Modulation space estimates for multilinear pseudodifferential operators

Abstract

Abstract. We prove that for symbols in the modulation spaces M p,q , p ≥ q, the associated multilinear pseudodifferential operators are bounded on products of appropriate modulation spaces. In particular, the symbols we study here are defined without any reference to smoothness, but rather in terms of their time-frequency behavior.

Bilinear Localization Operators on Modulation Spaces

“…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.…”

“…Sjöstrand's results were thereafter extended in [22,26,39,[43][44][45]. Moreover, we refer to [10][11][12] for the multilinear Kohn-Nirenberg ΨDOs.…”

“…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.…”

Pseudodifferential operators on locally compact abelian groups and Sjöstrand's symbol class

“…Analogous to [19] we define the modulation space M -norm, we obtain a class of symbols that leads to compact operators, see [3].…”

Sharpness of some properties of weighted modulation spaces

“…Nowadays, the theory has played more and more notable roles. Among numerous references, one can see [9] for the historical perspectives and background on the motivations which led to the invention of the modulation spaces, see [4,12,21,22,23,24] for understanding many characterizations and fundamental properties of the modulation space, see [1,2,3,10] for the study of relevant operators on modulation space, and see [5,26,27,28] for the study of nonlinear evolution equations related to the modulation space.…”