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

TL;DR

This paper establishes kernel theorems for $oldsymbol{ extalpha}$-modulation spaces, characterizing boundedness and compactness of operators via their kernels in mixed $oldsymbol{ extalpha}$-modulation spaces, using atomic decompositions.

Contribution

It provides new characterizations of boundedness and compactness of linear operators on $ extalpha$-modulation spaces through kernel membership in mixed spaces.

Findings

Boundedness characterized by kernel membership in mixed $ extalpha$-modulation spaces.

Compactness characterized by kernel in a closed subspace of mixed $ extalpha$-modulation spaces.

Proofs based on atomic decomposition and reduction to atom actions.

Abstract

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 $M^p_{\alpha}(\mathbb{R}^d)$ into another $\alpha$-modulation space $M^q_{\alpha}(\mathbb{R}^d)$, 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.