A metric approach to sparse domination
José M. Conde Alonso, Francesco Di Plinio, Ioannis Parissis, Manasa N. Vempati
aa r X i v : . [ m a t h . C A ] S e p A METRIC APPROACH TO SPARSE DOMINATION
JOSÉ M. CONDE-ALONSO, FRANCESCO DI PLINIO, IOANNIS PARISSIS, AND MANASA N. VEMPATIAbstract. We present a general approach to sparse domination based on single-scale L p -improving as a key property. The results are formulated in the setting of metric spaces ofhomogeneous type and avoid completely the use of dyadic-probabilistic techniques as well asof Christ-Hytönen-Kairema cubes. Among the applications of our general principle, we recoversparse domination of Dini-continuous Calderón-Zygmund kernels on spaces of homogeneoustype, we prove a family of sparse bounds for maximal functions associated to convolutions withmeasures exhibiting Fourier decay, and we deduce sparse estimates for Radon transforms alongpolynomial submanifolds of R n . Introduction
The prototypical example of a singular integral operator of interest in Harmonic Analysis,the Hilbert transform, may be decomposed into the ℓ -superposition over scales of convolu-tions with a suitably chosen and rescaled smooth function. This paradigm of superpositionof single-scale operators is most general, and also extends in particular to Radon transforms,defined by convolution with a measure supported on a lower dimensional set.Sparse domination theory rose to prominence in the pursuit of sharp weighted norm in-equalities for Calderón-Zygmund operators, through the seminal works of Lerner [22, 23, 24],and Lacey [20]. Its main thrust is to estimate, pointwise, in dual form or in norm, the singularintegral T f by a sparse operator, that is a tamer, positive and localized multiscale operator
S f which is a superposition of averages of f on a sparse – i.e. having pairwise disjoint majorsubsets– collection of cubes. This control is performed via some type of high-low, localizedcancellation enjoyed by T , and has since been carried out for much more singular operatorsthan those of Caldéron-Zygmund type: a non-exhaustive list includes modulation invariantoperators [9], non-integral operators [4], rough kernels [8, 17] oscillatory integrals [19], dis-crete singular integrals [10, 13], and, most importantly for the present article, Radon trans-forms, beginning with the work of Lacey [21] on the spherical maximal operator Af ( x ) = sup t > | A t f ( x )| , x ∈ R d , Mathematics Subject Classification.
Primary: 42B20. Secondary: 42B25.
Key words and phrases.
Sparse domination, quasi-metric spaces, singular integral operators, Radon transform,weighted norm inequalities.J. M. Conde-Alonso was partially supported by ERC Grant 32501.F. Di Plinio was partially supported by the National Science Foundation under the grants NSF-DMS-1650810,NSF-DMS-1800628, NSF-DMS-2000510, by the Severo Ochoa Program SEV-2013-0323 and by Basque GovernmentBERC Program 2014-2017.I. Parissis is partially supported by the project PGC2018-094528-B-I00 (AEI/FEDER, UE) with acronym “IHAIP”,grant T1247-19 of the Basque Government and IKERBASQUE.. where A t is the spherical average on x + t S d − . Lacey showed that a high-low cancellationtype strengthening of the well known L p -improving property of the single scale operator f sup t ∼ | A t f | may be upgraded into a sparse domination type result by a slick refor-mulation of the high-low scheme employed in [8]. The work [21] was followed by the mo-ment curve analogue of Cladek and Ou [7], see also [26], and by the general result of Hu [15],which achieves a sparse domination type result, and consequent weighted norm inequalities,for singular integrals on finite type submanifolds in the generality of Christ, Nagel, Stein andWainger [6]. The works [7, 15, 26] operate at different levels of generality within the footprintof [21]: in particular, the integral representation of the kernel is relied upon at different points,and the iterative argument leading to sparse domination involves a discretization of the op-erator which is made possible by variants of the Christ-Hytönen-Kairema dyadic systems inspaces of homogeneous type [5, 16]. In fact, the dominating sparse operator involves averagesover these dyadic cubes.In this article, we set forth a general sparse domination principle formulated in the general-ity of homogeneous measure metric spaces under natural minimal structural assumptions. Tostart with, the operators of interest can be written as sums of single-scale operators, and eachsuch piece is well localized. We do not assume any integral representation for our operators, inparticular the kernel estimates of Calderón-Zygmund theory are not available. Instead theseare replaced by postulating a version of suitably normalized L p -improving estimates for eachsingle-scale piece of the operator in accordance with the approach of Lacey in [21]. We thenobtain a corresponding sparse domination result for both ℓ -multiscale, and maximal opera-tors of this nature. The dominating sparse form involves averaging over sparse collections ofquasimetric balls, and no appeal to dyadic systems in the vein of [5, 16] is needed. Our resultsappear to be optimal in that sense: if one assumes a sparse domination result in an open rangeof indices then the scale-invariant L p -improving property follows.The main results of this article, Theorem A for the ℓ sum and Theorem B for the maximaloperator associated to a sequence of single scale operator T ( s ) , may be in fact loosely describedas follows. In the general context of spaces of homogeneous type, in addition to the structuralsingle scale localization property of each T ( s ) , it suffices to have uniform L p -boundedness ofpartial sums or of the maximal operator and an L p → L ( p ) ′ -improving property with log-Dinitype modulus of continuity to ensure a ( p , p ) sparse bound. In fact, Proposition 2.5 providesa converse to Theorems A, B in an open range of exponent. The main results are stated inSection 2, together with laying out the framework of spaces of homogeneous type we workwith.We provide in Section 3 several applications of our main theorems. The first two are ofclassical nature. Theorem C is a new form of the well-known sparse domination for Dini-continuous Calderón-Zygmund operators on spaces of homogeneous type. This result wasfirst obtained in [18, 30], extending to Dini moduli of continuity the A theorem in homoge-neous spaces of [25]. Unlike [18, 25, 30], our proof does not rely in any way on the dyadicsystems constructed in [5, 16]. Corollary C.1 is instead a deduction of the sparse bound forgeometric maximal operators in spaces of homogeneous type.The most conspicuous applications are provided in the context of maximal and singularRadon transforms. Our general setting is the Euclidean space R n equipped with a quasi-normwhich is homogeneous with respect to a dilation semigroup { δ r : r > } . In Theorem D, which METRIC APPROACH TO SPARSE DOMINATION 3 is suitably deduced from Theorem B, we obtain a sparse estimate for the maximal operator T ⋆ f ≔ sup s | f ∗ d µ s | where each µ s is the δ s -pushforward of some Borel measure m s supported in the unit metricball, provided that the m s have uniform algebraic Fourier decay rate at ∞ . An analogousestimate is established for the ℓ -sum if m s are of cancellative nature, using Theorem A instead.Theorem D is a sparse version of the stalwart result of Duoandikoetxea and Rubio de Francia[11, Theorems A and B], and it has not appeared in previous literature. As a further application,we derive from it a sparse domination theorem for singular Radon transform along polynomialsubvarieties of R n , stated in Corollary D.1. Results of this type for the ℓ -sum are contained inthe recent article by Hu [15], in fact within the more general framework of singular integralson finite type submanifolds in the vein of [6]. While it is plausible that the arguments of [15]may likely be adapted to cover the maximal function case, the maximal case of Corollary D.1has not appeared before.The structure of the remaining sections of the paper is as follows: Section 4 contains somenecessary preliminaries about Whitney covering in geometrically doubling metric spaces,which are relevant in the proof of the main theorems. In fact, part of the interest of this papercomes from demonstrating that the Whitney covering properties are sufficient to generate thepacking of the collections of balls constructed during the sparse domination algorithm. Thelatter, and thus the proofs of Theorems A, B and C, are carried out in Section 5 to 7. Remark.
Before this article was made publicly available, David Beltran, Joris Roos and An-dreas Seeger kindly shared with us their upcoming preprint [3] on multi-scale sparse dom-ination. Although in different settings, both papers use scale-invariant versions of the L p -improving property as a standing assumption and cover some classes of singular Radon trans-forms such as the one in [26]. We thank the authors of [3] for sharing their preprint. Acknowledgments.
This research project originated during the workshop on
Sparse Dom-ination of Singular Integrals , held at the American Institute of Mathematics (AIM), October9–13, 2017. The authors want to express their gratitude to the personnel and staff of AIM. Preliminaries and main results
In the next paragraphs we explain in more detail the setting in which we work before statingour main result.2.1.
The space of homogeneous type.
Let X be a set equipped with a quasi-metric d. Hereand in what follows a function d : X × X → [ , ∞) will be called a quasi-metric if for all x , y , z ∈ X we have that d ( x , y ) = ⇔ x = y and there exist constants c d , ˜ c d ≥ ( x , y ) ≤ ˜ c d d ( y , x ) and d ( x , y ) ≤ c d ( d ( x , z ) + d ( z , y )) .We denote by B ( x , r ) ≔ { y ∈ X : d ( x , y ) < r } the quasi-metric ball with radius r , centered at x , and assume without loss of generality thatthe balls B ( x , r ) are open in the sense that for all x ∈ X , r > x ′ ∈ B ( x , r ) there exists r ′ > B ( x ′ , r ′ ) ⊂ B ( x , r ) . Throughout this paper, if B = B ( x , r ) , we denote by αB the ball with same center and α -times the radius, namely αB ≔ B ( x , αr ) . We will assume thateach ball B in X comes with a fixed center c B , and radius r B , although these are not necessarily J. M. CONDE-ALONSO, F. DI PLINIO, I. PARISSIS, AND M. N. VEMPATI uniquely determined by B . We say that a Borel measure | · | on X is ( α , β ) - doubling for some α , β > B | αB | ≤ β | B | . If | · | is ( α , β ) - doubling for some α , β > ( X , ρ , | · |) as a space ofhomogeneous type . We simply write L p for L p ( X , ρ , | · |) and of course all d x integrations thatappear in this paper are with respect to | · | . Throughout the paper we will write L p -averages,with respect to the underlying measure and some (metric) ball B as h f i p , B ≔ (cid:16) | B | ∫ B | f | p (cid:17) p ≕ (cid:16) ∫ B | f | p (cid:17) p . Without loss of generality we can and will assume throughout the paper that all our doublingmeasures are ( , β ) -doubling for some β >
1. We will always assume that X is geometricallydoubling, see below for details.2.2. An operator which is a sum of single scale pieces.
In this paragraph we describethe environment and main assumptions for our main results. We fix ( X , d , | · |) a space ofhomogeneous type and write Lip ( X ) for the Lipschitz functions on X . We consider a linearoperator T , initially defined on all f ∈ Lip ( X ) with compact support, and assume that T canbe written formally as a sum T ∼ Í s ∈ Z T ( s ) where each T ( s ) is a linear operator; the reader isencouraged to think of T ( s ) as being a possibly singular average at scale 2 s . For σ , τ ∈ Z weset T τσ f ( x ) ≔ Õ σ ≤ s < τ [ T ( s ) f ]( x ) , x ∈ X , with the understanding that T τσ f ≡ σ ≥ τ . We assume that d localizes the operators T τσ in the sense that there exist a constant c o ≥ L in X with r L = s L and σ ≤ s L we have(2.1) supp ( T s L σ [ f L ]) ⊂ c o L . We make the quantitative assumption that(2.2) sup σ < τ k T τσ k p → p ≕ C p < ∞ for some 1 < p < ∞ . Remark 2.1.
In this abstract setup the operator T is not currently well defined which is whywe use the vague notation T ∼ Í s T ( s ) ; the question of whether and how the infinite sumconverges to T is unspecified. In applications we will actually start with a concrete operator T ,discretize it as a sum over scales Í s T ( s ) , and try to recover T as a weak limit of the truncatedsums. Typically what happens is that the weak limit of the truncated sums can differ fromthe original operator T by a pointwise multiplication operator; see [28, §I.7.2]. Thus, for thepurposes of formulating an abstract theorem we will assume the uniform bound (2.2) for thetruncations and realize T in the form(2.3) h T f , д i = h T f , д i + h m f , д i , METRIC APPROACH TO SPARSE DOMINATION 5 where m ∈ L ∞ ( X ) and there exist sequences σ j → −∞ , τ k → + ∞ such that for all f , д ∈ Lip ( X ) with compact support we have lim j , k h T τ k σ j f , д i = h T f , д i . Thus up to taking subsequences we can always think of T as being the weak limit of thetruncations T τσ modulo a pointwise multiplication operator by a bounded function. We willcome back to that point in the proof of the main theorem, Theorem A, in §5.4.We now come to the reformulation of the L p -improving assumption in the context of metricmeasure spaces of homogeneous type. This involves the definition of r -molecules. Definition 2.2.
We say that b ∈ L p is a ( p , r ) -molecule, with p ≥ , r >
0, if there exists afinite or countable family of balls { B j } such that b = Í j b j with r B j = r , supp ( b j ) ⊆ B j , ∫ B j b j ( x ) d x = , Õ j B j ≤ M . Here M > X butnot on b , r , or { B j } j .With this definition in hand we can now formulate the notion of ( p , p ′ ) -improving in ametric way. We will use the term modulus of continuity to refer to an increasing continuousfunction ω : [ , ) → [ , ∞) such that lim t → + ω ( t ) = ω ( ) =
0. In practice we will use twodifferent types of conditions, the log-Dini moduli of continuity , which additionally satisfy k ω k ℓ − Dini ≔ ∫ ω ( δ ) log ( + δ − ) d δδ < ∞ , and the Dini moduli of continuity which satisfy the standard Dini condition k ω k Dini ≔ ∫ ω ( δ ) d δδ < ∞ . Definition 2.3.
Let p , p ∈ [ , ∞] with p ′ ≥ p and s ∈ Z . We say that T ∼ Í s T ( s ) is ( p , p ′ ) -improving at scale s with modulus ω if there are constants γ , γ ≥
1, depending only on theconstants of the space of homogeneous type, and possibly on T , such that for all s ∈ R , allballs L with r L h s , all ( p , r ) -molecules b = Í j b j with 0 < r ≤ s and all f ∈ L p loc we have(2.4) |h T ( s )( f L ) , b i| ≤ ω (cid:0) r s (cid:1) | L |h f i p , γ L (cid:10) Õ j | b j | (cid:11) p , γ L . Remark 2.4.
In several applications, we will need a symmetric version of the definition abovewhich amounts to saying that T ∗ ∼ Í s T ( s ) ∗ is ( p , p ′ ) -improving at scale s . This means that T satisfies(2.5) |h T ( s ) b , f L i| ≤ ω (cid:0) r s (cid:1) | L |h f i p , γ L (cid:10) Õ j | b j | (cid:11) p , γ L whenever b = Í j b j is a ( p , r ) -molecule and r ≤ s h r L . J. M. CONDE-ALONSO, F. DI PLINIO, I. PARISSIS, AND M. N. VEMPATI
Main results.
Given η ∈ ( , ) we say that the collection of measurable sets A is η -sparse if it is countable and for all A ∈ A there exists a set E A ⊂ A with | E A | > η | A | and A , A ′ ∈ A , E A ∩ E A ′ , ∅ = ⇒ A = A ′ . We will say that a collection A is sparse if it is η -sparsefor some fixed η ∈ ( , ) .After establishing the framework, we are ready to state the first main result of the article. Theorem A.
Let ( X , d , | · |) be a space of homogeneous type. Let < p ≤ p ′ < ∞ and let T bea linear operator on ( X , d , | · |) satisfying structural assumption (2.1) . Furthermore, assume: estimate (2.2) holds for p = p , p = p ′ with constants C p , C p ′ , respectively; T is ( p , p ′ ) -improving at every scale s ∈ Z with a log-Dini modulus ω ; T ∗ is ( p , p ′ ) -improving at every scale s ∈ Z with a log-Dini modulus ω .Then, for all f , f ∈ Lip ( X ) with compact support and every σ , τ ∈ Z with σ < τ there exists asparse collection B σ , τ consisting of d -balls B with σ ≤ r B ≤ τ such that |h T τσ f , f i| . ( C p + C p ′ + k ω k ℓ − Dini ) Õ B ∈B σ , τ | B |h f i p , B h f i p , B . Furthermore if T is defined through (2.3) then for all f , f ∈ Lip ( X ) there exists a sparse collection B consisting of d -balls such that |h T f , f i| . ( C p + C p ′ + k ω k ℓ − Dini ) Õ B ∈B | B |h f i p , B h f i p , B . The implicit constants depend on the homogeneous metric structure of ( X , d , | · |) and the constantin (2.1) but are independent of σ , τ , f , f . A maximal version.
We want to describe below a variation of Theorem A which providesa sparse domination result for abstract maximal operators in metric spaces of homogeneoustype. In order to set it up we consider again an abstract sequence of linear operators { T ( s )} s ∈ Z .We assume the localization condition: there exists a constant c ≥ L with r L = s L (2.6) supp ( T ( s )[ f L ]) ⊆ c o L ∀ s ≤ s L . We consider the maximal operator T ⋆ f ( x ) ≔ sup s ∈ Z | T ( s ) f ( x )| , x ∈ X . Theorem B.
Let { T ( s )} s ∈ Z be a sequence of linear operators satisfying (2.6) and such that T ⋆ isbounded on L ∞ ( X ) . We assume that for some ≤ p ≤ p ′ ≤ ∞ and for each s ∈ Z the operator T ∗ is ( p , p ′ ) -improving at scale s in the sense that T ( s ) satisfies (2.5) with a Dini modulus ofcontinuity. Then, for all f , f bounded functions with compact support and σ , τ ∈ Z with σ < τ there exists a sparse collection B σ , τ consisting of d -balls B with σ ≤ r B ≤ τ such that |h sup σ ≤ s < τ | T ( s ) f | , f i| . (cid:0) k T ⋆ k L ∞ ( X ) + k ω k Dini (cid:1) Õ B ∈B σ , τ | B |h f i p , B h f i p , B . Furthermore for all f , f bounded functions with compact support there exists a sparse collection B consisting of d -balls such that |h T ⋆ f , f i| . (cid:0) k T ⋆ k L ∞ ( X ) + k ω k Dini (cid:1) Õ B ∈B σ , τ | B |h f i p , B h f i p , B . METRIC APPROACH TO SPARSE DOMINATION 7
The implicit constants above depend on the homogeneous metric structure of ( X , d , | · |) and theconstant in (2.6) , but are independent of σ , τ , f , f . Theorems A and B above have a partial converse which in several concrete realizationsbecomes a full converse. For the abstract setup we content ourselves with stating the followingproposition with a stronger statement coming up in Lemma 3.3 of the next section.
Proposition 2.5.
Suppose that for each σ , τ ∈ Z with σ < τ the operator T ∼ Í s T ( s ) satisfiesthe localization property (2.1) and the conclusion of Theorem A or (2.6) and the conclusion ofTheorem B. Then for every ball L with r L = s we have h T ( s )( f L )i p ′ , L ≤ h f i p , L with implicit constant depending on the localization properties of T and the constants in thesparse domination assumption but not on L or s .Proof. Using the existence of a sparse bound in the form of either Theorem A or Theorem Bwe conclude that for each s ∈ Z there is a sparse collection B consisting of balls of radius 2 s such that |h T ( s )( f L ) , д i| . Õ B ∈B | E B |h f L i p , B h д c o L i p , B with { E B } B ∈B disjoint. The above estimate and the doubling assumptions on ( X , d , | · |) thenimply that |h T ( s )( f L ) , д i| . | L |h f L i p , L h д c o L i p , L which by duality yields the desired conclusion. (cid:3) We postpone concrete realizations of Theorem A and Theorem B to Section 3, where the ( p , p ′ ) improving condition is suitably reinterpreted in a more familiar form. Here, we pointout that Theorem A yields as a corollary quantitative weighted norm inequalities of A p ∩ RH q type for the operator T : this theme has recently been pursued for several classes of operatorswithin and beyond the scope of Calderón-Zygmund theory in the Euclidean setting, see forinstance [4, 8, 9, 21, 22].We briefly recall the definition of A p weights in the context of metric measure space ofhomogeneous type ( X , d , | · |) ; these are locally integrable non-negative functions w such that [ w ] A p ≔ sup B h w i B h w − p − i p − B < ∞ , < p < ∞ , where the supremum is taken over all d-balls and all the integrations are with respect to thedoubling measure | · | . For p = [ w ] A to be the smallest constant c > B we have h w i , B ≤ c inf B w The
Reverse Hölder class RH p is defined for 1 < p < ∞ as the class of non-negative locallyintegrable functions w on X such that [ w ] RH p ≔ sup B h w i p , B h w i , B < ∞ . The proof of the following weighted estimate is an easy consequence of the sparse dominationresult of Theorem A; see for example [4, §6].
J. M. CONDE-ALONSO, F. DI PLINIO, I. PARISSIS, AND M. N. VEMPATI
Corollary B.1.
Let T ∼ Í s ∈ Z T ( s ) in the sense of (2.3) and assume that T satisfies the assump-tions of Theorem A for ≤ p < p ′ < ∞ . Then for any p < p < p ′ and w ∈ A p / p ∩ RH p ′ / p wehave the following weighted norm estimate k T : L p ( w ) → L p ( w )k . (cid:18) [ w ] A pp [ w ] RH (cid:0) p ′ p (cid:1) ′ (cid:19) max (cid:0) p − p , p ′ − p ′ − p (cid:1) . The implicit constant depends on the assumptions for T , the homogeneous structure of ( X , d , | · |) and the indices p , p , p .
3. Applications: the L p -improving property revisited Calderón-Zygmund theory.
In this subsection we digress a bit in order to describeclassical Calderón-Zygmund operators in the homogeneous setup. These are themselves L p -improving operators par excellence and help illustrate and contextualize the definitions in thispaper. We make this precise below.3.1.1. Calderón-Zygmund operators.
We begin by giving the formal definition of Calderón-Zygmund operator on a space of homogeneous type ( X , d , | · |) . We provide the definitionbelow for Dini continuous operators but of course more general definitions are possible. Definition 3.1.
Let ( X , d , | · |) be a space of homogeneous type. We say that T is a Calderón–Zygmund operator on X if T is bounded on L p ( X ) for some p ∈ ( , ∞) and there exists a kernel K : X × X \ { x , y ∈ X : x = y } → C such that for all f ∈ Lip ( X ) with compact support wehave T ( f )( x ) = ∫ X K ( x , y ) f ( y ) d y , ∀ x < supp f . The kernel K ( x , y ) is assumed to satisfy the following size and regularity conditions: thereexist constants C T , A > x , y ,(3.1) | K ( x , y )| ≤ C T V ( x , y ) , and for pairwise different x , x ′ , y ∈ X with d ( x , x ′ ) ≤ A − d ( x , y ) we have that(3.2) | K ( x , y ) − K ( x ′ , y )| + | K ( y , x ) − K ( y , x ′ )| ≤ C T V ( x , y ) ω (cid:18) d ( x , x ′ ) d ( x , y ) (cid:19) . In the estimates above we have used V ( x , y ) ≔ | B ( x , d ( x , y ))| , and ω is a Dini modulus ofcontinuity. Note that by the doubling condition on the measure | · | we have that V ( x , y ) ≃ V ( y , x ) .We first recall an easy decomposition of Calderón-Zygmund operators into local pieces.Given T a Calderón-Zygmund operator on ( X , d , | · |) associated with a kernel K we define T ( f )( x ) = Õ s ∈ Z [ T ( s ) f ]( x ) ≔ Õ s ∈ Z ∫ s ≤ d ( x , y ) < s + K ( x , y ) f ( y ) d y , x ∈ X . We set K s ( x , y ) ≔ K ( x , y ) {( x , y )∈ X × X : 2 s ≤ d ( x , y ) < s + } ( x , y ) METRIC APPROACH TO SPARSE DOMINATION 9 so that for s ∈ Z T ( s ) f ( x ) = ∫ K s ( x , y ) f ( y ) d y , x ∈ X . Note that the formula above makes sense for functions f which are Lipschitz with compactsupport as we restrict ( x , y ) away from the diagonal and K satisfies the size condition (3.1).Furthermore it is well known that the maximal truncations of Calderón-Zygmund operatorson metric spaces of homogeneous type are uniformly bounded, see for example [28, §I.7].With these definitions in hand one easily verifies that the truncations of T satisfy the local-ization properties (2.1). Furthermore we can readily see that T is ( , ∞) improving. Lemma 3.2.
Let T = Í s T ( s ) be a Calderón-Zygmund operator on ( X , d , | · |) as defined above.Then T and T ∗ are ( , ∞) improving in the sense of Definition 2.3.Proof. Let s ∈ Z and f be a Lipschitz function with compact support. Let L be a ball with r L ≃ s and b = Í j b j where each b j is supported in some ball B j = B j ( c j , r ) and the collection { B j } j has finite overlap. Then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Õ j ∫ [ T ( s )[ f L ]( x ) b j ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Õ j ∫ L ∫ c o L | K s ( x , y ) − K s ( c j , y )] f ( y ) b j ( x )| d x d y . ω (cid:16) r s (cid:17) ∫ L ∫ c o L V ( x , y ) | f ( y )| Õ j | b j | { s ≤ d ( x , y ) < s + } ( x , y ) d y . Now V ( x , y ) = | B ( x , d ( x , y ))| ≥ | B ( x , s )| & | L | by the doubling assumption and the obser-vation that L ⊆ B ( x , c s ) for some suitable constant c >
0. This proves the ( , ∞) -improvingproperty according to Definition 2.3 with the same modulus of continuity as in the definitionof T ; in particular here ω is assumed to satisfy the Dini condition. The conclusion for T ∗ follows since T is essentially self adjoint. (cid:3) Note that combining Lemma 3.2 with Theorem A we can immediately conclude a sparsedomination theorem for log-Dini Calderón-Zygmund operators. However, given the moreprecise kernel assumption we will prove in §6 the stronger theorem below, valid for Calderón-Zygmund operators which only satisfy a Dini smoothness condition. Of course this result isknown, see for example [18, 30]. However, our proof bypasses the usage of dyadic systems inspaces of homogeneous type, unlike previous approaches.
Theorem C.
Let ( X , d , | · |) be a space of homogeneous type and T be a Calderón–Zygmundoperator on ( X , d , | · |) which is bounded on some L p ( X ) , < p < ∞ , with Dini modulus ofcontinuity. Then, for all f , f ∈ Lip ( X ) with compact support there exists a sparse collection B consisting of d -balls such that |h T f , f i| . ( C p + k ω k Dini ) Õ B ∈B | B |h f i , B h f i , B . The implicit constant depends on the homogeneous metric structure of ( X , d , | · |) and on theconstants in the kernel assumptions for T . Geometric maximal operators.
A somewhat trivial application of Theorem B provides asparse domination theorem for geometric maximal operators in metric spaces. For this con-sider the maximal operatorM f ( x ) ≔ sup x ∈ B ∫ B | f ( y )| d y , x ∈ X . First of all note that if B = B ( c B , r B ) with 2 s < r B ≤ s + then the doubling property of themeasure | · | implies that ∫ B | f ( y )| d y . ∫ B ( c B , s + ) | f ( y )| d y and so we can assume that all the balls in the definition of M have dyadic radii. Now we candefine the single scale average T ( s ) f ( x ) ≔ sup B ∋ xr B = s ∫ B | f | , M f . T ⋆ f ≔ sup s ∈ Z | T ( s ) f | , x ∈ X . A well known procedure allows us to approximate T ( s ) by a smoother operator. Take a func-tion f which is bounded and compactly supported in some ball B with r B = s B and fix somescale s ∈ Z with s ≤ s B . Since we are working on a homogeneous space we can cover B bya union of balls L τ ≔ B ( c τ , s ) so that for every ρ > Í τ ρL τ .
1; see Lemma 4.3.Then one easily constructs a ∼ / s -Lipschitz partition of unity { ψ τ } τ , 0 ≤ ψ τ ≤
1, subordi-nate to the cover { L τ } τ so that ψ τ & c L τ for c > ψ τ ⊆ c L τ foreach τ and some structural constants c > c >
1. Then the single scale operator T ( s ) can beapproximated T ( s ) f ( x ) . Õ τ ψ τ ( x ) ∫ B ( c τ , c s ) | f ( y )| d y . Õ τ ψ τ ( x )| B ( c τ , c s )| ∫ | f ( y )| ψ τ ( y ) d y ≕ A ( s )| f | . The process above is a version of discrete convolution which is a standard tool in harmonicanalysis on homogeneous spaces; see for example [1]. We consider for each s the duality form h f L , A ( s ) b i = Õ τ | B ( c τ , c s )| ∫ ∫ ψ τ ( x ) ψ τ ( y ) f ( x ) L ( x ) b ( y ) d y d x where b = Í j b j is a ( , r ) molecule with r L h s and r ≤ s . Then one easily verifies the ( , ∞) -improving property (2.5) |h A ( s ) ∗ ( f L ) , b i| . Õ τ Õ j | B ( c τ , c s )| ∫ (cid:18) ∫ ψ τ ( x ) f ( x ) L ( x ) d x (cid:19) ψ τ ( y ) b j ( y ) d y . Õ τ : c L τ ∩ L , ∅ Õ j : B j ∩ c L τ , ∅ | B ( c τ , c s )| ∫ (cid:18) ∫ L | ψ τ ( x ) f ( x )| d x (cid:19) | ψ τ ( y ) − ψ τ ( c τ )|| b j ( y )| d y . r s h f i , γ L h Õ j | b j |i , γ L for some constants γ , γ > X ; passing to the last line of the estimate above weused the Lipschitz condition on the functions ψ τ , the fact that supp ( b j ) ⊆ B j , the finite overlapof the balls { ρL τ } and the fact that each ball L τ has radius comparable to r L . This shows the METRIC APPROACH TO SPARSE DOMINATION 11 ( , ∞) -improving property (2.5) and by Theorem B and the fact that M f . sup s | A s (| f |)| weget the following. Corollary C.1.
For every f , f bounded with compact support there exists a sparse collection B such that |h M f , f i| . Õ B ∈B | B |h f i , B h f i , c o B . Singular Radon transforms along polynomial manifolds.
As anticipated in the in-troduction, our focus is on Radon transforms as examples of L p -improving operators, in par-ticular singular integrals along free monomial varieties. For this reason we focus on the nextparagraph on metric spaces of the form ( R n , d , | · |) , where | · | is the Lebesgue measure.3.2.1. Homogeneous norms on R n . Consider the metric space ( R n , d , | · |) where | · | denotes theLebesgue measure. In particular the underlying space is a vector space and we have transla-tions. Furthermore we will assume that the metric d is given by a quasi-norm ρ : R n → [ , ∞) and that there exists a dilation structure δ t : R n → R n , t >
0, with respect to which thequasi-norm ρ is homogeneousd ( x , y ) ≔ ρ ( x − y ) , ρ ( δ t x ) = t ρ ( x ) , x , y ∈ R n , t > . For the purposes of this paragraph it will be enough to consider the special case that thereexist α , ... , α n > δ t ( x , ... , x n ) ≔ ( t α x , ... , t α n x n ) , ( x , ... , x n ) ∈ R n , t > . One of many equivalent quasi-norms compatible with δ t can be defined as(3.3) ρ ( x ) ≔ (cid:16) n Õ j = | x j | αj (cid:17) , x = ( x , ... , x n ) ∈ R n , and ρ is homogeneous with respect to δ t . Clearly ρ is symmetric and satisfies a quasi-triangleinequality. Furthermore for any ball B ( x , r ) given by ρ we have | B ( x , r )| = | B ( , r )| h n r α where α ≔ α + ··· + α n will be referred to as the homogeneous dimension of ( R n , ρ , | · |) . Inthis context will will write ( R n , ρ , | · |) for the homogeneous metric structure on R n describedby these definitions. We note that this setup is classical and further details can be found inseveral references, see for example [27, 28, 29] and the references therein.This scenario is particularly useful for the applications to singular operators given by inte-gration against a measure supported on appropriate sub-manifolds of R n . For this reason weshall show an alternative way to deduce the ( p , p ′ ) -improving property, which is arguably themost crucial assumption in Theorem A given above.As mentioned in the introduction, several operators of interest, such as singular integralsgiven by convolution with measures supported on lower dimensional sets, satisfy a stronger ( p , p ′ ) -improving property described in Lemma 3.3 below. Our first task here is to deduce the ( p , p ′ ) -improving property of Definition 2.3 in that case. Lemma 3.3.
Consider the space ( R n , ρ , | · |) and a dilation semi-group { δ r } r > such that thequasi-norm ρ is homogeneous with respect to δ r . For each s ∈ Z let m s be a Borel probabilitymeasure supported on the unit ρ -ball of R n and define the Borel measures b µ s ( ξ ) ≔ c m s ( δ s ξ ) and T ( s ) f ≔ f ∗ d µ s . The following hold. (i) Suppose that | c m ( ξ )| . | ξ | − β for some β > and that T ( ) : L p → L p ′ for ( p − , p − ) ∈ Ω for some open set Ω . Then T ∼ Í s T ( s ) is ( p , p ′ ) -improving and T ∗ ∼ s T ( s ) ∗ is ( p , p ′ ) -improving in the sense of Definition 2.3 for indices in the same open set Ω . (ii) If the conclusion of either Theorem A or Theorem B hold for the truncations T τσ = Õ σ ≤ s < τ T ( s ) or for the maximal operator T ⋆ f = sup s ∈ Z | T ( s ) f | and ( p − , p − ) ∈ Ω for some open set Ω , then T ( ) : L p → L p ′ for indices in the sameopen set Ω .Proof. We begin with (i). The first step is to note the following scale-invariant continuityestimate: for every scale s ∈ Z and y ∈ B ( , c s ) and every ρ -ball L or radius r L h s (3.4) (cid:10) [ T ( s ) − Tr y T ( s )]( f L ) (cid:11) p ′ , c o L ≤ ω (cid:0) ρ ( δ − s y ) (cid:1) h f i p , L , ( p − , p − ) ∈ Ω , where [ Tr y д ]( x ) ≔ д ( x − y ) for x , y ∈ R n , ω ( t ) = t ε for t small and ε depends on the assumptionsfor T , the quasi-metric structure implied by ρ and the exponents p , p . Now by translationinvariance and scale invariance it suffices to prove (3.4) for s = L = B ( , ) .This estimate for T ( ) without the decay term ω (cid:0) ρ ( δ − s y ) (cid:1) is then a direct consequence of theassumption in (i). On the other hand we have that k T ( s ) − Tr y T ( s ) : L ( R n ) → L ( R n )k ≤ sup ξ ∈ R n |( e − iy · ξ − ) c m ( ξ )| . ρ ( y ) ε ′ for some ε ′ > β in the decay assumption for c m and the constants involvedin the definition of ρ ; this last dependence comes from the fact that we implicitly used theestimate | y | ≤ ρ ( y ) c for | y | .
1. Since Ω is open we can use interpolation to conclude (3.4)with a power modulus of continuity for the same open set of indices Ω .In order to complete the proof of (i) we show that (3.4) implies Definition 2.3. To that endlet b = Í j b j be a ( p , r ) -molecule where b j ∈ L p ( B j ) , B j = B ( c j , r j ) are balls, ∫ B j b j =
0, and r j = r ≤ s . Let L be a ball of radius r L h s . By the mean zero condition on the b j ’s we have h T ( s )( f L ) , b i = Õ j h T ( s )( f L ) , b j i = Õ j | B j | ∫ B j (cid:18) ∫ B j [ T ( s )( f L )( x ) − Tr x − x ′ T ( s )( f L )( x )] b j ( x ) d x ′ (cid:19) d x = Õ j | B j | ∫ B j (cid:18) ∫ x − B j [ T ( s )( f L )( x ) − Tr y T ( s )( f L )( x )] b j ( x ) d y (cid:19) d x . Now we remember that | · | is Lebesgue measure and each ρ -ball B j has radius r so that | B j | = | B ( , r )| h r α for every j . Furthermore, for x ∈ B j we have that x − B j ⊆ B j − B j ⊆ B ( , c r ) for METRIC APPROACH TO SPARSE DOMINATION 13 some constant c depending on the quasi-metric d. These remarks and the previous calculationshow that h T ( s )( f L ) , b i = | B ( , r )| ∫ (cid:18) ∫ B ( , c r ) [ T ( s )( f L )( x ) − Tr y T ( s )( f L )( x )] , Õ j b j ( x ) x − B j ( y ) d y (cid:19) d x = | B ( , r )| ∫ B ( , c r ) D [ T ( s ) − Tr y T ( s )]( f L ) , c L Õ j b j y + B j E d y with c depending on the quasi-metric constant. Note that in the last line we are allowedto insert the indicator c L . Indeed, since T satisfies (2.1) and r L h s , r ≤ s , we have for y ∈ B ( , c r ) ⊆ B ( , c s ) thatsupp [ T ( s ) − Tr y T ( s )] ⊆ c o L ∪ ( c o L + y ) ⊆ c L for some constant c depending on c , c o , the implicit constants in r L h s , and the quasi-metricconstant of d. Using (3.4) we can now conclude |h T ( s )( f L ) , b i| . X ∫ B ( , c r ) ω ( ρ ( δ − s y ))| L | p ′ − p k f L k p (cid:13)(cid:13)(cid:13) Õ j c L | b j | (cid:13)(cid:13)(cid:13) p d y . ω ( c r / s )| L |h f i p , L (cid:10) Õ j | b j | (cid:11) p , c L . This proves that T is ( p , p ′ ) -improving with modulus ω ( t ) = t ε for some ε > ( p − , p − ) ∈ Ω .We now prove (ii). By Proposition 2.5 we can conclude the scale-invariant estimate h T ( )( f L )i p ′ , L . h f i p , L whenever L is a ball of radius r L h
1. But then one can decompose R n into a finitely overlappingcollection of balls { L τ } τ with r L τ h τ which yields k T ( ) f k p ′ L p ′ . Õ τ ∫ c o L τ | T ( )[ f L τ ]| p ′ . Õ τ | L τ | − p ′ p h f i p ′ p p , L τ h Õ τ (cid:16) ∫ L τ | f | p (cid:17) p ′ p . k f k p ′ p since | L τ | ≃ p ′ ≥ p . (cid:3) L p -improving properties for kernels with Fourier decay. With Lemma 3.3 in hand we nowderive as an application the L p -improving property for a class of operators which are givenby convolution with measures supported on lower dimensional manifolds on R n . The mainassumption is the suitable Fourier decay of the measures, which ultimately relies on suitablecurvature assumptions on their support. Lemma 3.4.
Consider the space ( R n , ρ , | · |) where ρ is the quasi-norm given in (3.3) and | · | denotes the Lebesgue measure. For each s ∈ Z let m s be a Borel measure supported on the metricball B ( , ) . Assume that there exists β > such that uniformly in s ∈ Z we have (3.5) (cid:12)(cid:12)c m s ( ξ ) (cid:12)(cid:12) . | ξ | − β , ξ ∈ R n , ∫ R n | d m s | ≤ . For each s ∈ Z define the scaled measure µ s as b µ s ( ξ ) ≔ c m s ( s ξ ) for ξ ∈ R n and T ( s ) f ≔ f ∗ d µ s .Then there exists a modulus of continuity ω ( t ) = | t | ε for some ε > depending only on n , β and the metric ρ such that T ∼ Í s T ( s ) is ( p , p ′ ) -improving at scale s and T ∗ ∼ Í s T ∗ ( s ) is ( p , p ′ ) -improving in the sense of Definition 2.3 whenever < p , p < and β + nn > p + p .Proof. By Lemma 3.3 it suffices to prove that k T ( ) : L p → L p ′ k . p , p as in theconclusion of the lemma. This in turn will follow by interpolation with the easy L → L bound and the estimate(3.6) k T ( ) д k L q ′ ( R n ) . k д k L q ( R n ) , < q ≤ , ≤ q ′ < ∞ , q + q = + βn . Assume first that q ′ >
2. Using the Hardy-Littlewood-Sobolev inequality we get the followingwhenever β ≥ γ > = q ′ + γn k[ T ( ) д ]k L q ′ ( R n ) = k |∇| − γ (|∇| γ [ T ( ) д ])k L q ′ ( R n ) h s , n k | x | γ − n ∗ (|∇| γ [ T ( ) д ])k L q ′ ( R n ) . k |∇| γ [ T ( ) д ]k L ( R n ) . Now the right hand side in the display above can be further estimated as follows using Plancherel’stheorem k |∇| γ [ T ( ) д ]k L ( R n ) = ∫ R n | ˆ д ( ξ )| | ξ | γ | b µ ( ξ )| d ξ ≤ ∫ | ξ |≤ | ˆ д ( ξ )| d ξ + ∫ | ξ | > | ˆ д ( ξ )| | ξ | ( γ − β ) d ξ . The first summand above can be estimated by Hölder’s inequality combined with the Hausdorff-Young inequality by k д k L q for any 1 ≤ q ≤
2. Furthermore, if q = β = γ . Now suppose that 1 < q < < γ < β so that β − γ = n ( / q − / ) which is always possible. Then we can estimate ∫ | ξ | > | ˆ д ( ξ )| | ξ | ( β − γ ) d ξ . ∫ R n | ˆ д ( ξ )| ( + | ξ | ) ( β − γ ) d ξ . k д k L q ( R n ) by the dual Sobolev embedding theorem. Finally if q ′ = L -estimate above and the dual form of the Sobolev embedding theorem which embeds H − β into L q for β = n ( / q − / ) . This proves (3.6) and thus the conclusion for T ∼ Í s T ( s ) . As T isessentially self-adjoint we get for free that T also satisfies (2.5) and the proof is complete. (cid:3) We recall the known fact that operators given at each dyadic scale by a convolution with ameasure that has Fourier decay as in Lemma 3.4 are bounded on L p ( R n ) . Lemma 3.5.
Let { m s } s ∈ Z be a sequence of Borel measures all supported on the metric ball B ( , ) and such that ∫ R n | d m s | ≤ . Assume that there exists β > such that for every s ∈ Z (cid:12)(cid:12)c m s ( ξ ) (cid:12)(cid:12) . | ξ | − β , ξ ∈ R n . For each s ∈ Z define the scaled measure µ s as b µ s ( ξ ) ≔ c m s ( s ξ ) for ξ ∈ R n . (i) The operator T ⋆ f ≔ sup s | f ∗ d µ s | extends to a bounded operator on L p ( R n ) for all < p < ∞ . METRIC APPROACH TO SPARSE DOMINATION 15 (ii)
If in addition we have that ∫ R n d m s = for all s ∈ Z then T f ≔ Í s ∈ Z f ∗ d µ s extendsto a bounded operator on L p ( R n ) for all < p < ∞ . The same holds uniformly for thetruncations T τσ f = Í σ ≤ s < τ T ( s ) f . We omit the well known proof of Lemma 3.5 and refer the reader to [28, §XI.2.5] for themaximal version of (i), and to [28, §XI.4.4] for the singular integral version of (ii) above.With the ingredients above it is now easy to conclude a sparse domination theorem foroperators given by convolutions with suitable measures possessing Fourier decay as above.
Theorem D.
Consider the space ( R n , ρ , | · |) where ρ is a quasi-norm and | · | denotes the Lebesguemeasure. For each s ∈ Z let m s be a Borel measure supported on the metric ball B ( , ) . Assumethat there exists β > such that sup s ∈ Z ∫ R n | d m s | ≤ , sup s ∈ Z sup ξ ∈ R n | ξ | β (cid:12)(cid:12)c m s ( ξ ) (cid:12)(cid:12) ≤ For each s ∈ Z define the scaled measure µ s as b µ s ( ξ ) ≔ c m s ( δ s ξ ) and let T ⋆, σ , τ f ≔ sup σ < s ≤ τ | f ∗ d µ s | , T τσ f = Õ σ ≤ s < τ f ∗ dµ s . (i) For all f , f ∈ S( R n ) with compact support and σ , τ ∈ Z with σ < τ there exists a sparsecollection B ⋆, σ , τ consisting of balls B with σ ≤ r B ≤ τ such that |h T ⋆, σ , τ f , f i| . Õ B ∈B ⋆, σ , τ | B |h f i p , B h f i p B whenever β + nn > p + p ≥ . (ii) If in addition we have that ∫ d m s = ∀ s ∈ Z , then for all σ , τ ∈ Z with σ < τ and for all f , f ∈ S( R n ) with compact support thereexists a sparse collection B σ , τ consisting of balls B with σ ≤ r B ≤ τ such that |h T τσ f , f i| . Õ B ∈B | B |h f i p , B h f i p B whenever ≤ p , p ≤ and β + nn > p + p ≥ .The corresponding conclusions hold for the untruncated versions with sparse collections consistingof balls of all radii. Sparse domination for singular Radon transforms.
We culminate the considerations ofthis section by describing a class of singular Radon transforms given by convolution withmeasures supported on polynomial subvarieties of R n . To make this specific we fix somepositive integer d and consider the polynomial map(3.7) γ : R k → R N , γ ( t ) = ( t α ) ≤| α |≤ d , where N is the number of monomials t α = t α ··· t α k with | α | = α + ··· + α k ≤ d . It is convenientto describe points x ∈ R N in the form x = ( x α ) ≤| α |≤ d . With these conventions in hand wedefine dilations δ r (( x α ) α ) ≔ (( r | α | x α ) α ) and ρ by ρ ( x ) = (cid:18) Õ ≤| α |≤ d | x α | | α | (cid:19) , x = ( x α ) α ∈ R N , which is just formula (3.3) in current notation. We can always compare the quasi-norm ρ withthe Euclidean one by means of(3.8) ( | x | d . ρ ( x ) . | x | if | x | > , | x | . ρ ( x ) . | x | d if | x | ≤ . The homogeneous dimension of ( R N , ρ , | · |) is ∆ ≔ Õ ≤| α |≤ d | α | . Note the following basic behavior of ρ ( γ ( t )) with respect to dilations: for every r > r ρ ( γ ( t )) = ρ ( δ r γ ( t )) = ρ ( γ ( rt )) , rt = ( rt , ... , rt k ) ∈ R k . Now let Ω : S k − → R be a 0-homogeneous function with mean zero on S k − and Ω ∈ C ∞ ( S k − ) . We define the singular Radon transform(3.10) T γ f ( x ) ≔ p . v . ∫ R k f ( x − γ ( t )) Ω ( t )| t | k d t , x ∈ R N , and we have that T γ f ( x ) = Õ s ∈ Z [ T ( s ) f ]( x ) ≔ Õ s ∈ Z ∫ R k f ( x − γ ( t )) ψ (cid:18) | t | s (cid:19) Ω ( t ) d t | t | k , x ∈ R N , with ψ ∈ S( R ) , 0 ≤ ψ ≤ ψ is compactly supported in [ / , ] and identically one in [ , ] and such that Í s ∈ Z ψ (cid:0) | t | s (cid:1) h
1. Consider the Borel measure m defined as ∫ R N ϕ ( y ) d m ( y ) = ∫ R k ϕ ( γ ( t )) Ω ( t )| t | k d t ∀ ϕ ∈ S( R n ) , Ω ( t ) ≔ Ω ( t ) ψ (cid:0) | t | (cid:1) , which by (3.8) is compactly supported in some ρ -ball in R N of fixed radius and centered at 0.By (3.9) we have that T γ f = Õ s ∈ Z T ( s ) f , T ( s ) f = f ∗ d µ s , c d µ s ( ξ ) ≔ b m ( δ s ξ ) , ξ ∈ R N . For each s ∈ Z the measure d µ s is a rescaling of the measure d µ = m ∫ R N ϕ ( x ) d µ s ( x ) = ∫ R k ϕ ( δ s γ ( t )) Ω ( t )| t | k d t = ∫ R k ϕ ( γ ( t )) Ω ( t )| t | k ψ (cid:0) | t | s (cid:1) d t ∀ ϕ ∈ S( R N ) . With these definitions in mind and using (3.9) and (3.8) it is routine to verify that T obeys (2.1).We also record the basic calculation c d µ s ( ξ ) = ∫ R k e − iξ · γ ( s t ) Ω ( t )| t | k ψ (| t |) d t = ∫ S k − (cid:18) ∫ ∞ e − iξ · γ ( s rt ′ ) ψ ( r ) d rr (cid:19) Ω ( t ′ ) d σ k − , ξ ∈ R N , METRIC APPROACH TO SPARSE DOMINATION 17 whence ∫ R N d µ s = ∫ R N d m = b m ( ) h ∫ S k − Ω ( t ′ ) d σ k − ( t ′ ) = Ω . The previous calculation also implies that for all s ∈ Z we have k m k = k d µ s k . k Ω k L ( S k − ) . The companion maximal operator is given as M f ( x ) ≔ sup r > r k ∫ | t |≤ r | f ( x − γ ( t ))| d t . As before letting ψ s (| t |) ≔ − ks ψ (| t |/ s ) we can bound M γ f ( x ) . sup s ∈ Z sk ∫ s − ≤| t | < s | f ( x − γ ( t )| d t . ∫ | f ( x − γ ( t )| ψ s (| t |) d t ≕ sup s | f | ∗ d ν s with ∫ R n ϕ ( t ) d ν s = sk ∫ R k ϕ ( γ ( t )) ψ s (| t |) d t ∀ ϕ ∈ S( R n ) . As before it can be easily seen that d ν s is the δ s scaling of the measure ν = ν so that c d ν s ( ξ ) = c d ν ( δ s ξ ) and d ν is a compactly supported Borel measure with k d ν k = k d ν s k . s ∈ Z .We only miss one main ingredient in order to apply Theorem D for the singular and maximalRadon transforms, T and M , respectively; that is, the Fourier decay of the generating measure.However such estimates are standard in the context above since the polynomial map γ is offinite type d . Lemma 3.6.
Let d µ , d ν denote the Borel measures defined above. Then | c d ν ( ξ )| . | ξ | − d , | c d µ ( ξ )| . | ξ | − d , ξ ∈ R N . The proof of the lemma above is classical and relies on the fact that the smooth polynomialmap R k ∋ t γ ( t ) is of finite type (at most d ) at each point; see [28, §XI.2.2] and [28, §VII.3.2].Combining the estimates of Lemma 3.6 and the properties of the operators T γ , M γ yieldsthe following sparse domination result. Corollary D.1.
Let γ : R k → R N be the map γ ( t ) = ( t α ) ≤| α |≤ d with N denoting the dimensionof the space spanned by the monomials of degree at most d , and let ρ be given by (3.7) . For every f , f ∈ S( R n ) with compact support there exists sparse collections B ⋆ , B such that |hM γ f , f i| . Õ B ∈B ⋆ | B |h f i p , B h f i p , B , |h T γ f , f i| . Õ B ∈B | B |h f i p , B h f i p , B , whenever ≤ p , p ≤ and ≤ p + p ≤ + Nd . Corresponding statements hold for thetruncated versions as above. Variations of the sparse domination result are possible with weaker conditions on Ω forexample but we do not pursue those here. Furthermore one can provide a sparse dominationtheorem whenever some L p → L p ′ improving property is known. We give one such examplebelow. Corollary D.2.
Let γ : R → R denote the polynomial map γ ( t ) = ( t , t , t , t t , t ) anddefine the singular Radon transform T τσ f ( x ) ≔ ∫ σ ≤| t | < τ f ( x − γ ( t )) Ω ( t )| t | d t , with ∫ S Ω ( t ) d σ ( t ) = and Ω ∈ L ∞ ( S ) . For every f , f with compact support and every σ , τ ∈ Z with σ , τ there exists a sparse collection B σ , τ consisting of ρ -balls B with σ ≤ r B ≤ τ such that h T τσ f , f i . Õ B ∈B σ , τ | B |h f i B , p h f i B , p whenever ( p , p ) is in the interior of the triangle with vertices ( , ) , ( , ) , and ( , ) . A similarestimate holds for the maximal operator M f ( x ) ≔ sup r > r ∫ | t |≤ r | f ( x − γ ( t ))| d t . Furthermore no such sparse bound can hold outside the closed triangle with vertices as above.Proof.
Let T ( s ) denote the single scale operator T s + s . Observe that | T ( s ) f | . k Ω k L ∞ ( S ) s ∫ s ≤| t | < s + | f ( x − γ ( t ))| d t By [12] we know that T ( ) maps L p → L p ′ whenever ( p , p ) is in the open triangle of thestatement. It is also well known that the measure d m ∫ R ϕ ( x ) d m = ∫ ≤| t | < ϕ ( γ ( t )) d tt satisfies | b m ( ξ )| . | ξ | − as in Lemma 3.6. Since T ( s ) f = f ∗ d µ s with c d µ s ( ξ ) ≔ b m ( δ s ξ ) ,Lemma 3.3 shows that T ∼ Í s T ( s ) is ( p , p ′ ) improving in the sense of Definition 2.3. Fur-thermore these operators are singular Radon transforms along polynomial varieties of finitetype so they are known to be bounded on L p ( R ) for 1 < p < ∞ ; see for example [28, §XI].The sparse domination follows by an application of Theorem A.We prove the sharpness of the sparse region for T τσ by recalling a well known example. Let f δ ≔ B ( c o , δ ) with δ small and c o ≔ γ ( ) . Consider also Ω bounded and with mean zero on S and such that Ω ≡ R . We can then easily calculate that | T ( s ) f | = T ( s ) f & δ on the set of x in the positive quadrant of R such that | x − γ ( t )| ≤ δ / t ∈ ( / , ) . The set of such x has measure & δ and so T ( ) : L p → L p ′ implies that δ δ p ′ . k T ( )k L p ′ . k f δ k L p δ p which together with the symmetric estimate which follows by self-duality yields the restric-tions 2 + p ′ ≥ p , + p ′ ≥ p . The restrictions above describe the closure of the triangle in the statement so for ( p − , p − ) out-side the closed triangle the sparse domination result of the corollary has to fail. The exampleproving the sharpness of the sparse form for M is similar but simpler. (cid:3) METRIC APPROACH TO SPARSE DOMINATION 19
4. Whitney covers in geometrically doubling metric spaces
In this section we describe the covering argument that will be employed in the proof of themain theorem as a way to obtain appropriate stopping balls. Before stating it we first recallthe notion of a geometrically doubling metric space . Definition 4.1.
We will say that the quasi-metric space ( X , d ) is geometrically doubling if thereexists some positive integer N such that every ball of radius r may be covered by at most N balls of radius r / ( X , d , | · |) is a doubling quasi-metric measure space, then X isautomatically geometrically doubling. Secondly, we note that the definition of the geometricdoubling property does not really depend on the quasi-metric. Indeed, if ( X , d ) has the geo-metric doubling property then so does ( X , d ′ ) for any quasi-metric d ′ which is equivalent tod. In that case the number N appearing in the definition of geometric doubling will dependon the choice of quasi-metric; see [2, §2.1] for an extensive discussion on the geometry ofquasi-metric spaces. We shall not pursue these subtle issues in the current paper as for usthe consideration of a single quasi-metric in X will be sufficient, and the precise value of therelevant constant is unimportant.We state below the Whitney-type covering result that will be used throughout the paper. Inthe formulation below this Whitney decomposition is contained in [2, Theorem 2.4]. Theorem (Whitney-type decomposition, [2]) . Let ( X , d ) be a geometrically doubling quasi-metric space. Then for every η ∈ ( , ∞) there exist Λ ∈ ( η , ∞) and M ∈ N , both depending on d , η , and the geometric doubling constant of ( X , d ) , and which have the following significance.For each proper, nonempty, open subset Ω ⊂ X there exists a sequence of points { c j } j ∈ N in Ω and a sequence of positive radii { r j } j ∈ N , such that the following hold: (i) Ω = ∪ j ∈ N B ( c j , r j ) . (ii) We have that Í j ∈ N B ( c j , ηr j ) ≤ M . (iii) For each j ∈ N we have that B ( c j , ηr j ) ⊆ Ω and B ( c j , Λ r j ) ∩ ( X \ Ω ) , ∅ . (iv) If B ( c j , ηr j ) ∩ B ( c i , ηr i ) , ∅ for i , j ∈ N then r i h r j , with implicit constants independentof i , j ∈ N . Notice that (iv) above follows easily from (iii). Whenever we apply the Whitney decompo-sition above for some value of η ∈ ( , ∞) in order to produce a covering { B j } j of some openset Ω we will say that { B j } j is an η - Whitney covering of Ω , and we will use properties (i)-(iv)above with no particular mention. Another small reduction we will use is that, applying the2 η -Whitney-type covering we can always assume that the radii r j of the Whitney covering aredyadic. This provides an η -Whitney cover of Ω with balls of dyadic radii. We will always usethis reduction in what follows.Finally we note the following well known property of the Whitney covering in the case thatthe metric space supports a doubling measure. Lemma 4.2.
Let ( X , d , | · |) be a space of homogeneous type, η ∈ ( , ∞) , and let { B j } j be an η -Whitney cover of an open set Ω . Then for any j we have ♯ { ℓ : B j ∩ B ℓ , ∅} . X , η M , with the implicit constant depending on the doubling constant of | · | and the chosen η of theWhitney cover.Proof. Let J ≔ { ℓ : B j ∩ B ℓ , ∅} . Then ∫ Õ ℓ ∈ J B ℓ = Õ ℓ ∈ J | B ℓ | h X , η ♯ J | B j | using (iv) of the Whitney decomposition together with the fact that B ℓ ∩ B j , ∅ for all ℓ ∈ J and that | · | is doubling. On the other hand we have that ∪ ℓ B ℓ ⊆ cB j for some constant c depending on X , as r j h r ℓ , uniformly in ℓ, j . We conclude that ♯ J | B j | h ∫ Õ ℓ ∈ J B ℓ ≤ M (cid:12)(cid:12)(cid:12) Ø ℓ ∈ J B ℓ (cid:12)(cid:12)(cid:12) ≤ M | cB j | . X M | B j | and the lemma follows. (cid:3) In what follows we will need to split the support of our functions into essentially disjointballs of fixed scale. This is done in the following lemma which follows by more or less standardarguments in spaces of homogeneous type. In fact it is essentially contained in [14, Theorem1.16].
Lemma 4.3.
Let ( X , d , | · |) be a space of homogeneous type and let B be a ball. For each s ∈ Z with s ≤ r B there exists a countable collection of balls { L τ } τ with r L τ = s such that L τ ⊆ c B for each τ , and for every ρ > we have Í τ ρL τ . ρ , X . The constant c > and the implicitconstant depend on the homogeneous metric structure of X and on ρ .Proof. Let B be the collection of balls { B ( x , s ) : x ∈ B } . Obviously this collection covers B and sup B ∈B r B < ∞ . By the 5 R -covering lemma, see for example [14, Theorem 1.2], thereexists a disjoint subcollection B ′ = { B τ } τ ⊂ B such that Ø B ∈B B ⊆ Ø B ∈B ′ B . Here one can easily check that B ′ is necessarily finite. Let L τ ≔ B τ for each τ . Obviouslywe have that L τ ⊆ c B for some c > r L τ = s ≤ r B . Itremains to show the bounded overlap property. This follows by a well known argument thatwe include here for completeness.For x ∈ X and ρ > I x ≔ { τ : x ∈ ρB τ } . We have that B ( x , − s ) ⊆ cB τ ⊆ ˜ cB ( x , − s ) for all τ ∈ I x , where c , ˜ c depend on the quasi-metric of X and on ρ . Since | · | is doubling and theballs B ( x , − s ) and 5 ρB τ intersect and have comparable radii we get that | B ( x , − s )| h ρ | B τ | for all τ ∈ I x , with implicit constants depending on the homogeneous metric structure of X and on ρ . Since the balls B τ are disjoint we now have | B ( x , − s )| & ρ , X (cid:12)(cid:12)(cid:12) Ø τ ∈ I x B τ (cid:12)(cid:12)(cid:12) = Õ τ ∈ I x | B τ | & ρ , X ♯ I x | B ( x , − s )| and since | · | is doubling we get ♯ I x . ρ , X x , with implicit constants dependingon the homogeneous metric structure of ( X , d , | · |) and on ρ . Thus for x ∈ X we have Õ τ L τ ( x ) = Õ τ B τ ( x ) = ♯ I x . ρ , X METRIC APPROACH TO SPARSE DOMINATION 21 and the proof is complete. (cid:3)
Finally we record a standard estimate for doubling measures that allows us to compare theratio of radii of nested balls by the corresponding ratio of their measures; see for example[14, (4.16)].
Lemma 4.4.
Let ( X , | · | , d ) be a quasi-metric space of homogeneous type, that is, | · | is doubling.Then there exist constants ∆ X , δ X > depending only on the homogeneous metric structure of X such that for every pair of metric balls B ( x , r ) ⊆ B ( z , R ) we have | B ( x , r )|| B ( z , R )| ≥ ∆ X (cid:0) rR (cid:1) δ X . In fact one can take ∆ X h / β and δ X h log β with β the doubling constant of | · | and theimplicit constants depending on the quasi-metric constant of d .
5. Proof of Theorem A
Throughout the section we fix a space of homogeneous type ( X , d , | · |) . In all the estimatesbelow there are implicit constants depending on the constants c d , ˜ c d of the quasi-metric d, andon the doubling constant β of | · | . We abbreviate these dependencies by writing A . X B inorder to simplify the notation.As the underlying measure is doubling, the maximal operatorsM p f ≔ sup B h f i p , B B satisfy k M p k p → p , ∞ . X , k M p k q → q . X , q , q > p ≥ . An iteration procedure similar to that carried out in the proof of [8, Theorem C] yields Theo-rem A from the following Lemma. The iteration procedure is described in detail in §5.3.
Lemma 5.1.
Let κ ∈ ( , ) . We assume that T is an operator satisfying the assumptions ofTheorem A. Then there exist constants C h X C p + C p ′ and η = η ( X ) > such that the followinghold: let σ ∈ Z , B be a fixed ball with r B = s B ∈ Z , and f , f be bounded functions. Then thereexists η & X , a finite Whitney-type collection of balls B = { B j } j with r B j = s Bj and functions { f , j } j with | f , j | ≤ | f | B j for each j , such that (5.1) F B ≔ Ø j B j , e F B ≔ Ø j ηB j ⊂ ηB , | F B | ≤ κ | B | and (5.2) |h T s B σ ( f B ) , f i| ≤ Cκ − | B |h f i p , B h f i p , c o B + Õ j |h T s Bj σ ( f , j ) , f i| . Furthermore we have that inf j r B j ≥ σ and sup j r B j < β κ β r B for some constants β , β > depending on the homogeneous metric structure of ( X , d , | · |) , but not on κ . Proof of Lemma 5.1.
We begin by observing that, in view of the localization property(2.1) there is no loss in generality in assuming that supp f ⊂ B , supp f ⊂ c o B and normalizing h f i p , B = h f i p , c o B = Definition of the auxiliary set F B . With κ ∈ ( , ) fixed throughout the proof we firstconstruct the auxiliary set F B . This is done as follows: F B ≔ (cid:8) x ∈ X : max i = , M p i f i ( x ) > Θ / κ (cid:9) where Θ is a constant depending on our structural assumptions, to be chosen later. By ele-mentary considerations we have for x < ηB thatM p j f j ( x ) ≤ (cid:18) sup L ∋ xL ∩ c o B , ∅ | c o B |/| L | (cid:19) pj , where the supremum above is taken over metric balls L . For a ball L in the supremum abovewe have that r L & X r B if η is chosen to be sufficiently large depending on the constants of X and the doubling condition implies that | c o B | ≤ C X | L | and thus max i M p i f i ( x ) ≤ C X < C X / κ for x < ηB . Choosing Θ so that 2 Θ > C X guarantees that F B ⊂ ηB as desired. Then the maximaltheorem provides the estimate | F B | . X κ − Θ | B | and thus | F B | < κ | B | assuming Θ sufficiently large depending only upon X . This concludes theconstruction of the set F B . Observe that η ≥ X .5.1.2. Construction of the Whitney-type collection B . We consider the η -Whitney covering B = { B j } j = { B ( c j , r B j )} j of the open set F B given by Theorem 4; we will write r B j = r j = s j for therest of the proof in order to clean up the notation. We remember that the balls in B can bechosen to have dyadic radii. Furthermore we have that ηB j = B ( c j , ηr j ) ⊂ F B and thus ηB j ⊆ ηB for all j . We remember also that Í j ηB j ≤ M and that if ηB ∩ ηB ′ , ∅ then r B h X r B ′ .We now get an easy estimate for the radii { r j } j . Since | · | is doubling we have κ | ηB | h X κ | B | > | F B | = (cid:12)(cid:12)(cid:12) Ø j ηB j (cid:12)(cid:12)(cid:12) ≥ sup j | ηB j | . Since ηB j ⊂ ηB and | · | is doubling we get by Lemma 4.4 that for every j we have (cid:16) r j r B (cid:17) δ X . X | ηB j || ηB | . X κ and thus there exists constants β , β , depending only on X , such that sup j r j ≤ β κ β r B .5.1.3. Proof of (5.2) . All the implicit constants throughout this section may depend on theparameters of the space of homogeneous type c d , ˜ c d , β and on the constants c o , c of (2.1). Inorder to clean up the presentation we suppress these dependencies in what follows and we onlykeep track of the dependence on the norm bounds C p , C p ′ for T and on κ in the assumptionof the lemma. The first observation we make is that k f i ( F B ) c k ∞ . κ − . Since we assume the uniform boundedness of the truncations T τσ on L p and L p ′ we have |h T s B σ f ( ( F B ) c ) , f i| ≤ C p ′ k f ( F B ) c k p ′ k f c o B k p . κ − C p ′ | B | , |h T s B σ f , f ( ( F B ) c )i| . C p k f k p k f ( ( F B ) c )k p ′ . κ − C p | B | . (5.3) METRIC APPROACH TO SPARSE DOMINATION 23
By (5.3) we may assume in what follows that f , f are supported on F B . Using the Whitneycollection { B j } j we construct a –non-smooth, since we do not need smoothness for anything–partition of unity on F B by writing w j ≔ B j Í j B j , supp ( w j ) ⊆ B j , Õ j w j = F B . As we assume that f , f are supported on F B we then have f i = Õ j f i w j ≕ Õ j f i , j , | f i , j | ≤ | f i | B j , i ∈ { , } . We can then perform a Calderón-Zygmund decomposition f i = b i + д i where b i = Õ j b i , j , b i , j ≔ f i , j − (cid:16)∫ B j f i , j (cid:17) B j ≔ f i , j − д i , j , д i = Õ j д i , j , i ∈ { , } , and we note the following estimates for future use. Firstly we remember that, by the Whitneydecomposition, there exists Λ > η such that Λ B j ∩( X \ F B ) , ∅ . This together with the doublingproperty of | · | and the uniform bound on the overlap of the Whitney balls { B j } imply that(5.4) k д i , j k ∞ . κ − B j ∀ j , k д i k ∞ . κ − , i ∈ { , } . For the bad parts we have by definition that ∫ B j b j = j while for the averages we usethe Λ -stopping condition on the Whitney balls to get h b i i p i , B . (cid:16) | B | − pi Õ j ∫ B j | b i , j | p i (cid:17) pi . κ − (cid:16) | B | − Õ j | B j | (cid:17) pi . κ − . In the penultimate estimate we also used (5.1); thus(5.5) h b i , j i p i , B j . κ − , h b i i p i , B . κ − . We now split the duality form for our operator by means of the main estimate below h T s B σ f , f i = Õ j h T s B σ f , j , f i = Õ j : s j > σ h T s B σ f , j , f i + Õ j : s j ≤ σ h T s B σ f , j , f i = Õ j : s j > σ h T s j σ f , j , f i + Õ j : s j > σ h T s B s j f , j , f i + Õ j : s j ≤ σ h T s B σ f , j , f i = Õ j : s j > σ h T s j σ f , j , f i + Õ j h T s B max ( s j , σ ) f , j , f i . (5.6)Observe that the first summand above can be estimated by the form required in the secondsummand if the conclusion of the lemma. Note also that there are only finitely many termsappearing in the first summand of the identity above. Thus the proof will be complete oncewe show that(5.7) (cid:12)(cid:12)(cid:12) Õ j h T s B σ j f , j , f i (cid:12)(cid:12)(cid:12) . κ − ( C p + C p ′ )| B | , σ j ≔ max ( s j , σ ) . The rest of this paragraph is devoted to the proof of (5.7). We use the Calderón-Zygmunddecomposition to split the left hand side of (5.7) into four terms as follows Õ j h T s B σ j f , j , f i = Õ j h T s B σ j д , j , д i + Õ j h T s B σ j b , j , д i + Õ j h T s B σ j д , j , b i + Õ j h T s B σ j b , j , b i ≕ GG + BG + GB + BBand we proceed to estimate each one of these terms.The estimate for the term GG is the easiest one: using for example the L p -boundedness ofthe truncations (2.2) together with the fact that each д i , j is supported on B j , we get by (2.1),(5.4) and (5.5) that we have | GG | ≤ |h T s B σ д , д i| + Õ j : s j > σ |h T s j σ д , j , д c o B j i| . C p k γ B k p k д c o B k p ′ + C p Õ j k д B j k p k д c o B j k p ′ . κ − | B | since the collection { B j } j has finite overlap. The estimates for terms BG , GB are similar so weonly present the estimate for the term BG. Using (5.4) and (5.5) and the uniform boundednessof the truncations of T on L p we get | BG | ≤ |h T s B σ b , д i| + Õ j : s j > σ |h T s j σ b , j , д i| . C p k b k p k д c o B k p ′ + Õ j : s j > σ (cid:12)(cid:12) h T s j σ b , j , c o B j д i (cid:12)(cid:12) . κ − C p | B | + C p Õ j k b , j k p k c o B j д k p ′ . C p κ − | B | , where in the last approximate inequality we used again the uniform bound on the overlap ofthe Whitney balls { B j } j . The estimate for the term GB is symmetric, producing an estimate ofthe form | GB | . κ − C p ′ | B | .It remains to show the estimate for the term BB which is slightly more involved. To thatend we introduce some additional notation: Given k ∈ Z we let for i ∈ { , } b ki ≔ Õ j : s j = k b i , j where we remember that the radii s j were chosen to be dyadic and so b i = Í k ∈ Z b ki for i = , = Õ k ∈ Z s B Õ s = max ( k , σ ) h T ( s ) b k , b i = Õ k ≤ σ (cid:10) T s B σ b k , b (cid:11) + Õ k ∈ Z k > σ Õ j : s j = k (cid:10) T s B s j b , j , b (cid:11) ≕ BB ≤ σ + BB > σ . METRIC APPROACH TO SPARSE DOMINATION 25
We have implicitly defined | BB > σ | ≔ (cid:12)(cid:12)(cid:12) Õ k ∈ Z k > σ s B Õ s = k (cid:10) T ( s ) b k , b (cid:11)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Õ k > σ Õ ℓ ∈ Z s B Õ s = k (cid:10) T ( s ) b k , b ℓ (cid:11)(cid:12)(cid:12)(cid:12) ≤ Õ σ < s ≤ s B Õ σ < k ≤ s Õ ℓ (cid:12)(cid:12)(cid:10) T ( s ) b k , b ℓ (cid:11)(cid:12)(cid:12) = Õ σ < s ≤ s B Õ k ≥ Õ ℓ (cid:12)(cid:12)(cid:10) T ( s ) b s − k , b s − ℓ (cid:11)(cid:12)(cid:12) = Õ σ < s ≤ s B Õ k ≥ Õ ℓ ≥ k (cid:12)(cid:12)(cid:10) T ( s ) b s − k , b s − ℓ (cid:11)(cid:12)(cid:12) + Õ σ < s ≤ s B Õ k ≥ Õ ℓ< k (cid:12)(cid:12)(cid:10) T ( s ) b s − k , b s − ℓ (cid:11)(cid:12)(cid:12) ≕ BB > σ + BB > σ . The estimate for the bad-bad term and large balls.
We write the estimate for the firstterm BB > σ . For fixed s ∈ ( σ , s B ) let us cover the ball B by a countable collection of balls { L τ } τ with r L τ = s and finite overlap and such that L τ ⊆ cB , as provided by Lemma 4.3. Using thecollection { L τ } τ and invoking the ( p , p ′ ) -improving property of T we have (cid:12)(cid:12)(cid:10) T ( s ) b s − k , b s − ℓ (cid:11)(cid:12)(cid:12) ≤ Õ τ (cid:12)(cid:12)(cid:10) T ( s )( b s − k L τ ) , b s − ℓ (cid:11)(cid:12)(cid:12) ≤ ω ( − ℓ ) Õ τ | L τ |h| b s − k |i p , γ L τ D Õ j : r j = s − ℓ | b , j | E p , γ L τ . Now for any i ∈ { , } and n ∈ { s − k , s − ℓ } we use that supp b i , j ⊆ B j together with the finitenumber of neighbors property of Lemma 4.2 for the Whitney balls { B j } j to estimate ∫ γ i L τ (cid:16) Õ j : s j = n | b i , j | (cid:17) p ≤ Õ m : s m = n ∫ γ i L τ ∩ B m (cid:16) Õ j : s j = sB j ∩ B m , ∅ | b i , j | (cid:17) p . Õ m : s m = n ∫ γ i L τ ∩ B m Õ j : s j = nB j ∩ B m , ∅ | b i , j | p . ∫ γ i L τ Õ j : s j = n | b i , j | p ;passing to the last line we used the finite overlap property of the Whitney balls { B j } j . Summingin s we get for fixed k , ℓ ∈ Z that Õ σ < s ≤ s B (cid:12)(cid:12)(cid:10) T ( s ) b s − k , b s − ℓ (cid:11)(cid:12)(cid:12) . ω ( − ℓ ) Õ τ Õ σ < s ≤ s B | L τ | Õ j : s j = s − k ∫ γ L τ | b , j | p ! p Õ j : s j = s − ℓ ∫ γ L τ | b , j | p ! p ≕ ω ( − ℓ ) Σ k ,ℓ . We now claim that Σ k ,ℓ . | B | . Indeed remembering that p ′ ≥ p and setting δ ≔ / p − / p ′ ≥ Σ k ,ℓ ≤ sup s , τ , k (cid:18) Õ j : s j = s − k ∫ γ L τ | b , j | p (cid:19) δ Σ ′ k ,ℓ where Σ ′ k ,ℓ ≔ Õ τ Õ σ < s ≤ s B | L τ | Õ j : s j = s − k ∫ γ L τ | b , j | p ! p ′ Õ j : s j = s − ℓ ∫ γ L τ | b , j | p ! p . We estimate Σ ′ k ,ℓ by Hölder’s inequality Σ ′ k ,ℓ . Õ τ Õ σ < s ≤ s B Õ j : s j = s − k ∫ γ L τ | b , j | p ! p ′ Õ τ Õ σ < s ≤ s B Õ j : s j = s − ℓ ∫ γ L τ | b , j | p ! p . Õ τ ∫ γ L τ Õ j | b , j | p ! p ′ Õ τ ∫ γ L τ Õ j | b , j | p ! p . ∫ Õ j | b , j | p ! p ′ ∫ Õ j | b , j | p ! p where, in the last approximate inequality we used the property Í τ γ i L τ . j we conclude that Σ ′ k ,ℓ . κ −( + p / p ′ ) | B | .For the term with the supremum in (5.8) we note that for each s , τ and δ ≥ (cid:18) Õ j : s j = s − k ∫ γ L τ | b , j | p (cid:19) δ . (cid:18) Õ j : B j ∩ γ L τ , ∅ s j = s − k ∫ γ L τ | b , j | p (cid:19) δ . κ − p δ (cid:18) Õ j : B j ∩ γ L τ , ∅ s j = s − k | B j || L τ | (cid:19) δ . κ − p δ . In the last estimate we used the fact that r B j = r j = s j = s − k ≤ s = r L τ , since k ≥
0, togetherwith the doubling hypothesis and fact that the balls B j have finite overlap.Combining the estimates above and plugging them into (5.8) yields Σ . κ − | B | . Using thelog-Dini condition on ω we can thus concludeBB > σ . κ − | B | Õ ℓ : ℓ ≥ ω ( − ℓ ) ℓ . k ω k ℓ − Dini κ − | B | . The corresponding bound for BB > σ with ℓ < k follows by a similar argument, using the ( p , p ′ ) -improving property of T ∗ , according to Remark 2.4. More precisely writing B j = B ( c j , r j ) weneed to estimate Õ σ < s ≤ s B Õ k ≥ Õ j : s j = s − k Õ ℓ< k Õ m : s m = s − ℓ (cid:12)(cid:12)(cid:10) T ( s ) b , j , b , m (cid:11)(cid:12)(cid:12) = Õ σ < s ≤ s B Õ k ≥ Õ j : s j = s − k Õ ℓ< k Õ m : s m = s − ℓ (cid:12)(cid:12)(cid:10) T ( s ) b , j , B ( c j , c o s ) b , m (cid:11)(cid:12)(cid:12) By the Whitney property (iv) of the balls { B j } we only need to consider the range − C ≤ ℓ < k in the above sum, where the constant C > ℓ ≤ r m ≥ s > r j and notice that for a term in the sum above to be non-zero we needthat B ( c j , c o s ) ∩ B ( c m , r m ) , ∅ which implies that B ( c j , ηr j ) ∩ B ( c m , ηr m ) , ∅ if η was chosensufficiently large, depending only on the homogeneous structure of X . By property (iv) of theWhitney decomposition we then must have that r j h r m which is impossible if ℓ < − C forsome sufficiently large constant C >
0, depending again only on X . With this in mind theestimate for the term BB > σ follows by an argument similar to the one for BB > σ , by choosing adecomposition of the ball B into balls L τ of radius r L τ ≃ s , with implicit constants dependingon C above. This completes the proof of the desired estimate in the form of (5.7) for BB > σ . METRIC APPROACH TO SPARSE DOMINATION 27
The estimate for the bad-bad term and small balls.
We do a similar splitting as for theterm BB ≤ σ | BB ≤ σ | ≔ (cid:12)(cid:12)(cid:12) Õ k ∈ Z k ≤ σ Õ ℓ ∈ Z s B Õ s = σ (cid:10) T ( s ) b k , Õ ℓ ∈ Z b ℓ (cid:11)(cid:12)(cid:12)(cid:12) = s B Õ s = σ Õ k ≥ Õ ℓ ≥ k (cid:12)(cid:12)(cid:10) T ( s ) b σ − k , b σ − ℓ (cid:11)(cid:12)(cid:12) + s B Õ s = σ Õ k ≥ Õ ℓ< k (cid:12)(cid:12)(cid:10) T ( s ) b σ − k , b σ − ℓ (cid:11)(cid:12)(cid:12) ≕ BB ≤ σ + BB ≤ σ . The estimate for each of the two terms above follows exactly the same proof as for BB > σ ,partitioning the ball B into a finitely overlapping collection of balls { L τ } τ with r L τ = s foreach s ∈ [ σ , s B ] . We omit the details. Summing the estimates for the terms GG , BG , GB , BBand taking into account the splitting (5.6) completes the proof of Lemma 5.1.5.2.
Finite overlap implies sparse.
In order to complete the proof of Theorem A we willneed to iterate Lemma 5.1 until we exhaust all the available scales. Each time we applyLemma 5.1 we produce a finite family of balls B = { B j } j with finite overlap Í j B j ≤ M . Ourfirst task is to show that the finite overlap condition implies that the collection B is sparse,which is the content of the following lemma. Lemma 5.2.
Let { A j } j ≥ be a finite collection of sets such that Í j A j ≤ M , for some absoluteconstant M ≥ . Then for any measure | · | without point-masses we have that { A j } j is M − -sparse.Proof. Consider the σ -algebra Σ generated by the collection { A j } j , that is, Σ ≔ σ ({ A j }) . Then Σ is an atomic σ -algebra so that Σ = σ ({ α m }) where α m are the atoms of Σ . Now since | · | hasno point-masses there exist, for every m , partitions { α im } i such that α m = M Ø i = α im and | α im | = | α m |/ M . Let the the sets of indices Γ j be defined by means of A j ≕ Ø m ∈ Γ j α m . Then we can define major subsets E A j inductively as follows. For j = E A ≔ Ø m ∈ Γ α m , A ≔ { α im } m , i \ { α m } m ∈ Γ . Define for each m ∈ Γ the index n ( m , ) ≔ ℓ ≤ j and all m ∈ ∪ ℓ ≤ j Γ ℓ we have defined the indices n ( m , ℓ ) , andthen we set E A ℓ ≔ Ø m ∈ Γ ℓ α n ( m ,ℓ ) m , A ℓ ≔ A ℓ − \ { α n ( m ,ℓ ) m } m ∈ Γ ℓ . Then for each m ∈ Γ ℓ + we define the index n ( m , ℓ + ) ≔ min { i : α im ∈ A ℓ } and remarkthat n ( m , ℓ + ) is always well defined. Indeed, for ℓ + ≤ M this is obvious as α ℓ + m ∈ A ℓ sosuppose that ℓ + > M . Suppose, for the sake of contradiction, that for some m ∈ Γ ℓ + we havethat { i : α im ∈ A ℓ } = ∅ . Since ℓ + > M this means that α m ∩ A j , ∅ for at least M indices j with j ≤ ℓ . As α m ∩ A ℓ + , ∅ this means that α m ∩ A j for at least M + j , contradictingthe assumption Í j A j ≤ M . It remains to show that | E A j | & | A j | . Indeed we have | E A j | = Õ m ∈ Γ j | α n ( m , j ) m | = M Õ m ∈ Γ j | α m | = M | A m | so { A j } j is M − -sparse. (cid:3) Iterating Lemma 5.1.
The proof of Theorem A will follow by inductively applyingLemma 5.1 in order to prove a recurrence estimate, which is contained in the following lemma.
Lemma 5.3.
Let T ∼ Í s T ( s ) satisfy the assumptions of Theorem A and let f , f be Lipschitzfunctions with supp f ⊆ B o for some fixed ball B o with r B o = s Bo . There exist constants c X , ˜ c X and η = η X > , depending only on X , with the following meaning:There exist finite collections of balls T k and S k , k = { , , ... } , such that (5.9) |h T s Bo σ f , f i| ≤ C Õ B ∈T k − | L |h f i p , B h f i p , c o B + Õ L ∈S k |h T s L σ ( f , L ) , f i , r L = s L , where C . κ − ( C p + C p ′ + k ω k ℓ − Dini ) with implicit constant depending on X and the constant in (2.1) and (i) For every B ∈ T k − there exists E B ⊂ B \∪ L ∈S k L with | E B | > ˜ c X ( − κc X )| B | , and E B ∩ E B ′ = ∅ if B , B ′ ∈ T k with B , B ′ . (ii) The collection S k is a η -Whitney type collection consisting of balls B with radius r B = s B ≥ σ . In particular this means that the balls in S k have finite overlap and the finitenumber of neighbors property, and B , B ′ ∈ S k with ηB ∩ ηB ′ , ∅ implies that r B h X r B ′ ; allthe constants, explicit and implicit depend only upon X and the operator T . Furthermorewe have | f , L | ≤ | f | L for all L ∈ S k . (iii) For every positive integer k we have that sup L ∈S k r L = s L < β κ β sup B ∈T k − r B , where β , β > are the constants provided by Lemma 5.1.Proof. We prove the base step of the induction corresponding to k =
1. In this case we justapply Lemma 5.1 a single time to get (5.9) with T ≔ { B o } and S ≔ B( B o ) to be the Whitney-type collection of Lemma 5.1. Letting E B o ≔ B o \ ∪ L ∈B( B o ) L we note that | E B o | > ( − κ )| B o | .Statements (i), (ii) and (iii) are satisfied by Lemma 5.1.Now assume that (5.9) and (i),(ii), and (iii), hold for some k . We prove the inductive step. Forthis, we apply Lemma (5.1) to each ball L of the collection S k . This together with the inductivehypothesis provides the estimate |h T s Bo σ ( f ) , f i| ≤ C Õ B ∈T k − | B |h f i p , B h f i p , c o B + Õ B ∈S k | B |h f i p , B h f i p , c o B + Õ L ∈S k Õ L ′ ∈B( L ) |h T s L ′ σ ( f , L ′ ) , f i , where we also used the fact that | f , L | ≤ | f | L , provided by (ii) of the inductive hypothesis.For the constant C > C . X κ − ( C p + C p ′ ) . It is essential to note that if L ′ ∈ B( L ) then ηL ′ ⊆ ηL as proved in Lemma 5.1. We now set T k ≔ T k − ∪ S k , S k + ≔ {B( L ) : L ∈ S k } . METRIC APPROACH TO SPARSE DOMINATION 29
Conditions (ii) and (iii) are automatically satisfied for the collections T k and S k + by Lemma 5.1.With these definitions (5.9) is proved for k + B ∈T k \ T k − ⊆ S k . We set ˜ E B ≔ B \ Ø L ′ ∈S k + L ′ = B \ Ø L ∈S k Ø L ′ ∈B( L ) L ′ . Then we have (cid:12)(cid:12)(cid:12)(cid:12) B ∩ Ø L ∈S k Ø L ′ ∈B( L ) L ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Õ L ∈S k ηL ∩ ηB , ∅ (cid:12)(cid:12)(cid:12) Ø L ′ ∈B( L ) B ∩ L ′ (cid:12)(cid:12)(cid:12) ≤ κ Õ L ∈S k ηL ∩ ηB , ∅ | L | . Now we remember that, by the inductive hypothesis, we have that if L , B ∈ S k and ηL ∩ ηB , ∅ then r L h r B , and every ball in S k has a uniform bound, depending on X , on the number of itsneighbours. These facts together with the doubling property of | · | show that Õ L ∈S k ηL ∩ ηB , ∅ | L | . X | B | which in turn implies that | ˜ E B | & X ( − c X κ )| B | , with c X ≥ X , where wetake κ sufficiently small. Observe also that by the inductive hypothesis S k has finite overlap, sothe collection of sets { ˜ E B } B ∈S k also has finite overlap. By Lemma 5.2 the collection { ˜ E B } B ∈S k issparse so for every B ∈ S k there exist E B ⊆ ˜ E B ⊆ B \∪ L ∈S k + L with | E B | & X | ˜ E B | ≥ ˜ c X ( − κc X )| B | and { E B } B ∈S k is a pairwise disjoint collection. Furthermore, for any B ∈ S k and B ′ ∈ T k − we have that E B ′ ∩ E B = ∅ as E B ′ ⊆ B ′ \ ∪ L ∈S k L , again by the inductive hypothesis. Theseconsiderations show that (i) is also satisfied for k + (cid:3) Compiling the proof of Theorem A.
It is now relatively simple to put together all thepieces needed for the complete proof of Theorem A. We explain in detail the steps need toprove the conclusion for the untruncated version T ∼ Í s T ( s ) ; the proof for the truncatedversion is similar but simpler.Let T ∼ Í s T ( s ) be a linear operator satisfying the assumptions of the theorem and f , f Lipschitz functions supported on some ball B o with r B o = s Bo . By the assumption there exists r ∈ ( , ∞) such that sup σ < τ k T τσ k L r → L r = C r < ∞ . By Remark 2.1 we have h T f , f i = h m f , f i + lim j →∞ h T τ j σ j f , f i for some m ∈ L ∞ . The first term is controlled by the following lemma. Lemma 5.4.
Let f , f be Lipschitz functions with compact support. Then there exists a sparsecollection B , consisting of d -balls, such that |h f , f i| ≤ C Õ B ∈B | B |h f i , B h f i , B . Proof.
There are many ways to do this. For example, let supp ( f ) , supp ( f ) ⊆ B o and F B ≔ { x ∈ X : f ( x ) > C h f i , B } . Since f is continuous we have that F B is open and Theorem 4 gives a Whitney collection ofballs { B j } such that F B = Ø j B j , ηB j ⊆ B o . Then | F B | ≤ C − | B | and |h f , f i| ≤ C h f i , B h f i , B | B | + |h F B f , f i| . One then iterates thisestimate, exactly as in the proof of Lemma 5.3 to get |h f , f i| ≤ C Õ L ∈T k − | L |h f i , L h f i , L + Õ L ∈S k |h f , L , f i , where the collection T k − is sparse with major subsets avoiding S k and each f , L satisfies | f , L | ≤ | f | L . The only difference compared to Lemma 5.3 is the termination procedure.Observe that Õ L ∈S k | L | ≤ ( M / C ) k | B | where M is the overlap constant of the Whitney balls, which depends only on X . Choosing C = M say gives that | ∪ L ∈S k L | ≤ − k | B | . Then Õ L ∈S k |h f , L , f i ≤ k f k ∞ , B k f k ∞ , B − k | B | → as k → + ∞ . If we insist on a finite sparse collection we can stop the iteration once the term above becomesless than |h f , f i| and absorb it in the left hand side. (cid:3) For the second term we now choose σ o , τ o ∈ Z with σ o < τ o such that | lim j →∞ h T τ j σ j f , f i| ≤ |h T τ o σ o f , f i| . Note that it is without loss of generality to assume s B o > τ o by taking a bigger ball B ′ o ⊃ B o , ifnecessary. Now let κ ∈ ( , ) such that κ < ( c X ) − and β κ β < /
2, where c X is as in (i), and β , β are as in (iii), of Lemma 5.3. By considering k large enough in Lemma 5.3 we get that Õ L ∈S k |h T s L σ o ( f , L ) , f i| = . Indeed, since sup L ∈S k r L = s L < − k r B o we will have that s L < σ o and thus h T s L σ o ( f , L ) , f i = k is sufficiently large so that s B o − k < σ o . For this value of k we set B ≔ T k − and then |h T τ o σ o f , f i| . X Õ B ∈B | B |h f i p , B h f i p , c o B , where the collection B is 1 / c X -sparse. Clearly the collection { c o B} B ∈B is also sparse and theproof of Theorem A is complete.
6. Proof of Theorem C
The proof of Theorem C is similar the proof of Theorem A but using the stronger kernelassumptions for T , which by Lemma 3.2 also entail the ( , ∞) -improving property. Using asimilar iterative algorithm as in Lemma 5.3 yields Theorem C by recursion on the estimate ofthe following lemma. METRIC APPROACH TO SPARSE DOMINATION 31
Lemma 6.1.
Let κ ∈ ( , ) and T satisfy the assumptions of Theorem C. There exists constants C and η = η ( X ) > such that the following hold: let σ ∈ Z , B be a fixed ball with r B = s B , and f , f be bounded functions. Then there exists a finite Whitney-type collection of balls B = { B j } j with r B j = s Bj and functions { f , j } j with | f , j | ≤ | f | B j for each j , such that (6.1) F B ≔ Ø j B j , e F B ≔ Ø j ηB j ⊂ ηB , | F B | ≤ κ | B | and (6.2) |h T s B σ ( f B ) , f i| ≤ Cκ − | B |h f i , B h f i , c o B + Õ j |h T s Bj σ ( f , j ) , f i| . Furthermore we have that inf j r B j ≥ σ and sup j r B j < β κ β r B for some constants β , β > depending on the homogeneous metric structure of ( X , d , | · |) , but not on κ . The constant C > depends on the constants in the kernel assumptions on T and the constants in the homogeneousstructure of ( X , d , | · |) .Proof. We recall the splitting of T into localized pieces T f = Õ s ∈ Z T ( s ) f = Õ s ∈ Z ∫ s ≤ d ( x , y ) < s + K ( x , y ) f ( y ) d y , x ∈ X . We use a similar strategy and notation as in the proof of Lemma 5.1 with the difference thatnow the truncations of T are assumed to be uniformly bounded on L p ( X ) for some fixed p ∈( , ∞) and p = p =
1; see Lemma 3.2. Thus in the language of the proof of Lemma 5.1 weconsider the set F B and for i ∈ { , } the functions { f i , j } j , { b i , j } j and { д i , j } j , where we havenormalized h f i , B = h f i , c o B =
1. Remember the notation r B = s B and r B j = s Bj = s j .Because of the assumption of L p -boundedness of the truncations we can assume, as before,that the functions f , f are supported on the set F B . We repeat the splitting h T s B σ f , f i = Õ j : s j > σ h T s j σ f , j , f i + Õ j h T s B σ j f , j , f i , σ j ≔ max ( s j , σ ) . We estimate the second summand above Õ j h T s B σ j f , j , f i = h T s B σ j д , j , д i + Õ j h T s B σ j b , j , д i + Õ j h T s B σ j д , j , b i + Õ j h T s B σ j b , j , b i ≔ GG + BG + GB + BB . The estimate for the term GG is identical to the corresponding estimate in the proof of Lem-ma 5.1, using the L p -boundedness of the truncations of T on L p ( X ) and the localization as-sumption (2.1).Since we have kernel estimates for T we do more precise calculations for the terms BG , GBbelow instead of explicitly using the ( , ∞) -improving property or the L p -boundedness. Moreprecisely, using the fact that ∫ B j b j ( x ) =
0, remembering the notation σ j = max ( s j , σ ) , we havethe following where B j = B ( c j , s j ) | BG | ≤ Õ j |h T s B σ j b , j , д i| = Õ j ∫ sB > d ( x , y )≥ σj | K ( x , y ) − K ( x , c B j )|| b , j ( y )|| д ( x )| d y d x . κ − Õ j ∫ B j | b , j ( y )| (cid:18) ∫ sB > d ( x , y )≥ σj ω ( s j / d ( x , y )) V ( x , y ) d x (cid:19) d y . κ − Õ j s B Õ s = σ j ∫ B j | b , j ( y )| (cid:18) ∫ s + > d ( x , y )≥ s ω ( s j − s )| B ( y , d ( x , y )| d x (cid:19) d y . κ − k T k Dini | B | . The estimate for GB is similar so it remains to estimate the term BB. In fact, repeating thecalculation for the BB term in the proof of Lemma 5.1 we getBB = Õ j s B Õ s = σ j Õ m : s m ≤ σ j h T ( s ) b , j , b , m i + Õ j s B Õ s = σ j Õ m : s m ≤ σ j h T ( s ) b , j , b , m i ≕ BB + BB . As in the estimate for the the BG-term, we get | BB | ≤ Õ j s B Õ s = σ j Õ m : s m ≤ σ j ∫ B j | b , j ( y )| (cid:18) ∫ s + > d ( x , y )≥ s ω ( s j − s )| b , m ( x )|| B ( y , d ( x , y )| d x (cid:19) d y ≤ Õ j s B Õ s = σ j Õ m : s m ≤ σ j ω ( s j − s ) ∫ B j | b , j ( y )|| B ( y , s )| (cid:18) ∫ B ( y , s + ) | b , m ( x )| d x (cid:19) d y . Fixing j , y ∈ B j and s ≥ σ j we have that Õ m : s m ≤ σ j | B ( y , s )| ∫ B ( y , s + ) | b , m ( x )| d x . Õ m : s m ≤ σ j B m ∩ B ( y , s + ) , ∅ | B m || B ( y , s )| ∫ B m | b , m ( x )| d x . κ − taking into account the stopping condition for the collection { b m } m and the doubling propertyof the measure. This implies the estimate | BB | . κ − Õ j s B Õ s = σ j ω ( s j − s ) ∫ B j | b , j ( y )| d y . κ − k ω k Dini | B | . The estimate for BB is almost symmetric where now we have that s m > σ j . Note that if B j = ( c j , s j ) then by (2.1) we haveBB = Õ j s B Õ s = σ j Õ m : s m > σ j h T ( s ) b , j , b , m B ( x j , c s ) i since for all the terms in the sum there holds s ≥ σ j ≥ s j . We claim that there exists someconstant C depending only on the space such that s m < s + C for all the non-zero terms in thesums above. Indeed, assume that s m > s + C ; then the non-zero terms in the sum above mustsatisfy that B ( x j , c s ) ∩ B m , ∅ . Since r m = s m > C s > C σ j ≥ C s j = r j we gather that B ( c m , η s m ) ∩ B ( c j , η s j ) , ∅ for the non-zero terms of the sum, if η was chosen large enoughdepending only on ( X , d , | · |) . By the properties of the Whitney balls B ( c j , r j ) we then havethat we must have 2 s m = r m h r j = s j which is impossible if C was chosen sufficiently large. METRIC APPROACH TO SPARSE DOMINATION 33
With this in mind the reader will have no trouble repeating the argument for BB in order toconclude the estimate for BB , and the proof of the lemma is complete. (cid:3)
7. Proof of Theorem B
In this last section we provide the proof of the maximal version of our sparse dominationresult as formulated in Theorem B. Again the strategy is similar to the one that led to the proofof Theorem A.
Lemma 7.1.
Let κ ∈ ( , ) and T satisfy the assumptions of Theorem B. Assume that { E s } s ∈ Z is apairwise disjoint collection of measurable sets contained in c o B . Then there exists constants C and η = η ( X ) > such that the following hold: let σ ∈ Z , B be a fixed ball with r B = s B ∈ Z , and f , f be bounded functions. Then there exists a finite Whitney-type collection of balls B = { B j } j with r B j = s Bj and functions { f , j } j with | f , j | ≤ | f | B j for each j , such that F B ≔ Ø j B j , e F B ≔ Ø j ηB j ⊂ ηB , | F B | ≤ κ | B | and (cid:12)(cid:12)(cid:12) Õ σ ≤ s ≤ s B (cid:10) T ( s )( f B ) , f E s (cid:11)(cid:12)(cid:12)(cid:12) ≤ Cκ − | B |h f i p , B h f i p , c o B + Õ j (cid:12)(cid:12)(cid:12) s Bj Õ s = σ (cid:10) T ( s )( f , j ) , f E s (cid:11)(cid:12)(cid:12)(cid:12) . Furthermore we have that inf j r j ≥ σ , sup j r B j < β κ β r B for some constants β , β > dependingon the homogeneous metric structure of ( X , d , | · |) , but not on κ .Proof. We repeat the initial steps in the proof of Lemma 5.3. Fixing B with r B = s B and func-tions f , f supported in B and c o B , respectively. We normalize so that h f i p , B = h f i p , c o B = F B and its Whitney decomposition into finitely overlapping balls { B j } j are ex-actly as in the proof of Lemma 5.3 with r B j = r j = s j . Since T is assumed to be bounded on L ∞ we can repeat the calculations in (5.3) with exponents p = p = f , f are supported on the set F B . Defining the good and bad parts { д i , j } j and { b i , j } j for i ∈ { , } as before we then have that f i = Õ j f i , j ≔ Õ j b i , j + Õ j д i , j ≕ b i + д i , supp ( b i , j ) ∪ supp ( д i , j ) ⊆ B j ∀ j , i ∈ { , } . First of all notice that Õ σ ≤ s ≤ s B h T ( s ) f , f E s i = Õ j : s j ≥ σ s j Õ s = σ h T ( s ) f , j , f E s i + Õ j s B Õ s = σ j h T ( s ) f , j , f E s i , σ j ≔ max ( σ , s j ) . The first summand above matches the term in the conclusion of the lemma so we proceed toestimate the second summand above. There is an easy estimate that is available Õ j (cid:12)(cid:12)(cid:12) s B Õ s = σ j h T ( s ) д , j , f E s i (cid:12)(cid:12)(cid:12) . Õ j Õ s κ − ∫ c o B j | f | E s . κ − | B | relying on the disjointness of the collections { E s } s ∈ Z , the finite overlapping property for thecollection { B j } j , the L ∞ -bound for T together with (5.4), and the normalization h f i p , c o B = We now prove the following estimate for the remaining term(7.1) (cid:12)(cid:12)(cid:12) Õ j s B Õ s = σ j h T ( s ) b , j , f E s i (cid:12)(cid:12)(cid:12) . κ − k ω k Dini | B | . We repeat the calculations in the proof of Lemma 5.3: choosing a finitely overlapping partitionof B into balls { L τ } τ of size r L τ = s and such that Í τ ρL τ . ρ ρ >
1, asprovided by Lemma 4.3, we have Õ j s B Õ s = σ j h T ( s ) b , j , f E s i = Õ j s B Õ s = σ j Õ τ h T ( s ) b , j , f E s L τ i = Õ j : s j ≤ σ s B Õ s = σ Õ τ h T ( s ) b , j , f E s L τ i + Õ j : s j > σ s B Õ s = s j Õ τ h T ( s ) b , j , f E s L τ i ≕ B ≤ σ + B > σ . We have | B > σ | ≤ Õ σ < s ≤ s B Õ k ≥ Õ τ |h T ( s ) b s − k , f E s L τ i| . Õ k ≥ ω ( − k ) Õ σ < s ≤ s B Õ τ | L τ |h| b s − k |i p , γ L τ h| f | E s L τ i p , γ L τ by the ( p , p ′ ) -improving assumption for T ∗ . Now writing δ ≔ / p − / p ′ ≥ = p / p ′ + δp we estimate for fixed k ≥ Õ s , τ | L τ |h| b s − k |i p , γ L τ h| f | E s L τ i p , γ L τ ≤ sup s , τ h b s − k i δp , γ L τ Õ s , τ | L τ |h b s − k i p p ′ p , γ L τ h f E s L τ i p , γ L τ . The sum on the right hand side of the estimate above is estimated by κ − | B | by Hölder’s in-equality, using the disjointness of the sets { E s } s ∈ Z , the finite overlap of the sets { γ i L τ } τ , to-gether with the stopping condition (5.5) for b . On the other hand we have for fixed s , τ and δ ≥ ∫ γ L τ | b s − k | p ! p . κ − Õ j : B j ∩ γ L τ , ∅ s j ≤ s | B j || L τ | ! p . κ − . Combining the last two facts proves the desired estimate for the term B > σ . The proof of thecorresponding estimate for B ≤ σ is almost identical hence we omit the details. This proves (7.1)and hence the lemma. (cid:3) The rest of the proof of Theorem 7 is routine. Given bounded functions f , f supported insome ball B o we note that there exists σ o , τ o ∈ Z with σ o < τ o such that h T ⋆ f , f i . h sup σ o ≤ s ≤ τ o | T ( s ) f | , f i . Õ s h T ( s ) f , f E s i for some pairwise disjoint collection of measurable sets { E s } s ∈ Z ; the choice of this collectiondepends on f , f but then so does our sparse collection so we suppress this fact. The proof METRIC APPROACH TO SPARSE DOMINATION 35 of the theorem now follows by a straightforward repetition of the arguments in §A, usingLemma 7.1 in place of Lemma 5.3.
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2, 9(J. M. Conde-Alonso)
UAM - Departamento de Matemáticas, 7 Francisco Tomás y Valiente, 28049 Madrid,Spain
E-mail address : [email protected](F. Di Plinio, M. N. Vempati)
Department of Mathematics, Washington University in Saint Louis,1 Brookings Drive, Saint Louis, Mo 63130, USA
E-mail address : [email protected] , [email protected](I. Parissis) Departamento de Matemáticas, Universidad del Pais Vasco, Aptdo. 644, 48080 Bilbao, Spainand Ikerbasqe, Basqe Foundation for Science, Bilbao, Spain