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.
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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: 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)
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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