Bilinear Localization Operators on Modulation Spaces

Abstract

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.

The Grossmann–Royer Transform, Gelfand–Shilov Spaces, and Continuity Properties of Localization Operators on Modulation Spaces

“…In many situations such results overlap. For example, Proposition 10 in [72] coincides with certain sufficient conditions from [15, Theorem 1.1] when restricted to R(p) = 0, t 0 = −t 1 , and t 2 = |t 0 |. For our purposes it is convenient to rewrite [15, Theorem 1.1] in terms of the Grossmann-Royer transform.…”

“…Then RA ϕ,φ a is the multilinear operator given in [17,Definition 2.2]. The approach to multilinear localization operators related to Weyl pseudodifferential operators is given in [72,73]. Both Weyl and Kohn-Nirenberg correspondences are particular cases of the so-called τ−pseudodifferential operators, τ ∈ [0, 1] (τ = 1/2 gives Weyl operators, and τ = 0 we reveals Kohn-Nirenberg operators).…”

Continuity Properties of Multilinear Localization Operators on Modulation Spaces

“…Remark 3.2. When n = 2 in Definition 3.1 we obtain the bilinear localization operators studied in [33]. (There is a typo in [33, Definition 1]; the integration in (9) should be taken over R 4d .)…”

“…The formal expressions given below are justified due to the absolute convergence of the involved integrals and the standard interpretation of oscillatory integrals in distributional setting. We refer to [33,Section 5] for this and for a detailed calculations. The calculations from the proof of [33, Theorem 4] yield the following kernel representation of (3.3):…”

“…In this paper we first recall the basic facts on modulation spaces in Section 2. Then, in Section 3, following the definition of bilinear localization operators given in [33] we introduce multilinear localization operators, Definition 3.1. Then we define the multilinear Weyl pseudodifferential operators and give their weak formulation in terms of the multilinear Wigner transform (Lemma 3.3).…”