Linear and bilinear theorems à la Stein

By Árpád Bényi and Tadahiro Oh

Abstract

In this work, we state and prove versions of the linear and bilinear theorems involving quantitative estimates, analogous to the quantitative linear theorem due to Stein.

1. Introduction

The impact of the classical Calderón-Zygmund theory permeates through analysis and PDEs. Nowadays, both the linear and multilinear aspects of this theory are well understood and continue to be intertwined with aspects of analysis that are beyond their reach, such as those considering the bilinear Hilbert transform.

Two fundamental results in the linear theory from the 1980s are the celebrated theorem of David and Journé Reference 4 and theorem of David, Journé, and Semmes Reference 5. Both results were strongly motivated by the study of the Cauchy integral on a Lipschitz curve and the related Calderón commutators. Their gist lies in understanding the boundedness of a singular operator via appropriate simpler testing conditions.

In the theorem, one needs to test a singular operator and its transpose on the constant function 1. If both the operator and its transpose were bounded, then by duality and interpolation Reference 6, the operator would be bounded on . The remarkable aspect of the theorem is that one does not need to test the operator on the whole , but just on one special element in it. Going back to the Cauchy integral operator associated to a Lipschitz function , it turns out that it is not necessarily easy to test the operator on 1. It is, however, much easier to test the operator on the function . Thus, as the name suggests, the theorem extends the theorem by replacing the constant function 1 with a suitable function or, to be more precise, by replacing 1 with two suitable functions and in on which we test an operator and its transpose. The bilinear Calderón-Zygmund theory has its own versions of the and theorems, such as those proved by Grafakos and Torres Reference 8 and by Hart Reference 12, respectively. See Theorems D and E below.

In this work, we revisit the theorem, both in the linear and bilinear settings, through the lens of a gem due to Stein Reference 15. We are alluding to his formulation of the theorem involving quantitative estimates for a singular operator and its transpose when tested now on normalized bump functions. Our goal is to prove that an analogous natural formulation à la Stein can be given for the theorems in the linear and bilinear settings. We note that, while for the sake of clarity in our presentation we have chosen to delineate the linear and bilinear settings, a unified discussion is certainly possible under the encompassing more general multilinear setting.

2. Linear Calderón-Zygmund theory

In this section, we consider a linear singular operator a priori defined from into of the form

Here, we assume that, away from the diagonal , the distributional kernel of coincides with a function that is locally integrable on . The formal transpose of is defined similarly with the kernel given by .

Definition 2.1.

A locally integrable function on is called a (linear) Calderón-Zygmund kernel if it satisfies the following conditions:

(i)

For all , we have .

(ii)

There exists such that

for all satisfying .

We say that a linear singular operator of the form Equation 2.1 with a Calderón-Zygmund kernel is a linear Calderón-Zygmund operator if extends to a bounded operator on for some . It is well known Reference 14 that if is a linear Calderón-Zygmund operator, then it is bounded on for all . Hence, in the following, we restrict our attention to the -boundedness of such linear operators. We point out that the Calderón-Zygmund operator is also bounded. Here, denotes the space of functions of bounded mean oscillation, which we now recall.

Definition 2.2.

Given a locally integrable function on , define the -seminorm by

where the supremum is taken over all cubes and

Then, we say that is of bounded mean oscillation if and we define by

2.1. Classical linear and theorems

In this subsection, we provide a brief discussion of the classical and theorems proved in Reference 4 and Reference 5, respectively. In order to do so, we need to define a few more notions.

Definition 2.3.

We say that a function is a normalized bump function of order if and for all multi-indices with .

Here, denotes the ball of radius centered at . Given and , we set

Definition 2.4.

We say that a linear singular integral operator has the weak boundedness property if there exists such that we have

for all normalized bump functions and of order , , and .

We note that it suffices to verify Equation 2.4 for ; see Reference 11. The statement of the theorem of David and Journé Reference 4 is the following.

Theorem A ( theorem).

Let be a linear singular integral operator with a Calderón-Zygmund kernel. Then, can be extended to a bounded operator on if and only if

(i)

satisfies the weak boundedness property,

(ii)

and are in .

Since is a priori defined only in , the expressions and are, of course, not well defined and need to be interpreted carefully. The same comment applies to the corresponding theorems in the bilinear setting.

The main concept needed in extending the theorem to the theorem is that of para-accretive functions.

Definition 2.5.

We say that a function is para-accretiveFootnote1 if there exists such that, for every cube , there exists a subcube such that

1

An extra condition that is sometimes included in the definition of para-accretivity. This, however, is not necessary. Indeed, it follows from Equation 2.6 and the Lebesgue differentiation theorem that almost everywhere. In particular, we have .

It follows from Equation 2.5 that

In particular, the function 1 is automatically para-accretive. It is also worth pointing out that the definition of para-accretivity in Definition 2.5 is not the same as the one used in the classical theorem of David, Journé, and Semmes Reference 5. The notion of para-accretivity stated here is borrowed from Reference 10Reference 12; for a similar definition in which cubes are replaced by balls, see Christ’s monograph Reference 2. The two definitions of para-accretivity are nevertheless equivalent. Since this natural observation seems to be missing from the literature, for the convenience of the reader we have included its proof in the appendix.

Before giving a meaning to operators to which the theorem applies, we need one more definition.

Definition 2.6.

Given , let be the collection of all functions from such that , where the -norm is given by

We also denote by the subspace of all compactly supported functions in .

Definition 2.7.

Let and be para-accretive functions. A linear singular operator is called a linear singular integral operator of Calderón-Zygmund type associated to and if is continuous from into for some and there exists a Calderón-Zygmund kernel such that

for all such that . Here, denotes the operation of multiplication by .

With these preparations, we are now ready to state the classical theorem Reference 5.

Theorem B ( theorem).

Let and be para-accretive functions. Suppose that is a linear singular integral operator of Calderón-Zygmund type associated to and . Then, can be extended to a bounded operator on if and only if the following conditions hold:

(i)

satisfies the weak boundedness property,

(ii)

and are in .

In the special case when and are accretive⁠Footnote2 and , the theorem was independently proved by McIntosh and Meyer Reference 13.

2

A function is called accretive if there exists such that for all . Note that an accretive function is para-accretive.

Remark 2.8.

In Reference 5, condition (ii) of Theorem B is stated slightly differently; it was assumed that . We note that this is just a matter of notation. For example, the condition in Reference 5 means that that there exists such that

This is clearly equivalent to

Here, we used the fact that is a one-to-one correspondence since is para-accretive and thus, in particular, is bounded away from zero almost everywhere. In Theorem B, we followed the notation from Reference 12 to signify the fact that the condition indeed depends on both and , and what we mean by condition (ii) in Theorem B is precisely the statement Equation 2.7. See also Theorem E below in the bilinear setting.

Lastly, note that, as in the theorem, expressions and are not a priori well defined, and thus some care must be taken.

2.2. Formulations of the and theorems à la Stein

There is another formulation of the theorem due to Stein Reference 15 in which conditions (i) and (ii) in Theorem A are replaced by the quantitative estimate Equation 2.8 involving normalized bump functions.

Theorem C ( theorem à la Stein).

Let be as in Theorem A. Then, can be extended to a bounded operator on if and only if there exists such that we have

for any normalized bump function of order , , and .

By viewing the expressions and as and , it is natural to extend this result by replacing the constant function 1 by para-accretive functions and . This is the first result of our paper.

Theorem 1 ( theorem à la Stein).

Let , , and be as in Theorem B. Then, can be extended to a bounded operator on if and only if there exists such that the following two inequalities hold for any normalized bump function of order , , and :

We present the proof of Theorem 1 in Section 4.

As an application of this result, one could recover the well-known fact that the commutator of a pseudodifferential operator with symbol in the Hörmander class and the multiplication operator of a Lipschitz function is bounded on . Indeed, suppose that for all and all multi-indices we have

and let

be the corresponding pseudodifferential operator. Also, given such that for , let

be the commutator of and the multiplication operator . It is straightforward to check that the kernel of is Calderón-Zygmund and, by a similar computation to the one in Reference 15, pp. 309-310, Equation 2.9 and Equation 2.10 hold as well, thus proving .

3. Bilinear Calderón-Zygmund theory

Next, we turn our attention to the bilinear setting and consider the corresponding extensions of the results in Section 2. Namely, we consider a bilinear singular operator a priori defined from into of the form

where we assume that, away from the diagonal , the distributional kernel coincides with a function that is locally integrable on . The formal transposes and are defined in an analogous manner with the kernels and given by

Definition 3.1.

A locally integrable function on is called a (bilinear) Calderón-Zygmund kernel if it satisfies the following conditions:

(i)

For all , we have

(ii)

There exists such that

for all satisfying . Moreover, we assume that the formal transpose kernels and also satisfy the regularity condition Equation 3.2.

We say that a bilinear singular operator of the form Equation 3.1 with a bilinear Calderón-Zygmund kernel is a bilinear Calderón-Zygmund operator if extends a bounded operator on into for some with .

Similarly to the linear case, the crux of the bilinear Calderón-Zygmund theory is contained in the fact that if is a bilinear Calderón-Zygmund operator, then it is bounded on into for all with (with the appropriate statements at the endpoints); see Grafakos and Torres Reference 8. Therefore, the main question is to prove that there exists at least one triple ) with and such that is bounded from into .

The weak boundedness property for bilinear singular operators has a similar flavor as the one in the linear case.

Definition 3.2.

We say that a bilinear singular integral operator has the (bilinear) weak boundedness property if there exists such that we have

for all normalized bump functions of order , , and .

Remark 3.3.

It follows from Reference 1, Lemma 9 that it suffices to verify Equation 3.3 for .

3.1. Bilinear and theorems

We now state the bilinear theorem in the form given by Hart Reference 11.

Theorem D (Bilinear theorem).

Let be a bilinear singular integral operator with a standard Calderón-Zygmund kernel. Then, can be extended to a bounded operator on for all with if and only if

(i)

satisfies the weak boundedness property,

(ii)

and are in .

We chose this formulation since it closely follows the statement of the classical linear theorem given in the previous section. Further, note that Theorem D is equivalent to the formulation of Grafakos and Torres Reference 8; see also Christ and Journé Reference 3.

Next, we turn our attention to the bilinear version of the theorem.

Definition 3.4.

Let , , and be para-accretive functions. A bilinear singular operator is called a bilinear singular integral operator of Calderón-Zygmund type associated to , , and if is continuous from