Full Characterization of embedding relations between alpha modulation spaces

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

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