Full characterization of the embedding relations between α-modulation spaces

Abstract

In this paper, we consider the embedding relations between any two $\alpha$-modulation spaces. Based on an observation that the $\alpha$-modulationspace with smaller $\alpha$ can be regarded as a corresponding decomposition spaceassociated with $\alpha$-covering for larger $\alpha$, we give a complete characterizationof the Fourier multipliers between $\alpha$-modulation spaces with different$\alpha$. Then we establish a full version of optimal embedding relationsbetween $\alpha$-modulation spaces.