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Journal of Function Spaces
Volume 2018, Article ID 7560870, 10 pages
Research Article

Bilinear Localization Operators on Modulation Spaces

Department of Mathematics and Informatics, University of Novi Sad, Novi Sad, Serbia

Correspondence should be addressed to Nenad Teofanov;

Received 30 October 2017; Accepted 17 December 2017; Published 15 January 2018

Academic Editor: Kasso A. Okoudjou

Copyright © 2018 Nenad Teofanov. 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.


We introduce a class of bilinear localization operators and show how to interpret them as bilinear Weyl pseudodifferential operators. Such interpretation is well known in linear case whereas in bilinear case it has not been considered so far. Then we study continuity properties of both bilinear Weyl pseudodifferential operators and bilinear localization operators which are formulated in terms of a modified version of modulation spaces.

1. Introduction

Localization operators were introduced by Berezin in the study of general Hamiltonians related to quantization problem in quantum mechanics [1]. In signal analysis they are related to the localization technique proposed by Slepian-Polak-Landau; see, for example, a survey article [2]. Thereafter, a more detailed study of localization in phase space together with basic facts on localization operators and their applications in optics and signal analysis was given by Daubechies in [3]. That paper initiated further study of the topic. Daubechies studied localization operators on (see Section 2 for the definition) with Gaussian windows Such operators are named Daubechies’ operators afterwards. The eigenfunctions of Daubechies’ operators are Hermite functions: and the eigenvalues can be explicitly calculated. This is an important issue in applications (cf. [4]).

For related inverse problem in the case of simply connected localization domain , we refer to [5] where it is proved that if one of the eigenfunctions of Daubechies’ operator is a Hermite function, then is a disc centered at the origin.

An exposition of different quantization theories and connection between localization operators and Toeplitz operators is given in, for example, [6, 7]. The problem of quantization served as a motivation for the study of localization operators on , , where is a locally compact group; see [8]. There one can also find a product formula and Schatten-von Neumann properties of localization operators; see also [9].

Since the beginning of the XXI century, localization operators in the context of modulation spaces were studied by many authors (cf. [1023]). See also the references given there.

Our results are related to, but different from, investigations given in [15] due to the difference in definitions of bilinear localization operators; see Remark 2. Moreover, instead of standard modulation spaces observed in [15], continuity properties are here formulated in terms of a modified version of modulation spaces denoted by ; see Definition 5. Also, in contrast to [15] in this paper we do not observe multilinear version of localization operators, since we use sharp convolution estimates for modulation spaces given in [24]; see Theorems 8 and 9. These results are well suited for the study of bilinear operators, but their extension to multilinear case is a challenging problem.

One of the key ingredients in our investigations is the interpretation of bilinear localization operators as bilinear Weyl pseudodifferential operators, DO for short. Indeed, results on multilinear Kohn-Nirenberg DOs from [1012] served as a background for the continuity properties of multilinear Kohn-Nirenberg localization operators given in [15]. Here we consider the so-called Weyl correspondence instead and obtain continuity properties (Theorems 15 and 16) analogous to [15, Theorems and ] when restricted to bilinear operators.

The paper can be summarized as follows. We first introduce bilinear localization operators, bilinear Weyl DOs, and bilinear Wigner transform and show how they are connected; see Section 2. In particular, we prove that bilinear localization operator can be interpreted as bilinear DO, Theorem 4. In Section 3 we recall the definition and basic properties of modulation spaces and introduce their convenient modified version (see Definition 5 for details). Then we recall convolution estimates from [24] which will be used in the proof of main results of the paper given in Section 4. These results are boundedness of bilinear DOs on (Theorem 14) and sufficient and necessary conditions for boundedness of bilinear localization operators on (Theorems 15 and 16, resp.).

Notation. The Schwartz class is denoted by and the space of tempered distributions by . We use the brackets to denote the extension of the inner product on to any pair of dual spaces. The Fourier transform is normalized to be . The involution is , and the convolution of and is given by , when the integral exists.

We denote by the polynomial weights and , when

We use the notation to indicate that for a suitable constant , whereas means that for some .

2. Bilinear Localization Operators

To define localization operators we start with the short-time Fourier transform, a time-frequency representation related to Feichtinger’s modulation spaces [25, 26].

Let be the Gelfand-Shilov type space of analytic functions given by

If , then extends to a holomorphic function in the strip for a suitable [27, 28]. The dual space of will be denoted by

Translation and modulation operators are given by

The short-time Fourier transform (STFT in the sequel) of with respect to the window is given by

The map from to extends uniquely to a continuous operator from to by duality.

Moreover, for a fixed the following characterization holds:

We refer to [20, 21, 2931] for the proof and more details on STFT in other spaces of Gelfand-Shilov type.

The localization operator with symbol and windows is given by

Next, we define bilinear localization operators as follows.

Definition 1. Let and . The bilinear localization operator with symbol and window , where , , is given by where , , , and

Remark 2. Let denote the trace mapping that assigns to each function defined on a function defined on by the formula Then, is the bilinear operator given in [15, Definition ].

In order to give an interpretation of bilinear operators in the weak sense, we introduce the following notation. Let there be given Then where , , , and Thus, the weak definition of (9) is given by and . The brackets can be interpreted as suitable duality between a pair of dual spaces. Thus, is a well-defined continuous operator from to

Next, we introduce a class of bilinear Weyl pseudodifferential operators and use the Wigner transform to provide appropriate interpretation of bilinear localization operators as bilinear Weyl pseudodifferential operators.

Let . Then the Weyl pseudodifferential operator with the Weyl symbol can be defined as the oscillatory integral: This definition extends to each , so that is a continuous mapping from to . If denotes the Wigner transform, also known as the cross-Wigner distribution, then the following formula holds: for each ; see, for example, [26, 32, 33].

By analogy with (13) we define the bilinear Weyl pseudodifferential operator as follows: where , , and . Here denotes the identity matrix in (e.g., if , then ).

To give the interpretation of (15) in the context of bilinear DOs we introduce the bilinear version of (14) as follows. Let , , and . Then the bilinear Wigner transform is given by and It is easy to see that .

Lemma 3. Let and . Then given by (16) extends to a continuous map from to and the following formula holds:

Proof. The proof follows by the straightforward calculation: where we used and the change of variables and This extends to each , since when .

The so-called Weyl connection between the set of linear localization operators and Weyl DOs is well known; we refer to, for example, [21, 32, 34]. The proof of the following Weyl connection between the set of bilinear localization operators and corresponding bilinear Weyl DOs is based on the kernel theorem for Gelfand-Shilov spaces (see, e.g., [20, 35]) and direct calculations. Since the proof is quite technical we present it in the separate Section 5. The conclusion of Theorem 4 is that, as in the linear case, the bilinear localization operators can be viewed as a subclass of the bilinear Weyl DOs.

Theorem 4. Let there be given and let and , where , Then the localization operator is the Weyl pseudodifferential operator with the Weyl symbol Therefore, if , , and , , we have

3. Modulation Spaces

Since we essentially use the convolution estimates for polynomially weighted modulation spaces (Theorems 8 and 9), by Theorem 7 below it is enough to use the duality between and instead of the more general duality between and . We refer to [16, 17] for investigations in the framework of subexponential and superexponential weights and leave the study of bilinear localization operators in that case for a separate contribution.

Modulation spaces [25, 26] are defined through decay and integrability conditions on STFT, which makes them suitable for time-frequency analysis and for the study of localization operators in particular. Their definition is given in terms of weighted mixed-norm Lebesgue spaces.

In general, a weight on is a nonnegative and continuous function. By and , we denote the weighted Lebesgue space defined by the norm with the usual modification when . When , , we use the notation instead.

Similarly, the weighted mixed-norm space , , consists of (Lebesgue) measurable functions on such that where is a weight on .

In particular, when , , we use the notation .

Now, modulation space consists of distributions whose STFT belong to .

Definition 5. Let , , and . The modulation space consists of all such that (with obvious interpretation of the integrals when or ).

For the consistency, and according to (11), we denote by the set of , , such that

In special cases we use the usual abbreviations: , , and so on.

Remark 6. Notice that the original definition given in [25] contains more general submultiplicative weights. We restrict ourselves to , , since the convolution and multiplication estimates which will be used later on are formulated in terms of weighted spaces with such polynomial weights. As already mentioned, weights of exponential type growth are used in the study of Gelfand-Shilov spaces and their duals in [16, 2931]. We refer to [36] for a survey on the most important types of weights commonly used in time-frequency analysis.

The following theorem lists some of the basic properties of modulation spaces. We refer to [25, 26] for its proof.

Theorem 7. Let and , . Then (1) are Banach spaces, independent of the choice of ;(2)if , , , and , then (3) and

Modulation spaces include the following well-known function spaces:(a)(b)(c)Sobolev spaces: (d) Shubin spaces: (cf. [37]).

For the results on multiplication and convolution in modulation spaces and in weighted Lebesgue spaces we first introduce the Young functional:

When , the Young inequality for convolution reads as

The following theorem is an extension of the Young inequality to the case of weighted Lebesgue spaces and modulation spaces when .

Theorem 8. Let and , . Assume that , , andwith strict inequality in (30) when and for some .
Then on extends uniquely to a continuous map from (1) to ;(2) to .

For the proof we refer to [24]. It is based on the detailed study of an auxiliary three-linear map over carefully chosen regions in ; see Sections and in [24]. This result extends multiplication and convolution properties obtained in [38]. Moreover, the result is sharp in the following sense.

Theorem 9. Let and , . Assume that at least one of the following statements holds true: (1)The map on is continuously extendable to a map from to .(2)The map on is continuously extendable to a map from to ; Then (29) and (30) hold true.

4. Continuity Properties

We start estimates of the modulation space norm of the cross-Wigner distribution (cf. [21]). We refer to [39, Theorem ] for more refined estimates, and note that in [40] the sufficient conditions for the continuity of the cross-Wigner distribution on modulation spaces are proved to be necessary too (in the unweighted case). Proposition 10 coincides with certain sufficient conditions from [40, Theorem ] when restricted to , , and .

Proposition 10. Let the assumptions of Theorem 8 hold. If , then the map where is the cross-Wigner distribution given by (14) extends to sesquilinear continuous map from to .

Proof. We give a short version of the proof for the sake of completeness and refer to [21] for details. Let . Then where , since, by Theorem 7, modulation spaces are independent of the choice of the window function (from ).
From see [26, Lemma ], and from the proof of [26, Lemma (b)] it follows that Hence the norms in (32) are equivalent to where the convolution is obtained from the integration over and after the change of variables ; see also [13, Proposition ]. Therefore where the last estimate follows from Theorem 8.

We refer to [10] for the multilinear version of (32), which in turn gives the multilinear version of Corollary 11 (in unweighted case). In fact, from the inspection of the proof of Proposition 10, the definition of given by (17), and (25) we conclude the following.

Corollary 11. Let the assumptions of Theorem 8 hold. If , and , then the map , where is the cross-Wigner distribution given by (17) which extends to a continuous map from to .

Remark 12. Proposition 10, when restricted to , , and gives that is, [13, Proposition ] (with a slightly different notation).
For a certain choice of the parameters (in particular when ) we obtain estimates sharper than (37). In particular, if , , and , we obtain with when and for some .

Next we prove an extension of [26, Theorem ] to the bilinear Weyl DOs. Recall, if is the Weyl symbol of , then [26, Theorem ] says that is bounded on , . This result has a long history starting from the Calderon-Vaillancourt theorem on boundedness of the pseudodifferential operators with smooth and bounded symbols on , [41]. It is generalized by Sjöstrand in [42] where is used as appropriate symbol class. Sjöstrand’s results were thereafter extended in [22, 26, 39, 4345]. Moreover, we refer to [1012] for the multilinear Kohn-Nirenberg DOs.

To deal with duality when we observe that, by a slight modification of [10, Lemma ], the following is true.

Lemma 13. Let denote the space of bounded, measurable functions on which vanish at infinity and put equipped with the norms of , , and , respectively. Then, (a) is -closure of in , hence is a closed subspace of . Likewise for and (b)The following duality results hold for : , , and

From now on, we will use these duality relations in the cases and/or without further explanations.

Theorem 14. Let and let be given by (16). The operator is bounded from to , , with a uniform estimate for the operator norm.

Proof. Note that the integrals here below are well defined and that the order of integration is irrelevant. We have where is a window function. By slight modifications of the proof of [26, Theorem ] we obtain the following estimate: uniformly in Therefore, for all It follows that and Hence the operator is bounded on and as claimed.

Finally, we use the relation between the Weyl pseudodifferential operators and the localization operators (Theorem 4) and the convolution estimates for modulation spaces (Theorem 8) to obtain continuity results for for different choices of windows and symbols.

Theorem 15. Let the assumptions of Theorem 8 hold and let , and . If , , and , where , , and with when , then is bounded on for all , and the operator norm satisfies the uniform estimate

Proof. Let and From Corollary 11 it follows that Now, the calculation of from the proof of Theorem 4 together with Theorem 8 implies that if the involved parameters fulfill the conditions of the theorem.
On the one hand, for the Lebesgue parameters it is easy to see that is equivalent to and that is equivalent to . On the other hand, by the choice of the weight parameters and it follows that In particular, if then .
Put . Then Theorems 14, 4, and 8 imply that and the Theorem is proved.

We remark that a modification of Theorem 15 can be obtained by using [26, Theorem ] instead. That result allows symbols from weighted modulation spaces. We leave for the reader to check how to change the conditions on weight parameters in Theorem 15 in that case.

We finish with a necessary condition. The proof of Theorem 16 is a slight modification of the proof [13, Theorem ], and we leave it for the reader. Here below denotes the norm of a bounded operator.

Theorem 16. Let the assumptions of Theorem 8 hold and let . If there exists a constant depending only on the symbol and for all and , with , then .

5. The Proof of Theorem 4

Note first that the integrals here below are absolutely convergent and that changing the order of integration is allowed. Moreover, certain oscillatory integrals are meaningful in in a suitable interpretation. For example, if denotes the Dirac distribution, then the Fourier inversion formula in the sense of distributions gives , wherefrom , when .

We first rewrite (12) in the form of a kernel operator. where the kernel , , , and , is given by

Next, we calculate the convolution . We will use and the following covariance property of the Wigner transform:

Let , , and Then where we have used the commutation relation