Global, Local and Vector-Valued Tb Theorems on Non-Homogeneous Metric Spaces

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GLOBAL, LOCAL AND VECTOR-VALUED T b THEOREMS ON NON-HOMOGENEOUS METRIC SPACES

HENRI MARTIKAINEN

Academic dissertation To be presented for public examination

with the permission of the Faculty of Science of the University of Helsinki in Auditorium B123, Exactum, on 9 December 2011 at 2 p.m.

Department of Mathematics and Statistics Faculty of Science

University of Helsinki 2011

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ISBN 978-952-10-7365-6 (Paperback) ISBN 978-952-10-7366-3 (PDF) http://ethesis.helsinki.fi

Unigrafia Oy Helsinki 2011

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C^ONTENTS

Acknowledgements v

List of contributed articles vii

1. Overview 1

2. Non-homogeneous Calderón–Zygmund theory on metric spaces 3

2.1. Geometrically doubling metric spaces . . . 3

2.2. Upper doubling measures . . . 3

2.3. Standard kernels and Calderón–Zygmund operators . . . 3

2.4. Accretivity . . . 4

2.5. Weak boundedness property . . . 4

2.6. BMO and RBMO . . . 4

3. Global non-homogeneousT btheorems 5 3.1. Background . . . 5

3.2. Article [A] . . . 5

3.3. Applications . . . 6

3.4. Related developments . . . 7

4. Vector-valued Calderón–Zygmund theory 7 4.1. Background . . . 7

4.2. Article [B] . . . 8

5. LocalT btheorems 8 5.1. Background . . . 8

5.2. Article [C] . . . 10

6. Sharp weighted bounds for maximal truncationsT_\ 11 6.1. Background . . . 11

6.2. Article [D] . . . 12

6.4. Related developments . . . 14

References 15

Included papers [A, B, C, D]

iii

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ACKNOWLEDGEMENTS

I wish to express my deepest gratitude to my advisor Tuomas Hytönen for his expert guidance, collaboration, admirable efficiency and for always having the best interest of his students in mind.

I am grateful to the pre-examiners Brett Wick and Stefan Geiss. It is my plea- sure to thank Pascal Auscher for accepting to act as the opponent during the public examination of this dissertation.

For financial support I am indebted to the Academy of Finland.

I would like to thank the people at the Department of Mathematics for such a great working environment. Special thanks are due to Pertti Mattila for help- ing me during my first steps in the academic world and Tuomas Orponen for his friendship, many interesting mathematical discussions and collaboration. I thank Alexander Volberg for hosting my visit to Michigan State University during November–December 2010. I would also like to thank Michael Lacey, Maria Carmen Reguera, Eric Sawyer and Ignacio Uriarte–Tuero for collaboration.

I thank Risto Tahvanainen for being such a good friend for over a decade. Fi- nally, I am deeply grateful to my parents and my brother for their support and company.

Helsinki, November 2011 Henri Martikainen

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LIST OF INCLUDED ARTICLES

This thesis consists of an introductory part and the following four articles:

[A] T. Hytönen and H. Martikainen, Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces, J. Geom. Anal., to appear; preprint (2009).

[B] H. Martikainen,Vector-valued non-homogeneous Tb theorem on metric measure spaces, preprint (2010).

[C] T. Hytönen and H. Martikainen,On general local Tb theorems, Trans. Amer.

Math. Soc., to appear; preprint (2010).

[D] T. Hytönen, M. Lacey, H. Martikainen, T. Orponen, M. C. Reguera, E.

Sawyer, and I. Uriarte–Tuero, Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on Ap weighted spaces, J. Anal.

Math., to appear; preprint (2011).

In the introductory part these articles are referred to as [A], [B], [C], [D] and other references will be numbered as [1],[2],...

AUTHOR’S CONTRIBUTION

The research presented in this dissertation has been mainly done at the De- partment of Mathematics and Statistics at the University of Helsinki during the period 2009–2011. Part of the analysis and writing of [D] was done when the author was visiting Michigan State University during November–December 2010.

In [B] the author’s independent research is reported. In [A] and [C] the author has had a central role in the analysis and writing. Article [D] is a large project with many collaborators and there the author has had a key role in the analysis and writing of Chapter 9, which presents one of the main components of the proof scheme.

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T bTHEOREMS ON NON-HOMOGENEOUS METRIC SPACES 1

1. O^VERVIEW

VariousT b theorems play a key role in the modern harmonic analysis. They provide characterizations for the boundedness of Calderón–Zygmund type singular integral operators. The general philosophy is that to conclude the boundedness of an operatorT on some function space, one needs only to test it on some suitable functionb. One particular allowed choice is b = 1 – such theorems are calledT1theorems. The originalT1theorem is by David and Journé [14]. Gener- alizations which allow one to use more general test functionsb, where only some suitable notion of non-degeneracy is required, soon followed – see McIntosh and Meyer [35] and David, Journé and Semmes [15]. Testing conditions also involve the adjointT^∗ ofT. One can use different test functions, sayb₁ andb₂, forT and T^∗ respectively.

One may also use several test functions – one for each cube. This has some advantages on some situations, and constitutes the crux of the localT btheorems.

One assumes that to every cubeQthere exist functionsb¹_Q andb²_Q, supported on Q, so that we also know something about T b¹_Q and T^∗b²_Q. The aim is to demon- strate thatT mapsL²toL² with the operator norm having a natural dependence on the assumptions. First such theorem, withL^∞control ofb¹_Q, b²_Q,T b¹_QandT^∗b²_Q, is by Christ [10]. One may also consider more generalL^p type control like pio- neered by Auscher, Hofmann, Muscalu, Tao and Thiele [2].

The vector-valued theory of Calderón–Zygmund operators has greatly bene- fited from certainT1(andT btheorems). Given a Banach space, one is interested in the question whether or not the tensor extension of a scalar-valued Calderón–

Zygmund operator extends to a bounded Banach space valued operator. It is known by results of Bourgain and Burkholder (see [6], [9]) that the greatest generality in which this could be true is in the category of the so called UMD spaces.

However, it seems difficult to exploit the scalar-valued boundedness directly to establish the vector-valued boundedness. In fact, one does not generally know how to do this. Instead, the idea is to rely on the characterization given byT b theorems: one proves that the hypothesis ofT btheorems are enough to actually guarantee the UMD-valued boundedness. The vector-valued T1theorem is by Figiel [16].

One more twist, which is very pertinent for this dissertation, is the choice of the domain and the underlying measure. For example, the above mentioned theorems work mainly in the context of Rⁿ equipped with the Lebesgue measure, or some metric space equipped with a doubling measure. Theory that can handle more general measures, like those satisfying only the upper bound µ(B(x, r)) ≤ Cr^m, is of utmost importance too. Such results were most no- tably pursued by Nazarov–Treil–Volberg and Tolsa culminating in the global non-homogenousT btheorem [40] and the resolution of Painlevé’s problem [46].

The local non-homogenousT btheorem is also due to Nazarov, Treil and Volberg [39]. In the vector-valued situation a global non-homogeneousT btheorem is due

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2 HENRI MARTIKAINEN

to Hytönen [19]. The aforementioned non-homogenous results are tied toRⁿ. Be- fore this dissertation no equally satisfying theory of non-homogeneous analysis in metric spaces existed.

We studyT b theorems, global and local, which may be very general with respect to the underlying measure (upper doubling), the domain (geometrically doubling (quasi-)metric space) and the range (UMD space). These notions are defined carefully in the subsequent chapter. Also, in the final paper [D] we study weighted estimates for maximal truncations of Calderón–Zygmund operators.

This consists of reducing to certain Sawyer-type testing conditions, which are very much so in the spirit ofT b theorems and thus of this thesis. Moreover, the paper [D] is dyadic analysis using the randomization of dyadic grids – an underlying theme of the papers [A], [B] and [C] as well. The mentioned testing conditions must be verified with good enough dependency on certain parame- ters. In particular, this final paper extends the sharp bounds previously known only for untruncated operators, and also proves sharp weak type results, which are new even for untruncated operators.

The particular goal of [A] is to extend the non-homogeneous T b theorem of Nazarov, Treil and Volberg [40], which is valid inRⁿwith the standard Euclidian metric, to metric spaces. One of the difficulties of lifting the proof strategy of [40]

to metric spaces is to construct a randomization procedure for the metric dyadic cubes of Christ [10] which would be valid in metric spaces lacking a translation group. TheT btheorem of [A] is also more general with respect to the measures considered – and this generality is genuinely important for some interesting applications.

In [B] a vector-valued extension of the main result of [A] is considered. As indicated above, the point is to be able to establish that Calderón–Zygmund operators have bounded extensions to UMD-valued operators. Here we want to allow metric domains with non-homogeneous measures. Suppose, for example, that one would have a Calderón–Zygmund operator on the Heisenberg group. In order to know that this extends to a bounded UMD-valued operator, one would need a vector-valuedT1theorem valid on metric spaces. This is the general mo- tivation for [B], and it indeed solves such questions.

In [C] we study localT btheorems both in the doubling and non-doubling situation. There we extend a modified version of the general non-homogeneous proof technique of Nazarov, Treil and Volberg [39] to the case of upper doubling measures. We show that this technique can also be used in the homogeneous setting to prove local T b theorems with L^p type testing conditions. The rather large open problem regarding the question whether one can use test functions with L^p type testing conditions with p > 1butp ≈ 1, remains open. However, we give a completely new proof of the results originating from Auscher, Hof- mann, Muscalu, Tao and Thiele [2] and Hofmann [18] utilizing the full force of non-homogeneous analysis. Such methods could also turn out to be useful in the resolution of the full conjecture.

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In the next section we shall give some of the most central definitions related to this work. Sections following that will detail more on the papers [A]–[D] and discuss their relevance to previously known results. We will also discuss some applications and connections.

2. N^ON-HOMOGENEOUS C^ALDERÓN–ZYGMUND THEORY ON METRIC SPACES

2.1. Geometrically doubling metric spaces. We consider metric spaces (X, d) satisfying the following well known geometric condition: every ball B(x, r) = {y ∈ X : d(x, y) < r} can be covered by at most N balls of radius r/2. We call such metric spaces geometrically doubling. Our theory remains valid on most quasi-metric spaces as well. That is, we could also consider quasi-metrics ρsatisfying the following additional regularity condition: for every > 0there existsA()<∞so that

ρ(x, y)≤(1 +)ρ(x, z) +A()ρ(z, y)

for everyx, y, z ∈ X. However, we shall not insist on this detail in this introductory part. We just mention that the generality of quasi-metric spaces is needed in the natural applications.

2.2. Upper doubling measures. Consider a function λ: X × (0,∞) → (0,∞) such that r 7→ λ(x, r) is non-decreasing and λ(x,2r) ≤ C_λλ(x, r). We say that a Borel measure µ on X is upper doubling with a dominating function λ, if µ(B(x, r))≤λ(x, r)for allx∈Xandr >0. Of course, many measures are upper doubling – like all the bounded ones. But one should realize that the choice ofλ is important, because we will tie our standard kernel estimates to it. So choosing a trivialλwill yield very restrictive kernel estimates.

This theory is rich enough to encapsulate doubling measures (chooseλ(x, r) = µ(B(x, r))) and measures satisfying the power bound µ(B(x, r)) ≤ Cr^m (choose λ(x, r) =Cr^m), and then our kernel estimates will agree with the traditional ones in these particular cases. The framework of geometrically doubling metric spaces equipped with upper doubling measures was first considered by Hytönen in [20], but only in the context of BMO spaces.

2.3. Standard kernels and Calderón–Zygmund operators. Define∆ = {(x, x) : x∈ X}. A standard kernel is a mappingK: X² \∆ →Cfor which we have for someα >0andC <∞that

|K(x, y)| ≤Cmin 1

λ(x, d(x, y)), 1 λ(y, d(x, y))

, x6=y,

|K(x, y)−K(x⁰, y)| ≤C d(x, x⁰)^α

d(x, y)^αλ(x, d(x, y)), d(x, y)≥2d(x, x⁰), and

|K(x, y)−K(x, y⁰)| ≤C d(y, y⁰)^α

d(x, y)^αλ(y, d(x, y)), d(x, y)≥2d(y, y⁰).

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4 HENRI MARTIKAINEN

The smallest admissibleCwill be denoted bykKk_CZ_α.

Let T: L²(µ) → L²(µ)be a linear operator. It is called a Calderón–Zygmund operator with kernelK if

T f(x) = Z

X

K(x, y)f(y)dµ(y)

for x outside the support off. We assume the boundedness of T a priori, but we are interested in quantitative bounds for kTk, which only depend on some specified information.

2.4. Accretivity. A functionb is called accretive if Reb ≥ a > 0 almost every- where. We can also make do with the following weaker form of accretivity:

|R

Ab dµ| ≥ aµ(A) for all Borel setsA which satisfy the condition that B ⊂ A ⊂ 500B for some ball B = B(A). In the Euclidian space with the usual metric it suffices to consider only the case whenAis a cube. Generally, the point is to have the above estimate wheneverAis one of the metric dyadic cubes, but there is no easy explicit description of what kind of sets they actually are.

2.5. Weak boundedness property. An operator T is said to satisfy the weak boundedness property if|hT χ_B, χ_Bi| ≤Aµ(ΛB)for all ballsBand for some fixed constantsA >0andΛ>1. Hereh·, ·iis the bilinear dualityhf, gi=R

f g dµ. Let us denote the smallest admissible constant above bykTkW BP_Λ. Many variants of the weak boundedness property exist. For example, it can sometimes be of interest to replace the rough test functionsχ_B by some more regular ones. Such theory is also established in [A] but we will not discuss this in the introduction.

2.6. BMO and RBMO. We say thatf ∈ L¹_loc(µ) belongs to BMO^p_κ(µ), if for any ballB ⊂Xthere exists a constantf_Bsuch that

Z

B

|f −f_B|^pdµ1/p

≤Lµ(κB)^1/p, where the constantLdoes not depend onB.

Let% >1. A functionf ∈L¹_loc(µ)belongs to RBMO(µ)if there exists a constant L, and for every ballB, a constantf_B, such that one has

Z

B

|f−fB|dµ≤Lµ(%B), and, wheneverB ⊂B₁are two balls,

|fB−fB1| ≤L

1 + Z

2B1\B

dµ(x) λ(c_B, d(x, c_B))

. We do not demand that fB be the average hfiB = _µ(B)¹ R

Bf dµ. This is relevant in the RBMO(µ)-condition. The space RBMO(µ)is independent of the choice of parameter% > 1and satisfies the John–Nirenberg inequality. For these results in our setting, see [20]. The norms in these spaces are defined in the obvious way as the best constantL.

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3. G^{LOBAL NON}-HOMOGENEOUST b THEOREMS

3.1. Background. The originalT1 theorem by David and Journé [14] is clearly only in the Euclidian context. The arguments of the T b article [15] by David, Journé and Semmes carry over to a metric space – however, only if one is working with doubling measures.

In [40, p. 153], Nazarov, Treil and Volberg point out that “a (more or less) complete theory of Calderón–Zygmund operators on non-homogeneous spaces – – can be developed in an abstract metric space with measure.” It is the purpose of the article [A] to develop the needed machinery, including the randomization of M. Christ’s cubes in metric spaces, and to prove the analog of the main result of [40] in the general framework of geometrically doubling metric spaces equipped with upper doubling measures introduced above. It would have perhaps been also possible to extend the techniques of David [12] into metric spaces (this is another non-homogenousT b theorem). Some related metric theory already existed before [A]: the weak-typeL¹ inequality under a prioriL² boundedness by Nazarov, Treil and Volberg [38], estimates on Lipschitz spaces by García-Cuerva and Gatto [17], and a certain quite restricted version of theT1theorem by Bra- manti [5].

3.2. Article [A]. Our main result of [A] for metric spaces reads as follows.

3.1.Theorem. Let (X, d) be a geometrically doubling metric space which is equipped with an upper doubling measure µ. Let T be an L²(µ)-bounded Calderón–Zygmund operator with a standard kernel K, let b₁ and b₂ be two essentially bounded accretive functions, letα >0andκ,Λ>1be constants. Then

kTk.kT b1k_{BM O}²_κ_(µ)+kT^∗b2k_{BM O}²_κ_(µ)+kMb2T Mb1kW BP_Λ +kKkCZα. HereM_b: f 7→bf is the multiplication operator with symbolb.

We remark that one can replace BMO²_κ(µ)by BMO¹_κ(µ). Also, one can deal with quasi-metric spaces and somewhat more general notions of weak boundedness property.

The basic idea of the proof is to somehow develop an adaptation of the proof strategy of Nazarov, Treil and Volberg [40] which would work in metric spaces.

Also, one needs new estimates to deal with the more general measures and kernel estimates. To give some idea, we give a very brief outline now illustrating some aspect and difficulties of the proof.

We start by constructing a slight modification of the cubes of Christ [10] using just the condition of geometrically doubling – no auxiliary doubling measure is needed for this (we do not even assume our space to be complete so no such measure needs to even exist). Then it is noted that this construction depends only on some fixed center points – the choice of which we may appropriately randomize.

This new randomization then provides us with the absolutely essential property that for a fixed point it is unlikely to end up too close to the boundary of some cube.

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6 HENRI MARTIKAINEN

Having the notion of two independent randomized gridsD and D⁰ in metric spaces allows us to use the idea of good and bad cubes by Nazarov, Treil and Volberg (which have been succesfully used for example in [40] and [39]). In our setting this reads as follows. Letδ ≤ 1/1000, r be a fixed large parameter, d :=

log₂C_λ (an upper bound for the dimension ofµ) and γ := α

2(α+d).

Consider a cubeQ ∈ D. We say that Qis good if for any cube R ∈ D⁰ for which we have`(Q) ≤δ^r`(R), we have eitherd(Q, R)≥ `(Q)^γ`(R)^1−γord(Q, X \R)≥

`(Q)^γ`(R)^1−γ. Here`(Q)denotes the side length of a cubeQand is always of the formδ^kfor somek ∈Z.

One then takes two functionsf andgand expands them tob_i-adapted martingale differences using the dyadic gridsDand D⁰. This gives us a decomposition forhT f, gi– a huge sum which is then split into several pieces. One follows the outline of [40] again. One difficulty is that the upper boundλ(x, r)depends on the center x. Moreover, in the diagonal part a significant difficulty arises. We need to employ a construction of random almost-coverings of the space by balls of comparable radius, by which we mean that any given point has a small proba- bility of not being covered. This is because it is far more natural to formulate the weak boundedness property in terms of balls rather than metric cubes, and so we essentially need to cut our dyadic cubes into comparable balls when estimating the diagonal part of the operator.

We feel that we have achieved a quite satisfactory analog of the main result of [40] and, in the process of doing so, also developed some interesting geometrical methods in metric spaces. One slightly unsatisfactory part is the notion of accretivity – we would like to be able to state the weakest formulation concerning averages using only balls, just like we managed to do with the weak boundedness property.

3.3. Applications. One interesting application, already presented in [A], is to note that the upper doubling formalism is exceptionally well suited to give a new proof of the main result of [49]. There Volberg and Wick obtained a characterization of measuresµin the unit ballB2nofCⁿfor which the analytic Besov–Sobolev spaceB₂^σ(B2n)embeds continuously intoL²(µ). The measuresµin [49] satisfy the upper power boundµ(B(x, r)) ≤ r^m, except possibly whenB(x, r) ⊆ H, where H = B2n. One notes that this means that their measures are actually upper doubling with

µ(B(x, r))≤max(δ(x)^m, r^m) =: λ(x, r), whereδ(x) =d(x, H^c). The kernel in [49] has the specific form

K(x, y) = (1−x¯·y)^−m, x, y ∈B¯²ⁿ ⊂Cⁿ.

One can check that the standard estimates of our theory are verified. Our metric T1 theorem then applies to give the same characterization as in [49]. We

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also mention the article [50] by Volberg and Wick where a metric version of the Bergman typeT1theorem from [49] is considered – the proof uses randomized metric grids from [A].

In [1, Theorem 2.6] Adams and Eiderman study certain capacities γ_K,op and γK,∗ in some metric spaces. The equivalence of these capacities is established in [1] using the non-homogeneous metric T1 theorem of [A]. A very nice trick of F.

Nazarov is also presented in [1], the philosophy of which is that some operators which are not initially Calderón–Zygmund can be made into such by a suitable choice of the underlying metric. This means that metricT btheorems can be useful even inRⁿwith the usual metric.

3.4. Related developments. By now, a lot is known about Calderón–Zygmund operators on non-homogeneous metric spaces. We mention some results that complement the theory above. In [24] Hytönen, Liu, Yang and Yang generalize the results of Nazarov, Treil and Volberg [38] to the upper doubling setting.

Namely, it is shown that the boundedness of a Calderón-Zygmund operator T onL²(µ)is equivalent to that ofT on L^p(µ)for somep∈ (1,∞), and to that ofT fromL¹(µ)toL^1,∞(µ). Moreover, the boundedness of the corresponding maximal operatorT_\is established. The non-atomicity conditionµ({x}) = 0for allx ∈X appears, but this restriction is likely just a consequence of the proof technique.

Some similar results are also considered by Bui and Duong in [8].

Hytönen, Yang and Yang consider in [27] the atomic Hardy spaceH¹(µ) and show that its dual is the space RBMO(µ) in this context. It is also shown that Calderón–Zygmund operators are bounded fromH¹(µ)toL¹(µ).

One can also consider Calderón–Zygmund commutators[b, T]f :=bT f−T(bf) with a fixedb ∈RBMO(µ). Their boundedness and the inequalityk[b, T]fkL^p(µ)≤ Ckbk_RBMO(µ)kfk_L^p_(µ) for everyp∈ (1,∞)in the case of power bounded measures in Rⁿ was shown by Tolsa in [45]. This was extended to metric spaces by Bui and Duong in [8] (still in the case λ(x, r) = r^m). We have proved that they are also bounded in the case of general upper doubling measures in geometrically doubling metric spaces (unpublished note).

4. V^ECTOR-^VALUED C^ALDERÓN–ZYGMUND THEORY

4.1. Background. Typical singular integral operators will not extend boundedly to Banach space valued operators if the Banach space is arbitrary. Indeed, the Hilbert transformH extends boundedly to L^p(R, Y) if and only if (Burkholder [9] and Bourgain [6]) the target space Y has the so called UMD property: there holds that

n

X

k=1

_kd_k

L^p(Ω,Y) ≤C

n

X

k=1

d_k L^p(Ω,Y)

whenever(dk)ⁿ_k=1is a martingale difference sequence inL^p(Ω, Y)andk =±1are constants. This property does not depend on the parameter1< p <∞.

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8 HENRI MARTIKAINEN

As was explained in the overview of this introductory part, the way to prove the existence of such an extensions in the full generality is via a proper vector- valuedT1theorem. The classical version was proved by Figiel [16]. In the non- homogeneous situation inRⁿthis was settled by Hytönen [19]. In the recent work of Müller and Passenbrunner [36], a UMD-valuedT1 theorem is established in metric spaces – however, only with Ahlfors-regular measuresµ(i.e.µ(B(x, r))∼ r^m). The purpose of article [B] is to extend the main result of [A] into the UMD- valued setting, and thus generalize all of the aforementioned results.

4.2. Article [B]. Our main theorem of [B] states that the hypotheses of Theorem 3.1 are enough to guarantee that T: L^p(X, Y) → L^p(X, Y) boundedly for any p∈(1,∞)and any UMD-spaceY. Let us still repeat this in the form of a theorem.

4.1.Theorem. Let(X, d)be a geometrically doubling metric space, and assume that this space is equipped with an upper doubling measureµ. Let Y be a UMD space and1 <

p < ∞. Let T be an L^p(X, Y)-bounded Calderón–Zygmund operator with a standard kernelK,b₁andb₂ be two bounded accretive functions,α >0andκ,Λ >1. Then

kTk.kT b₁k_{BM O}¹_κ_(µ)+kT^∗b₂k_{BM O}¹_κ_(µ)+kM_b₂T M_b₁k_{W BP}_Λ +kKk_CZ_α, wherekTk=kTk_L^p_(X,Y_)→L^p_(X,Y₎.

Notice that this gives the boundedness with any p ∈ (1,∞). Getting this boundedness with any p (in the scalar-valued case) using the weak-type (1,1) result from [24] seems to require the artificial non-atomicity condition. In the general vector-valued case the choicep = 2 does not simplify anything, so this kind of generality is basically forced.

The techniques required to prove Theorem 4.1 are mostly combinations of ideas from [A] and from [19]. Many of the technical problems that arise have to do with the fact that the metric way to randomize cubes is quite different from the way it is done inRⁿ. In fact, several modifications of the randomization procedure introduced in [A] were absolutely necessary for the proof. In one aspect the modified randomization presented by Hytönen and Kairema [22] turned out to be useful.

4.3. Applications. One application, already presented in [B], is as follows: the Cauchy–Szegö projectionC (see Stein [44, pp. 532–543]) defined on the Heisen- berg groupHⁿmapsL^p(Hⁿ, Y)toL^p(Hⁿ, Y)for every UMD spaceY and for every indexp∈(1,∞). This is in the doubling case but in a metric domain – which goes to show that it is convenient to get the doubling theory as a by-product of the upper doubling theory. This question was asked through a private communication by Tao Mei. He had been able to show this in the special case thatY is a so-called non-commutativeL^p space.

5. L^OCALT bTHEOREMS

5.1. Background. First local T b theorem, with L^∞ control of the test functions and their images, is by Christ [10]. This was proven for doubling measures

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(even in metric spaces). Nazarov, Treil and Volberg [39] then obtained a non- homogeneous version of this theorem inRⁿ which also allows BMO control of the images of the test functions underT if the kernel ofT is antisymmetric.

One can also consider more generalL^p type testing conditions. Such a theory is still only available in the homogeneous setting (even though we shall employ many techniques of the non-homogeneous analysis). This branch of localT btheorems was introduced by Auscher, Hofmann, Muscalu, Tao and Thiele [2], but the proof was only for special operators, the so-called perfect dyadic singular integral operators. The assumptions are of the formR

Q|b¹_Q|^p ≤ |Q|, R

Q|b²_Q|^q ≤ |Q|, R

Q|T b¹_Q|^q⁰ ≤ |Q| and R

Q|T^∗b²_Q|^p⁰ ≤ |Q|, where s⁰ denotes the dual exponent of s and1< p, q ≤ ∞.

It has turned out to be surprisingly difficult to extend these theorems to the general Calderón–Zygmund operators – at least if one wishes to maintain the full range of exponentsp, q ∈ (1,∞]. Hofmann [18] was able to extend to general Calderón–Zygmund operators but at the cost of demanding the following stronger set of assumptions: R

Q|b¹_Q|^s ≤ |Q|, R

Q|b²_Q|^s ≤ |Q|, R

Q|T b¹_Q|² ≤ |Q| and R

Q|T^∗b²_Q|² ≤ |Q|for somes > 2. Auscher and Yang [4] establish, by reducing the question to the known case of perfect dyadic operators, the theorem for standard Calderón–Zygmund operators in the case1/p+ 1/q≤1.

There are various other points of interest in this theory. One is the fact that the above proofs are all in some sense indirect. For one, they all rely on the standard localT1theorem. Also, the proof by Auscher and Yang [4] is a reduction to the case of perfect dyadic operators. One of our aims in [C] was to give a new direct proof. Analysis of Hofmann’s proof [18] shows that some of the most delicate problems arise from certain boundary regions of cubes. Such problems are conveniently avoided with the use of non-homogeneous analysis. Actually, our proof gives two results simultaneously. It also generalizes the result of Nazarov, Treil and Volberg [39] to the case of upper doubling measures µ with a quite heavily modified proof. Indeed, regarding non-homogeneous analysis in this local situation, there turned out to to be a previously unnoticed problem with the use of goodness and the implication it may have on the collapse of certain para- products. Our proof circumvents this rather subtle issue.

While giving finishing touches to [C] another direct proof [3] by Auscher and Routin appeared. It also sheds some light to the general case of exponents, however, not giving a definite answer. It involves several new technical conditions, the necessity of which are still unknown. Some skepticism towards the conjecture concerning the full range of exponents is expressed there. Also, this proof operates in the metric space setting. This is related to an interesting feature of localT btheorems with general assumptions: extending them to metric spaces in a completely satisfactory and natural way seems to be a bit of a problem. Indeed, it is unnatural to demand the existence of functionsb_Q in metric cubes, since we have no idea how they concretely look like. Moreover, with theL^p type testing

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10 HENRI MARTIKAINEN

conditions the passage from functions defined in balls to functions defined in cubes is not as easy as one would perhaps think. Such a thing was claimed by Tan and Yan in [47] but, according to Auscher and Routin [3], this issue has been overlooked there.

In any case, in [C] we decided to work in Rⁿ. However, lifting our proof to metric spaces in complete generality using the techniques of [A] would not be difficult if only one would know a way to circumvent the above problem concerning the passage from balls to metric cubes. Related to this, in [3] Auscher and Routin consider certain Hardy type inequalities in metric spaces and some sufficient geometric conditions for the metric space to satisfy these are given.

5.2. Article [C]. With a general upper doubling measure µ, we assume that to every cubeQ⊂Rⁿthere exist two functionsb¹_Qandb²_Qso that there holds

(i) sptb¹_Q⊂Q, sptb²_Q⊂Q;

(ii) kb¹_Qk_L^∞_(µ) ≤C,kb²_Qk_L^∞_(µ) ≤C;

(iii) kT b¹_Qk_L^∞_(µ)≤C,kT^∗b²_Qk_L^∞_(µ) ≤C; (iv) R

Qb¹_Qdµ=µ(Q) =R

Qb²_Qdµ.

We call theseaccretiveL^∞systems. One manages with BMO control in the operator side if the operator has an antisymmetric kernel (see [39] for the case of power bounded measures). We do not detail on this, however.

With a doubling measureν, we use the following weaker set of assumptions:

to every cubeQ⊂Rⁿthere exist two functionsb¹_Qandb²_Q so that there holds (i) sptb¹_Q⊂Q, sptb²_Q⊂Q;

(ii) R

Q|b¹_Q|²dν ≤Cν(Q),R

Q|b²_Q|²dν ≤Cν(Q); (iii) R

Q|T b¹_Q|^sdν≤Cν(Q),R

Q|T^∗b²_Q|^sdν ≤Cν(Q)for some fixeds >2; (iv) R

Qb¹_Qdν =ν(Q) = R

Qb²_Qdν.

We call these accretive L² systems (suppressing from the name the fact that we actually impose the somewhat strongerL^scondition on the operator side).

Our main theorem of [C] may now be stated.

5.1.Theorem. Letµbe an upper doubling measure with a dominating functionλand T: L²(µ) → L²(µ) be a Calderón–Zygmund operator with a standard kernel K. As- suming the existence of accretiveL^∞systems(b¹_Q)and(b²_Q), we havekTk ≤C, whereC depends on the dimensionnand on the explicit constants in the definitions ofλ,K,(b¹_Q) and(b²_Q).

Ifµ=ν for some doubling measureν, then the same conclusion holds assuming only the existence of accretiveL² systems(b¹_Q)and(b²_Q).

So for one thing, the previous theorem is a generalization of the localT b theorem by Nazarov, Treil and Volberg [39] to upper doubling measures. This has some relevance – for example, in [1] Adams and Eiderman state that this theorem could be used in the proof of one their main results (however, they opted for a different proof). Also, the proof is quite different from [39] in parts, and corrects

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what we believe to be a bit of a technical problem in the proof of [39]. The related delicate detail is fully explained in [C]. Moreover, [C] gives a completely new and direct proof of the localT btheorem with L^p type testing conditions. We are pretty sure that with some modifications our techniques could be sharpened to yield the casep=q = 2, but we have not yet checked this carefully. As it stands, this part of our theorem is basically the theorem of Hofmann [18], but with a general doubling measure and with the extra integrability assumptions pushed into the operator side, allowing to demand less integrability from the test functions.

Certainly some new ideas are still needed to establish the theorem with general pand q. We feel that it is aesthetically pleasing to get these two theorems with only one proof.

Moreover, the proof reveals some interesting details of how the martingale difference decomposition (for precise definitions see [C]) works with test functions satisfying onlyL² type testing conditions. A curious detail and a source of trou- ble is that the estimate

X

Q∈D

k(∆_Q)^∗fk²₂ .kfk²₂

is not, in general, true for accretiveL²systems. This is demonstrated by a counter example.

6. SHARP WEIGHTED BOUNDS FOR MAXIMAL TRUNCATIONS T_\ 6.1. Background. TheA₂ conjecture states that the sharp bound

kT fk_L²_(w) ≤C_Tkwk_A₂kfk_L²_(w)

holds for everyL²-bounded Calderón–Zygmund operatorT and weightw∈A₂. The corresponding sharpL^p bound

kT fkL^p(w) ≤CTkwkmax{1,(p−1)⁻¹}

Ap kfkL^p(w)

forp∈(1,∞)follows from the casep= 2by the sharp form of Rubio de Francia’s extrapolation theorem due to Dragiˇcevi´c, Grafakos, Pereyra and Petermichl [13].

Here

[w]_A_p = sup

Q

w(Q)

|Q|

σ(Q)

|Q|

p−1

, 1< p <∞,

where the supremum is taken over all the cubesQ⊂Rⁿandσis the dual weight defined by settingσ=w^−1/(p−1).

After the work of several authors the A₂ conjecture was finally settled in full generality by Hytönen [21]. Let us briefly recall some of the recent developments that lead to the full solution (we also refer the reader to the survey by Lacey [29]

and the introductory parts of the papers by Hytönen [21] and Hytönen, Pérez, Treil and Volberg [26]). Buckley [7] started the sharp weighted theory by proving the corresponding sharp estimate for the Hardy–Littlewood maximal operator.

The case of the Beurling–Ahlfors operator was first dealt with by Petermichl and

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Volberg [43]. Petermichl also settled the case of the Hilbert transform [41] and the Riesz transform on all dimensions [42].

Unified theory encompassing the transformations of Beurling–Ahlfors, Hilbert and Riesz was introduced by Lacey, Petermichl and Reguera [30]. The technique there is to verify the sharp weighted bound for certain Haar shifts. Indeed, this suffices as all of the above operators can be obtained from them by averaging as in the work of Petermichl [41], for example. Vagharshakyan was able to recover all smooth enough, odd, convolution-type Calderón–Zygmund operators in R from dyadic shifts. Using [30] the sharp weighted bound followed for all such operators. We still mention the somewhat different route that is tied to Lerner’s formula [32]. For results and techniques related to this see the papers by Lerner [33] and Cruz–Uribe, Martell and Pérez [11].

After this listing of results, let us attempt to formulate the moral of the story.

A key point is to reduce to dyadic theory by representing a Calderón–Zygmund operator as some average of dyadic operators. However, for example the representation for the Hilbert transform in [41] is based on specific symmetries of the Hilbert transform, and such is the story for all of the representations in the articles prior to [21]. The key step by Hytönen in [21] was to establish a representation for an arbitrary Calderón–Zygmund operator. However, this representation involves Haar shifts of arbitrary complexity (this notion is defined below) un- like the previous representations. Thus, the article [30] by Lacey, Petermichl and Reguera was not enough to conclude as it gives exponential dependence on complexity. The final required step, which is also accomplished in [21], is to sharpen this dependence on the complexity to be good enough to be used in conjunction with the new representation theorem.

Our aim here is to consider the extension of the sharp weighted theory to maximal truncationsT_\defined by

T_\f(x) = sup

>0

Z

|x−y|>

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

Initially, this might sound to be some easy tweak of the above theory. However, this is not the case. Note in particular that the passage toT_\is certainly not going to be some triviality like applying Cotlar’s inequality – indeed, such techniques have no hope to yield the sharp bound. Instead, one has to directly work withT_\ and this is actually quite challenging in many respects. Also, the corresponding sharp weak type bound is considered in [D], and this is new even in the untruncated case.

6.2. Article [D]. We state the main theorem of [D].

6.1.Theorem. ForT anL²(R^d)bounded Calderón-Zygmund operator, there holds kT_\fk_L^p,∞_(w) ≤C_Tkwk_A_pkfk_L^p_(w) , 1< p <2,

kT_\fk_L^p_(w) ≤C_Tkwkmax{1,(p−1)⁻¹}

Ap kfk_L^p_(w) , 1< p < ∞.

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The basic idea of the proof follows the story from above. Indeed, the proof works by reducing to the Haar shifts, which have the form

Sf(x) = X

Q∈D

SQf(x) = X

Q∈D

1

|Q|

Z

Q

s_Q(x, y)f(y)dy,

wheres_Q, the kernel of the componentSQ, is supported onQ×Qandks_Qk_∞≤1. Also, there is a numberκ so thats_Q is constant on the dyadic sub-cubesR ⊂ Q for which `(R) < 2^−κ`(Q). The smallest such κ is called the complexity of the shift. The maximal truncationS\is defined by

S\f(x) = sup

>0

X

Q∈D

`(Q)>

SQf(x) .

The representation of a general Calderón–Zygmund operator in the weak sense as an average of certain Haar shifts [21, Theorem 4.2], or more precisely the somewhat modified version [26, Theorem 4.1], can be shown to imply the pointwise bound

T_\f(x)≤C(T)Eβ

X

(m,n)∈Z²+

2^−(m+n)δ/2(S^βm,n)_\f(x) +C(T)M f(x).

Here the parameterβ is related to the randomization – for everyβ there is a different random dyadic gridD^β to which the corresponding shiftsS^βm,nare related.

ShiftS^βm,n has complexityκ = max(m, n,1). Also, the components(S^βm,n)Qof our shiftsS^βm,n satisfy the important cancellationR

(S^βm,n)_Q = 0except in one particular case of a dual paraproduct. In any case, recalling Buckley’s sharp estimate for the maximal function, things are reduced to the dyadic case: one needs to establish the analog of Theorem 6.1 for Haar shifts with good enough dependence (at most polynomial) on the complexity. Actually, linear dependence is established in [D].

The first part of [D] is then dedicated to proving certain sufficient conditions for the weak and strong type two-weight inequalities for a maximal truncationS^\ of a Haar shiftS. These Sawyer-type conditions (and one new non-standard one) are theT btype part of this theory. We point out that the usage of maximal truncations seems important in these testing conditions. Indeed, even if one would only be interested in the boundedness ofS, one does not, at least presently, know a way to prove testing conditions that do not also involveS\. After these Sawyer type testing conditions have been proved, we are reduced to verifying them with good enough dependence on the complexity and kwk_A_p. One point of interest here is that the proof works with anyp ∈ (1,∞). In particular, no extrapolation is used and the casep= 2is not in a special role.

The verification of these conditions requires certain distributional estimates with exponential decay. In turn, the proof of these distributional estimates uses an unweighted weak type(1,1)estimate for duals L^∗ of all the linearizationsL

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of the maximal truncationS\ofS. Namely, one needs to prove that

|{|L^∗f|>1}|.κkfk₁,

where the inequality should hold uniformly in the choice of the linearizationLof S\. This inequality for the dual of linearizations is much more difficult than the corresponding result forS^\.

We still detail a little on the proof of this weak type (1,1) result. It is based on the size and density kind of thinking just like in the proof of the fact that the Carleson operator (that is related to the pointwise convergence of Fourier series) is bounded onL² (see, for example, the article by Lacey and Thiele [31]).

However, the argument is not equally difficult and some nice simplifications can be made. Forf ∈L²andg ∈L¹ one defines

size_f(Q) := sup

Q∈Q

1

|Q|

Z

Q

|f|²1/2

, dense_g(Q) := sup

Q∈Q

sup

Q⁰⊃Q

1

|Q⁰| Z

Q⁰

|g|

for a collection of dyadic cubesQ. The proof is then based on the restricted weak type formalism splitting certain relevant collections of dyadic cubes inductively into trees with controlled size and density.

6.3. Applications. In [D] the following immediate corollary of the main theorem, which was conjectured by Lerner and Ombrosi [34, Conjecture 1.3] for untruncated operatorsT, is recorded:

6.2.Corollary. For T anL²(R^d) bounded Calderón-Zygmund Operator and1 < q <

p <∞,

kT\fk_L^p_(w) ≤CT ,p,qkwkAqkfk_L^p_(w).

6.4. Related developments. A new twist in the study of the sharp weighted estimates is to involve the smaller characteristic[w]A∞. This is considered by Hytö- nen and Pérez [25], where it is obtained that

kT fk_L²_(w) ≤C_Tkwk^1/2_A

2 max(kwk^1/2_A_∞,kw⁻¹k^1/2_A_∞)kfk_L²_(w).

Extrapolation of this estimate seems unfavourable, though. Thus, the case of a generalpis not so simple. It turns out that there holds

kT_\fk_L^p_(w)≤C_Tkwk^1/p_A

p max(kwk^1/p_A_∞⁰,kw^1−p⁰k^1/p_A_∞)kfk_L^p_(w).

In some special cases this was first proved by Lacey [28], and the general case was settled by Hytönen and Lacey [23]. This sees the return of Lerner’s formula [32], which is used to reduce to positive shifts. For these, the non-standard testing condition of [D] can be controlled with better bounds.

As pointed out by Hytönen, the techniques of [21], where theA₂ conjecture is proved inRⁿ, are such that they, in light of the techniques of [A], should extend to metric spaces. One detail is that the representation theorems [21, Theorem 4.2]

and [26, Theorem 4.1] use only one grid instead of two. A modification of the randomization presented in [A] is considered by Nazarov, Reznikov and Volberg in [37], and there the metricA2is also proved.

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