Domination of multilinear singular integrals
by positive sparse forms
We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one-dimensional subspace by positive sparse forms involving -averages. This class includes the adjoint forms to the bilinear Hilbert transforms. Our result strengthens the -boundedness proved by Muscalu, Tao and Thiele, and entails as a corollary a novel rich multilinear weighted theory. A particular case of this theory is the -boundedness of the bilinear Hilbert transform when the weights belong to the class . Our proof relies on a stopping time construction based on newly developed localized outer- embedding theorems for the wave packet transform. In an Appendix, we show how our domination principle can be applied to recover the vector-valued bounds for the bilinear Hilbert transforms recently proved by Benea and Muscalu.
Key words and phrases:Positive sparse operators, bilinear Hilbert transform, weighted norm inequalities
2010 Mathematics Subject Classification:Primary: 42B20. Secondary: 42B25
A. Culiuc]email@example.com F. Di Plinio]firstname.lastname@example.org Y. Ou]email@example.com
1. Introduction and main results
The -boundedness theory of Calderón-Zygmund operators, whose prototype is the Hilbert transform, plays a central role in harmonic analysis and in its applications to elliptic partial differential equations, geometric measure theory and related fields.
A recent remarkable discovery is that the action of a singular integral operator on a function can be dominated in a pointwise sense by the averages of over a sparse, i.e. essentially disjoint, collection of cubes in . This control is much stronger than -norm bounds and carries significantly more information on the operator itself. As of now, the most striking consequence is that sharp weighted norm inequalities for follow from the corresponding, rather immediate estimates for the averaging operators. Such a pointwise domination principle, albeit in a slightly weaker sense, appears explicitly for the first time in the proof of the theorem by Lerner . We also point out the recent improvements by Lacey  and Lerner , and the analogue for multilinear Calderón-Zygmund operators by Lerner and Nazarov . Most recently, Bernicot, Frey and Petermichl  extend this approach to non-integral singular operators associated with a second-order elliptic operator, lying outside the scope of classical Calderón-Zygmund theory.
The main focus of the present article is to formulate a similar principle for the class of multilinear multiplier operators, invariant under simultaneous modulations of the input functions, which includes the bilinear Hilbert transforms. Besides their intrinsic interest, our results yield a rich, and sharp in a suitable sense, family of multilinear weighted bounds for this class of operators. In fact, Theorem 3 below is the first result of this kind. Weighted estimates for the bilinear Hilbert transforms have been mentioned as an open problem in several related works [8, 9, 12].
Let and be a fixed unit vector, nondegenerate in the sense that
We are concerned with the trilinear forms
acting on triples of Schwartz functions on , where is a Fourier multiplier satisfying, in multi-index notation,
The one-parameter family (with respect to ) of trilinear forms adjoint to the bilinear Hilbert transforms is obtained by choosing
Not unexpectedly, a pointwise domination principle for this class of bilinear operators is not allowed to hold, as we elaborate in Remark 1.3 below. This obstruction is overcome by introducing the closely related notion of domination by sparse positive forms of the adjoint trilinear form, which we turn to in what follows.
We say that is a -sparse collection of intervals if for every there exists a measurable with such that are pairwise disjoint. The positive sparse trilinear form of type associated to the sparse collection is defined by
we omit the subscript and write when . A rather immediate consequence of the Hardy-Littlewood maximal theorem is the following proposition.111 We omit the proof, which is a simplified version of the proof of Corollary 6 given in the appendix
Let be a bilinear operator. Suppose that for all tuples there holds
Then for all there holds
provided that for and .
Our main result is a strengthening of Theorem 1 to a domination by positive sparse forms. To formulate it, we need one more notion. We say that is an admissible tuple if
If all the constraints hold with strict inequality, we say that is an open admissible tuple.
Let be an open admissible tuple. There exists such that the following holds. For any tuple there exists a -sparse collection such that
where the supremum is being taken over the family of multipliers satisfying (1.2).
We stress that the constants and depend only on the exponent tuple , and the choice of the sparse collection depends only on and and is, in particular, independent of the multiplier .
Remark 1.2 (Sharpness of Theorem 2).
Let be an exponent pair with . Then there exists an open admissible tuple with if and only if (1.4) holds for . This observation, coupled with Proposition 1.1, yields Theorem 1 as a corollary of Theorem 2.
On the other hand, let be an even Schwartz function with be an orthonormal basis of . Define the family of multipliers on
where , . The same argument as in [17, Section 2.2] yields
while the family satisfies (1.2) uniformly. This implies that the range (1.4) of Theorem 1 is sharp up to equality holding in (1.4) and, in turn, that (1.8) cannot hold for any tuple violating (1.7). Hence, Theorem 2 is sharp up to possibly replacing the assumption open admissible with the stronger admissible. The behavior of the forms for tuples at the boundary of the admissible region is studied in detail in .
Remark 1.3 (No uniform control by a bilinear positive sparse operator).
For bilinear Calderón-Zygmund operators , there holds a pointwise domination by sparse operators of the type
One can take : see . Essentially self-adjoint operators enjoying such pointwise domination inherit the boundedness property
which, as described in the previous Remark 1.2, fails for the generic of the class (1.2) when . In fact, no -boundedness properties are expected to hold even for the bilinear Hilbert transforms. Summarizing, no such pointwise domination principle can be obtained for when and, most likely, neither for the case when . Our formulation in terms of positive sparse forms overcomes this obstacle: a similar idea, albeit not explicit, appears in the linear setting in .
Theorem 2 implies multilinear weighted bounds for the forms . Our main weighted theorem will involve multilinear Muckenhoupt constants. Given any tuple , a Hölder tuple and a weight vector satisfying
these are defined as
For , these weight classes have been introduced in , to which we send for an exhaustive discussion of their properties. A particular case of (1.11) (where ) can be found in  as a necessary and sufficient condition for weighted -boundedness of the bilinear fractional integrals. Furthermore, the classes (1.11) appear in ongoing work on multilinear Calderón-Zygmund operators satisfying Hörmander type conditions .
One is usually interested in weighted estimates involving Muckenhoupt and reverse Hölder constants of each single weight. Recall that the and constant of a weight on are defined as
A suitable choice of admissible tuple in Theorem 3 yields the following corollary.222We have come to know that Xiaochun Li  has some unpublished results about weighted estimates for the bilinear Hilbert transforms.
and be given weights with . Then the operator norms
of the family of multipliers satisfying (1.2) with uniform constants are uniformly bounded above by a positive constant depending on only.
We refer to the recent monograph  for details on the and classes. Here we remark that if then [5, Section 3.8] if and only if We mention that a theory of linear extrapolation for weights in the classes has been introduced in ; see also the already mentioned monograph .
As a further application of Corollary 4, weighted, vector-valued estimates for multipliers satisfying condition (1.2), extending the results of [2, 26] can be obtained by a multilinear version of the extrapolation theory of . These extensions are the object of an upcoming companion article by the same authors. However, Theorem 2 can be employed to recover the unweighted vector-valued estimates of [2, 26] in a rather direct fashion. In order to keep our outline as simple as possible, we postpone the complete statement and proof of the vector-valued estimates to Appendix A.
Structure of the article and proof techniques
The class of multipliers (1.2), in addition to the familiar invariances under isotropic dilations and translations proper of Coifman-Meyer type multipliers, enjoys a one-parameter invariance under simultaneous modulation of the three input functions along the line . The invariance properties of the class (1.2) are essentially shared by a family of discretized trilinear forms involving the maximal wave packet coefficients of the input functions parametrized by rank 1 collection of tritiles, which we call tritile form.
The first step in the proof of Theorem 2, carried out in Section 2, is to establish that for any multiplier satisfying (1.2), the form lies in the convex hull of finitely many tritile forms. This discretization procedure is largely the same as the one employed in . Theorem 2 then reduces to the analogous result for tritile forms, Theorem 5. It is of paramount importance here that the sparse collection constructed in Theorem 5 is independent of the particular tritile form.
The explicit construction of the collection , and in fact the proof of Theorem 5, is performed in Section 5 by means of an inductive argument. The intervals of are, roughly speaking, the stopping intervals of the -Hardy-Littlewood maximal function of the -th input. At each stage of the argument, the contribution of those wave packets localized within one of the stopping intervals will be estimated at the next step of the induction, after a careful removal of the tail terms. The main term, which is the contribution of the wave packets whose spatial localization is not contained in the union of the stopping interval is estimated by means of a localized outer embedding Theorem for the wave packet transform.
This outer embedding, which is the concern of Proposition 4.1, is a close relative of the main result of  by two of us, namely, a localized embedding theorem for the continuous wave packet transform. In fact, while Proposition 4.1 is proved here via a transference argument based upon [6, Theorem 1], a direct proof can be given by repeating the arguments of  in the discrete setting. The construction of the outer spaces on rank 1 collections, which parallels the outer theory introduced by Do and Thiele in , is performed in Section 3.
Let . For an interval centered at and of length , we write
We will make use of the weighted spaces
with positive integer. We write
for the -Hardy Littlewood maximal functions. Finally, the constants implied by almost inequality sign and the comparability sign are meant to be absolute throughout the article.
The authors want to thank David Cruz-Uribe, Kabe Moen and Rodolfo Torres for providing additional insight on multilinear weighted theory. The authors are grateful to Gennady Uraltsev for fruitful discussions on the notion of localized outer embeddings.
2. Tritile maps
In this section, we reduce Theorem 2 to the corresponding statement for a class of multilinear forms which we call tritile maps. Throughout, we assume that the nondegenerate unit vector is fixed and let be a unit vector perpendicular to , spanning the singular line of the multipliers from (1.2).
2.1. Rank 1 collection of tri-tiles
A tile is the cartesian product of two intervals with . A tri-tile is an ordered triple of tiles with the property that
we denote by the frequency cube corresponding to and by the convex hull of the intervals We say that the collection of tri-tiles is of rank 1 if
and , are scale-separated dyadic grids;
if are such that then for each ;
if are such that for some then ;
if are such that for some then for .
We can take .
2.2. Tritile forms
Let be a fixed increasing sequence of positive constants. For each tile we define the adapted family to be the collection of Schwartz functions satisfying
Let be a rank 1 collection of tritiles and . We define the tritile maps by
and the trisublinear tritile form associated to by
2.3. Reduction to uniform bounds for tritile forms
The following lemma is a reformulation of the well-known discretization procedure from . Several versions of this procedure have since appeared, see for instance the monographs [24, 27]. We omit the standard (by now) proof.
There exists a finite collection of rank 1 collections of tritiles such that, for any multiplier satisfying (1.2) and any tuple of Schwartz functions , there holds
and the adaptation constants of the adapted families defining depend on only. Furthermore, the character depends only on the nondegeneracy constant of .
Let be an open admissible tuple. There exists such that the following holds. For any tuple with and compactly supported there exists a -sparse collection such that
where the supremum is being taken over all rank 1 collections of tritiles of finite cardinality and adaptation sequence . In particular, the collection depends only on and the tuple .
3. Outer spaces of tritiles
In this section, we formulate the outer measure space that is needed for our proof, which is based on a finite rank 1 collection of tritiles . Recall that . The generating collection is the set of trees . The set is a tree with top data if
By property d. of the rank 1 collections, we have that each tree can be written as the union
where each is a tree with the same top data as and has the additional property
The premeasure is given by
We now define a tuple of sizes on , that is, homogeneous and quasi-subadditive maps . For each we define the corresponding size on functions by
and denote the corresponding outer measure spaces as and outer spaces as
Here we recall that for ,
where the super level measure is defined to be the infimum of all values ( being the outer measure generated by the premeasure ), for running through all Borel subset of such that
We also note that there holds the following Hölder’s inequality:
be a Hölder tuple. Let , . Then
with absolute implicit constant.
Define another size
Then it is obvious that for any there holds
which by the Radon Nikodym proposition in  implies that
Furthermore, according to (3.1) and the classical Hölder’s inequality, one can easily check that for any fixed ,
Hence the outer Hölder inequality in  yields that
which completes the proof. ∎
4. Localized Carleson embeddings
In this section, when we write dyadic interval, we mean intervals , where is a fixed dyadic grid on . Fix a dyadic interval and with . We define the -stopping intervals of on by
Notice that is a pairwise disjoint collection of dyadic intervals and that the maximal theorem guarantees the sparseness condition
provided is chosen large enough. Furthermore, from the very definition of , there holds
In what follows, we fix a finite collection of rank 1 tritiles whose intervals are dyadic. We introduce the notation and the set of good tritiles
Let be a dyadic interval and be a Schwartz function. For any , there exists and such that
4.1. Proof of Proposition 4.1
The Proposition will be proved by a transference argument using the main result of , which is the continuous parameter version recalled below. However, Proposition 4.1 may also be obtained directly, by repeating the arguments of  in the (simpler, in fact) discrete parameter setting. We leave the details to the interested reader.
4.1.1. A continuous parameters version of Proposition 4.1
We need to define the continuous outer measure space on the base set
we are using that is a finite set. Let be an interval and . The corresponding generalized tent and its lacunary part, with fixed geometric parameters , are defined by
We use the superscript to distinguish discrete trees with top data from continuous tents with same top data . It will also be convenient to use the notation
for the projection of on the first two components.
An outer measure on , with
as generating collection is then defined via the premeasure . For Borel measurable, we define the size
Denoting by the corresponding strong and weak outer spaces, we turn to the reformulation of the main result of . A family of Schwartz functions
is said to be an adapted system with adaptation constants if
for all nonnegative integers and furthermore
The wave packet transform of a Schwartz function is then a function on defined by
With the same notation as in (4.1) for , and introducing the corresponding good set of parameters
we have the following continuous parameter version of Proposition 4.1.
[6, Theorem 1] Let be a dyadic interval and be a Schwartz function. For any , there exists and such that
The above proposition is obtained by choosing in [6, Theorem 1]. There are, however, two minor discrepancies between the result of  and the one recalled above. The first one is that, in definition (4.7), the intervals are used in place of . This change is necessary in order to perform a reduction argument to compact support in of see [6, Section 7.3.1], and can thus be avoided in the setup of Proposition 4.2 since the parameter is already in a compact interval. The second difference is that the adapted family used in  to define the wave packet transform is obtained by applying dilation, translation and modulation symmetries to a fixed mother wave packet. However, the arguments of  adapt naturally to the more general transform obtained from (4.6). We leave the details for the interested reader.
For each define
Up to possibly splitting into finitely many subcollections the sets are pairwise disjoint subsets of . Furthermore, the -measure of is comparable to up to a constant factor. Let be a fixed Schwartz function and be chosen such that
Then the family defined by for all , if does not belong to any is an adapted system. We claim that, if is the corresponding wave packet transform