Abstract
Let be Banach spaces of measurable functions in and let be a locally integrable function in . We say that if defined for and with compactly supported Fourier transform, extends to a bounded bilinear operator from to . In this paper we investigate some properties of the class for general spaces which are invariant under translation, modulation, and dilation, analyzing also the particular case of r.i. Banach function spaces. We shall give some examples in this class and some procedures to generate new bilinear multipliers. We shall focus on the case and find conditions for these classes to contain nonzero multipliers in terms of the Boyd indices for the spaces.
1. Introduction
Throughout the paper stands for the space of complex valued measurable functions defined on , and for the spaces of continuous function with compact support and vanishing at infinity, respectively, for the Schwartz class on , and for the set of functions in such that is compact. The Fourier transform of is defined by . For and , we denote , and are the translation, modulation, and dilation operators given by , by , and by for . We also recall the notation .
Throughout the paper we shall be considering such that is a Banach space and satisfies
We denote by the class of Banach spaces satisfying (1), (2), and (3).
We say that is homogeneous, to be denoted , whenever is dense in and, for any , the maps and are continuous from into .
If then is dense in . Indeed, using Minkowski’s inequality, for ) and one has Hence given we first approximate by and then, by a standard argument, we approximate by using the continuity of the map .
Let and ; we write
For instance, in the case one has and for , where is a submultiplicative Young functions with , one has (see [1, Remark 2.6]).
If and are Banach spaces in , we denote by the space of “pointwise” multipliers; that is,This becomes a Banach space under the norm
For one obtains the Köethe dual . Also notice that Hölder’s inequality gives for and . Also for Orlicz spaces (see [2], [3, page 64]) if , is Young functions satisfying then .
It is straightforward to see that if then and that
Given a couple we shall use the notation for the space of locally integrable functions defined on such thatwell defined for , satisfying that We endow the space with the norm .
The reader should be aware that sometimes is defined as the space of distributions such that for all . We are only restricting to those distributions such that .
For we shall write for the space of “convolution” multipliers; that is,This becomes a Banach space under the normOf course , i.e., for any and .
Using that , , and one obtains that is stable under convolution; that is, whenever . Moreover,
On the other hand Young’s inequality gives for with and . Also for Orlicz spaces (see [2], [3, page 64]) we have that if , are Young functions satisfying then .
From (4) we see that for any . Actually, using approximations of the identity, one has whenever .
With the notation for and and , we recall some well-known properties of the space of linear multipliers (see [4, 5]): whenever , for and for ,
In this paper we shall be concerned with the bilinear analogues and extensions of the above formulas for general function spaces. We shall extend several results shown by the author in the setting of Lebesgue and Orlicz spaces ([1, 6]). We present now the definition of a bilinear multiplier we shall be dealing with.
Definition 1. Let be a locally integrable function on . Definefor .
Let and . A locally integrable function is said to be a bilinear multiplier on of type if there exists such that for any and .
We write for such a space and where stands for the norm of the bounded bilinear map .
The theory of multilinear multipliers acting on Lebesgue spaces for “nice” symbols was originated in the work by R. Coiffman and C. Meyer [7] in the eighties and continued by L. Grafakos and R. Torres [8] and many others (see [9, 10]). The theory was retaken and pushed in the nineties after the celebrated result by M. Lacey and C. Thiele, solving the old standing conjecture of Calderón on the boundedness of the bilinear Hilbert transform (see [11, 12]). The bilinear versions of several classical linear operators appearing in Harmonic Analysis, such as the Hilbert transform or the fractional integral, are the motivation for the class of bilinear multipliers that we shall analyze in the paper. Recall that the bilinear Hilbert transform and the bilinear fractional integral are defined byand where .
It is easy to see that (19) and (20) correspond to the bilinear multipliers given by the symbols and , respectively; i.e.,This motivates the following particular class of bilinear multipliers.
Definition 2. We denote by the space of measurable functions such that We keep the notation
The boundedness results on -spaces for the bilinear and took long time to be achieved. In particular it was shown that for , , and ; i.e., there exists such that (Lacey-Thiele, [11–13]) and that for , , and ; i.e., there exists such that (Kenig-Stein [10], Grafakos-Kalton [9]).
The case of more general nonsmooth symbols was also analyzed by J. Gilbert and A. Namod (see [14, 15]).
The study of bilinear multipliers acting on other function spaces has been addressed in the literature. Lorentz spaces have been studied mainly by O. Blasco and F. Villarroya (see [16, 17]), weighted Lebesgue spaces or Lebesgue spaces with variable exponent by T. Gürkanli and O. Kulak [9], rearrangement invariant quasi-Banach spaces by S. Rodriguez-López [18], and more recently Orlicz spaces by O. Blasco and A. Osancliol [1].
Our objective is to study the basic properties of the classes and , to find examples of bilinear multipliers in these classes, and to get methods to produce new ones. We shall restrict ourselves to rearrangement Banach function spaces to recover some known results under some conditions on the Boyd indices. The results presented in what follows could be formulated for any , but we shall write our results only for for simplicity.
2. Bilinear Multipliers: The Basics
Throughout this section and . Let us start with some elementary properties of the bilinear multipliers when composing with translations, modulations, and dilations. Next result, already established in [6] for Lebesgue spaces and in [1] for Orlicz spaces, follows easily from the basic formulas
Proposition 3. Let . (a) for each and (b) for each and (c) for each and
Proof. (a) Let . It is easily seen that Hence and (b) If then one has Therefore, and (c) Let . We first observe that for each . Indeed, This gives which shows that and the desired estimate for the norm.
We start presenting an elementary example of bilinear multipliers. Recall that if is a Borel regular measure in then is a bounded measurable function in .
Proposition 4. Let be a Borel regular measure in , , and set . If with norm then . Moreover, .
Proof. Let us first rewrite the value as follows: Hence, using Minkowski’s inequality, one has This completes the proof.
Remark 5. Selecting in Proposition 4 one obtains , selecting one obtains , and selecting one has that
Proposition 6. Let and and , where .(a)If then . Moreover (b)If then . Moreover (c)If is a Borel regular measure on then . Moreover
Proof. (a) It follows trivially from (b) It was shown ([6, Proposition 2.5, (b)]) that From the vector-valued Minkowski inequality and part (a) in Proposition 3, we have (c) Observe that Argue as above, using now part (b) in Proposition 3, to conclude
With all these procedures we have several useful methods to produce examples of multipliers in , which extend those provided in particular cases in [1, 6].
Corollary 7. Let . (a)If and then (b)If and is a bounded measurable set in then(c)If then .(d)Let and and assume that is integrable in for each . Define Then and
Proof. (a), (b), and (c) follow trivially from Proposition 6.
(d) It is immediate to observe thatHence Minkowski’s inequality together with (1) and part (c) in Proposition 3 lead to the desired result and estimate.
3. The Case
As mentioned in the introduction a number of important bilinear multipliers, such as the bilinear fractional integral, the bilinear Hilbert transform, and other bilinear singular integrals, are defined for symbols for a given measurable function defined in . Let us restrict ourselves to this family of multipliers. As in the previous section we always assume and . We denote by the space of locally integrable functions such that , that is to say,defined for and compactly supported, satisfies the inequality We keep the notation
This class does have much richer properties than . Since the symbol is also defined on we can establish the following behaviour of the bilinear map under translations, modulations, and dilations:
Proposition 8. .
Proof. Let and define . We have that , , and for any and . Hence from the formulas for we obtain the following ones for : From them properties (1), (2), and (3) in are easily shown.
For symbols for a given we have the following expression.
Proposition 9. Let for a Borel regular measure . Then
Proof. Given , we can write and the proof is finished.
Note that, selecting and in Proposition 4, we obtain next example, but we would like to point out that it also follows from Proposition 9 even for spaces in .
Proposition 10. Let such that with norm and let . Then with .
We now produce a method to get multipliers in from those in .
Proposition 11. Let , , , and set . Then and
Proof. We use now the following formula for : Now recall that we actually have that for any . Using that we conclude the result.
As in the previous section we can generate new multipliers in extending [6, Proposition 3.5].
Proposition 12. Let and . Then (a) and (b) and (c)Let and and assume that is integrable in for each . Define Then and .
Proof. (a) Apply Minkowski’s inequality to the formula ([6, Proposition 3.5. (a)]: (b) Use ([6, Proposition 3.5. (b)] which establishes that together with Minkowski’s inequality and (51).
(c) Write making use of (50) the following formula: Therefore, from Minkowski’s again one getswhich finishes the proof.
Let us show that the classes are reduced to for some values of the parameters. We follow the approach used first in [6] and later in [1].
Lemma 13. Assume that and let such that for all . Then there exists a constant such that
Proof. It is known (see [6, Proposition 3.3]) that for we can write Let . One has that and for certain constant . Making use of (50) and (63) we have that Since we haveUsing that we have .
Hence The proof is then complete.
Theorem 14. Assume that and . Thenand
Proof. Using Proposition 12 we may assume that there exists a nonzero continuous and integrable function belonging to . Let such that . By using Lemma 13 for the function we obtain Since and continuous and in particular , through the convolution with an approximation of the identity and taking limits as one obtains This gives (67).
Since there exists such that . Using again Lemma 13, applied now to , we obtainTherefore, taking limits as we getHence we get (68) and the proof is finished.
Corollary 15. Let and let us writeIf then .
In the cases for the constants and can be explicitly computed. Therefore one recovers the following result.
Corollary 16 (see [6, 17]). Let and Then
4. Bilinear Multipliers on Rearrangement Invariant Banach Function Spaces
In this section we shall restrict our study to Banach function spaces. A space is called a “Banach function space” (see [19]), in short , if is a Banach space which satisfies(1) and a.e. implies that and .(2)If a. e. then .(3) whenever is measurable and .(4)For each there exists such that .
We shall denote and . It is clear that for any and that is dense in whenever .
Recall that is said to have “absolutely continuous norm”, in short , if for every and every sequence of measurable sets with a.e.
Proposition 17. If then .
Proof. Let . The fact that is dense in follows since bounded functions compactly supported are dense (see [19, Theorem 3.11]). To show that and are continuous for any we shall make use of the Lebesgue dominated theorem (see [19, Proposition 3.6]) which holds because has absolutely continuous norm. Now given and a sequence one has and what gives that . Therefore is continuous at the origin and hence at any point. To study the translation we first assume that is a bounded function supported on a finite set and with . In such a case and with . This gives that and therefore is continuous for any bounded function with finite support. Using the density of such functions in one gets the result for any .
Proposition 18. Assume that and . If a.e. where with then and .
Proof. For each and one has that a.e. and . Hence using Fatou’s lemma (see [19, Theorem 1.7]), one has This gives the result.
Recall that is said to be invariant under rearrangement, in short , whenever it satisfies the following.(5)If and is equimeasurable to then and .
Recall that if one defineswhere and is the r.i. space defined on with the same distribution function. In particular .
We observe that rearrangement invariant Banach function spaces preserve translations, modulations, and dilations; that is, Indeed, it follows using that , is equimeasurable to for any and for any (see [19, Proposition 5.11]), since . In particular if then . We shall write the class of rearrangement invariant Banach function spaces with absolutely continuous norm.
Taking into account that is equimeasurable with thenfor any and .
In the setting of Banach function spaces we can always consider the associate space , corresponding to the Köthe dual . It is well-known that is isometrically embedded into the dual (see [19, Lemma 2.8]) and that actually (see [19, Theorem 2.7]). This allows us to give a characterization of bilinear multipliers in in terms of the duality.
Proposition 19. Let for and such that , and let be a locally integrable function in . Then if and only if there exists such thatfor all and .
Corollary 20. Let . Then if and only if .
Proof. Due to Proposition 19, for some implies that Changing the variables and implies that
Let us give now a necessary condition for bilinear multipliers homogeneous of degree in the setting of rearrangement invariant Banach function spaces. We need to recall the definition of Boyd indices (see [19, page 149]): these are given by
Proposition 21. Let and , and assume that be a nonzero multiplier such that for any . Then and
Proof. From assumption for . Using now Proposition 3 we can write Since we haveTherefore,This shows thatHence making limits as and one obtains (81) and (82), respectively.
Corollary 22. Let . If is a nonzero and homogeneous of degree then .
In particular the bilinear Hilbert transform and the fractional integral can only be bounded whenever and , respectively.
We use now our general approaches to get concrete examples of multipliers in and .
Proposition 23. Let and satisfyLet and then and
Proof. We invoke firt Boyd’s result (see [19, Theorem 5.18]) which establishes that if and only if . Since and for one has that and . Now applying part (a) in Proposition 6 one obtains the result.
Proposition 24. Let . Then
Proof. Let and . We shall see that and We use the formulation (see [6, Proposition 3.3]) given byHence, since whenever it is well defined, for we can write This gives the result.
We now shall combine our results with the method of interpolation for Banach lattices due to Calderón (see [4, 20]) to get some sufficient conditions on multipliers in . Recall that for and we can define the Banach function space
Proposition 25. Let satisfying that and let . Set and assume that . Then
Proof. Consider the trilinear form From Proposition 9, assuming that , we have for . Now from Proposition 10 we conclude that is bounded from into and it has norm bounded by .
On the other hand, if , using Hölder’s inequality,This shows that is also bounded from into . Therefore, by interpolation, for each one obtains that is bounded from into . This shows that for any .
Let us apply the previous proposition for , , and with with and .
Corollary 26. Let and with . If thenwhere is such that .
Proof. Since invoking Hausdorff-Young, . Select such that for and for . Hence . The result now follows from Proposition 25.
Proposition 27. Let satisfying that and let . Setand assume that . Then
Proof. Note that . Hence each . Therefore, Hence, denoting as above , one obtains that is bounded from into .
Using duality, , where , becauseTherefore is also bounded from into .
Now the result follows again by interpolation.
Let us apply the previous proposition for , , and with with with (in particular ).
Corollary 28. Let with and . If thenwhere is such that .
Proof. As above . Select such that for and for . Hence and the result follows from Proposition 27.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that they have no conflicts of interest.
Acknowledgments
This study is partially supported by Proyecto MTM2014-53009-P.