Full Characterization of embedding relations between alpha modulation spaces

TL;DR

This paper provides a comprehensive analysis of the embedding relations between alpha-modulation spaces, including Fourier multipliers and operator characterizations, advancing the understanding of their structure and interactions.

Contribution

It offers a complete characterization of Fourier multipliers and optimal embedding relations between alpha-modulation spaces for different alpha values.

Findings

Complete Fourier multiplier characterization between alpha-modulation spaces.

Established optimal embedding relations for different alpha values.

Identified translation-commuting operators as convolution operators.

Abstract

In this paper, we consider the embedding relations between any two $\alpha$% -modulation spaces. Based on an observation that the $\alpha$-modulation space with smaller $\alpha$ can be regarded as a corresponding $\alpha$% -modulation space with larger $\alpha$, we give a complete characterization of the Fourier multipliers between $\alpha$-modulation spaces with different $\alpha$. Then we establish a full version of optimal embedding relations between $\alpha$-modulation spaces. As an application, we determine that the bounded operators commuting with translations between $\alpha$-modulation spaces are of convolution type.