Boundedness and compactness kernel theorem for $\alpha$-modulation spaces

This paper is devoted to establishing the kernel theorems for $\alpha$-modulation spaces in terms of boundedness and compactness. We characterize the boundedness of a linear operator $A$ from an $\alpha$-modulation space $M_{\alpha }^p({\mathbb{R}}^d)$ into another $\alpha$-modulation space $M_{\alpha }^q({\mathbb{R}}^d)$, by the membership of its distributional kernel in mixed $\alpha$-modulation spaces. We also characterize the compactness of $A$ by means of the kernel in a certain closed subspace of mixed $\alpha$-modulation spaces. The proofs are based on the viewpoint that the action of the linear operator on certain function space can be reduced to the action on the suitable atoms of this function space.