Bilinear Localization Operators on Modulation Spaces

We introduce a class of bilinear localization operators and showhow to interpret them as bilinearWeyl 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.


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    =   1 , 2  on  2 (R  ) (see Section 2 for the definition) with Gaussian windows radial symbol  ∈  1 (R  ) . ( Such operators are named Daubechies' operators afterwards.The eigenfunctions of Daubechies' operators are Hermite functions: 2 ) (exp (− 2 )) () ,  ∈ R  ,  = 0, 1, . . ., 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 Ω, we refer to [5] where it is proved that if one of the eigenfunctions of Daubechies' operator is a Hermite function, then Ω 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   (), 1 ≤  ≤ ∞, 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].
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 M , , ; 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, ΨDO for short.Indeed, results on multilinear Kohn-Nirenberg ΨDOs from [10][11][12] 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 4.5 and 4.8] when restricted to bilinear operators.
The paper can be summarized as follows.We first introduce bilinear localization operators, bilinear Weyl ΨDOs, 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 ΨDO, Theorem 4. In Section 3 we recall the definition and basic properties of modulation spaces and introduce their convenient modified version M , , (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 ΨDOs on M , , (Theorem 14) and sufficient and necessary conditions for boundedness of bilinear localization operators on M , , (Theorems 15 and 16, resp.).
Notation.The Schwartz class is denoted by S(R  ) and the space of tempered distributions by S  (R  ).We use the brackets ⟨, ⟩ to denote the extension of the inner product ⟨, ⟩ = ∫ ()() on  2 (R  ) to any pair of dual spaces.
We denote by ⟨⋅⟩  the polynomial weights and We use the notation  ≲  to indicate that  ≤  for a suitable constant  > 0, whereas  ≍  means that  −1  ≤  ≤  for some  ≥ 1.

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].
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  ∈ S (1) (R 2 ).Then the Weyl pseudodifferential operator   with the Weyl symbol  can be defined as the oscillatory integral: This definition extends to each  ∈ S (1)  (R 2 ), so that   is a continuous mapping from S (1) denotes the Wigner transform, also known as the cross-Wigner distribution, then the following formula holds: for each  ∈ S (1)  (R 2 ); see, for example, [26,32,33].
By analogy with (13) we define the bilinear Weyl pseudodifferential operator as follows: where  ∈ S (1)  (R 4 ),  1 ,  2 ∈ S (1) (R  ), and  →  = ( 1 ,  2 ).Here I denotes the identity matrix in 2 (e.g., if  = 2, then To give the interpretation of ( 15) in the context of bilinear ΨDOs we introduce the bilinear version of ( 14) as follows.Let  1 ,  2 ,  1 , and Then the bilinear Wigner transform (  → ,  →  ) is given by and ) and  1 ,  2 ,  1 , and  2 ∈ S (1) (R  ).Then   given by ( 16) extends to a continuous map from S (1) (R  ) ⊗ S (1) (R  ) to S (1)  (R 4 ) and the following formula holds: Proof.The proof follows by the straightforward calculation: where we used (  →  ,  → ) = (  → ,  →  ) and the change of variables  =  + /2 and V =  − /2.This extends to each  ∈ S The so-called Weyl connection between the set of linear localization operators and Weyl ΨDOs 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 ΨDOs 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 ΨDOs.

Modulation Spaces
Since we essentially use the convolution estimates for polynomially weighted modulation spaces (Theorems 8 and 9), by Theorem 7(3) below it is enough to use the duality between S and S  instead of the more general duality between S (1) and S (1)  .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 R  is a nonnegative and continuous function.By    (R  ) and  ∈ [1, ∞], we denote the weighted Lebesgue space defined by the norm with the usual modification when  = ∞.When () = ⟨⟩  ,  ∈ R, we use the notation Similarly, the weighted mixed-norm space  ,  (R 2 ), ,  ∈ [1, ∞], consists of (Lebesgue) measurable functions on R 2 such that where (, ) is a weight on R 2 .
For the consistency, and according to (11), we denote by M , , (R 2 ) the set of In special cases we use the usual abbreviations:  , 0,0 =   ,  , , =    , and so on.Remark 6.Notice that the original definition given in [25] contains more general submultiplicative weights.We restrict ourselves to (, ) = ⟨⟩  ⟨⟩  , ,  ∈ R, 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,[29][30][31].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.
For the results on multiplication and convolution in modulation spaces and in weighted Lebesgue spaces we first introduce the Young functional: When R() = 0, 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 0 ≤ R() ≤ 1/2.
Then ( 1 ,  2 )  →  1 *  2 on  ∞ 0 (R  ) extends uniquely to a continuous map from 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 R  ; see Sections 3.1 and 3.2 in [24].This result extends multiplication and convolution properties obtained in [38].Moreover, the result is sharp in the following sense.Theorem 9. Let   ,   ∈ [1, ∞] and   ,   ∈ R,  = 0, 1, 2. Assume that at least one of the following statements holds true: Then ( 29) and ( 30) hold true.

Continuity Properties
We start estimates of the modulation space norm of the cross-Wigner distribution (cf.[21]).We refer to [39, Theorem 4.2] 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 1.1] when restricted to R() = 0,  0 = − 1 , and  2 = | 0 |.

Proposition 10. Let the assumptions of Theorem 8 hold. If
,  ∈ S(R  ), then the map (, )  → (, ) where  is the cross-Wigner distribution given by ( 14) extends to sesquilinear continuous map from Proof.We give a short version of the proof for the sake of completeness and refer to [21] for details.Let ,  ∈ S(R  ).Then where ,  ∈ R 2 , since, by Theorem 7(1), modulation spaces are independent of the choice of the window function (from see [26,Lemma 4.3.1],and from the proof of [26, Lemma 14.5.1 (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 ( where the last estimate follows from Theorem 8 (1).
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.(17) which extends to a continuous and that is, [13, Proposition 2.5] (with a slightly different notation).
To deal with duality when  = ∞ we observe that, by a slight modification of [10, Lemma 2.2], the following is true.Lemma 13.Let  0 (R 4 ) denote the space of bounded, measurable functions on R 4 which vanish at infinity and put equipped with the norms of M ∞, , M ,∞ , and M ∞,∞ , respectively.Then, . Likewise for M ,0 and M 0,0 .
From now on, we will use these duality relations in the cases  = ∞ and/or  = ∞ without further explanations.Theorem 14.Let  ∈  ∞,1 (R 4 ) and let   be given by (16).The operator 1 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 Φ ∈ S(R 2 ) \ 0 is a window function.By slight modifications of the proof of [26, Theorem 14.5.2]we obtain the following estimate: uniformly in  ∈ R 4 .Therefore, Hence the operator   is bounded on M , (R 2 ) and ‖  ‖  ≤ ‖‖  ∞,1 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.
On the other hand, by the choice of the weight parameters  and V it follows that  * ( ( and the Theorem is proved.
We remark that a modification of Theorem 15 can be obtained by using [26,Theorem 14.5.6]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 4.3], and we leave it for the reader.Here below ‖ ⋅ ‖  ∞ denotes the norm of a bounded operator.

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 S (1)  (R  ) in a suitable interpretation.For example, if  denotes the Dirac distribution, then the Fourier inversion formula in the sense of distributions gives ∫  2  = (), wherefrom ∬ () 2(−)   = (), when  ∈ S (1) (R  ).
We first rewrite (12) in the form of a kernel operator.