ADJACENT AND RANDOM DYADIC SYSTEMS AND THEIR APPLICATIONS TO METRIC, EUCLIDEAN AND
VECTOR-VALUED ANALYSIS
OLLI TAPIOLA
Academic dissertation
To be presented for public examination,
with the permission of the Faculty of Science of the University of Helsinki, in Auditorium XIV, Fabianinkatu 33, on 21st May, 2016, at 10.00
Department of Mathematics and Statistics Faculty of Science
University of Helsinki 2016
ISBN 978-951-51-2118-9 (paperback) ISBN 978-951-51-2119-6 (PDF) Unigraa Oy
Helsinki 2016
Contents
Acknowledgements i
List of included articles iii
1. Overview 1
2. Preliminaries on dyadic analysis on metric spaces 2
2.1. Background . . . 2
2.2. Quasimetric spaces . . . 3
2.3. Geometrically doubling metric spaces . . . 3
2.4. Spaces of homogeneous type and doubling measures . . . 4
2.5. Christ-type dyadic systems in metric spaces . . . 5
3. Random dyadic systems in metric spaces 7 3.1. Background . . . 7
3.2. Paper [A] . . . 7
3.2.1. General randomization procedure for metric dyadic systems 7 3.2.2. Independent random dyadic systems . . . 8
3.2.3. Almost Lipschitz-continuous wavelets in metric measure spaces . . . 9
3.3. Related developments and open problems . . . 10
4. Adjacent dyadic systems in metric spaces 11 4.1. Background . . . 11
4.2. Paper [C], and Paper [A] revisited . . . 12
4.2.1. Metric adjacent dyadic systems via randomization . . . 12
4.2.2. Tools for decomposing dyadic systems in metric spaces . . 13
4.2.3. TheLp-boundedness of shift operators in metric spaces . . 14
4.3. Related developments and open problems . . . 14
5. Muckenhoupt weights and quantitative weighted norm estimates 14 5.1. Background . . . 14
5.2. Paper [B] . . . 17
5.2.1. WeakA∞weights via adjacent dyadic systems . . . 18
5.2.2. Related developments and open problems . . . 18
5.3. Paper [D] . . . 19
5.3.1. TheA2 theorem and domination by sparse operators . . . 19
5.3.2. Weighted estimates for rough homogeneous singular integrals 21 5.3.3. Related developments and open problems . . . 22
References 23
Acknowledgements
The past is a strange place. About three weeks before I started to work on this project, I was planning to leave the academic world. Instead, I ended up wandering around in weird good and bad rooms full of weights, singular integrals and of course dyadic cubes. The last four years were not easy but I am glad that I got to experience them.
This dissertation would not exist without my advisor Tuomas Hytönen. He intro- duced me to the fantastic world of dyadic harmonic analysis, taught me a great deal about being a mathematician and remained positive about this project even when I was feeling insecure. He always seemed to have time for my questions and he never made me feel embarrassed about my mistakes but encouraged me to learn from them.
For all of this, I am deeply grateful to him. I consider myself highly privileged having been one of his students.
I wish to thank my pre-examiners, Professor María Cristina Pereyra and Assistant Professor Kabe Moen, for taking the time to read through the manuscript and for all of their encouraging words and helpful comments. I would also like to thank Luz Roncal and Tess Anderson for mathematical collaboration and Professor David Cruz-Uribe for accepting to act as the opponent during the public examination of this dissertation.
Financially, this work was supported by The Academy of Finland through the con- sortium 133264 Stochastic and Harmonic Analysis: Interactions and Applications in 2012 and the European Union through T. Hytönen's ERC Starting Grant Analytic- probabilistic methods for borderline singular integrals in 2013-2016. During this time, I worked in the Helsinki Harmonic Analysis Research Group which is a part of the Finnish Centre of Excellence in Analysis and Dynamics Research.
Over these last eight years in the academic world, I have met dozens of amazing people, many of whom have helped me with various things related to my studies, research and everyday life. In particular, I wish to thank my 'brother-in-arms' Esko Heinonen for several hundred lunches and countless conversations about mathematics and life in general. Rami Luisto and Terhi Hautala helped me with mathematical and practical matters so many times that I lost the count years ago; I thank both of them for this. Also, I want to thank Professor Steve Hofmann for hosting my visit to University of Missouri during January - March 2016.
Outside the academic world, my family and friends helped me to get through stressful times and to recover from academic setbacks and disappointments. I thank all of them for their support.
Lastly, I want to thank my dear friend Emilia Järvelä for wine, tortillas and brunch- ners, and for helping me to stay sane enough over the last 14 years.
Helsinki, April 2016
Olli Tapiola
List of included articles
This thesis consists of an introductory part and the following four research articles:
[A] T. Hytönen and O. Tapiola, Almost Lipschitz-continuous wavelets in met- ric spaces via a new randomization of dyadic cubes. J. Approx. Theory, 185:1230, 2014.
[B] T. C. Anderson, T. Hytönen and O. Tapiola, Weak A∞ weights and weak Reverse Hölder property in a space of homogeneous type. To appear in J.
Geom. Anal. (2016), arXiv:1410.3608.
[C] O. Tapiola, Adjacent dyadic systems and the Lp-boundedness of shift op- erators in metric spaces revisited. To appear in Colloq. Math. (2016), arXiv:1504.01596.
[D] T. Hytönen, L. Roncal and O. Tapiola, Quantitative weighted estimates for rough homogeneous singular integrals. To appear in Israel J. Math. (2016), arXiv:1510.05789.
Author's contribution
The research presented in this dissertation was done at the Department of Mathe- matics and Statistics at the University of Helsinki during the period 2012 - 2015.
The Article [A] is based on the author's master's thesis which was completed in 2012 under the supervision of T. Hytönen. The authors had equal roles in the analysis and writing of the paper. In Article [B], the author had the central role in the analysis and he wrote the article aside from parts of Section 7. The Article [D] is an equal collaboration of three people. The author had a central role in the analysis and writing of Section 2.
1. Overview
Active research of harmonic analysis started as the study of questions related to expressing a real-valued function living in the Euclidean space with the help of suitable building blocks. Originally, in the groundbreaking work of J. Fourier (1768 - 1830), these building blocks were sine and cosine waves, but in later extensions of Fourier's theory various other types of base functions have been used alongside these waves.
Since these types of questions are intertwined with dierent decomposition techniques, it has been crucial to nd customizable ways to partition the set Rn and its subsets into smaller sets. For this, the standard system of dyadic cubes D,
D:=2−k([0,1)n+m) :k∈Z, m∈Zn ,
has been a simple yet powerful tool: with its help it has been easy to construct such fundamental objects as Haar functions and to prove various important results like the Calderón-Zygmund decomposition. The system provides a trivial way to partition Rn into sets that are shifted copies of each other and that can be partitioned further into smaller sets from the same system.
In modern-day harmonic analysis, similar questions are still valid but they are un- derstood in a broader sense. Developments in the eld have led to e.g. considering vector-valued versions of many classical results, studying both qualitative and quan- titative mapping properties of singular integral operators and exploring connections between harmonic analysis and other branches of mathematics, such as geometric measure theory. The standard system of dyadic cubes has had an important role also in these developments, but many modern problems have seen the need to develop generalized versions of the collection D and techniques that involve several or even innitely many dierent systems of dyadic cubes. Three types of techniques have been under particular interest:
(i) generalized dyadic techniques suitable for analysis outside the Euclidean set- ting;
(ii) adjacent dyadic techniques, i.e. techniques involving a bounded number of dierent dyadic systems;
(iii) random dyadic techniques.
During the last couple of decades, all of these techniques have established a place in the toolkit of harmonic analysis and are the only known or, by far, the simplest known techniques to solve some types of problems. Adjacent dyadic techniques have been useful in homogeneous analysis since they oer an eective way to approximate balls with cubes; they have been used e.g. to simplify the proof of the A2 theorem in Rn [71] and to develop Sawyer-type testing conditions for certain integral operators in spaces of homogeneous type [62]. Random dyadic techniques, in turn, have been irreplaceble in non-homogeneous analysis of both Rn and metric spaces due to the way they can be used to control chaotic boundary eects [89, 54]. We discuss these techniques in Sections 3 and 4.
In this dissertation, we both improve random and adjacent dyadic techniques in metric spaces and apply these types of techniques to metric, Euclidean and vector- valued analysis. In Paper [A], we present a general randomization procedure for dyadic systems in metric spaces which can be customized for dierent purposes. This procedure both unies the construction of metric random dyadic systems of T. Hytö- nen and H. Martikainen [54] and metric adjacent dyadic systems of T. Hytönen and A. Kairema [51] while improving the properties of both of these systems. As an ap- plication of the new random systems, we improve the continuity properties of metric
wavelets of P. Auscher and T. Hytönen [5] by exploiting the improved smallness of boundary property of our random cubes. We prove some additional properties for our adjacent dyadic systems in Paper [C] to prove a decomposition result for dyadic systems in metric spaces. This result allows us to split a dyadic system into a bounded number of subcollections that can be conveniently embedded inside some other dyadic systems. With the help of this decomposition result, we give an al- ternative proof for the quantitative bound of the Lp norm of shift operators acting on vector-valued functions in metric spaces. This Lp bound was rst proved by T.
Figiel in the Euclidean setting [34, 35] and P. Müller and M. Passenbrunner in normal spaces of homogeneous type [86].
Adjacent dyadic systems are central also in Papers [B] and [D]. In Paper [B], we explore certain properties of the MuckenhouptA∞ weight class, the class of Reverse Hölder weights and their weakened versions in spaces of homogeneous type. In the Euclidean setting, theA∞class has numerous dierent equivalent denitions, one of which is the Reverse Hölder property, but in spaces of homogeneous type some of those equivalences break down. We show that although certain denitions are no longer equivalent in spaces of homogeneous type, their weakened versions still dene the same weight classes. In particular, we show that every Fujii-Wilson-type weak A∞ weight satises a weak Reverse Hölder property and that all the weak Reverse Hölder weights have a self-improving property. In the literature, these types of weak inequalities have been used especially in the theory of partial dierential equations.
Although the denition of the weak Fujii-Wilson A∞weights is purely non-dyadic, we can use adjacent dyadic systems to give an alternative characterization for these weights. Covering arguments are not as trivial with several dierent dyadic systems as they are with a single one but since the number of the systems is bounded and each of the systems is nested, the adjacent dyadic techniques work almost as well as the corresponding single-system techniques. Using this approach not only shortens the proofs but it also makes them more straightforward.
We consider Muckenhoupt weights also in Paper [D] in the form of weighted norm inequalities in the Euclidean space. To be more precise, we prove quantitative weighted bounds for so called rough homogeneous singular integrals. In qualitative sense, the weighted theory of these operators has been studied by J. Duoandikoetxea, J. L. Rubio de Francia and D. K. Watson in the 1980's and 1990's [32, 31, 104]. We use modied and applied versions of the techniques of Duoandikoetxea and Watson and combine them with a fully quantitative version of M. Lacey's recent extension of theA2theorem [68]. The proof of this extension is based on the domination technique of A. Lerner [70, 71] which provides a way to dominate Calderón-Zygmund operators pointwise with the help of a nite number of simple sparse operators associated with adjacent dyadic systems. We use the original ideas of Lacey as a starting point of the proof of the extension but we have to impose additional ideas to prove it in a fully quantitative form. We present a careful proof of the theorem but apply it as a black box for the rough homogeneous singular integrals.
2. Preliminaries on dyadic analysis on metric spaces
2.1. Background. Nested structures and partitions of sets are important in mathe- matical analysis since they allow us to analyze the global behaviour of some objects via their local behaviour on dierent types of sets. For example, a function may behave very dierently in dierent parts of the space so partitioning the space to dierent parts according to the behaviour of the function helps us to analyze its
2
properties. This is why the standard system of dyadic cubes D, D={2−k([0,1)n+m) :k∈Z, m∈Zn},
is an inseparable part of modern-day harmonic analysis of the Euclidean space Rn. It is countable and nested, cubes of a xed side length partition the space and the sim- plicity of the system makes it customizable for many types of situations. It has been applied thoroughly in classical and modern analysis for dierent types of questions related to e.g. harmonic analysis, functional analysis and geometric measure theory.
Since the collection D is not suitable for analysis outside the space Rn, there has been a need to develop similar tools for other spaces as well. However, this has not been a trivial task since the structure of the collection D depends so highly on the geometry of the Euclidean space. Constructing systems that resemble the collection D is not elementary but they can be constructed in very general spaces, namely geometrically doubling quasimetric spaces. We discuss these spaces and the general systems in the subsequent sections.
2.2. Quasimetric spaces. We say that a nonempty set Xequipped with a function d:X ×X → [0,∞) a quasimetric space if the function d satises the axioms of a metric except for the triangle inequality which is assumed to hold in the following weaker form: for some κ≥1 and everyx, y, z∈X we have
d(x, y)≤κ(d(x, z) +d(z, y)).
From many technical points of view, quasimetric spaces behave similarly to metric spaces and we only need to be careful about tracking the dependencies of certain bounds on the constantκ. However, their structure is not suitable for dierent types of delicate estimates due to some undesired properties: if B(x, r)is an open ball in a quasimetric space (X, d), then
• there may exist pointsy, z∈B(x, r)such thatd(y, z)>2r;
• B(x, r)is not necessarily an open set.
We can modify the example given in [51, Section 2.1] to construct a quasimetric space that satises both of these properties. Let us set X = (−∞,−1]∪ {0} ∪[1,∞), d(−1,0) = d(0,1) = 1/4 and d(x, y) = |x−y| for all other pairs (x, y) ∈ X ×X. Then the open ball B(0,1/2) ={−1,0,1}is not an open set asB(1, ε)⊂B(0,1/2) for every ε >0 andd(−1,1) = 2>2·1/2.
Despite these problems, the behaviour of quasimetric spaces cannot be too wild compared to the behaviour of metric spaces. To be more precise, the classical results of R. Macías and C. Segovia [78] (see also [82, Theorem 1.2] for more remarks about these results) say that the topology a quasimetric induces is always metrizable and for every quasimetricdthere exists a quasimetricdsuch that opend-balls are open sets andc1d(x, y)≤d(x, y)≤c2d(x, y)for some constantsc1 andc2 and everyx, y∈X. These results are standard tools that have been used widely in analysis of spaces of homogeneous type (see e.g. [17, 86, 56, 3]).
2.3. Geometrically doubling metric spaces. We say that a (quasi)metric space (X, d)is geometrically doubling if there exists a constant M ≥1(called the geomet- rical doubling constant of X) such that every ball of radius 2rcan be covered by at most M balls of radius r. In the literature, these spaces are often referred to simply as doubling metric spaces [43, Section 10.13], the geometrical doubling property is sometimes called the homogeneity property of X [20, Chapitre III] or nite doubling property [60], and numerous alternative characterizations for these spaces exist (e.g.
spaces with nite Assouad dimension, (C, s)-homogeneous spaces and HTB-spaces
Rn
S
e S e(x, y) n x
S
y n x ∈S
k∈N #B(x,2k)>2k·#B(x, k) k∈N
2k k 2k
(S, e)
(S, e) (N,| · |)
Rn
(X, d) μ
D ≥1 r > 0 x ∈ X
0< μ(B(x,2r))≤Dμ(B(x, r))<∞.
We call these types of measures doubling measures.
Coifman and Weiss introduced these spaces to give a natural extension to the Euclidean Calderón-Zygmund operator theory since the condition (2.2) allows us to use many Euclidean-like techniques in this context2. However, this does not mean that all spaces of homogeneous type behave essentially like subsets of the Euclidean space, but the class of these spaces contains e.g. certain Riemannian manifolds and locally compact groups [21, Section 2]. A characterization lemma by A. Kairema gives some kind of idea about the general structure of these spaces:
Lemma 2.3 ([63, Lemma 4.2]). Suppose that (X, μ) is a space of homogeneous type and #X =∞. Then precisely one of the following is satised:
1) μ(X)<∞;
2) X is countably innite and μ({x})≥δ >0 for allx∈X;
3) μ(B)can have arbitrarily small and large values with balls B.
For example, ([0,1], dx)is a space of type 1) and (R, dx)is a space of type 3). As for type 2), every nitely generated group with polynomial growth equipped with the counting measure is a space of homogeneous type. A famous theorem by M. Gromov characterizes these groups as the groups which have nilpotent subgroups of nite index [40]. We also note that if the space (X, d, μ)is complete, then μis a Radon measure and(X, d)is locally compact [101, Theorem 2.3 (1)].
Spaces of homogeneous type are geometrically doubling simply by the condition (2.2) [20, Chapitre III], but geometrical doubling condition is genuinely more general than the doubling condition of the measure: there exist geometrically doubling metric spaces that do not carry any doubling measures [43, Remark13.20 (d)]. However, if a geometrically doubling metric space is complete, then it does carry a doubling measure [77].
In the literature, there are also dierent types of other doubling conditions imposed on spaces or measures, such as the normality condition of spaces of homogeneous type [78] or the reverse doubling condition: for some C≥1 andε >0we have
μ(B(x, Cr))≥(1 +ε)μ(B(x, r))
for every x ∈X andr∈ (0,diam(X)/C). The reverse doubling condition is useful with various dierent techniques (see e.g. [107]) but it is restrictive since it excludes the presence of point masses and empty annuli [102, Chapter I, Lemma 20]. It is also often unnecessary (compare e.g. [99, 103] and [62]). For more information and references, see e.g. the introduction of [5].
In our results, the only doubling conditions that we consider are the geometrical doubling condition and the measure doubling condition (2.2). Also, we do not impose any additional common assumptions on our spaces, such as μ(X) = ∞, no point masses or small boundary property for balls.
2.5. Christ-type dyadic systems in metric spaces. After Coifman and Weiss introduced spaces of homogeneous type, many of the existing Euclidean results were generalized to these spaces but the lackof dyadic systems in these settings was a challenge (see e.g. the introduction of Chapter I in [102]). This inspired various authors to construct similar nested systems in more abstract settings for dierent purposes in the 1980's.
2This theory has an extension also for non-doubling measures (see e.g. [53] for some extensions, basic denitions and many references). In this dissertation, we do not consider this theory in detail.
Denition 2.4. Let(X, d)be a geometrically doubling (quasi)metric space. A col- lection of Borel setsD,
D :=
k∈Z
Dk, Dk:={Qkα:k∈Z, α∈Ak}, Ak⊆N,
is a dyadic system (of parameters δ∈(0,1) and 0< c1 ≤C1 < ∞) if the following properties hold:
• the collectionD is nested: ifQ, P ∈D, then Q∩P ∈ {∅, Q, P},
• X =
Q∈DkQfor every k∈Zand the union is disjoint,
• the setsQkαare not too thin or wide:
B(zkα, c1δk)⊆Qkα⊆B(zkα, C1δk) =:BQkα, (2.5)
• for every setQkα there exists at mostN setsQk+1βi such thatQkα=N
i=1Qk+1βi , where the constantNdepends only on the spaceX. We call the setQkαa dyadic cube with a center pointzαk and side length (Qkα) = δk. If Qkα ⊂Qlβ, then we call Qkα a dyadic descendant of Qlβ of side length δkandQlβ the dyadic ancestor of Qkα of side lengthδl. Ifl=k−1, then we callQkαa dyadic child ofQlβandQlβ the dyadic parent ofQkα.
The geometrical doubling property is an important assumption since it makes sure that the numberN is bounded.
First constructions of this type were due to H. Aimar and R. Macías [1] and G.
David [26] but the rst complete construction for spaces of homogeneous type was due to M. Christ [17]; see also an alternative construction by E. Sawyer and R.
Wheeden [96]. T. Hytönen and H. Martikainen [54] and T. Hytönen and A. Kairema [51] generalized the result later for geometrically doubling quasimetric spaces with no measurability assumptions (see also [60]). We note that metric dyadic construc- tions have been useful also outside purely metric settings with questions related to Riemannian manifolds [84] and uniform rectiability inRn [45].
None of the constructions mentioned above are canonical since they are based on choosing rst the center points of the cubes and then creating a suitable non-unique parent-child relation for the center points. This changes the nature of dyadic cubes when comparing them in the Euclidean setting and a metric setting. In the Euclidean setting, cubes are, in a way, more natural objects than balls: for example, the be- haviour of the Lebesgue measure and then-dimensional Riemann integral is simple on cubes and requires some additional work on balls. In a metric setting, the situation is vice versa: even in Rn, the metric construction of dyadic cubes produces abstract sets that are usually very dicult to express with the help of unions, intersections and Cartesian products. While working with metric dyadic cubes, it is important to be careful that the result does not depend on the choice of the particular dyadic system.
One of the most useful properties of dyadic cubes is that many covering arguments related to them are completely trivial just by the nestedness property. This and other properties can be used in many powerful techniques, such as the stopping cube technique. Usually dyadic denitions of dierent objects are weaker than their general counterparts (compare e.g. the Hardy-Littlewood maximal operator and the dyadic Hardy-Littlewood maximal operator) but in some cases the general results can be recovered from the dyadic cases (see e.g. the well-known article by J. Garnett and P.
Jones [39]).
6
3. Random dyadic systems in metric spaces
3.1. Background. Like we mentioned earlier, dierent types of decomposition tech- niques are essential in harmonic analysis since presenting sets and functions with the help of simple or reasonably well-behaving building blocks makes it considerably easier to use simultaneously various dierent types of techniques to analyze many objects. In particular, dierent Calderón-Zygmund-type decompositions, which help us to present dierent objects as a combination of a good part and a bad part, have been widely explored and applied in the literature. However, in some situa- tions (particularly in non-homogeneous analysis) the analysis of the bad parts is very challenging using classical techniques. To overcome this problem, F. Nazarov, S. Treil and A. Volberg approached it from a surprising angle: instead of analyzing the bad parts, they showed that in certain situations they occur so rarely that we can basically ignore them [89, 90]. Their approach was based on randomizing dyadic systems, i.e. constructing inntely many dierent dyadic systems and equipping the sample space with a suitable probability measure. This technique allows us to use probabilistic arguments in the proofs of many non-probabilistic results. During the last two decades, this technique has been crucial in many advances in harmonic anal- ysis, particularly in the development of non-homogeneous theory of singular integrals and theory of weighted norm inequalities both in the Euclidean setting and metric spaces (see e.g. [49, 79]).
A simple way to randomize the standard system of dyadic cubes is to equip the set Ω := ({0,1}n)Zwith the natural probability measure Pand set
D(ω) :=
2−k([0,1)n+m) +
j>k
2−jωj: k∈Z, m∈Zn
(3.1) for every ω := (ωj)j∈Z ∈ Ω. However, since this construction is based on shifting arguments, it is dicult to apply it for a setting where the underlying space is not a vector space. T. Hytönen and H. Martikainen, who were the rst to present a randomization technique for general metric spaces in [54], created an analogue for shifting by randomizing the choice of the center points of the dyadic cubes. Their construction was simplied in [51] and modied in [91] but these constructions use the same basic idea of Hytönen and Martikainen.
3.2. Paper [A]. In Paper [A], we provide a new way to randomize dyadic systems in metric spaces and apply these systems to improve the continuity properties of the metric wavelets constructed by Auscher and Hytönen [5]. We have kept our presentation simple but it is still easy to apply our randomization techniques for other situations, such as the construction of adjacent dyadic systems that we will discuss in Section 4.
3.2.1. General randomization procedure for metric dyadic systems. The central idea behind our randomization technique is to keep the center points of the cubes xed and randomize the sizes of the cubes instead. To be more precise, we randomize the constants c1 and C1 in Denition 2.4 in the following way: we write the condition (2.5) in a form
B(zαk, rak)⊆Qkα⊆B(zαk, Rak),
whererak :=14(δk+akδk+1),Rak := 4rkandak: Ω→Y ⊂R+is a random variable for everyk∈Z. This does not only change the sizes of the cubes but also the parent-child
relation changes drastically, which can be seen by going through the construction in a little more detail.
This type of randomization leaves several parameters free: we can choose the prob- ability space(Ω,P), the random variablesakand the values of the random variables the way we want. In [A], we only need two variants of discrete randomization; we apply the randomization only for Y = {0,1, . . . ,1/δ}. We discuss these variants below and in Section 4.2.1.
3.2.2. Independent random dyadic systems. We use our randomization procedure to construct random dyadic systems that have a better smallness of boundary property than the systems in [54, 51, 91]. We choose
Ω =0,1, . . . ,1δZ
, Y =0,1, . . . ,1δ
, ak((ωi)i∈Z) =ωk,
and we letPbe the natural product probability measure of Ω. This gives us dyadic systemsD(ω) :={Qkα(ω): k∈Z, α∈Ak}that have been randomized independently for each level k∈Z. These independent random dyadic systems have the following property: denoting
∂εA:={x∈A:d(x, Ac)< ε} ∪ {x∈Ac:d(x, A)< ε}, we have
Theorem 3.2 (Theorem 5.2 in [A]). Let (X, d) be a geometrically doubling metric space with a doubling constantM and letε >0. Then for independent random dyadic systems with δ <84M1 8 it holds that
Pω x∈
α
∂εQ¯kα(ω)
≤Cδ
ε δk
ηδ
, where
Cδ:= 1
δ and ηδ:= 1−log 84M8 log1δ . In particular,limδ→0ηδ= 1.
The value of the boundary exponent ηδ has not had an important role in most applications related to random dyadic cubes, but in the wavelet constructions of P. Auscher and T. Hytönen it is directly linked with the continuity properties of the wavelets (see Section 3.2.3). We note that there cannot exist random dyadic systems with the same properties as in Theorem 3.2 in general quasimetric spaces, since otherwise the construction of Auscher and Hytönen would imply the existence of η-Hölder-continuous wavelet bases in these spaces for every η ∈ (0,1). This is impossible due the next lemma:
Lemma 3.3. For every η∈(0,1), there exists such a quasimetric space (X, ρ)that for everyν∈(η,1)every ν-Hölder-continuous function f:X→R is constant.
Proof. Notice thatν-Hölder-continuous functions f:R → R are constant for every ν > 1 since they are dierentiable with zero derivative. Thus, if we equip R with a quasimetricρ,ρ(x, y) = |x−y|η, η ≥1, then every ν-Hölder-continuous function
f: (R, ρ)→Ris constant forν > 1η.
8
3.2.3. Almost Lipschitz-continuous wavelets in metric measure spaces. Wavelets are fundamental objects in both theoretical and practical mathematics since they can be used e.g. to explore various questions related to function spaces [81] as well as to solve many problems related to image processing. In the Euclidean case, the strong geometrical structure of the underlying space is a signicant aid in wavelet constructions that is not available in more general settings. However, in many cases dyadic systems oer a convenient starting point for wavelet constructions since the nestedness property and the property (2.5) makes it simple to construct functions with bounded support that can be expressed as a sum of functions with disjoint supports, e.g. 1Q =
i1Qi where the cubes Qi are the dyadic children of the cube Q. Yet, if we want to construct functions with some additional continuity properties, then the question becomes signicantly more complicated.
In [5] (see also the addendum [6]), P. Auscher and T. Hytönen construct orthonor- mal Hölder-continuous wavelet bases for general spaces of homogeneous type by con- structing rst systems of Hölder-continuous spline functions. Spline functions have been used in Euclidean analysis for a long time but since their construction has relied strongly on the properties of Rn, not many generalizations have been available in literature outside some specic settings like S2 and S3 [24]. If the spline functions are available, then a general procedure of Meyer [81] can be applied also in spaces of homogeneous type to construct spline wavelets. Let us recall the denitions from [5]:
Denition 3.4. A set of continuous functions skα: X →[0,1] is a system of spline functions if the following properties are satised for some points Zk:={zαk}α, where Zk⊆Zk+1⊆X for allk, and for constants δ∈(0,1)andC > c >0:
1) 1B(zk
α,cδk)(x)≤skα(x)≤1B(zk
α,Cδk)(x),
2)
αskα(x) = 1, 3) skα(x) =
βpkαβ·sk+1β (x), 4) skα(zβk) =δαβ,
where {pkαβ}β is a nite set of nonnegative coecients with
αpkαβ= 1. The indices k andαrun respectively over ZandNifX is unbounded, or over{k∈Z:k≥k0} and {0,1, . . . , nk}for some nitek0 ∈Zand nk ∈NifX is bounded. We say that the splines are Hölder-continuous of exponent η if
|skα(x)−skα(y)| ≤C
d(x, y) δk
η . Ifη= 1, we call the splines Lipschitz-continuous.
Denition 3.5. A set of functions ψkα : X → R is a basis of wavelets with - localization, where : [0,∞) → [0,1] is a non-increasing function, if the following properties are satised for some points Yk :={ykα}α ∈ X and constants δ∈ (0,1) andC >0:
ˆ
ψαk(x)dμ(x) = 0, |ψkα(x)| ≤ C
μ(B(yαk, δk))d(x, yαk) δk
, and the functions ψαk form an orthonormal basis ofL20(μ), where
L20(μ) :=
L2(μ), ifX is unbounded,
f ∈L2(μ) :´
Xf(x)dμ(x) = 0
, ifX is bounded.
The indiceskandα run over similar sets as in the case of splines. We say that the -localized wavelets are Hölder-continuous of exponent η if
|ψkα(x)−ψkα(y)| ≤ C
μ(B(ykα, δk))d(x, ykα) δk
d(x, y) δk
η
. Ifη= 1, we call the wavelets Lipschitz-continuous.
The clever idea of Auscher and Hytönen is to use random dyadic cubes to smooth the behaviour of indicator functions near the boundaries of cubes: they set
skα(x) :=P(x∈Q¯kα(ω)),
where Qkα(ω), ω ∈ Ω, are the cubes from the random dyadic systems D(ω). These splines satisfy properties 1)-4) in Denition 3.4 and moreover, they are η-Hölder- continuous for the exponentηgiven by Theorem 3.2. In the construction of Auscher and Hytönen, theη-Hölder-continuity of the splines implies the η-Hölder-continuity of wavelets. Thus, the construction and Theorem 3.2 give us the following result:
Theorem 3.6 (Corollaries 6.8 and 6.13 in [A]). Let(X, d)be a geometrically doubling metric space and letμ be a doubling Borel measure on X. Then for every η∈[0,1)
• there exists a system of η-Hölder-continuous spline functions in (X, d),
• there exists a basis of η-Hölder-continuous wavelets in (X, d, μ)such that we have (x) = exp(−γx)for someγ >0.
We want to emphasize that our contribution in this result is related to the exponent η: the results of Auscher and Hytönen says that Theorem 3.6 holds in quasimetric spaces for some (usually small) η ∈ [0,1). Thus, we strenghten properties of the metric wavelets without working with the reverse doubling condition that is present in many articles related to abstract wavelet theory, e.g. [41]. Our improvement should be useful for wavelet theory and its applications in metric spaces like we explain in the introduction of the Paper [A]. We note that our improvements cannot hold in an arbitrary quasimetric space due to Lemma 3.3.
3.3. Related developments and open problems. Although we strengthen the smallness of boundary property of metric random dyadic systems in the Paper [A], our results leave one obvious question open:
Open problem 3.7. Does there exist random dyadic systems with boundary exponent 1in every geometrically doubling metric space?
If the answer to this question is yes, then we can use the construction of Auscher and Hytönen to give a positive answer to another open question: Does there exist a basis of Lipschitz-continuous wavelets in every geometrically doubling metric space?
However, even if no such metric random dyadic systems exist, there may exist another way to construct a system of Lipschitz-continuous spline functions or some non-spline- type Lipschitz-continuous wavelets. In their original form, the wavelets of Auscher and Hytönen have been applied for various questions e.g. in [16] and [36].
Our randomization procedure was already found useful in [42], where the authors applied the randomization only for a nite number of generations to study the con- nection between BMO and dyadic BMO in metric spaces in the same sense as J.
Garnett and P. Jones [39].
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4. Adjacent dyadic systems in metric spaces
4.1. Background. Let Dα, α = 1,2, . . . , K = K(X)be a nite number of dyadic systems in a geometrically doubling (quasi)metric space (X, d). We say that the systems Dα are adjacent dyadic systems if for every ball B := B(x, r)there exists a cube QB ∈
αDα such that B ⊆ QB and (QB) ≈r. More generally put, the systemsDαare adjacent dyadic systems if they approximate the collection of all balls well. Since every system Dαis countable, the union of these systems is countable as well.
The philosophy behind using techniques involving adjacent dyadic systems is not new but it is familiar from both formal and natural sciences: instead of working with a complicated problem, we show that the problem is approximately the same as some simpler problem, i.e. we only lose a small amount of information if we work with a simpler problem. In our situation, the complicated element is the collection of all balls. This collection is usually uncountable and quite chaotic with respect to intersections, which makes covering and decomposition arguments both necessary and tricky in many situations related to this collection. Adjacent dyadic systems can help us to simplify these problems since they allow us to use modied dyadic techniques to solve them.
These types of adjacent systems have appeared in various forms in the analysis of the Euclidean space since the early 1980's but their origin is not widely known.
In some papers they are attributed to J. Garnett and P. Jones [67, Section 2.2], M.
Christ [88, Section 5] or both [22] but these papers do not refer to any specic articles of these authors. According to the investigations of D. Cruz-Uribe [23, Section 3], these systems appear in a weaker form in [15, Lemma 3.2] and the underlying tech- nique called the one-third trick appears in [106, Lemma 1.4], but the rst complete construction seems to be due to K. Okikiolu in [92, Lemma 1] for K= 2n. Later, T.
Mei [80] and J. Conde [22] have shown that it suces to take n+ 1carefully chosen systems to obtain the desired approximation properties. In geometrically doubling quasimetric spaces, T. Hytönen and A. Kairema were the rst ones to construct these types of systems [51, Theorem 4.1].
In Rn, it is simple to construct adjacent dyadic systems that satisfy even some additional approximation properties: we can simply set
Dα:=2−k[0,1)n+m+ (−1)kα:k∈Z, m∈Zn
(4.1) for every α∈ {0,1/3,2/3}n. These systemsDα satisfy the following result:
Lemma 4.2 ([52, Lemma 2.5]). For every ball B:=B(x, r)and every m∈Nthere exists a cube QB∈
αDα such that i) B⊂QB;
ii) 2mB⊂Q(m)B ; iii) 3r≤(QB)<6r,
where Q(m)B is the dyadic ancestor of QB of side length (Q(m)B ) = 2m·(QB). There are couple of things worth noticing about this result. First, the properties i) and ii) can be achieved with only one dyadic system (we can simply choose Dαfor α= (1/3,1/3, . . . ,1/3)) but these kinds of systems are rarely useful for applications.
Second, it is simple to modify the delightfully elementary proof of [52, Lemma 2.5] to prove that the properties i) and iii) hold for the collections Dα,α∈ {0,1/3}n. Third, the property ii) is not usually needed in the applications [67, 68, 71, 87, 88, 92] but in some situations it is very convenient or even crucial (see [50, Section 6], [52, Section
2] and Section 4.2.2). Lastly, although it is not immediate to see, the systemsDαare a special case of the random dyadic systems (3.1). This can be seen by constructing sequences(xk)and(yk)of powers of2 such that
kxk= 13 and
kyk= 23.
In metric spaces, constructing adjacent dyadic systems is far more complicated than constructing them inRnfor the same reason we mentioned earlier: it is dicult to shift objects when the underlying space is not a vector space. However, the idea of T. Hytönen and H. Martikainen that we discussed in Section 3 is valid also with this problem: we can construct sets of points that act, in a way, as the center points of the original and shifted cubes. T. Hytönen and A. Kairema [51] used this idea and created suitable selection rules to construct the desired adjacent systems. They also show that their systems are closely related to the metric random dyadic systems of T. Hytönen and H. Martikainen [51, Theorem 6.1] [54].
Metric adjacent dyadic systems have been applied for e.g. one- and two-weight norm inequalities [63, 62, 3] and questions related to tent spaces [2, 65]. See also [64].
4.2. Paper [C], and Paper [A] revisited. In Papers [A] and [C], we present a new way to construct adjacent dyadic systems in metric spaces and show that these systems can be used easily to prove decomposition results for dyadic systems. As an application, we present an alternative proof for theLp-boundedness of shift operators acting on vector-valued functions in metric spaces. In the following results, let (X, d) be a geometrically doubling metric space with a doubling contant M.
4.2.1. Metric adjacent dyadic systems via randomization. Let us recall the random- ization procedure from Section 3.2.1 and choose
Ω =Y =0,1, . . . ,1δ
, ak(ω) =a(ω) =ω.
Thus, we only use one random variable a. These random systems D(ω), ω ∈ Ω, satisfy the following result:
Theorem 4.3 (Theorem 5.9 in [A]). Suppose that δ < 1/168M8 and let p ∈ N be xed. Then for any ballB:=B(x, r)there exists a cubeQB∈
ωD(ω)such that i) B⊆QB;
ii) (QB)≤δ−2r;
iii) δ−pB⊆Q(p)B ,
whereQ(p)B is the unique dyadic ancestor of QB of side length(Q(p)B ) =δ−p·(QB). Hence, the systems D(ω)satisfy similar properties as the systems Dα in Lemma 4.2. In particular, we can construct adjacent dyadic systems by keeping the center points of the cubes xed and constructing additional systems by changing the sizes of the cubes. Since we construct the systems via a randomization procedure, we can use probabilistic arguments in the proof.
By examining the proof of Theorem 4.3, it is simple to generalize the result:
Theorem 4.4 (Theorem 2.6 in [C]). Let n, p ∈ N be xed. If δ <1/(n·168M8), then for any collection of ballsB1, B2, . . . , Bn there exist cubesQB1, QB2, . . . , QBn in
ωD(ω)such that properties i) - iii) in Theorem 4.3 hold for each pair (Bi, QBi). We note that this result has not appeared in Euclidean analysis for n≥2but the Euclidean version can be proved easily with same techniques as the case n= 1.
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4.2.2. Tools for decomposing dyadic systems in metric spaces. Probabilistic methods are a standard tool in modern-day vector-valued analysis. Classical results of D.
Burkholder [14], J.-P. Kahane [61], J. Bourgain [12], and many others form a basis for harmonic analysis related to functions that take their values in a UMD (unconditional martingale dierence) space. These are exactly the Banach spaces E for which the Hilbert transform is bounded on Lp(Rn;E)[14, 12], where Lp(Rn;E)consist of the functions f: Rn →E such that theLp-Bochner norm off is nite.
Analysis in UMD spaces is centered around such fundamental concepts as con- ditional expectations and martingales. Martingales are sequences of functions (fi) related to a sequence of increasing σ-algebras (Fi)such that certain conditional ex- pectations are zero. A common technique in this eld is to modify the underlying sequence of σ-algebras in such a way that the modied sequence is still increasing.
This way we can access the probabilistic machinery since these types of modications give us tools for approximating certain functions by the conditional expectations of other functions. In Paper [C], we use this technique for indicator functions. More precisely, we use Theorem 4.4 to decompose a xed dyadic system D into a bounded number of subsystems that we can use to modify the sequence of σ-algebras generated by the dyadic generations D(ω)k.
Our decomposition result is the following. Let Dbe a dyadic system and{D(ω)}ω
be adjacent dyadic systems given by Theorem 4.4 for the number δ <1/(2·168M8). Suppose that m≥1 andτ: D→D is an injective function such that τ(Q)⊆mBQ
for every Q∈D andτDk⊆Dkfor every k∈Z. Then the following result holds:
Proposition 4.5 ([C, Proposition 3.1]). The system D is a disjoint union of a bounded number of subcollections Dλ ⊆D, λ= (i, j, ω), with the following property:
for every Q∈Dλ∩Dk there exist cubes PQ, Pτ(Q)∈D(ω)k−3 andPQ∗ ∈D(ω)k−3−T, where 2mδT ≤1, such that
Q⊆PQ, τ(Q)⊆Pτ(Q), PQ∪Pτ(Q)∪2mBQ⊆PQ∗; ifQ1, Q2∈Dλ∩Dk, then (PQ1∪Pτ(Q1))∩(PQ2∪Pτ(Q2)) =∅; ifQ1, Q2∈Dλ, Q1Q2, thenPQ∗1⊆PQ2.
Thus, this result allows us to split the collection D into a bounded number of subcollections with respect to the mapping τ so that
• each cube Q and its image τ(Q) can be embedded inside larger cubes PQ
andPτ(Q)from a same dyadic system so that the cubes PQ andPτ(Q)have a mutual dyadic ancestor;
• ifQ1 andQ2 belong to the same subcollection, Q1= Q2, thenQ1 cannot be the image of Q2 and vice versa;
• ifQ1 andQ2 belong to the same subcollection and Q1 Q2, then both PQ1
andPτ(Q1)are contained in PQ2.
In particular, we can remove the cubes PQ and Pτ(Q) from the collection D(ω)k−3 and replace them with the union PQ∪Pτ(Q) so that the σ-algebra generated by this modied version of D(ω)k−3still contains the σ-algebra generated by D(ω)k−3−T. In a way, we deliberately lose some information but this is actually benecial when we combine it with some classical results (see the proof of [C, Theorem 4.7]).
Once the adjacent dyadic systems of Theorem 4.4 are available, the proof of Propo- sition 4.5 is actually quite elementary. We can use adjacent dyadic systems as a sort of forcing tool in the sense that since we can approximate balls with cubes from the adjacent systems, for every cube Q∈D and every constant C≥1there has to exist
at least one cubeP ∈
ωD(ω)such thatQ⊆CBQ ⊆P and(P)≈C·(Q). In par- ticular, we can embed dyadic cubes inside other dyadic cubes so that the embedded cube is far away from the boundary of the larger cube. This is essential in various dyadic techniques.
4.2.3. The Lp-boundedness of shift operators in metric spaces. The idea of (Haar) shift operators Tm, m > 0, is simple: they map the generating Haar functions in- jectively to other Haar functions so that the supports of hQ and TmhQ depend on the numbermand belong to the same generation of dyadic cubes. In the Euclidean case, they are the linear extensions of the mappings hI→hI+m|I| form∈Z. These operators were considered by T. Figiel [34, 35] when he proved a generalization of the famous T(1)theorem of G. David and J.-L. Journé [27] for UMD-valued functions f:Rn→E. His key observation was that every Calderón-Zygmund operator can be decomposed into sums and products of Haar shifts or rearrangements, Haar multipli- ers and paraproducts. One of the steps of the proof of his generalization was to show that the shift operators satisfy the following quantitative Lp-bound forp∈(1,∞):
TmfLp(Rn;E) ≤ C·log(2 +|m|)αfLp(Rn;E), (4.6) whereα <1 depends only onE andp, and the constantC depends onE,pandα.
Figiel's technique was extended to normal spaces of homogeneous type by P. F.
X. Müller and M. Passenbrunner to prove a T(1) theorem in this setting [86]. For this, they generalized the denition of a shift operator and proved a bound similar to (4.6). Their proof was simplied by R. Lechner and M. Passenbrunner [69] who proved the bound in a more general setting by providing a deterministic way to modify a nite collection of Christ-type dyadic cubes [69, Theorem 3.1]. In Paper [C], we use this same starting point and prove the bound (4.6) in metric spaces with the help of Proposition 4.5 which we can use as a black box. We present a fairly short and straightforward proof with the help of some classical results.
4.3. Related developments and open problems. Another way to construct ad- jacent dyadic systems in the Euclidean setting is to simply consider the concentric 3-enlargements of the standard dyadic cubes, i.e. the collection D:={3Q:Q∈D}.
This collection is a disjoint union of 3n nested subcollections Di that can be used similarly as the collections Dα [50, Section 6]. These collections have the following additional property: every cube Q ∈ Di is a disjoint union of 3n cubes from the original collection D and every cube Q∈ D can be trivially embedded well inside the cube3Q ∈D. It is an open problem to construct similar collections in metric spaces.
Figiel's decomposition technique has also been used for proving an operator-valued T btheorem [58]. Alternative proofs for the UMD-valued T(1)theorem can be found from [57] and [59].
5. Muckenhoupt weights and quantitative weighted norm estimates 5.1. Background. Dierent types of weighted norm inequalities are an integral part of modern-day harmonic analysis. They are centered around the following fundamen- tal question. Given an integral operatorTacting on functions from the measure space (X, μ), what are the sucient and necessary conditions for weights (i.e. non-negative locally integrable functions)ω andσ such thatT is bounded fromLp(ω)toLq(σ)in
14
the following precise form ˆ
X|T f(x)|qσ(x)dμ(x) 1/q
≤ C ˆ
X|f(x)|pω(x)dμ(x) 1/p
(5.1) for some constant C = C(T, p, q) and allf ∈Lp(ω)? These inequalities became an active area of research in the 1970's through the pioneering work of B. Muckenhoupt, R. Hunt, R. Coifman, C. Feerman and R. Wheeden [85, 48, 19]. Their theory gave a qualitative answer to the question in the Euclidean setting for the Hardy-Littlewood maximal operator, the Hilbert transform and the class of Calderón-Zygmund opera- tors in the one-weight case where ω=σ andp=q. To be more precise, they showed that the inequality (5.1) holds in the one-weight case if and only if ωbelongs to the Muckenhoupt Ap class, i.e.
[w]Ap := sup
Q Q
w
·
Q
w−1/(p−1) p−1
< ∞. (5.2)
Questions related to two-weight norm inequalities, whereω =σ, are generally much more complicated and usually there does not exist a simple elegant characterization like (5.2). In this dissertation, we only consider one-weight norm inequalities for p=q and the properties of the weight classes, but we note that a common strategy to approach the two-weight problem is the testing condition approach rst studied by E. Sawyer [97].
The question about the optimal constant Cin (5.1) is a natural follow-up question to the qualitative characterizations. Starting from the work of S. Buckley [13] in the 1990's, the quantitative dependence of the constant C on theAp constant [w]Ap has been a widely explored topic. It was further motivated by the conjecture of K. Astala, T. Iwaniec and E. Saksman [4] that showed the important and surprising consequences of nding the optimal power s such that C ≤ C[w]sA
p, where C is independent of [w]Ap. In Buckley's work with the Hardy-Littlewood maximal operator M, it w as shown that
MLp(Rn,w)→Lp(Rn,w) ≤ Cn,p[w]1/(p−1)A
p , 1< p <∞, (5.3) and the result is sharp in the following sense: the constant [w]1/(p−1)A
p cannot be replaced by ω([w]Ap) for any non-negative non-decreasing map ω: [0,∞) → [0,∞) with slower growth-rate than t→t1/(p−1).
Finding similar sharp weighted estimates for Calderón-Zygmund operators remained an open problem for a long time. Developments in extrapolation theory [95, 30] re- duced the question to nding the optimal constant in the case p= 2, but even this reduced question (that became known as the A2 conjecture) remained open for sev- eral years. It was rst solved for various specic operators like the Beurling-Ahlfors transform [94], the Hilbert transform [93] and the dyadic paraproduct [11] before T. Hytönen solved it in full generality [49] using the random dyadic techniques we discussed in Section 3. During recent years, numerous further developments related to this A2 theorem have appeared in the literature. For example, the proof of the theorem has been shortened and simplied by A. Lerner [71], and the theorem has been proved in metric spaces by F. Nazarov, A. Reznikov and A. Volberg [91] and ex- tended for the class of Calderón-Zygmund operators with Dini-continuous kernels by M. Lacey [68]. In Paper [D], we revisit and rene this recent extension by Lacey and prove quantitative weighted norm estimates for rough homogeneous singular intergral as an application.
