Adjacent dyadic systems in metric spaces 11

4.1. Background. Let D^α, 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 D^α are adjacent dyadic systems if for every ball B := B(x, r)there exists a cube QB ∈

αD^α such that B ⊆ QB and (QB) ≈r. More generally put, the systemsD^αare adjacent dyadic systems if they approximate the collection of all balls well. Since every system D^αis 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

D^α:=2^−k[0,1)ⁿ+m+ (−1)^kα:k∈Z, m∈Zⁿ

(4.1) for every α∈ {0,1/3,2/3}ⁿ. These systemsD^α 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∈

αD^α such that i) B⊂QB;

ii) 2^mB⊂Q^(m)_B ; iii) 3r≤(QB)<6r,

where Q^(m)_B is the dyadic ancestor of QB of side length (Q^(m)_B ) = 2^m·(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 D^αfor 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 D^α,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 systemsD^αare 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 theL^p-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, . . . ,¹_δ

, 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 D^α 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.

12

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 L^p(Rⁿ;E)[14, 12], where L^p(Rⁿ;E)consist of the functions f: Rⁿ →E such that theL^p-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 andτD^k⊆D^kfor 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∩D^k there exist cubes PQ, Pτ(Q)∈D(ω)^k−3 andP_Q^∗ ∈D(ω)^k−3−T, where 2mδ^T ≤1, such that

Q⊆PQ, τ(Q)⊆Pτ(Q), PQ∪Pτ(Q)∪2mBQ⊆P_Q^∗; ifQ1, Q2∈Dλ∩D^k, then (PQ1∪Pτ(Q1))∩(PQ2∪Pτ(Q2)) =∅; ifQ1, Q2∈Dλ, Q1Q2, thenP_Q^∗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 L^p-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 L^p-bound forp∈(1,∞):

TmfL^p(Rⁿ;E) ≤ C·log(2 +|m|)^αfL^p(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 D^α [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