Research Article | Open Access
On a Class of Bilinear Pseudodifferential Operators
We provide a direct proof for the boundedness of pseudodifferential operators with symbols in the bilinear Hörmander class , . The proof uses a reduction to bilinear elementary symbols and Littlewood-Paley theory.
1. Introduction: Main Results and Examples
Coifman and Meyer’s ideas on multilinear operators and their applications in partial differential equations (PDEs) have had a great impact in the future developments and growth witnessed in the topic of multilinear singular integrals. One of their classical results [1, Proposition 2, p. 154] is about the boundedness of a class of translation invariant bilinear operators (bilinear multiplier operators) given by We have the following.
Theorem A. If for all and all multi-indices , then has a bounded extension from into , for all , such that .
In fact, Coifman and Meyer’s approach yields Theorem A only for . The optimal extension of their result to the range (as implied in the theorem above) can be obtained using interpolation arguments and an end-point estimate into in the works of Grafakos and Torres  and Kenig and Stein .
Bilinear pseudodifferential operators are natural nontranslation invariant generalizations of the translation invariant ones; they allow symbols to depend on the space variable as well. Let us then consider bilinear operators a priori defined from into of the form Perhaps unsurprisingly, we impose then similar conditions on the derivatives of the symbol with the expectation that they would yield indeed bounded operators on appropriate spaces of functions. The estimates that we have in mind define the so-called bilinear Hörmander classes of symbols, denoted by . We say that if for all and all multi-indices . Note that we need smoothness in as in the linear Hörmander classes. As usual, the notation means that there exists a positive constant (independent of ) such that .
With this terminology, we can restate Theorem A as follows.
If the -independent symbol belongs to the class , then is bounded from into for all such that .
The condition of translation invariance (equivalently, the -independence of the symbol) is superfluous. Moreover, the previous boundedness result can be shown to hold for the larger class of symbols , where . This is a known fact that is tightly connected to the bilinear Calderón-Zygmund theory developed by Grafakos and Torres in  and the existence of a transposition symbolic calculus proved by Bényi et al. . Let us briefly give an outline of how this follows. First, we note that the bilinear kernels associated to bilinear operators with symbols in , , are bilinear Calderón-Zygmund operators in the sense of . Second, we recall that [2, Corollary 1], which is an application of the bilinear theorem therein, states the following.
Theorem B. If and its transposes, and , have symbols in , then they can be extended as bounded operators from into for , and .
Third, by [4, Theorem 2.1], we have the following.
Theorem C. Assume that , , and . Then, for ,, where .
Finally, since , we can directly combine Theorems B and C to recover the following optimal extension of the Coifman-Meyer result; note that now the symbol is allowed to depend on while is still allowed to be in the optimal interval .
Theorem 1. If is a symbol in , , then has a bounded extension from into for all , such that .
Once we have the boundedness of the class on products of Lebesgue spaces, a “reduction method” allows us to deduce also the boundedness of the class on appropriate products of Sobolev spaces. Moreover, our estimates in this case come in the form of Leibniz-type rules; for more on these kinds of properties, see the work of Bernicot et al. . In the particular case when the bilinear operator is just a differential operator, the Leibniz-type rules are referred to as Kato-Ponce’s commutator estimates and are known to play a significant role in the study of the Euler and Navier-Stokes equations, see ; see also Kenig et al.  for further applications of commutators to nonlinear Schrödinger equations. Let denote the linear Fourier multiplier operator with symbol , where . By definition, we say that belongs to the Sobolev space if . We have the following.
Theorem 2. Let be a symbol in , , , and let be its associated operator. Then there exist symbols and in such that, for all , In particular, then one has that has a bounded extension from into , provided that , , . Moreover,
The proof of Theorem 2 follows a similar path as the one in the work of Bényi et al. [8, Theorem 2.7]. For the convenience of the reader, we sketch here the main steps in the argument. Let be a -function on such that , supp , and on . Then (4) holds if we let Now, straightforward calculations that take into account the support condition on given that and belong to . The Leibniz-type estimate (5) follows now from Theorem 1 and (4).
It is also worthwhile to note that we can replace (5) with a more general Leibniz-type rule of the form where , , .
One of the main reasons for the study of the Hörmander classes of bilinear pseudodifferential operators is the fact that the conditions imposed on the symbols arise naturally in PDEs. In particular, the bilinear Hörmader classes model the product of two functions and their derivatives.
Example 3. Consider first a bilinear partial differential operator with variable coefficients Note that , where the bilinear symbol is given by Assuming that the coefficients have bounded derivatives, it is easy to show that .
Example 4. The symbol in the previous example is almost equivalent to a multiplier of the form Indeed, this symbol belongs to . We can also think of this symbol as the bilinear counterpart of the multiplier that defines the linear operator .
Example 5. With the notation in Example 4, the multipliers and belong to . In general, the multipliers belong to .
Example 6. One of the recurrent techniques in PDE estimates is to truncate a given multiplier at the right scale. Consider now where and are smooth “cutoff” functions supported in the annulus , and the coefficients satisfy derivative estimates of the form Elementary calculations show that .
Remark 7. Theorems 1 and 2 lead to the natural question about the boundedness properties of other Hörmander classes of bilinear pseudodifferential operators. An interesting situation arises when we consider the bilinear Calderón-Vaillancourt class . A result of Bényi and Torres  shows that, in this case, the boundedness fails. One can impose some additional conditions (besides being in ) on a symbol to guarantee that the corresponding bilinear pseudodifferential operator is bounded; see, for example,  and the recent work of Bernicot and Shrivastava . However, there is a nice substitute for the Lebesgue space estimates. If we consider instead modulation spaces (see the excellent book by Gröchenig  for their definition and basic properties), we can show, for example, that if then (which contains ). This and other more general boundedness results on modulation spaces for the class were obtained by Bényi et al. . Then, this particular boundedness result with the reduction method employed in Theorem 2 allows us to also obtain the boundedness of the class from into . Interestingly, we can also obtain the boundedness of the class , but we have to require in this case the order to depend on the Lebesgue exponents; see the work of Bényi et al. , also Miyachi and Tomita  for the optimality of the order and the extension of the result in  below . The most general case of the classes is also given in .
In the remainder of the paper we will provide an alternate proof of Theorem 1 that does not use sophisticated tools such as the symbolic calculus. The proof is in the original spirit of the work of Coifman and Meyer that made use of the Littlewood-Paley theory. As such, we will only be concerned here with the boundedness into the target space with . Of course, obtaining the full result for is then possible because of the bilinear Calderón-Zygmund theory, which applies to our case. We will borrow some of the ideas from Bényi and Torres , which in turn go back to the nice exposition (in the linear case) by Journé , by making use of the so-called bilinear elementary symbols.
2. Proof of Theorem 1
We start with two lemmas that provide the anticipated decomposition of our symbol into bilinear elementary symbols. Since they are the immediate counterparts of [15, Lemma 1 and Lemma 2] to our class , we will skip their proofs; see also [16, pp. 72–75] and [1, pp. 55–57]. The first reduction is as follows.
Lemma 8. Fix a symbol in the class , , and an arbitrary large positive integer . Then, for any , can be written in the form where is an absolutely convergent sequence of numbers, with each a -function supported on the set , and is a bounded operator from into , for , .
Now, if is any of the symbols in (14) and we knew a priori that are bounded from into with operator norms depending only on the implicit constants from (15), the fact that the sequence is absolutely convergent immediately implies the boundedness of . Our first step has thus reduced the study of generic symbols in the class to symbols of the form where , and is supported in .
Our second step is to further reduce the simpler looking symbol given in (16) to a sum of bilinear elementary symbols.
Lemma 9. Let be as in (16). One can further reduce the study to symbols of the form where the elementary symbols , , are defined via with , , , , , and .
Therefore, we are now only faced with the question of boundedness for the two operators and , with and defined in Lemma 9; boundedness of follows from that of by symmetry. In the following, for , we write In this case, for , we can write
Claim 1. is bounded from into .
Proof. By (20) and the Cauchy-Schwarz inequality, we can write We used here the fact that the coefficients are bounded, see Lemma 9. Using now Hölder’s inequality, we have Finally, the conditions on the supports of and allow us to make use of the Littlewood-Paley theory and conclude that
Obtaining the boundedness of the operator is a bit more delicate due to the support condition on , specifically having contained in a disk rather than in an annulus. Nevertheless, we still claim the following.
Claim 2. is bounded from into .
Proof. Note first that we can still use the Littlewood-Paley theory on the part of the sum that defines . We have the following inequalities: At this point, however, we must proceed more cautiously. Observe that Denoting then , we now have , where , and satisfies the support condition (25). Assume for the moment that the following inequality holds: Then, the boundedness of the operator can be obtained as follows:
Claim 3. Assume that and . Then, for all , we have
In our proof of this claim, we will make use of Journé’s lemma [16, p. 69].
Lemma 10. There exists a constant such that, for all , , where , , and .
Proof of Claim 3. First, we consider the case . With the notation in Lemma 10, it suffices to estimate and .
The estimate on the “bad part” follows from the triangle and Cauchy-Schwarz inequalities and the control ; recall that : Let us now look at the “good part”. We start by noticing that, given the two summation indices , we have Thus, for , we have . In view of this orthogonality, Plancherel’s theorem with the estimate gives This completes the proof of the case . In the general case , we again seek the control of the “bad” and “good” parts. The estimate on the “bad” part follows virtually the same as in the case : where we used Minkowski’s integral inequality in the last step.
For the “good” part, we can think of as being dyadic blocks in the Littlewood-Paley decomposition of the sum . Thus, it will be enough to control uniformly (in the norm) the sums , , in order to obtain the same bound on . The control on however follows from the uniform estimate on the ’s and an immediate application of Littlewood-Paley theory.
Á. Bényi’s work is partially supported by a Grant from the Simons Foundation (no. 246024). T. Oh acknowledges support from an AMS-Simons Travel Grant.
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Copyright © 2013 Árpád Bényi and Tadahiro Oh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.