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 quesconstruc-tions 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: 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].
10
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
4.2. Paper [C], and Paper [A] revisited. In Papers [A] and [C], we present a