The inclusion relation between Sobolev and modulation spaces

TL;DR

This paper explicitly characterizes the inclusion relations between Sobolev and modulation spaces and explores the mapping properties of certain Fourier multipliers between these spaces.

Contribution

The paper provides explicit criteria for the inclusion relations between Sobolev and modulation spaces and analyzes the boundedness of specific Fourier multipliers.

Findings

Explicit inclusion relations between Sobolev and modulation spaces

Mapping properties of the Fourier multiplier $e^{i|D|^eta}$

Conditions for boundedness between these function spaces

Abstract

The inclusion relations between the $L^p$-Sobolev spaces and the modulation spaces is determined explicitly. As an application, mapping properties of unimodular Fourier multiplier $e^{i|D|^\alpha}$ between $L^p$-Sobolev spaces and modulation spaces are discussed.