto use the language of tangent spaces rather than that of spaces of directions.The length space setting is only a small generalisation of the manifold setting,due to the fact that complete non-negatively curved length spaces are manifoldsalmost everywhere.

Given parameters – Ø 1 and ⁄ œ (0, 1), we define the extension of an openset E µ X by

Eú–,⁄ :=

€ ;B œ B– : “(B fl E)

“(B) > ⁄<

.

Note that we can write

Eú–,⁄ = {x œ X : M–1E(x) > ⁄},

where M– is the –-local maximal operator from Appendix 2.5, and so Eú–,⁄ is

open. Furthermore, since for each – Ø 1 the local maximal function is of weaktype (1, 1) with respect to “, we have

“(Eú–,⁄) Æ C–

⁄“(E)

for all ⁄ œ (0, 1).

61

For all x œ X, for all unit tangent vectors v œ TxX (recalling that we haveassumed that X is a manifold), and for all t > 0, define the sector

R(v, t) :=€

0ÆsÆt

B(fl(s), s/4)

opening from x in the direction of v along the unit speed geodesic fl with flÕ(0) = v.

Lemma 2.6.3. Let — Ø 1. There exists – Ø 1 and ⁄ œ (0, 1) such that thefollowing holds: if E µ X is open and y œ R(v, t) µ E, with v œ TxX and0 < t Æ —m(x), then B(y, 2t) µ Eú

–,⁄.

Proof. Suppose that E µ X is open and y œ R(v, t) µ E, with v œ TxX and0 < t Æ —m(x). We search for – and ⁄ so that

B(y, 2t) œ B– and “(B(y, 2t) fl E)“(B(y, 2t)) > ⁄.

Denote by fl the unit speed geodesic determined by v and begin by observing thatB(fl(t), t/4) µ R(v, t) µ B(y, 2t) fl E, while B(y, 2t) µ B(fl(t), 4t), so that

“(B(y, 2t) fl E)“(B(y, 2t)) Ø “(B(fl(t), t/4))

“(B(fl(t), 4t)) .

Now d(x, fl(t)) Æ t Æ —m(x), and by (C) we have m(x) Æ c—m(fl(t)), so t Æ—m(x) Æ —c—m(fl(t)). This means that B(fl(t), t/4) is —c—/4-admissible, so thatby (A),

“(B(fl(t), 4t)) Æ A—“3

B3

fl(t), t

4

44

for some constant A—. We may now choose ⁄ < 1/A— to get

“(B(y, 2t) fl E)“(B(y, 2t)) > ⁄.

To choose –, note that since d(x, y) Æ 2t Æ 2—m(x), we have m(x) Æ c2—m(y),and so t Æ —c2—m(y). In order to have B(y, 2t) œ B–, we choose – = 2—c2—. Bythe definition of the extension, we now have B(y, 2t) µ Eú

–,⁄.

Dictated by the final paragraph in the proof of the following lemma, we nowfix — = c1, and choose – and ⁄ in accordance with Lemma 2.6.3. We also writeEú = Eú

–,⁄. Recall that the admissible tent T (O) over an open set O µ X is givenby

T (O) := D \ �(Oc),

where �(Oc) := fixœOc�(x).

62

Lemma 2.6.4 (Cone covering lemma). Assume that X is non-negatively curved,and let E µ X be a bounded open set. Then for every x œ E there exist finitelymany points x1, . . . , xN œ X \ E, with N depending only on the dimension of X,such that

�(x) \ T (Eú) µN€

m=1�(xm).

Proof. Let x œ E and pick unit vectors v1, . . . , vN œ TxX so that every v œ TxX

has \(v, vm) Æ tan≠1(1/4) for some m = 1, . . . , N . For each m, denote by flm theunit speed geodesic determined by vm, and let tm > 0 be the minimal number (Eis bounded) for which B(flm(tm), tm/4) intersects X \ E, so that we may choosean xm œ (X \ E) fl B(flm(tm), tm/4). Note that now R(vm, tm) µ E for each m.

Letting (y, t) œ �(x) \ T (Eú), we need to show that d(y, xm) < t for somem. By completeness of X, we may choose a unit speed minimising geodesic fl

from x to y and then fix an m so that \(flÕ(0), vm) Æ tan≠1(1/4). Corollary 2.6.1guarantees that y œ R(vm, d(x, y)).

Suppose first that x is close to Ec in the direction of vm, in the sense thattm Æ —m(x). If d(x, y) > tm, then by Corollary 2.6.1 fl(tm) is in B(flm(tm), tm/4),and so

d(y, xm) Æ d(y, fl(tm)) + d(fl(tm), xm)

Æ d(y, fl(tm)) + tm

2Æ d(y, fl(tm)) + d(fl(tm), x)= d(y, x) < t.

On the other hand, if d(x, y) Æ tm, then y œ R(vm, tm)—that is, y œ B(flm(s), s/4)for some 0 Æ s Æ tm—and so

d(y, xm) Æ d(y, flm(s)) + d(flm(s), flm(tm)) + d(flm(tm), xm)

Æ s

4 + tm ≠ s + tm

4 Æ 2tm.

According to Lemma 2.6.3, B(y, 2tm) µ Eú, but since (y, t) ”œ T (Eú) implies thatB(y, t) ”µ Eú, we must have 2tm < t.

Second, we show that it is not possible to have tm > —m(x) with — = c1. Notefirst that since d(x, y) < t < m(y), we have by (C) that t < m(y) Æ c1m(x). Ifindeed we had tm > c1m(x), then y œ R(vm, c1m(x)) µ R(vm, tm) µ E. InvokingLemma 2.6.3 gives B(y, c1m(x)) µ B(y, 2c1m(x)) µ Eú, while B(y, t) ”µ Eú andso c1m(x) < t, which is a contradiction.

63

The cone covering lemma allows stronger pointwise estimation of the func-tional Aq when q Ø 1 (cf. Lemma 2.4.4):

Corollary 2.6.5. Assume that X is non-negatively curved. Suppose 1 Æ q < Œ,and let f be a function on D with bounded support. Let ⁄ > 0 and write E ={x œ X : Aqf(x) > ⁄}. Then

Aq(f1D\T (Eú))(x) .dim X ⁄ for all x œ X.

Proof. If x œ X \ E, then

Aq(f1D\T (Eú))(x) Æ Aqf(x) Æ ⁄

by the definition of E. So let x œ E. Since E is a bounded open set, we mayuse Lemma 2.6.4 to pick x1, . . . , xN œ X \ E (with N depending only on thedimension of X) such that

�(x) \ T (Eú) µN€

m=1�(xm).

We can then estimate

Aq(f1D\T (Eú))(x) =A¨

�(x)\T (Eú)|f(y, t)|q d“(y)

“(B(y, t))dt

t

B1/q

ÆNÿ

m=1

A¨�(x

m

)|f(y, t)|q d“(y)

“(B(y, t))dt

t

B1/q

Æ N⁄,

proving the corollary.

Remark 2.6.6. At the time of writing we do not know of any doubling Riemannianmanifolds (equipped with „ and m) for which the cone covering lemma fails.It would be interesting to determine more precisely which spaces admit conecoverings of the type above.

64

Chapter 3

Interpolation and embeddings ofweighted tent spaces

Abstract

Given a metric measure space X, we consider a scale of function spaces T p,qs (X),

called the weighted tent space scale. This is an extension of the tent space scale ofCoifman, Meyer, and Stein. Under various geometric assumptions on X we iden-tify some associated interpolation spaces, in particular certain real interpolationspaces within the reflexive range. These are identified with a new scale of func-tion spaces, which we call Z-spaces, that have recently appeared in the work ofBarton and Mayboroda on elliptic boundary value problems with boundary datain Besov spaces. We also prove Hardy–Littlewood–Sobolev-type embeddings be-tween weighted tent spaces.

3.1 Introduction

The tent spaces, denoted T p,q, are a scale of function spaces first introducedby Coifman, Meyer, and Stein [32, 33] which have had many applications inharmonic analysis and partial di�erential equations. In some of these applications‘weighted’ tent spaces have been used implicitly. These spaces, which we denoteby T p,q

s , seem not to have been considered as forming a scale of function spaces intheir own right until the work of Hofmann, Mayboroda, and McIntosh [50, §8.3],in which factorisation and complex interpolation theorems are obtained for them.

In this article we further explore the weighted tent space scale. In the in-terests of generality, we consider weighted tent spaces T p,q

s (X) associated with a

65

metric measure space X, although our theorems are new even in the classical casewhere X = Rn equipped with the Lebesgue measure. Under su�cient geometricassumptions on X (ranging from the doubling condition to the assumption thatX = Rn), we uncover two previously unknown novelties of the weighted tentspace scale.

First, we identify some real interpolation spaces between T p0,qs0 and T p1,q

s1 when-ever s0 ”= s1. In Theorem 3.3.4 we prove that

(T p0,qs0 , T p1,q

s1 )◊,p◊

= Zp◊

,qs

◊

(3.1)

for appropriately defined parameters, where the scale of ‘Z-spaces’ is defined inDefinition 3.3.3. We require p0, p1, q > 1 in this identification, but in Theorem3.3.9 we show that in the Euclidean setting the result holds for all p0, p1 > 0 andq Ø 1. In the Euclidean setting, Z-spaces have appeared previously in the work ofBarton and Mayboroda [21]. In their notation we have Zp,q

s (Rn) = L(p, ns+1, q).Barton and Mayboroda show that these function spaces are useful in the studyof elliptic boundary value problems with boundary data in Besov spaces. Theconnection with weighted tent spaces shown here is new.

Second, we have continuous embeddings

T p0,qs0 Òæ T p1,q

s1

whenever the parameters satisfy the relation

s1 ≠ s0 = 1p1

≠ 1p0

. (3.2)

This is Theorem 3.3.19. Thus a kind of Hardy–Littlewood–Sobolev embeddingtheorem holds for the weighted tent space scale, and by analogy we are justifiedin referring to the parameter s in T p,q

s as a regularity parameter.We also identify complex interpolation spaces between weighted tent spaces

in the Banach range. This result is already well-known in the Euclidean setting,and its proof does not involve any fundamentally new arguments, but we includeit here for completeness.

These results in this paper will play a crucial role in forthcoming work,1 inwhich we will use weighted tent spaces and Z-spaces to construct abstract ho-mogeneous Hardy–Sobolev and Besov spaces associated with elliptic di�erentialoperators with rough coe�cients. This will be an extension of the abstract Hardyspace techniques initiated independently by Auscher, McIntosh, and Russ [13] andHofmann and Mayboroda [49].

1See Part II of this thesis.

66

Notation

Given a measure space (X, µ), we write L0(X) for the set of µ-measurable func-tions with values in the extended complex numbers C fi {±Œ, ±iŒ}. As usual,by a ‘measurable function’, we actually mean an equivalence class of measurablefunctions which are equal except possibly on a set of measure zero. We will saythat a function f œ L0(X) is essentially supported in a subset E µ X if we haveµ{x œ X \ E : f(x) ”= 0} = 0.

A quasi-Banach space is a complete quasi-normed vector space; see for exam-ple [56, §2] for further information. If B is a quasi-Banach space, we will writethe quasi-norm of B as either ||·||B or ||· | B||, according to typographical needs.

For 1 Æ p Æ Œ, we let pÕ denote the Hölder conjugate of p, which is definedby the relation

1 = 1p

+ 1pÕ ,

with 1/Œ := 0. For 0 < p, q Æ Œ, we define the number

”p,q := 1q

≠ 1p

,

again with 1/Π:= 0. This shorthand will be used often throughout this article.We will frequently use the the identities

”p,q + ”q,r = ”p,r,

”p,q = ”qÕ,pÕ ,

1/q = ”Œ,q = ”qÕ,1.

As is now standard in harmonic analysis, we write a . b to mean that a Æ Cb

for some unimportant constant C Ø 1 which will generally change from line toline. We also write a .c1,c2,... b to mean that a Æ C(c1, c2, . . .)b.

Acknowledgements

I thank Pierre Portal and Pascal Auscher for their comments, suggestions, cor-rections, and support. I also thank Yi Huang, Gennady Uraltsev, and ChristianZillinger for interesting discussions regarding technical aspects of this work.

Some of this work was carried out during the Junior Trimester Program onOptimal Transportation at the Hausdor� Research Institute for Mathematics inBonn. I thank the Hausdor� Institute for their support, as well as the participants

67

of the trimester seminar for their comments. I also thank Lashi Bandara and thePréfecture de Bobigny for indirectly influencing my participation in this program.

Finally, I acknowledge financial support from the Australian Research Coun-cil Discovery Grant DP120103692, and also from the ANR project “Harmonicanalysis at its boundaries” ANR-12-BS01-0013-01.

3.2 Preliminaries

3.2.1 Metric measure spacesA metric measure space is a triple (X, d, µ), where (X, d) is a nonempty metricspace and µ is a Borel measure on X. For every x œ X and r > 0, we writeB(x, r) := {y œ X : d(x, y) < r} for the ball of radius r, and we also writeV (x, r) := µ(B(x, r)) for the volume of this ball. The generalised half-spaceassociated with X is the set X+ := X ◊R+, equipped with the product topologyand the product measure dµ(y) dt/t.

We say that (X, d, µ) is nondegenerate if

0 < V (x, r) < Œ for all x œ X and r > 0. (3.3)

This immediately implies that the measure space (X, µ) is ‡-finite, as X may bewritten as an increasing sequence of balls

X =€

nœNB(x0, n) (3.4)

for any point x0 œ X. Nondegeneracy also implies that the metric space (X, d) isseparable [24, Proposition 1.6]. To rule out pathological behaviour (which is notparticularly interesting from the viewpoint of tent spaces), we will always assumenondegeneracy.

Generally we will need to make further geometric assumptions on (X, d, µ).In this article, the following two conditions will be used at various points. Wesay that (X, d, µ) is doubling if there exists a constant C Ø 1 such that

V (x, 2r) Æ CV (x, r) for all (x, r) œ X+.

A consequence of the doubling condition is that there exists a minimal numbern Ø 0, called the doubling dimension of X, and a constant C Ø 1 such that

V (x, R) Æ C(R/r)nV (x, r)

68

for all x œ X and 0 < r Æ R < Œ.For n > 0, we say that (X, d, µ) is AD-regular of dimension n if there exists

a constant C Ø 1 such that

C≠1rn Æ V (x, r) Æ Crn (3.5)

for all x œ X and all r < diam(X). One can show that AD-regularity (of somedimension) implies doubling. Note that if X is unbounded and AD-regular ofdimension n, then (3.5) holds for all x œ X and all r > 0.

3.2.2 Unweighted tent spaces

Throughout this section we suppose that (X, d, µ) is a nondegenerate metric mea-sure space. We will not assume any further geometric conditions on X withoutexplicit mention. All of the results here are known, at least in some form. Weprovide statements for ease of reference and some proofs for completeness.

For x œ X we define the cone with vertex x by

�(x) := {(y, t) œ X+ : y œ B(x, t)},

and for each ball B µ X we define the tent with base B by

T (B) := X+ \Q

a€

x/œB

�(x)R

b .

Equivalently, T (B) is the set of points (y, t) œ X+ such that B(y, t) µ B. Fromthis characterisation it is clear that if (y, t) œ T (B), then t Æ rB, where we define

rB := sup{r > 0 : B(y, r) µ B for some y œ X}.

Note that it is possible to have rB(y,t) > t.Fix q œ (0, Œ) and – œ R. For f œ L0(X+), define functions Aqf and Cq

–f onX by

Aqf(x) :=A¨

�(x)|f(y, t)|q dµ(y)

V (y, t)dt

t

B1/q

(3.6)

and

Cq–f(x) := sup

B–x

1µ(B)–

A1

µ(B)

¨T (B)

|f(y, t)|q dµ(y) dt

t

B1/q

(3.7)

for all x œ X, where the supremum in (3.7) is taken over all balls B µ X

containing x. We abbreviate Cq := Cq0 . Note that the integrals above are always

69

defined (though possibly infinite) as the integrands are non-negative, and so weneed not assume any local q-integrability of f . We also define

AŒf(x) := ess sup(y,t)œ�(x)

|f(y, t)| (3.8)

andCŒ

– f(x) := supB–x

1µ(B)1+–

ess sup(y,t)œT (B)

|f(y, t)|.

Lemma 3.2.1. Suppose that q œ (0, Œ], – œ R, and f œ L0(X+). Then thefunctions Aqf and Cq

–f are lower semicontinuous.

Proof. For q ”= Œ see [3, Lemmas A.6 and A.7]. It remains only to show thatAŒf and CŒ

– f are lower semicontinuous for f œ L0(X+).For each s > 0 write

�(x) + s := {(y, t) œ X+ : (y, t ≠ s) œ �(x)} = {(y, t) œ X+ : y œ B(x, t ≠ s)}.

Geometrically �(x) + s is a ‘vertically translated’ cone, and �(x) + s ∏ �(x) + r

for all r < s. The triangle inequality implies that

�(x) + s µ �(xÕ) for all xÕ œ B(x, s).

To show that AŒf is lower semicontinuous, suppose that x œ X and ⁄ > 0are such that (AŒf)(x) > ⁄. Then the set O := {(y, t) œ �(x) : |f(y, t)| > ⁄}has positive measure. We have

O =Œ€

n=1O fl (�(x) + n≠1).

Since the sequence of sets O fl (�(x) + n≠1) is increasing in n, and since O haspositive measure, we find that there exists n œ N such that O fl (�(x) + n≠1) haspositive measure. Thus for all xÕ œ B(x, n≠1),

{(y, t) œ �(xÕ) : |f(y, t)| > ⁄} ∏ O fl (�(x) + n≠1)

has positive measure, and so (AŒf)(xÕ) > ⁄. Therefore AŒf is lower semicon-tinuous.

The argument for CŒ– is simpler. We have (CŒ

– f)(x) > ⁄ if and only if thereexists a ball B – x such that

1µ(B)1+–

ess sup(y,t)œT (B)

|f(y, t)| > ⁄.

This immediately yields (CŒ– f)(xÕ) > ⁄ for all xÕ œ B, and so CŒ

– f is lowersemicontinuous.

70

Definition 3.2.2. For p œ (0, Œ) and q œ (0, Œ], the tent space T p,q(X) is theset

T p,q(X) := {f œ L0(X+) : Aqf œ Lp(X)}

equipped with the quasi-norm

||f ||T p,q(X) := ||Aqf ||Lp(X) .

We define T Œ,q(X) by

T Œ,q(X) := {f œ L0(X+) : Cqf œ LŒ(X)}

equipped with the corresponding quasi-norm. We define T Œ,Œ(X) := LŒ(X+)with equal norms.

For the sake of notational clarity, we will write T p,q rather than T p,q(X) unlesswe wish to emphasise a particular choice of X. Although we will always refer totent space ‘quasi-norms’, these are norms when p, q Ø 1.

Remark 3.2.3. Our definition of AŒf gives a function which is less than or equalto the corresponding function defined by Coifman, Meyer, and Stein [33], whichuses suprema instead of essential suprema. We also do not impose any continuityconditions in our definition of T p,Œ. Therefore our space T p,Œ(Rn) is strictlylarger than the Coifman–Meyer–Stein version.

By a cylinder we mean a subset C µ X+ of the form C = B(x, r) ◊ (a, b)for some (x, r) œ X+ and 0 < a < b < Œ. We say that a function f œ L0(X+)is cylindrically supported if it is essentially supported in a cylinder. In generalcylinders may not be precompact, and so the notion of cylindrical support is moregeneral than that of compact support. For all p, q œ (0, Œ] we define

T p,q;c := {f œ T p,q : f is cylindrically supported}.

andLp

c(X+) := {f œ Lp(X+) : f is cylindrically supported}.

A straightforward application of the Fubini–Tonelli theorem shows that forall q œ (0, Œ) and for all f œ L0(X+),

||f ||T q,q

= ||f ||Lq(X+) ,

and so T q,q = Lq(X+). When q = Πthis is true by definition.

71

Proposition 3.2.4. For all p, q œ (0, Œ), the subspace T p,q;c µ T p,q is dense inT p,q. Furthermore, if X is doubling, then for all p, q œ (0, Œ], T p,q is complete,and when p, q ”= Œ, Lq

c(X+) is densely contained in T p,q.

Proof. The second statement has already been proven in [3, Proposition 3.5],2 sowe need only prove the first statement. Suppose f œ T p,q and fix a point x0 œ X.For each k œ N, define

Ck := B(x0, k) ◊ (k≠1, k) and fk := 1Ck

f.

Then each fk is cylindrically supported. We have

limkæŒ

||f ≠ fk||pT p,q

= limkæŒ

ˆX

Aq(1Cc

k

f)(x)p dµ(x)

=ˆ

X

limkæŒ

Aq(1Cc

k

f)(x)p dµ(x)

=ˆ

X

A

limkæŒ

¨�(x)

|(1Cc

k

f)(y, t)|q dµ(y)V (y, t)

dt

t

Bp/q

dµ(x)

=ˆ

X

A¨�(x)

limkæŒ

|(1Cc

k

f)(y, t)|q dµ(y)V (y, t)

dt

t

Bp/q

dµ(x)

= 0.

All interchanges of limits and integrals follow from monotone convergence. Hencewe have f = limkæŒ fk, which completes the proof.

Recall the following duality from [3, Proposition 3.10].

Proposition 3.2.5. Suppose that X is doubling, p œ [1, Œ), and q œ (1, Œ).Then the L2(X+) inner product

Èf, gÍ :=¨

X+f(x, t)g(x, t) dµ(x) dt

t(3.9)

identifies the dual of T p,q with T pÕ,qÕ.

Suppose that p œ (0, 1], q œ [p, Œ], and B µ X is a ball. We say thata function a œ L0(X+) is a T p,q atom (associated with B) if a is essentiallysupported in T (B) and if the size estimate

||a||T q,q

Æ µ(B)”p,q

holds (recall that ”p,q := q≠1 ≠ p≠1). A short argument shows that if a is aT p,q-atom, then ||a||T p,q

Æ 1.2The cases where q = Œ are not covered there. The same proof works—the only missing

ingredient is Lemma 3.4.1, which we defer to the end of the article.

72

Theorem 3.2.6 (Atomic decomposition). Suppose that X is doubling. Let p œ(0, 1] and q œ [p, Œ]. Then a function f œ L0(X+) is in T p,q if and only if thereexists a sequence (ak)kœN of T p,q-atoms and a sequence (⁄k)kœN œ ¸p(N) such that

f =ÿ

kœN⁄kak (3.10)

with convergence in T p,q. Furthermore, we have

||f ||T p,q

ƒ inf ||⁄k||¸p(N) ,

where the infimum is taken over all decompositions of the form (3.10).

This is proven by Russ when q = 2 [81], and the same proof works for generalq œ [p, Œ). For q = Œ we need to combine the original argument of Coifman,Meyer, and Stein [33, Proposition 2] with that of Russ. We defer this to Section3.4.2.

3.2.3 Weighted tent spaces: definitions, duality, and atomsWe continue to suppose that (X, d, µ) is a nondegenerate metric measure space,and again we make no further assumptions without explicit mention.

For each s œ R, we can define an operator V s on L0(X+) by

(V sf)(x, t) := V (x, t)sf(x, t)

for all (x, t) œ X+. Note that for r, s œ R the equality V rV s = V r+s holds, andalso that V 0 is the identity operator. Using these operators we define modifiedtent spaces, which we call weighted tent spaces, as follows.

Definition 3.2.7. For p œ (0, Œ), q œ (0, Œ], and s œ R, the weighted tent spaceT p,q

s is the setT p,q

s := {f œ L0(X+) : V ≠sf œ T p,q}

equipped with the quasi-norm

||f ||T p,q

s

:=------V ≠sf

------T p,q

.

We also define T Œ,Œs in this way. For q ”= Œ, and with an additional parameter

– œ R, we define T Œ,qs;– by the quasi-norm

||f ||T Œ,q

s;–:=

------Cq

–(V ≠sf)------LŒ(X)

.

Note that T Œ,q0;0 = T Œ,q. We write T Œ,q

s := T Œ,qs;0 .

73

Remark 3.2.8. The weighted tent space quasi-norms of Hofmann, Mayboroda,and McIntosh [50, §8.3] (with p ”= Œ) and Huang [51] (including p = Œ with– = 0) are given by

||f ||T p,q

s

(Rn) :=------(y, t) ‘æ t≠sf(y, t)

------T p,q(Rn)

, (3.11)

which are equivalent to those of our spaces T p,qs/n(Rn). In general, when X is

unbounded and AD-regular of dimension n, the quasi-norm in (3.11) (with X

replacing Rn) is equivalent to that of our T p,qs/n. We have chosen the convention of

weighting with ball volumes, rather than with the variable t, because this leads tomore geometrically intrinsic function spaces and supports embedding theoremsunder weaker assumptions.

For all r, s œ R, the operator V r is an isometry from T p,qs to T p,q

s+r. Theoperator V ≠r is also an isometry, now from T p,q

s+r to T p,qs , and so for fixed p and q

the weighted tent spaces T p,qs are isometrically isomorphic for all s œ R. Thus by

Proposition 3.2.4, when X is doubling, the spaces T p,qs are all complete.

Recall the L2(X+) inner product (3.9), which induces a duality pairing be-tween T p,q and T pÕ,qÕ for appropriate p and q when X is doubling. For all s œ Rand all f, g œ L2(X+) we have the equality

Èf, gÍ = ÈV ≠sf, V sgÍ, (3.12)

which yields the following duality result.

Proposition 3.2.9. Suppose that X is doubling, p œ [1, Œ), q œ (1, Œ), ands œ R. Then the L2(X+) inner product (3.9) identifies the dual of T p,q

s withT pÕ,qÕ

≠s .

Proof. If f œ T p,qs and g œ T pÕ,qÕ

≠s , then we have V ≠sf œ T p,q and V sg œ T pÕ,qÕ , soby Proposition 3.2.5 and (3.12) we have

|Èf, gÍ| .------V ≠sf

------T p,q

||V sg||T p

Õ,q

Õ = ||f ||T p,q

s

||g||T p

Õ,q

Õ≠s

.

Conversely, if Ï œ (T p,qs )Õ, then the map f ‘æ Ï(V sf) determines a bounded

linear functional on T p,q with norm dominated by ||Ï||. Hence by Proposition3.2.5 there exists a function g œ T pÕ,qÕ with ||g||T p

Õ,q

Õ . ||Ï|| such that

Ï(f) = Ï(V s(V ≠sf)) = ÈV ≠sf, gÍ = Èf, V ≠sgÍ

for all f œ T p,qs . Since

------V ≠sg

------T p

Õ,q

Õ≠s

= ||g||T p

Õ,q

Õ . ||Ï|| ,

we are done.

74

There is also a duality result for p < 1 which incorporates the spaces T Œ,qs;–

with – > 0. Before we can prove it, we need to discuss atomic decompositions.Suppose that p œ (0, 1], q œ [p, Œ], s œ R, and B µ X is a ball. We say that

a function a œ L0(X+) is a T p,qs -atom (associated with B) if V ≠sa is a T p,q-atom.

This is equivalent to demanding that a is essentially supported in T (B) and that

||a||T q,q

s

Æ µ(B)”p,q .

The atomic decomposition theorem for unweighted tent spaces (Theorem 3.2.6)immediately implies its weighted counterpart.

Proposition 3.2.10 (Atomic decomposition for weighted tent spaces). Supposethat X is doubling. Let p œ (0, 1], q œ [p, Œ], and s œ R. Then a functionf œ L0(X+) is in T p,q

s if and only if there exists a sequence (ak)kœN of T p,qs -atoms

and a sequence (⁄k)kœN œ ¸p(N) such that

f =ÿ

kœN⁄kak (3.13)

with convergence in T p,qs . Furthermore, we have

||f ||T p,q

s

ƒ inf ||⁄k||¸p(N) ,

where the infimum is taken over all decompositions of the form (3.13).

Using this, we can prove the following duality result for p < 1.

Theorem 3.2.11. Suppose that X is doubling, p œ (0, 1), q œ [1, Œ), and s œ R.Then the L2(X+) inner product (3.9) identifies the dual of T p,q

s with T Œ,qÕ

≠s;”1,p

.

Proof. First suppose that a is a T p,qs -atom associated with a ball B µ X, and

that g œ T Œ,qÕ

≠s,”1,p

. Then we have

|Èa, gÍ| ƨ

T (B)|V ≠sa(y, t)||V sg(y, t)| dµ(y) dt

t

Æ ||a||T q,q

s

µ(B)1/qÕµ(B)”1,p ||g||

T Œ,q

Õ≠s,”1,p

Æ µ(B)”p,q

+”q,1+”1,p ||g||

T Œ,q

Õ≠s,”1,p

= ||g||T Œ,q

Õ≠s,”1,p

.

For general f œ T p,qs we write f as a sum of T p,q

s -atoms as in (3.13) and get

|Èf, gÍ| Æ ||g||T Œ,q

Õ≠s,”1,p

||⁄||¸1 Æ ||g||T Œ,q

Õ≠s,”1,p

||⁄||¸p

75

using that p < 1. Taking the infimum over all atomic decompositions completesthe argument.

Conversely, suppose that Ï œ (T p,qs )Õ. Exactly as in the classical duality proof

(see [3, Proof of Proposition 3.10]), using the doubling assumption, there existsa function g œ LqÕ

loc(X+) such that

Ï(f) = Èf, gÍ

for all f œ T p,q;cs . To show that g is in T Œ,qÕ

≠s,”1,p

, we estimate ||V sg||Lq

Õ (T (B)) for eachball B µ X by duality:

||V sg||Lq

Õ (T (B)) = supfœLq(T (B))

|Èf, V sgÍ| ||f ||≠1Lq(T (B))

= supfœLq

c

(T (B))|ÈV sf, gÍ| ||f ||≠1

Lq(T (B)) .

Hölder’s inequality implies that

||V sf ||T p,q

s

Æ µ(B)”q,p ||f ||Lq(T (B))

when f is essentially supported in T (B), so we have

||V sg||Lq

Õ (T (B)) Æ µ(B)”q,p ||Ï||(T p,q

s

)Õ ,

and therefore

||g||T Œ,q

Õ≠s,”1,p

= supBµX

µ(B)”p,1≠(1/qÕ) ||V sg||Lq

Õ (T (B))

Æ ||Ï||(T p,q

s

)Õ supBµX

µ(B)”p,1+”1,q

+”q,p

= ||Ï||(T p,q

s

)Õ ,

which completes the proof.

Remark 3.2.12. Note that q = 1 is included here, and excluded in the otherduality results of this article. Generally the spaces T p,q with p Æ q are easier tohandle than those with p > q.

We end this section by detailing a technique, usually referred to as ‘convexreduction’, which is very useful in relating tent spaces to each other. Supposef œ L0(X+) and M > 0. We define a function fM œ L0(X+) by

(fM)(x, t) := |f(x, t)|M

76

for all (x, t) œ X+. For all q œ (0, Œ] and s œ R we then have

Aq(V ≠sfM) = AMq(V ≠s/Mf)M ,

and for – œ R we also have

Cq–(V ≠sfM) = CMq

–/M(V ≠s/Mf)M .

Therefore, for p œ (0, Œ) we have------fM

------T p,q

s

=------AMq(V ≠s/Mf)M

------Lp(X)

=------AMq(V ≠s/Mf)

------M

LMp(X)

=------f | T Mp,Mq

s/M

------M

,

and likewise for p = Πand q < Πwe have------fM

------T Œ,q

s,–

=------f | T Œ,Mq

s/M,–/M

------M

The case p = q = Πbehaves in the same way:------fM

------T Œ,Œ

s

=------(V ≠s/Mf)M

------LŒ(X+)

= ||f ||MT Œ,Œs/M

.

These equalities often allow us to deduce properties of T p,qs from properties of

T Mp,Mqs/M , and vice versa. We will use them frequently.

3.3 Interpolation and embeddingsAs always, we assume that (X, d, µ) is a nondegenerate metric measure space.We will freely use notation and terminology regarding interpolation theory; theuninitiated reader may refer to Bergh and Löfström [22].

3.3.1 Complex interpolationIn this section we will make the following identification of the complex inter-polants of weighted tent spaces in the Banach range of exponents.

Theorem 3.3.1. Suppose that X is doubling, p0, p1 œ [1, Œ] (not both Œ),q0, q1 œ (1, Œ), s0, s1 œ R, and ◊ œ (0, 1). Then we have the identification

[T p0,q0s0 , T p1,q1

s1 ]◊ = T p◊

,q◊

s◊

where p≠1◊ = (1 ≠ ◊)p≠1

0 + ◊p≠11 , q≠1

◊ = (1 ≠ ◊)q≠10 + ◊q≠1

1 , and s◊ = (1 ≠ ◊)s0 + ◊s1.

77

Remark 3.3.2. In the case where X = Rn with the Euclidean distance andLebesgue measure, this result (with p0, p1 < 1 permitted) is due to Hofmann,Mayboroda, and McIntosh [50, Lemma 8.23]. A more general result, still withX = Rn, is proven by Huang [51, Theorem 4.3] with q0, q1 = Œ also permitted,and with Whitney averages incorporated. Both of these results are proven bymeans of factorisation theorems for weighted tent spaces (with Whitney averagesin the second case), and by invoking an extension of Calderón’s product formulato quasi-Banach spaces due to Kalton and Mitrea [58, Theorem 3.4]. We havechosen to stay in the Banach range with 1 < q0, q1 < Œ for now, as establishinga general factorisation result would take us too far afield.

Note that if p0 = Œ (say) then we are implicitly considering T Œ,q0s0;– with – = 0;

interpolation of spaces with – ”= 0 is not covered by this theorem. This is becausethe method of proof uses duality, and to realise T Œ,q0

s0;– with – ”= 0 as a dual spacewe would need to deal with complex interpolation of quasi-Banach spaces, whichadds di�culties that we have chosen to avoid.

Before moving on to the proof of Theorem 3.3.1, we must fix some notation.For q œ (1, Œ) and s œ R, write

Lqs(X+) := Lq(X+, V ≠qs≠1) := Lq

A

X+, V ≠qs(y, t) dµ(y)V (y, t)

dt

t

B

(3.14)

(this notation is consistent with viewing the function V ≠qs≠1 as a weight on theproduct measure dµ dt/t).

An important observation, originating from Harboure, Torrea, and Viviani[44], is that for all p œ [1, Œ), q œ (1, Œ) and s œ R, one can write

||f ||T p,q

s

=------Hf | Lp(X : Lq

s(X+))------

for f œ L0(X+), whereHf(x) = 1�(x)f.

Hence H is an isometry from T p,qs to Lp(X : Lq

s(X+)). Because of the restrictionon q, the theory of Lebesgue spaces (more precisely, Bochner spaces) with valuesin reflexive Banach spaces is then available to us.

This proof follows previous arguments of the author [3], which are based onthe ideas of Harboure, Torrea, and Viviani [44] and of Bernal [23], with only smallmodifications to incorporate additional parameters. We include it to show wherethese modifications occur: in the use of duality, and in the convex reduction.

78

Proof of Theorem 3.3.1. First we will prove the result for p0, p1 œ (1, Œ). SinceH is an isometry from T

pj

,qj

sj

to Lpj (X : L

qj

sj

(X+)) for j = 0, 1, the interpolationproperty implies that H is bounded (with norm Æ 1 due to exactness of thecomplex interpolation functor)

[T p0,q0s0 , T p1,q1

s1 ]◊ æ Lp◊

1X : [Lq0

s0(X+), Lq1s1(X+)]◊

2.

Here we have used the standard identification of complex interpolants of Banach-valued Lebesgue spaces [22, Theorem 5.1.2]. The standard identification of com-plex interpolants of weighted Lebesgue spaces [22, Theorem 5.5.3] gives

[Lq0s0(X+), Lq1

s1(X+)]◊ = Lq◊

s◊

(X+),

and we conclude that

||f ||T p

◊

,q

◊

s

◊

=------Hf | Lp

◊(X : Lq◊

s◊

(X+))------

Æ------f | [T p0,q0

s0 , T p1,q1s1 ]◊

------

for all f œ [T p0,q0s0 , T p1,q1

s1 ]◊. Therefore

[T p0,q0s0 , T p1,q1

s1 ]◊ µ T p◊

,q◊

s◊

. (3.15)

To obtain the reverse inclusion, we use the duality theorem for complex inter-polation [22, Theorem 4.5.1 and Corollary 4.5.2]. Since X is doubling and by ourrestrictions on p and q, at least one of the spaces T p0,q0

s0 and T p1,q1s1 is reflexive (by

Proposition 3.2.9) and their intersection is dense in both spaces (as it contains thedense subspace Lmax(q0,q1)

c (X+) by Proposition 3.2.4). Therefore the assumptionsof the duality theorem for complex interpolation are satisfied, and we have

T p◊

,q◊

s◊

= (T pÕ◊

,qÕ◊

≠s◊

)Õ

µ [T pÕ0,qÕ

0≠s0 , T

pÕ1,qÕ

1≠s1 ]Õ◊

= [T p0,q0s0 , T p1,q1

s1 ]◊

where the first two lines follow from Proposition (3.2.9) and (3.15), and the thirdline uses the duality theorem for complex interpolation combined with Proposition3.2.9.

We can extend this result to p0, p1 œ [1, Œ] using the technique of [3, Proposi-tion 3.18]. The argument is essentially identical, so we will not include the detailshere.

79

3.3.2 Real interpolation: the reflexive rangeIn order to discuss real interpolation of weighted tent spaces, we need to introducea new scale of function spaces, which we denote by Zp,q

s = Zp,qs (X).3

Definition 3.3.3. For c0 œ (0, Œ), c1 œ (1, Œ), and (x, t) œ X+, we define theWhitney region

�c0,c1(x, t) := B(x, c0t) ◊ (c≠11 t, c1t) µ X+,

and for q œ (0, Œ), f œ L0(X+), and (x, t) œ X+ we define the Lq-Whitneyaverage

(Wqc0,c1f)(x, t) :=

Aˆ�

c0,c1 (x,t)|f(›, ·)|q dµ(›) d·

B1/q

.

For p, q œ (0, Œ), s œ R, c0 œ (0, Œ), c1 œ (1, Œ), and f œ L0(X+), we thendefine the quasi-norm

||f ||Zp,q

s

(X;c0,c1) :=------Wq

c0,c1(V ≠sf)------Lp(X+)

.

and the Z-space

Zp,qs (X; c0, c1) := {f œ L0(X+) : ||f ||Zp,q

s

(X;c0,c1) < Œ}.

In this section we will prove the following theorem, which identifies real inter-polants of weighted tent spaces in the reflexive range. We will extend this to thefull range of exponents in the Euclidean case in the next section.

Theorem 3.3.4. Suppose that X is AD-regular and unbounded, p0, p1, q œ (1, Œ),s0 ”= s1 œ R, and ◊ œ (0, 1). Then for any c0 œ (0, Œ) and c1 œ (1, Œ) we havethe identification

(T p0,qs0 , T p1,q

s1 )◊,p◊

= Zp◊

,qs

◊

(X; c0, c1) (3.16)

with equivalent norms, where p≠1◊ = (1 ≠ ◊)p≠1

0 + ◊p≠11 and s◊ = (1 ≠ ◊)s0 + ◊s1.

As a corollary, in the case when X is AD-regular and unbounded, and whenp, q > 1, the spaces Zp,q

s (X; c0, c1) are independent of the parameters (c0, c1) withequivalent norms, and we can denote them all simply by Zp,q

s .4 We remark thatmost of the proof does not require AD-regularity, but in its absence we obtainidentifications of the real interpolants which are less convenient.

3We use this notation because almost every other reasonable letter seems to be taken.4One can prove independence of the parameters (c

0

, c1

) directly when X is doubling, butproving this here would take us even further o� course.

80

The proof relies on the following identification of real interpolants of weightedLq spaces, with fixed q and distinct weights, due to Gilbert [39, Theorem 3.7].The cases p Æ 1 and q < 1 are not considered there, but the proof still workswithout any modifications in these cases. Note that the original statement of thistheorem contains a sign error in the expression corresponding to (3.17).

Theorem 3.3.5 (Gilbert). Suppose (M, µ) is a ‡-finite measure space and let w

be a weight on (M, µ). Let p, q œ (0, Œ) and ◊ œ (0, 1). For all r œ (1, Œ), andfor f œ L0(M), the expressions

----

----

3r≠k◊

------1x:w(x)œ(r≠k,r≠k+1]f

------Lq(M)

4

kœZ

----

----¸p(Z)

(3.17)

----

----s1≠◊

------1x:w(x)Æ1/sf

------Lq(M,wq)

----

----Lp(R+,ds/s)

(3.18)

and ----

----s≠◊

------1x:w(x)>1/sf

------Lq(M)

----

----Lp(R+,ds/s)

(3.19)

define equivalent norms on the real interpolation space

(Lq(M), Lq(M, wq))◊,p.

The first step in the proof of Theorem 3.3.4 is a preliminary identification ofthe real interpolation norm.

Proposition 3.3.6. Let all numerical parameters be as in the statement of The-orem 3.3.4. Then for all f œ L0(X+) we have the equivalence

------f | (T p0,q

s0 , T p1,qs1 )◊,p

◊

------ ƒ

------x ‘æ

------1�(x)f | (Lq

s0(X+), Lqs1(X+))◊,p

◊

------------Lp

◊ (X).

(3.20)

Proof. We use the notation of the previous section. We have already noted thatthe map H : T p,q

s æ Lp(X : Lqs(X+)) with Hf(x) = 1�(x)f is an isometry. Fur-

thermore, as shown in [3] (see the discussion preceding Proposition 3.12 there),H(T p,q

s ) is complemented in Lp(X : Lqs(X+)), and there is a common projection

onto these spaces. Therefore we have (by [89, Theorem 1.17.1.1] for example)------f | (T p0,q

s0 , T p1,qs1 )◊,p

◊

------ ƒ

------Hf | (Lp0(X : Lq

s0(X+)), Lp1(X : Lqs1(X+)))◊,p

◊

------ .

The Lions–Peetre result on real interpolation of Banach-valued Lebesgue spaces(see for example [76, Remark 7]) then implies that

------f | (T p0,q

s0 , T p1,qs1 )◊,p

◊

------ ƒ

------Hf | (Lp

◊(X : (Lqs0(X+), Lq

s1(X+))◊,p◊

)------ .

Since Hf(x) = 1�(x)f , this proves (3.20).

81

Having proven Proposition 3.3.6, we can use Theorem 3.3.5 to provide someuseful characterisations of the real interpolation norm. For f œ L0(X+) anda, b œ [0, Œ], we define the truncation

fa,b := 1X◊(a,b)f.

Note that in this theorem we allow for p0, p1 Æ 1; we will use this range ofexponents in the next section.

Theorem 3.3.7. Suppose p0, p1, q œ (0, Œ), s0 ”= s1 œ R, and ◊ œ (0, 1), andsuppose that X is AD-regular of dimension n and unbounded. Let r œ (1, Œ).Then for f œ L0(X+) we have norm equivalences

------x ‘æ

------1�(x)f | (Lq

s0(X+), Lqs1(X+))◊,p

◊

------------Lp

◊ (X)

ƒ----

----·n(s1≠s0)(1≠◊) ||f·,Œ||T p

◊

,q

s1

----

----Lp

◊ (R+,d·/·)(3.21)

ƒ----

----·≠n(s1≠s0)◊ ||f0,· ||T p

◊

,q

s0

----

----Lp

◊ (R+,d·/·)(3.22)

ƒr

----

----(r≠nk◊(s1≠s0)------fr≠k,r≠k+1

------T

p

◊

,q

s0)kœZ

----

----¸p

◊ (Z). (3.23)

Proof. First assume that s1 > s0. Let µqs0 be the measure on X+ given by

dµqs0(y, t) := t≠qs0n dµ(y) dt

V (y, t)t .

Since X is AD-regular of dimension n and unbounded, we have that ||f ||Lq(µq

s0 ) ƒ||f ||Lq

s0 (X+). Also define the weight w(y, t) := t≠(s1≠s0)n, so that wqµqs0 = µq

s1 .We will obtain the norm equivalence (3.23). For 1 < r < Œ and k œ Z, we

have r≠k < w(y, t) Æ r≠k+1 if and only if t œ [r(k≠1)/n(s1≠s0), rk/n(s1≠s0)) (here weuse s1 > s0). Using the characterisation (3.17) of Theorem 3.3.5, and replacing r

with rn(s1≠s0), for f œ L0(X+) we have------x ‘æ

------1�(x)f | (Lq

s0(X+), Lqs1(X+))◊,p

◊

------------Lp

◊ (X)

ƒAˆ

X

------1�(x)f

------p

◊

(Lq(µq

s0 ),Lq(wqµq

s0 ))◊,p

◊

dµ(x)B1/p

◊

ƒQ

aˆ

X

ÿ

kœZr≠n(s1≠s0)k◊p

◊

------1�(x)frk≠1,rk

------p

◊

Lq(µq

s0 )dµ(x)

R

b1/p

◊

ƒQ

aÿ

kœZr≠n(s1≠s0)k◊p

◊

ˆX

Aq(V ≠s0frk≠1,rk)(x)p◊ dµ(x)

R

b1/p

◊

=----

----(r≠n(s1≠s0)k◊------frk≠1,rk

------T

p

◊

,q

s0)kœZ

----

----¸p

◊ (Z).

82

This proves the norm equivalence (3.23) for all f œ L0(X+) when s1 > s0. Ifs1 < s0, one simply uses that (Lq

s0(X+), Lqs1(X+))◊,p

◊

= (Lqs1(X+), Lq

s0(X+))1≠◊,p◊

[22, Theorem 3.4.1(a)] to reduce the problem to the case where s0 < s1.The equivalences (3.21) and (3.22) follow from the characterisations (3.18) and

(3.19) of Theorem 3.3.5 in the same way, with integrals replacing sums through-out. We omit the details here.

Finally we can prove the main theorem: the identification of the real inter-polants of weighted tent spaces as Z-spaces.

Proof of Theorem 3.3.4. Suppose f œ L0(X+). Using the characterisation (3.23)in Theorem 3.3.7 with r = c1 > 1, and using aperture c0/c1 for the tent space(making use of the change of aperture theorem [3, Proposition 3.21]), we have

------f | (T p0,q

s0 , T p1,qs1 )◊,p

◊

------p

◊

ƒÿ

kœZc≠n(s1≠s0)k◊p

◊

1

ˆX

Q

aˆ ck

1

ck≠11

ˆB(x,c0t/c1)

|t≠ns0f(y, t)|q dµ(y)V (y, t)

dt

t

R

bp

◊

/q

dµ(x)

ƒˆ

X

ÿ

kœZc≠n(s1≠s0)k◊p

◊

1 ·

·ˆ ck

1

ck≠11

Q

aˆ ck

1

ck≠11

ˆB(x,c0t/c1)

|t≠ns0f(y, t)|q dµ(y)V (y, t)

dt

t

R

bp

◊

/qdr

rdµ(x)

.ˆ

X

ÿ

kœZc≠n(s1≠s0)k◊p

◊

1

ˆ ck

1

ck≠11

Aˆ�

c0,c1 (x,r)|r≠ns0f(y, t)|q dµ(y) dt

Bp◊

/qdr

rdµ(x)

ƒˆ

X

ˆ Œ

0r≠n(s1≠s0)◊p

◊

Aˆ�

c0,c1 (x,r)|r≠ns0f |q

Bp◊

/qdr

rdµ(x)

=¨

X+

Aˆ�

c0,c1 (x,r)|r≠ns

◊f |qBp

◊

/q

dµ(x) dr

r

ƒ ||f ||p◊

Zp

◊

,q

s

◊

(X;c0,c1) ,

using that B(x, c0t/c1) ◊ (ck≠11 , ck

1) µ �c0,c1(x, r) whenever r œ (ck≠11 , ck

1).To prove the reverse estimate we use the same argument, this time using

that for r, t œ (2k≠1, 2k) we have �c0,c1(x, t) µ B(x, 2c0t) ◊ (c≠11 2k≠1, c12k). Using

83

aperture 2c0 for the tent space, we can then conclude that

||f ||p◊

Zp

◊

,q

s

◊

(X;c0,c1)

ƒˆ

X

ÿ

kœZ2≠n(s1≠s0)k◊p

◊

ˆ 2k

2k≠1

Aˆ�

c0,c1 (x,r)|r≠ns0f |q

Bp◊

/qdr

rdµ(x)

.ˆ

X

ÿ

kœZ2≠n(s1≠s0)k◊p

◊ ·

·ˆ 2k

2k≠1

Q

aˆ c12k

c≠11 2k≠1

ˆB(x,2c0t)

|r≠ns0f(y, t)|q dµ(y)V (y, t)

dt

t

R

bp

◊

/qdr

rdµ(x)

ƒˆ

X

ÿ

kœZ2≠n(s1≠s0)k◊p

◊ ·

·ˆ c12k

c≠11 2k≠1

Q

aˆ c12k

c≠11 2k≠1

ˆB(x,2c0t)

|r≠ns0f(y, t)|q dµ(y)V (y, t)

dt

t

R

bp

◊

/qdr

rdµ(x)

ƒ------f | (T p0,q

s0 , T p1,qs1 )◊,p

◊

------p

◊

.

This completes the proof of Theorem 3.3.4.

Remark 3.3.8. Note that this argument shows that----

----(r≠nk◊(s1≠s0)------fr≠k,r≠k+1

------T

p

◊

,q

s0)kœZ

----

----¸p

◊ (Z)ƒ ||f ||Zp

◊

,q

s

◊

(X;c0,c1)

whenever X is AD-regular of dimension n and unbounded, for all p0, p1 œ (0, Œ),c0 œ (0, Œ), and c1 œ (1, Œ). Therefore, since Theorem 3.3.7 also holds for thisrange of exponents, to establish the identification (3.16) for p0, p1 œ (0, Œ) itsu�ces to extend Proposition 3.3.6 to p0, p1 œ (0, Œ). We will do this in the nextsection in the Euclidean case.

3.3.3 Real interpolation: the non-reflexive rangeIn this section we prove the following extension of Theorem 3.3.4. In what follows,we always consider Rn as a metric measure space with the Euclidean distance andLebesgue measure.

Theorem 3.3.9. Suppose that p0, p1 œ (0, Œ), q œ [1, Œ), s0 ”= s1 œ R, and◊ œ (0, 1). Then for any c0 œ (0, Œ) and c1 œ (1, Œ) we have the identification

(T p0,qs0 (Rn), T p1,q

s1 (Rn))◊,p◊

= Zp◊

,qs

◊

(Rn; c0, c1) (3.24)

with equivalent quasi-norms, where p≠1◊ = (1≠◊)p≠1

0 +◊p≠11 and s◊ = (1≠◊)s0+◊s1.

84

The main di�culty here is that vector-valued Bochner space techniques arenot available to us, as we would need to use quasi-Banach valued Lp spaceswith p < 1, and such a theory is not well-developed. Furthermore, although theweighted tent spaces T p,q

s embed isometrically into Lp(X : Lqs(X+)) in this range

of exponents, their image may not be complemented, and so we cannot easilyidentify interpolants of their images.5 We must argue directly.

First we recall the so-called ‘power theorem’ [22, Theorem 3.11.6], which allowsus to exploit the convexity relations between weighted tent spaces. If A is a quasi-Banach space with quasi-norm ||·|| and if fl > 0, then ||·||fl is also a quasi-normon A, and we denote the resulting quasi-Banach space by Afl.

Theorem 3.3.10 (Power theorem). Let (A0, A1) be a compatible couple of quasi-Banach spaces. Let fl0, fl1 œ (0, Œ), ÷ œ (0, 1), and r œ (0, Œ], and definefl := (1 ≠ ÷)fl0 + ÷fl1, ◊ := ÷fl1/fl, and ‡ := rfl. Then we have

((A0)fl0 , (A1)fl1)÷,r = ((A0, A1)◊,‡)fl

with equivalent quasi-norms.

Before proving Theorem 3.3.9 we must establish some technical lemmas. Re-call that we previously defined the spaces Lq

s(X+) in (3.14).

Lemma 3.3.11. Suppose x œ X, – œ (0, Œ), and let all other numerical parame-ters be as in the statement of Theorem 3.3.9. Then for all cylindrically supportedf œ L0(X+) we have

K(–, 1�(x)f ; Lqs0(X+), Lq

s1(X+))= inf

f=Ï0+Ï1

1Aq(V ≠s0Ï0)(x) + –Aq(V ≠s1Ï1)(x)

2(3.25)

and

K(–, 1�(x)f ; Lqs0(X+)p0 , Lq

s1(X+)p1)= inf

f=Ï0+Ï1

1Aq(V ≠s0Ï0)(x)p0 + –Aq(V ≠s1Ï1)(x)p1

2(3.26)

where the infima are taken over all decompositions f = Ï0 + Ï1 in L0(X+) withÏ0, Ï1 cylindrically supported.

5Harboure, Torrea, and Viviani [44] avoid this problem by embedding T 1 into a vector-valued Hardy space H1. If we were to extend this argument we would need identifications ofquasi-Banach real interpolants of certain vector-valued Hardy spaces Hp for p Æ 1, which isvery uncertain terrain (see Blasco and Xu [25]).

85

Proof. We will only prove the equality (3.25), as the proof of (3.26) is essentiallythe same.

Given a decomposition f = Ï0 + Ï1 in L0(X+), we have a corresponding de-composition 1�(x)f = 1�(x)Ï0 + 1�(x)Ï1, with

------1�(x)Ï0

------Lq

s0 (X+)= Aq(V ≠s0Ï0)(x)

and likewise for Ï1. This shows that

K(–, 1�(x)f ; Lqs0(X+), Lq

s1(X+)) Æ inff=Ï0+Ï1

1Aq(V ≠s0Ï0)(x) + –Aq(V ≠s1Ï1)(x)

2.

For the reverse inequality, suppose that 1�(x)f = Ï0 + Ï1 in L0(X+), andsuppose f is essentially supported in a cylinder C. Multiplication by the char-acteristic function 1�(x)flC does not increase the quasi-norms of Ï0 and Ï1 inLq

s0(X+) and Lqs1(X+) respectively, so without loss of generality we can assume

that Ï0 and Ï1 are cylindrically supported in �(x). Now let f = Â0 + Â1 be anarbitrary decomposition in L0(X+), and define

ÊÂ0 := 1�(x)Ï0 + 1X+\�(x)Â0,

ÊÂ1 := 1�(x)Ï1 + 1X+\�(x)Â1.

Then f = ÊÂ0 + ÊÂ1 in L0(X+), and we have

Aq(V ≠s0 ÊÂ0)(x) = Aq(V ≠s0Ï0)(x) =------1�(x)Ï0

------Lq

s0 (X+)

and likewise for ÊÂ1. The conclusion follows from the definition of the K-functional.

Lemma 3.3.12. Suppose f œ Lqc(X+). Then Aqf is continuous.

Proof. Let f be essentially supported in the cylinder C := B(c, r) ◊ (Ÿ0, Ÿ1).First, for all x œ X we estimate

Aqf(x) ÆA¨

C

|f(y, t)|q dµ(y)V (y, t)

dt

t

B1/q

Æ3

infyœB

V (y, Ÿ0)4≠1/q

||f ||Lq(X+)

. ||f ||Lq(X+) ,

using the estimate (3.40) from the proof of Lemma 3.4.1.For all x œ X we thus have

limzæx

|Aqf(x) ≠ Aqf(z)| Æ limzæx

A¨X+

|1�(x) ≠ 1�(z)||f(y, t)|q dµ(y)V (y, t)

dt

t

B1/q

= 0

86

by dominated convergence, since 1�(x) ≠ 1�(z) æ 0 pointwise as z æ x, and sinceA¨

X+|1�(x) ≠ 1�(z)||f(y, t)|q dµ(y)

V (y, t)dt

t

B1/q

. ||f ||Lq(X+) .

Therefore Aqf is continuous.

Having established these lemmas, we can prove the following (half-)extensionof Proposition 3.3.6.

Proposition 3.3.13. Let all numerical parameters be as in the statement ofTheorem 3.3.9. Then for all f œ Lq

c(X+) the function

x ‘æ------1�(x)f | (Lq

s0(X+), Lqs1(X+))◊,p

◊

------ (3.27)

is measurable on X (using the discrete characterisation of the real interpolationquasi-norm), and we have

------f | (T p0,q

s0 , T p1,qs1 )◊,p

◊

------ &

------x ‘æ

------1�(x)f | (Lq

s0(X+), Lqs1(X+))◊,p

◊

------------Lp

◊ (X).

(3.28)We denote the quantity on the right hand side of (3.20) by

------f | Ip

◊

,qs0,s1,◊

------.

Proof. First we take care of measurability. Using Lemma 3.3.11, for x œ X wewrite

------1�(x)f | (Lq

s0(X+), Lqs1(X+))◊,p

◊

------p

◊

=ÿ

kœZ2≠kp

◊

◊K12k, 1�(x)f ; Lq

s0(X+), Lqs1(X+)

2p◊

=ÿ

kœZ2≠kp

◊

◊ inff=Ï0+Ï1

1Aq(V ≠s0Ï0)(x) + 2kAq(V ≠s1Ï1)(x)

2p◊

where the infima are taken over all decompositions f = Ï0 + Ï1 in L0(X+)with Ï0 œ Lq

s0(X+) and Ï1 œ Lqs1(X+) cylindrically supported. By Lemma

3.3.12, we have that Aq(V ≠s0Ï0) and Aq(V ≠s1Ï1) are continuous. Hence foreach k œ Z and for every such decomposition f = Ï0 + Ï1 the function x ‘æAq(V ≠s0Ï0)(x) + 2kAq(V ≠s1Ï1)(x) is continuous. The infimum of these functionsis then upper semicontinuous, therefore measurable.

Next, before beginning the proof of the estimate (3.28), we apply the powertheorem with A0 = T p0,q

s0 , A1 = T p1,qs1 , fl0 = p0, fl1 = p1, and ‡ = p◊. Then we

have fl = p◊, ÷ = ◊p◊/p1, r = 1, and the relation p◊ = (1 ≠ ÷)p0 + ÷p1 is satisfied.We conclude that

((T p0,qs0 , T p1,q

s1 )◊,p◊

)p◊ ƒ ((T p0,q

s0 )p0 , (T p1,qs1 )p1)◊p

◊

/p1,1.

87

Thus it su�ces for us to prove------f | ((T p0,q

s0 )p0 , (T p1,qs1 )p1)◊p

◊

/p1,1------ &

------f | Ip

◊

,qs0,s1,◊

------p

◊ (3.29)

for all f œ Lqc(X+).

We write------f | ((T p0,q

s0 )p0 , (T p1,qs1 )p1)◊p

◊

/p1,1------

=ÿ

kœZ2≠k◊p

◊

/p1K12k, f ; (T p0,q

s0 )p0 , (T p1,qs1 )p1

2

=ÿ

kœZ2≠k◊p

◊

/p1 inff=Ï0+Ï1

3||Ï0||p0

Tp0,q

s0+ 2k ||Ï1||p1

Tp1,q

s1

4

=ÿ

kœZ2≠k◊p

◊

/p1 inff=Ï0+Ï1

ˆX

Aq(V ≠s0Ï0)(x)p0 + 2kAq(V ≠s1Ï1)(x)p1 dµ(x)

Øÿ

kœZ2≠k◊p

◊

/p1

ˆX

inff=Ï0+Ï1

1Aq(V ≠s0Ï0)(x)p0 + 2kAq(V ≠s1Ï1)(x)p1

2dµ(x)

=ÿ

kœZ2≠k◊p

◊

/p1

ˆX

K12k, 1�(x)f(x); Lq

s0(X+)p0 , Lqs1(X+)p1

2dµ(x) (3.30)

=ˆ

X

------1�(x)f | (Lq

s0(X+)p0 , Lqs1(X+)p1)◊p

◊

/p1,1------ dµ(x)

ƒˆ

X

------1�(x)f | (Lq

s0(X+), Lqs1(X+))◊,p

◊

------p

◊

dµ(x) (3.31)

=------f | Ip

◊

,qs0,s1,◊

------p

◊

where again the infima are taken over cylindrically supported Ï0 and Ï1. Theequality (3.30) is due to Lemma 3.3.11. The equivalence (3.31) follows from thepower theorem. This completes the proof of Proposition 3.3.13.

As a corollary, we obtain half of the desired interpolation result.

Corollary 3.3.14. Let all numerical parameters be as in the statement of The-orem 3.3.9, and suppose that X is AD-regular of dimension n and unbounded.

(T p0,qs0 , T p1,q

s1 )◊,p◊

Òæ Zp◊

,qs

◊

(X; c0, c1). (3.32)

Proof. This follows from Theorem 3.3.7, Remark 3.3.8, and the density of Lqc(X+)

in (T p0,qs0 , T p1,q

s1 )◊,p◊

(which follows from the fact that Lqc(X+) is dense in both T p0,q

s0

and T p1,qs1 , which is due to Lemma 3.2.4).

We now prove the reverse containment in the Euclidean case. This rests ona dyadic characterisation of the spaces Zp,q

s (Rn; c0, c1). A standard (open) dyadic

88

cube is a set Q µ Rn of the form

Q =nŸ

i=1(2kxi, 2k(xi + 1)) (3.33)

for some k œ Z and x œ Zn. For Q of the form (3.33) we set ¸(Q) := 2k (thesidelength of Q), and we denote the set of all standard dyadic cubes by D. Forevery Q œ D we define the associated Whitney cube

Q := Q ◊ (¸(Q), 2¸(Q)),

and we define G := {Q : Q œ D}. We write Rn+1+ := (Rn)+ = Rn ◊ (0, Œ). Note

that G is a partition of Rn+1+ up to a set of measure zero.

The following proposition is proven by a simple covering argument.

Proposition 3.3.15. Let p, q œ (0, Œ), s œ R, c0 > 0 and c1 > 1. Then for allf œ L0(Rn+1

+ ),

||f ||Zp,q

s

(Rn;c0,c1) ƒc0,c1

Q

aÿ

QœG

¸(Q)n(1≠ps)[|f |q]p/q

Q

R

b1/p

,

where

[|f |q]Q :=ˆ

Q

|f(y, t)|q dy dt.

As a consequence, we gain a convenient embedding.

Corollary 3.3.16. Suppose q œ (0, Œ), p œ (0, q], and s œ R. Then

Zp,qs (Rn) Òæ T p,q

s (Rn).

89

Proof. We have

||f ||T p,q

s

(Rn) ƒQ

aˆRn

A¨�(x)

|t≠nsf(y, t)|q dy dt

tn+1

Bp/q

dx

R

b1/p

Æ

Q

caˆRn

Q

aÿ

QœG

1Qfl�(x) ”=?(Q)¨

Q

|t≠nsf(y, t)|q dy dt

tn+1

R

bp/q

dx

R

db

1/p

ƒ

Q

caˆRn

Q

aÿ

QœG

1Qfl�(x) ”=?(Q)¸(Q)≠nsq[|f |q]Q

R

bp/q

dx

R

db

1/p

ÆQ

aˆRn

ÿ

QœG

1Qfl�(x) ”=?(Q)¸(Q)≠nps[|f |q]p/q

Qdx

R

b1/p

(3.34)

=Q

aÿ

QœG

¸(Q)≠nps[|f |q]p/q

Q|{x œ Rn : �(x) fl Q ”= ?}|

R

b1/p

.Q

aÿ

QœG

¸(Q)n(1≠ps)[|f |q]p/q

Q

R

b1/p

(3.35)

ƒ ||f ||Zp,q

s

(X;c0,c1) ,

where (3.34) follows from p/q Æ 1, (3.35) follows from

|{x œ Rn : �(x) fl Q ”= ?}| = |B(Q, 2¸(Q))| . |Q| ƒ ¸(Q)n,

and the last line follows from Proposition 3.3.15. This proves the claimed em-bedding.

It has already been shown by Barton and Mayboroda that the Z-spaces forma real interpolation scale [21, Theorem 4.13], in the following sense. We willstop referring to the parameters c0 and c1, as Proposition 3.3.15 implies that theassociated quasi-norms are equivalent.

Proposition 3.3.17. Suppose that all numerical parameters are as in the state-ment of Theorem 3.3.9. Then we have the identification

(Zp0,qs0 (Rn), Zp1,q

s1 (Rn))◊,p◊

= Zp◊

,qs

◊

(Rn).

Now we know enough to complete the proof of Theorem 3.3.9.

Proof of Theorem 3.3.9. First suppose that p0, p1 œ (0, 2]. By Corollary 3.3.16we have

Zpj

,qs

j

(Rn) Òæ T pj

,qs

j

(Rn),

90

for j = 0, 1, and so

(Zp0,qs0 (Rn), Zp1,q

s1 (Rn))◊,p◊

Òæ (T p0,qs0 (Rn), T p1,q

s1 (Rn))◊,p◊

.

Therefore by Proposition 3.3.17 we have

Zp◊

,qs

◊

(Rn) Òæ (T p0,qs0 (Rn), T p1,q

s1 (Rn))◊,p◊

,

and Corollary 3.3.14 then implies that we in fact have equality,

Zp◊

,qs

◊

(Rn) = (T p0,qs0 (Rn), T p1,q

s1 (Rn))◊,p◊

.

This equality also holds for p0, p1 œ (1, Œ) by Theorem 3.3.4. By reiteration,this equality holds for all p0, p1 œ (0, Œ). The proof of Theorem 3.3.9 is nowcomplete.

Remark 3.3.18. This can be extended to general unbounded AD-regular spacesby establishing a dyadic characterisation along the lines of Proposition 3.3.15(replacing Euclidean dyadic cubes with a more general system of ‘dyadic cubes’),and then proving analogues of Corollary 3.3.16 and Proposition 3.3.17 using thedyadic characterisation. The Euclidean applications are enough for our plannedapplications, and the Euclidean argument already contains the key ideas, so weleave further details to any curious readers.

3.3.4 Hardy–Littlewood–Sobolev embeddingsIn this section we prove the following embedding theorem.

Theorem 3.3.19 (Weighted tent space embeddings). Suppose X is doubling.Let 0 < p0 < p1 Æ Œ, q œ (0, Œ] and s0 > s1 œ R. Then we have the continuousembedding

T p0,qs0 Òæ T p1,q

s1

whenever s1 ≠s0 = ”p0,p1. Furthermore, when p0 œ (0, Œ], q œ (1, Œ), and – > 0,we have the embedding

T p0,qs0 Òæ T Œ,q

s1;–

whenever (s1 + –) ≠ s0 = ”p0,Œ.

These embeddings can be thought of as being of Hardy–Littlewood–Sobolev-type, in analogy with the classical Hardy–Littlewood–Sobolev embeddings of ho-mogeneous Triebel–Lizorkin spaces (see for example [55, Theorem 2.1]).

The proof of Theorem 3.3.19 relies on the following atomic estimate. Notethat no geometric assumptions are needed here.

91

Lemma 3.3.20. Let 1 Æ p Æ q Æ Œ and s0 > s1 œ R with s1 ≠ s0 = ”1,p.Suppose that a is a T 1,q

s0 -atom. Then a is in T p,qs1 , with ||a||T p,q

s1Æ 1.

Proof. Suppose that the atom a is associated with the ball B µ X. When p ”= Œ,using the fact that B(x, t) µ B whenever (x, t) œ T (B) and that ≠”1,p > 0, wehave

||a||T p,q

s1=

------Aq(V ≠s1a)

------Lp(B)

Æ------V ≠”1,p

------LŒ(T (B))

------Aq(V ≠s0a)

------Lp(B)

Æ µ(B)”p,1µ(B)”

q,p ||a||T q,q

s0

Æ µ(B)”p,1+”

q,p

+”1,q

= 1,

where we used Hölder’s inequality with exponent q/p Ø 1 in the third line.When p = q = Œ the argument is simpler: we have

||a||T Œ,Œs1

=------V ≠s0≠”1,Œa

------LŒ(T (B))

Æ------V ≠”1,Œ

------LŒ(T (B))

------V ≠s0a

------LŒ(T (B))

Æ µ(B)”Œ,1µ(B)”1,Œ

= 1

using the same arguments as before (without needing Hölder’s inequality).

Now we will prove the embedding theorem. Here is a quick outline of theproof. First we establish the first statement for p0 = 1 and 1 < p1 Æ q by usingpart (1) of Lemma 3.3.20. A convexity argument extends this to 0 < p0 < p1 Æ q,with q > 1. Duality then gives the case 1 < q Æ p0 < p1 Æ Œ, including whenp1 = Œ and – ”= 0. A composition argument completes the proof with q > 1.Finally, we use another convexity argument to allow for q œ (0, 1] (with p1 < Œ).To handle the second statement, we argue by duality again.

Proof of Theorem 3.3.19. The proof is split into six steps, corresponding to thoseof the outline above.

Step 1. First suppose that f œ T 1,qs0 and 1 Æ p1 Æ q. By the weighted atomic

decomposition theorem, we can write f = qk ⁄kak where each ak is a T 1,q

s0 -atom,with the sum converging in T 1,q

s0 . By Lemma 3.3.20 we have

||f ||T p1,q

s1Æ ||⁄k||¸1(N) .

92

Taking the infimum over all atomic decompositions yields the continuous embed-ding

T 1,qs0 Òæ T p1,q

s1 (1 < p1 Æ q Æ Œ, s1 ≠ s0 = ”1,p1). (3.36)

Step 2. Now suppose 0 < p0 < p1 Æ q, s1 ≠ s0 = ”p0,p1 , and f œ T p0,qs0 . Using

(3.36) and noting that q/p0 > 1 and

p0s1 ≠ p0s0 = p0”p0,p1 = ”1,p1/p0 ,

we have

||f ||T p1,q

s1=

------fp0 | T p1/p0,q/p0

p0s1

------1/p0

.------fp0 | T 1,q/p0

p0s0

------1/p0

= ||f ||T p0,q

s0,

which yields the continuous embedding

T p0,qs0 Òæ T p1,q

s1 (0 < p0 < p1 Æ q Æ Œ, q > 1, s1 ≠ s0 = ”p0,p1). (3.37)

Step 3. We now use a duality argument. Suppose 1 < q Æ p0 < p1 Æ Œ.Define fi0 := pÕ

1, fi1 := pÕ0, fl := qÕ, ‡0 := ≠s1, and ‡1 := ≠s0, with s1 ≠ s0 = ”p0,p1 .

Then‡1 ≠ ‡0 = ≠s0 + s1 = ”p0,p1 = ”fi0,fi1 ,

and so (3.37) gives the continuous embedding

T fi0,fl‡ Òæ T fi1,fl

‡1 .

Taking duals results in the continuous embedding

T p0,qs0 Òæ T p1,q

s1 (1 < q Æ p0 < p1 Æ Œ, s1 ≠ s0 = ”p0,p1). (3.38)

Step 4. Now suppose that 0 < p0 Æ q Æ p1 Æ Œ and q > 1, again withs1 ≠ s0 = ”p0,p1 . Then combining (3.37) and (3.38) gives continuous embeddings

T p0,qs0 Òæ T q,q

s0+”p0,q

Òæ T p1,qs0+”

p0,q

+”q,p1

= T p1,qs1 . (3.39)

Step 5. Finally, suppose q Æ 1, and choose M > 0 such that q/M > 1.Then using a similar argument to that of Step 2, with Ms1 ≠ Ms0 = M”p0,p1 =”p0/M,p1/M ,

||f ||T p1,q

s1=

------fM | T p1/M,q/M

Ms1

------1/M

.------fM | T p0/M,q/M

Ms0

------1/M

= ||f ||T p0,q

s

.

93

All possible positions of q relative to 0 < p0 < p1 Æ Œ have thus been covered,so the proof of the first statement is complete.

Step 6. For the second statement, we let (s1 + –) ≠ s0 = ”p0,Œ, and first wesuppose that p0 œ (1, Œ]. Let

fi0 := (1 + –)≠1 œ (0, 1),fi1 := pÕ

0 œ (1, Œ],fl = qÕ, ‡0 = ≠s1, ‡1 = ≠s0.

Then – = ”1,fi0 = ”p1,Œ and so we have

‡1 ≠ ‡0 = ”p0,Œ ≠ – = ”1,fi1 ≠ ”1,fi0 = ”fi0,fi1 ,

which yieldsT fi0,fl

‡0 Òæ T fi1,fl‡1 .

Taking duals yieldsT p0,q

s0 Òæ T Œ,qs1,– ,

which completes the proof when p0 œ (1, Œ]. One last convex reduction argument,as in Step 2, completes the proof.

We remark that this technique also yields the embedding T Œ,qs0,–0 Òæ T Œ,q

s1,–1 when(s1 + –1) ≠ (s0 + –0) = 0, s0 > s1, and 0 Æ –0 < –1.

Remark 3.3.21. The embeddings of Theorems 3.3.19, at least for p, q œ (1, Œ),also hold with Zp,q

s replacing T p,qs on either side (or both sides) of the embed-

ding. This can be proven by writing Zp,qs as a real interpolation space between

tent spaces T p,qs with p near p and s near s, applying the tent space embedding

theorems, and then interpolating again. These embeddings can also be proven‘by hand’, even for p, q Æ 1. We leave the details to any curious readers.

3.4 Deferred proofs

3.4.1 T p,Œ–LŒ estimates for cylindrically supported func-tions

The following lemma, which extends [3, Lemma 3.3] to the case q = Œ, is usedin the proof that T p,Œ is complete (see Proposition 3.2.4).

94

Lemma 3.4.1. Suppose that X is doubling and let K µ X+ be cylindrical. Thenfor all p œ [1, Œ],

||1Kf ||T p,Œ .K ||f ||LŒ(K) .K ||f ||T p,Œ .

Proof. When p = Πthis reduces to

||1Kf ||LŒ(X+) = ||f ||LŒ(K) Æ ||f ||LŒ(X+) ,

which is immediate. Thus it su�ces to prove the result for p = 1, for the generalcase will then follow by interpolating between the L1(K) æ L1(X) and LŒ(K) æLŒ(X) boundedness of the sublinear operator AŒ. Write K µ BK ◊ (Ÿ0, Ÿ1) forsome ball BK = B(cK , rK) µ X and 0 < Ÿ0 < Ÿ1 < Œ.

To prove that ||1Kf ||T 1,Œ .K ||f ||LŒ(K), observe that

||1Kf ||T 1,Œ Æ ||f ||LŒ(K) µ{x œ X : �(x) fl K ”= ?}

Æ ||f ||LŒ(K) V (cK , rK + Ÿ1)

because if x /œ B(cK , rK + Ÿ1) then �(x) fl (BK ◊ (Ÿ0, Ÿ1)) = ?. Note also thatV (cK , rK + Ÿ1) is finite and depends only on K.

Now we will prove that ||f ||LŒ(K) .K ||f ||T 1,Œ . First note that the doublingproperty implies that for all R > 0 and for all balls B µ X,

infxœB

µ(B(x, R)) &X,R,rB

µ(B). (3.40)

Indeed, if x œ B and R Æ 2rB then

µ(B) Æ µ(B(x, R(2rBR≠1))) .X (2rBR≠1)nµ(B(x, R)).

where n Ø 0 is the doubling dimension of X. If R > 2r(B) then since 2rBR≠1 < 1,we have µ(B) Æ µ(B(x, R)).

Let (xj)jœN be a countable dense subset of BK . Then we have

K =€

jœN(�(xj) + Ÿ0) fl K.

By definition the set {(y, t) œ K : |f(y, t)| > 2≠1 ||f ||LŒ(K)} has positive measure,so there exists j œ N such that |f(y, t)| > 2≠1 ||f ||LŒ(K) for (y, t) in some subsetof (�(xj) + Ÿ0) fl K with positive measure. Since (�(xj) + Ÿ0) fl K µ �(x) fl K forall x œ B(xj, Ÿ0), we have that AŒ(f)(x) Ø 2≠1 ||f ||LŒ(K) for all x œ B(xj, Ÿ0).

95

Therefore, using (3.40),

||AŒf ||L1(X) Ø 12µ(B(xj, Ÿ0)) ||f ||LŒ(K)

&X,K µ(BK) ||f ||LŒ(K)

ƒK ||f ||LŒ(K) .

This completes the proof of the lemma.

3.4.2 T p,Œ atomic decompositionAs stated above, the atomic decomposition theorem for T p,Œ can be proven bycombining the arguments of Coifman–Meyer–Stein (who prove the result in theEuclidean case) and Russ (who proves the atomic decomposition of T p,2(X) for0 < p Æ 1 when X is doubling).

First we recall a classical lemma (see for example [81, Lemma 2.2]), whichcombines a Vitali-type covering lemma with a partition of unity. This is provenby combining the Vitali-type covering of Coifmann–Weiss [34, Théorème 1.3] withthe partition of unity of Macías–Segovia [65, Lemma 2.16].

Lemma 3.4.2. Suppose that X is doubling, and let O be a proper subset of X

of finite measure. For all x œ X write r(x) := dist(x, Oc)/10. Then there existsM > 0, a countable indexing set I, and a collection of points {xi}iœI such that

• O = fiiœIB(xi, r(xi)),

• if i, j œ I are not equal, then B(xi, r(xi)/4) and B(xj, r(xj)/4) are disjoint,and

• for all i œ I, there exist at most M indices j œ I such that B(xj, 5r(xj))meets B(xi, 5r(xi)).

Moreover, there exist a collection of measurable functions {Ïi : X æ [0, 1]}iœI

such that

• supp Ïi µ B(xi, 2r(xi)),

• qi Ïi = 1O (for each x œ X the sum q

i Ïi(x) is finite due to the thirdcondition above).

Now we can follow a simplified version of the argument of Russ, which isessentially the argument of Coifman–Meyer–Stein with the partition of unity ofLemma 3.4.2 replacing the use of the Whitney decomposition.

96

Proof of Theorem 3.2.6, with q = Œ. Suppose f œ T p,Œ, and for each k œ Zdefine the set

Ok := {x œ X : AŒf(x) > 2k}.

The sets Ok are open by lower semicontinuity of AŒf (Lemma 3.2.1), and thefunction f is essentially supported in fikœZT (Ok) \ T (Ok+1). Thus we can write

f =ÿ

kœZ1T (O

k

)\T (Ok+1)f. (3.41)

Case 1: µ(X) = Œ. In this case we must have µ(Ok) < Œ for each k œ Z, forotherwise we would have ||AŒf ||Lp(X) = Œ and thus f /œ T p,Œ. Hence for eachk œ Z there exist countable collections of points {xk

i }iœIk µ Ok and measurablefunctions {Ïk

i }iœIk as in Lemma 3.4.2. Combining (3.41) with qiœIk

Ïki = 1Ok

and T (Ok) µ Ok ◊ R+, we can write

f(y, t) =ÿ

kœZ

ÿ

iœIk

Ïki (y)1T (Ok)\T (Ok+1)(y, t)f(y, t)

=ÿ

kœZ

ÿ

iœIk

aki (y, t).

Note that------ak

i

------LŒ(X+)

Æ ess sup(y,t)/œT (Ok+1)

|f(y, t)| Æ 2k+1, (3.42)

the second inequality following from T (Ok+1) = X+ \ (fix/œOk+1�(x)) and the factthat |f(y, t)| Æ AŒf(x) Æ 2k+1 for all x /œ Ok+1 and (y, t) œ �(x).

Defineak

i := 2≠(k+1)µ(Bki )≠1/pak

i ,

where Bki := B(xk

i , 14r(xki )). We claim that ak

i is a T p,Œ-atom associated withthe ball Bk

i . The estimate (3.42) immediately implies the size condition------ak

i

------T Œ,Œ

Æ µ(Bki )”

p,Π,

so we need only show that aki is essentially supported in T (Bk

i ). To show this, it issu�cient to show that if y œ B(xk

i , 2r(xki )) and d(y, (Ok)c) Ø t, then d(y, (Bk

i )c) Øt. Suppose z /œ Bk

i (such a point exists because µ(Bki ) < µ(X) = Œ), Á > 0 and

u /œ Ok such that

d(xki , u) < d(xk

i , (Ok)c) + Á = 10r(xki ) + Á.

97

Then we have

d(y, z) + Á Ø d(z, xki ) ≠ d(xk

i , y) + Á

Ø 12r(xki ) + Á

= 2r(xki ) + 10r(xk

i ) + Á

> d(y, xki ) + d(xk

i , u)Ø d(y, u)Ø t,

where the last line follows from u /œ Ok and d(y, (Ok)c) Ø t. Since z /œ Bki and

Á > 0 were arbitrary, this shows that d(y, (Bki )c) Ø t as required, which proves

that aki is a T p,Œ-atom associated with Bk

i .Thus we have

f(y, t) =ÿ

kœZ

ÿ

iœIk

⁄ki ak

i ,

where⁄k

i = 2k+1µ(Bki )1/p.

It only remains to show thatÿ

kœZ

ÿ

iœIk

|⁄ki |p . ||f ||pT p,Œ .

We estimateÿ

kœZ

ÿ

iœIk

|⁄ki |p =

ÿ

kœZ2(k+1)p ÿ

iœIk

µ(Bki )

.X

ÿ

kœZ2(k+1)p ÿ

iœIk

µ(B(xki , r(xk

i )/4)) (3.43)

Æÿ

kœZ2(k+1)pµ(Ok) (3.44)

. pÿ

kœZ

ˆ 2k

2k≠1tp≠1µ({x œ X : AŒf(x) > t}) dt

= ||AŒf ||pLp(X)

= ||f ||T p,Π,

using doubling in (3.43) and pairwise disjointness of the balls B(xki , r(xk

i )/4) in(3.44). This completes the proof in the case that µ(X) = Œ.

Case 2: µ(X) < Œ. In this case we may have Ok = X for some k œ Z, sowe cannot apply Lemma 3.4.2 as before. One can follow the argument of Russ

98

[81, page 131], which shows that the partition of unity is not required for such k.With this modification, the argument of the previous case still works. We omitthe details.

99

100

Part II

Abstract Hardy–Sobolev andBesov spaces for elliptic

boundary value problems withcomplex LŒ coe�cients

101

Abstract

We establish a theory of Besov–Hardy–Sobolev spaces adapted to operators whichare bisectorial on L2, with bounded HŒ functional calculus on their ranges, andsatisfying o�-diagonal estimates. We apply these spaces to the study of well-posedness of boundary value problems associated with elliptic systems div AÒu =0 with complex t-independent coe�cients on the upper half-space, and withboundary data in classical Besov–Hardy–Sobolev spaces.

In the range of exponents for which the Besov–Hardy–Sobolev spaces adaptedto the perturbed Dirac operator DB are equal to those adapted to the unper-turbed operator D (where B is a bounded multiplier associated with A), we showthat well-posedness of a boundary value problem is equivalent to an associatedprojection being an isomorphism. This is done by classifying all solutions toCauchy–Riemann systems associated with DB, or equivalently all conormal gra-dients to solutions of div AÒu = 0, within certain weighted tent spaces and theirreal interpolants. Our approach uses minimal assumptions on the coe�cients A,and in particular does not require De Giorgi–Nash–Moser estimates.

As an application, for real coe�cient scalar equations, we extend known well-posedness results for the Regularity problem with data in Hardy and Lebesguespaces to a large range of Besov–Hardy–Sobolev spaces by interpolation and du-ality.

103

104

Chapter 4

Introduction

4.1 Introduction and contextThis main focus of this article is the well-posedness of boundary value problemsassociated to divergence-form elliptic systems

LAu := div AÒu = 0, (4.1)

where the unknown is a Cm-valued function u on the upper half-space R1+n+ :=

{(t, x) œ R1+n : t > 0}. We work in ambient dimension 1 + n Ø 2, with m Ø 1.The special case m = 1 corresponds to a scalar equation rather than a system.

The gradient operator Ò sends Cm-valued functions f to Cm(1+n)-valued func-tions (Cm-valued vector fields) Òf by considering f = (f j)m

j=1 as an m-tuple ofC-valued functions, and acting as the usual gradient operator componentwise.The divergence operator div is defined similarly, sending Cm(1+n)-valued func-tions to Cm-valued functions. These di�erential operators are interpreted in theweak (distributional) sense. Vectors v œ Cm(1+n) are split into transversal andtangential parts v = (v‹, vÎ) according to the splitting

Cm(1+n) = Cm ü Cmn, (4.2)

and likewise functions f with codomain Cm(1+n) can be split into transversal andtangential parts f = (f‹, fÎ), with codomains Cm and Cmn respectively. We writeÒÎ and divÎ for the corresponding tangential restrictions of Ò and div.

Throughout the entire article we assume (unless explicitly stated otherwise)that the coe�cient matrix A œ LŒ(Rn+1

+ : L(Cm(1+n))) is bounded, measur-able, complex, and t-independent, meaning that A(t, x) = A(x) for almost every(t, x) œ R1+n

+ . Thus we may identify A as an element of LŒ(Rn : L(Cm(1+n))).

105

Furthermore we assume that A is strictly accretive on curl-free vector fields, inthe sense that there exists Ÿ > 0 such that

ReˆRn

(A(x)f(x), f(x)) dx Ø Ÿ ||f ||22 (4.3)

for all f œ L2(Rn : Cm(1+n)) such that curlÎ(fÎ) = 0. The round bracket in theintegrand above is the usual Hermitean inner product on Cm(1+n). By curlÎ(fÎ) =0 we mean that

ˆjfk = ˆkfj (1 Æ k, j Æ n, k ”= j),

with (weak) partial derivatives acting componentwise on Cm-valued functions.The strict accretivity condition (4.3) is weaker than the usual notion of pointwisestrict accretivity

Re(A(x)v, v) Ø Ÿ|v|2 (v œ Cm(1+n), x œ Rn)

unless m = 1, in which case these two notions are equivalent (see [8, §2]).We always consider weak solutions to (4.1). That is, we say that a function

u œ W 21,loc(Rn : Cm) solves (4.1) if for all Ï œ CŒ

0 (R1+n+ : Cm) we have

¨R1+n

+

(A(x)Òu(t, x), ÒÏ(t, x)) dx dt = 0.

4.1.1 Formulation of boundary value problemsOne can formulate various boundary value problems associated with the equationLAu = 0. First, for 1 < p < Œ, we formulate the Lp-Dirichlet problem for LA,denoted by (DH)p

0,A:

(DH)p0,A :

Y__]

__[

LAu = 0 in R1+n+ ,

limtæ0 u(t, ·) = f œ Lp(Rn : Cm),||Núu||Lp

. ||f ||Lp

,

This should be read:

for all f œ Lp(Rn : Cm),there exists u œ W 2

1,loc(R1+n+ : Cm) solving LAu = 0,

with u æ f in Lp (the boundary condition),such that ||Núu||p . ||f ||p (the interior estimate).

Here Nú is the non-tangential maximal function

Núu(x) := sup(t,y)œ�(x)

|u(t, y)|,

106

where �(x) is the cone in R1+n+ based at x (defined in Subsection 5.1.2). We say

that the problem (DH)p0,A is well-posed if for all f œ Lp(Rn : Cm) there exists a

unique u satisfying these conditions.For all of the boundary value problems that we consider, well-posedness is

defined analogously: for all boundary data, there must exist a unique solution(modulo constants, for Regularity and Neumann problems) which satisfies thestated conditions.

Next, for n/(n+1) < p < Œ, we formulate the Hp-Regularity problem for LA:

(RH)p0,A :

Y___]

___[

LAu = 0 in R1+n+ ,

limtæ0 ÒÎu(t, ·) = ÒÎf œ Hp(Rn : Cmn),------ÊNú(Òu)

------Lp

.------ÒÎf

------Hp

,

where ÊNúu is the modified non-tangential maximal function

ÊNúu(x) := sup(t,y)œ�(x)

Aˆ�(t,y)

|u(·, ›)|2 d· d›

B1/2

(4.4)

(the Whitney region �(t, y) is defined in Subsection 5.1.3), and where Hp(Rn :Cmn) is the (Cmn-valued) real Hardy space, which may be identified with Lp(Rn :Cmn) when p > 1.

Remark 4.1.1. If f is a distribution with ÒÎf œ Hp(Rn : Cmn), then f may beidentified with an element of Hp

1 (Rn : Cm) (the Cm-valued homogeneous Hardy–Sobolev space of order 1, defined in Subsection 5.1.5), and the boundary conditionlimtæ0 ÒÎu(t, ·) = ÒÎf œ Hp(Rn : Cmn) is equivalent to the condition

limtæ0

u(t, ·) = f œ Hp1 (Rn : Cm).

Therefore, by considering potentials rather than tangential gradients, we can seethe Hp-Regularity problem as a kind of Hp

1 -Dirichlet problem. Conversely, byshifting viewpoint from functions to their tangential gradients, the Lp-Dirichletproblem (DH)p

0,A can be seen as a kind of Hp≠1-Regularity problem. It will be

technically convenient for us to consider Regularity problems rather than Dirichletproblems.

For n/(n + 1) < p < Œ, we also formulate the Hp-Neumann problem for LA,

(NH)p0,A :

Y___]

___[

LAu = 0 in R1+n+ ,

limtæ0 ˆ‹A

u(t, ·) = ˆ‹A

f œ Hp(Rn : Cm),------ÊNú(Òu)

------Lp

. ||ˆ‹A

f ||Hp

,

107

where the A-conormal derivative ˆ‹A

of u is given by

ˆ‹A

u(t, ·) = ≠e0 · AÒu(·, t), (4.5)

where ≠e0 is the normal vector to Rn µ R1+n, relative to R1+n+ .

The boundary value problems (DH)p0,A, (RH)p

0,A, and (NH)p0,A are all problems

of order zero:1 in each of these problems, the interior estimates are in terms ofboundary data in either the Lebesgue space Lp or the Hardy space Hp. One canalso formulate Regularity and Neumann problems of order ≠1.

For 1 < p < Œ, the Hp≠1-Regularity problem, which is similar to the Lp-

Dirichlet problem but with a di�erent interior estimate2 and a decay conditionat infinity (see Remark 4.1.1), is

(RH)p≠1,A :

Y______]

______[

LAu = 0 in R1+n+ ,

limtæ0 ÒÎu(t, ·) = ÒÎf œ Hp≠1(Rn : Cmn),

limtæŒ ÒÎu(t, ·) = 0 in Z Õ(Rn : Cmn)||Òu||T p

≠1.

------ÒÎf

------Hp

≠1.

Here Z Õ(Rn : Cmn) is the space of Cmn-valued tempered distributions modulopolynomials; this is the natural space in which all homogeneous Hardy–Sobolevand Besov spaces are embedded. We can enlargen the range of exponents to‘p Ø Œ’; this is done rigorously by using BMO and the homogeneous Hölderspaces �–. For 0 < – < 1 we define

(RH)(Œ,–)≠1,A :

Y______]

______[

LAu = 0 in R1+n+ ,

limtæ0 ÒÎu(t, ·) = ÒÎf œ �–≠1(Rn : Cmn),limtæŒ ÒÎu(t, ·) = 0 in Z Õ(Rn : Cmn)||Òu||T Œ

≠1;–.

------ÒÎf

------�

–≠1,

and furthermore, with – = 0,

(RH)(Œ,0)≠1,A :

Y______]

______[

LAu = 0 in R1+n+ ,

limtæ0 ÒÎu(t, ·) = ÒÎf œ ˙BMO≠1(Rn : Cmn),limtæŒ ÒÎu(t, ·) = 0 in Z Õ(Rn : Cmn)||Òu||T Œ

≠1;0.

------ÒÎf

------ ˙BMO≠1

.

1We use the term ‘order’ here, since there is no confusion with this ‘order’ and the factthat these are boundary value problems for ‘second-order’ elliptic equations. We could use theterm ‘regularity’ instead, but this would probably cause more ambiguity with the Regularityproblem.

2It is known that------ ÂNúu

------Lp

. ||Òu||T p≠1

in the range of p that we shall deal with, but theconverse is in general not known.

108

The spaces ˙BMO≠1 and �–≠1 are best considered as the homogeneous Triebel–Lizorkin space F Œ,2

≠1 and Besov spaces BŒ,Œ–≠1 respectively, as their negative or-

ders prevent traditional (i.e. non-Littlewood–Paley) characterisations in terms ofsmoothness. For these problems the limit in the boundary condition is imposedin the weak-star topology.

With the same ranges of p and –, we also define order ≠1 Neumann problems(NH)p

≠1,A, (NH)(Œ,–)≠1,A , and (NH)(Œ,0)

≠1,A in the same way, with tangential gradientsÒÎ replaced by A-conormal derivatives ˆ‹

A

in the boundary condition (we keepÒÎ in the decay condition at infinity).

Note that in the ‘order ≠1’ problems above, we impose a tent space estimateon Òu rather than a nontangential maximal function estimate. The weightedtent spaces T p

≠1 and T Œ≠1;0 are defined in Subsection 5.1.2. We also impose a

decay condition on the tangential gradient ÒÎu at infinity. For p su�cientlysmall this is implied by the other conditions; we remark that if LA satisfies a DeGiorgi–Nash–Moser condition (see (7.51)) then it is implied for all p < Œ, andalso for some range of – > 0. (see Lemma 7.2.1).

Remark 4.1.2. We have not imposed any nontangential convergence of solutionsto boundary data in the problems above. This is because the classification theo-rems of Auscher and Mourgoglou, in particular [15, Corollaries 1.2 and 1.4], auto-matically yield almost everywhere (a.e.) non-tangential convergence of Whitneyaverages (of either the solution or its conormal gradient, whichever is relevant)to the boundary data. When the operator LA satisfies a De Giorgi–Nash–Mosercondition (see (7.51)) this can be improved to a.e. non-tangential convergencewithout Whitney averages.

Let us summarise the problems we have introduced so far. There are Dirichletproblems of order 0 and 1 (seeing the Hp-Regularity problem as a Hp

1 -Dirichletproblem), Regularity problems of order 0 and ≠1, and Neumann problems oforder 0 and ≠1.

In their recent monograph [21], Barton and Mayboroda consider problemsof intermediate order. They formulate Dirichlet problems of order ◊ œ (0, 1)and Neumann problems of order ◊ œ (≠1, 0) as follows.3 For 0 < ◊ < 1 and

3We have a di�erent indexing convention, where we index our problems according to theorder of the boundary function space used in the interior estimate. Barton and Mayborodarefer to (NB)p

◊≠1,A as (N)p◊,A. Also, Barton and Mayboroda only consider scalar equations, i.e.

the case m = 1.

109

n/(n + ◊) < p Æ Œ,

(DB)p◊,A :

Y___]

___[

LAu = 0 in R1+n+ ,

Tr u = f œ Bp,p◊ (Rn : Cm)

||Òu||L(p,◊,2) . ||f ||Bp,p

◊

,

and

(NB)p׭1,A :

Y___]

___[

LAu = 0 in R1+n+ ,

ˆ‹A

u|ˆR1+n

+= ˆ‹

A

f œ Bp,p◊≠1(Rn : Cm)

||Òu||L(p,◊,2) . ||ˆ‹A

f ||Bp,p

׭1.

The Besov spaces Bp,p◊ are defined in Subsection 5.1.5. The spaces L(p, ◊, 2) are

defined by the norms

||F ||L(p,◊,2) :=Q

a¨

R1+n

+

Aˆ�(t,x)

|· 1≠◊F (·, ›)|2 d› d·

Bp/2

dxdt

t

R

b1/p

with the usual modification when p = Œ. We will refer to these spaces as Z-spaces starting from Subsection 5.1.3 (the letter L already being overused), withan indexing convention such that Zp

◊ = L(p, ◊+1, 2). The boundary condition for(DB)p

◊,A is phrased in terms of the trace operator, which is shown to be boundedfrom W (p, ◊, 2) (the space of functions whose gradients are in L(p, ◊, 2)) to Bp,p

◊

when p > n/(n + ◊) [21, Theorem 3.9]. A similar argument is used to define theboundary conormal derivative ˆ‹

A

u|ˆR1+n

+.

As we stated earlier, it will be technically convenient for us to consider Reg-ularity problems rather than Dirichlet problems. We would also prefer to stickwith problems of order between ≠1 and 0. To this end we define, for ≠1 < ◊ < 0and p such that n/(n + ◊ + 1) < p Æ Œ,

(RB)p◊,A :

Y______]

______[

LAu = 0 in R1+n+ ,

limtæ0 ÒÎu(t, ·) = ÒÎf œ Bp,p◊ (Rn : Cmn)

limtæŒ ÒÎu(t, ·) = 0 in Z Õ(Rn : Cmn)||Òu||Zp

◊

.------ÒÎf

------Bp,p

◊

and

(NB)p◊,A :

Y______]

______[

LAu = 0 in R1+n+ ,

limtæ0 ˆ‹A

u(t, ·) = ˆ‹A

f œ Bp,p◊ (Rn : Cm)

limtæŒ ÒÎu(t, ·) = 0 in Z Õ(Rn : Cmn)||Òu||Zp

◊

. ||ˆ‹A

f ||Bp,p

◊

,

110

replacing the trace conditions with limiting conditions for consistency with the‘endpoint order’ problems that we have already defined,4 writing Zp

◊ instead ofL(p, ◊+1, 2), and including a decay condition at infinity. When p = Œ we imposethe boundary condition in the weak-star topology. If we omit the decay conditionat infinity, the Regularity problem (RB)p

◊,A is equivalent to the Dirichlet problem(DB)p

◊+1,A defined above by an argument similar to that of Remark 4.1.1, andthe Neumann problem (NB)p

◊,A is simply a rewriting of the previously-definedNeumann problem.

The Besov spaces Bp,p◊ with ◊ œ (≠1, 0) are not the only function spaces

situated between Hp0 and Hp

≠1. One can also consider the Hardy–Sobolev spacesHp

◊ with ◊ œ (≠1, 0). These are defined in Subsection 5.1.5; they may be identifiedwith the homogeneous Triebel–Lizorkin spaces F p,2

◊ , whereas the Besov spacesBp,p

◊ may be identified with F p,p◊ when p < Œ. We use Hardy–Sobolev spaces to

formulate the following Regularity and Neumann problems, with ≠1 < ◊ < 0 andn/(n + ◊ + 1) < p < Œ,

(RH)p◊,A :

Y______]

______[

LAu = 0 in R1+n+ ,

limtæ0 ÒÎu(t, ·) = ÒÎf œ Hp◊ (Rn : Cmn)

limtæŒ ÒÎu(t, ·) = 0 in Z Õ(Rn : Cmn)||Òu||T p

◊

.------ÒÎf

------Hp

◊

and

(NH)p◊,A :

Y______]

______[

LAu = 0 in R1+n+ ,

limtæ0 ˆ‹A

u(t, ·) = ˆ‹A

f œ Hp◊ (Rn : Cm)

limtæŒ ÒÎu(t, ·) = 0 in Z Õ(Rn : Cmn)||Òu||T p

◊

. ||ˆ‹A

f ||Hp

◊

.

Furthermore, for ≠1 < ◊ < 0 we formulate ‘endpoint’ problems (RH)Œ◊,A and

(NH)Œ◊,A by replacing Hp

◊ with the homogeneous BMO-Sobolev space ˙BMO◊,which may be identified with the homogeneous Triebel–Lizorkin spaces F Œ,2

◊ .In this case the boundary condition is imposed in the weak-star topology. Incontrast with the boundary value problems with Besov space data, in these casesthere is no trace theorem for the function space defined by Òu œ T p

◊ (this isalso the case for ◊ = ≠1 and ◊ = 0 with the spaces defined by Òu œ T p

≠1 andÊNú(Òu) œ Lp respectively).

Let us briefly summarise the Regularity and Neumann problems that we haveintroduced. At order zero we have problems (RH)p

0,A and (NH)p0,A, which have

4The trace conditions may be removed by invoking [21, Theorem 6.3], the trace theorem forfunctions with gradients in Zp

◊ .

111

boundary data in Hp and a modified non-tangential maximal estimate on theinterior. At order ≠1 we have (RH)p

≠1,A and (NH)p≠1,A, with boundary data in

Hp≠1 and a T p

≠1 interior estimate, and also (RH)(Œ,–)≠1,A and (NH)(Œ,–)

≠1,A with boundarydata in �–≠1 (or ˙BMO≠1 when – = 0) and a T Œ

≠1;– interior estimate. In between,i.e. for order ◊ œ (≠1, 0), we have (RB)p

◊,A and (NB)p◊,A with boundary data in

Bp,p◊ , and (RH)p

◊,A and (NH)p◊,A with boundary data in Hp

◊ . In these cases theinterior estimates are in Zp

◊ and T p◊ respectively. For all problems of negative

order we also impose a decay condition on ÒÎu(t, ·) as t æ Œ in the space Z Õ oftempered distributions modulo polynomials. In many cases this decay conditionis redundant (see Lemma 7.2.1).

Note that for p = 2 and for all s, the problems (RH)2s,A and (RB)2

s,A (andlikewise for Neumann problems) coincide, since H2

s = B2,2s and Z2

s = T 2s .

4.1.2 The first-order approach: perturbed Dirac opera-tors and Cauchy–Riemann systems

Let D denote the di�erential operator on Cm(1+n)-valued functions given by

D :=S

U 0 divÎ

≠ÒÎ 0

T

V

with respect to the transversal/tangential splitting (4.2) of Cm(1+n). We refer to D

as a Dirac operator, because D2 acts as the tangential Laplacian �Î on transversalfunctions. Suppose that B œ LŒ(Rn : L(Cm(1+n))) is a bounded coe�cientmatrix satisfying the same assumptions as those we previously assumed on A:boundedness, measurability, complexity, t-independence, and strict accretivityon curl-free vector fields. We refer to the operator DB as a perturbed Diracoperator.

The Cauchy–Riemann system associated with DB is the first-order partialdi�erential system

(CR)DB :Y]

[ˆtF + DBF = 0 in R1+n

+ ,curlÎ FÎ = 0 in R1+n

+(4.6)

interpreted in the weak (L2loc) sense: that is, we say that F œ L2

loc(R1+n+ : Cm(1+n))

solves (CR)DB if for all test functions Ï œ CŒc (R1+n

+ : Cm(1+n)),¨

R1+n

+

(F (t, x), ˆtÏ(t, x)) dx dt =¨

R1+n

+

(F (t, x), Bú(x)DÏ(t, x)) dx dt

112

and for all  œ CŒc (R1+n

+ : Cm) and 1 Æ j, k Æ n, j ”= k,¨

R1+n

+

(Fk(t, x), ˆjÂ(t, x)) dx dt = ≠¨

R1+n

+

(Fj(t, x), ˆkÂ(t, x)) dx dt.

The condition curlÎ FÎ = 0 is equivalent to the condition F œ R(D), the rangeof D (considered as acting on Cm(1+n)-valued distributions modulo polynomials),and so the Cauchy–Riemann system (CR)DB may be considered as an evolutionequation in the space R(D).

The first-order approach to boundary value problems for elliptic systemsLAu = 0 exploits a correspondence between these elliptic systems and Cauchy–Riemann systems (CR)DB. Recall that A œ LŒ(Rn : L(Cm(1+n))). Write A inmatrix form with respect to the transversal/tangential splitting (4.2) of Cm(1+n)

as

A =S

UA‹‹ A‹Î

A΋ AÎÎ

T

V , (4.7)

and using this representation of A define auxiliary matrices

A :=S

UA‹‹ A‹Î

0 I

T

V and A :=S

U I 0A΋ AÎÎ

T

V

in LŒ(Rn : L(Cm(1+n))). Strict accretivity of A implies that A‹‹ is invertible inLŒ(Rn : L(Cm)), and so A is invertible in LŒ(Rn : L(Cm(1+n))). Thus we maydefine

A := AA≠1

.

The transformed coe�cient matrix A is bounded and strictly accretive on curl-free vector fields as in (4.3), and ˆA = A [8, Proposition 3.2].

The A-conormal gradient ÒAu of a function u : R1+n+ æ Cm is defined by

ÒAu =S

Uˆ‹A

u

ÒÎu

T

V , (4.8)

where the A-conormal derivative ˆ‹A

is defined in (4.5). Notice that the compo-nents of ÒAu are exactly the quantities appearing in the boundary conditions ofthe Regularity and Neumann problems. This explains our preference for Regu-larity problems over Dirichlet problems.

The following theorem, due to Auscher, Axelsson (Rosén), and McIntosh, pro-vides a bridge between elliptic equations LAu = 0 and Cauchy–Riemann systems(CR)DB. See [8, §3], [7, Proposition 4.1], [79, §2], and [15, Lemma 7.1] for proofsand discussions.

113

Theorem 4.1.3 (Auscher–Axelsson–McIntosh). Let A be as above, and let B =A. If u solves LAu = 0, then the conormal gradient ÒAu solves the Cauchy–Riemann system (CR)DB. Conversely, if F solves (CR)DB, then there exists afunction u, unique up to an additive constant, such that LAu = 0 and F = ÒAu.

Therefore in our consideration of elliptic systems we may focus on Cauchy–Riemann systems if they are more useful. The principal advantage of Cauchy–Riemann systems over elliptic equations is that the Cauchy equation ˆtF +DBF = 0 can be solved by semigroup methods. We will sketch how this isdone, following Auscher, Axelsson, and McIntosh [8] and Auscher and Axelsson[7]. This approach is the foundation for the rest of the article.

Consider D as an unbounded operator on L2 := L2(Rn : Cm(1+n)) with naturaldomain, and consider B as a multiplication operator on L2. Then, still assumingstrict accretivity of B on R(D),5 the composition DB is bisectorial and hasbounded HΠfunctional calculus on its range [8, Proposition 3.3 and Theorem3.4].6 This is a highly non-trivial fact: it is part of the framework developed byAxelsson, Keith, and McIntosh [19], which encompasses the solution of the Katosquare root problem [9].

Using the direct sum decomposition

L2 = N (DB) ü R(DB)

which follows from bisectoriality of DB, along with the bounded HΠfunctionalcalculus associated with DB on R(DB), we obtain a decomposition

L2 = N (DB) ü R(DB)+ ü R(DB)≠.

The positive and negative spectral subspaces R(DB)± are the images of R(DB)under the projections ‰±(DB), which are defined via the functions ‰± : C\ iR æ{≠1, 1} given by

‰±(z) := 1z:± Re(z)>0;

‰+ and ‰≠ are the characteristic functions of the right and left half-plane respec-tively. They are bounded and holomorphic on every bisector, so they fall withinthe scope of the HŒ functional calculus.

On the positive spectral subspace R(DB)+ we can construct a strongly con-tinuous semigroup (e≠tDB)t>0 via the family of functions (z ‘æ e≠tz)t>0, which

5R(D) denotes the closure of the range of D in L2(Rn : Cm(1+n)). We can obviously restrictattention to such functions when defining ‘strict accretivity on curl-free vector fields’.

6These notions are properly discussed in Section 5.2.1.

114

are holomorphic and bounded on the right half-plane. For each f œ R(DB)+ wemay construct a generalised Cauchy operator C+

DBf , defined by

(C+DBf)(t, x) := (e≠tDBf)(x).

The following theorem is a combination of parts of [8, Theorem 2.3] and [7,Corollary 8.4].

Theorem 4.1.4 (Auscher–Axelsson–McIntosh). If f œ R(DB)+, then C+DBf

solves (CR)DB, with------ÊNú(C+

DBf)------2

ƒ ||f ||2 and limtæ0

(C+DBf)(t, ·) = f in L2.

Conversely, if F solves (CR)DB and ÊNú(F ) œ L2, then F = C+DBf for a unique

f œ R(DB)+.

By combining this with Theorem 4.1.3, we obtain a new characterisation ofwell-posedness of the boundary value problems (RH)2

0,A and (NH)20,A. Consider

the H2-Regularity problem (RH)20,A and let B = A. A function u solves LAu = 0

with ÊNú(Òu) œ L2 (Òu and ÒAu are interchangeable in this assumption) if andonly if ÒAu = C+

DBg for some g œ R(DB)+, and therefore ÒÎu(t, ·) = (C+DBg)(t)Î

and limtæ0 ÒÎu(t, ·) = gÎ. Hence (RH)20,A is well-posed if and only if g ‘æ gÎ is an

isomorphism from R(DB)+ to L2(Rn : Cmn) fl N (curlÎ). By the same argument,(NH)2

0,A is well-posed if and only if g ‘æ g‹ is an isomorphism from R(DB)+ toL2(Rn : Cm).

By characterising solutions to (CR)DB within various function spaces, we canreduce well-posedness of corresponding Regularity and Neumann problems toproving that the transversal and tangential restriction maps are isomorphismsbetween certain function spaces ‘on the boundary’. However, in this section weonly described how to handle boundary value problems of order 0 with L2 bound-ary data. We shall extend this technique to boundary value problems of moregeneral order, and beyond L2.

Adapted function spaces

‘Adapted’ Hardy spaces HpL, with respect to which some operator L has good

properties (such as bounded HŒ functional calculus), have been developed invarious contexts. For example, Hardy spaces of di�erential forms on Rieman-nian manifolds were constructed by Auscher, McIntosh, and Russ [13] (these areadapted to the Hodge-Dirac operator d + dú on the de Rham complex); Hardy

115

spaces adapted to non-negative self-adjoint operators satisfying Davies–Ga�neyestimates on spaces of homogeneous type were studied by Hofmann, Lu, Mitrea,Mitrea, and Yan [48] (generalising the work of Auscher, McIntosh, and Russ);Hardy spaces adapted to divergence-form elliptic operators on Rn were devel-oped by Hofmann and Mayboroda [49] and also McIntosh [50]. This is a verysmall sample of the work that has been done.

Hardy spaces HpDB and Sobolev spaces Wp

≠1,DB adapted to perturbed Diracoperators DB were introduced by Auscher and Stahlhut [16] (see also Stahlhut’sthesis [84]).7 These spaces consist of Cm(1+n)-valued functions (at least formally);the simplest case is

H2DB = R(DB) µ L2 = L2(Rn : Cm(1+n)).

The bounded HΠcalculus of DB on H2DB extends by boundedness to Hp

DB andWp

≠1,DB, yielding spectral decompositions

HpDB = Hp,+

DB ü Hp,≠DB, Wp

≠1,DB = Wp,+≠1,DB ü Wp,≠

≠1,DB.

Furthermore, the Cauchy operator C+DB on R(DB)+ extends to operators on

Hp,+DB and Wp,+

≠1,DB, both of which we denote by C+DB.

The main application of these spaces, which incorporates results from both[16] and the subsequent work of Auscher and Mourgoglou [15], is a classification ofsolutions to the Cauchy–Riemann system (CR)DB with various Lp-type interiorestimates, for p such that certain DB-adapted spaces may be identified withD-adapted spaces.8

Theorem 4.1.5 (Auscher–Mourgoglou–Stahlhut). Let 1 < p < Œ be such thatHp

DB ƒ HpD.

(i) If f œ Hp,+DB, then C+

DBf solves (CR)DB, with------ÊNú(C+

DBf)------p

ƒ ||f ||Hp

and limtæ0

C+DBf(t) = f in Hp.

Conversely, if F solves (CR)DB and ÊNúF œ Lp, then F = C+DBf for some

f œ Hp,+DB.

7We will not define HpDB here, but only mention that it is defined, along with more general

spaces, in Section 6.1.8For simplicity we only state results for 1 < p < Œ here. Corresponding results for p Æ 1

and p = Œ (BMO and Hölder spaces) are also available.

116

(ii) If f œ Wp,+≠1,DB, then C+

DBf solves (CR)DB, with------C+

DBf------T p

≠1ƒ ||f ||W p

≠1and lim

tæ0C+

DBf(t) = f in W p≠1.

Conversely, if F œ T p≠1 solves (CR)DB and limtæŒ F (t)Î = 0 in Z Õ(Rn),

then F = C+DBf for some f œ Wp,+

≠1,DB.

Furthermore, Auscher and Stahlhut [16, Theorem 5.1] show that for everyB there exists an open interval I0(H, DB) – 2 such that Hp

DB ƒ HpD for all

p œ I0(H, DB), thus yielding a nontrivial range of exponents for which Theorem4.1.5 applies.

As we described in the p = 2 case, Theorem 4.1.5 implies a characterisation ofwell-posedness of various Regularity and Neumann problems, both of order 0 andorder ≠1, in terms of certain transversal and tangential restriction maps beingisomorphisms. We will state our extension of this result in Theorem 4.1.7.

The main goal of this article is to extend Theorem 4.1.5 to orders œ (≠1, 0), incorporating both Hardy–Sobolev spaces and Besov spaces.To this end, we introduce Hardy–Sobolev spaces Hp

s,L and Besov spaces Bps,L

adapted to operators L satisfying ‘standard assumptions’, which are satisfied inparticular by the perturbed Dirac operators DB and BD. We define extensionoperators

(QÏ,Lf)(t) = Ï(tL)f (t > 0, f œ R(L))

for appropriate holomorphic functions Ï, and the adapted Hardy–Sobolev andBesov norms are then, roughly speaking, defined by

||f ||Hp

s,L

:= ||QÏ,Lf ||T p

s

, ||f ||Bp

s,L

:= ||QÏ,Lf ||Zp

s

.

These definitions are reminiscent of the Ï-transform characterisations of Triebel–Lizorkin and Besov spaces due to Frazier and Jawerth [38], with functional cal-culus and tent/Z-spaces taking the place of discretised Littlewood–Paley decom-positions and sequence spaces.

Chapters 5 and 6 are occupied with setting up a su�ciently rich generaltheory of adapted Hardy–Sobolev and Besov spaces. The theory is relativelystraightforward once enough preliminaries have been collected, but this takessome time. We point out in particular the amount of work needed to establishindependence on Ï of the spaces Hp

s,L and Bps,L (essentially all of Sections 5.2.3

and 5.2.4) and the care which must be taken in discussing completions (Subsection6.1.3), which is necessary to discuss interpolation.

117

4.1.3 Characterisation of solutions to CR systems, andapplications to well-posedness

The main theorem of this article is the following classification of solutions tothe Cauchy–Riemann system (CR)DB. In this statement we restrict ourselves to1 < p < Œ. Our theorem allows for p Æ 1 and p = Œ, but the correspondingresults are better stated in terms of the ‘exponent notation’ that we introducein Subsection 5.1.1. See Theorems 7.3.1 and 7.3.2 for the full statements of thisresult.

Theorem 4.1.6. Let ≠1 < s < 0 and 1 < p < Œ.

(i) Suppose that Hps,DB = Hp

s,D. If f œ Hp,+s,DB, then C+

DBf solves (CR)DB, with------C+

DBf------T p

s

ƒ ||f ||Hp

s

and limtæ0

C+DBf(t) = f in Hp

s ,

and furthermore limtæŒ C+DBf(t)Î = 0 in Z Õ(Rn). Conversely, if F œ T p

s

solves (CR)DB and limtæŒ F (t)Î = 0 in Z Õ(Rn), then F = C+DBf for some

f œ Hp,+s,DB.

(ii) Suppose that Bps,DB = Bp

s,D. If f œ Bp,+s,DB, then C+

DBf solves (CR)DB, with------C+

DBf------Zp

s

ƒ ||f ||Bp,p

s

and limtæ0

C+DBf(t) = f in Bp,p

s .

and furthermore limtæŒ C+DBf(t)Î = 0 in Z Õ(Rn). Conversely, if F œ Zp

s

solves (CR)DB and limtæŒ F (t)Î = 0 in Z Õ(Rn), then F = C+DBf for some

f œ Bp,+s,DB.

Parts (i) and (ii) of this theorem are essentially identical, the only modifica-tions being the replacement of (adapted) Hardy–Sobolev spaces with (adapted)Besov spaces, and of tent spaces with Z-spaces. In fact, our arguments applyequally to both parts, and we prove them simultaneously. Although the theo-rem can be thought of as ‘intermediate to’ Theorem 4.1.5, it does not simplyfollow by any interpolation procedure. It is proven similarly, but the underlyingtechniques must be generalised, and this takes a considerable amount of work.Neither direction is easy, but the ‘converse’ direction is certainly the harder one.

Starting from information on the intervals I0(H, DB), I0(H, DBú) – 2 (asgiven by Auscher and Stahlhut), a procedure of ‘¸-duality’ and interpolationallows us to find non-trivial regions I(H, DB) and I(B, DB) of exponents (p, s)for which Theorem 4.1.6 applies (this is done in Subsection 6.2.1).

118

With Theorem 4.1.6 as a springboard, we are able to extend the characterisa-tion of well-posedness of Regularity and Neumann problems, described for p = 2after the statement of Theorem 4.1.3 and then extended to p ”= 2 and s œ {≠1, 0}by Auscher, Mourgoglou, and Stahlhut, as follows.

When ≠1 Æ s Æ 0 and 1 < p < Œ (and in fact for a slightly larger range ofexponents), Hp

s,D is equal to the set of those f œ Hps (Rn : Cm(1+n)) with curlÎ fÎ =

0. Let N‹ and NÎ denote the projections from Hps,D onto Hp

s,‹ := Hps (Rn : Cm)

and Hps,Î := Hp

s (Rn : Cmn) fl N (curlÎ) respectively. For (p, s) as in Theorem 4.1.6we have an identification of Hp,+

s,DB as a subset of Hps,D, and so we can use the

projections N‹ and NÎ to define

N (p,s)H,DB,Î : Hp,+

s,DB æ Hps,Î and N (p,s)

H,DB,‹ : Hp,+s,DB æ Hp

s,‹

Corresponding definitions of N (p,s)B,DB,Î and N (p,s)

B,DB,‹ are also made for Besov spaces.It is these operators that carry the well-posedness of Regularity and Neumannproblems, as shown by the following theorem.9 The s œ {≠1, 0} endpoints followfrom Theorem 4.1.5.

Theorem 4.1.7. Let B = A, ≠1 Æ s Æ 0, and 1 < p < Œ. Suppose thatHp

s,DB = Hps,D. Then (RH)p

s,A (resp. (NH)ps,A) is well-posed if and only if N (p,s)

H,DB,Î

(resp. N (p,s)H,DB,‹) is an isomorphism. The same results hold mutatis mutandi for

Besov spaces.

For all coe�cients A, the Lax–Milgram theorem guarantees well-posedness ofthe problems (RH)2

≠1/2,A and (NH)2≠1/2,A (see [12, Theorems 3.2 and 3.3]). We

refer to solutions of these boundary value problems as energy solutions. Thereare certain situations where (RH)p

s,A is well-posed for some (p, s), but where theunique solution to (RH)p

s,A with boundary data f œ H2≠1/2 fl Hp

s (energy data)is not the corresponding energy solution. This is shown in [18] for the Dirichletproblems. This behaviour shows why we insist on specifying an interior estimatein the definitions of our boundary value problems.

We say that a boundary value problem (as above) is compatibly well-posed if itis well-posed, and if in addition the unique solution to the boundary value problemwith energy data is the energy solution. By Theorem 4.1.7, (RH)p

s,A is compatiblywell-posed if N (p,s)

H,DB,Î is an isomorphism, and if the inverses (N (p,s)H,DB,Î)≠1 and

(N (2,≠1/2)H,DB,Î )≠1 are consistent, in the sense that they are equal on the intersection

9The full theorem (Theorem 7.4.4) allows for p Æ 1 and p = Œ (and again, uses new‘exponent notation’).

119

Hps,Î flH2

≠1/2,Î (and likewise for Neumann problems, and with Besov spaces). Thisallows us to interpolate compatible well-posedness as a straightforward corollaryof Theorem 4.1.7.10 Furthermore, by using real interpolation, we can deducecompatible well-posedness of boundary value problems with Besov boundary datafrom that of those with Hardy–Sobolev boundary data.

Theorem 4.1.8. Suppose ≠1 Æ s0, s1 Æ 0, 1 < p0, p1 < Œ, and – œ (0, 1), andlet

1p

= 1 ≠ –

p0+ –

p1and s = (1 ≠ –)s0 + –s1.

(i) If Hpj

sj

,DB = Hpj

s0,D for j = 0, 1, and if (RH)p0s0,A and (RH)p0

s1,A are compatiblywell-posed, then (RH)p

s,A is compatibly well-posed, and furthermore if s0 ”= s1

then (RB)ps,A is compatibly well-posed.

(ii) If Bpj

sj

,DB = Bpj

s0,D for j = 0, 1, and if (RB)p0s0,A and (RB)p0

s1,A are compatiblywell-posed, then (RB)p

s,A is compatibly well-posed.

Corresponding results are also true for Neumann problems.

Since invertibility is stable in complex interpolation scales, well-posedness ofour boundary value problems is also stable, in the following sense.11

Theorem 4.1.9. Let ≠1 < s < 0 and 1 < p < Œ, and suppose that Hp0s0,DB =

Hp0s0,D for all (p0, s0) in some neighbourhood of (p, s) (in the usual topology on

R2). Suppose also that (RH)ps,A is (compatibly) well-posed. Then (RH)p1

s1,A is(compatibly) well-posed for all (p1, s1) in some neighbourhood of (p, s). Similarresults hold for Neumann problems and with Besov spaces.

Note that well-posedness extrapolates to well-posedness, and compatible well-posedness extrapolates to compatible well-posedness.

Finally, we have the duality result for well-posedness.12

Theorem 4.1.10. Let ≠1 Æ s Æ 0 and 1 < p < Œ. Then (RH)ps,A is (compati-

bly) well-posed if and only if (RH)pÕ

≠s≠1,Aú is (compatibly) well-posed, and similarresults hold for Neumann problems and with Besov spaces.

10As with the other theorems, we have not stated this in full generality. The full result isTheorem 7.4.5.

11The full result here is Theorem 7.4.6.12See Theorem 7.4.8.

120

Note that the mapping (p, s) ‘æ (pÕ, ≠s ≠ 1) can be seen as a reflection aboutthe point (1/2, ≠1/2) in the (1/p, s)-plane. This corresponds to what we willlater refer to as ‘¸-duality’.

These theorems can be used to derive new well-posedness results for Regularityproblems (RH)p

s,A with fractional order s œ (≠1, 0), and also to derive knownresults for (RB)p

s,A which were recently obtained by di�erent methods by Bartonand Mayboroda [21]. For details see Subsection 7.4.2.

4.2 Summary of the article

In Section 5.1 we introduce the various function spaces that we use, their basicproperties, and their interrelations. There are two types of function spaces thatwe consider. First, the ‘ambient spaces’: tent spaces, Z-spaces, and slice spaces.Many of the results here are new, or have not been used in this context, so wemake ourselves well acquainted with these spaces. The second type of space thatwe consider are the homogeneous ‘smoothness spaces’: Hardy–Sobolev spaces,Besov spaces, and so on. Since we do not establish any new properties of thesespaces, we restrict ourselves to a quick review. We also introduce a new system ofnotation for exponents. This is not strictly necessary, but it greatly cleans up theexposition of later parts of the article and makes the flow of ideas more apparent.

In Section 5.2 we discuss the basic operator-theoretic notions that we willneed. The operators that we use in applications (i.e. the perturbed Dirac oper-ators DB and BD) are bisectorial, with bounded HŒ functional calculi on theirranges, and satisfying certain o�-diagonal estimates. Most of the abstract theorywe develop works for any operator A satisfying these ‘standard assumptions’, sowe work with such operators until we are forced to use more specific properties ofperturbed Dirac operators. We establish the boundedness of certain integral op-erators between tent spaces and Z-spaces. Particular examples of these operatorsare given in terms of ‘extension’ and ‘contraction’ operators QÏ,A and SÂ,A, whichwe will introduce and discuss. This section culminates in Theorem 5.2.20, whichquantifies when operators of the form QÂ,A÷(A)SÏ,A are bounded between di�er-ent tent/Z-spaces, where ÷ is a holomorphic function on an appropriate bisectorwhich is not necessarily bounded.

In Section 6.1 we define and investigate Hardy–Sobolev and Besov spacesadapted to an operator A satisfying the aforementioned standard assumptions.First we introduce ‘pre’-Besov–Hardy–Sobolev spaces Hp

A and BpA and establish

121

their basic properties in Subsection 6.1.1. Mapping properties of the holomorphicfunctional calculus between these spaces, including boundedness for HŒ functionsof A and ‘regularity shifting’ estimates for operators such as powers of A, arecollected in Subsection 6.1.2. These all follow from Theorem 5.2.20. In Subsection6.1.3 we discuss completions. This is more subtle than it initially seems. We define‘canonical completions’ ÂHp

A and ÂBpA in terms of an auxiliary functions Â, and

show how these can be used to formulate satisfactory duality and interpolationresults (Proposition 6.1.19 and Theorem 6.1.23). Finally, in Subsection 6.1.4 weshow that the Cauchy operators C±

A produce strong solutions of the Cauchy–Riemann equation (CR)A with initial data in any completion of any pre-Besov–Hardy–Sobolev space, and we also show the quasi-norm equivalence

||f ||HpA

ƒ------C±

A f------T p (f œ Hp,±

A ) (4.9)

when p = (p, s) with p Æ 2 and s < 0, and likewise for Besov spaces and Z-spaces (Theorem 6.1.25). This is important because it implies that the Cauchyoperators can be used to construct solutions of (CR)A which satisfy good tent/Z-space estimates, at least for this range of exponents p = (p, s).

Up until this point, we work with CN -valued functions for an arbitrary N œN, as in this abstract setting we gain nothing from the transversal/tangentialstructure of Cm(1+n).

In Section 6.2 we consider the case when A is a perturbed Dirac operator ofthe form DB or BD (and we finally specialise to Cm(1+n)-valued functions). Weshow that for a large range of exponents p the spaces Hp

D and BpD may be realised

as projections of classical smoothness spaces (Theorem 6.2.1). Then we define‘identification regions’ I(H, DB) and I(B, DB), consisting of exponents p forwhich we can identify Hp

D and BpD as completions of Hp

DB and BpDB respectively.

These regions turn out to be stable under interpolation and ¸-duality (in a sensewhich interchanges B and Bú). Finally, in Theorem 6.2.12 we show that forp = (p, s) œ I(H, DB) with s < 0 we have boundedness of the Cauchy operatorC+

DB from HpDB to T p, extending the ‘abstract’ estimate (4.9) (and likewise for

Besov spaces and Z-spaces). This is a long argument which requires various ad-hoc estimates. The result is known to fail for s = 0, so it does not follow byinterpolation.

After presenting some basic properties of gradients of solutions of LAu = 0(or equivalently solutions of (CR)DB) we prove Theorems 7.3.1 and 7.3.2, theclassification of solutions to (CR)DB in tent/Z-spaces with a decay condition at

122

infinity.13 The argument is quite long, particularly for exponents p = (p, s) withp > 2, and uses all the preceding material. We have been (perhaps excessively)pedantic in citing dependence on previous results, so it should be possible to treatcertain technical lemmas as ‘black boxes’ in initial readings. We point out thatalthough these results are ‘intermediate to’ the Auscher–Mourgoglou–Stahlhuttheorem 4.1.5, and although it is proven with a similar argument, it does notfollow by any interpolation procedure. The results must be reproven manually.14

In Section 7.4 we present straightforward (but still somewhat technical) ap-plications to well-posedness and compatible well-posedness of Regularity andNeumann problems. These have already been summarised in the introduction(Subsection 4.1.3). In particular, we derive a range of well-posedness for theRegularity problem for real coe�cient scalar equations in Subsection 7.4.2. ForHardy–Sobolev boundary data, this seems to be new. In Subsection 7.4.3 westate (without proof) a convergence result for Whitney averages of solutions toLAu = 0 within tent spaces and Z-spaces. Finally, we sketch the relationshipbetween our approach and the method of layer potentials in Subsection 7.4.4. Inthe range of exponents p for which our results hold, the solutions to boundaryvalue problems are all given by (generalised) layer potentials.

4.3 Notation

The following notational conventions, some of them non-standard, will be usedthroughout the article.

For a, b œ R and t > 0 we write

mba(t) :=

Y]

[ta (t Æ 1)t≠b (t Ø 1).

For 0 < p, q Æ Œ, we define the number

”p,q := 1q

≠ 1p

,

with the interpretation 1/Œ = 0.We write the Euclidean distance on Rn as d(x, y) = d(y, x) := |x≠y|, the open

ball with centre x œ Rn and radius r > 0 by B(x, r) := {y œ Rn : d(x, y) < r},

13The decay condition is removed for certain exponents in Section 7.2.14Of course, we do manage to recycle some arguments from [16] and [15].

123

and the (half closed, half open) annulus with centre x œ Rn, inner radius r0 > 0,and outer radius r1 > r0 by

A(x, r0, r1) := B(x, r1) \ B(x, r0) = {y œ Rn : r0 Æ d(x, y) < r1}.

For subsets E, F µ Rn, we write

d(E, F ) := dist(E, F ) = inf{d(x, y) : x œ E, y œ F}.

We let L0(� : E) denote the set of strongly measurable functions from ameasure space � to a Banach space E. For two quasi-Banach spaces X and Y ,we write X Òæ Y to mean that X µ Y (possibly after some identification hasbeen made) and that the identity map is bounded. Often we will refer to normsas ‘quasinorms’ even though they are actually norms; for example, we will referto the Lp quasinorm when p œ (0, Œ], even though this is a norm when p Ø 1.For a quick introduction to quasi-Banach spaces the reader can consult the earlysections of [56].

When necessary, we will label dual pairings by the space on the left: forexample, by Èf, gÍLp , we will mean the canonical duality pairing between Lp andLpÕ , with f œ Lp and g œ LpÕ .

AcknowledgementsThis work forms part of a doctoral thesis under the supervision of Pierre Por-tal and Pascal Auscher. I am greatly indebted to Pascal in particular, whodirected this portion of the thesis, and who read the manuscript with painstak-ing care and provided invaluable improvements. For enlightening discussions Ithank Michael Cwikel, Moritz Egert, Yi Huang, Andrew Morris, Jill Pipher, andChristian Zillinger. I also thank Sakis Chatzikaleas, Shaoming Guo, GennadyUraltsev, and Micha≥ Warchalski for allowing me to use their o�ce and its co�eemachine. Finally I thank Lashi Bandara and Ziping Rao, who have endured morethan their fair share of complaining about the writing of this article.

Parts of this work were carried out at the Junior Trimester Program on Op-timal Transportation at the Hausdor� Research Institute for Mathematics inBonn, and I thank the Hausdor� Institute for their support. I also acknowl-edge financial support from the Australian Research Council Discovery GrantDP120103692, and from the ANR project “Harmonic analysis at its boundaries”ANR-12-BS01-0013-01.

124

Chapter 5

Technical preliminaries

5.1 Function space preliminariesThroughout this entire section we will consider CN -valued functions for somefixed N œ N, but since nothing really changes whether we choose N = 1 orN ”= 1 (see Remark 5.1.13), we will not refer to CN in the notation. So we willwrite L2(Rn) = L2(Rn : CN), T p(Rn) = T p(Rn : CN), and so on. For z œ CN wewill write |z| in place of ||z||CN

.

5.1.1 ExponentsThis work makes heavy use of the relationship between di�erent exponents forfunction spaces. The most e�cient way to do this, balancing economy of notationand clarity of ideas, is to introduce a new formalism for exponents right at thebeginning, and work with it consistently.

Fix n œ N+ corresponding to the dimension in which we will work. Thefollowing system of notation depends implicitly on n.

The set of exponents is the disjoint union

E := Efin Û EŒ

where Efin := {(p, s) : p œ (0, Œ), s œ R} and EŒ := {(Œ, s; –) : s œ R, – Ø 0}.We say that an exponent is finite if it is in Efin, and infinite if it is in EŒ.

We define two functions i : E æ (0, Œ], r : E æ R, representing integrabilityand a kind of regularity, by

i(p, s) := p, i(Œ, s; –) := Œ,

r(p, s) := s, r(Œ, s; –) := s + –.

125

We also define functions j, ◊ : E æ R by

j(p, s) := 1/p, j(Œ, s; –) := ≠–/n

◊(p, s) := s, ◊(Œ, s; –) := s.

Note that p is finite if and only if j(p) is positive, and furthermore everyexponent p is determined by the pair (j(p), ◊(p)).

For r œ R and p œ E, define p + r to be the unique exponent satisfying

j(p + r) = j(p) and ◊(p + r) = ◊(p) + r.

We similarly define p ≠ r.For every exponent p, we define the dual exponent pÕ to be the unique ex-

ponent satisfying j(pÕ) + j(p) = 1 and ◊(pÕ) + ◊(p) = 0. Concretely, for finiteexponents we have

(p, s)Õ :=Y]

[(pÕ, ≠s) (p > 1)(Œ, ≠s; n(1

p ≠ 1)) (p Æ 1)

where pÕ is the usual Hölder conjugate of p. Clearly pÕÕ = p. We also define the¸-dual exponent

p¸ := pÕ ≠ 1,

and a quick computation shows that p¸¸ = p.For two exponents p, q œ E, we write p Òæ q to mean that

◊(p) Ø ◊(q) and ◊(q) ≠ ◊(p) = n(j(q) ≠ j(p)).

We always have p Òæ p. Observe that p Òæ q and q Òæ r implies p Òæ r, andp Òæ q if and only if qÕ Òæ pÕ. We define the Sobolev exponent pú be the uniqueexponent satisfying p Òæ pú and ◊(pú) = ◊(p) ≠ 1.

For ÷ œ R, define [p, q]÷ to be the unique exponent satisfying

j([p, q]÷) = (1 ≠ ÷)j(p) + ÷j(q),◊([p, q]÷) = (1 ≠ ÷)◊(p) + ÷◊(q).

Note that [p, q]0 = p and [p, q]1 = q. Note also that p Òæ q if and only ifq = [p, pú]÷ for some ÷ Ø 0.

Lemma 5.1.1. Suppose p and q are exponents with p Òæ q. Then [p, q]÷0 Òæ[p, q]÷1 whenever ÷0 Æ ÷1.

126

Proof. Write

◊([p, q]÷1) ≠ ◊([p, q]÷0) = ((1 ≠ ÷1)◊(p) + ÷1◊(q)) ≠ ((1 ≠ ÷0)◊(p) + ÷0◊(q))= (÷1 ≠ ÷0)(◊(q) ≠ ◊(p)) (5.1)= n(÷1 ≠ ÷0)(j(q) ≠ j(p)) (5.2)= n (((1 ≠ ÷1)j(p) + ÷1j(q)) ≠ ((1 ≠ ÷0)j(p) + ÷0j(q)))= n(j([p, q]÷1) ≠ j([p, q]÷0)).

Where line (5.2) follows from p Òæ q. Furthermore, line (5.1), ÷1 ≠ ÷0 Ø 0, and◊(p) Ø ◊(q) imply that ◊([p, q]÷0) Ø ◊([p, q]÷1). Thus [p, q]÷0 Òæ [p, q]÷1 .

A straightforward computation shows the following lemma.

Lemma 5.1.2. Suppose p Òæ q and ÷0, ÷1, ⁄ œ R. Then

[[p, q]÷0 , [p, q]÷1 ]⁄ = [p, q](1≠⁄)÷0+⁄÷1 .

In particular this implies

p = [[p, q]≠1, q]1/2

q = [p, [p, q]2]1/2.

The most convenient way of visualising exponents and relations between themis as points in the (j, ◊) plane. In Figure 5.1 we show two exponents p and q withp Òæ q, their dual exponents, their ¸-duals, and various other exponents whichmay be constructed from them. The operations p ‘æ pÕ and p ‘æ p¸ are givenby reflection about the marked points at (1/2, 0) and (1/2, ≠1/2) respectively.1

Observe that we have p Òæ q if and only if the line segment from p to q is parallelthe line from ((n + 1)/n, 0) to (1, ≠1) with the same orientation.

5.1.2 Tent spacesThe most fundamental function spaces in this work are the tent spaces. Thesewere first introduced by Coifman, Meyer, and Stein [32, 33], and they have sinceproven their worth in harmonic analysis and PDE. The other ‘ambient spaces’that we will use, namely Z-spaces and slice spaces, are closely related to tentspaces, so a solid knowledge of tent spaces will be useful.

1The exponent (1/2, ≠1/2) is special: in Section 7.4 we introduce it as the ‘energy exponent’.Certain boundary value problems associated with this exponent are automatically well-poseddue to the Lax–Milgram theorem.

127

Figure 5.1: Various exponents in the (j, ◊) plane.

p

q

pÕ

qÕ

p¸

q¸

[p, q]≠1

[p, q] 32

◊

12

≠12

≠1

12

1 n+1n

j

For x œ Rn we define the cone with vertex x by

�(x) := {(t, y) œ R1+n+ : y œ B(x, t)}

where B(x, t) is the open ball with centre x and radius t, and for each open ballB µ X we define the tent with base B by

T (B) := R1+n+ \

Q

a€

x/œB

�(x)R

b .

Equivalently, T (B) is the set of points (y, t) œ R1+n+ such that B(y, t) µ B.

The tent space quasinorms are defined in terms of the Lusin operator A andCarleson operators C–. These are defined as follows. For all – Ø 0, f œ L0(R1+n

+ )and x œ Rn, we define

Af(x) :=A¨

�(x)|f(t, y)|2 dy dt

tn+1

B1/2

(5.3)

and

C–f(x) := supB–x

1r–

B

A1

rnB

¨T (B)

|f(y, t)|2 dydt

t

B1/2

.

For s œ R, we define an operator Ÿs on L0(R1+n+ ) by

(Ÿsf)(t, x) := tsf(t, x)

128

for all (t, x) œ R1+n+ .

Now we will start using the exponent notation of Subsection 5.1.1, althoughit will not be truly useful just yet.

Definition 5.1.3. For a finite exponent p, the tent space T p = T p(Rn) is the set

T p = T ps := {f œ L0(R1+n

+ ) : A(Ÿ≠sf) œ Lp(Rn)}

equipped with the quasinorm

||f ||T p

s

:=------A(٭sf)

------Lp(Rn)

.

For an infinite exponent p = (Œ, s; –) we define T p by

T p = T Œs;– := {f œ L0(R1+n

+ ) : C–(Ÿ≠sf) œ LŒ(Rn)}

with its natural norm.

Remark 5.1.4. The spaces T ps agree with those defined by Hofmann, Mayboroda,

and McIntosh [50, §8.3], and with the spaces T p,22,s of Huang [51]. Our spaces T Œ

s;0

agree with Huang’s spaces T Œ,22,s .

All tent spaces are quasi-Banach spaces (Banach when i(p) Ø 1). For a finiteexponent p the subspace T p;c µ T p of compactly supported functions is dense inT p, and L2

c(R1+n+ ) is densely contained in T p.

Definition 5.1.5. Let p be an exponent with i(p) Æ 1, and suppose B µ Rn isa ball. We say that a function a œ L0(R1+n

+ ) is a T p-atom (associated with B) ifa is essentially supported in T (B) and if

||a||T 2s

Æ |B|”p,2 .

where ”p,2 = 12 ≠ 1

p (as defined in Section 4.3).

Theorem 5.1.6 (Atomic decomposition). Let p be an exponent with i(p) Æ 1.Then a function f œ L0(R1+n

+ ) is in T p if and only if there is a sequence (ak)kœN

of T p-atoms and a sequence ⁄ œ ¸p(N) such that

f =ÿ

kœN⁄kak (5.4)

with convergence in T p. Furthermore, we have

||f ||T p ƒ inf ||⁄||¸p(N)

where the infimum is taken over all such decompositions.

129

This is simply derived from the usual atomic decomposition theorem [33,Theorem 1c].

Note that the following duality theorem includes all finite exponents, withoutneeding to separate the cases i(p) Æ 1 and i(p) > 1. This is the first justificationof our exponent notation.

Theorem 5.1.7 (Duality). Suppose that p is a finite exponent. Then for allf, g œ L0(R1+n

+ ) we have¨

R1+n

+

|(f(t, x), g(t, x))| dxdt

t. ||f ||T p ||g||T pÕ , (5.5)

and the pairingÈf, gÍ :=

¨R1+n

+

(f(t, x), g(t, x)) dxdt

t(5.6)

identifies the Banach space dual of T p with T pÕ.

Note in particular that the integral in (5.5) converges absolutely.

Remark 5.1.8. Throughout this article we will refer to the duality pairing appear-ing in (5.6) as the L2 duality pairing.

When p is finite and i(p) Ø 2, T p may also be characterised in terms of theCarleson operator C0. This is a straightforward extension of [33, Theorem 3].

Theorem 5.1.9 (Carleson characterisation of T p). Suppose p is a finite exponentwith i(p) > 2. Then for all f œ L0(R1+n

+ ) we have

||f ||T p ƒ------C0(Ÿ≠◊(p)f)

------Lp(Rn)

.

Theorem 5.1.10 (Change of aperture). For — œ (0, Œ) and x œ Rn define

�—(x) := {(t, y) œ R1+n+ : y œ B(x, —t)},

and for f œ L0(R1+n+ ) define A—f(x) as in (5.3), with �—(x) in place of �(x). Then

for — œ (0, Œ) and each finite exponent p we have an equivalence of quasinorms

||f ||T p ƒ------A—(Ÿ≠◊(p)f)

------Li(p)(Rn)

.

This was proven by Coifman, Meyer, and Stein for T p0 [33, Proposition 4]

and Harboure, Torrea, and Viviani for T p,q with q œ (1, Œ) [44, Proposition2.3].2 This can be simply extended to p, q œ (0, Œ) [3, Proposition 3.21], and

2We have not defined the spaces T p,q with q ”= 2 here, because we will not use them.

130

the extension to the more general tent spaces here is immediate. Note that themethod of proof in [3] requires knowledge of the result for q ”= 2.

The following embedding theorem, which can be seen as a tent space analogueof the Hardy–Littlewood–Sobolev embedding theorem, is proven in [4, Theorem2.19].3

Theorem 5.1.11 (Embeddings). Let p and q be exponents with p Òæ q. Thenwe have the embedding

T p Òæ T q.

The following complex interpolation theorem was proven by Hofmann, May-boroda, and McIntosh for finite exponents [50, Lemma 8.23], and the extensionto one infinite exponent follows by duality [4, Theorem 2.1].

Theorem 5.1.12 (Complex interpolation). Suppose p and q are exponents withj(p), j(q) Ø 0 (with equality for at most one exponent), and 0 < ◊ < 1. Then wehave the identification

[T p, T q]◊ = T [p,q]◊ .

Remark 5.1.13. In contrast with the article [4], we define the operator Ÿs in termsof powers of t rather than powers of ball volumes, and so our tent spaces T p

s (Rn)correspond to the tent spaces T p,2

s/n(Rn) of [4]. We also use CN -valued functionsinstead of C-valued functions. This does not change the validity of previousresults, as one can always split T p

s (Rn : CN) ƒ üNj=1T

ps (Rn : C) and apply the

results to each summand individually. This reduction would fail if we were toreplace CN with a general Banach space, but thankfully we have no need for suchgenerality.

5.1.3 Z-spaces

We now introduce a class of function spaces, called Z-spaces, which are relatedto tent spaces by real interpolation. The Z-spaces play the role for Besov spacesBp,p

s that the tent spaces play for Hardy–Sobolev spaces Hps .

Definition 5.1.14. We refer to a pair

c = (c0, c1) œ (0, Œ) ◊ (3/2, Œ)3The case where p and q are both infinite is not explicitly proven there, but it follows by

the same argument.

131

as a Whitney parameter. To each Whitney parameter c and each (t, x) œ R1+n+

we associate the Whitney region

�c(t, x) := (c≠11 t, c1t) ◊ B(x, c0t) µ R1+n

+ ,

and for f œ L0(R1+n+ ) we define the L2-Whitney averages

Wcf(t, x) :=Aˆ

�c

(t,x)|f(·, ›)|2 d› d·

B1/2

.

For an exponent p and a Whitney parameter c, and for all f œ L0(R1+n+ ), we

define the quasinorm

||f ||Zpc

:=------Wc(٭r(p)f)

------Li(p)(R1+n

+ )(5.7)

(note the appearance of r(p) here) and a corresponding function space

Zpc = Zp

c (Rn) := {f œ L0(R1+n+ ) : ||f ||Zp

c

< Œ}.

For simplicity we write �(t, x) := �(1,2)(t, x).

Remark 5.1.15. The spaces Zpc coincide with the spaces L(i(p), r(p) + 1, 2) in-

troduced by Barton and Mayboroda [21]. In our applications these spaces willplay the same role as they do in [21] - namely that of an ambient space for thegradient of a solution to an elliptic BVP with boundary data in a Besov space.The connection with tent spaces presented here (extending that of [4]) is new.

Remark 5.1.16. The restriction c1 > 3/2 is for technical reasons. The first timethat it is actually needed is in our proof of the atomic decomposition theorem. Itit possible to extend everything to c1 > 1 by a straightforward covering argument,but this would take extra work, and c1 > 3/2 is su�cient for our applications.

The following real interpolation theorem appears in [4, Theorem 2.9]. InTheorem 5.1.30 we will extend it to infinite exponents.

Theorem 5.1.17 (Real interpolation for tent spaces with finite exponents). Sup-pose that p and q are finite exponents with ◊(p) ”= ◊(q), and 0 < ◊ < 1. Thenfor all Whitney parameters c we have the identification

(T p, T q)◊,p◊

= Z [p,q]◊

c

with equivalent quasinorms, where p◊ = i([p, q]◊).

132

Consequently, when p is finite, the Z-spaces Zpc are complete and independent

of c (up to equivalence of quasinorms). Hence we may simply write Zp in placeof Zp

c . We will soon extend this to infinite exponents.We will establish further properties of the Z-spaces ‘by hand’ rather than

arguing by interpolation, because this yields stronger results. In particular, ityields absolute convergence of L2 duality pairings, while interpolation would onlyprove this on dense subspaces. This will be important in applications (Chapter7). Our main tool is an equivalent dyadic characterisation of the Zp-quasinorm.4

To establish this characterisation we will need some notation and a preliminarycounting lemma.

For a standard (open) dyadic cube Q œ Q(Rn),5 and for k œ Z, define theWhitney cube

Qk := (2k¸(Q), 2k+1¸(Q)) ◊ Q,

and the Whitney gridGk := {Q

k : Q œ Q(Rn)}.

For each k œ Z, Gk is a partition of R1+n+ up to a set of measure zero.

For each Whitney parameter c, each k œ Z, and each Whitney cube Qk œ Gk,

we define

Gc(Qk) := {R

k œ Gk : Rk fl �c(t, x) ”= ? for some (t, x) œ Q

k}.

Lemma 5.1.18. Let c be a Whitney parameter and k œ N. Then for all Qk œ Gk

we have|Gc(Q

k)| .c,k,n 1

(where | · | denotes cardinality).

Proof. The condition Rk fl �c(t, x) ”= ? may be rewritten as

¸(R) œ [t/2k+1c1, 2≠kc1t] and dist(R, x) < c0t.

By rescaling and translating, the number of R œ Q(Rn) such that this conditionis satisfied is equal to the number of R œ Q(Rn) such that

¸(R) œ (1/2k+1c1, 2≠kc1) and dist(R, 0) < c0,

which is finite and depends only on c, k, and n.4This characterisation is stated and used by Barton and Mayboroda [21, Proof of Theorem

4.13], but without proof.5Any system of dyadic cubes will work here.

133

Proposition 5.1.19 (Dyadic characterisation). Let p be a finite exponent, c aWhitney parameter, and k œ Z. Then we have

||f ||Zpc

ƒc,k,p

----

----¸(Q)≠r(p)[|f |2]1/2Q

k

----

----¸p(Gk,¸(Q)n)

,

where [|f |2]1/2Q

k

=------f | L2(Qk

, d·d›/· 1+n)------.

Proof. Write p = (p, s) and estimate

||f ||pZpc

=ÿ

QkœGk

¨Q

k

Wc(٭sf)(t, x)p dt

tdx

ƒÿ

QkœGk

(2k¸(Q))≠ps

¨Q

k

------f | L2(�c(t, x), d·d›/· 1+n)

------p dt

tdx (5.8)

.ÿ

QkœGk

(2k¸(Q))≠ps

¨Q

k

ÿ

RkœG

c

(Qk)

------f | L2(Rk

, d·d›/· 1+n)------p dt

tdx (5.9)

ƒk,p,s

ÿ

QkœGk

RkœG

c

(Qk)

¸(R)n≠ps------f | L2(Rk

, d·d›/· 1+n)------p

(5.10)

ƒÿ

RkœGk

¸(R)n1¸(R)≠s

------f | L2(Rk

, d·d›/· 1+n)------2p

. (5.11)

The equivalence (5.8) comes from the fact that · ƒ 2k¸(Q) when (·, ›) œ �c(t, x)and (t, x) œ Q

k. The upper bound (5.9) comes from covering �c(t, x) with theWhitney cubes R

k œ Gc(Q), of which there are boundedly many by Lemma 5.1.18.The equivalence (5.10) comes from noting that ¸(R) ƒ ¸(Q) when R

k œ Gc(Qk).

Finally, (5.11) follows from the fact that every cube Rk œ Gk appears at least once,

and at most a bounded number of times, in the multiset {Rk œ Gc(Q

k) : Qk œ G}.

To prove the converse statement, we need only prove the converse directionof (5.9). To do this we note that there exists a Whitney parameter Âc such thatwhenever R

k œ Gc(Qk) and (t, x) œ Q

k, we have Rk µ �Âc(t, x). Indeed, one can

take Âc0 = 2(c0 + 2≠kÔn(c1 + 1)) and Âc1 = 4c1. This, along with the independence

of Zpc on c, completes the proof.

Remark 5.1.20. The same proof will work for infinite exponents once we showthat the corresponding Z-space norms are independent of c.

The dyadic characterisation of the Z-space quasinorm can be used to prove aduality theorem when i(p) > 1. As with the corresponding result for tent spaces,this is not just an abstract identification of dual spaces (which could be deducedby real interpolation), but also includes absolute convergence of the L2 dualitypairing.

134

Proposition 5.1.21 (Duality: reflexive range). Suppose i(p) œ (1, Œ). Then forall f, g œ L0(R1+n

+ ) we have¨R1+n

+

|(f(t, x), g(t, x))| dxdt

t. ||f ||Zp ||g||ZpÕ , (5.12)

and the L2 duality pairing identifies the Banach space dual of Zp with ZpÕ.

Proof. Let Q0 = (1, 2) ◊ (0, 1)n equipped with the measure dx dt/t1+n, and foreach function f œ L0(R1+n

+ ) and each cube Q œ G, let fQ be the function on Q0

which is the a�ne reparametrisation of 1Qf , so that [|f |2]1/2Q

=------fQ

------L2(Q0)

. Thenby Proposition 5.1.19, writing p = (p, s), we have

||f ||Zp ƒ------¸(Q)≠s[|f |2]1/2

Q

------¸p(G,¸(Q)n)

ƒ------¸(Q)≠sfQ

------¸p(G,¸(Q)n:L2(Q0))

=:------fQ

------¸p

s

(G,¸(Q)n:L2(Q0)).

Evidently the map f ‘æ (fQ)QœG is an isomorphism between Zp and ¸ps(G, ¸(Q)n :

L2(Q0)).Furthermore, for all f, g œ L0(R1+n

+ ) we haveˆR1+n

+

|(f(t, x), g(t, x))| dxdt

tƒ

ÿ

QœG

¸(Q)n

¨Q0

|(fQ(t, x), gQ(t, x))| dx dt

t1+n,

and so the mapping f ‘æ (fQ)QœG identifies the L2(R1+n+ ) duality pairing with the

¸2(G, ¸(Q)n : L2(Q0)) duality pairing (up to a constant).Since we have

ÿ

QœG

¸(Q)n|(fQ, gQ)L2(Q0)| .------fQ

------¸p

s

(G,¸(Q)n:L2(Q0))

------gQ

------¸p

Õ≠s

(G,¸(Q)n:L2(Q0))

and since the ¸2(G, ¸(Q)n : L2(Q0)) duality pairing identifies ¸pÕs (G, ¸(Q)n : L2(Q0))

as the dual of ¸p≠s(G, ¸(Q)n : L2(Q0)), the corresponding results for Zp follow.

The dyadic characterisation can also be used to prove an atomic decompositiontheorem for Z-spaces.

Definition 5.1.22. Let p = (p, s) be a finite exponent and c a Whitney param-eter. We say that a function a œ L0(R1+n

+ ) is a Zpc -atom associated with the point

(t, x) œ R1+n+ if a is essentially supported in �c(t, x) and if

------٭sa

------L2(�

c

(t,x),dx dt/t)Æ tn”

p,2 .

(recall that ”p,2 = 12 ≠ 1

p is defined in Section 4.3).

135

Lemma 5.1.23. Let p be a finite exponent and suppose a is a Zpc -atom associated

with (t0, x0) œ R1+n+ . Then

||a||Zp .c,p 1.

Proof. A reasonably quick computation shows that

{(t, x) œ R1+n+ : �c(t, x) fl �c(t0, x0) ”= ?} µ �Âc(t0, x0)

where Âc0 = c0(1 + c21) and Âc1 = c2

1. Hence we can estimate, using the assumedsupport and size conditions for a and writing p = (p, s),

||a||Zpc

ÆQ

a¨

�Âc(t0,x0)

A1

t1+n

¨�

c

(t0,x0)· |·≠sa(›, ·)|2 d›

d·

·

Bp/2

dxdt

t

R

b1/p

.c tn”

p,20

Q

a¨

�Âc(t0,x0)t≠np/20 dx

dt

t

R

b1/p

ƒc,p tn”

p,2≠ n

2 + n

p

0

= 1

as required.

Theorem 5.1.24 (Atomic decomposition of Z-spaces). Suppose p = (p, s) isa finite exponent with p Æ 1 and c is a Whitney parameter. Then a functionf œ L0(R1+n

+ ) is in Zp if and only if there exists a sequence (ak)k œ N of Zpc -

atoms and a scalar sequence ⁄ œ ¸p(N) such thatÿ

kœN⁄kak = f

with convergence in Zp. Furthermore, we have

||f ||Zp ƒ inf ||⁄||¸p(N) ,

where the infimum is taken over all such decompositions.

Proof. Given such a decomposition of f , we have

||f ||pZp =------

------

ÿ

kœN⁄kak

------

------

p

Zp

. ||⁄||p¸p(N)

136

by Lemma 5.1.23, and so ||f ||Zp . inf ||⁄||¸p(N) . It remains to prove the reverseestimate. For each k œ Z we can write

f =ÿ

QkœGk

fQ

k (5.13)

where fQ

k = 1Q

kf ,6 and by Proposition 5.1.19 this sum converges in Zp.7 Ifk Ø log2(c≠1

0Ô

n/3) + 1 and if c1 > 3/2 (this is the first place where we actuallyuse this assumption) then we have

Qk µ �c(cQ, tQ)

for all Q œ Q(Rn), where cQ is the center of Q and tQ is the midpoint of 2k¸(Q)and 2k+1¸(Q). Therefore, under this condition on k, each f

Qk satisfies the support

condition required of a Zpc -atom. The norms

------٭sf

Qk

------L2(�

c

(cQ

,tQ

),dx dt/t)are all

finite by Proposition 5.1.19, so we can define

⁄Q

k := t≠n”

p,2

Qk

------٭sf

Qk

------L2(�

c

(cQ

,tQ

),dx dt/t)

and

aQ

k :=Y]

[⁄≠1

Qk

fQ

k (fQ

k ”= 0)0 (f

Qk = 0).

Then each aQ

k is a Zpc -atom and

f =ÿ

QkœGk

⁄Q

kaQ

k

with convergence in Zp, and furthermore

------(⁄

Qk)

------¸p(Gk)

=-----

-----t≠n”

p,2

Qk

------٭sf

Qk

------L2(�

c

(cQ

k

,tQ

k

),dx dt/t)

-----

-----¸p(Gk)

ƒ----

----(2k¸(Q))≠n”p,2≠s|Q|1/2[|f |2]1/2

Qk

----

----¸p(Gk)

ƒ----

----¸(Q)(n/p)≠s[|f |2]1/2Q

k

----

----¸p(Gk)

ƒ ||f ||Zp

again using Proposition 5.1.19.6Note that this notation di�ers from that in the proof of Proposition 5.1.21.7Convergence in Zp does not follow immediately: one must write the series (5.13) as a limit

of partial sums and argue via dominated convergence.

137

Remark 5.1.25. In contrast to the setting of tent spaces, it is very easy to constructatomic decompositions of functions f œ Zp: as in the proof of the theorem, simplydecompose f via the Whitney grid Gk for su�ciently large k. This works for allfinite p, even if i(p) > 1. Abstract decompositions will be used to prove Zp-ZpÕ

duality when i(p) Æ 1.

Lemma 5.1.26. For all Whitney parameters c and all f œ L0(R1+n+ ), the function

Wcf is lower semicontinuous.

Proof. Fix M > 0 and suppose that Wcf(t, x) > M . Then there exists a small Á >

0 such that W(c0≠Á,c1≠Á)f(t, x) > M also. A short computation shows that if x œB(x, Át/2) and if |t≠t| < (c1/(c1≠Á)≠1)t, then �c(t, x) contains �(c0≠Á,c1≠Á)(t, x),so for all such (t, x) we have

Wcf(t, x) Ø W(c0≠Á,c1≠Á)f(t, x) > M.

Therefore the set {(t, x) œ R1+n+ : Wcf(t, x) > M} is open.

Corollary 5.1.27. Let p be an infinite exponent. Then

||f ||Zpc

= sup(t,x)œR1+n

+

Wc(٭r(p)f)(t, x),

i.e. the essential supremum in the definition of the Zpc -norm can be replaced with

a supremum.

Proof. Lower semicontinuity of the function Wc(٭r(p)f) implies that if

Wc(٭r(p)f)(t, x) > M

for some M < Πat one point (t, x), then it continues to hold in an open neigh-bourhood of (t, x), and in particular on a set of positive measure.

We can finally prove a duality theorem for Zp with i(p) Æ 1. As with theother duality results so far, note that this includes absolute convergence of theL2 duality pairing.

Theorem 5.1.28 (Duality: non-reflexive range). Suppose i(p) Æ 1 and let c bea Whitney parameter. Then for all f, g œ L0(R1+n

+ ) we have¨

R1+n

+

|(f(t, x), g(t, x))| dxdt

t. ||f ||Zp

c

||g||ZpÕ

c

, (5.14)

and the L2 duality pairing identifies the Banach space dual of Zpc with ZpÕ

c .

138

Proof. Write p = (p, s), so that pÕ = (Œ, ≠s, n”p,1). First suppose a is a Zpc -atom

associated with a point (t0, x0) œ R1+n+ . Then we have

¨R1+n

+

|(a(t, x), g(t, x))| dxdt

t

Æ------Ÿ≠sa

------L2(R1+n

+ ,dx dt/t)||Ÿsg||L2(�

c

(t0,x0),dx dt/t)

. tn”

p,2+n”1,p

+(n/2)0

------Ÿs≠n”1,pg

------L2(�

c

(t0,x0),dx dt/t1+n)

Æ ||g||Zpc

by Corollary 5.1.27. For general f œ Zp, write f as the sum of Zpc -atoms as in

Theorem 5.1.24, so that¨

R1+n

+

|(f(t, x), g(t, x))| dxdt

tÆ

ÿ

kœN|⁄k|¨

R1+n

+

|(ak(t, x), g(t, x))| dxdt

t

. ||g||ZpÕ

c

||⁄||¸p(N)

since p Æ 1. Taking the infimum over all atomic decompositions of f proves(5.14).

Now suppose that „ œ (Zpc )Õ. By the same technique as in the proof of

Proposition 5.1.21, we find that there exists a sequence (gQ) œ ¸Œ≠s(G : L2(Q0))

corresponding to the induced action of „ on ¸ps(G, ¸(Q)n : L2(Q0)) (since ¸p(N)Õ =

¸Œ(N) for p Æ 1). Hence there exists a function G„ œ L0(R1+n+ ) corresponding to

the action of „ on Zpc . We need to show that G„ is in ZpÕ

c .Suppose (t, x) œ R1+n

+ . Then we can estimate

Wc(Ÿs≠n”1,pG„)(t, x) ƒ t≠n”1,p ||G„||L2≠s

(�c

(t,x),d› d·/·1+n)

= t≠n”1,p supF œL2

s

(�c

(t,x),d› d·/·1+n)||F ||Æ1

|(F, G„)|

. t≠n”1,p

≠(n/2)≠n”p,2 ||„||(Zp

s

)Õ supF œL2

s

(�c

(t,x),d› d·/·)||F ||Ætn”

p,2

||F ||Zp

s

Æ ||„||(Zp)Õ ,

using n”1,p + (n/2) + n”p,2 = 0, the fact that the condition in the final supremumimplies that F is a Zp

c -atom, and Lemma 5.1.23. Therefore we have

||G„||ZpÕ

c

= sup(t,x)œR1+n

+

Wc(Ÿs≠n”1,pG„)(t, x) . ||„||(Zpc

)Õ

as desired.

139

Corollary 5.1.29. For all infinite exponents p and all Whitney parameters c,the Zp

s norms are mutually equivalent.

Hence for all exponents p we write Zp in place of Zpc .

Now that we have identified the duals of all Zp spaces for finite p, we cangive a full interpolation theorem.

Theorem 5.1.30 (Real interpolation of tent spaces: full range). Suppose thatp and q are exponents with ◊(p) ”= ◊(q), and 0 < ◊ < 1. Then we have theidentification

(T p, T q)◊,p◊

= Z [p,q]◊

with equivalent quasinorms, where p◊ = i([p, q]◊).

Proof. For finite exponents this is precisely Theorem 5.1.17. If 1 < i(p), i(q) ƌ, this follows by writing

(T p, T q)◊,p◊

= ((T pÕ)Õ, (T qÕ)Õ)◊,p◊

= (T pÕ, T qÕ)Õ

◊,pÕ◊

via the duality theorem for real interpolation [22, Theorem 3.7.1], using thatT pÕ fl T qÕ is dense in both T pÕ and T qÕ , and then noting that

pÕ◊ = i([p, q]◊)Õ = i([pÕ, qÕ]◊).

The full result follows by Wol� reiteration [92, Theorem 1].

Proposition 5.1.31 (Interpolation of Z-spaces). Let p and q be exponents whichare not both infinite, and let ◊ œ (0, 1). Then we have

(Zp, Zq)◊,p◊

= Z [p,q]◊

with equivalent quasinorms, where p◊ = i([p, q]◊). Furthermore if i(p), i(q) Ø 1,then we have

[Zp, Zq]◊ = Z [p,q]◊ .

Proof. The real interpolation result follows from Theorem 5.1.30 along with thereiteration theorem for real interpolation [22, Theorem 5.2.4]. The complex in-terpolation result is proven by Barton and Mayboroda via the dyadic character-isation of the norm [21, Theorem 4.13].

Remark 5.1.32. The Z-spaces can be seen as Wiener amalgam spaces W (L2, Lpw)

associated to the semidirect product R+ nRn coming from the dilation action of

140

the multiplicative group R+ on Rn. Topologically R+nRn = R1+n+ , and the group

operation is given by (t, x) · (s, y) := (t + s, x + ty). Thus many of the propertiesabove can be deduced from properties of abstract Wiener amalgam spaces. For areview of these spaces, see [45] and the references therein. However, if we were touse Wiener amalgam space arguments, we would not obtain any results for quasi-Banach Z-spaces (as the abstract theory of quasi-Banach Wiener amalgam spacesseems not to have been su�ciently developed), and we would not obtain absoluteconvergence of L2 duality pairings (only abstract duality pairings). Furthermore,these arguments would not show the connection with tent spaces.

5.1.4 Unification: tent spaces, Z-spaces, and slice spacesTent spaces and Z-spaces share the same fundamental properties. To make thistotally explicit, we will write X as a placeholder for either T or Z when a state-ment holds for tent spaces and Z-spaces. When considering two di�erent spaces,either of which can be a tent space or a Z-space independently, we will use sub-scripts X0, X1. For example, one can concisely write the conclusions of Theorem5.1.30 and Proposition 5.1.31 as

(Xp, Xq)◊,p◊

= Z [p,q]◊ ,

and the tent space and Z-space duality results can be written extremely conciselyas

(Xp)Õ = XpÕ.

In this section we establish further properties of tent spaces and Z-spaces, in-cluding some interrelations between the two.

First, we simply point out that for all s œ R we have

X2s = L2

s(R1+n+ ),

where||f ||L2

s

:=------٭sf

------L2(R1+n

+ ). (5.15)

The following embedding theorem extends Theorem 5.1.11 not only to Z-spaces, but also to combinations of tent and Z-spaces.

Theorem 5.1.33 (Mixed embeddings). Let X0, X1 œ {T, Z} and let p Òæ q withp ”= q. Then we have the embedding

(X0)p Òæ (X1)q.

141

Proof. When X0 = X1 = T , this is Theorem 5.1.11.Let r = [p, q]2, so that p Òæ r and [p, r]1/2 = q (by Lemmas 5.1.1 and 5.1.2).

Then we have embeddings T p Òæ T p (trivially) and T p Òæ T r (Theorem 5.1.11).Therefore we have

T p Òæ (T p, T r)1/2 = Z [p,r]1/2 = Zq

by Theorem 5.1.30, using that p ”= q and p Òæ q imply ◊(p) ”= ◊(q). Similarly,putting s = [p, q]≠1, we have T s Òæ T q and T q Òæ T q, so

Zp = (T s, T q)1/2 Òæ T q.

Finally, putting t = [p, q]1/2 and using the previous results, we have

Zp Òæ T t Òæ Zq,

which completes the proof.

We also have a convenient mixed embedding which only holds for infiniteexponents.

Lemma 5.1.34. Suppose that p is infinite. Then T p Òæ Zp.

Proof. Let f œ L0(R1+n+ ) and write p = (Œ, s; –). For ⁄ > 0 su�ciently large

and for all (t, x) œ R1+n+ ,

Q

aˆ

�(1,2)(t,x)|·≠(–+s)f(·, ›)|2 d› d·

R

b1/2

ƒQ

at≠n≠2–

¨�(1,2)(t,x)

|·≠sf(·, ›)|2 d›d·

·

R

b1/2

. t≠–

A

t≠n

¨T (B(x,⁄t))

|·≠sf(·, ›)|2 d›d·

·

B1/2

.

Taking suprema over (t, x) œ R1+n+ yields

||f ||Zp . ||f ||T p .

Proposition 5.1.35 (Density of intersections). Let p and q be exponents, and letX1, X2 œ {T, Z}. If p is finite then (X1)p fl (X2)q is dense in (X1)p. Otherwise,(X1)p fl (X2)q is weak-star dense in (X1)p.

142

Proof. This follows immediately from the fact that L2c(R1+n

+ ) is (weak-star) densein T r for (infinite) exponents r, and likewise in Zr (this can be proven directly,or by real interpolation, or by the embeddings of Theorem 5.1.33).

For all r œ R+, define a ‘downward shift’ operator Sr on L0(R1+n+ ) by

(Srf)(t, y) := f(t + r, y)

for all f œ L0(R1+n+ ). These operators are well-behaved on certain tent spaces

and Z-spaces, as shown in the following proposition.

Proposition 5.1.36 (Uniform boundedness of downward shifts). Let p be anexponent.

(i) If i(p) Æ 2 and ◊(p) < ≠1/2, then the operators (Sr)rœR+ are uniformlybounded on Xp.

(ii) If i(p) œ (2, Œ] and r(p) < ≠(n + 1)/2, then the operators (Sr)rœR+ areuniformly bounded on Xp.

Remark 5.1.37. Note that the assumptions for i(p) Æ 2 and i(p) > 2 are quitedi�erent: there is a sudden jump in dimensional dependence at i(p) > 2. Wedo not currently have a good explanation for this behaviour, and there is nointerpolation procedure to obtain stronger results when 2 < i(p) < Œ. Notethat we can include endpoints when considering tent spaces (i.e. we can include◊(p) = ≠1/2 or r(p) = ≠(n + 1)/2 respectively). However, to realise the spacesZp as interpolants of tent spaces, we need to interpolate between tent spacesT p0 and T p1 with ◊(p0) ”= ◊(p1), and so the endpoint Z-space results cannot beproven by this argument.

Proof. It su�ces to prove the tent space results; the Z-space results follow byreal interpolation.

First we will prove boundedness on tent spaces for i(p) Æ 1 and for i(p) = 2;the rest of part (i) follows by complex interpolation. Suppose p = (p, s) withp Æ 1 and let a be a T p-atom associated with a ball B of radius rB. Then Sra is

143

supported on T (B), and we have

||Sra||T 2s

ƒAˆ r

B

≠r

0

ˆRn

t≠2s≠1|a(t + r, x)|2 dx dt

B1/2

ÆAˆ r

B

≠r

0

ˆRn

(t + r)≠2s≠1|a(t + r, x)|2 dx dt

B1/2

=Aˆ r

B

r

ˆRn

|·≠sa(·, x)|2 dxd·

·

B1/2

Æ ||a||T 2s

Æ |B|”p,2

using that ≠2s ≠ 1 > 0. Therefore Sra is, up to a uniform constant, a T p-atomassociated with B. Hence if f = q

kœN ⁄kak is an atomic decomposition of f inT p, then Srf = q

kœN ⁄k(Srak) is an atomic decomposition of Srf in T p up toa uniform constant. Therefore the operators (Sr)rœR+ are uniformly bounded onT p. A similar argument (without needing atoms) works for p = (2, s) provideds < ≠1/2.

Now let p = (p, s) with p œ (2, Œ) and s < ≠(n + 1)/2, and fix f œ L0(R1+n+ ).

First we estimate Srf in T p:

||Srf ||T p =Q

aˆRn

A¨�(x)

t≠2s≠n≠1|f(t + r, y)|2 dy dt

Bp/2

dx

R

b1/p

ÆQ

aˆRn

A¨�(x)

(t + r)≠2s≠n≠1|f(t + r, y)|2 dy dt

Bp/2

dx

R

b1/p

=Q

aˆRn

A¨�(x)+r

·≠2s≠n≠1|f(·, y)|2 dy d·

Bp/2

dx

R

b1/p

Æ ||f ||T p

using that ≠2s ≠ n ≠ 1 > 0 and �(x) + r µ �(x), where �(x) + r is the ‘verticallytranslated cone’

�(x) + r := {(t, y) œ R1+n+ : (t ≠ r, y) œ �(x)}.

This proves part (ii) in the case where p is finite.Now suppose p = (Œ, s; –) with s+– < ≠(n+1)/2, and let B = B(c, R) µ Rn

144

be a ball. If r Æ R, then we can write

R≠–≠n/2A¨

T (B)t≠2s≠1|f(t + r, y)|2 dy dt

B1/2

Æ R≠–≠n/2A¨

T (B)(t + r)≠2s≠1|f(t + r, y)|2 dy dt

B1/2

. (R + r)≠–≠n/2A¨

T (B(c,R+r))·≠2s≠1|f(·, y)|2 dy d·

B1/2

. ||f ||T Œs

using that ≠2s ≠ 1 > 0. If r > R then instead we write

R≠–≠n/2A¨

T (B)t≠2s≠1|f(t + r, y)|2 dy dt

B1/2

= R≠–≠n/2A¨

T (B)+r

·≠2s≠13

· ≠ r

·

4≠2s≠1|f(·, y)|2 dy d·

B1/2

Æ R≠–≠n/23

R

r

4≠s≠1/2 A¨T (B)+r

·≠2s≠1|f(·, y)|2 dy d·

B1/2

Æ3

R + r

R

4–+n/2 3R

r

4≠s≠1/2||f ||T Œ

–

. ||f ||T Œ–

using that s+– Æ ≠(n+1)/2 in the last line, where T (B)+r is defined analogouslyto �(x)+r. These estimates imply that ||Srf ||T Œ

–

. ||f ||T Œ–

as desired, completingthe proof.

Now we shall define the slice spaces. These were introduced in connectionwith tent spaces and boundary value problems by Auscher and Mourgoglou [15].The name comes from the fact that functions in slice spaces are, roughly speak-ing, horizontal ‘slices’ of functions in tent or Z-spaces (this is made precise inProposition 5.1.39).

Definition 5.1.38. Suppose p is an exponent and t > 0. For f œ L0(Rn) wedefine

||f ||Ep(t) := t≠r(p)------x ‘æ ||f ||L2(B(x,t),dy/tn)

------Li(p)(Rn)

.

These quasinorms define the slice spaces

Ei(p)r(p)(t) = Ep(t) = Ep(t)(Rn) := {f œ L0(Rn) : ||f ||Ep(t) < Œ}.

145

For t > 0, h > 3/2 (this technical restriction corresponds to that in thedefinition of Whitney parameter) and f œ L0(Rn), define ÿt,h(f) œ L0(R1+n

+ ) bysetting

ÿt,h(f)(s, x) := f(x)1[t,ht](s)

for all (s, x) œ R1+n+ , and for g œ L0(R1+n

+ ) define fit(g) œ L0(Rn) by

fit,h(g)(x) :=ˆ ht

t

g(s, x) ds

s.

for all x œ Rn.

Proposition 5.1.39. For all exponents p, the operators

Ep(t) ÿt,h≠æ Xp fi

t,h≠æ Ep(t),

are bounded uniformly in t. Furthermore, the compositions of these operators areidentity maps.

Proof. The tent space results with ◊(p) = 0 are already stated in [15, §3]; theextension to all tent spaces is simple. Likewise, the composition statement isclear. The proof for Z-spaces is a straightforward (one page) argument that weomit.

Therefore we can view the spaces Ep(t) as retracts of Xp. Consequently,properties of tent spaces and Z-spaces descend to slice spaces.

Proposition 5.1.40. If 0 < t0, t1 < Πand p, q are exponents with i(p) = i(q),then Ep(t0) = Eq(t1) with equivalent quasinorms.

This follows from change of aperture for tent spaces (see [15, Lemma 3.5]). Forp œ (0, Œ] we write Ep := Ep(1) for any p with i(p) = p; all Ep(t) quasinormsare equivalent to the Ep quasinorm (but not uniformly in t or p).

We have a duality theorem for slice spaces, and of course one should noticeonce more that this includes absolute convergence of the L2 duality pairing (nowon Rn rather than R1+n

+ ). This is proven in [15, Lemma 3.2].

Proposition 5.1.41 (Duality). Fix t > 0 and let p be a finite exponent.. Thenwe have ˆ

Rn

|(f(x), g(x))| dx . ||f ||Ep ||g||EpÕ , (5.16)

and the L2(Rn) duality pairing identifies the Banach space dual of Ep with EpÕ.

146

The tent space and Z-space embedding results also descend to slice spaces,though for slice spaces the ‘regularity’ parameters are not so important.

Proposition 5.1.42 (Embeddings). Suppose 0 < p0 Æ p1 Æ Œ. Then Ep0 ÒæEp1.

Proof. Fix p0 and p1 with i(p0) = p0, i(p1) = p1, and p0 Òæ p1. Then we havebounded operators

Ep0 ÿ1,2≠æ Xp0 Òæ Xp1 fi1,2≠æ Ep1

whose composition is the identity map, with the inclusion following from Theorem5.1.33.

Slice spaces contain the Schwartz functions, and are contained in the space oftempered distributions. This is contained in [15, Lemma 3.6].8

Proposition 5.1.43. For all p œ (0, Œ] we have S µ Ep µ S Õ.

We also have a straightforward integration by parts formula for functions inslice spaces. This is proven in [15, Lemma 3.8].

Proposition 5.1.44 (Integration by parts in slice spaces). Let p be a finiteexponent and suppose that ˆ is a first-order di�erential operator with constantcoe�cients, and let ˆú be the adjoint operator. If f, ˆf œ Ep and g, ˆúg œ EpÕ,then ˆ

Rn

(ˆf(x), g(x)) dx =ˆRn

(f(x), ˆúg(x)) dx.

Finally, we have an equivalent dyadic quasinorm for the slice spaces. Thisfollows from the dyadic characterisation of Z-spaces (Proposition 5.1.19 and theremark following it) and Proposition 5.1.39.

Proposition 5.1.45 (Dyadic characterisation). For all p œ (0, Œ] we have

||f ||Ep

ƒ------(||f ||L2(Q))QœD1

------¸p(D1)

where D1 is the grid of standard dyadic cubes in Rn with sidelength 1.

Remark 5.1.46. The slice spaces Ep are equal to the Wiener amalgam spaces

W (L2, Lp)(Rn)

when p Ø 1 (see [45] and the references therein). Therefore, as with Z-spaces,many properties of slice spaces can be deduced from properties of Wiener amal-gam spaces. In order to emphasise the connection with tent spaces and Z-spaces,we have proven these results in this context.

8Only the case p < Œ is included there, but everything (except the density statement)extends to p = Œ.

147

5.1.5 Homogeneous smoothness spacesWe will only give a quick definition of these, and state a few properties that wewill need. These definitions are special cases of the Littlewood–Paley definitionsof Triebel–Lizorkin and Besov spaces; we will not need these in full generality. Formore information the reader can consult Grafakos [42, Chapter 6] or the manyworks of Triebel (for example [90, §5]).

Let Z(Rn) µ S(Rn) be the set of Schwartz functions f such that D–f(0) = 0for every multi-index –, and let Z Õ(Rn) be the topological dual of Z(Rn). Thespace Z Õ(Rn) can be identified with the quotient space S Õ(Rn) \ P(Rn), whereP(Rn) is the space of polynomials on Rn.

Definition 5.1.47. Let � œ S(Rn) be a radial bump function with

� Ø 0, supp � µ A(0, 6/7, 2), and �|A(0,1,12/7) = 1

(of course these precise parameters are not so important), and for j œ Z let �j

denote the associated Littlewood–Paley operators.For f œ Z Õ(Rn), – œ R, and 0 < p < Œ define

||f ||Hp

–

:=----

----------j ‘æ 2j–(�jf)(·)

------¸2(Z)

----

----Lp(Rn)

,

and for 0 < p Æ Œ define

||f ||Bp,p

–

:=------j ‘æ 2j– ||�j(f)||Lp(Rn)

------¸p(Z)

.

The homogeneous Hardy–Sobolev spaces Hp– = F p,2

– and Besov spaces Bp,p– are

then the sets of those f œ Z Õ(Rn) for which the corresponding quasinorms arefinite.

These quasinorms are independent of the choice of � (up to equivalence), andHp

– and Bp.p– are Banach spaces (quasi-Banach when p < 1).

For f œ Z Õ(Rn) and – œ R, the Riesz potential I–f œ Z Õ(Rn), defined by

I–f(x) := (|›|sf(›))‚(x),

is well-defined. These operators can be used to characterise the Hardy–Sobolevspaces when p > 1.

Theorem 5.1.48. Suppose 1 < p < Œ and – œ R. Then f œ Z Õ(Rn) is Hp– if

and only if I–f œ Lp, and ||I–f ||Lp

is an equivalent norm on Hp–. Furthermore,

for all s œ R, I– is an isomorphism from Hps to Hp

s+–.

148

We will need characterisations of the Hardy–Sobolev and Besov spaces byintegrals of di�erences. For all p œ [1, Œ], g œ L0(Rn), and s œ R, define

Dpsg(x) :=

AˆRn

|g(x + y) ≠ g(x)|p|y|n+ps

dy

B1/p

(x œ Rn).

Lemma 5.1.49. Suppose – œ (0, 1) and p œ (2n/(n + –), Œ). Then for allf œ L2(Rn) we have

||f ||Hp

–

ƒ------D2

–f------Lp

. (5.17)

Proof. Whenever f = I–Ï for some Ï œ CŒc , the estimate (5.17) follows from a

lemma of Stein [85, Lemma 1] combined with the Riesz potential characterisationof Hp

– (Theorem 5.1.48). A density argument, using the fact that elements of Hp–

may be represented as L2loc functions when – œ (0, 1), completes the proof.

The corresponding characterisation for Besov spaces can be found in [90, The-orem 5.2.3.2].

Theorem 5.1.50. Suppose – œ (0, 1) and p œ [1, Œ). Then for all f œ L2(Rn)we have

||f ||Bp,p

–

ƒ ||Dp–f ||Lp

.

For – > 0, the Besov space BŒ,Œ– can be identified with the more familiar

homogeneous Hölder–Lipschitz space �–(Rn). We will only use this space when– œ (0, 1), and in this range �–(Rn) has a simple characterisation: it is the spaceof functions f on Rn such that

||f ||�–

:= supx,yœRn

|f(x) ≠ f(y)||x ≠ y|– < Œ,

modulo constants. Such functions are automatically continuous.We will also need to consider the Triebel–Lizorkin spaces F Œ,2

– for – œ R,which are the subspaces of Z Õ(Rn) determined by the quasinorms

||f ||F Œ,2–

:= inf----

----------j ‘æ 2j–|fj(·)|

------¸2(Z)

----

----LŒ(Rn)

,

with infima taken over all decompositions

f =ÿ

jœZ�jfj

with each fj œ LŒ(Rn), where �j are Littlewood–Paley operators as in Definition5.1.47. When – Ø 0 these may be identified with the homogeneous BMO-Sobolev

149

spaces ˙BMO–(Rn), which are defined as the images of BMO(Rn) under the Rieszpotentials I– defined above, as subspaces of Z Õ(Rn), with a corresponding norm.Of course ˙BMO0(Rn) = BMO(Rn). Information on these spaces can be foundin [87] and [90, §5.1.4]. In particular, we have the following characterisation of

˙BMO–(Rn) for – œ (0, 1) due to Strichartz [87, Theorem 3.3].

Theorem 5.1.51. Suppose – œ (0, 1). Then for all f œ L2(Rn) we have

||f || ˙BMO–

ƒ supQ

A1

|Q|

ˆQ

ˆQ

|f(x) ≠ f(y)|2|x ≠ y|n+2–

dy dx

B1/2

,

where the supremum may be taken over all cubes or all balls.

We will introduce some unconventional but useful notation for these spaces.For a finite exponent p = (p, s), define

Hp := Hps = F p,2

s and Bp := Bp,ps .

For p = (Œ, s; 0), define

Hp := F Œ,2s and Bp := BŒ,Œ

s .

When s > 0 we have Hp = ˙BMOs and Bp = �s. Finally, for p = (Œ, s; –) with– > 0, define

Hp := Bp := BŒ,Œs+– .

As a consequence of these definitions and the various duality identifications forclassical smoothness spaces, for all finite exponents p we have

(Xp)Õ = XpÕ

whenever X denotes either H or B.We also have the following interpolation theorem. This is a combination of

standard results (see for example Mendez and Mitrea [73, Theorem 11], Triebel[88, Theorems 8.1.3 and 8.3.3a]), and Bergh and Löfström [22, Theorem 6.4.5]).9

Theorem 5.1.52. Let p and q be finite exponents, and suppose ◊ œ (0, 1) andp◊ := i([p, q]◊). Then we have

[Hp, Hq]◊ = H[p,q]◊

9The results cited in [88] and [22] are for inhomogeneous spaces. As always, essentially thesame technique proves the result for homogeneous spaces. To obtain the stated results for Besovspaces with ◊(p) = ◊(q), write Bp,p

◊ = F p,p◊ and use the interpolation results for Triebel-Lizorkin

spaces.

150

and (also allowing infinite exponents)

[Bp, Bq]◊ = B[p,q]◊ and (Bp, Bq)◊,p

◊

= B[p,q]◊ .

Furthermore if ◊(p) ”= ◊(q), then we have

(Hp, Hq)◊,p◊

= B[p,q]◊ .

5.2 Operator-theoretic preliminaries

5.2.1 Bisectorial operators and holomorphic functional cal-culus

The material of this section is not new, but we present it here to fix notation.Useful standard references are [70, 71, 2, 43], and a particularly nice recent ex-position which focuses on bisectorial operators on Banach spaces is contained inthe thesis of Egert [36, Chapter 3].

Let 0 < Ê < fi/2. The open bisector of angle Ê is the set

SÊ := {z œ C \ {0} : | arg(z)| < Ê or | arg(≠z)| < Ê} µ C,

where the argument arg(z) takes values in (≠fi, fi]. The closed bisector of angleÊ is the topological closure SÊ of SÊ in C.

Throughout this section we will write L2 = L2(Rn).

Definition 5.2.1. Let 0 Æ Ê < fi/2. A closed linear operator A on L2 is calledbisectorial of angle Ê if ‡(A) µ SÊ, and if for all µ œ (Ê, fi/2) and all z œ C \ Sµ

we have the resolvent bound------(z ≠ A)≠1

------L(L2)

.µ |z|≠1. (5.18)

Note that closedness of A is included in this definition. This is not standard,but it is convenient. Generally the precise angle Ê is not important, in whichcase we will simply refer to A as bisectorial.

The following proposition is proven in [36, Proposition 3.2.2] (except for theadjoint statement, which is a simple computation).

Proposition 5.2.2. Let A be a bisectorial operator on L2. Then A is densely-defined, and we have a topological (not necessarily orthogonal) splitting

L2 = N (A) ü R(A). (5.19)

Furthermore, Aú is also bisectorial.

151

The procedure of constructing an operator Ï(A) from a given bisectorial oper-ator A and holomorphic function Ï on an appropriate bisector, known as holomor-phic functional calculus, plays a central role in this work. In order to introduceholomorphic functional calculus properly, we must first define some classes ofholomorphic functions.

For an angle µ œ (0, fi/2), the set of holomorphic functions Ï : Sµ æ C isdenoted by H(Sµ). For ‡, · œ R and Ï œ H(Sµ) we define

||Ï||�·

‡

(Sµ

) = ||z ‘æ Ï(z)/m·‡(|z|)||LŒ(S

µ

)

(the function m·‡ is defined in Section 4.3) and

�·‡(Sµ) := {Ï œ H(Sµ) : ||Ï||�·

‡

(Sµ

) < Œ}.

Each �·‡(Sµ) is a Banach space when normed by ||·||�·

‡

(Sµ

), and consists of thoseholomorphic functions on Sµ which decay of order ‡ at 0 and of order · at Œ.10

An important special case is �00(Sµ) = HŒ(Sµ), the set of bounded holomorphic

functions on Sµ. We will usually surpress reference to Sµ in this notation, as therelevant bisector is generally clear from context.

The spaces �·‡ are decreasing in ‡ and · , in the sense that if ‡ < ‡Õ and

· < · Õ, then �· Õ‡Õ Òæ �·

‡. For ‡, · œ R we define the set

�·+‡ :=

€

· Õ>·

�· Õ

‡ ,

and we define the sets �·‡+ and �·+

‡+ analogously. The set �++ := �0+

0+ is particu-larly important: it is the set of holomorphic functions (on the relevant bisector)with polynomial decay of some positive order at 0 and Œ. We also define

�Œ‡ :=

‹

·

�·‡,

the set of functions with polynomial decay of arbitrarily large order at Œ. Simi-larly we can define �·

Œ, �ŒŒ, �Œ

‡+, and so on.There are a few holomorphic functions which we will use extensively. We

define ‰+, ‰≠ œ HŒ by

‰+(z) := 1z:Re(z)>0(z) and ‰≠(z) := 1z:Re(z)<0(z) (z œ Sµ); (5.20)

these are the indicator functions of the two halves of the bisector Sµ. We alsodefine

[z] :=Y]

[z (Re(z) > 0)

≠z (Re(z) < 0)= (‰+(z) ≠ ‰≠(z))z.

10Decay of negative order is interpreted as growth.

152

This lets us define a bounded version of the exponential map,

sgp := [z ‘æ e≠[z]] œ �Œ0 , (5.21)

which will be used characterise solutions to Cauchy–Riemann systems in Chapter7. For ⁄ œ R \ {0} we may also define the power function

[z ‘æ z⁄] œ �≠⁄⁄

via a branch cut on the half-line i(≠Œ, 0] µ C.We say that a function Ï œ H(Sµ) is nondegenerate if it does not vanish on any

open subset of Sµ. All the holomorphic functions defined above are nondegenerateexcept for ‰+ and ‰≠.

Let us introduce some useful operations on holomorphic functions. Let Ï œH(Sµ). There is a natural involution Ï ‘æ Ï on H(Sµ) defined by

Ï(z) := Ï(z) (z œ Sµ).

This involution is isometric on �·‡ for all ‡, · œ R. For t > 0 we define the dilation

Ït œ H(Sµ) byÏt(z) := Ï(tz).

The following lemma is a simple consequence of the above definitions.

Lemma 5.2.3. Let ‡ œ R. Then for all t > 0 we have

||Ït||�≠‡

‡

= t‡ ||Ï||�≠‡

‡

.

Fix an angle Ê œ [0, fi/2) and let A be an Ê-bisectorial operator on L2. Ifµ œ (Ê, fi/2) and Ï œ �+

+(Sµ), then we can define an operator Ï(A) on L2 by theCauchy integral

Ï(A)f := 12fii

ˆˆS

‹

Ï(z)(z ≠ A)≠1f dz (f œ L2) (5.22)

for any choice of ‹ œ (Ê, µ), where ˆS‹ is oriented counterclockwise. Then theintegral (5.22) is well-defined and independent of the choice of ‹, and we have

||Ï(A)||L(L2) .A,‡,·,µ ||Ï||�·

‡

(Sµ

) .

The proof is straightforward; we mention only that independence of ‹ fol-lows from Cauchy’s integral theorem. The following homomorphism propertyalso holds: when Ï,  œ �+

+, we have (ÏÂ)(A) = Ï(A)Â(A). Straightforward

153

manipulations show that for all  œ �++, the adjoint operator Â(A)ú is given by

Â(Aú).Often it is convenient to assume that the operator A is injective and has

dense range. In the context in which we work this generally does not hold, butthe splitting L2 = N (A)üR(A) from Proposition 5.2.2 shows that the restrictionA|R(A), acting on the Hilbert space R(A) (with inner product induced by thatof L2(Rn)), is injective and has dense range. One can also show that A|R(A) isbisectorial.

The integral in (5.22) converges whenever Ï œ �++, but if Ï œ HŒ is merely

bounded, convergence is not guaranteed. We would like to be able to constructoperators Ï(A) when Ï œ HŒ. For certain operators this is possible.

Definition 5.2.4. Let A be a bisectorial operator on L2. We say that A hasbounded HŒ functional calculus on R(A) if for all Ï œ �+

+ and all f œ R(A), wehave the estimate

||Ï(A)f ||L2 . ||Ï||Œ ||f ||L2(Rn) .

The property of having bounded HŒ functional calculus on R(A) is equivalentto certain quadratic estimates being satisfied; this is an important theorem dueto McIntosh (see [70, §7 and §8] and the other references at the start of thissection).

Theorem 5.2.5 (McIntosh). Let A be a bisectorial operator on L2. Then A hasbounded HΠfunctional calculus on R(A) if and only if the estimate

||f ||L2 ƒÂ

Aˆ Œ

0||Ït(A)f ||2L2

dt

t

B1/2

(5.23)

holds for all f œ R(A) and some (equivalently, all) nondegenerate Ï œ �++.

Note that the quadratic estimate (5.23) need not hold for Ï œ HŒ.If A has bounded HŒ functional calculus on R(A), then for all Ï œ HŒ we

can define a bounded operator Ï(A) on R(A) by

Ï(A)f := lim–

Ï–(A)f (f œ R(A)), (5.24)

where (Ï–) is a net in �++ which converges to Ï in HŒ. We then have

||Ï(A)||L(R(A)) . ||Ï||HŒ .

Furthermore, for Ï,  œ HŒ, we have the homomorphism property Ï(A)Â(A) =(ÏÂ)(A). For further details of this construction see [70, 71]. Thus we may define

154

bounded operators ‰±(A) and e≠t[A] = sgpt(A) (for all t > 0) on R(A), using thecorresponding HŒ functions defined in (5.20) and (5.21).

If Ï œ �0+, then we can extend Ï(A) from R(A) to all of L2 by

Ï(A)f := Ï(A)PR(A)f,

where PR(A) is the projection onto R(A) associated with the decomposition L2 =N (A) ü R(A). The operator Ï(A) then maps L2 into R(A). We have

||Ï(A)||L(L2) .------PR(A)

------L(L2)

||Ï(A)||L(R(A)) ,

and furthermore the homomorphism property Â(A)Ï(A) = (ÂÏ)(A) continues tohold for all  œ HŒ.

We refer to the operators ‰+(A) and ‰≠(A) as the positive and negative spec-tral projections associated with A. From the identities (‰±)2 = ‰±, ‰+‰≠ = 0,and ‰+ + ‰≠ = 1S

µ

for the functions ‰±, we deduce the identities

(‰±(A))2 = ‰±(A), ‰+(A)‰≠(A) = ‰≠(A)‰+(A) = 0, IR(A) = ‰+(A)+‰≠(A)

for the spectral projections as operators on R(A). Therefore ‰+(A) and ‰≠(A)are complementary projections in R(A) onto the positive and negative spectralsubspaces

R(A)± := ‰±(A)R(A),

and we have a topological direct sum decomposition

R(A) = R(A)+ ü R(A)≠.

We define the Cauchy operators C±A : R(A) æ LŒ(R± : R(A)±) by

C±A f(t) := e≠t[A]‰±(A)f. (5.25)

These are solution operators for the Cauchy problems associated with A on theupper half-space and lower half-space, in the following sense (see [8]).

Proposition 5.2.6. Suppose that A has bounded HŒ functional calculus onR(A). If f œ R(A)±, then F := C±

A f solves the Cauchy problem

ˆtF (t) ± AF (t) = 0, F (0) = f

in CŒ(R± : R(A)).

155

We end this section with a discussion of unbounded operators arising fromholomorphic functional calculus, and some situations where their compositionsare bounded.

Suppose that Ï œ �·‡ with min(‡, ·) Æ 0, so that the integral (5.22) need not

be absolutely convergent. We can define an unbounded operator Ï(A) on R(A)as follows. Fix ” > max(≠‡, ≠·) Ø 0 and define ÷” œ �”

” by

÷”(z) :=A

z

(1 + z)2

B”

.

Then ÷”Ï œ �++, so that the operator (÷”Ï)(A) is defined by (5.22). We also

have ÷” œ �++, so ÷”(A) is also defined by (5.22), and since A is injective with

dense range on R(A), so is ÷”(A). Therefore the unbounded operator ÷”(A)≠1 isdefined, with D(÷”(A)≠1) := R(÷”(A)). We then define the unbounded operator

Ï(A) := ÷”(A)≠1(÷”Ï)(A) (5.26)

with domain

D(Ï(A)) := {f œ R(A) : (÷”Ï)(A)f œ D(÷”(A)≠1)}.

The operator Ï(A) is closed, densely-defined, and independent of the choice of ”.Of course, if min(‡, ·) > 0, then we can take ” = 0 in the definition (5.26) andrecover the original definition of Ï(A) by the Cauchy integral (5.22).

Now suppose  œ �·1‡1 and Ï œ �·2

‡2 . Then a quick computation shows thatÏ(A)Â(A) ™ (ÏÂ)(A). Note that if ‡1 +‡2 > 0 and ·1 +·2 > 0, then the operator(ÏÂ)(A) is bounded and given by the Cauchy integral (5.22), while the operatorÏ(A)Â(A) is not a priori given by such a representation. This observation willbe convenient in what follows.

5.2.2 O�-diagonal estimates and the Standard Assump-tions

For x œ R, write ÈxÍ := max{1, |x|}. We continue to write L2 = L2(Rn).

Definition 5.2.7. Suppose � µ C\{0}, and let (Sz)zœ� be a family of operatorsin L(L2). Let M Ø 0. We say that (Sz) satisfies o�-diagonal estimates of orderM if for all Borel subsets E, F µ Rn, all z œ �, and all f œ L2,

||1F Sz(1Ef)||2 .K

d(E, F )|z|

L≠M

||1Ef ||2 . (5.27)

156

Many families of operators constructed from first-order di�erential operators(in particular, certain families of resolvents) satisfy o�-diagonal estimates of someorder. The following theorem shows that certain families constructed in terms ofholomorphic functional calculus of a bisectorial operator A satisfy o�-diagonal es-timates, under the assumption that a certain resolvent family satisfies o�-diagonalestimates. This is a slight extension of [84, Proposition 2.7.1]

Theorem 5.2.8 (O�-diagonal estimates for families constructed by functionalcalculus). Fix 0 Æ Ê < ‹ < µ < fi/2, M Ø 0, and ‡, · > 0. Let A bean Ê-bisectorial operator on L2 with bounded HŒ functional calculus on R(A),such that ((I +⁄A)≠1)⁄œC\S

‹

satisfies o�-diagonal estimates of order M . Supposethat (÷(t))t>0 is a continuous family of functions in HŒ(Sµ) which is uniformlybounded.11 If  œ �·

‡(Sµ), then the family of operators (÷(t)(A)Ât(A))t>0 satisfieso�-diagonal estimates of order min{‡, M}, with constants depending linearly on||Â||�·

‡

||÷||, where ||÷|| := supt>0 ||÷(t)||Œ, and also depending on A, M , ‡, and· .

Proof. Fix Borel sets E, F µ Rn. Because Ât(A) maps into R(A) for each t, wecan apply ÷(t)(A) to Ât(A)(1Ef) for each t > 0. We need to prove the estimate

||1F ÷(t)(A)Ât(A)(1Ef)||2 .A,M,‡,· ||÷|| ||Â||�·

‡

(Sµ

)

Kd(E, F )

t

L≠ min{‡,M}

||1Ef ||2

for all f œ L2. Fix ‹ Õ œ (‹, µ) throughout the proof.If d(E, F ) Æ t, then Èd(E, F )/tÍ ƒ 1, and so we have

||1F ÷(t)(A)Ât(A)(1Ef)||2 ƈ

ˆS‹

Õ

|÷(t)(z)||Â(tz)|------1F (z ≠ A)≠1(1Ef)

------2

|dz|

(5.28)

.A ||÷|| ||Â||�·

‡

||1Ef ||2ˆ

ˆS‹

Õ

m·‡(|z|) |dz|

|z| (5.29)

ƒ‡,·,M ||÷|| ||Â||�·

‡

Kd(E, F )

t

L≠ min{‡,M}

||1Ef ||2 ,

where we used the resolvent bound coming from bisectoriality of A in (5.29).Now suppose that d(E, F ) > t. Then, rearranging (5.28) and using the as-

11Continuity isn’t really needed here - we only assume it to avoid measurability issues.

157

sumed o�-diagonal estimates for ((I + ⁄A)≠1)⁄œC\S‹

, we have

||1F ÷(t)(A)Ât(A)(1Ef)||2

.A ||÷|| ||Â||�·

‡

||1Ef ||2ˆ

ˆS‹

Õ

m·‡(|z|)

Kd(E, F )

t/|z|

L≠M |dz||z|

. ||÷|| ||Â||�·

‡

||1Ef ||2 (I0 + IŒ), , (5.30)

whereI0 :=

ˆ td(E,F )≠1

0m·

‡(⁄) d⁄

⁄

and

IŒ :=ˆ Œ

td(E,F )≠1m·

‡(⁄)A

⁄

td(E, F )

B≠Md⁄

⁄.

The integral I0 is estimated by

I0 ƈ td(E,F )≠1

0⁄‡ d⁄

⁄ƒ‡

At

d(E, F )

B‡

.M

Kd(E, F )

t

L≠ min(‡,M)

. (5.31)

To estimate IŒ, we use that td(E, F )≠1 Æ 1 to write

IŒ ƒM

Kd(E, F )

t

L≠M Aˆ 1

td(E,F )≠1⁄‡≠M d⁄

⁄+ C(·, M)

B

whereC(·, M) =

ˆ Œ

1⁄≠·≠M d⁄

⁄.

If ‡ Æ M , then we haveˆ 1

td(E,F )≠1⁄‡≠M d⁄

⁄.‡,M

At

d(E, F )

B‡≠M

,

and in this case

IŒ .‡,M

Kd(E, F )

t

L≠MQ

aK

d(E, F )t

LM≠‡

+ C(·, M)R

b

.·,M

Kd(E, F )

t

L≠ min{‡,M}

. (5.32)

Otherwise, we have ˆ 1

td(E,F )≠1⁄‡≠M d⁄

⁄.‡,M 1,

and this also yields (5.32). Putting the estimates (5.31) and (5.32) into (5.30)completes the proof.

158

O�-diagonal estimates can also be used to deduce uniform boundedness andconvergence results for families of operators on slice spaces. These propositionsare proven in [15, §4].

Proposition 5.2.9 (Uniform boundedness of families on slice spaces). Let p œ(0, Œ]. If (Ts)s>0 is a family of operators on L2 satisfying o�-diagonal estimatesof order greater than n min(|”p,2|, 1/2), then Ts extends to a bounded operator onEp

0(t) uniformly in 0 < s Æ t.

Proposition 5.2.10 (Strong convergence in slice spaces). Let p œ (0, Œ). Sup-pose (Ts)s>0 is a family of operators on L2 satisfying o�-diagonal estimates oforder greater than n min(1/p, 1/2), and such that limsæ0 Ts = I strongly in L2.Then limsæ0 Ts = I strongly in Ep.

Throughout the ‘abstract’ part of this work, the following assumptions willbe su�cient. They can be a bit of a mouthful if stated in full, so we give them aname.

Definition 5.2.11. We say that an operator A satisfies the Standard Assumptionsif

• A is a Ê-bisectorial operator on L2 for some Ê œ [0, fi/2),

• A has bounded HŒ functional calculus on R(A), and

• for all ‹ œ (Ê, fi/2) the family ((I + ⁄A)≠1)⁄œC\S‹

satisfies o�-diagonalestimates of arbitrarily large order.

The main examples we have in mind are perturbed Dirac operators.

Theorem 5.2.12. Suppose D and B are as in Subsection 4.1.2 of the intro-duction. Then the perturbed Dirac operators DB and BD satisfy the standardassumptions (see Definition 5.2.11).

See [17, Proposition 2.1] and [16, Lemma 2.3, Propositions 3.1 and 3.2]; theo�-diagonal estimates stated there are in a di�erent but equivalent form.

5.2.3 Integral operators on tent spacesLet (St,· )t,·>0 be a continuous two-parameter family of bounded operators onL2 = L2(Rn), and for all f œ L2

c(R+ : L2) define Sf œ L0(R+ : L2) by

Sf(t) :=ˆ Œ

0St,· f(·) d·

·. (5.33)

159

Since f is compactly supported in R+, the Cauchy–Schwarz inequality shows thatthe integral (5.33) is absolutely convergent. We write S ≥ (St,· )t,·>0 to say thatS is given by the kernel (St,· )t,·>0.

We would like know when S can be extended from L2c(R+ : L2) to an operator

between various tent spaces and Z-spaces. A first step is given by the followingSchur-type lemma. Recall that L2

s is defined in (5.15) and coincides with X2s .

Lemma 5.2.13. Let s, ” œ R, and let S ≥ (St,· )t,·>0 on L2 as above. Supposethat there exists “ œ L1(R+ : R) (where R+ is equipped with the Haar measuredt/t) such that for all t, · > 0,

------·≠”St,·

------L(L2)

Æ “(t/·)(t/·)s+”.

Then for all f œ L2c(R+ : L2),

||Sf ||L2s+”

. ||“||L1(R+) ||f ||L2s

. (5.34)

Proof. We argue by duality. For all g œ L2≠(s+”), we can estimate

|ÈSf, gÍ|

ƈ Œ

0

ˆ Œ

0||St,· f(·)||2 ||g(t)||2

d·

·

dt

t

ƈ Œ

0

ˆ Œ

0“(t/·)

------·≠sf(·)

------2

------ts+”g(t)

------2

d·

·

dt

t

ÆAˆ Œ

0

ˆ Œ

0“(t/·) dt

t

------·≠sf(·)

------2

2

d·

·

B1/2

·

·Aˆ Œ

0

ˆ Œ

0“(t/·) d·

·

------ts+”g(t)

------2

2

dt

t

B1/2

Æ ||“||L1(R+) ||f ||L2s

||g||L2s+”

,

which implies (5.34).

For certain kernels (St,· ), assuming an L2s æ L2

s+” estimate (such as that whichcould be derived from the lemma above) and some o�-diagonal estimates, we areable to deduce the boundedness of S from T p to T p+” for some exponents p withi(p) œ (0, 1]. This is a generalisation of an argument of Auscher, McIntosh, andRuss [13].

Theorem 5.2.14 (Extrapolation of L2 boundedness to tent spaces). Let p =(p, s) be an exponent with p Æ 1, let ” œ R, and let (St,· )t,·>0 be a continuous

160

two-parameter family of bounded operators on L2 such that for all t0, ·0 > 0the one-parameter families (t≠”St,·0)tœ(·0,Œ) and (·≠”St0,· )·œ(t0,Œ) both satisfy o�-diagonal estimates of order M , with implicit constant K uniform in ·0 and t0

respectively. Suppose a, b œ R, and let S ≥ (mba(t/·)St,· )t,·>0.

If we have the norm estimate

||Sf ||L2s+”

. ||f ||L2s

(5.35)

for all f œ L2c(R+ : L2), with

≠ n”p,2 < b + s < M and a > s + ”, (5.36)

then||Sf ||T p+”

. ||f ||T p

for all f œ L2c(R+ : L2), and the implicit constant is a linear combination of K

and ||S|| := ||S||L2s

æL2s+”

.

Proof. Step 1: an estimate for compactly supported atoms. Suppose thatf is a compactly-supported T p-atom associated with a ball B = B(c, r) µ Rn.Then f œ L2

c , and so Sf is defined. We will show that Sf is in T p+” withquasinorm bounded independently of f . To do this we will exhibit an atomicdecomposition of Sf , and we will estimate ||Sf ||T p+”

using the coe�cients of thisdecomposition.

Let T1 := T (4B) and Tk := T (2k+1B) \ T (2kB) for all integers k Ø 2. Thendefine Fk := 1T

k

Sf for all k œ N, so that we have Sf = qŒk=1 Fk pointwise almost

everywhere. For each k œ N the function Fk is supported in a tent, so we canrenormalise by writing Fk = ⁄kfk for some ⁄k œ C and some T p+”-atom fk. Weneed only estimate the coe�cients ⁄k.

Estimate for the local part. For k = 1, since F1 is supported in T (4B),we must estimate ||F1||L2

s+”

in terms of |4B|”p,2 . It follows from (5.35) and thefact that f is a T p-atom that

||F1||L2s+”

Æ ||S|| |B|”p,2 ƒn,p ||S|| |4B|”p,2 ,

and so we can set ⁄1 ƒn,p ||S||.Estimate for the global parts. Suppose k Ø 2. Since Fk is supported in

the tent T (2k+1B), we must estimate ||Fk||L2s+”

in terms of |2k+1B|”p,2 . We use

161

Minkowski’s integral inequality to estimate

||Fk||L2s+”

=Q

aˆ 2k+1r

0

-----

-----t≠(s+”)1B(c,2k+1r≠t)

ˆ r

0mb

a(t/·)St,· f(·) d·

·

-----

-----

2

L2

R

b1/2

ÆQ

aˆ 2k+1r

0

Aˆ r

0t≠(s+”)mb

a(t/·)------1A(c,2kr≠t,2k+1r≠t)St,· f(·)

------L2

d·

·

B2dt

t

R

b1/2

ƈ r

0

Q

aˆ 2k+1r

0t≠2(s+”)mb

a(t/·)2------1A(c,2kr≠t,2k+1r≠t)St,· f(·)

------2

L2

dt

t

R

b1/2

d·

·.

Note that f(·) is supported in B(c, r ≠ ·). We have

d(supp f(·), A(c, 2kr ≠ t, 2k+1r ≠ t)) Ø d(B(c, r),Rn \ B(c, 2kr ≠ t))= ((2k ≠ 1)r ≠ t)+,

so we split the region of integration (0, 2k+1r) ◊ (0, r) into three subregions,

R1 := {(t, ·) : t < · < r}

R2 :=I

(t, ·) : · < t <2k ≠ 1

2 r

J

R3 :=I

(t, ·) : t >2k ≠ 1

2 r

J

,

and denote the corresponding integrals by I1, I2, and I3.12

On R3, where t > · and where there is no spatial separation, we have

I3 . K

ˆ r

0

Q

aˆ 2k+1r

2k≠12 r

t≠2(s+”)(t/·)≠2bt2” ||f(·)||2L2dt

t

R

b1/2

d·

·

.b,s K

ˆ r

0

------·≠sf(·)

------L2 · b+s(2kr)≠(b+s) d·

·

Æ K2≠k(b+s)Aˆ r

0

------·≠sf(·)

------2

L2

d·

·

B1/2 Aˆ r

0

3·

r

42(b+s) d·

·

B1/2

ƒb,s K2≠k(b+s) ||f ||L2s

(5.37)

Æ K2≠k(b+s)|B|”p,2

= K2≠k(b+s)|2k+1B|”p,2

A|2k+1B|

|B|

B≠”p,2

ƒ K2≠k(b+s+n”p,2)|2k+1B|”p,2 ,

12The reason for using the factor (2k≠1)/2 rather than 2k≠1 will be apparent when estimatingI

2

.

162

where we used b + s > ≠n”p,2 > 0 in (5.37).On R1, where t < · and where the o�-diagonal estimates for (·≠”St,· )·œ(t,Œ)

involve spatial separation,

I1 . K

ˆ r

0

Q

caˆ ·

0t≠2(s+”)(t/·)2a

Q

aA

(2k ≠ 1)r ≠ t

·

B≠M

||f(·)||L2

R

b2

dt

t

R

db

1/2

· ” d·

·

ƒa,s,” K2≠kM

ˆ r

0

3·

r

4M ------·≠sf(·)

------L2

d·

·(5.38)

Æ K2≠kM ||f ||L2s

Aˆ r

0

3·

r

42M d·

·

B1/2

.M K2≠k(M+n”p,2)|2k+1B|”p,2 (5.39)

using that a > s + ” in (5.38), and deducing (5.39) from the same argument usedfor I3.

On R2, we have t > · and the o�-diagonal estimates for (t≠”St,· )tœ(·,Œ) againinvolve spatial separation, and the restrictions on t imply (2k ≠ 1)r ≠ t > 2k≠1

2 r &2kr. Therefore

I2 . K

ˆ r

0

Q

caˆ 2k≠1

2 r

·

t≠2s(t/·)≠2b

A(2k ≠ 1)r ≠ t

t

B≠2M

||f(·)||2L2dt

t

R

db

1/2d·

·

. K

ˆ r

0

------·≠sf(·)

------L2 · b+s

Q

caˆ 2k≠1

2 r

·

t≠2(b+s)A

2kr

t

B≠2Mdt

t

R

db

1/2d·

·

.b,s,M K2≠kM

ˆ r

0

------·≠sf(·)

------L2 · b+sr≠M(2kr)≠(b+s≠M) d·

·(5.40)

.b,s K2≠k(b+s+n”p,2)|2k+1B|”p,2 (5.41)

using that M > s0 + b in (5.40) and arguing as before to conclude (5.41).Summing up, we have

||Fk||L2s+”

Æ I1 + I2 + I3

.a,b,M,p,s,” K12≠k(M+n”

p,2) + 2≠k(b+s+n”p,2)

2|2k+1B|”p,2 ,

and so for k Ø 2 we can set

⁄k ƒ K12≠k(M+n”

p,2) + 2≠k(b+s+n”p,2)

2,

163

which implies that

||(⁄k)||p¸p(N) ƒ ||S||p + KpŒÿ

k=2

12≠k(M+n”

p,2) + 2≠k(b+s+n”p,2)

2p

Æ ||S||p + KpŒÿ

k=22≠kp(M+n”

p,2) + 2≠kp(b+s+n”p,2),

which is finite because of the assumption (5.36). The implicit constants do notdepend on the atom f . Therefore Sf is in T p

s1 , with quasinorm bounded indepen-dently of f and controlled by a linear combination of ||S|| and K.

Step 2: from compactly supported atoms to T ps fl L2

c(R+ : L2). Thisfinal part of the argument exactly follows [13, Proof of Theorem 4.9, Step 3]. Onemust show that every function in T p

s fl L2c(R+ : L2) may be decomposed into a

sum of compactly supported atoms, and that such decompositions converge inboth T p

s (which is automatic) and in L2s. We omit further details.

5.2.4 Extension and contraction operators

Throughout this section we assume that A is an operator which satisfies thestandard assumptions (see Definition 5.2.11).

Definition 5.2.15. For all  œ HŒ define the extension operator

QÂ,A : R(A) æ LŒ(R+ : L2)

by(QÂ,Af)(t) := Ât(A)f (f œ R(A), t œ R+).

If in addition  œ �++, then QÂ,A is defined on all of L2, and by Theorem 5.2.5

we have boundedness QÂ,A : L2 æ L2(R1+n+ ).

Definition 5.2.16. For all Ï œ �++ define the contraction operator

SÏ,A : L2(R1+n+ ) æ R(A)

bySÏ,A := (QÏ,Aú)ú.

Note that QÂ,A = (SÂ,Aú)ú when  œ �++.

A quick computation yields the following representation of SÏ,A. The integralin (5.42) converges absolutely since f œ L1(R+ : L2(Rn)).

164

Proposition 5.2.17. Suppose Ï œ �++. Then for all f œ L2

c(R+ : L2) we have

SÏ,Af :=ˆ Œ

0Ït(A)f(t) dt

t. (5.42)

Fix ” œ R, and suppose ÷ œ �”≠”,  œ HŒ, and Ï œ �≠”

” fl �++. Then for all

f œ L2c(R+ : L2) we have SÏ,Af œ D(÷(A)) and the integral representation

(QÂ,A÷(A)SÏ,Af)(t) =ˆ Œ

0(Ât(A)÷(A)Ï· (A))f(·) d·

·. (5.43)

Therefore we can write

QÂ,A÷(A)SÏ,A ≥ ((Ât÷Ï· )(A))t,·>0.

Our goal now is to check when the results of Section 5.2.3 apply to this operator.In fact, we will be able to draw some conclusions even when ÷ is not bounded, aslong as Ât÷Ï· œ �+

+. This will ultimately lead to Theorem 6.1.11.

Lemma 5.2.18. Suppose ‡ + · Ø 0 and ” œ R. Let  œ �·‡, Ï œ �‡≠”

·+” , and÷ œ �”

≠”, and define the operator

St,r := m·+”‡ (t/r)≠1(Ât÷Ïr)(A). (5.44)

Then for all t0, r0 > 0 the operator families (t≠”St,r0)tœ(r0,Œ) and (r≠”St0,r)rœ(t0,Œ)

satisfy o�-diagonal estimates of order ‡ + · , uniformly in r0 and t0 respectively.The implicit constants in these o�-diagonal estimates depend linearly on ||÷||�”

≠”

.

This is a variation of [13, Lemma 3.7].

Proof. If t0 Æ r we can write

r≠”St0,r = r≠”(t0/r)≠‡[Ât0(z)÷(z)Ïr(z)](A)= [(t0z)≠‡Ât0(z)÷”(z)(rz)‡≠”Ï· (z)](A)

where ÷” œ HŒ is defined by ÷”(z) := z”÷(z). Note that------÷”

------Œ

= ||÷||�”

≠”

. Since œ �·

‡ and ‡ + · Ø 0, the function

“(t0) : z ‘æ (t0z)≠‡Ât0(z)÷”(z)

is in HŒ with bound uniform in t0, linear in ||÷||�”

≠”

, and clearly independent ofr. Furthermore, the function ◊ : z ‘æ z‡≠”Ï(z) is in �0

‡+· , and so we can write

r≠”St0,r = “(t0)(A)◊· (A)

165

where “(t) is uniformly in HŒ and ◊ œ �0‡+· . Theorem 5.2.8 then implies that

the family (St0,r)rœ(t0,Œ) satisfies o�-diagonal estimates of order ‡ + · uniformlyin t0 > 0, with implicit constants linear in ||÷||Œ.

Likewise, if r0 Æ t we can write

t≠”St,r0 = t≠”(t/r0)·+”[Ât(z)÷(z)Ïr0(z)](A)= [(r0z)≠(·+”)Ïr0(z)÷”(z)(tz)· Ât(z)](A)

and proceed in the same way, the consequence being that (t≠”St,r0)tœ(r0,Œ) satisfieso�-diagonal estimates of order ‡ + · uniformly in ·0 > 0, with implicit constantslinear in ||÷||�”

≠”

.

Lemma 5.2.19. Fix s, ” œ R. Suppose  œ �≠(s+”)+(s+”)+ , Ï œ �s+

≠s+, and ÷ œ �”≠”.

Then the operator S ≥ ((Ât÷Ïr)(A))t,r>0 extends to a bounded operator L2s æ

L2s+”.

Proof. Fix Á > 0 such that  œ �Á≠(s+”)Á+s+” and Ï œ �Á+s

Á≠s. First note that Ât÷Ïr œ�+

+, so the operators St,r := (Ât÷Ïr)(A) are all bounded and defined by theintegral (5.22) on L2. We will make use of Lemma 5.2.13, so we write r = Ÿt andbegin by estimating

------r≠”St,r

------L(L2(Rn))

.Â,Ï (Ÿt)≠”------÷1/t

------�”

≠”

ˆ Œ

0mÁ≠s

Á+s(t⁄)mÁ+sÁ≠s(Ÿt⁄) d⁄

⁄

Æ Ÿ≠” ||÷||�”

≠”

ˆ Œ

0mÁ≠s

Á+s(⁄)mÁ+sÁ≠s(Ÿ⁄) d⁄

⁄. (5.45)

using Lemma 5.2.3 to eliminate the powers of t in (5.45). If Ÿ Æ 1, we have

Ÿ≠”

ˆ Œ

0mÁ≠s

Á+s(⁄)mÁ+sÁ≠s(Ÿ⁄) d⁄

⁄

= Ÿ≠”

A

ŸÁ≠s

ˆ 1

0⁄2Á d⁄

⁄+ ŸÁ≠s

ˆ 1/Ÿ

1

d⁄

⁄+ Ÿ≠Á≠s

ˆ Œ

1/Ÿ

⁄≠2Á d⁄

⁄

B

. ŸÁ≠s≠”(2 + log(1/Ÿ)).

If Ÿ Ø 1, then by the same argument we have

Ÿ≠”

ˆ Œ

0mÁ≠s

Á+s(⁄)mÁ+sÁ≠s(Ÿ⁄) d⁄

⁄. Ÿ≠Á≠s≠”(2 + log(Ÿ)).

Since the function

“(Ÿ) :=Y]

[ŸÁ(2 + log(1/Ÿ)) (Ÿ Æ 1)Ÿ≠Á(2 + log(Ÿ)) (Ÿ Ø 1)

is in L1(R+), Lemma 5.2.13 completes the proof.

166

The following theorem is the basis of Chapter 6. From the viewpoint ofapplications, the important part of this theorem is the decay condition on  at0. We would particularly like to take  = sgp œ �Œ

0 and ” = 0, which is possibleprovided that i(p) Æ 2 and ◊(p) < 0.

Theorem 5.2.20 (Boundedness of contraction/extension compositions). Sup-pose p is an exponent, ” œ R and ÷ œ �”

≠”. Suppose that either

• i(p) Æ 2 and

 œ �(≠(◊(p)+”)+n| 12 ≠j(p)|)+

(◊(p)+”)+ fl HŒ and Ï œ �◊(p)+(≠◊(p)+n| 1

2 ≠j(p)|)+ fl �++,

(5.46)or

• i(p) Ø 2 and

 œ �(≠◊(p))+(◊(p)+n| 1

2 ≠j(p)|)+ fl �++ and Ï œ �(◊(p)≠”+n| 1

2 ≠j(p)|)+(≠◊(p)+”)+ fl �+

+.

then QÂ,A÷(A)SÏ,A extends to a bounded operator Xp æ Xp+” (by duality wheni(p) = Œ), with bounds linear in ||÷||�”

≠”

.

Proof. We will only prove the result for tent spaces. The Z-space result canbe deduced by real interpolation, or alternatively it can be proven directly viathe dyadic characterisation of Proposition 5.1.19. Furthermore, the result fori(p) Ø 2 follows from the result for i(p) Æ 2 by duality, so we need only provethe result for i(p) Æ 2. Note that (5.43) and the assumptions on  and Ï implythat QÂ,A÷(A)SÏ,A contains the integral operator with kernel ((Ât÷Ï· )(A))t,·>0,so it su�ces to work with this operator. Furthermore, the assumptions (5.46) and(5.2.19) imply that this operator is bounded from T 2

◊(p) to T 2◊(p)+”, which yields

the result for i(p) = 2.Step 1: i(p) Æ 1. The assumptions (5.46) imply that there exists Á > 0 such

that œ �·+Á

‡+Á and Ï œ �(‡+Á)≠”(·+Á)+” ,

where ‡ := ◊(p) + ” and · := ≠◊(p + ”) + n|(1/2) ≠ j(p)|. Therefore by Lemma5.2.18, the operator families (t≠”St,r0)tœ(r0,Œ) and (r≠”St0,r)rœ(t0,Œ), where St,r isdefined as in (5.44), satisfy o�-diagonal estimates of order n|(1/2) ≠ j(p)| + 2Á.Theorem 5.2.14 then applies with a = ‡+Á, b = · +Á+”, and M = 2Á+n|(1/2)≠j(p)|, and we can conclude that QÂ,A÷(A)SÏ,A is bounded from T p to T p+”.

167

Step 2: i(p) œ (1, 2). The following argument originates from the thesis ofStahlhut [84, Step 4, proof of Lemma 3.2.6]. For ⁄ œ C, define functions Â⁄ andÏ⁄ by

Â⁄(z) :=A

[z]1 + [z]

B⁄

Â(z), Ï⁄(z) :=A

11 + [z]

B⁄

Ï(z).

If Re ⁄ Ø n(1 ≠ j(p)), then Step 1 applies with exponent (1, ◊(p)), and we findthat QÂ⁄,A÷(A)SÏ⁄,A is bounded from T 1

◊(p) to T 1◊(p)+”. Furthermore, if Re ⁄ Ø

≠n|(1/2) ≠ j(p)|, then the discussion of the first paragraph of the proof applies,and we find that QÂ⁄,A÷(A)SÏ⁄,A is bounded from T 2

◊(p) to T 2◊(p)+”. By Stein

interpolation in tent spaces (see [10, Proof of Lemma 3.4]), when Re ⁄ = 0, wehave that QÂ⁄,A÷(A)SÏ⁄,A is bounded from T p

◊(p) to T p◊(p)+” when p œ (1, 2) and

◊ œ (0, 1) satisfy

1p

= (1 ≠ ◊) + ◊

2 , 0 = (1 ≠ ◊)(1 ≠ j(p)) + ◊31

2 ≠ j(p)4

.

This occurs when p = i(p). Applying this with ⁄ = 0 yields boundedness ofQÂ,A÷(A)SÏ,A from T p to T p+”.

Finally, we shall discuss an abstract form of the Calderón reproducing formula,which is ubiquitous in the study of abstract Hardy spaces, and which will playan important role in what follows.

Whenever  œ �++ and Ï œ HŒ, we can define a bounded holomorphic

function�Â,Ï(z) :=

ˆ Œ

0Ât(z)Ït(z) dt

t, z œ Sµ.

This integral converges absolutely because ÂÏ œ �++. It is not hard to show that

SÂ,AQÏ,A = �Â,Ï(A)

as operators on R(A).In [16, Proposition 4.2] it is shown that if Ï œ HŒ is nondegenerate, then

there exists  œ �ŒŒ such that �Ï, © 1. This implies the following abstract

Calderón reproducing formula.

Theorem 5.2.21. Suppose Ï œ HŒ is nondegenerate. Then there exists  œ �ŒŒ

such thatSÂ,AQÏ,A = IR(A) (5.47)

as operators on R(A). Furthermore, if Ï œ �++, then the operator QÂ,ASÏ,A is a

projection from L2(R1+n+ ) onto QÂ,AR(A).

168

We refer to a pair (Ï, Â), with Ï œ HŒ,  œ �++ and satisfying (5.47), as

Calderón siblings.Here is a simple example of the use of the abstract Calderón reproducing

formula.

Corollary 5.2.22. Suppose Ï œ HŒ is nondegenerate. Then the extension op-erator QÏ,A : R(A) æ L2(R+ : R(A)) is injective.

Proof. Let  œ �++ be a Calderón sibling of Ï, and suppose f œ R(A) with

QÏ,Af = 0. Then by (5.47) we have

f = SÂ,AQÏ,Af = 0,

and so QÏ,A is injective.

169

170

Chapter 6

Adapted function spaces

6.1 Adapted Hardy–Sobolev and Besov spacesThroughout this section we will fix an operator A satisfying the standard assump-tions (see Definition 5.2.11). As in the previous chapter, we will implicitly workwith CN -valued functions without referencing this in the notation.

6.1.1 Initial definitions, equivalent norms, and dualityThe adapted Hardy–Sobolev and Besov spaces are, defined, roughly speaking,by measuring extensions by QÂ,A in tent spaces and Z-spaces respectively. Wewill soon show that the resulting function space is independent of  for  withsu�cient decay at 0 and Œ depending on p.

Definition 6.1.1. Let  œ HŒ and let p be an exponent. We define the sets

HpÂ,A := {f œ R(A) : QÂ,Af œ T p},

BpÂ,A := {f œ R(A) : QÂ,Af œ Zp},

equipped with quasinorms1

------f | Hp

Â,A

------ := ||QÂ,Af ||T p ,

------f | Bp

Â,A

------ := ||QÂ,Af ||Zp .

We call these spaces pre-Hardy–Sobolev and pre-Besov spaces associated with A

(respectively), and we call  an auxiliary function.1These are shown to be quasinorms in Proposition 6.1.2. Of course, they are actual norms

when i(p) Ø 1.

171

Generally we will want to refer to the pre-Hardy–Sobolev and pre-Besov spacessimultaneously. In this case we will write

XpÂ,A := {f œ R(A) : QÂ,Af œ Xp},

where the pair (X,X) is either (T,H) or (Z,B). This follows the conventioninitiated in Subsection 5.1.4.

Proposition 6.1.2. Let  œ HŒ and let p be an exponent. Then------· | Xp

Â,A

------ is a

quasinorm on XpÂ,A.

Proof. The only quasinorm property which does not follow directly from linearityof QÂ,A and the corresponding quasinorm properties of Xp is positive definiteness.To show this, suppose f œ Xp

Â,A and------f | Xp

Â,A

------ = 0. Then we have QÂ,Af = 0 in

Xp, and hence also in L2(R+ : R(A)). By injectivity of QÂ,A : R(A) æ L2(R+ :R(A)) (Corollary 5.2.22), we conclude that f = 0.

The following proposition quantifies the amount of decay needed on the aux-iliary function  in order to ensure that the Xp

Â,A quasinorm is equivalent to theXp

Ï,A quasinorm whenever Ï has decay of arbitrarily high order at 0 and Œ.

Proposition 6.1.3 (Independence on auxiliary function). Let Ï œ �ŒŒ and  œ

HΠbe nondegenerate, let p be an exponent, and suppose that either

• i(p) Æ 2 and  œ �(≠◊(p)+n| 12 ≠j(p)|)+

◊(p)+ , or

• i(p) Ø 2 and  œ �≠◊(p)+(◊(p)+n| 1

2 ≠j(p)|)+ fl �++.

Then we have XpÂ,A = Xp

Ï,A with equivalent quasinorms.

Proof. First, let ‹ œ �ŒŒ be a Calderón sibling of Â. Then for f œ Xp

Â,A we have------f | Xp

Ï,A

------ = ||QÏ,Af ||Xp

= ||QÏ,AS‹,AQÂ,Af ||Xp

. ||QÂ,Af ||Xp (6.1)=

------f | Xp

Â,A

------ ,

where (6.1) follows from Theorem 5.2.20, by the standard assumptions along withÏ, ‹ œ �Œ

Œ.Now let ‹ œ �Œ

Œ be a Calderón sibling of Ï. Then we can repeat the previousargument with the roles of Ï and  reversed, using the additional assumptionson  to apply Theorem 5.2.20. This leads to the reverse estimate

------f | Xp

Â,A

------ .

------f | Xp

Ï,A

------

172

which completes the proof.

Definition 6.1.4. For an exponent p, we define the spaces

XpA := Xp

Â,A,

where any auxiliary function  œ �ŒŒ may be used to define the space and its

corresponding quasinorm. We also define �(XpA) to be the set of all nondegenerate

Ï œ HŒ such that XpÏ,A = Xp

A with equivalent quasinorms.

With this notation at hand, Proposition 6.1.3 tells us that

�(≠◊(p)+n| 12 ≠j(p)|)+

◊(p)+ fl HŒ µ �(XpA) (i(p) Æ 2),

�≠◊(p)+(◊(p)+n| 1

2 ≠j(p)|)+ fl �++ µ �(Xp

A) (i(p) > 2).

Recall that the positive and negative spectral subspaces

R(A)± := ‰±(A)R(A)

were defined and discussed in Section 5.2.1. These can be used to define corre-sponding positive and negative spectral subspaces of Xp

A.

Definition 6.1.5. Let p be an exponent. Then we define the positive and neg-ative pre-Hardy–Sobolev and Besov spaces by

Xp,±A := Xp

A fl R(A)±,

equipped with any of the equivalent XpA quasinorms. Often we will just refer to

these as the spectral subspaces.

In Corollary 6.1.13 we will characterise the positive and negative spaces Xp,±A

as images of the spectral projections ‰±(A).The spaces Xp

A may also be characterised in terms of the contraction mapsSÂ,A for any  œ �+

+. Recall that X2 = T 2 = Z2 = L2(R1+n+ ).

Proposition 6.1.6 (Characterisation by contraction maps). Let p be an exponentand let  œ �+

+ be nondegenerate. Then we have

XpA = SÂ,A

1X2 fl Xp

2, (6.2)

and the mapping

f ‘æ inf{||F ||Xp : F œ X2 fl Xp, SÂ,AF = f}

is an equivalent quasinorm on XpA.

173

Proof. Fix a Calderón sibling Ï œ �ŒŒ of Â. First we will show the equality

(6.2). Suppose f œ XpA. Then QÏ,Af œ X2 fl Xp, and by Theorem 5.47 we have

f = SÂ,A(QÏ,Af). Conversely, suppose that f = SÂ,AF for some F œ X2 fl Xp.Then f œ R(A), and Theorem 5.47 implies that QÏ,Af = F œ Xp, which showsthat f œ Xp

A. This proves (6.2).Now prove the quasinorm equivalence. Suppose f œ Xp

A. Then f = SÂ,AQÏ,Af

with QÏ,Af œ X2 fl Xp, and so

inf{||F ||Xp : F œ X2 fl Xp, SÂ,AF = f} Æ ||QÏ,Af ||Xp ƒ ||f ||XpA

.

Conversely, suppose F œ X2 fl Xp and SÂ,AF = f . Then

||F ||Xp & ||QÏ,ASÂ,AF ||Xp

= ||QÏ,Af ||Xp

ƒ ||f ||XpA

,

completing the proof.

Corollary 6.1.7 (Density of intersections). Let p and q be exponents, and sup-pose X1,X2 œ {H,B}. If p is finite then (X1)p

A fl (X2)qA is dense in (X1)p

A. Oth-erwise, (X1)p

A fl (X2)qA is weak-star dense in (X1)p

A.

Proof. We will suppose that p is finite; the same argument works for infinitep, replacing limits with weak-star limits and norms with appropriate dualitypairings.

Suppose f œ (X1)pA, and fix  œ �+

+. By Proposition 6.1.6 we can writef = SÂ,AF for some F œ T 2 fl (X1)p, and by Proposition 5.1.35 we can writeF = limkæŒ Fk (limit in (X1)p) for some sequence (Fk)kœN in T 2 fl (X1)p fl (X2)q.For all k œ N define

fk := SÂ,AFk œ (X1)pA fl (X2)q

A.

Then we have, again using Proposition 6.1.6,

limkæŒ

||f ≠ fk||(X1)pA

. limkæŒ

||F ≠ Fk||(X1)p = 0.

This proves the claimed density.

The pre-Hardy–Sobolev and pre-Besov spaces inherit a duality pairing fromR(A) µ L2(Rn). However, we cannot say that XpÕ

Aú is the dual of XpA, because

in general these spaces are incomplete, while the dual of a quasinormed space isalways complete. We will deal with completions in Subsection 6.1.3.

174

Proposition 6.1.8 (Duality estimate). Let p be an exponent. Then for all f œXp

A and g œ XpÕ

Aú we have

|Èf, gÍ| . ||f ||XpA

||g||XpÕA

ú, (6.3)

where È·, ·Í is the inner product on L2(Rn).

Proof. Let Ï,  œ �ŒŒ be nondegenerate and suppose Á > 0. By Proposition 6.1.6

there exist F œ X2 flXp and G œ X2 flXpÕ such that SÏ,AF = f and SÂ,AúG = g,with

||F ||Xp . (1 + Á) ||f ||XpA

and ||G||XpÕ . (1 + Á) ||g||XpÕA

ú.

Since SúÏ,A = QÏ,Aú , and using that the L2(R1+n

+ ) inner product yields a dualitypairing for tent and Z-spaces, we thus have

|Èf, gÍ| = |ÈF,QÏ,AúSÂ,AúGÍL2(R1+n

+ )|

. ||F ||Xp ||QÏ,AúSÂ,AúG||XpÕ

. ||F ||Xp ||G||XpÕ (6.4)

. (1 + Á)2 ||f ||XpA

||g||XpÕA

ú,

where (6.4) follows from Theorem 5.2.20. Since Á > 0 was arbitrary we obtain(6.3).

The tent space and Z-space embeddings of Section 5.1 immediately yieldcorresponding embeddings of the pre-Hardy–Sobolev and pre-Besov spaces.

Proposition 6.1.9 (Mixed embeddings). Let p and q be exponents with p ”= qand p Òæ q. Then we have the continuous embedding

(X0)pA Òæ (X1)q

A,

where X0,X1 œ {H,B}, and the corresponding embedding holds for positive andnegative versions.

Proof. This follows directly from the definition of the spaces XpA and from Theo-

rem 5.1.33. The spectral subspace versions follow by intersecting with R(A)±.

Remark 6.1.10. For p = (p, s) we will sometimes write XpA = Xp

s,A, and forp = (Œ, s; –) we may write Xp

A = Xps;–,A. This notation is a bit heavy, so

we avoid it whenever possible, except in the case of X20,A, which we can simply

abbreviate as X2A (as is standard).

175

6.1.2 Mapping properties of holomorphic functional cal-culus

In the same way that we proved independence on auxiliary functions in the previ-ous section, we can prove various mapping properties (including boundedness) ofthe holomorphic functional calculus between pre-Hardy–Sobolev and pre-Besovspaces.

The first result says heuristically that an operator of homogeneity ” decreasesregularity by ”.

Theorem 6.1.11. Let p be an exponent and ” œ R. Suppose ÷ œ �”≠”. Then the

operator ÷(A) maps D(÷(A)) fl XpA into Xp+”

A , and the quasinorm estimate

||÷(A)f ||Xp+”

A

. ||÷||�”

≠”

||f ||XpA

holds for all f œ D(÷(A)) fl XpA. The same results hold for spectral subspaces.

Proof. Let Ï,  œ �ŒŒ and let ‹ œ �Œ

Œ be a Calderón sibling of Â. Then for allf œ D(÷(A)) fl Xp

A we have

||÷(A)f ||Xp+”

A

ƒ ||QÏ,A÷(A)S‹,AQÂ,Af ||Xp+”

. ||÷||�”

≠”

||QÂf ||Xp (6.5)

ƒ ||÷||�”

≠”

||f ||XpA

where (6.5) follows from Theorem 5.2.20. To incorporate spectral subspaces inthis argument, write f œ D(÷(A)) fl Xp,±

A as f = ‰±(A)f and

÷(A)f = ÷(A)‰±(A)f = ‰±(A)÷(A)f,

and note that this shows that ÷(A) maps D(÷(A)) fl R(A)± into R(A)±.

Because the spaces XpA may be incomplete, we cannot extend the operators

÷(A) by boundedness without introducing completions. This is done in Subsection6.1.3. Of course, when ÷ œ HŒ we have D(÷(A)) = R(A), and so we obtainbounded holomorphic functional calculus in the following sense.

Corollary 6.1.12. Let p be an exponent and ÷ œ HŒ. Then the operator ÷(A)is bounded on Xp

A, with

||÷(A)f ||XpA

. ||÷||Œ ||f ||XpA

for all f œ XpA, and likewise for spectral subspaces.

176

This allows us to characterise the positive and negative subspaces Xp,±A as

images of spectral projections.

Corollary 6.1.13. Let p be an exponent. Then we have

Xp,±A = ‰±(A)Xp

A.

Proof. If f œ Xp,±A then by definition we have f = ‰±(A)f œ ‰±(A)Xp

A. Con-versely, if f œ ‰±(A)Xp

A, then f is in R(A)±, and by Corollary 6.1.12 we havef œ Xp

A. Therefore f œ R(A)± fl XpA = Xp,±

A .

The power functions A⁄ := [z ‘æ z⁄](A) for ⁄ œ R \ {0} (see Section 5.2.1)are generally unbounded on R(A), but they do map between our adapted spaceswith a shift in regularity (when we intersect with the domain). This is a directconsequence of Theorem 6.1.11 since [z ‘æ z⁄] œ �≠⁄

⁄ ; the norm equivalence isobtained by applying Theorem 6.1.11 with both ⁄ and ≠⁄.

Corollary 6.1.14. Let p be an exponent and ⁄ œ R \ {0}. Then A⁄ mapsD(A⁄) fl Xp

A into Xp≠⁄A with the quasinorm estimate

------A⁄f

------Xp≠⁄

A

ƒ ||f ||XpA

for all f œ D(A⁄) fl XpA.

Since the operators A⁄ are all densely defined in R(A), and since A⁄0A⁄1 =A⁄0+⁄1 whenever this is meaningful, we have almost proven that A⁄ is an iso-morphism from Xp

A to Xp≠⁄A . We need to extend everything by boundedness to

make this rigorous. As previously mentioned, this requires the introduction ofcompletions.

6.1.3 Completions and interpolationThe spaces Xp

A defined in the previous section are called pre-Hardy–Sobolev andpre-Besov spaces because, with the exception of X2

0,A = R(A), they need not becomplete. One could try to solve this problem by taking arbitrary completions(Xp

A)c of XpA and declaring these to be the Hardy–Sobolev and Besov spaces as-

sociated with A. However, if we take this approach, then for di�erent exponentspi, there may not exist a natural topological vector space in which the comple-tions (Xp

i

A )c both embed.2 This prevents us from discussing interpolants of these2Stahlhut takes this approach in his thesis [84, §4.1], but his ambient space - a product space

of abstract completions indexed over all exponents - is not as natural as the one we are aboutto propose.

177

completions. The impact of this problem on abstract Hardy space theory seemsto have first been discussed by Auscher, McIntosh, and Morris [11]. We avoidthis issue by introducing certain canonical completions within tent and Z-spaces(and hence within L0(R1+n

+ )). If another completion is possible - for example,within the space Z Õ(Rn) of distributions modulo polynomials, in which the classi-cal Hardy–Sobolev and Besov spaces are embedded - then we are free to identifythis with our canonical completion.

By a completion of a quasinormed space Q we mean a continuous injectivemap ÿ : Q æ Q, where Q is a complete quasinormed space and ÿ(Q) is dense inQ. By a weak-star completion of Q, we mean ÿ : Q æ Q as above, where Q is adual space and where ÿ(Q) is weak-star dense in Q. Eventually we will refer toQ itself as the completion, with the associated inclusion being implicit.

In this section, whenever p is infinite, we will interpret ‘completion’ to mean‘weak-star completion’.

Definition 6.1.15. For an exponent p and an auxiliary function  œ �(XpA),

define the canonical completion

ÂXpA := QÂ,AXp

A µ Xp

and likewiseÂXp,±

A := QÂ,AXp,±A µ ÂXp

A

where the closures are taken in the Xp quasinorm when p is finite, and in theweak-star topology on Xp when p is infinite. We equip ÂXp

A and ÂXp,±A with

the Xp quasinorm, so that ÂXpA and ÂXp,±

A become quasi-Banach spaces.

Proposition 6.1.16. Fix p and  as in Definition 6.1.15. Then QÂ,A : XpA æ

ÂXpA and QÂ,A : Xp,±

A æ ÂXp,±A are completions of Xp

A and Xp,±A .

Proof. By construction the spaces ÂXps,A and ÂXp,±

s,A are complete and containQÂ,AXp

A and QÂ,AXp,±A respectively as dense subspaces (in the weak-star topology

when p is infinite). The map QÂ,A : XpA æ ÂXp

A is continuous since  œ �(XpA),

and injective by Corollary 5.2.22. These properties automatically continue tohold for the restrictions of QÂ,A to the spectral subspaces Xp,±

A .

Of course, completions are always unique, and so any completion may be iden-tified with any canonical completion. It will be useful to make this identificationprecise.

178

Proposition 6.1.17 (Identification of completions). Fix p and  as in Definition6.1.15 and suppose that ÿ : Xp

A æ X is a completion of XpA. Then the unique map

QÿÂ,A : X æ Xp such that the triangle

XQÿ

Â,A

!!Xp

A

ÿ

OO

QÂ,A

// Xp

commutes, is an isomorphism between X and ÂXpA. Its inverse is given by the

map SÿÂ,A : Xp æ X, which is the unique continuous extension of the map ÿ ¶

S‹,A : X2 fl Xp æ X (using the weak-star topology on Xp when p is infinite) forany ‹ œ �Œ

Œ which is a Calderón sibling of Â. The same results hold if we replaceall spaces with corresponding positive and negative subspaces.

Proof. Since QÂ,A : XpA æ ÂXp

A is a completion of XpA, by the universal property

of completions there exists a unique map ÊQÿÂ,A : X æ ÂXp

A such that the triangle

XÂQÿ

Â,A

""Xp

A

ÿ

OO

QÂ,A

// ÂXpA

commutes. Hence we have a commutative diagram

XÂQÿ

Â,A

""

Qÿ

Â,A

&&Xp

A

ÿ

OO

QÂ,A

// ÂXpA id

// Xp.

Since(id ¶ÊQÿ

Â,A) ¶ ÿ = QÂ,A = QÿÂ,A ¶ ÿ,

by uniqueness we must have QÿÂ,A = id ¶ ÊQÿ

Â,A = ÊQÿÂ,A. Therefore it su�ces to

show that ÊQÿÂ,A satisfies the desired properties.

To show that SÿÂ,A

ÊQÿÂ,A = idX, observe that we have a commutative diagram

XÂQÿ

Â,A // ÂXpA� � // Xp Sÿ

Â,A //XÂQÿ

Â,A // ÂXpA

XpA

ÿ

OO

QÂ,A//

id

66Â(XpA)?�

OO

� � //

id

66X2 fl Xp?�

OO

S‹,A // Xp

A

ÿ

OO

QÂ,A// Â(Xp

A).?�

OO(6.6)

179

Thus we haveSÿ

Â,AÊQÿ

Â,Aÿ = ÿ ¶ id = ÿ

andÊQÿ

Â,ASÿÂ,A|Â(Xp

A

) = id,

so by uniqueness of extensions we must have that ÊQÿÂ,A and Sÿ

Â,A are mutualinverses.

The corresponding proofs for positive and negative spectral subspaces areidentical.

As a corollary of this argument we can show that ÂXpA is a retract of Xp.

This will be crucial in identifying interpolants.

Corollary 6.1.18. Fix p, Â, and ‹ as in Proposition 6.1.17, and let ÿ : XpA æ X

be a completion of XpA. Then the map Qÿ

Â,ASÿÂ,A : Xp æ ÂXp

A is the extension ofthe projection QÂ,AS‹,A : X2 fl Xp æ QÂ,AXp

A in the appropriate topology (henceindependent of ÿ), and it is a projection onto ÂXp

A. The same statements holdfor spectral subspaces.

Therefore we can write QÂ,AS‹,A := QÿÂ,ASÿ

Â,A to denote this extension.Now that we have thought hard enough about completions, we can extend

the duality and boundedness results of the previous sections.

Proposition 6.1.19 (Duality). Let p be a finite exponent, and let Â, ‹ œ �ŒŒ

be Calderón siblings. Then the X2 inner product identifies ‹XpÕ

Aú as the Banachspace dual of ÂXp

A, and also identifies ‹XpÕ,±Aú as the Banach space dual of ÂXp,±

A .

Proof. If f œ ÂXpA and g œ ‹XpÕ

Aú , then we immediately have

|Èf, gÍX2| Æ ||f ||Xp ||g||XpÕ = ||f ||ÂXpA

||g||‹XpÕ

A

ú,

so every g œ ‹XpÕ

Aú induces a bounded linear functional on XpA.

Conversely, suppose Ï œ (ÂXpA)Õ. Then we can define a bounded linear func-

tional � œ (Xp)Õ by�(F ) := Ï(QÂ,AS‹,AF )

for all F œ Xp. By X-space duality, there exists a function G� œ XpÕ such that

ÈF, G�ÍX2 = �(F )

for all F œ Xp, which satisfies

||G�||XpÕ ƒ ||�||(Xp)Õ . ||Ï||(ÂXpA

)Õ .

180

Hence for all f œ ÂXpA we have

Èf, (QÂ,AS‹,A)úG�ÍX2 = Èf, G�ÍX2 = �(f) = Ï(f)

since f = QÂ,AS‹,Af . Since QÂ,AS‹,A is the continuous extension of QÂ,AS‹,A

from X2 fl Xp to Xp, and since (QÂ,AS‹,A)ú = Q‹,AúSÂ,Aú on X2, we find that(QÂ,AS‹,A)ú = Q‹,AúSÂ,Aú . Therefore we have

Ï(f) = Èf, GÏÍX2

for all f œ ÂXpA, where GÏ = Q‹,AúSÂ,AúG� œ ‹XpÕ

Aú . Furthermore we have

||GÏ||‹XpÕ

A

ú=

------Q‹,AúSÂ,AúG�

------XpÕ

. ||Ï||(ÂXpA

)Õ .

As with every other result in this section, the same proof works for spectralsubspaces.

Proposition 6.1.20 (Boundedness of functional calculus). Let p be an exponent,” œ R, and ÷ œ �”

≠”. Suppose ÿ1 : XpA æ X and ÿ2 : Xp+”

A æ Y are completions.Then ÷(A) extends to a bounded operator ]÷(A) : X æ Y, in the sense that thediagram

D(÷(A)) fl XpA

ÿ1✏✏

÷(A) // Xp+”A

ÿ2✏✏

X]÷(A)

//Y

commutes, and that ----

----]÷(A)f----

----Y. ||÷||�”

≠”

||f ||X . (6.7)

for all f œ X. Similar results hold for spectral subspaces.

Proof. Since D(÷(A)) is dense in X2A = R(A) and since X2

A fl XpA is dense in

XpA (Corollary 6.1.7 for finite exponents, duality for infinite exponents using the

weak-star topology), we have that D(÷(A)) flXpA is dense in Xp

A. The result thenfollows from Theorem 6.1.11 and the universal property of completions.

Remark 6.1.21. Evidently ‘completed’ versions of Corollaries 6.1.12, 6.1.13, and6.1.14 can be formulated.Remark 6.1.22. In the situation of Proposition 6.1.20 we will use the symbol ÷(A)to denote both the original operator D(÷(A)) fl Xp

A æ Xp+”A and its extension to

completions X æ Y. This will not cause any ambiguity, but one should becareful.

181

Finally we can state the interpolation theorem for canonical completions ofpre-Hardy–Sobolev and pre-Besov spaces. Having established so much abstracttheory, this is now a simple consequence of the interpolation results for tent spacesand Z-spaces.

Theorem 6.1.23 (Interpolation of completions). Fix 0 < ◊ < 1 and  œ �ŒŒ.

Let p and q be exponents.

(i) Suppose j(p), j(q) Ø 0, with equality for at most one exponent. Then wehave the identification

[ÂHpA, ÂHq

A]◊ = ÂH[p,q]◊

A .

(ii) Suppose i(p), i(q) Ø 1, with p and q not both infinite. Then we have theidentification

[ÂBpA, ÂBq

A]◊ = ÂB[p,q]◊

A .

(iii) Suppose ◊(p) ”= ◊(q). Then we have the identification

(ÂXpA, ÂXq

A)◊,p◊

= ÂB[p,q]◊

A

where p◊ = i([p, q]◊).

Proof. Fix a Calderón sibling ‹ œ �ŒŒ of Â. By Corollary 6.1.18 the map QÂ,AS‹,A

extends to a map QÂ,AS‹,A : Xp + Xq æ ÂXpA + ÂXq

A which restricts to projec-tions Xp æ ÂXp

A and Xq æ ÂXqA. Therefore by the retraction/coretraction

interpolation theorem (see [89, §1.2.4]),3, for all interpolation functors F we have

F(XpA, Xq

A) = QÂ,AS‹,AF(Xp, Xq).

The results then follow from Corollary 6.1.18 and the interpolation theorems5.1.12, 5.1.30, and 5.1.31.

6.1.4 The Cauchy operator on general adapted spacesRecall the function sgp = [z ‘æ e≠[z]]. This function is in �Œ

0 µ HŒ, and thereforefor all t > 0 the operator e≠t[A] is defined and bounded on R(A). By Corollary6.1.12, for all exponents p we have that e≠t[A] is bounded on Xp

A. Furthermore,3This is only stated for Banach spaces in the given reference. The only property specific to

Banach spaces which is needed is the validity of the closed graph theorem, which also holds forquasi-Banach spaces [56, §2], so the proof goes through even for quasi-Banach spaces.

182

when restricted to the positive or negative spectral subspace, the operator Qsgp,A

coincides with the Cauchy operator C±A , which produces solutions to the Cauchy

problem associated with A on R+ or R≠ respectively.Given a completion X of Xp

A, each of the operators e≠t[A] : XpA æ Xp

A and thespectral projections ‰±(A) extend to maps X æ X (in the sense of Proposition6.1.20), and by means of these maps we can extend the Cauchy operators C±

A tomaps

C±A : X æ LŒ(R± : X±).

Note that we construct these operators by extending each operator e≠t[A]‰±(A) byboundedness, rather than by extending the Cauchy operators directly. Similarlywe can define

Qsgp,A : X æ LŒ(R+ : X).

Proposition 6.1.24 (Properties of Cauchy extensions). Let p be an exponent,and fix a completion X of Xp

A.4 Then for all f œ X the extension Qsgp,Af is inCŒ(R+ : X), and if f œ ‰±(A)X, then the Cauchy extension C±

Af solves theCauchy equation

ˆtC±Af ± AC±

Af = 0

strongly in CŒ(R± : X). Furthermore for all f œ X we also the limits

limtæ0

Qsgp,Af(t) = f and limtæŒ

Qsgp,Af(t) = 0. (6.8)

Proof. First we will prove the limit results. These reduce to the case of finiteexponents p, as for infinite p we can deduce the limits (6.8) by testing againstXpÕ

Aú . Furthermore, by density, it su�ces to prove the limits

limtæ0

e≠t[A]f = f and limtæŒ

e≠t[A]f = 0

for f œ XpA.

For f œ HpA, these follow from arguments almost identical to those of [16,

Propositions 4.5 and 4.6], the only di�erence being the presence of the weightŸ≠◊(p), which does not change the argument. Now fix an exponent q ”= p suchthat q Òæ p, so that Hq

A Òæ BpA (Proposition 6.1.9). For f œ Hq

A, we then have

limtæ0

------e≠t[A]f ≠ f

------Bp

A

. limtæ0

------e≠t[A]f ≠ f

------Hq

A

= 0

4Recall that we mean a weak-star completion when p is infinite, and in this case we use theweak-star topology on X.

183

andlimtæŒ

------e≠t[A]f

------Bp

A

. limtæŒ

------e≠t[A]f

------Hq

A

= 0.

Since HqA is dense in Bp

A (Corollary 6.1.7), these limits hold for all f œ BpA.

Now we will prove the smoothness result. It su�ces to work with f œ ‰+(A)Xhere, as the result for ‰≠(A)X uses the same argument, and the general resultfollows from the decomposition X = ‰+(A)X ü ‰≠(A)X. First observe that thefunction � : R+ æ HŒ defined by �(t) = [z ‘æ e≠t[z]] is smooth with Fréchetderivative Dt� : R æ HŒ given by Dt�(·) = [z ‘æ ≠· [z]e≠t[z]]. Next, note thatthe map �A : HŒ æ L(‰+(A)X) with �A(f) = f(A) is linear and bounded inthe strong topology (Proposition 6.1.20). By the chain rule, the composition ofthese maps is smooth, with Fréchet derivative

Dt(�A ¶ �)(·) = �A ¶ Dt�(·) = ≠·Ae≠tA.

We can then write

ˆtC+Af(t) = Dt(�A ¶ �)(1)f = ≠Ae≠tAf = ≠AC+

Af(t),

which completes the proof.

Now we must address the question of whether or not CA maps XpA into Xp.

This would imply that one can construct CŒ(R± : XpA) solutions5 to (CR)A which

are in Xp with given initial data in Xp,±A . It turns out that this is only reasonable

when ◊(p) < 0. For i(p) Æ 2 we already know everything we need to prove this;for i(p) > 2 we need more information (see Subsection 6.2.2).

Theorem 6.1.25 (Cauchy characterisation of adapted spaces, i(p) Æ 2). Let pbe an exponent with i(p) Æ 2 and ◊(p) < 0. Then for all f œ R(A),

||f ||XpA

ƒ ||Qsgp,Af ||Xp .

Proof. By Proposition 6.1.3, we have

sgp œ �Œ0 µ �(≠◊(p)+n| 1

2 ≠j(p)|)+◊(p)+ fl HŒ µ �(Xp

A),

which yields the result.5This solution concept does not always agree with the L2

loc

solution concept that we arereally interested in. This is discussed further in Subsection 7.3.1.

184

Remark 6.1.26. The estimate

||f ||XpA

. ||Qsgp,Af ||Xp

holds for all p, as can be shown by a Calderón reproducing argument as in theproof of Proposition 6.1.3. The reverse estimate need not hold in general.

We will need the following technical lemmas in Section 7.3.

Lemma 6.1.27. For every M > 0, there exist functions Ï+, Ï≠ œ HŒ suchthat (ϱ

s (A))s>0 satisfies o�-diagonal estimates of order M , ϱs (A) = e≠s[A] on

the corresponding spectral subspace X2,±A , and limsæ0 ϱ

s (A) = I in the L2-strongoperator topology.

For a proof, see [15, Lemma 15.1], noting that H2,±A = B2,±

A . Although thisresult is stated for A œ {DB, BD} there, the proof only uses the standard as-sumptions.

Corollary 6.1.28. Let p œ (0, Œ]. Suppose f œ X±A fl Ep. Then C±

A f(t) œ Ep

for each t œ R±, and if p < Œ then limtæ0 C±A f(t) = f in Ep.

Proof. Choose functions ϱ as in Lemma 6.1.27, such that (ϱs (A))s>0 satisfies

o�-diagonal estimates of large order. By Proposition 5.2.9, the operators ϱt (A)

are bounded on Ep. Since ϱt (A) = e≠t[A] on X2,±

A , we have C±A f(t) = e≠t[A]f œ Ep

for all t œ R±. The limit statement follows from Lemma 6.1.27 (which gives strongconvergence in L2) and Proposition 5.2.10 (which improves this to Ep).

6.2 Spaces adapted to perturbed Dirac opera-tors

We now begin to work with Cm(1+n)-valued functions for some fixed m œ N.When applying results of the previous sections, we implicitly take N = m(1 + n).

In this section, we fix the Dirac operator D and consider multipliers B as inSubsection 4.1.2 of the introduction. Recall that the perturbed Dirac operatorsDB and BD then satisfy the Standard Assumptions (Theorem 5.2.12). Further-more, R(DB) = R(D), and the restrictions DB|R(DB) and BDR(BD) are similarunder conjugation by B|R(DB) [17, Proposition 2.1]. Consequently, wheneverf œ D(D) fl R(BD) and Ï œ HŒ we have

DÏ(BD)f = Ï(DB)Df.

We will refer to this principle as similarity of functional calculi and use it repeat-edly.

185

6.2.1 Identification of spaces adapted to D, DB, and BD

The operators DB and BD satisfy the Standard Assumptions, so we can definepre-Besov–Hardy–Sobolev spaces Xp

DB. The case B = I yields XpD.

For a certain range of exponents p that we denote by Imax, the spaces XpD

may be identified as projections of classical smoothness spaces (to be shownin Proposition 6.2.1). Recall that we use the notation Xp to denote classicalsmoothness spaces, as in Subsection 5.1.5. These spaces are all contained inZ Õ(Rn), the space of tempered distributions modulo polynomials.

In [16, Theorem 4.16] it is shown that for p œ (n/(n + 1), Œ) we have anidentification

H(p,0)D ƒ PD(H(p,0) fl L2) µ Z Õ(Rn)

where PD is the bounded projection from L2(Rn) onto R(D). Since PD extendsboundedly to the spaces H(p,0) by virtue of being a Fourier multiplier withinthe scope of the Mikhlin multiplier theorem (see [90, Theorem 5.2.2] and [53,Proposition 4.4]), we may write

H(p,0)D = PD(H(p,0)) = H(p,0) fl DZ Õ µ Z Õ(Rn),

thus providing a completion of H(p,0)D within the space of distributions modulo

polynomials. Hence if we have an identification H(p,0)DB ƒ H(p,0)

D , we can find acompletion of H(p,0)

DB in Z Õ.Furthermore, combining [16, Lemma 11.6] with Corollary 6.1.14 shows that

for all p œ (1, Œ) we have

H(p,≠1)D ƒ PD(H(p,≠1) fl L2) fl DZ Õ µ Z Õ(Rn),

and for all – œ [0, 1) we have

H(Œ,0;–)D ƒ PD(H(Œ,0;–) fl L2) µ Z Õ(Rn),

so by the same argument we may write

H(p,≠1)D ƒ PD(H(p,≠1)) = H(p,≠1) fl DZ Õ µ Z Õ(Rn)

andH(Œ,0;–)

D ƒ PD(H(Œ,0;–)) = H(Œ,0;–) fl DZ Õ µ Z Õ(Rn)

(where PD is extended by duality).We can interpolate between these observations to yield an identification of the

spaces HpD in a restricted (but for our applications, su�ciently large) range of p.

186

Figure 6.1: The region Imax on which HpD ƒ Hp fl DZ Õ.

12

1 n+1n

j(p)0≠1

n

◊(p)

0

≠1

Theorem 6.2.1 (Identification of D-adapted spaces). Suppose p is in the regionI

max

pictured in Figure 6.1. Then

HpD ƒ PD(Hp fl L2).

Furthermore, if p is in the interior of Imax

, then

BpD ƒ PD(Bp fl L2).

We abuse notation by writing XpD = PD(Xp).

To prove this we will need the following lemma.

Lemma 6.2.2. Suppose that f œ R(D) (note that we do not take the closure ofthe range here). Then there exists g œ D(D) fl R(D) such that f = Dg and

||f ||Xp ƒ ||g||Xp+1 .

for all exponents p.

Proof. Since f œ R(D) there exists g œ D such that f = Dg. Let g = PDg. Sincethe projection PD is along N (D), we have f = Dg also. The estimate

||f ||Xp . ||g||Xp+1

187

then follows since D is a first-order homogeneous di�erential operator. To obtainthe reverse estimate, we need to invert D on R(D). Let

T = D�≠1PD + (≠�)≠1/2(I ≠ PD).

One can show that T is a homogeneous Fourier multiplier of order ≠1, and hencemaps Xp to Xp+1. Furthermore, TPD inverts D on R(D), and so we have theestimate

||g||Xp+1 . ||f ||Xp

which completes the proof.

Proof of Theorem 6.2.1. When p is finite, this follows directly from the identifi-cation of complex and real interpolants of the spaces Hp (Theorem 5.1.52).

Now suppose p is infinite. Observe that the subregion of Imax consistingof infinite exponents (the lightest shaded region, including the dashed line) isprecisely the ¸-dual region of the darkest shaded region, including the solid line.Therefore for all infinite p œ Imax, p¸ is finite and in Imax. If f = Dg for someg œ D(D) fl R(D) as in Lemma 6.2.2, then we have

||f ||XpD

ƒ ||g||Xp+1D

(6.9)

ƒ suphœXp¸

D

|Èg, hÍ|

ƒ suphœXp¸ flL2

|Èg,PDhÍ|

= suphœXp¸ flL2

|ÈPDg, hÍ| (6.10)

ƒ ||g||Xp+1 (6.11)ƒ ||f ||Xp . (6.12)

with all suprema taken over appropriately normalised elements. Line (6.9) followsfrom Corollary 6.1.14. In (6.10) we use orthogonality of the decomposition L2 =N (D) ü R(D). In line (6.11) we remove the projection by using that g œ R(D).In (6.12) we use the conclusion of Lemma 6.2.2, which follows from our choiceof g. Therefore by weak-star density, we get Xp

D ƒ PD(Xp fl L2) (the projectioncomes from the fact that f œ R(D) in this estimate).

Now we shall discuss spaces adapted to DB, and the range of exponents forwhich they may be identified with spaces adapted to D. As shown in the intro-duction, the following ‘identification region’ plays a central role in the theoremsof Chapter 7.

188

Definition 6.2.3. We define

I(X, DB) :=Óp œ Imax fl Efin : ||f ||Xp

DB

ƒ ||f ||XpD

for all f œ R(DB) = R(D)Ô

.

(6.13)and for s œ R,

Is(X, DB) := {i(p) : p œ I(X, DB) : ◊(p) = s} µ (0, Œ).

Note that I(X, DB) is defined to be a set of finite exponents. We couldinclude infinite exponents in this definition, but it is technically more convenientto restrict ourselves to finite exponents.6 It is also defined to be contained inImax, so not only do we have Xp

DB = XpD, but we also have the identification of

XpD as the projection of a classical space.

We recall a key result of Auscher and Stahlhut, which follows from [16, The-orem 5.1 and Remark 5.2].

Theorem 6.2.4 (Auscher–Stahlhut). There exists Á = Á(B) > 0 such that(2n/(n + 2) ≠ Á, 2 + Á) µ I0(H, DB). Furthermore, if n = 1, then I0(H, DB) =(1/2, Œ).

We will extend this result to allow for more general exponents of order ◊(p) œ[≠1, 0], and also to incorporate Besov spaces.

The ¸-duality operation on exponents provides a link between I(X, DB) andI(X, DBú) (Proposition 6.2.7). We need some preliminary results to establishthis link. First we state a local coercivity property of B, which is proven in [16,Lemma 5.14].

Lemma 6.2.5. For any u œ L2loc

with Du œ L2loc

and any ball B(x, t) œ Rn, wehave ˆ

B(x,t)|Du|2 .B,n,N

ˆB(x,2t)

|BDu|2 + t≠2ˆ

B(x,2t)|u|2.

Proposition 6.2.6 (Intertwining and regularity shift). Let p be an exponent,and suppose f œ Xp

BD fl D(D). Then Df œ Xp≠1DB and

||Df ||Xp≠1DB

ƒ ||f ||XpBD

.

6More precisely: in applications, whenever we deal with infinite exponents, we always con-sider the space in question as the dual space, and the predual exponent will be in I(X, DBú)(see for example Theorem 7.3.2).

189

Proof. We will only prove the result for T p with p = (p, s) finite; all other casesare proven by the same argument.

Let  œ �ŒŒ be nondegenerate and define  œ �Œ

Œ by Â(z) = zÂ. Then Â(DB)maps R(DB) into D((DB)≠1). Since f œ D(D) we have Df œ R(D) = R(DB).Using similarity of functional calculi, write

||Df ||Hp≠1DB

ƒ ||t ‘æ Â(tDB)Df ||T p

s≠1

= ||t ‘æ DÂ(tBD)f ||T p

s≠1

=------t ‘æ D(BD)≠1Â(tBD)f

------T p

s

.

For all t > 0 we have

(BD)≠1Â(tBD)f œ L2, D(BD)≠1Â(tBD)f = Â(tDB)Df œ L2,

so we can apply Lemma 6.2.5 for each t > 0 with u = (BD)≠1Â(tBD)f as follows:for all x œ Rn,

A(t ‘æ t≠sD(BD)≠1Â(tBD)f)(x)

=Aˆ Œ

0t≠2s

ˆB(x,t)

|(D(BD)≠1Â(tBD)f)(y)|2 dydt

tn+1

B1/2

.Aˆ Œ

0t≠2s

CˆB(x,2t)

|(Â(tBD)f)(y)|2 dy

+ˆ

B(x,2t)|(Â(tBD)f)(y)|2 dy

Ddt

tn+1

B1/2

. A(Ÿ≠sQÂ,BDf)(x) + A(Ÿ≠sQÂ,BDf)(x).

Therefore

||Df ||Hp≠1DB

.------QÂ,BDf

------T p + ||QÂ,BDf ||T p ƒ ||f ||Hp

BD

.

To prove the reverse estimate, using that f œ D(D) = D(BD), write

||f ||HpBD

=------(BD)≠1BDf

------Hp

BD

. ||BDf ||Hp≠1BD

ƒ ||t ‘æ Â(tBD)BDf ||T p≠1

= ||t ‘æ BÂ(tDB)Df ||T p≠1

. ||QÂ,DBDf ||T p≠1

ƒ ||Df ||Hp≠1DB

190

using (6.1.14), boundedness of the multiplier B, and similarity of functional cal-culi.

Proposition 6.2.7 (¸-duality of identification regions). If p œ I(X, DB), then||f ||

Xp¸DB

úƒ ||f ||

Xp¸D

for all f œ R(D). In particular, if p¸ is also finite, thenp¸ œ I(X, DBú).

Proof. Suppose g œ D(D)flXpÕ

BúD. Then arguing similarly to the proof of Theorem6.2.1,

||Dg||Xp¸

DB

úƒ sup

hœD(D)flX(p¸)ÕB

úD

|ÈDg, hÍ|

= suphœD(D)flXp+1

B

úD

|Èg, DhÍ|

ƒ suphœXp

DB

ú

|Èg, hÍ| (6.14)

ƒ suphœXp

D

|Èg, hÍ| (6.15)

ƒ ||g||XpÕD

ƒ ||Dg||Xp¸

D

(6.16)

with all suprema taken over appropriately normalised elements. The equivalence(6.14) uses Proposition 6.2.6, and then (6.15) uses the assumption on p. Thefinal equivalence (6.16) uses Corollary 6.1.14. Since D(D(D) fl XpÕ

BúD) is dense7

in Xp¸

DBú (by density of R(D) in X2D and Corollary 6.1.7), we are done.

The following result then follows immediately from Theorem 6.2.4.

Corollary 6.2.8. There exists Á > 0 such that

I≠1(H, DB) ∏

Y__]

__[

(1, Œ) (n = 1)(2 ≠ Á, Œ) (n = 2)(2 ≠ Á, 2n/(n ≠ 2) + Á) (n Ø 3)

The main application of our discussion of interpolations and completions ofadapted spaces, particularly Theorem 6.1.23, is in showing that I(X, DB) isclosed under interpolation, and also that information on I(H, DB) implies infor-mation on I(B, DB).

Proposition 6.2.9 (Convexity of identification regions). Let ◊ œ [0, 1].7Weak-star dense when pÕ is infinite.

191

(i) If p, q œ I(H, DB), then [p, q]◊ is in I(H, DB). Furthermore, if ◊(p) ”=◊(q), then [p, q]◊ is in I(B, DB).

(ii) If p, q œ I(B, DB), then [p, q]◊ œ I(B, DB).

Proof. We will only prove the first part, as the proof of the other parts areidentical but with real interpolation replacing complex interpolation.

Suppose p, q œ I(H, DB). Let Â, Ï œ �ŒŒ be Calderón siblings. First note

that we have a map

QÂ,DBXpDB + QÂ,DBXq

DB

SÏ,DB≠æ Xp

DB + XqDB

idÒæ Xp

D + XqD

(here we use Theorem 6.2.1), which restricts appropriately and which extends byboundedness to

SÏ,DB : ÂXpDB + ÂXq

DB æ XpD + Xq

D.

By Proposition 6.1.17, the restrictions of SÏ,DB to ÂXpDB and ÂXq

DB are isomor-phisms, and their inverses both extend QÂ,DB : X(2,0)

DB æ QÂ,DBX(2,0)DB . Therefore,

by complex interpolation (Theorem 6.1.23) we have an isomorphism

SÏ,DB : ÂX[p,q]◊

DB æ X[p,q]◊

D

which extends SÏ,DB : QÂ,DBX(2,0)DB æ X(2,0)

DB = X(2,0)D . Hence for all f œ R(DB) =

X(2,0)DB we have

||f ||X[p,q]◊

DB

ƒ ||QÂ,DBf ||X[p,q]◊

ƒ ||SÏ,DBQÂ,DBf ||X[p,q]◊

D

= ||SÏ,DBQÂ,DBf ||X[p,q]◊

D

= ||f ||X[p,q]◊

D

,

and therefore [p, q]◊ œ I(H, DB).

Therefore for every B we have a region Imin such that Imin µ I(H, DB) andIo

min µ I(B, DB), pictured in Figure 6.2, where lower bounds on I0(H, DB) andI≠1(H, DB) can be found in Theorem 6.2.4 and Corollary 6.2.8.

Remark 6.2.10. If p œ I(X, DB), then we identify the projected classical Besov–Hardy–Sobolev space PD(Xp) = Xp flDZ Õ as a completion of Xp

DB via the exten-sion of the identity map Xp

DB æ XpD. If p is infinite and p¸ œ I(X, DBú), then

by Proposition 6.2.7 we may identify PD(Xp) as a weak-star completion of XpDB.

We abuse notation by writing XpDB for these completions.

192

Figure 6.2: The region Imin µ I(H, DB).

12

1 n+1n

j(p)0≠1

n

◊(p)

0

≠1

I0(H, DB)

I≠1(H, DB)

Having made these identifications, note that we do not have equality of Xp,+DB

and Xp,+D . The first of these spaces is defined via the spectral projection ‰+(DB),

while the second is defined via ‰+(D). However, we do of course have Xp,+DB µ Xp

D.This will be important in applications to boundary value problems.

Remark 6.2.11. For a coe�cient matrix A as in the introduction, if B = A, then„Aú = B := NBúN , where

N :=S

UI 00 ≠I

T

V .

Since DN = ≠ND and N acts on R(D), the operators DBú and ≠DB aresimilar on R(D) = R(DBú) = R(DB). Thus all functional calculus propertiesof DBú can be transferred to DB, and vice versa. This gives natural isomorphismsbetween Xp

DBú and XpDB

, and in particular we have I(X, DBú) = I(X, DB). Forfurther details see [16, §12.2].

6.2.2 The Cauchy operator on DB-adapted spaces

This section is devoted to the proof of the following theorem. Recall that theregion Imax is introduced in Theorem 6.2.1.

193

Theorem 6.2.12 (Cauchy characterisation of adapted spaces, i(p) > 2). Letp be such that i(p) > 2, ◊(p) œ (≠1, 0) and p¸ œ I(X, DBú). Then for allf œ R(DB), ---

---C+DBf

------Xp . ||f ||Xp

D

.

Remark 6.2.13. The condition p¸ œ I(X, DBú) is equivalent to p œ I(X, DB)when p is finite (Proposition 6.2.7).

Remark 6.2.14. The reverse estimate

||f ||XpA

.------C+

A f------Xp

holds for general operators A (satisfying the standard assumptions) and for allp (see Remark 6.1.26). However, we do not know whether Theorem 6.2.12 holdswith DB replaced by A and without the assumption on p¸.

In contrast with Theorem 6.1.25, the proof of this theorem is quite long. Wethank Pascal Auscher for suggesting this argument.

Before proving the theorem, we establish a technical lemma.

Lemma 6.2.15. Suppose ◊ œ (≠1, 0), g œ D(D), and f = Dg. Then for all› œ Rn, · > 0, and M œ N, we have

¨T (B(›,·))

|t≠◊(I + itDB)≠2f(x)|2 dx dt

t

.M

ˆB(›,4·)

Q

aˆ

B(›,4·)+

Œÿ

j=22≠2j(M≠ n

2 ≠(1+◊))ˆ

A(›,2j≠1·,2j+2·)

R

b G(x, y) dx dy

whereG(x, y) := |g(x) ≠ g(y)|2

|x ≠ y|n+2(1+◊) .

Proof. Fix ‰1, ‰ œ CŒc (Rn) such that

supp ‰1 µ B(›, 4·), ‰1|B(›,2·) © const,supp ‰ µ A(›, ·/2, 4·) ‰|A(›,·,2·) © const

For all j Ø 2 define ‰j(x) := ‰(2≠jx), so that supp ‰j µ A(›, 2j≠1·, 2j+2·) and‰j|A(›,2j·,2j+1·) © 1. We can choose the functions ‰1 and ‰ such that qŒ

j=1 ‰j © 1.Let

c :=ˆ

B(›,·/2)g.

194

Then we have

f = D(g ≠ c) =Œÿ

j=1D(g ≠ c)‰j.

First we will prove that

¨T (B(›,·))

|t≠◊(I + itDB)≠2D(g ≠ c)‰1(x)|2 dx dt

t

.ˆ

B(›,4·)

ˆB(›,4·)

G(x, y) dx dy. (6.17)

Since �20 œ �≠◊

◊+ fl HŒ µ �(X(2,◊)DB ) and since (2, ◊) œ I(H, DB), we have

¨T (B(›,·))

|t≠◊(I + itDB)≠2D(g ≠ c)‰1(x)|2 dx dt

t

ƨ

R1+n

+

|t≠◊(I + itDB)≠2D(g ≠ c)‰1(x)|2 dx dt

t

ƒ ||D(g ≠ c)‰1||2H(2,◊)DB

ƒ ||(g ≠ c)‰1||2H2◊+1

ƒˆRn

|D2◊+1(g ≠ c)‰1(x)|2 dx

using Lemma 5.1.49 (which is valid since 2 > 2n/(n + 1 + ◊)) in the last line.

We claim that

ˆRn

ˆRn

|(g ≠ c)‰1(z) ≠ (g ≠ c)‰1(y)|2|z ≠ y|n+2(◊+1) dz dy .

¨B(›,4·)2

|g(z) ≠ g(y)|2|z ≠ y|n+2(◊+1) dz dy,

(6.18)from which estimate (6.17) will follow. First observe that if y œ B(›, ·) andz œ B(›, ·/2), then ‰1(z) = ‰1(y) = 1 and the estimate (6.18) (restricted to such

195

y and z) follows immediately. Next, we can estimateˆ

B(›,·)c

ˆB(›,·/2)

|(g ≠ c)‰1(z) ≠ (g ≠ c)‰1(x)|2|z ≠ x|n+2(◊+1) dz dx

.ˆ

A(›,·,4·)

ˆB(›,·/2)

|(g ≠ c)(z)(1 ≠ ‰1(x))|2|z ≠ x|n+2(◊+1) dz dx + A

.‰

ˆA(›,·,4·)

ˆB(›,·/2)

|z ≠ x|≠n≠2(◊+1)A

B(›,·/2)|g(z) ≠ g(y)| dy

B2

dz dx + A

. rn

ˆB(›,·/2)

|z ≠ y|≠n≠2(◊+1)A

B(›,·/2)|g(z) ≠ g(y)| dy

B2

dz + A

. rn

ˆB(›,·/2)

ˆB(›,·/2)

|g(z) ≠ g(y)|2|z ≠ y|n+2(◊+1) dz dy + A

.¨

B(›,4·)2

|g(z) ≠ g(y)|2|z ≠ y|n+2(◊+1) dz dy,

where

A .¨

B(›,4·)2

|g(z) ≠ g(y)|2|z ≠ y|n+2(◊+1) dz dy,

and where we used |z ≠ x| & |z ≠ y| on the region of integration.Finally, we estimateˆRn

ˆB(›,·/2)c

|(g ≠ c)‰1(z) ≠ (g ≠ c)‰1(x)|2|z ≠ x|n+2(◊+1) dz dx

.ˆ

B(›,4·)

ˆA(›,·/2,4·)

|(g ≠ c)(x)(‰1 ≠ 1)(z) ≠ (g ≠ c)(x)(‰1 ≠ 1)(x)|2|z ≠ x|n+2(◊+1) dz dx + A

.Ò›

ˆB(›,4·)

ˆA(›,·/2,4·)

|z ≠ x|≠n≠2◊

A

B(›,·/2)|g(x) ≠ g(y)| dy

B2

dz dx + A

.ˆ

B(›,4·)r≠2◊

A

B(›,·/2)|g(x) ≠ g(y)| dy

B2

dx + A

.¨

B(›,4·)2

|g(x) ≠ g(y)|2|x ≠ y|n+2(◊+1) dx dy

with A as before, using r & |x ≠ y| and |x ≠ y|n+2◊ & |x ≠ y|n+2(◊+1) on the regionof integration.

Now we will handle the remaining ‰j terms. For j Ø 2, by local coercivity(Lemma 6.2.5), the equality

BD(I + itBD)≠1 = (I ≠ (I + itBD)≠1)/it,

196

and o�-diagonal estimates of the families (I + itBD)≠1 and (I + itBD)≠2 of orderM , we can estimate¨

T (B(›,·))|t≠◊(I + itDB)≠2D(g ≠ c)‰j(x)|2 dx dt

t

.ˆ ·

0t≠2◊

ˆB(›,·)

|D(I + itBD)≠2(g ≠ c)‰j(x)|2 dxdt

t

.ˆ ·

0t≠2◊

C ˆB(›,2·)

|(BD(I + itBD)≠2(g ≠ c)‰j(x)|2 dx

+ ·≠2ˆ

B(›,2·)|(I + itBD)≠2(g ≠ c)‰j(x)|2 dx

Ddt

t

.ˆ ·

0t≠2◊

A2j·

t

B≠2M 1t≠2 + ·≠2

2||(g ≠ c)‰j||22

dt

t

. 2≠2jM

· 2(1+◊) ||(g ≠ c)‰j||22 .

Furthermore, for each j Ø 2 we have

||(g ≠ c)‰j||22 ƈ

A(›,2j≠1·,2j+2·)

A

B(›,·/2)|g(x) ≠ g(y)| dy

B2

dx

Æ ·≠n

ˆB(›,·/2)

ˆA(›,2j≠1·,2j+2·)

|g(x) ≠ g(y)|2 dx dy

. · 2(1≠◊)2j(n+2(1+◊))ˆ

B(›,4·)

ˆA(›,2j≠1·,2j+2·)

|g(x) ≠ g(y)|2|x ≠ y|n+2(1+◊) dx dy.

Putting these estimates together completes the proof of the lemma.

Proof of Theorem 6.2.12. Step 1: Reduction to a resolvent estimate.As stated in [15, Proof of Lemma 15.1], there exists fl œ HŒ of the form

fl(z) =Nÿ

m=1cm(1 + imz)≠2

for some scalars c1, . . . , cN œ C, and  œ �2N nondegenerate, such that

e≠z = fl(z) + Â(z) for all z œ S+µ .

We thus have------C+

DBf------Xp .N

------t ‘æ (I + itDB)≠2‰+(DB)f

------Xp +

------QÂ,DB‰+(DB)f

------Xp .

(6.19)For N su�ciently large we have

 œ �2N µ �≠◊(p)+

(◊(p)+n| 12 ≠j(p)|)+ fl �+

+ µ �(XpDB),

197

and so ------QÂ,DB‰+(DB)f

------Xp . ||f ||Xp

DB

ƒ ||f ||XpD

by Proposition 6.2.7 and p¸ œ I(X, DBú). Therefore it su�ces to prove theestimate ---

---t ‘æ (I + itDB)≠2f------Xp . ||f ||Xp

D

(6.20)

for all f œ R(DB). Applying this inequality to ‰+(DB)f and invoking theboundedness of ‰+(DB) on Xp

DB will yield------C+

DBf------Xp . ||f ||Xp

DB

ƒ ||f ||XpD

.

To prove (6.20), by density (Corollary 6.1.7 and density of R(D) in X2D),8 it

su�ces to consider f = Dg for g œ D(D) fl R(D) such that

||f ||XpD

ƒ ||f ||Xp ƒ ||g||Xp+1

as in Lemma 6.2.2.Step 2a: Completing the proof for Hardy–Sobolev spaces.Suppose i(p) < Œ and (X,X) = (T,H). Lemma 6.2.15 and a crude estimate

give ¨T (B(›,·))

|t≠◊(p)(I + itDB)≠2f(x)|2 dx dt

t

.M

Q

a1 +Œÿ

j=22≠2j(M≠ n

2 ≠(1+◊(p)))

R

bˆ

B(›,4·)|D2

1+◊(p)g(x)|2 dx,

and so by taking M > n2 ≠ (1 + ◊(p)) we get¨

T (B(›,·))|t≠◊(p)(I + itDB)≠2f(x)|2 dx dt

t.ˆ

B(›,4·)|D2

1+◊(p)g(x)|2 dx.

Hence for all › œ Rn we have

C(t ‘æ t≠◊(p)(I + itDB)≠2f)(›)2 . sup·>0

ˆB(›,4·)

|D21+◊(p)g|2 = M2(D2

1+◊(p)g)(›)2,

and so by Theorem 5.1.9, boundedness of M2 on Li(p) (since i(p) > 2), Lemma5.1.49 (using 1 + ◊(p) œ (0, 1)), Dg = f , and p œ I(H, DB), we get

------t ‘æ (I + itDB)≠2f

------T p .

------M2(D2

1+◊(p)g)------Li(p)

.------D2

1+◊(p)g------Li(p)

ƒ ||g||Hp+1

ƒ ||f ||HpD

,

8When p is infinite, we use weak-star density.

198

which completes the proof in the Hardy–Sobolev case.Step 2b: Completing the proof for BMO-Sobolev spaces. Suppose

p = (Œ, ◊; 0) and (X,X) = (T,H). For all (t, x) œ R1+n+ , Lemma 6.2.15 and the

Strichartz characterisation of ˙BMO1+◊ (Theorem 5.1.51) yieldA

t≠n

¨T (B(x,t))

|·≠◊(I + i·DB)≠2f(›)|2 d›d·

·

B1/2

.M t≠n/2

Q

atn ||g||2 ˙BMO1+◊

+Œÿ

j=22≠2j(M≠ n

2 ≠(1+◊))(2j+2t)n ||g||2 ˙BMO1+◊

R

b1/2

ƒ ||g|| ˙BMO1+◊

Q

a1 +Œÿ

j=22≠2j(M≠n≠(1+◊))

R

b1/2

ƒ ||g|| ˙BMO1+◊

provided that M is su�ciently large. Therefore we have as in the previous step------· ‘æ (I + i·DB)≠2f

------T p . ||g||Hp+1 ƒ ||f ||Hp

D

,

which completes the proof in the BMO-Sobolev case.Step 2c: Completing the proof for Hölder spaces. Let p = (Œ, ◊; –).

First we prove the result for X = T . By the definition of the Hölder norm wehave

G(x, y) Æ ||g||2�1+◊+–

|x ≠ y|2–≠n,

and so by Lemma 6.2.15,

t≠–

A

t≠n

¨T (B(x,t))

|·≠◊(I + i·DB)≠2f(›)|2 d›d·

·

B1/2

.M t≠–≠ n

2 ||g||�1+◊+–

A¨B(x,4t)2

d› d÷

|› ≠ ÷|n≠2–

+Œÿ

j=22≠2j(M≠ n

2 ≠(1+◊))ˆ

B(x,4t)

ˆA(x,2j≠1t,2j+2t)

d› d÷

|› ≠ ÷|n≠2–

B1/2

. t≠–≠ n

2 ||g||�1+◊+–

Q

atn+2– +Œÿ

j=22≠2j(M≠ n

2 ≠(1+◊))2≠j(n≠2–)tn+2–

R

b1/2

= ||g||�1+◊+–

for M su�ciently large. Therefore, by the same concluding argument as in theprevious steps, ---

---· ‘æ (I + itDB)≠2f------T p . ||f ||�◊+–

.

199

In the case that X = Z, since p is infinite, Lemma 5.1.34 yields T p Òæ Zp,and so by previous estimate we have

------· ‘æ (I + itDB)≠2f

------Zp .

------· ‘æ (I + itDB)≠2f

------T p . ||f ||�◊+–

.

This completes the proof in the Hölder space case.Step 2d: Completing the proof for Besov spaces.Let p = (p, ◊). We use a slightly di�erent argument here. Fix cuto� functions

‰1, ‰ œ CŒc (Rn) with

supp ‰1 µ B(0, 4), ‰1|B(0,2) © const,supp ‰ µ A(0, 1/2, 4) ‰|A(0,1,2) © const,

for all integers j Ø 2 define ‰j(x) := ‰(2≠jx), and for all j Ø 1 define

÷j(t, x, ›) := ‰j

Ax ≠ ›

t

B

((t, x) œ R1+n+ , › œ Rn);

as before, these functions can be chosen such that qŒj=1 ÷j = 1. Also define

g(t, x, ›) := g(›) ≠ˆ

B(x,t)g(’) d’ ((t, x) œ R1+n

+ , › œ Rn)).

By the triangle inequality we have------t ‘æ (I + itDB)≠2f

------p

Zp

◊

.Œÿ

j=1

¨R1+n

+

Aˆ�(t,x)

|·≠◊(I + i·DB)≠2D(g÷j(t, x, ›))|2 d› d·

Bp/2

dxdt

t, (6.21)

where the operators involving D and B act in the › variable. By using localcoercivity (Lemma 6.2.5) as in the proof of Lemma 6.2.15, the j-th term in (6.21)can be estimated by

¨R1+n

+

Aˆ�(t,x)

|·≠◊(I + i·DB)≠2D(g÷j(t, x, ›))|2 d› d·

Bp/2

dxdt

t

.¨

R1+n

+

Aˆ�

c

(t,x)|·≠◊≠1(I + i·BD)≠1(g÷j(t, x, ›))|2 d› d·

Bp/2

dxdt

t

+¨

R1+n

+

Aˆ�

c

(t,x)|·≠◊≠1(I + i·BD)≠2(g÷j(t, x, ›))|2 d› d·

Bp/2

dxdt

t

with Whitney parameter c = (2, 2). The two terms in this sum di�er only in thepower of the resolvent. The resolvent families (I + i·DB)≠1 and (I + i·DB)≠2

200

both satisfy o�-diagonal estimates of arbitrarily large order M (as o�-diagonalestimates may be composed); we will use this to estimate the terms above, makingreference only to (I + i·DB)≠1.

For j = 1 we estimate

¨R1+n

+

Aˆ�

c

(t,x)|·≠◊≠1(I + i·BD)≠1(g÷1(t, x, ›))|2 d› d·

Bp/2

dxdt

t

.¨

R1+n

+

A

t≠2◊≠2≠n

ˆB(x,4t)

|g(t, x, ›)|2 d›

Bp/2

dxdt

t

.¨

R1+n

+

Q

aˆ

B(x,4t)

ˆB(x,t)

-----g(›) ≠ g(’)

t◊+1

-----

2

d’ d›

R

bp/2

dxdt

t

ƨ

R1+n

+

ˆB(x,4t)

ˆB(x,t)

-----g(›) ≠ g(’)

t◊+1

-----

p

d’ d› dxdt

t

=ˆRn

ˆRn

ˆ Œ

0

ˆB(›,4t)flB(’,t)

dx1

t2n+p(◊+1)dt

t|g(›) ≠ g(’)|p d’ d›

ƈRn

ˆRn

ˆ Œ

|’≠›|/5

1tn+p(◊+1)

dt

t|g(›) ≠ g(’)|p d’ d›

ƒˆRn

ˆRn

|g(›) ≠ g(’)|p|’ ≠ ›|n+p(◊+1) d’ d›

ƒ ||g||Bp,p

◊+1

ƒ ||f ||Bp,p

◊

,

using that ÷1(t, x, ·) is supported in B(x, 4t), that p/2 > 1, that B(›, 4t) fl B(’, t)is nonempty only if t > |’ ≠ ›|/5, and the Besov norm characterisation fromTheorem 5.1.50.

201

For j Ø 2 we have, using o�-diagonal estimates,¨

R1+n

+

Aˆ�

c

(t,x)|·≠◊≠1(I + i·BD)≠1(g÷j(t, x, ›))|2 d› d·

Bp/2

dxdt

t

. 2≠j(Mp≠(np)/2)¨

R1+n

+

A

t≠2◊≠2ˆ

A(x,2j≠1t,2j+2t)

ˆB(x,t)

|g(›) ≠ g(’)|2 d’ d›

Bp/2

dxdt

t

Æ 2≠j(Mp≠(np)/2)¨

R1+n

+

ˆA(x,2j≠1t,2j+2t)

ˆB(x,t)

-----g(›) ≠ g(’)

t◊+1

-----

p

d’ d› dxdt

t

= 2≠j(Mp≠(np)/2+n)·

·ˆRn

ˆRn

ˆ Œ

0

1t2n+p(◊+1)

ˆB(’,t)flA(›,2j≠1t,2j+1t)

dxdt

t|g(›) ≠ g(’)|p d› d’

. 2≠j(Mp≠(np)/2+2n)ˆRn

ˆRn

ˆ Œ

2≠j |’≠›|

1tn+p(◊+1)

dt

t|g(›) ≠ g(’)|p d› d’

ƒ 2≠j(pM≠(np)/2+n≠p(◊+1)) ||f ||Bp,p

◊

arguing similarly to before. For M su�ciently large, we can thus estimate (6.21)by summing a geometric series, yielding

------t ‘æ (I + itDB)≠2f

------p

Zp

◊

. ||f ||Bp,p

◊

as required. This completes the proof.

202

Chapter 7

Elliptic equations,Cauchy–Riemann systems, andboundary value problems

In this section we implicitly work with a fixed m œ N, meaning that we considerLAu = 0 with u a Cm-valued function. All of our arguments are independent ofm. As in the previous section, we fix the Dirac operator D and multipliers B

from Subsection 4.1.2.

7.1 Basic properties of solutions

We will use the following properties of conormal gradients of solutions to LAu = 0(or equivalently, of solutions to (CR)DA; see Theorem 4.1.3 in the introduction).

Proposition 7.1.1. Suppose that u solves LAu = 0. Then the following are true.

(1) The transversal derivative ˆtu solves LA(ˆtu) = 0.

(2) The function t ‘æ ÒAu(t, ·) is in CŒ(R+ : L2loc

(Rn)), and for all Whitneyparameters c = (c0, c1) and t œ R+ we have

ˆB(x,c0t)

|ÒAu(t, x)|2 dx .ˆ

�c

(t,x)|ÒAu(s, y)|2 ds dy.

203

(3) For all exponents p, all k œ N, and all C Ø 1 we have

supt,tÕœR+

C≠1Æt/tÕÆC

------ˆk

t ÒAu(t, ·)------Ep≠k(tÕ)

.C

------ˆk

t ÒAu------Xp≠k

=------ÒAˆk

t u------Xp≠k

. ||ÒAu||Xp .

In particular, if ÒAu is in Xp, then the function t ‘æ ÒAu(t, ·) is in CŒ(R+ :Ep)

Proof. (1) follows from t-independence of the coe�cients. The remaining state-ments are consequences of the classical Caccioppoli inequality, and are proven in[15, §5] for tent spaces. The corresponding Z-space statements are proven in thesame way.

Remark 7.1.2. By Theorem 4.1.3, if F is a solution to the Cauchy–Riemannsystem (4.6), then parts (2) and (3) of Proposition 7.1.1 hold with ÒAu replacedby F .

Furthermore, suppose that G solves the anti-Cauchy–Riemann system

(a CR)DB :Y]

[ˆtG ≠ DBG = 0 in R1+n

+ ,curlÎ GÎ = 0 in R1+n

+(7.1)

defined analogously to (CR)DB but with a sign change. Then the reflectionF (t) := G(≠t) solves (CR)DB on the lower half-space R1+n

≠ . By using X-spacesassociated with the lower half-space rather than the upper half-space, parts (2)and (3) of Proposition 7.1.1 hold with ÒAu replaced by F and with R+ replacedby R≠. A simple reflection argument then shows parts (2) and (3) of Proposition7.1.1 hold for G.

The following technical lemma is analogous to [15, Lemma 10.2].

Lemma 7.1.3. Fix p with i(p) < 2 and ◊(p) < 0, suppose M œ N, and letf œ Xp

DB. Then for all t > 0 we have that (tDB)Me≠t[DB]‰±(DB)f œ Ep, with

supt>0

------(tDB)Me≠t[DB]‰±(DB)f

------Ep(t)

. ||f ||XpDB

.

Proof. We estimate

supt>0

------(tDB)Me≠t[DB]‰±(DB)f

------Ep(t)

.------t ‘æ (tDB)Me≠t[DB]‰±(DB)f

------Xp

ƒ ||f ||XpDB

.

204

The first line comes from Proposition 7.1.1, using that (DB)Me≠t[DB]‰±(DB)fsolves either (CR)DB or (a CR)DB. The second line is due to the fact that [z ‘æzMe≠[z]] œ �(Xp

DB) when i(p) < 2 and ◊(p) < 0.

7.2 Decay of solutions at infinity

In the boundary value problems introduced in Subsection 4.1.1, we have imposedthe decay condition

limtæŒ

ÒÎu(t, ·) = 0 in Z Õ(Rn : Cmn)

for a solution u to LAu with Òu in Xp. In this section we will show that thiscondition is redundant for certain p (quantified in terms of A). In fact, our resultsgive not just decay in Z Õ, but in the slice space EŒ (in the setting of Lemma7.2.1) or in L2 (in Lemma 7.2.4).1

Classical elliptic theory implies that there exists a number ⁄(A) œ (0, n + 1)such that for all ⁄ œ [0, ⁄(A)), for all (t0, x0) œ Rn+1

+ and 0 < r < R < Œ, andfor all weak solutions u to LAu = 0, we have

¨B((t0,x0),r)

|Òu(t, x)|2 dx dt .⁄

3r

R

4ڬ

B((t0,x0),R)|Òu(t, x)|2 dx dt, (7.2)

where B((t0, x0), r) and B((t0, x0), R) denote open balls in R1+n, with B((t0, x0), R)contained in R1+n

+ . These balls can be taken with respect to any norm on R1+n,keeping in mind that the implicit constant in (7.2) will depend on the chosennorm. By ellipticity we may replace the gradient Ò with the conormal gradientÒA in (7.2).

Lemma 7.2.1. Suppose that the exponent p lies in the shaded region pictured inFigure 7.1, which depends on ⁄(A). Let u be a solution to LAu = 0 on R1+n

+ suchthat ÒAu œ Xp. Then limtæŒ ÒAu(t, ·) = 0 in EŒ (and therefore also in Z Õ).

Remark 7.2.2. The shaded region in Figure 7.1 is the open half-plane determinedby the equation j(p) > ◊(p)

n ≠ n+1≠⁄(A)2n . Note that n+1≠⁄(A)

2n Ø 12 when ⁄(A) Æ

1. In Lemma 7.2.4 we will handle exponents p with i(p) Æ 2 and ◊(p) < 0independently of ⁄(A).

1Demanding decay in Z Õ is really just an artefact of having identified the classical smoothnessspaces Xp as subspaces of Z Õ.

205

Figure 7.1: The exponent region in Lemma 7.2.1.

◊

⁄(A)≠(n+1)2

12

n+1≠⁄(A)2n

j

Proof. The region pictured in Figure 7.1 is precisely the set of exponents p suchthat there exists an infinite exponent q with p Òæ q and r(q) < ⁄(A)≠(n+1)

2 . Fixsuch a q. For all ⁄ < ⁄(A) we can estimate

||ÒAu(t, ·)||EŒ0 (1) ƒ sup

xœRn

AˆB(x,1)

|ÒAu(t, y)|2 dy

B1/2

. supxœRn

Q

aˆ t+ 1

2

t≠ 12

ˆB(x,1)

|ÒAu(s, y)|2 dy ds

R

b1/2

(7.3)

. supxœRn

Q

at≠⁄

ˆ t+ t

2

t≠ t

2

ˆB(x,t)

|ÒAu(s, y)|2 dy ds

R

b1/2

(7.4)

. t(n+1)≠⁄

2 +r(q)

Q

aˆ t+ t

2

t≠ t

2

||ÒAu(s, ·)||2Eq(s) ds

R

b1/2

. t(n+1)≠⁄

2 +r(q) ||ÒAu||Xp

where (7.3) follows from Proposition 7.1.1, (7.4) follows from (7.2),2 and thelast line follows from the embeddings Ep(s) Òæ Eq(s) and another application ofProposition 7.1.1. For ⁄ su�ciently close to ⁄(A) we have (n+1≠⁄)/2+r(q) < 0,and so we find that limtæŒ ÒAu(t, ·) = 0 in EŒ.

Remark 7.2.3. It is known that ⁄(A) > n ≠ 1 if and only if A satisfies the DeGiorgi–Nash–Moser condition (7.51) of all exponents less than – = (⁄(A) ≠ (n ≠1))/2. In this case we have ⁄(A)≠(n+1)

2 = – ≠ 1 and n+1≠⁄(A)2n = 1≠–

n , and Lemma7.2.1 then holds for the shaded region pictured in Figure 7.2. Evidently thisregion increases as the De Giorgi–Nash–Moser exponent – increases.

2Here we use the balls B((t, x), r) := (t ≠ r/2, t + r/2) ◊ B(x, r).

206

Figure 7.2: The region in Lemma 7.2.1 in the case that A satisfies the De Giorgi–Nash–Moser condition with exponent –.

◊

– ≠ 1

12

1≠–n

≠1

j

A di�erent argument can be used to deduce decay in L2 for exponents p withi(p) Æ 2 and ◊(p) < 0.

Lemma 7.2.4. Let p = (p, s) with i(p) Æ (0, 2] and ◊(p) < 0, and supposeF œ Xp solves (CR)DB or (a CR)DB. Then limtæ0 F (t) = 0 in L2.

Proof. By Proposition 7.1.1, for all t œ R+ we have

||F (t)||Ep(t) . ||F ||Xp ,

and so

||F (t)||L2 . ||F (t)||E

i(p)0 (t) = t◊(p) ||F (t)||Ep(t) . t◊(p) ||F ||Xp

using the embedding Ei(p)0 (t) Òæ E2

0(t) = L2.3 Since ◊(p) < 0, we have

limtæŒ

F (t) = 0

in L2.

7.3 Classification of solutions toCauchy–Riemann systems

In this section we will prove the following classification theorems for solutions to(CR)DB (as formulated in Subsection 4.1.2 of the introduction).

3The equality E2

0

(t) = L2 is a consequence of Fubini’s theorem.

207

Theorem 7.3.1 (Classification of solutions to (CR)DB, i(p) Æ 2). Let p = (p, s)with p Æ 2 and s < 0, and fix a completion Xp

DB of XpDB.

(i) For all F0 œ Xp,+DB , C+

DBF0 solves (CR)DB, and------C+

DBF0------Xp . ||F0||Xp

DB

.

(ii) Conversely, if F œ Xp solves (CR)DB, then there exists a unique F0 œ Xp,+DB

such that F = C+DBF0. Furthermore, ||F0||Xp

DB

. ||F ||Xp.

When p > 2 the argument is much more complicated, we must restrict atten-tion to exponents p such that the adapted space Xp¸

DB may be identified with theclassical space Xp¸

D , and we need an additional decay condition on F .

Theorem 7.3.2 (Classification of solutions to (CR)DB, i(p) > 2). Let p be anexponent with i(p) > 2 and ◊(p) œ (≠1, 0), and such that p¸ œ I(X, DBú). Inparticular, for such p we have identified Xp,+

DB as a subspace of XpD.

(i) If F0 œ Xp,+DB , then C+

DBF0 solves (CR)DB, limtæŒ C+DBF0(t)Î = 0 in Z Õ(Rn :

Cnm), and------C+

DBF0------Xp . ||F0||Xp

DB

.

(ii) Conversely, if F œ Xp solves (CR)DB and limtæŒ F (t)Î = 0 in Z Õ(Rn :Cnm), then there exists a unique F0 œ Xp,+

DB = XpD such that F = C+

DBF0.Furthermore, ||F0||Xp

DB

. ||F ||Xp.

Note that if p is finite, then p¸ œ I(X, DBú) if and only if p œ I(X, DB)(Proposition 6.2.7). Note also that if p is in the region given by Lemma 7.2.1,then the decay condition on F is redundant. In particular, this holds for all p asin Theorem 7.3.1, so the decay condition need not be included there.

7.3.1 Construction of solutions via Cauchy extension

Here we will prove part (i) of Theorems 7.3.1 and 7.3.2. We will deal with boththeorems simultaneously

Let F0 œ Xp,+DB . Then the estimate

------C+

DBF0------Xp . ||F0||Xp

DB

follows fromeither Theorem 6.1.25 or Theorem 6.2.12.

In Proposition 6.1.24 we showed that C+DBF0 solves (CR)DB strongly in Xp,+

DB .Generally Xp,+

DB need not be contained in L2loc(Rn), and so these two solution

concepts need not coincide. We must argue di�erently here. If F0 œ R(DB), thenProposition 5.2.6 implies that C+

DBF0 solves (CR)DB strongly in CŒ(R+ : L2), andthis implies that C+

DBF0 solves (CR)DB. It remains to deal with F0 œ XpDB \Xp

DB.For such an F0, let (F k

0 )kœN be a sequence in XpDB which converges to F0 as k æ Œ

208

(in the weak-star topology when p is infinite). Then, again using either Theorem6.1.25 or Theorem 6.2.12, we have

limkæŒ

C+DBF k

0 = C+DBF0 in Xp,

and hence also in L2loc(R1+n

+ ). It follows that C+DBF0 solves (CR)DB.

It remains to show that limtæŒ C+DBF0(t)Î = 0 in Z Õ(Rn : Cnm) when i(p) > 2.

This follows from Proposition 6.1.24, since we have limtæŒ C+DBF0(t) = 0 in

XpDB Òæ Z Õ.

7.3.2 Initial limiting arguments

We now begin preparation for the proof of part (ii) of Theorems 7.3.1 and 7.3.2.This section is a rephrasing of the start of [15, §8]. There are no fundamentallynew ideas, but the notation and the flow of ideas are simplified.

For t0 œ R+ we write Rt0 := R \ {t0} and R+,t0 := R+ \ {t0}.

Definition 7.3.3. For t0 œ R+ and Ï œ L2(Rn), we define the test functionGt0,Ï œ CŒ(R+,t0 : D(BúD)) by

Gt0,Ï(t) := sgn(t0 ≠ t)e≠[(t0≠t)BúD]‰sgn(t0≠t)(BúD)PR(BúD)Ï

for all t œ R+,t0 .

Note that ˆtGt0,Ï = BúDGt0,Ï. Also observe that since D annihilates thenullspace N2(BúD) and since L2(Rn) = N2(BúD) ü R(BúD), whenever Ï œD(D),

DGt0,Ï(t) = sgn(t0 ≠ t)e≠[(t0≠t)DBú]‰sgn(t0≠t)(DBú)DÏ. (7.5)

The following lemma is a rewording of [15, Lemma 7.4].

Lemma 7.3.4. Let F solve (CR)DB. Fix Ï œ L2(Rn), t0 œ R+, and let ÷ œLip(R+ : R) and ‰ œ Lip(Rn : R) be compactly supported in R+,t0 and Rn respec-tively. Then we have, with absolutely convergent integrals,

¨R1+n

+

È÷Õ(t)‰(x)BúDGt0,Ï(t, x), F (t, x)Í dx dt

=¨

R1+n

+

È÷(t)Bú[D, m‰] ˆtGt0,Ï(t, x), F (t, x)Í dx dt (7.6)

where m‰ denotes the multiplication operator on L2(Rn) with symbol ‰.

209

As a corollary, under an integrability condition involving F and Ï, we canobtain the following.

Corollary 7.3.5. Let F , Ï, and t0 be as in the statement of Lemma 7.3.4. Sup-pose also that for all compact K µ R+,t0 we have

1K(t)|BúDGt0,Ï(t, x)||F (t, x)| œ L1(R1+n+ ). (7.7)

Then for all ÷ œ Lip(R+ : R) compactly supported in R+,t0, we have the absolutelyconvergent integral¨

R1+n

+

È÷Õ(t)BúDGt0,Ï(t, x), F (t, x)Í dt dx = 0. (7.8)

Proof. Fix ‰ œ Lip(Rn : R) with ‰(x) = 1 for all x œ B(0, 1), and for R > 0define ‰R(x) := ‰(x/R). Then ‰R æ 1 and [D, m‰

R

] æ 0 pointwise as R æ Œ,4

since ||[D, m‰R

]||Œ . R≠1 ||Ò‰||Œ. Condition (7.7) applied with K = supp ÷, thefact that ˆtGt0,Ï = BúDGt0,Ï, and boundedness of ÷ and ÷Õ imply

|÷Õ(t)BúDGt0,Ï(t, x)||F (t, x)| œ L1(R1+n+ ) and

|÷(t)ˆtGt0,Ï(t, x)||F (t, x)| œ L1(R1+n+ ).

This allows us to deduce (7.8) from the equality of Lebesgue integrals (7.6) anddominated convergence.

Now, assuming that (7.7) holds, we can conclude the following.

Corollary 7.3.6. Let F , Ï, and t0 be as in the statement of Lemma 7.3.4. As-sume also that condition (7.7) is satisfied. Then for su�ciently small Á > 0 wehave ˆ t0+2Á

t0+Á

ˆRn

ÈBúDGt0,Ï(t, x), F (t, x)Í dx dt

=ˆ t0+Á≠1

t0+(2Á)≠1

ˆRn

ÈBúDGt0,Ï(t, x), F (t, x)Í dx dt (7.9)

and

≠ˆ 2Á

Á

ˆRn

ÈBúDGt0,Ï(t0 ≠ t, x), F (t0 ≠ t, x)Í dx dt

=ˆ 2Á

Á

ˆRn

ÈBúDGt0,Ï(t, x), F (t, x)Í dx dt. (7.10)

These are all absolutely convergent integrals.4More precisely, [D, m‰R ] is given by multiplication with a function that tends to 0 pointwise.

210

Figure 7.3: The functions ÷1 and ÷2.

t

÷1

(t)

0

1

t0

+ Á t0

+ 2Á t0

+ (2Á)≠1 t0

+ Á≠1

t

÷2

(t)

0

1

Á 2Á t0

≠ 2Á t0

≠ Á

Proof. As in [15, §8, Step 1b] this follows from applying Corollary 7.3.5 with thepiecewise linear functions ÷1, ÷2 œ Lip(R+ : R) drawn in Figure 7.3, where weimpose Á < min(t0/4, 1/4, 1/t0) (we have carried out a change of variables in theleft hand side of (7.10)).

7.3.3 Proof of Theorem 7.3.1Recall that part (i) has already been proven in Subsection 7.3.1; here we provepart (ii).

All of the results in this section are valid for p = (p, s) such that p Æ 2 ands < 0. We do not ‘fix’ such a p, however, because in the final step we will invokeprior results with a di�erent choice of p.

Step 1: Verification and application of initial limiting arguments.

Lemma 7.3.7. Let Ï œ L2(Rn). Then we have 1K◊RnBúDGt0,Ï œ XpÕ for allcompact K µ R+,t0, with

||1K◊RnBúDGt0,Ï||XpÕ . ||Ï||2 dist(K, t0)≠1Ks+n”

p,2≠ (7.11)

where K≠ = inf(K).

Proof. First we note that the estimate

||1K◊RnBúDGt0,Ï||X2≠s≠n”

p,2. ||Ï||2 dist(K, t0)≠1K

s+n”p,2

≠

can be shown by writing

||1K◊RnBúDGt0,Ï||X2≠s≠n”

p,2=

Aˆ K+

K≠

------ts+n”

p,2BúDGt0,Ï

------2

2

dt

t

B1/2

. ||Ï||2

Aˆ K+

K≠

t2(s+n”p,2) dist(K, t0)≠2 dt

t

B1/2

(7.12)

. ||Ï||2 dist(K, t0)≠1Ks+n”

p,2≠ . (7.13)

211

The estimate (7.12) follows by writing

BúDGt0,Ï = sgn(t0 ≠ t)t0 ≠ t

(t0 ≠ t)BúDe≠[(t0≠t)BúD]‰sgn(t0≠t)(BúD)PR(BúD)Ï

and noting that the operator

(t0 ≠ t)BúDe≠[(t0≠t)BúD]‰sgn(t0≠t)(BúD)PR(BúD)

is bounded on L2(Rn) uniformly in t œ R+,t0 , and that |(t0 ≠ t)≠1| . dist(K, t0)≠1

for t œ K. Then (7.13) follows because s + n”p,2 is negative whenever s < 0 andp < 2.

Now use the X-space embeddings to write

X2≠s≠n”

p,2 Òæ XpÕ,

from which follows (7.11).

Corollary 7.3.8. Let Ï œ L2(Rn), and suppose that F œ Xp solves (CR)DB.Then

limÁæ0

ˆ t0+2Á

t0+Á

ˆRn

ÈBúDGt0,Ï(t, x), F (t, x)Í dx dt = 0. (7.14)

Proof. For Á > 0 small the previous lemma yields------1[t0+(2Á)≠1,t0+Á≠1]◊RnBúDGt0,Ï

------XpÕ . ||Ï||2 (2Á)(t0 + (2Á)≠1)s+n”

p,2 ,

which decays as Á æ 0 since s+n”p,2 is negative when s < 0 and p Æ 2. Thereforein particular, by X-space duality, condition (7.7) is satisfied, and by boundednessof the above quasinorms as Á æ 0 we can take the Á æ 0 limit in (7.9) to obtain(7.14).

Step 2: Semigroup property of F .

Lemma 7.3.9. Suppose F œ Xp solves (CRDB). Then F œ CŒ(R+ : H2DB),

F (t) œ D(DB) for all t > 0, and ˆtF + DBF = 0 holds strongly in CŒ(R+ :H2

DB).

Proof. We already have that F œ CŒ(R+ : L2loc(Rn)) from Proposition 7.1.1,

and furthermore that ˆtF œ Xp≠1. Hence we have F (t0), (ˆtF )(t0) œ Ep for allt0 œ R+, and therefore by the slice space containments of Proposition 5.1.42 weobtain F (t0), (ˆtF )(t0) œ L2 for all t0 œ R+. Therefore F (t0) œ D(DB) for allt0 œ R+, and ˆtF + DBF = 0 holds in L2. We can iterate this argument by

212

reapplying ˆt, as this preserves the property of solving (CR)DB as well as thepreviously stated L2 containments, so we obtain F œ CŒ(R+ : L2).

Now since limt0æŒ F (t0) = 0 in L2 (Lemma 7.2.4), we can write

F (t0) = ≠ˆ Œ

t0

(ˆtF )(·) d· = ≠ˆ Œ

t0

DB(F (·)) d· œ R(DB)

by the fundamental theorem of calculus. Therefore F (t0) œ H2DB for all t0, and

since the H2DB-norm is equivalent to the L2-norm when restricted to R(DB), this

completes the proof.

Lemma 7.3.10. Suppose that F œ Xp solves (CR)DB. Then for all t0 > 0 and· Ø 0 we have F (t0) œ H2,+

DB = R(DB)+ and

F (t0 + ·) = e≠·DB(F (t0)). (7.15)

Proof. For all Ï œ L2(Rn), the function t ‘æ BúDGt0,Ï(t) is smooth in t œ R+,t0

with values in H2BúD and with

limt¿t0

BúDGt0,Ï(t) = ≠BúD‰≠(BúD)PR(BúD)Ï

in H2BúD. Furthermore, by Lemma 7.3.9, t ‘æ F (t) is smooth in t œ R+ with

values in H2DB. Therefore we may write for all Ï œ L2(Rn), using (7.14) from

Corollary 7.3.8,

0 = limÁæ0

ˆ t0+2Á

t0+Á

ˆRn

ÈBúDGt0,Ï(t, x), F (t, x)Í dx dt

= limÁæ0

ˆ t0+2Á

t0+Á

ÈBúDGt0,Ï(t), F (t)ÍH2B

úD

dt

= ≠ÈBúD‰≠(BúD)PR(BúD)Ï, F (t0)ÍH2B

úD

. (7.16)

Hence for all „ œ R(BúD) and all ” > 0, since e≠”[BúD] maps H2,≠BúD into itself,

applying (7.16) to Ï = e≠”[BúD]„ yields

ÈBúDe”BúD‰≠(BúD)„, F (t0)ÍH2B

úD

= 0. (7.17)

The subspace

{BúDe”BúD‰≠(BúD)„ : „ œ R(BúD)} µ L2(Rn)

is dense in H2,≠BúD (see [15, p. 28]), so (7.17) and the decomposition H2

DB =H2,+

DB ü H2,≠DB imply that F (t0) œ H2,+

DB.

213

Now we will derive the semigroup equation (7.15). For all ” Ø 0 and Ï œ H2BúD,

define

Ï” := e≠”[BúD]Ï

and

IÁ,”t0,Ï :=

ˆ 2Á

Á

ÈBúDGt0,Ï”

(t), F (t)ÍH2B

úD

dt.

Then by (7.10), using the same argument as before to write everything in termsof H2

BúD-duality, we have

limÁæ0

IÁ,”t0,Ï = ≠ lim

Áæ0

ˆ 2Á

Á

ÈBúDe≠tBúDe≠”BúD‰+(BúD)Ï, F (t0 ≠ t)ÍH2B

úD

dt

= ≠ÈBúDe≠”BúD‰+(BúD)Ï, F (t0)ÍH2B

úD

.

Therefore for all · Ø 0, ” Ø 0, and Ï œ H2BúD, using IÁ,”+·

t0,Ï = IÁ,·t0+”,Ï, we have

ÈBúDe≠”BúD‰+(BúD)Ï, e≠·DB(F (t0))ÍH2B

úD

= ÈBúDe≠(”+·)BúD‰+(BúD)Ï, F (t0)ÍH2B

úD

= ≠ limÁæ0

IÁ,”+·t0,Ï

= ≠ limÁæ0

IÁ,”t0+·,Ï

= ÈBúDe≠”BúD‰+(BúD)Ï, F (t0 + ·)ÍH2B

úD

.

As before, the subspace {BúDe≠”BúD‰+(BúD)Ï : Ï œ H2BúD} is dense in H2,+

BúD,so by duality we have F (t0 + ·) = e≠·DBF (t0) in H2,+

DB for all t0 > 0 and all· Ø 0.

Step 3: Completing the proof.

Proposition 7.3.11 (Existence of boundary trace). Suppose that F œ Xp solves(CR)DB, and let Xp

DB be a completion of XpDB. Then there exists a unique F0 œ

Xp,+DB such that F = C+

DBF0. Furthermore, ||F0||XpDB

. ||F ||Xp.

Proof. Fix an exponent p with i(p) œ (1, 2] and ◊(p) < 0 such that p Òæ p (whenp > 1 we may take p = p). By Lemma 7.3.10 we have F (t0) œ H2,+

DB fl D(DB) for

214

all t0 > 0. We can then estimate

||F (t0)||XpDB

ƒ ||DBF (t0)||Xp≠1DB

(7.18)

=------· ‘æ e≠·DB(DBF )(t0)

------Xp≠1 (7.19)

= ||· ‘æ DBF (t0 + ·)||Xp≠1 (7.20)= ||St0DBF ||Xp≠1

. ||DBF ||Xp≠1 (7.21)= ||ˆtF ||Xp≠1

. ||F ||Xp (7.22)

. ||F ||Xp . (7.23)

The first line (7.18) is from Corollary 6.1.14. Line (7.19) comes from Theorem6.1.25. Line (7.20) comes from Lemma 7.3.10, (7.21) comes from Proposition5.1.36 because i(p ≠ 1) Æ 2 and s(p ≠ 1) Æ ≠1/2, (7.22) comes from Proposition7.1.1, and finally (7.23) follows from X-space embeddings by p Òæ p. ThereforeF (t0) œ Xp,+

DB uniformly in t0 > 0.Since Xp,+

DB is the dual of XpÕ,+BúD for any completion XpÕ,+

BúD of XpÕ,+BúD, there exists

a sequence tk ¿ 0 and an F0 œ Xp,+DB such that F (tk) converges weakly to F0 in

Xp,+DB as k æ Œ. We thus have for all Ï œ XpÕ,+

BúD and for all · > 0,

ÈÏ, e≠·DBF0ÍXpÕB

úD

= Èe≠·BúDÏ, F0ÍXpÕB

úD

= limkæŒ

Èe≠·BúDÏ, F (tk)ÍXpÕB

úD

= limkæŒ

ÈÏ, e≠·DBF (tk)ÍXpÕB

úD

= limkæŒ

ÈÏ, F (tk + ·)ÍXpÕB

úD

(7.24)

= ÈÏ, F (·)ÍXpÕB

úD

(our notation for duality pairings is explained in Section 4.3), using Lemma 7.3.10in (7.24). Therefore by density we have C+

DBF0 = F .It only remains to show that F0 is in Xp,+

DB , with the right quasinorm estimate,and uniquely determined. Recall that C+

DB = Qsgp,DB when restricted to thepositive spectral subspace. Let Ï œ �Œ

Œ be a Calderón sibling of sgp. ThenF0 = SÏ,DBQsgp,DBF0 = SÏ,DBF , and so by Proposition 6.1.17 we have F0 œ Xp

DB

with ||F0||XpDB

. ||F ||Xp . In fact, since F (t0) œ H2,+DB for all t0 > 0, we find that

F is in the positive subspace Xp,+DB . Uniqueness follows by injectivity of Qsgp,DB

(Proposition 6.1.17).

This completes the proof of Theorem 7.3.1.

215

7.3.4 Proof of Theorem 7.3.2

Recall that part (i) has already been proven in Subsection 7.3.1; here we provepart (ii).

Our proof roughly follows that of [15, Theorem 1.3], arguing via a series ofrather technical lemmas. In this section we will continually assume that p satisfiesthe assumptions of Theorem 7.3.2. Most of the lemmas work without assumingp¸ œ I(X, DBú), but we gain nothing from dropping this assumption.

Step 1: Establishing a good class of test functions.

We define the following class of test functions for XpDB:

Dp(X) :=ÓÏ œ D(D) : DÏ œ Xp¸

DBú , ‰±(DBú)DÏ œ Ep¸Ô.

This is large enough to contain the Schwartz functions and to be stable underthe action of various operators, yet it is restrictive enough to let us exploit slicespace containments.

Lemma 7.3.12. The Schwartz class S(Rn : Cm(1+n)) is contained in Dp(X).

Proof. Suppose Ï œ S. Then Ï œ D(D) and DÏ œ Xp¸

D = Xp¸

DBú by the assump-tion on p. It remains to show that ‰±(DBú)DÏ œ Ep¸ , and this takes somework. This is a modification of the argument of [15, Lemma 8.10].

Since DÏ œ S µ Ep¸ (Proposition 5.1.43) and since

DÏ = ‰+(DBú)DÏ + ‰≠(DBú)DÏ

it su�ces to show that ‰+(DBú)DÏ is in Ep¸ .Define  œ �Œ

N , with N large to be chosen later, by

Â(z) := [z]Ne≠[z]

N ! .

Then for all t œ R \ {0} we haveˆ Œ

0Â(st) ds

s= 1

N !

ˆ Œ

0sNe≠s ds

s= 1,

so by holomorphy we have ˆ Œ

0Â(sz) ds

s= 1

216

for all z œ Sµ. By the same argument, along with integration by parts andinduction on N , for all z œ Sµ we have

ˆ Œ

1Â(sz) ds

s= P ([z])e≠[z]

where P is a real polynomial of degree N ≠ 1. Therefore by functional calculuson R(DBú) we may write

‰+(DBú)DÏ =ˆ 1

0(‰+)(sDBú)DÏ

ds

s+ P (DBú)e≠DBú

‰+(DBú)DÏ.

By Lemma 7.1.3 (using i(p¸) < 2 and ◊(p¸) < 0) we have

P (DBú)e≠DBú‰+(DBú)DÏ œ Ep¸

,

so it su�ces to show thatˆ 1

0(‰+)(sDBú)DÏ

ds

sœ Ep¸

.

For f œ L2(Rn) write

G(f) :=ˆ 1

0(‰+)(sDBú)f ds

s;

since ‰+ œ �Œ+ this is defined for all f œ L2(Rn) (not just f œ R(DBú)).

Note that the family ((‰+)(sDBú))s>0 satisfies o�-diagonal estimates of orderN (Theorem 5.2.8). For Q, R œ D1 (recall that D1 is the set of standard dyadiccubes in Rn with sidelength 1) with d(Q, R) Ø 1 we can estimate

||G(1RDÏ)||L2(Q) =Q

aˆ

Q

-----

ˆ 1

0(‰+)(sDBú)1RDÏ(x) ds

s

-----

2

dx

R

b1/2

ƈ 1

0

------(‰+)(sDBú)1RDÏ

------L2(Q)

ds

s

.ˆ 1

0

Ad(Q, R)

s

B≠Nds

s||1RDÏ||2

ƒ d(Q, R)≠N ||1RDÏ||2 .

For all other Q, R œ D1 we have instead

||G(1RDÏ)||L2(Q) . ||1RDÏ||2 .

217

Therefore by the discrete characterisation of slice spaces (Proposition 5.1.45),writing R ≥ Q to mean that dist(R, Q) = 0 and noting that dist(R, Q) Ø 1 ifR ”≥ Q,

||G(DÏ)||Ep¸ ƒQ

aÿ

QœD1

||G(DÏ)||i(p¸)

L2(Q)

R

b1/i(p¸)

.Q

aÿ

QœD1

Q

aÿ

R≥Q

+ÿ

R ”≥Q

R

b ||G(1RDÏ)||i(p¸)

L2(Q)

R

b1/i(p¸)

.

Q

ccaÿ

QœD1R≥Q

||DÏ||i(p¸)

L2(R)

R

ddb

1/i(p¸)

+

Q

ccaÿ

QœD1R ”≥Q

d(Q, R)Ni(p¸) ||DÏ||i(p¸)

L2(R)

R

ddb

1/i(p¸)

=: I1 + I2.

Since the number of cubes R œ D1 such that R ≥ Q is uniform in Q, we have

I1 ƒQ

aÿ

RœD1

||DÏ||i(p¸)

L2(R)

R

b1/i(p¸)

ƒ ||DÏ||Ep¸ .

To handle I2 write

I2 =Q

aÿ

RœD1

||DÏ||i(p¸)

L2(R)

Œÿ

k=1kNi(p¸)|{Q œ D1 : d(Q, R) = k}|

R

b1/i(p¸)

(7.25)

ƒN,p¸,n ||DÏ||Ep¸

using that the innermost sum in (7.25) is independent of R and convergent forN su�ciently large. Therefore

||G(DÏ)||Ep¸ . ||DÏ||Ep¸ < Œ

which shows that ‰+(DBú)DÏ œ Ep¸ and completes the proof.

Lemma 7.3.13. We have the following stability properties of Dp(X):

(i) for all ” > 0 we have e≠”[BúD]Dp(X) µ Dp(X),

(ii) ‰±(BúD)Dp(X) µ Dp(X),

218

Proof. (i) The function [z ‘æ e≠”[z]] is in HŒ and has a polynomial limit at 0,so e≠”[BúD]Ï may be defined for all Ï œ D(D) (not just those in R(BúD)).5

For all such Ï we can write using the similarity of functional calculi

D(e≠”[BúD]Ï) = e≠”[DBú]DÏ. (7.26)

Since DÏ is in Xp¸

DBú , so is D(e≠”[BúD]Ï). To see the slice space containmentsof spectral projections, write

‰±(DBú)D(e≠”[BúD]Ï) = e≠”[DBú]‰±(DBú)DÏ.

By assumption ‰±(DBú)DÏ is in Ep¸ fl Xp¸,±DBú µ Ep¸ fl X2,±

DBú , and byCorollary 6.1.28, e≠”[DBú]‰±(DBú)DÏ is in Ep¸ .

(ii) Similarly, we have ‰±(BúD)D2(D) µ D2(BúD), and by similarity of func-tional calculi

D‰±(BúD)Ï0 = ‰±(DBú)DÏ0 œ Xp¸

DBú

and

‰±(DBú)D‰±(BúD)Ï = ‰±(DBú)DÏ0 œ Ep¸,

‰û(DBú)D‰±(BúD)Ï = 0 œ Ep¸.

Lemma 7.3.14. Suppose that Ï œ XpÕ

BúD flD(BúD). Then ‰±(DBú)e≠t[DBú]/2DÏ

is defined and in EpÕ for all t > 0. Furthermore, e≠[BúD]/2Ï œ Dp(X).

Proof. Note that D(BúD) = D(D). Since DÏ œ Xp¸

DBú (Proposition 6.2.6), byLemma 7.1.3 we find that

‰±(DBú)e≠t[DBú]/2DÏ = e≠t[DBú]/2‰±(DBú)DÏ œ Ep¸ = EpÕ. (7.27)

To see that e≠[BúD]/2Ï is in Dp(X), note that

e≠[BúD]/2Ï œ D(BúD) = D(D),

thatDe≠[BúD]/2Ï = e≠[DBú]/2DÏ œ Xp¸

DBú ,

and that‰±(DBú)De≠[BúD]/2Ï = e≠t[DBú]/2‰±(DBú)DÏ œ Ep¸

by (7.27).5Although we did not discuss this in Subsection 5.2.1, this is a standard procedure. The

representation (7.26) is all we need.

219

Step 2: Verification and application of the initial limiting arguments.

Lemma 7.3.15. Define the operator

Gt0 : Ï ‘æ sgn(t0 ≠ t)e≠[(t0≠t)DBú]‰sgn(t0≠t)(DBú)Ï.

Let K µ R+,t0 be compact. Then for all k œ N, 1K◊RnGt0 is bounded from Xp¸

DBú

to Xp¸+k, and this boundedness is uniform in K provided K≠ > t0 + 1.

Proof. We will prove the result for tent spaces; the Z-space result then followsby real interpolation because the assumption on p is open in (j(p), ◊(p)).

Suppose Ï œ Xp¸

DBú and write K = K0 fi KŒ, where K0 µ (0, t0) and KŒ µ(t0, Œ). For all x œ Rn,

A2(Ÿ◊(p)+1≠k1K◊RnGt0(Ï))(x)2

=Aˆ

K0

ˆB(x,t)

+ˆ

KŒ

ˆB(x,t)

B

|t◊(p)+1≠kGt0(Ï)(t, y)|2 dy dt

t1+n

=: I0 + IŒ.

There exists – > 0 (depending on K0) such that if (t0 ≠ ·, y) œ (K0 ◊Rn) fl �(x),then (·, y) œ �–(x) (see Figure 7.4). Thus, using (7.5) and that t0 ≠· ƒK · whent0 ≠ · œ K0,

I0 ƨ

�–(x)1K0(t0 ≠ ·)

---(t0 ≠ ·)◊(p)+1≠kGt0(Ï)(t0 ≠ ·, y)---2 dy d·

(t0 ≠ ·)1+n

.K,k

¨�–(x)

---· ◊(p)+1e≠·DBú‰+(DBú)Ï(y)

---2 dy d·

· 1+n

Similarly, there exists — > 0 such that if (t0 + ‡, y) œ (K1 ◊ Rn) fl �(x), then(‡, y) œ �—(x), and using 2(◊(p) + 1) ≠ n ≠ 1 < 0 we have

IŒ ƨ

�—(x)1KŒ(t0 + ‡)

---(t0 + ‡)◊(p)+1≠ke‡DBú‰≠(DBú)Ï(y)

---2 dy d‡

(t0 + ‡)1+n

Æ (KŒ)≠2k≠

¨�—(x)

(t0 + ‡)2(◊(p)+1)≠n≠1---e‡DBú

‰≠(DBú)Ï(y)---2

dy d‡

Æ (KŒ)≠2k≠

¨�—(x)

‡2(◊(p)+1)≠n≠1---e‡DBú

‰≠(DBú)Ï(y)---2

dy d‡

= (KŒ)≠2k≠

¨�—(x)

---‡◊(p)+1e‡DBú‰≠(DBú)Ï(y)

---2 dy d‡

‡1+n.

220

Figure 7.4: Cones of large aperture, used in Lemma 7.3.15.

Rn

R+

x

t0

K0

KŒ

Therefore we can estimate------1K◊RnGt0(Ï)

------T p¸+k

. C(K, k)------e≠tDBú

‰+(DBú)Ï------T p¸ + (KŒ)≠2k

≠

------etDBú

‰≠(DBú)Ï------T p¸

. ||Ï||Hp¸

DB

ú

< Œ,

using the semigroup characterisation of the Hp¸,±DBú quasinorm (Theorem 6.1.25),

which is valid since i(p¸) < 2 and ◊(p¸) < 0. Note that if K≠ > t0 + 1 thenI0 = 0, and that the aperture — can remain fixed in this argument, which impliesthe claimed uniformity in K since K≠2k

≠ is bounded in K≠ > t0 + 1.

Corollary 7.3.16. Let Ï œ Dp(X) and k œ N. Then 1K◊RnDGt0,Ï œ Xp¸+k forall compact K µ R+,t0, with uniform boundedness in K provided K≠ > t0 + 1.

Proof. For Ï œ Dp(X) we have DÏ œ Xp¸

DBú and DGt0,Ï = Gt0(DÏ), so thisfollows from Lemma 7.3.15.

For k œ N, whenever F œ Xp≠k solves (CR)DB we can invoke Corollary 7.3.6when Ï œ Dp(X), yielding the equalities (7.9) and (7.10) for su�ciently smallÁ > 0.

Corollary 7.3.17. Let Ï œ Dp(X) and k œ N, and suppose that F œ Xp≠k solves(CR)DB. Then

limÁæ0

ˆ t0+2Á

t0+Á

ˆRn

ÈBúDGt0,Ï(t, x), F (t, x)Í dx dt = 0. (7.28)

221

Proof. For Á < 1/2 we have t0 + (2Á)≠1 > t0 + 1, and so by Corollary 7.3.16 wehave

1[t0+(2Á)≠1,t0+Á≠1]◊RnDGt0,Ï œ Xp¸+k+1

with uniformly bounded quasinorms. Since F œ Xp≠k, and since (p ≠ k)Õ =p¸ + k + 1, absolute convergence of the X-space duality integrals implies thatcondition (7.7) is satisfied, and also that

ˆ t0+Á≠1

t0+(2Á)≠1

ˆRn

|ÈBúDGt0,Ï(t, x), F (t, x)Í| dx dt . 1

for all Á < 1/2. Therefore we can take the limit as Á æ 0 of both sides of (7.9)using dominated convergence to conclude that the right hand side vanishes.

Step 3: Weak semigroup properties of solutions.

Lemma 7.3.18. Suppose that F œ Xp≠k solves (CR)DB for some k œ N. Whent0 > 0, · Ø 0, and Ï œ Dp(X), we have

ÈBúDÏ, F (t0 + ·)ÍEp¸ = ÈBúe≠·DBú‰+(DBú)DÏ, F (t0)ÍEp¸ . (7.29)

Proof. We need to rewrite the integrals in (7.28) and (7.10) in terms of dualityof slice spaces. By Proposition 7.1.1, F (t) is in Ep for each t œ R+. By Lemma7.1.3, since DÏ œ Xp¸

DBú , we have that BúDGt0,Ï(t) is in Ep¸ . HenceˆRn

ÈBúDGt0,Ï(t, x), F (t, x)Í dx = ÈBúDGt0,Ï(t), F (t)ÍEp¸

by the slice space duality identification of Proposition 5.1.41. Therefore (7.28)and (7.10) can be rewritten as

limÁæ0

ˆ t0+2Á

t0+Á

ÈBúDGt0,Ï(t), F (t)ÍEp¸ dt = 0 (7.30)

and

≠ limÁæ0

ˆ 2Á

Á

ÈBúDGt0,Ï(t0 ≠ t), F (t0 ≠ t)ÍEp¸ dt = limÁæ0

ˆ 2Á

Á

ÈBúDGt0,Ï(t), F (t)ÍEp¸ dt.

(7.31)We need to evaluate these limits by using continuity of the integrands. By

Proposition 7.1.1, we have F œ CŒ(R+ : Ep). By the definition of Dp(X), wehave that BúDGt0,Ï(t) œ Ep¸ for all t œ R+,t0 , with

limt√t0

BúDGt0,Ï(t) = ≠BúD‰≠(BúD)PR(BúD)Ï

= ≠Bú‰≠(DBú)DÏ

222

in Ep¸ by Corollary 6.1.28. Therefore (7.30) becomes

ÈBú‰≠(DBú)DÏ, F (t0)ÍEp¸ = 0. (7.32)

Next, we will prove

ÈBúe≠·DBú‰+(DBú)DÏ, F (t0)ÍEp¸ = ÈBú‰+(DBú)DÏ, F (t0 + ·)ÍEp¸ . (7.33)

by taking the limit of the left hand side of (7.31) and exploiting an algebraicproperty of the right hand side. Summing (7.32) (at t0 + ·) and (7.33) will yield(7.29) and complete the proof.

For Ï œ Dp(X) and ” Ø 0, define

IÁ,”t0,Ï :=

ˆ 2Á

Á

ÈBúDGt0,Ï”

(t), F (t)ÍEp¸ dt

where Ï” := e≠”[BúD]Ï. By Lemma 7.3.13, Ï” is in Dp(X), and so we can apply(7.31) to get

limÁæ0

IÁ,”t0,Ï = ≠ lim

Áæ0

ˆ 2Á

Á

ÈBúDGt0,Ï”

(t0 ≠ t), F (t0 ≠ t)ÍEp¸ dt

= ≠ limÁæ0

ˆ 2Á

Á

ÈBúe≠tDBúe≠”DBú

‰+(DBú)DÏ(t), F (t)ÍEp¸ dt

= ≠ÈBúe≠”DBú‰+(DBú)DÏ(t0), F (t0)ÍEp¸

using the same argument as in the previous paragraph to establish the finalequality. A simple computation shows that we have

IÁ,”t0,Ï = IÁ,0

t0+”,Ï,

and so we can conclude

ÈBúe≠·DBú‰+(DBú)DÏ, F (t0)ÍEp¸ = ≠ lim

Áæ0IÁ,·

t0,Ï

= ≠ limÁæ0

IÁ,0t0+·,Ï

= ÈBú‰+(DBú)DÏ, F (t0 + ·)ÍEp¸ ,

completing the proof.

We can use this lemma, using that Ep µ S Õ, to see what happens when wetest against Schwartz functions.

223

Corollary 7.3.19. Let F , t0, and · be as in Lemma 7.3.18, and suppose Ï œ S.Then

≠ ÈÏ, (ˆtF )(t0 + ·)ÍS = ÈBúe≠·DBú‰+(DBú)DÏ, F (t0)ÍEp¸ . (7.34)

Proof. By Lemma 7.3.12, S µ Dp(X), so we can apply Lemma 7.3.18 to Ï. SinceF (t0 + ·) and (ˆtF )(t0 + ·) are in Ep, and Ï and BúDÏ are in Ep¸ , we can applyintegration by parts in slice spaces (Proposition 5.1.44) to derive (7.34).

Step 4: A reproducing formula for (ˆtF )(t0) in terms of higher deriva-tives.

Lemma 7.3.20. Let t0 > 0, k œ N+, and suppose that F œ Xp solves (CR)DB.Then (ˆk

t F )(t0) œ XpD, with

------(ˆk

t F )(t0)------Xp

D

. t≠k0 ||F ||Xp . (7.35)

Proof. Suppose Ï œ S. First note that since ˆk≠1t F solves (CR)DB, and is in

Xp≠(k≠1) by Proposition 7.1.1, Corollary 7.3.19 yields

≠ ÈÏ, (ˆkt F )(t0/2 + ·)ÍS = ÈBúe≠·DBú

‰+(DBú)DÏ, (ˆk≠1t F )(t0/2)ÍEp¸ . (7.36)

Applying this with · = t0/2 and using the slice space estimates of Lemma 7.1.3and Proposition 7.1.1, slice space duality, and p¸ œ I(X, DBú), we have

-----

ˆRn

ÈÏ(x), ˆkt F (t0)(x)Í dx

-----

Æ------Búe≠t0DBú/2‰+(DBú)DÏ

------Ep¸ (t0/2)

------(ˆk≠1

t F )(t0/2)------Ep+1(t0/2)

. ||DÏ||Xp¸

DB

út≠k0

------(ˆk≠1

t F )(t0/2)------Ep≠(k≠1)(t0)

. ||DÏ||Xp¸D

t≠k0

------ˆk≠1

t F------Xp≠(k≠1)

. ||Ï||XpÕ t≠k0 ||F ||Xp .

Since Ï was arbitrary, this implies that (ˆkt F )(t0) œ (XpÕ)Õ = Xp with the norm

estimate (7.35). Furthermore, since ˆkt F solves (CR)DB, each (ˆk

t F )(t0) is inR(DB) = R(D), which implies membership in Xp

D.

We recall the following elementary lemma (see [15, Lemma 9.2]).

Lemma 7.3.21. Suppose k œ N and g œ Ck(R+ : C), with tjg(j)(t) æ 0 ast æ Œ for all integers 0 Æ j Æ k ≠ 1. Then for all t > 0 we have

g(t) = (≠1)k

(k ≠ 1)!

ˆ Œ

t

g(k)(·)(· ≠ t)k≠1 d·.

224

Corollary 7.3.22. Suppose that F œ Xp solves (CR)DB. Then for all t0 > 0and Ï œ S we have

ÈÏ, (ˆtF )(t0)ÍS = (≠1)k

(k ≠ 1)!

ˆ Œ

t0

ÈÏ, (ˆk+1t F )(t)ÍEpÕ (t ≠ t0)k≠1 dt.

Proof. By Lemma 7.3.20 we have that the function t0 ‘æ (ˆtF )(t0) is in CŒ(R+ :Xp

D). Therefore for all Ï œ S the function gÏ defined by

gÏ(t0) := ÈÏ, (ˆtF )(t0)ÍS

is in CŒ(R+ : C), and for k œ N+ we have

g(k)Ï (t0) = ÈÏ, (ˆk+1

t F )(t0)ÍS = ÈÏ, (ˆk+1t F )(t0)ÍEpÕ .

Furthermore, by the same lemma, we have

|tk0gÏ(t0)| = tk

0

----ÈÏ, (ˆtF )(t0)ÍXp

ÕD

----

.Ï,F t≠10 ,

so the hypotheses of Lemma 7.3.21 are satisfied, and the result follows.

Step 5: Construction of associated ‘nice’ solutions.In this step of the proof, given a solution F œ Xp of (CR)DB, we will construct

distributions modulo polynomials F (t0) œ XpD which satisfy the properties we

want to show for F (t0). In the remaining steps we will show that F (t0) = F (t0),which will complete the proof.

Lemma 7.3.23. Suppose F œ Xp solves (CR)DB. Then for all t0 œ [0, Œ) andfor su�ciently large N œ N we have

------(t, y) ‘æ tN(ˆN

t F )(t0 + t, y)------Xp . ||F ||Xp .

Proof. This is an immediate corollary of Propositions 5.1.36 and 7.1.1.

Let F œ Xp solve (CR)DB. For N œ N large enough that Lemma 7.3.23applies, define ’ œ �Œ

1 by’(z) := cNze≠[z]/2

where cN = (≠1)N+1/N !. For k œ N define ‰k := 1[k≠1,k]◊B(0,k), and for all t0 Ø 0define

Fk(t0) := S’,DB

5t ‘æ ‰ktN(ˆN

t F )3

t0 + t

2

46. (7.37)

225

By Lemma 7.3.23 we have [t ‘æ ‰ktNˆNt F

1t0 + t

2

2] œ Xp fl X2, and so Fk(t0) is a

well-defined element of XpDB. Furthermore, since ’ œ �0

+, Proposition 6.1.6 andLemma 7.3.23 tell us that

------Fk(t0)

------Xp

DB

.----

----t ‘æ ‰ktN(ˆNt F )

3t0 + t

2

4----

----Xp

. ||F ||Xp .

Since the functions [t ‘æ ‰ktN(ˆNt F )

1t0 + t

2

2] converge to [t ‘æ tN(ˆN

t F )1t0 + t

2

2]

in Xp as k æ Œ, given a completion6 XpDB of Xp

DB, we get an element F (t0) œXp

DB defined by

Ft0 := S’,DB

5t ‘æ tN(ˆN

t F )3

t0 + t

2

46

and satisfying------F (t0)

------Xp

DB

. ||F ||Xp . (7.38)

Since p¸ œ I(X, DBú), we can identify XpD as a completion of Xp

DB, and so inthis case each F (t0) œ Xp

D is a distribution modulo polynomials.

Lemma 7.3.24. Let t0 Ø 0. Suppose F œ Xp solves (CR)DB, let XpDB be

a completion of XpDB, and define F (t0) œ Xp

DB as in the previous paragraphs.Suppose also that „ œ XpÕ

BúD fl D(BúD). Then we have

È„, F (t0)ÍXpÕB

úD

= ≠cN

ˆ Œ

0

=tBúe≠ t

2 [DBú]D„, tN(ˆNt F )

3t0 + t

2

4>

EpÕ

dt

t(7.39)

Proof. First we show the the EpÕ duality pairing (7.39) makes sense. Since „ œXpÕ

BúD fl D(BúD), Lemma 7.3.14 yields e≠t[DBú]/2D„ œ EpÕ . Since each tBú is abounded operator on EpÕ (not uniformly in t of course) we have tBúe≠t[DBú]/2D„ œEpÕ . On the other hand, since t ‘æ (ˆN

t F )(t0+t/2) solves (CR)DB, by Proposition7.1.1 and Lemma 7.3.23 we have

------(ˆN

t F )(t0 + t/2)------Ep≠N (t)

.------t ‘æ (ˆN

t F )(t0 + t/2)------Xp≠N

. ||F ||Xp

for all t > 0. Therefore the slice space dual pairing in (7.39) is meaningful.

6Recall that when p is infinite we always use weak-star completions instead of ordinarycompletions.

226

Now write

È„, F (t0)ÍXpÕB

úD

= limkæŒ

È„, Fk(t0)ÍXpÕB

úD

= limkæŒ

ÈQ’,BúD„, [t ‘æ ‰ktN(ˆNt F )(t0 + t/2)]ÍXp

= ÈQÂ’,BúD„, [t ‘æ tN(ˆN

t F )(t0 + t/2)]ÍXp

= ≠cN

¨R1+n

+

1t(BúDe≠t[BúD]/2„)(x), tN(ˆN

t F )(t0 + t/2, x)2

dxdt

t

= ≠cN

ˆ Œ

0ÈtBúe≠t[DBú]/2D„, tN(ˆN

t F )(t0 + t/2)ÍEpÕdt

t

using Â’ = ’ and the slice space containments from the previous paragraph.

Now we will show that the distributions (modulo polynomials) (F (t0))t0Ø0 arein fact given by the Cauchy operator applied to F (0).

Proposition 7.3.25. Let F œ Xp solve (CR)DB, fix a completion XpDB of Xp

DB,and define F as above. Then for all t0 Ø 0 we have

F (t0) = e≠t0[DB]‰+(DB)F (0),

In particular, F (0) œ Xp,+DB , and so F = C+

DB(F (0)).

Proof. Since F (t0) œ XpDB and since XpÕ

BúD flD(BúD) is dense in XpÕ

BúD (Corollary6.1.7 and density of D(BúD) in X2

BúD), it su�ces to test against „ œ XpÕ

BúD flD(BúD). For all such „ write

È„, e≠t0[DB]‰+(DB)F (0)ÍXpÕB

úD

= Èe≠t0[BúD]‰+(BúD)„, F (0)ÍXpÕB

úD

= ≠cN

ˆ Œ

0

etBúe≠ t

2 [DBú]D1e≠t0[BúD]‰+(BúD)„

2, tN(ˆN

t F )(t/2)f

EpÕ

dt

t(7.40)

= ≠cN

ˆ Œ

0

etBúe≠t0[DBú]‰+(DBú)D

1e≠ t

2 [BúD]„2

, tN(ˆNt F )(t/2)

f

EpÕ

dt

t

= ≠cN

ˆ Œ

0

=tBúD

1e≠ t

2 [BúD]„2

, tN(ˆNt F )

3t0 + t

2

4>

EpÕ

dt

t(7.41)

= ≠cN

ˆ Œ

0

=tBúe≠ t

2 [DBú]D„, tN(ˆNt F )

3t0 + t

2

4>

EpÕ

dt

t

= È„, F (t0)ÍXpÕB

úD

(7.42)

In (7.40) we used that e≠t0[BúD]‰+(BúD) maps XpÕ

BúD fl D(BúD) into itself, andthe representation (7.39). In (7.41) we used Lemma 7.3.18, which is valid sincee≠t[BúD]/2„ œ Dp(X) (Lemma 7.3.14) and since [t ‘æ (ˆtF )(t/2)] œ Xp≠1 solves(CR)DB. We use the representation (7.39) once more in the last line.

227

This immediately implies the following corollary.

Corollary 7.3.26. Let F œ Xp solve (CR)DB. Then F (0) œ Xp,+D , F is equal to

the Cauchy extension C+DB(F (0)), and F œ Xp.

Proof. All we need to show is that F is in Xp. This follows from Theorem6.2.12.

Step 6: Equality of ˆtF and ˆtF .

By Corollary 7.3.26 and Proposition 6.1.24, for F œ Xp which solves (CR)DB, thefunction t0 ‘æ F (t0) is in CŒ(R+ : Xp

D). Therefore we can consider (ˆtF )(t0) œXp

D as a distribution modulo polynomials.

Lemma 7.3.27. Let F œ Xp solve (CR)DB. Then for all t0 > 0 we have(ˆtF )(t0) = (ˆtF )(t0) in Z Õ(Rn).

Proof. Fix Ï œ Z.7 For all k œ N we have already computed (using that every-thing is in L2)

ÈÏ, Fk(t0)ÍZ

= ≠cN

¨R1+n

+

1t(Búe≠t[DBú]/2DÏ)(x), ‰ktN(ˆN

t F )(t0 + t/2, x)2

dxdt

t. (7.43)

Since Ï œ Z we have DÏ œ XpÕ

D , so for each t > 0 we may apply the (extendedoperator) e≠t[DBú]/2 to DÏ. We then have

------t ‘æ tBúe≠t[DBú]/2DÏ

------XpÕ

=------t ‘æ Búe≠t[DBú]/2DÏ

------Xp¸

. ||DÏ||Xp¸D

(7.44)

ƒ ||Ï||XpÕ < Œ,

where (7.44) follows from Proposition 6.1.3 since [z ‘æ ze≠[z]/2] œ �(Xp¸

D ) (herewe use i(p¸) < 2 and ◊(p¸) < 0). Since [t ‘æ tNˆN

t F (t0 + t/2)] œ Xp (Lemma7.3.23), the integral (7.43) is uniformly bounded in k and so we can take the limit

ÈÏ, F (t0)ÍZ = limkæŒ

ÈÏ, Fk(t0)ÍZ

= ≠cN

¨R1+n

+

1t(Búe≠t[DBú]/2DÏ)(x), tN(ˆN

t F )(t0 + t/2, x)2

dxdt

t

7Recall that Z(Rn) is the space of Schwartz functions f with D–f(0) = 0 for every multi-index –.

228

by dominated convergence. Using dominated convergence again, we can take thederivative:

ÈÏ, (ˆtF )(t0)ÍZ

= ˆtÈÏ, F (t0)ÍZ

= ≠cN

¨R1+n

+

1t(Búe≠t[DBú]/2DÏ)(x), tN(ˆN+1

t F )(t0 + t/2, x)2

dxdt

t

= ≠cN

ˆ Œ

0

etBúe≠t[DBú]/2DÏ, tN(ˆN+1

t F )(t0 + t/2)f

EpÕ

dt

t

using that Ï œ Dp(X) (Lemma 7.3.12) to conclude that the slice space dualitypairing is meaningful as in the proof of Lemma 7.3.24.

Now we rearrange:etBúe≠t[DBú]/2DÏ, tN(ˆN+1

t F )(t0 + t/2)f

EpÕ

=etBúD

1e≠t[BúD]/2Ï

2, tN(ˆN+1

t F )(t0 + t/2)f

EpÕ (7.45)

=etBú‰+(DBú)De≠t[BúD]/2Ï, tN(ˆN+1

t F )(t0 + t/2)f

EpÕ (7.46)

= tN+1eBúe≠t[DBú]/2‰+(DBú)DÏ, (ˆN+1

t F )(t0 + t/2)f

EpÕ (7.47)

= tN+1eBúDÏ, (ˆN+1

t F )(t0 + t)f

EpÕ (7.48)

= ≠tN+1eÏ, (ˆN+2

t F )(t0 + t)f

EpÕ . (7.49)

The first line (7.45) uses that Ï œ D(D) = D(BúD), (7.46) uses (7.32) andthe fact that e≠t[BúD]/2Ï is in Dp(X) (Lemma 7.3.13), (7.47) is just similarity offunctional calculi and rearrangement, (7.48) uses the weak semigroup property(7.29), and (7.49) finishes with integration by parts in slice spaces (Proposition5.1.44) and (CR)DB.

Therefore we have

ÈÏ, (ˆtF )(t0)ÍZ = cN

ˆ Œ

0

eÏ, (ˆN+2

t F )(t0 + t)f

EpÕ tN+1 dt

t

= (≠1)N+1

N !

ˆ Œ

t0

eÏ, (ˆN+2

t F )(t)f

EpÕ (t ≠ t0)N dt.

Finally, applying Corollary 7.3.22 with k = N + 1, we get

ÈÏ, (ˆtF )(t0)ÍZ = ÈÏ, (ˆtF )(t0)ÍZ

for all Ï œ Z and all t0 > 0. Therefore we have (ˆtF )(t0) = (ˆtF )(t0) in Z Õ forall t0 > 0 as claimed.

229

Step 7: Completing the proof.

Lemma 7.3.28. Let F œ Xp solve (CR)DB with limtæŒ F (t)Î = 0 in Z Õ(Rn :Cnm). Then F = F .

Proof. By Lemma 7.3.27 we have ˆtF = ˆtF in Z Õ, so there exists G œ Z Õ

such that F (t0) = G + F (t0) for all t0 œ R+. Since limt0æŒ F (t0) = 0 in XpD

(Proposition 6.1.24, using the weak-star topology when p is infinite) and hencealso in Z Õ, we find that GÎ = 0. Following the argument of [15, Step 5, page50], we find that G = —a modulo polynomials, where a is invertible in LŒ and— œ Cm. To complete the proof it su�ces to show that — = 0.

Note that the constant function [t ‘æ G = F (t) ≠ F (t)] is in Xp. If p is finite,then G œ Ep (since [t ‘æ G] solves (CR)DB), and this forces — = 0. If p is infinite,then the argument completing the proof of [15, Case q Æ 1, Theorem 1.3] showsthat if — ”= 0 then [t ‘æ G] /œ T Œ

≠1;– for all – œ [0, 1). Since ◊(p) > ≠1 we have

G œ Xp Òæ T Œ≠1;1+–(p)+◊(p),

and since –(p)+◊(p) œ [≠1, 0) (this follows from p œ Imax), we must have — = 0.This completes the proof.

Therefore, by Corollary 7.3.26, under the assumptions of Theorem 7.3.2, wehave that F = F = C+

DB(F (0)), with F (0) œ Xp,+D such that

------F (0)

------Xp,+

D

.||F ||Xp (by (7.38)). Furthermore, if f œ Xp,+

DB and F = C+DBf , then by Proposi-

tion 6.1.24 we havef = lim

tæ0C+

DBF (t) = F (0)

with limit in XpDB. This completes the proof of Theorem 7.3.2.

7.4 Applications to boundary value problems

7.4.1 Characterisation of well-posedness and corollaries

First let us put the boundary value problems given in the introduction (Subsection4.1.1) in a more convenient form.

Fix m œ N and let p be an exponent. Consider the spaces (Xp fl DZ Õ)(Rn :Cm(1+n)), using the notation of Subsection 6.2.1. Making use of the naturalsplitting

Xp(Rn : Cm(1+n)) = Xp(Rn : Cm) ü Xp(Rn : Cmn)

230

and the corresponding splitting for Z Õ(Rn : Cm(1+n)), we can write

(Xp fl DZ Õ)(Rn : Cm(1+n)) = Xp(Rn : Cm) ü (Xp(Rn : Cmn) fl ÒÎZ Õ(Rn : Cmn))=: Xp

‹ ü XpÎ .

In particular, if p œ Imax, we can make the identification

XpD(Rn : Cm(1+n)) ƒ Xp

‹ ü XpÎ

(see Theorem 6.2.1).For p œ Imax with ◊(p) < 0, define

ÁXp :=;

F œ Xp : limtæŒ

F (t)Î = 0 in Z Õ(Rn)<

and when p œ Imax and ◊(p) = 0 (so p = (p, 0) with p œ (n/(n + 1), Œ)) define

ÁXp :=ÓF : Nú(F ) œ Li(p)

Ô

where Nú(F ) is defined in (4.4).8 We set ||F ||ÊXp to be ||F ||Xp or ||NúF ||Li(p)

respectively.

Definition 7.4.1. For p œ Imax we define the Regularity problem

(RX)pA :

Y__]

__[

LAu = 0 in R1+n+ ,

limtæ0 ÒÎu(t, ·) = f œ XpÎ ,

||Òu||ÊXp . ||f ||Xp ,

and the Neumann problem

(NX)pA :

Y__]

__[

LAu = 0 in R1+n+ ,

limtæ0 ˆ‹A

u(t, ·) = f œ Xp‹,

||Òu||ÊXp . ||f ||Xp .

By limtæ0 ÒÎu(t, ·) = f œ XpÎ we mean that f œ Xp

Î and that the limit is in theXp

Î topology, and likewise for the limit in the Neumann problem. We say thatsuch a problem is well-posed if for all boundary data f there exists a unique u

(up to additive constant) satisfying the conditions of the problem.

We will denote these problems simultaneously by (PX)pA, with P standing for

either R or N .8In the notation of Huang [51], ÂX(p,0) = T p,2

Π.

231

Remark 7.4.2. The boundary condition in (RX)pA is equivalent to the Dirichlet

conditionlimtæ0

u(t, ·) = g œ Xp+1‹

where ÒÎg = f , since ÒÎ is an isomorphism from Xp+1‹ onto Xp

Î . Therefore(RX)p

A could be thought of as a Dirichlet problem (DX)p+1A .

Remark 7.4.3. The problems (RX)pA and (NX)p

A include all Regularity and Neu-mann problems introduced in Subsection 4.1.1. The definition above is muchmore concise (but, initially, much less clear).

Now we will use Theorems 7.3.1 and 7.3.2 to characterise the well-posedness of(RX)p

A and (NX)pA. Let N‹ and NÎ denote the projections from Xp

D(Rn : Cm(1+n))onto Xp

‹ and XpÎ respectively. If p œ I(X, DB) or p¸ œ I(X, DBú), then we

can realise Xp,+DB(Rn : Cm(1+n)) as a subset of Xp

D(Rn : Cm(1+n)), and via thisidentification we define

NpX,DB,Î : Xp,+

DB æ XpÎ and Np

X,DB,‹ : Xp,+DB æ Xp

‹

Note again that the condition p¸ œ I(X, DBú) is equivalent to p œ I(X, DB)when i(p) œ (1, Œ).

Theorem 7.4.4 (Characterisation of well-posedness). Let B = A. Suppose psatisfies Y

]

[p œ I(X, DB) if i(p) Æ 2,p¸ œ I(X, DBú) if i(p) > 2.

Then (RX)pA (resp. (NX)p

A) is well-posed if and only if NpX,DB,Î (resp. Np

X,DB,‹)is an isomorphism.

Proof. The results for ◊(p) = 0 and ◊(p) = ≠1 correspond to [15, Theorems 1.5and 1.6], so we need only consider ◊(p) œ (≠1, 0). Since ÒA =

ˈ‹

A

, ÒÎÈ, The

boundary conditions for (RX)pA and (NX)p

A can be rewritten as

NÎ

3limtæ0

ÒAu(t, ·)4

= f œ XpÎ and

N‹

3limtæ0

ÒAu(t, ·)4

= f œ Xp‹

respectively. By Theorem 4.1.3, solutions u to LAu = 0 are in bijective corre-spondence (modulo additive constant) to solutions F to (CR)DB, with F = ÒAu.Furthermore, by Theorems 7.3.1 and 7.3.2 and by the assumptions of this theo-rem, every such F œ ÁXp is given by F = C+

DBF0 for a unique F0 œ Xp,+DB (and so

232

F (t) œ Xp,+DB for all t), every such F0 determines a solution F , and by continuity

of the semigroup on Xp,+DB (Proposition 6.1.24)

F0 = limtæ0

ÒAu(t, ·).

The result follows.

Define the energy exponent e = (2, ≠1/2). For all A, the Lax–Milgram theo-rem guarantees well-posedness of the problems (RX)e

A and (NX)eA (see [12, The-

orems 3.2 and 3.3]). We say that a problem (PX)pA is compatibly well-posed if

it is well-posed and if for all boundary data f œ Xp• fl Xe

• (where • is eitherÎ or ‹ depending on the choice of boundary condition), the solution to (PX)e

A

with boundary data f (the energy solution) coincides with the solution to (PX)pA

with boundary data f . If p satisfies the assumptions of Theorem 7.4.4, then thistheorem says that (PX)p

A is compatibly well-posed if and only if NpX,DB,• is an

isomorphism and (NpX,DB,•)≠1 = (Ne

X,DB,•)≠1 on Xp• fl Xe

•.For finite exponents we can interpolate compatible well-posedness; compati-

bility is required in order to interpolate invertibility.

Theorem 7.4.5 (Interpolation of compatible well-posedness). Fix ◊ œ (0, 1), andsuppose p and q are finite exponents satisfying the assumptions of Theorem 7.4.4.If (PX)p

A and (PX)qA are compatibly well-posed, then (PX)[p,q]

◊

A is compatibly well-posed. Furthermore, if ◊(p) ”= ◊(q) and X = H, then (PB)[p,q]

◊

A is also compatiblywell-posed.

Proof. We use interpolation result for smoothness spaces, Theorem 5.1.52. Bythe previous discussion, we have

(NpX,DB,•)≠1 = (Nq

X,DB,•)≠1 = (NeX,DB,•)≠1

on the intersection Xp• fl Xq

• fl Xe•. Since this intersection is dense in Xp

• and Xq•

(here is where we use finiteness of p and q), we have a well-defined operator

N : Xp• + Xq

• æ Xp,+DB + Xq,+

DB

which restricts to (NpX,DB,•)≠1 and (Nq

X,DB,•)≠1 on Xp• and Xq

• respectively. Bycomplex interpolation, N restricts to a bounded operator N◊ : X[p,q]

◊

• æ X[p,q]◊

,+DB .

Since N◊ is equal to (NeX,DB,•)≠1 on X[p,q]

◊

• fl Xe•, and since N [p,q]

◊

X,DB,• is equal toNe

X,DB,• on Xp,+DB fl Xe,+

DB, we find that N◊ is the inverse of N [p,q]◊

X,DB,•. ThereforeN [p,q]

◊

X,DB,• is an isomorphism, and since [p, q]◊ satisfies the assumptions of Theorem

233

7.4.4 (by Proposition 6.2.9), (PX)[p,q]◊

A is well-posed. Furthermore, since N◊ =(Ne

X,DB,•)≠1 on X[p,q]◊

• fl Xe•, (PX)[p,q]

◊

A is compatibly well-posed. When X = Hand ◊(p) ”= ◊(q), applying real interpolation with the same argument yieldscompatible well-posedness of (PB)[p,q]

◊

A .

Although well-posedness without compatibility cannot be interpolated, it canbe extrapolated by making use of a theorem of äneıberg [83].9 This extrapola-tion procedure also extrapolates compatible well-posedness, and works for infiniteexponents (excluding the BMO-Sobolev range of spaces).

Theorem 7.4.6 (Extrapolation of well-posedness). Let B = A, and let p satisfyY]

[p œ I(X, DB)o i(p) Æ 2p¸ œ I(X, DBú)o i(p) > 2

(note the appearance of the interior of the identification regions), and if X = Hthen further assume that j(p) ”= 0. Suppose also that (PX)p

A is (compatibly) well-posed. Then there exists a (j, ◊)-neighbourhood Op of p such that for all q œ Op,(PX)q

A is (compatibly) well-posed.

The restriction j(p) ”= 0 for X = H rules out BMO-Sobolev spaces, which arenot in the interior of any of our complex interpolation scales. Note that whenp œ (1, Œ), p¸ œ I(X, DBú)o is equivalent to p œ I(X, DB)o.

Proof. We will prove the result for i(p) œ (1, Œ) as the proof for general expo-nents follows the same argument.

Let • denote either ‹ or Î as before. By Theorem 7.4.4, NpX,DB,• : Xp,+

DB æ Xp•

is an isomorphism. Let Bp be a ball in the (j, ◊)-plane centred at p such thatBp µ I(X, DB). Fix r œ Bp. Then we have

Xp,+DB = [X[p,r]≠1,+

DB , Xr,+DB, ]1/2

since p = [[p, r]≠1, r]1/2. Since the spaces Xp• form a complex interpolation scale,10

by the extrapolation theorem of äneıberg,11 there exists Á > 0 such that

N [p,r]‹

X,DB,• : X[p,r]‹

,+DB æ X[p,r]

‹

• is an isomorphism for all ‹ œ (≠Á, Á).9This was extended to quasi-Banach spaces by Kalton and Mitrea [58, Theorem 2.7], and

elaborated upon by Kalton, Mayboroda, and Mitrea [57, Theorem 8.1].10This is immediate for Xp

‹, and for XpÎ this is because Xp

Î is the image of Xp(Rn : Cm(1+n))under the retraction NÎPD.

11See [58, Theorem 2.7] for a reference incorporating both quasi-Banach spaces and theEnglish language.

234

Furthermore, inspection of the Kalton–Mitrea proof of this result shows that Á isindependent of r. Therefore there exists a ball Op µ Bp centred at p such that

NqX,DB,• : Xq,+

DB æ Xq• is an isomorphism for all q œ Op.

In addition, the inverses of these maps are consistent ([57, Theorem 8.1]), and soif the inverse of Np

X,DB,• is consistent with that of NeX,DB,• (i.e. when (PX)p

A iscompatibly well-posed) then this also holds for all q œ Op. By Theorem 7.4.4,this completes the proof.

Remark 7.4.7. Note that this proof also shows that if (PX)pA is well-posed but not

compatibly well-posed, then the same is true for (PX)qA for nearby q (the inverses

of NqX,DB,• are consistent, so they are either all consistent with Ne

X,DB,• or all notconsistent with Ne

X,DB,•). Therefore, staying within the range of exponents forwhich Theorem 7.4.4 holds, the set of p such that (PX)p

A is compatibly well-posedis a connected component of the set of p such that (PX)p

A is well-posed.

Now we present a ¸-duality principle for well-posedness.

Theorem 7.4.8 (¸-duality of well-posedness). Let B = A, and suppose that p œI(X, DB). If (PX)p

A is (compatibly) well-posed, then (PX)p¸

Aú is also (compatibly)well-posed.

Of course, if i(p) œ (1, Œ), then this statement is an equivalence. We pointout the case where p = (1, s) with s œ (≠1, 0]: in this case the result says thatwell-posedness of a problem with coe�cients A and boundary data in the Hardy–Sobolev space H1

s (resp. the Besov space B1,1s ) implies well-posedness of the

corresponding problem for Aú with boundary data in the image of BMO–Sobolevspace ˙BMO≠s (resp. the Hölder space �≠s) under D.

Proof. We will be sketchy because all the important details of this argument arealready done by Auscher, Mourgoglou, and Stahlhut (see [16, §12.2] and [15,§13]). Recall from Remark 6.2.11 that „Aú = NBúN =: B. When p is finite, thepairing

Èf, gÍNXp

DB

:= Èf, NgÍXpDB

a duality pairing between XpDB and XpÕ

BD. We have that ||Dg||

Xp¸DB

ƒ ||g||XpÕBD

whenever g œ D(D) fl XpÕ

BD(Proposition 6.2.6), and so the pairing

Èf, g͸Xp

DB

:= Èf, ND≠1gÍXpDB

(7.50)

235

is a duality pairing between XpDB and R(D) fl Xp¸

DB. Since p œ I(X, DB), we

can identify XpD = Xp

DB and Xp¸

D = Xp¸

DBas completions of Xp

DB and Xp¸

DB

respectively (using a simple modification of Proposition 6.2.7 to make the secondidentification), and by density the pairing (7.50) extends to a duality pairingbetween Xp

DB and Xp¸

DB. As in the proof of [15, Lemma 13.3], this pairing realises

Xp¸,ûDB

as the dual of Xp,±DB , Xp¸

‹ as the dual of XpÎ , and Xp¸

Î as the dual ofXp

‹. The remainder of the argument precisely follows the proof of [15, Theorem1.6].

7.4.2 The regularity problem for real coe�cient scalarequations

The results above show that from compatible well-posedness of a boundary valueproblem for an exponent p œ I(X, DB) with B = A, we may deduce compatiblewell-posedness for a larger range of exponents by ¸-duality and interpolation. Asan application of this principle we consider the regularity problems (RX)p

A in thereal scalar case.

Suppose that m = 1 (so that LAu = 0 is a single equation rather than asystem) and that the entries of A are real. In this setting, there exists a number– œ (0, 1] such that for every Euclidean ball B = B(X0, 2r) in R1+n

+ and everysolution u to LAu = 0 in B, we have

|u(X) ≠ u(X Õ)| .A

|X ≠ X Õ|r

B– AˆB

|u|2B1/2

(7.51)

for all X, X Õ in the smaller ball B(X0, r). In this case say that the coe�cients A

satisfy the De Giorgi–Nash–Moser condition of exponent –. The adjoint matrixAú will also satisfy a De Giorgi–Nash–Moser condition of (possibly di�erent)exponent –ú.

Auscher and Stahlhut [16, Corollary 13.3] show that in this case12 we have3

n

n + –, p+(DB)

4µ I0(H, DB),

3n

n + –ú , p+(DB)4

µ I0(H, DB),

where B = Aú (note that B ”= Bú) and where p+(DB), p+(DB) > 2. Therefore

12In fact, a somewhat weaker assumption is needed there.

236

Figure 7.5: Exponents p œ I(H, DB), when m = 1 and A is real, with B = A.

12

1 n+1n

j(p)0

◊(p)

0

≠1

1/p+(DB) n+–n

1/p+(DB)Õ

xA

by ¸-duality (see Proposition 6.2.7 and Remark 6.2.11), we have

(p+(DB)Õ, Œ) µ I≠1(H, DB),(p+(DB)Õ, Œ) µ I≠1(H, DB).

By interpolation (Proposition 6.2.9) we then have that I(H, DB) contains theregion pictured in Figure 7.5, and I(B, DB) contains the interior of that region.The point xA here is defined as the pictured intersection, which is a function ofn, –, and p+(DB) that we need not compute explicitly.

There is also a corresponding diagram for B that we have not pictured, in-cluding a corresponding exponent xAú . By applying ¸-duality to the exponentsp œ I(H, DB) with i(p) œ (1, 2), and another application of interpolation, wecan increase these ranges to that pictured in Figure 7.6.

It has been shown that there exist pR(A) > 1 (possibly small) and 0 < –˘ Æmin(–, –ú) such that the Regularity problem (RH)(p,0)

A is compatibly well-posedfor all p œ (n/(n + –˘), pR(A)], and likewise for Aú (with the same –˘).13 Bythe results of the previous paragraph, we have (p, 0) œ I0(H, DB) fl I0(H, DB)for all such p,14 and so we may apply ¸-duality and interpolation as in the

13The p0

endpoint of this result is due to Kenig and Rule in dimension n+1 = 2 [62, Theorem1.4] and Hofmann, Kenig, Mayboroda, and Pipher in dimension n + 1 Ø 3 [46, Corollary 1.2].The other endpoint is an extrapolation result of Auscher and Mourgoglou [14, §10.1].

14It is possible that p > p+

(DB) or p > p+

(DB), in which case we have to restrict to smallp. In general p is small, so this is not a serious loss of generality.

237

Figure 7.6: More exponents p œ I(H, DB), when m = 1 and A is real, withB = A. The dark shaded region corresponds to Figure 7.5.

12

1 n+1n

j(p)0

◊(p)

0

≠1

1/p+(DB) n+–n

1/p+(DB)Õ

x¸Aú

xA

previous argument to deduce compatible well-posedness of (RH)pA for p in the

region pictured in Figure 7.7, and of (RB)pA in the interior of this region.15

We can expand this region slightly for Besov spaces: applying ¸-duality tocompatible well-posedness of (RB)p

Aú for p in the open triangle with vertices yAú ,(n + –˘/n, 0), and (1, 0), we find that (RB)(Œ,–;0)

A is compatibly well-posed for all– œ (≠1, ≠1 ≠ –˘). Therefore (after another iteration of interpolation) we havewell-posedness of (RB)p

A for all p in the shaded region of Figure 7.8. This is thesame region obtained by Barton and Mayboroda for compatible well-posednessof (RB)p

A in this setting [21, Figure 3.5].16 To recover the result of [21, Corollary3.24], one need only apply Lemma 7.2.1 (which is valid for this region of p, seeFigure 7.2) to remove the decay assumption at infinity from (RB)p

A, and thetrace theorem [21, Theorem 6.3] to replace our boundary condition with a tracecondition.

In the case that A is symmetric in addition to the above assumptions, thenresults of Kenig and Pipher [61] imply that we have the additional information

15We can also deduce results for BMO-Sobolev spaces, which correspond to the unpicturedj(p) = 0 range.

16The only di�erence is in the light shaded ‘region of applicability’: ours depends on theAuscher–Stahlhut exponent p+(DB), while that in Barton–Mayboroda is in terms of the expo-nent appearing in Meyers’ theorem [75, Theorem 2] (see also [21, Lemma 2.12]). It is not clearwhether there is any relationship between these exponents.

238

Figure 7.7: Exponents p for which (RH)pA is compatibly well-posed (the dark

shaded region). The light shaded region is from Figure 7.6.

12

1 n+1n

j(p)0

◊(p)

0

≠1

1/pR(A) n+–˘

n

1/pR(Aú)Õ

y¸Aú

yA

pR(A) = pR(Aú) > 2, and regions of compatible well-posedness of (RX)pA can

be expanded accordingly. Furthermore, in this case the corresponding Neumannproblems (NX)p

A are well-posed for the same range of p by repeating the argu-ments above (starting from the information given by [61]).

7.4.3 Additional boundary behaviour of solutions

It is possible to establish the following boundary behaviour of solutions to LAu =0.

Theorem 7.4.9. Let B = A and let p be an exponent with ◊(p) œ (≠1, 0). Letu solve LAu = 0, with ÒAu œ ÁXp.

(i) Suppose p is finite and p œ I(X, DB). Then there exists v œ Xp+1 suchthat

limRæ0

ˆ�(R,x)

u(·, ›) d› d· = v(x) a.e. x œ Rn,

with ÒÎu = ÒÎv in Z Õ.

(ii) Suppose p is infinite and p¸ œ I(X, DBú). Then u œ Xp+1(R1+n+ ).

The proof, which we do not provide here, requires a series of ad hoc arguments(much like the proof of Theorem 6.2.12) that exploit the semigroup representation

239

Figure 7.8: Exponents p for which (RB)pA is compatibly well-posed (the dark

shaded region); this includes no exponents with ◊(p) = 0 or ◊(p) = ≠1. Thelight shaded region is from Figure 7.6.

12

1 n+1n

j(p)0

◊(p)

0

≠1

1/pR(A) n+–˘

n

1/pR(Aú)Õ

(0, ≠1 + –˘)

(1, ≠–˘)

of the conormal gradient ÒAu provided by Theorems 7.3.1 and 7.3.2. Full details,which have been communicated to us by Pascal Auscher, will be provided in afuture version of this article.

7.4.4 Layer potentials

We conclude the article by briefly indicating the relation between the first-orderapproach and the method of layer potentials. Further information on this link isavailable in [79] and [16, §12.3].

Suppose, for the moment, that A and Aú both satisfy the De Giorgi–Nash–Moser condition (7.51) of some exponent. Then for all (t, x) œ R1+n there exists afundamental solution �(t,x) for LAú in R1+n with pole at (t, x).17 The fundamentalsolution �(t,x) is a Cm-valued function on R1+n

+ satisfying

div AúÒ�(t,x) = ”(t,x)1 in R1+n

in the usual weak sense, where ”(t,x) is the Dirac mass at (t, x) and 1 = (1, . . . , 1) œCm.

17Fundamental solutions were constructed in dimension n + 1 Ø 3 by Hofmann and Kim [47],and in dimension n + 1 = 2 by Rosén [79].

240

For a (reasonable) function h : Rn æ Cm and for (t, x) œ R1+n+ , define the

double layer potential

Dth(x)i :=ˆRn

1ˆ‹

A

ú �(t,x)(0, y)i, h(y)2

dy (i = 1, . . . , m).

and the single layer potential

Sth(x)i :=ˆRn

(�(t,x)(0, y)i, h(y)) dy. (i = 1, . . . , m).

One can solve Dirichlet problems for LA in R1+n+ with boundary data Ï by solving

the double layer equationlimt√0

Dth = Ï,

and likewise one can solve Neumann problems for LA in R1+n+ with boundary data

Ï by solving the single layer equation

limt√0

ˆ‹A

Sth = Ï.

The corresponding solutions u are then given by u(t, x) = Dth(x) and u(t, x) =Sth(x) respectively.

It was shown by Rosén [79] that these layer potential operators fall within thescope of the first-order framework. Keeping the De Giorgi–Nash–Moser assump-tion on A and Aú, and writing B = A as usual, for all f œ L2(Rn : Cm) and t œ Rwe have

Dtf = sgn(t)Q

ae≠|t|BD‰sgn(t)(BD)S

Uf

0

T

V

R

b

‹

and

ÒAStf = ≠ sgn(t)Q

ae≠|t|DB‰sgn(t)(DB)S

Uf

0

T

V

R

b

‹

,

where the vectorsS

Uf

0

T

V are in L2(Rn : Cm(1+n)), written with respect to the

transversal/tangential splitting. In terms of Cauchy operators, on R1+n± we can

write

Df = ±Q

aC±BD

S

Uf

0

T

V

R

b

‹

, ÒASf = ûQ

aC±DB

S

Uf

0

T

V

R

b

‹

. (7.52)

The right hand sides of these expressions are defined for all coe�cients A, whetheror not the De Giorgi–Nash–Moser assumptions are satisfied.

241

For all exponents p œ I(X, DB) (and for all infinite exponents p with p¸ œI(X, DBú)) we have the estimate

------C+

DBf------Xp . ||f ||Xp

D

(f œ R(DB)+)

(see Theorems 6.1.25 and 6.2.12), which immediately yields

||ÒASf ||Xp . ||f ||Xp . (7.53)

This can be seen as boundedness of S from the classical smoothness space Xp intoa Sobolev-type space built on Xp. On the other hand, because of the equalityÒÎg‹ = ≠(Dg)Î (for any g : Rn æ Cm(1+n)) and similarity of functional calculus,we can write

ÒÎDf = ≠Q

aC+DBD

S

Uf

0

T

V

R

b

Î

=Q

aC+DB

S

U 0ÒÎf

T

V

R

b

Î

,

which implies (for all f with ÒÎf œ L2)

------ÒÎDf

------Xp .

------

------C+

DB

Q

a

S

U 0ÒÎf

T

V

R

b

------

------Xp

.------ÒÎf

------Xp

ƒ ||f ||Xp+1 ,

and similarly

||ˆtDf ||Xp .------

------BDC+

BD

S

Uf

0

T

V

------

------Xp

.------

------C+

DBD

S

Uf

0

T

V

------

------Xp

. ||f ||Xp+1 ,

so that||ÒDf ||Xp . ||f ||Xp+1 . (7.54)

Bounds for layer potentials on the lower half-space corresponding to (7.53) and(7.54) can also be derived. Compare these results with those of Barton and May-boroda [21, Theorem 3.1]. Various other mapping properties of layer potentialsfollow from the identifications (7.52) and the mapping properties of functionalcalculus on the spaces Xp

DB, for example the uniform bounds

supt”=0

||ÒAStf ||Xp + ||Stf ||Xp+1 . ||f ||Xp (7.55)

242

(St can be defined via Cauchy operators as in [16, §12.3]) and

supt”=0

||ÒADtf ||Xp + ||Dtf ||Xp+1 . ||f ||Xp+1 . (7.56)

We also obtain limits for these operators as t æ 0± (in Xp or Xp+1 accordingly,and in the strong or the weak-star topology depending on whether p is finite). Inparticular we can also recover the jump relations with this formalism. We referthe reader to Auscher and Stahlhut [16, §12.3] for further details.

For p as above, Rosén’s identification of the layer potentials in terms of Cauchyoperators and the boundedness results above imply that the solutions to boundaryvalue problems that we construct via Cauchy operators coincide with solutionsconstructed by the method of layer potentials. It is possible that this fails outsidethis range of p.

243

This paper is already too long, so details will be left as a challenge tothe reader.

-Alan McIntosh, Operators which have an HΠfunctional calculus [70]

244

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