https://doi.org/10.1007/s12220-020-00514-y
On Pointwise
r-Sparse Domination in a Space of Homogeneous Type
Emiel Lorist1
Received: 23 April 2020 / Accepted: 3 September 2020 / Published online: 3 October 2020
© The Author(s) 2020
Abstract
We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual 1-sum in the sparse operator is replaced by an r-sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the A2-theorem for vector- valued Calderón–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.
Keywords Sparse domination·Space of homogeneous type·Muckenhoupt weight· Singular integral operator·Mihlin multiplier theorem·Rademacher maximal operator
Mathematics Subject Classification Primary: 42B20·Secondary: 42B15·42B25· 46E40
1 Introduction
The technique of controlling various operators by so-called sparse operators has proven to be a very useful tool to obtain (sharp) weighted norm inequalities in the past decade.
The author is supported by the VIDI subsidy 639.032.427 of the Netherlands Organization for Scientific Research (NWO).
B
Emiel Lorist e.lorist@tudelft.nl1 Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
The key feature in this approach is that a typically signed and non-local operator is dominated, either in norm, pointwise or in dual form, by a positive and local expression.
The sparse domination technique comes from Lerner’s work towards an alterna- tive proof of the A2-theorem, which was first proven by Hytönen in [38]. In [54]
Lerner applied his local mean oscillation decomposition approach to theA2-theorem, estimating the norm of a Calderón–Zygmund operator by the norm of a sparse opera- tor. This was later improved to a pointwise estimate independently by Conde-Alonso and Rey [15] and by Lerner and Nazarov [57]. Afterwards, Lacey [51] obtained the same result for a slightly larger class of Calderón–Zygmund operators by a stopping cube argument instead of the local mean oscillation decomposition approach. This argument was further refined by Hytönen, Roncal, and Tapiola [35] and afterwards made strikingly clear by Lerner [55], where the following abstract sparse domination principle was shown:
IfT is a bounded sublinear operator fromLp1(Rn)toLp1,∞(Rn)and thegrand maximal truncation operator
MT f(s):=sup
Qs
ess sup
s∈Q
|T(f1Rn\3Q)(s)|, s∈Rn,
is bounded from Lp2(Rn)to Lp2,∞(Rn)for some 1 ≤ p1,p2 < ∞, then there is an η ∈ (0,1) such that for every compactly supported f ∈ Lp(Rn)with p0 :=
max{p1,p2}there exists anη-sparse family of cubesSsuch that
|T f(s)|
Q∈S
|f|p0,Q1Q(s), s∈Rn. (1.1)
Herefpp,Q :=
Q fp:= |Q1|
Q fpfor p∈(0,∞)and positive f ∈ Llocp (Rn)and we call a family of cubesSη-sparse if for everyQ∈Sthere exists a measurable set EQ⊆Qsuch that|EQ| ≥η|Q|and such that theEQ’s are pairwise disjoint.
This sparse domination principle was further generalized in the recent paper [58]
by Lerner and Ombrosi, in which the authors showed that the weakLp2-boundedness of the more flexible operator
M#T,αf(s):=sup
Qs
ess sup
s,s∈Q
|T(f 1Rn\αQ)(s)−T(f 1Rn\αQ)(s)|, s∈Rn,
for someα ≥ 3 is already enough to deduce the pointwise sparse domination as in (1.1). Furthermore, they relaxed the weak Lp1-boundedness condition on T to a condition in the spirit of theT(1)-theorem.
1.1 Main Result
Our main result is a generalization of the main result in [58] in the following four directions:
(i) We replaceRnby a space of homogeneous type(S,d, μ).
(ii) We letT be an operator from Lp1(S;X)toLp1,∞(S;Y), where X andY are Banach spaces.
(iii) We use structure of the operatorTand geometry of the Banach spaceYto replace the1-sum in the sparse operator by anr-sum forr ≥1.
(iv) We replace the truncationT(f1Rn\αQ)in the grand maximal truncation operator by an abstract localization principle.
The extensions (i) and (ii) are relatively straightforward. The main novelty of this paper is (iii), which controls the weight characteristic dependence that can be deduced from the sparse domination. Generalization (iv) will only make its appearance in Theorem 3.2and can be used to make the associated grand maximal truncation operator easier to estimate in specific situations.
Let(S,d, μ)be a space of homogeneous type and letX andY be Banach spaces.
For a bounded linear operatorT fromLp1(S;X)toLp1,∞(S;Y)andα≥1 we define the followingsharp grand maximal truncation operator
M#T,αf(s):=sup
Bs
ess sup
s,s∈B
T(f 1S\αB)(s)−T(f 1S\αB)(s)
Y, s∈S, where the supremum is taken over all ballsB⊆Scontainings∈S. Our main theorem reads as follows.
Theorem 1.1 Let(S,d, μ)be a space of homogeneous type and let X and Y be Banach spaces. Take p1,p2,r ∈ [1,∞)and set p0:=max{p1,p2}. Takeα≥3c2d/δ, where cdis the quasi-metric constant andδis as in Proposition2.1. Assume the following conditions:
• T is a bounded linear operator from Lp1(S;X)to Lp1,∞(S;Y).
• M#T,αis a bounded operator from Lp2(S;X)to Lp2,∞(S).
• There is a Cr >0such that for disjointly and boundedly supported f1, . . . , fn ∈ Lp0(S;X)
Tn
k=1
fk
(s)
Y ≤Cr
n
k=1
T fk(s)r
Y
1/r
, s∈S.
Then there is anη ∈ (0,1)such that for any boundedly supported f ∈ Lp0(S;X) there is anη-sparse collection of cubesSsuch that
T f(s)Y S,αCTCr
Q∈S
fX
r
p0,Q1Q(s)1/r
, s∈S,
where CT = TLp1→Lp1,∞+ M#T,αLp2→Lp2,∞.
As the assumption in the third bullet of Theorem1.1expresses a form of sublinearity of the operatorTwhenr=1, we will call this assumptionr-sublinearity. Note that it is crucial that the constantCris independent ofn∈N. IfCr =1 it suffices to consider n=2.
1.2 Sharp Weighted Norm Inequalities
One of the main reasons to study sparse domination of an operator is the fact that sparse bounds yield weighted norm inequalities and these weighted norm inequalities are sharp for many operators. Here sharpness is meant in the sense that forp∈(p0,∞) we have aβ ≥0 such that
TLp(S,w;X)→Lp(S,w;Y)[w]βAp/p
0, w∈ Ap/p0, (1.2) and (1.2) is false for anyβ< β.
The first result of this type was obtained by Buckley [9], who showed thatβ =
1
p−1 for the Hardy–Littlewood maximal operator. A decade later, the quest to find sharp weighted bounds attracted renewed attention because of the work of Astala, Iwaniec, and Saksman [4]. They proved sharp regularity results for the solution to the Beltrami equation under the assumption thatβ = 1 for the Beurling–Ahlfors transform forp≥2. This linear dependence on theApcharacteristic for the Beurling–
Ahlfors transform was shown by Petermichl and Volberg in [72]. Another decade later, after many partial results, sharp weighted norm inequalities were obtained for general Calderón–Zygmund operators by Hytönen in [38] as discussed before.
In Sect.4, we will prove weightedLp-boundedness for the sparse operators appear- ing in Theorem1.1. As a direct corollary from Theorem1.1and Proposition4.1we have:
Corollary 1.2 Under the assumptions of Theorem1.1we have for all p∈(p0,∞)and w∈ Ap/p0
TLp(S,w;X)→Lp(S,w;Y)CT Cr[w]max 1
p−p0,1r
Ap/p0 , where the implicit constant depends on S,p0,p,r , andα.
As noted before, the main novelty in Theorem1.1is the introduction of the parameter r ∈ [1,∞). Ther-sublinearity assumption in Theorem1.1becomes more restrictive asr increases and the conclusions of Theorem1.1and Corollary1.2consequently become stronger. In order to check whether the dependence on the weight characteristic is sharp, one can employ, e.g., [65, Theorem 1.2], which provides a lower bound for the best possible weight characteristic dependence in terms of the operator norm ofT fromLp(S;X)toLp(S;Y). For some operators, like Littlewood–Paley or maximal operators, sharpness in the estimate in Corollary1.2is attained forr > 1 and thus Theorem1.1can be used to show sharp weighted bounds for more operators than precursors like [58, Theorem 1.1].
1.3 How to Apply Our Main Result
Let us outline the typical way how one applies Theorem1.1(or the local and more general version in Theorem3.2) to obtain (sharp) weightedLp-boundedness for an operatorT:
(i) IfT is not linear it is oftenlinearizable, which means that we can linearize it by putting part of the operator in the norm of the Banach spaceY. For example, if T is a Littlewood–Paley square function, we takeY =L2, and ifT is a maximal operator, we takeY =∞. Alternatively one can apply Theorem3.2, which is a local and more abstract version of Theorem1.1that does not assumeT to be linear.
(ii) The weakLp1-boundedness ofT needs to be studied separately and is often already available in the literature.
(iii) The operatorM#T,αreflects the non-localities of the operatorT. The weakLp2- boundedness ofM#T,αrequires an intricate study of the structure of the operator.
In many examplesM#T,αcan be pointwise dominated by the Hardy–Littlewood maximal operator Mp2, which is weak Lp2-bounded. This is exemplified for Calderón–Zygmund operators in the proof of Theorem6.1. Sometimes, one can choose a suitable localization in Theorem3.2such that the sharp maximal trun- cation operator is either zero (see Sect.8on the Rademacher maximal operator), or pointwise dominated byT.
(iv) Ther-sublinearity assumption onT is trivial forr=1, which suffices if one is not interested in quantitative weighted bounds. To check ther-sublinearity for somer > 1, one needs to use the structure of the operator and often also the geometric properties of the Banach spaceY like typer. See, for example, the proofs of Theorems8.1and [64, Theorem 6.4] how to checkr-sublinearity in concrete cases.
1.4 Applications
The main motivation to generalize the results in [58] comes from the application in the recent work [64] by Veraar and the author, in which Calderón–Zygmund theory is developed for stochastic singular integral operators. In particular, in [64, Theorem 6.4] Theorem1.1is applied with p1 = p2 = r = 2 to prove a stochastic version of the vector-valuedA2-theorem for Calderón–Zygmund operators, which yields new results in the theory of maximal regularity for stochastic partial differential equations.
The fact thatr=2 in [64, Theorem 6.4] was needed to obtain a sharp result motivated the introduction of the parameterrin this paper. In future work, further applications of Theorem1.1to both deterministic and stochastic partial differential equations will be given, for which it is crucial that we allow spaces of homogeneous type instead of justRn, as, in these applications,Sis typicallyR+×Rnwith the parabolic metric.
In this paper, we will focus on applications in harmonic analysis. We will provide a few examples that illustrate the sparse domination principle nicely and comment on further potential applications in Sect.9.
• As a first application of Theorem1.1, we prove anA2-theorem for vector-valued Calderón–Zygmund operators with operator-valued kernel in a space of homo- geneous type. The A2-theorem for vector-valued Calderón–Zygmund operators with operator-valued kernel in Euclidean space has previously been proven in [32]
and theA2-theorem for scalar-valued Calderón–Zygmund operators in spaces of homogeneous type in [3,69]. Our theorem unifies these two results.
• Using the A2-theorem, we prove a weighted, anisotropic, mixed-norm Mihlin multiplier theorem, which is a natural supplement to the recent results in [24] and is particularly useful in the study of spaces of smooth, vector-valued functions.
• In our second application of Theorem1.1, we study sparse domination and quanti- tative weighted norm inequalities for the Rademacher maximal operator, extending the qualitative bounds in Euclidean space in [50]. The proof demonstrates how one can use the geometry of the Banach space to deducer-sublinearity for an operator.
As a corollary, we deduce that the lattice Hardy–Littlewood and the Rademacher maximal operator are not comparable.
1.5 Outline
This paper is organized as follows: After introducing spaces of homogeneous type and dyadic cubes in such spaces in Sect.2, we will set up our abstract sparse domination framework and deduce Theorem1.1in Sect.3. We also give some further generaliza- tions of our main results. In Sect.4we introduce weights and state weighted bounds for the sparse operators in the conclusions of Theorem1.1, from which Corollary1.2 follows. To prepare for our application sections, we will discuss some preliminaries on, e.g., Banach space geometry in Sect.5. Afterwards we will use our main result to prove the previously discussed applications in Sects.6,7and8. Finally, in Sect.9we discuss some potential further applications of our main result.
2 Spaces of Homogeneous Type
A space of homogeneous type(S,d, μ), originally introduced by Coifman and Weiss in [14], is a setS equipped with a quasi-metricd and a doubling Borel measureμ.
That is, a metricdwhich instead of the triangle inequality satisfies d(s,t)≤cd
d(s,u)+d(u,t)
, s,t,u∈ S,
for somecd ≥1, and a Borel measureμthat satisfies the doubling property μ
B(s,2ρ)
≤cμμ
B(s, ρ)
, s∈S, ρ >0,
for somecμ≥ 1, where B(s, ρ):= {t ∈ S :d(s,t) < ρ}is the ball arounds with radiusρ. Throughout this paper, we will assume additionally that all ballsB⊆Sare Borel sets and that we have 0< μ(B) <∞.
It was shown in [78, Example 1.1] that it can indeed happen that balls are not Borel sets in a quasi-metric space. This can be circumvented by taking topological closures and adjusting the constantscd andcμaccordingly. However, to simplify matters we just assume all balls to be Borel sets and leave the necessary modifications if this is not the case to the reader. The size condition on the measure of a ball ensures that taking the averagefp,B of a positive function f ∈ Llocp (S)over a ball B ⊆ S is always well defined.
Asμis a Borel measure, i.e., a measure defined on the Borelσ-algebra of the quasi- metric space(S,d), the Lebesgue differentiation theorem holds and as a consequence the continuous functions with bounded support are dense inLp(S)for allp∈ [1,∞). The Lebesgue differentiation theorem and consequently our results remain valid ifμ is a measure defined on aσ-algebra that contains the Borelσ-algebra as long as the measure space(S, , μ)is Borel semi-regular. See [1, Theorem 3.14] for the details.
Throughout we will write that an estimate depends onSif it depends oncdandcμ. For a thorough introduction to and a list of examples of spaces of homogeneous type we refer to the monographs of Christ [12] and Alvarado and Mitrea [1].
2.1 Dyadic Cubes
Let 0<c0≤C0<∞and 0< δ <1. Suppose that fork∈Zwe have an index set Jk, pairwise disjoint collectionDk= {Qkj}j∈Jkof measurable sets and a collection of points{zkj}j∈Jk. We callD :=
k∈ZDk adyadic systemwith parametersc0,C0and δif it satisfies the following properties:
(i) For allk∈Zwe have
S=
j∈Jk
Qkj;
(ii) Fork≥l,Q∈DkandQ∈Dlwe either haveQ∩Q=∅orQ⊆Q; (iii) For eachk∈Zand j ∈ Jkwe have
B(zkj,c0δk)⊆Qkj ⊆B(zkj,C0δk);
We will call the elements of a dyadic systemD cubes and for a cube Q ∈ D we define therestricted dyadic systemD(Q):= {P ∈D: P⊆ Q}. We will say that an estimate depends onDif it depends on the parametersc0,C0andδ.
One can viewzkj andδk as the center and side length of a cubeQkj ∈ Dk. These have to be with respect to a specifick∈Z, as thiskmay not be unique. We therefore think of a cubeQ∈Dto also encode the information of its centerzand generationk.
The structure of individual dyadic cubesQ∈Din a space of homogeneous type can be very messy and consequently the dilations of such cubes do not have a canonical definition. Therefore for a cube Q ∈Dwith centerzand of generationkwe define thedilationsαQforα≥1 as
αQ:=B
z, α·C0δk ,
which are actually dilations of the ball that contains Qby property (iii) of a dyadic system.
When S = Rn andd is the Euclidean distance, the standard dyadic cubes form a dyadic system and, combined with its translates overα ∈ {0,13,23}n, it holds that any ball inRn is contained in a cube of comparable size from one of these dyadic systems (see, e.g., [43, Lemma 3.2.26]). We will rely on the following proposition for
the existence of dyadic systems with this property in a general space of homogeneous type. For the proof and a more detailed discussion, we refer to [40].
Proposition 2.1 Let(S,d, μ)be a space of homogeneous type. There exist0<c0≤ C0 < ∞, γ ≥ 1,0 < δ < 1 and m ∈ N such that there are dyadic systems D1, . . . ,Dmwith parameters c0, C0andδ, and with the property that for each s∈S andρ >0there is a j ∈ {1, . . . ,m}and a Q∈Dj such that
B(s, ρ)⊆Q, and diam(Q)≤γρ.
The following covering lemma will be used in the proof of our main theorem:
Lemma 2.2 Let (S,d, μ) be a space of homogeneous type andD a dyadic system with parameters c0, C0, andδ. Suppose thatdiam(S)= ∞, takeα≥3c2d/δ and let E ⊆S satisfy0<diam(E) <∞. Then there exists a partitionD⊆Dof S such that E ⊆αQ for all Q∈D.
Proof Fors ∈ S andk ∈ Zlet Qks ∈ Dk be the unique cube such thats ∈ Qks and denote its center byzks. Define
Ks :=
k∈Z: E2cdQks , wherecdis the quasi-metric constant. Ifk∈Zis such that
diam(2cdQks)≤4c2dC0δk <diam(E),
thenE 2cdQks, i.e.,k∈ Ks so isKsnon-empty. On the other hand, ifk∈Zis such thatC0δk >sups∈Ed(s,s),then
sup
s∈E
d(s,zks)≤cd
sup
s∈E
d(s,s)+d(s,zks)
≤2cdC0δk
soE ⊆2cdQks and thusk∈/ Ks. ThereforeKsis bounded from below.
Defineks := minKs and setD := {Qkss : s ∈ S}. ThenDis a partition of S.
Indeed, suppose that for s,s ∈ S we have Qkss ∩Qkss = ∅. Then using property (ii) of a dyadic system we may assume without loss of generality thatQkss ⊆ Qkss. Property (ii) of a dyadic system then implies thatks ≥ks. In particulars∈ Qkss , so by the minimality ofks we must haveks =ks. Therefore, since the elements ofDks
are pairwise disjoint, we can concludeQkss =Qkss .
To conclude note thatzkss ∈ Qkss ⊆Qkss−1by property (ii) of a dyadic system, so d(zkss−1,zkss)≤C0δks−1. Therefore using the minimality ofks we obtain
E⊆2cdQkss−1=B(zkss−1,2cdC0δks−1)⊆B
zkss,3cd2
δ ·C0δks
⊆αQkss,
which finishes the proof.
2.2 The Hardy–Littlewood Maximal Operator
On a space of homogeneous type(S,d, μ)with a dyadic systemD, we define the dyadic Hardy–Littlewood maximal operatorfor f ∈L1loc(S)by
MDf(s):= sup
Q∈D:s∈Q
|f|
1,Q, s∈S.
By Doob’s maximal inequality (see, e.g., [43, Theorem 3.2.2]) MD is strong Lp- bounded for allp∈(1,∞)and weakL1-bounded. We define the (non-dyadic)Hardy–
Littlewood maximal operatorfor f ∈ L1loc(S)by M f(s):=sup
Bs
|f|
1,Q, s∈ S,
where the supremum is taken over all balls B ⊆ Scontainings. By Proposition2.1 there are dyadic systemsD1, . . . ,Dm such that
M f(s)S
m j=1
MDf(s), s∈S,
so M is also strong Lp-bounded for p ∈ (1,∞)and weak L1-bounded. For p0 ∈ [1,∞)and f ∈Llocp0(S), we define
Mp0 f(s):=sup
Bs
|f|
p0,Q =M
|f|p0
(s)1/p0, s∈S,
which is strongLp-bounded for p ∈(p0,∞)and weak Lp0-bounded. This follows from the boundedness ofMby rescaling.
3 Pointwiser-Sparse Domination
In this section, we will prove a local version of the sparse domination result in Theorem 1.1, from which we will deduce Theorem1.1by a covering argument using Lemma 2.2. This local version will use an abstract localization of the operator T, since it depends upon the operator at hand as to the most effective localization. For example, in the study of a Calderón–Zygmund operator it is convenient to localize the function inserted intoT, for a maximal operator it is convenient to localize the supremum in the definition of the maximal operator and for a Littlewood–Paley operator it is most suitable to localize the defining integral.
Definition 3.1 Let(S,d, μ)be a space of homogeneous type with a dyadic systemD, letXandY be Banach spaces,p∈ [1,∞)andα≥1. For a bounded operator
T: Lp(S;X)→Lp,∞(S;Y),
we say that a family of operators {TQ}Q∈D from Lp(S;X) to Lp,∞(Q;Y)is an α-localization family of T if for allQ∈Dand f ∈Lp(S;X)we have
TQ(f1αQ)(s)=TQf(s), s∈ Q, (Localization) TQ(f 1αQ)(s)
Y ≤T(f 1αQ)(s)
Y, s∈ Q, (Domination) ForQ,Q∈DwithQ⊆Qwe define the difference operator
TQ\Qf(s):=TQf(s)−TQf(s), s∈ Q. and forQ∈Dthelocalized sharp grand maximal truncation operator
M#T,Qf(s):= sup Q∈D(Q):
s∈Q
ess sup
s,s∈Q(TQ\Q)f(s)−(TQ\Q)f(s)Y, s∈S.
In order to obtain interesting results, one needs to be able to recover the boundedness ofT from the boundedness ofTQ uniformly inQ∈D. The canonical example of an α-localization family is
TQf(s):=T(f1αQ)(s), s∈ Q.
for allQ∈Dand it is exactly this choice that will lead to Theorem1.1. We are now ready to prove our main result, which is a local, more general version of Theorem1.1.
Theorem 3.2 Let(S,d, μ)be a space of homogeneous type with dyadic systemDand let X and Y be Banach spaces. Take p1,p2,r ∈ [1,∞), set p0 :=max{p1,p2}and takeα≥1. Suppose that
• T is a bounded operator from Lp1(S;X)to Lp1,∞(S;Y)withα-localization family {TQ}Q∈D.
• M#T,Q is bounded from Lp2(S;X)to Lp2,∞(S)uniformly in Q∈D.
• For all Q1, . . . ,Qn∈Dwith Qn⊆ · · · ⊆Q1and any f ∈Lp(S;X) TQ1f(s)
Y ≤Cr
TQn f(s)r
Y +
n−1
k=1
TQk\Qk+1f(s)r
Y
1/r
, s∈ Qn.
Then for any f ∈ Lp0(S;X)and Q∈Dthere exists a 12-sparse collection of dyadic cubesS⊆D(Q)such that
TQf(s)
Y S,D,αCTCr
P∈S
fX
r
p0,αP1P(s)1/r
, s∈ Q,
with CT := TLp1→Lp1,∞+supP∈DM#T,PLp2→Lp2,∞.
The assumption in the third bullet in Theorem 3.2 replaces the r-sublinearity assumption in Theorem1.1. We will call this assumption alocalizedr-estimate.
Proof Fix f ∈Lp(S,X)andQ∈D. We will prove the theorem in two steps: we will first construct the 12-sparse family of cubesSand then show that the sparse expression associated toSdominatesTQf pointwise.
Step 1:We will construct the 12-sparse family of cubesSiteratively. Given a col- lection of pairwise disjoint cubesSk for somek ∈ Nwe will first describe how to constructSk+1. Afterwards we can inductively defineSk for allk ∈Nstarting from S1= {Q}and setS:=
k∈NSk.
Fix aP ∈Skand forλ≥1 to be chosen later define 1P:=
s∈ P : TPf(s)Y > λCT fX
p0,αP
2P:=
s∈ P :M#T,P(f)(s) > λCT fX
p0,αP
,
andP :=1P∪2P. Letc1≥1, depending onS,Dandα, be such thatμ(αP)≤ c1μ(P). By the domination property of theα-localization family we have
TPf(s)Y ≤ T(f1αP)(s)Y, s∈ P, and by the localization property
M#T,P(f)(s)=M#T,P(f 1αP)(s), s∈ P.
Thus by the weak boundedness assumptions onT andM#T,Pand Hölder’s inequality we have fori =1,2
μ(iP)≤f1αPLpi(S;X)
λ fX
p0,αP
pi
= fX
pi
pi,αP
λpi fX
pi p0,αP
μ(αP)≤ c1
λ μ(P). (3.1) Therefore it follows that
μ(P)≤ 2c1
λ μ(P). (3.2)
To construct the cubes inSk+1we will use a local Calderón–Zygmund decomposition (see, e.g., [26, Lemma 4.5]) on
P,ρ := {s∈ P :MD(P)(1P) > 1ρ}, ρ >0,
which will be a proper subset of P for our choice ofλandρ. Here MD(P) is the dyadic Hardy–Littlewood maximal operator with respect to the restricted dyadic sys- tem D(P). The local Calderón–Zygmund decomposition yields a pairwise disjoint collection of cubesSP ⊆D(P)and a constantc2≥2, depending onS andD, such thatP,c2 =
P∈SPPand
1
c2 μ(P)≤μ(P∩P)≤ 12μ(P), P∈SP. (3.3)
Then by (3.2), (3.3) and the disjointness of the cubes inSPwe have
P∈SP
μ(P)≤c2
P∈SP
μ(P∩P)≤c2μ(P)≤ 2c1c2
λ μ(P).
Therefore, by choosingλ = 4c1c2, we have
P∈SP μ(P)≤ 12μ(P). This choice ofλalso ensures thatP,c2 is a proper subset ofPby as claimed before. We define Sk+1:=
P∈SkSP.
Now takeS1= {Q}, iteratively defineSkfor allk∈Nas described above and set S:=
k∈NSk. ThenSis12-sparse family of cubes, since for anyP ∈Swe can set EP :=P\
P∈SP
P,
which are pairwise disjoint by the fact that
P∈Sk+1 P ⊆
P∈SkP for allk ∈ N and we have
μ(EP)=μ(P)−
P∈SP
μ(P)≥ 1 2μ(P).
Step 2:We will now check that the sparse expression corresponding toSconstructed in Step 1 dominatesTQf pointwise. Since
klim→∞μ
P∈Sk
P
≤ lim
k→∞
1
2k μ(Q)=0,
we know that there is a setN0of measure zero such that for alls∈ Q\N0there are only finitely manyk ∈ Nwiths ∈
P∈Sk P. Moreover by the Lebesgue differentiation theorem we have for anyP∈Sthat1P(s)≤MD(P)(1P)(s)for a.e.s∈ P. Thus
P\NP ⊆P,1⊆P,c2 =
P∈SP
P (3.4)
for some setNP of measure zero. We defineN :=N0∪
P∈SNP,which is a set of measure zero.
Fixs∈Q\Nand take the largestn∈Nsuch thats∈
P∈Sn P, which exists since s ∈/ N0. Fork=1, . . . ,nlet Pk ∈Sk be the unique cube such thats∈ Pk and note that by construction we havePn ⊆ · · · ⊆P1=Q.Using the localizedr-estimate of T we splitTQf(s)rY into two parts
TQf(s)r
Y ≤CrrTPnf(s)r
Y +
n−1
k=1
TPk\Pk+1f(s)r
Y
=:Crr
A + B
.
For A note thats ∈/ NPn ands ∈/
P∈Sn+1 P and therefore by (3.4) we know thats∈ Pn\Pn. So by the definition of1Pn
A ≤λrCrT fX
r p0,αPn.
For 1≤k≤n−1 we have by (3.2) and (3.3) that μ
Pk+1\(Pk+1∪Pk)
≥μ(Pk+1)−μ(Pk+1)−μ(Pk+1∩Pk)
≥μ(Pk+1)− 1
2c2μ(Pk+1)−1
2μ(Pk+1) >0, (3.5) soPk+1\(Pk+1∪Pk)is non-empty. Takes∈ Pk+1\(Pk+1∪Pk), then we have
TPk\Pk+1f(s)
Y ≤TPk\Pk+1f(s)−TPk\Pk+1f(s)
Y +TPk\Pk+1f(s)
Y
≤M#T,Pk f(s)+TPk(s)
Y +TPk+1(s)
Y
≤2λCT fX
p0,αPk + fX
p0,αPk+1
,
where we used the definition ofM#T,Pk and TPk+1\Pk in the second inequality and s ∈/ Pk+1 ∪Pk in the third inequality. Using(a+b)r ≤ 2r−1(ar +br)for any a,b>0 this implies that
B ≤
n−1
k=1
2r2r−1λrCrT fX
r
p0,αPk + fX
r p0,αPk+1
≤ n k=1
4rλrCTr fX
r p0,αPk.
Combining the estimates for A and B we obtain TQf(s)
Y ≤5λCTCr
n
k=1
fX
r p0,αPk
1/r
=5λCTCr
P∈S
fX
r
p0,αP1P(s)1/r
.
Sinces∈ Q\N was arbitrary andN has measure zero, this inequality holds for a.e.
s∈ Q. Noting thatλ=4c1c2andc1andc2only depend onS,αandDfinishes the
proof of the theorem.
As announced Theorem1.1now follows directly from Theorem3.2and a covering argument with Lemma2.2.
Proof of Theorem1.1 We will prove Theorem 1.1in three steps: we will first show that the assumptions of Theorem1.1imply the assumptions of Theorem3.2, then we will improve the local conclusion of Theorem3.2to a global one and finally we will replace the averages over the dilationαP in the conclusion of Theorem3.2by the average over larger cubesP.
To start letD1, . . . ,Dm be as in Proposition2.1with parametersc0,C0,δ, andγ, which only depend onS.
Step 1:For anyQ∈D1defineTQ byTQf(s):=T(f 1αQ)(s)fors∈Q. Then:
• {TQ}Q∈D1 is anα-localization family ofT.
• For anyQ∈D1and f ∈ Lp1(S;X)we have
M#T,Qf(s)≤M#T,α(f 1αQ)(s), s∈ Q.
So by the weakLp2-boundedness ofM#T,αit follows thatM#T,Qf is weak Lp2- bounded uniformly inQ∈D1.
• For any f ∈Lp(S;X)andQ1, . . . ,Qn∈D1withQn⊆ · · · ⊆Q1the functions fk := f 1αQk\αQk+1 for k = 1, . . . ,n −1 and fn := f 1αQn are disjointly supported. Thus by ther-sublinearity ofT
TQ1 f(s)
Y ≤Cr
TQn f(s)r
Y +
n−1
k=1
TQk\Qk+1f(s)r
Y
1/r
, s∈Qn.
So the assumptions of Theorem3.2follow from the assumptions of Theorem1.1.
Step 2:Let f ∈Lp(S;X)be boundedly supported. First suppose that diam(S)=
∞and letEbe a ball containing the support of f. By Lemma2.2there is a partition D ⊆ D1such that E ⊆ αQ for all Q ∈ D. Thus by Theorem3.2we can find a
1
2-sparse collection of cubesSQ ⊆D1(Q)for everyQ∈Dwith T f(s)
Y S,α CTCr P∈SQ
fX
r
p0,αP1P(s)1/r
, s∈Q,
where we used thatTQf =T(f1αQ)=T f as supp f ⊆αQ. SinceDis a partition, S:=
Q∈DSQis also a 12-sparse collection of cubes with T f(s)
Y S,α CT Cr
P∈S
fX
r
p0,αP1P(s)1/r
, s∈ S, (3.6)
If diam(S) < ∞, then (3.6) follows directly from Theorem3.2sinceS ∈D in that case.
Step 3:For anyP∈Swith centerzand sidelengthδk we can find aP∈Dj for some 1≤ j ≤msuch that
αP=B(z, αC0·δk)⊆P, diam(P)≤γ αC0·δk.
