Sparse bounds for maximal rough singular integrals
via the Fourier transform
We prove that the class of convolution-type kernels satisfying suitable decay conditions of the Fourier transform, appearing in the works of Christ , Christ-Rubio de Francia  and Duoandikoetxea-Rubio de Francia  gives rise to maximally truncated singular integrals satisfying a sparse bound by -averages for all , with linear growth in . This is an extension of the sparse domination principle by Conde-Alonso, Culiuc, Ou and the first author  to maximally truncated singular integrals. Our results cover the rough homogeneous singular integrals on
with angular part and having vanishing integral on the sphere. Consequences of our sparse bound include novel quantitative weighted norm estimates as well as Fefferman-Stein type inequalities. In particular, we obtain that the norm of the maximal truncation of depends quadratically on the Muckenhoupt constant , extending a result originally by Hytönen, Roncal and Tapiola . A suitable convex-body valued version of the sparse bound is also deduced and employed towards novel matrix weighted norm inequalities for the maximal truncated rough homogeneous singular integrals. Our result is quantitative, but even the qualitative statement is new, and the present approach via sparse domination is the only one currently known for the matrix weighted bounds of this class of operators.
Key words and phrases:Sparse domination, rough singular integrals, weighted norm inequalities
2010 Mathematics Subject Classification:Primary: 42B20. Secondary: 42B25
1. Introduction and main results
Let . A countable collection of cubes of is said to be -sparse if there exist measurable sets such that
Let be a sublinear operator mapping the space of complex-valued, bounded and compactly supported functions on into locally integrable functions. We say that has the sparse bound  if there exists a constant such that for all we may find a -sparse collection such that
in which case we denote by the least such constant . As customary,
Estimating the sparse norm(s) of a sublinear or multisublinear operator entails a sharp control over the behavior of such operator in weighted -spaces; this theme has been recently pursued by several authors, see for instance [1, 8, 18, 20, 21, 33]. This sharp control is exemplified in the following proposition, which is a collection of known facts from the indicated references.
In this article, we are concerned with the sparse norms (1.1) of a class of convolution-type singular integrals whose systematic study dates back to the celebrated works by Christ , Christ-Rubio de Francia , and Duoandikoetxea-Rubio de Francia , admitting a decomposition with good decay properties of the Fourier transform. To wit, let be a sequence of (smooth) functions with the properties that
for some . We consider truncated singular integrals of the type
and their maximal version
with absolute dimensional implicit constant, in particular uniform over families satisfying (1.2).
Theorem A entails immediately a variety of novel corollaries involving weighted norm inequalities for the maximally truncated operators . In addition to, for instance, those obtained by suitably applying the points of Proposition 1.1, we also detail the quantitative estimates below, whose proof will be given in Section 7.
Let be a sublinear operator satisfying the sparse bound (1.1) with .
For any ,
with implicit constant possibly depending on and dimension ; in particular,
The Fefferman-Stein type inequality
holds with implicit constant possibly depending on only.
The - estimate
holds for and , with implicit constant possibly depending on and only.
The following Coifman-Fefferman type inequality
holds for all with implicit constant possibly depending on and only.
Take with and having vanishing integral on , and consider the associated truncated integrals and their maximal function
It is well known– for instance, see the recent contribution [16, Section 3]– that
as well. The main result of  is the stronger control
The above estimate, in particular, is stronger that the uniform weak type for the operators , a result originally due to Seeger . As the weak type of under no additional smoothness assumption on is a difficult open question, estimating the sparse norm of as in (1.8) seems out of reach.
The study of sharp weighted norm inequalities for (the uniformity in is of course relevant here) was initiated in the recent article  by Hytönen, Roncal and Tapiola. Improved quantifications have been obtained in  as a consequence of the domination result (1.8), and further weighted estimates– including a Coifman-Fefferman type inequality, that is a norm control of by on all , when – have been later derived from (1.8) in the recent preprint by the third named author, Pérez, Roncal and Rivera-Rios .
Although (1.7) is a bit weaker than (1.8), we see from comparison of (1.4) from Theorem B with the results of [7, 28] that the quantification of the -norm dependence on entailed by the two estimates is the same– quadratic; on the contrary, for , (1.8) yields the better estimate We also observe that the proof of the mixed estimate (1.5) actually yields the following estimate for the non-maximally truncated operators, improving the previous estimate given in 
Finally, we emphasize that (1.7) also yields a precise dependence on of the unweighted operator norms. Namely, from the sparse domination, we get
with absolute dimensional implicit constant, which improves on the implicit constants in . Moreover, we note that the main result of  implies that if (1.9) is sharp, then our quantitative weighted estimate (1.4) is also sharp.
1.4. Matrix weighted estimates for vector valued rough singular integrals
Let , and be the canonical basis, scalar product and norm on over , where . A recent trend in Harmonic Analysis– see, among others, [2, 3, 9, 14, 30]– is the study of quantitative matrix weighted norm inequalities for the canonical extension of the (integral) linear operator
to -valued functions . In Section 6 of this paper, we introduce an , , version of the convex body averages first brought into the sparse domination context by Nazarov, Petermichl, Treil and Volberg , and use them to produce a vector valued version of Theorem A. As a corollary, we obtain quantitative matrix weighted estimates for the maximal truncated vector valued extension of the rough singular integrals from (1.6). In fact, the next corollary is a special case of the more precise Theorem E from Section 6.
Let be a positive semidefinite and locally integrable -valued function on and be as in (1.6). Then
with implicit constant depending on only, where the matrix constant is given by
As the left hand side of (1.10) dominates the matrix weighted norm of the vector valued maximal operator first studied by Christ and Goldberg in , the finiteness of is actually necessary for the estimate to hold. To the best of the authors’ knowledge, Theorem E has no precedessors, in the sense that no matrix weighted norm inequalities for vector rough singular integrals were known before, even in qualitative form. At this time we are unable to assess whether the power appearing in (1.10) is optimal. For comparison, if the angular part is Hölder continuous, the currently best known result  is that (1.10) holds with power ; see also .
1.5. Strategy of proof of the main results
We will obtain Theorem A by an application of an abstract sparse domination principle, Theorem C from Section 3, which is a modification of [7, Theorem C]. At the core of our approach lies a special configuration of stopping cubes, the so-called stopping collections , and their related atomic spaces. The necessary definitions, together with a useful interpolation principle for the atomic spaces, appear in Section 2. In essence, Theorem C can be summarized by the inequality
where the supremum is taken over all stopping collections and all measurable linearizations of the truncation parameters , and are suitably adapted localizations of (the adjoint form to the linearized versions of) . In Section 4, we prove the required uniform estimates for the localizations coming from Dini-smooth kernels. The proof of Theorem A is given in Section 5, relying upon the estimates of Section 4 and the Littlewood-Paley decomposition of the convolution kernels (1.2) whose first appearance dates back to .
With we indicate the Lebesgue dual exponent to , with the usual extension , . The center and the (dyadic) scale of a cube will be denoted by and respectively, so that . We use the notation
for the -Hardy Littlewood maximal function and write in place of . Unless otherwise specified, the almost inequality signs imply absolute dimensional constants which may be different at each occurrence.
This work was completed during F. Di Plinio’s stay at the Basque Center for Applied Mathematics (BCAM), Bilbao as a visiting fellow. The author gratefully acknowledges the kind hospitality of the staff and researchers at BCAM and in particular of Carlos Pérez. The authors also want to thank José Conde-Alonso, Amalia Culiuc, Yumeng Ou and Ioannis Parissis for several inspiring discussions on sparse domination principles.
2. Stopping collections and interpolation in localized spaces
The notion of stopping collection with top the (dyadic) cube has been introduced in [7, Section 2], to which we send for details. Here, we recall that such a is a collection of pairwise disjoint dyadic cubes contained in and satisfying suitable Whitney type properties. More precisely,
A consequence of (2.3) is that the cardinality of is bounded by an absolute constant.
The spaces have also been defined in [7, Section 2]: here we recall that is the subspace of of functions satisfying
where stands for the (non-dyadic) -fold dilate of , and that is the subspace of of functions satisfying
Finally, we write if and each has mean zero. We will omit from the subscript of the norms whenever the stopping collection is clear from context.
There is a natural interpolation procedure involving the -spaces. We do not strive for the most general result but restrict ourselves to proving a significant example, which is also of use to us in the proof of Theorem A.
Let be a bisublinear form and be positive constants such that the estimates
Then for all
We may assume , otherwise there is nothing to prove. We are allowed to normalize . Fixing now , so that , it will suffice to prove the estimate
for each pair with with implied constant depending on dimension only. Let to be chosen later. Using the notation , we introduce the decompositions
which verify the properties
We have used that is supported on the union of the cubes and has mean zero on each , and therefore has the same property, given that . Therefore
which yields (2.6) with the choice . ∎
3. A sparse domination principle for maximal truncations
We consider families of functions satisfying
and associate to them the linear operators
and their sublinear maximal versions
We assume that there exists such that
For pairs of bounded measurable functions , we also consider the linear operators
From the definition (3.2), it follows that
In consequence, for the linearized versions defined in (3.4) we have
A related word on notation: we will be using linearizations of the type and similar, where is the (dyadic) scale of a (dyadic) cube . With this we mean we are using the constant function equal to as our upper truncation function. Finally, we will be using the notations for the linearizing function and for the linearizing function .
Given two bounded measurable functions and a stopping collection with top , we define the localized truncated bilinear forms
Within the above framework, we have the following abstract theorem.
The proof follows essentially the same scheme of [7, Theorem C]; for this reason, we limit ourselves to providing an outline of the main steps.
Step 1. Auxiliary estimate
First of all, an immediate consequence of the assumptions of the Theorem is that the estimate
where , holds with uniform over bounded measurable functions . See [7, Lemma 2.7]. Therefore,
Step 2. Initialization
The argument begins as follows. Fixing with compact support, we may find measurable functions which are bounded above and below and a large enough dyadic cube from one of the canonical dyadic systems such that and
and we clearly can replace by in what follows.
Step 3. Iterative process
Then, the argument proceeds via iteration over of the following construction, which follows from (3.9) and the Calderón-Zygmund decomposition and is initialized by taking for . Given a disjoint collection of dyadic cubes with the further Whitney property that (2.3) holds for in place of , there exists a further collection of disjoint dyadic cubes such that
(2.3) for in place of continues to hold,
each subcollection is a stopping collection with top ,
and for which for all there holds
More precisely, is composed by the maximal dyadic cubes such that
for a suitably chosen absolute large dimensional constant . This construction, as well as the Whitney property (2.3) results into
guaranteeing that is a sparse collection for all . When is such that , the iteration stops and the estimate
is reached. This completes the proof of Theorem C.∎
4. Preliminary localized estimates for the truncated forms (3.5)
We begin by introducing our notation for the Dini constant of a family of kernels as in (3.1). We write
The estimates contained within the lemmata that follow are meant to be uniform over all measurable functions and all stopping collections . The first one is an immediate consequence of the definitions: for a full proof, see [7, Lemma 2.3].
Let . Then
The second one is a variant of [7, Lemma 3.2]; we provide a full proof.
We consider the family fixed and use the simplified notation in place of , and similarly for the truncated operators . By horizontal rescaling we can assume . Let . Recalling the definition (2.5) and using bilinearity of it suffices for each stopping cube to prove that
as , and conclude by summing up over the disjoint , whose union is contained in . We may further assume ; otherwise . In addition we can assume is positive, by repeating the same argument below with the real and imaginary, and positive and negative parts of . Using the definition of the truncated forms (3.5) and the disjointness of ,
Thus, if denotes the cube concentric to and whose sidelength is , using the support conditions and abbreviating a standard calculation
which is bounded by the right hand side of (4.2). ∎
The third localized estimate is new. However, its roots lie in the well-known principle that the maximal truncations of a Dini-continuous kernel to scales larger than do not oscillate too much on a ball of radius , see (4.7). This was recently employed, for instance, in [11, 16, 19].
We use similar notation as in the previous proof and again we rescale to , and work with positive . We can of course assume that . We begin by removing an error term; namely, referring to notation (2.1), if