# Kernel theorems for modulation spaces

## TL;DR

This paper characterizes the continuity of linear operators on modulation spaces using kernel membership in (mixed) modulation spaces, highlighting the advantages of the modulation space framework over distribution theory.

## Contribution

It provides a complete characterization of operator continuity on modulation spaces via kernel membership, extending Feichtinger's kernel theorem and emphasizing the strengths of modulation space formalism.

## Key findings

- Complete characterization of operator continuity on $M^p$ spaces.
- Recovers Feichtinger's kernel theorem as a special case.
- Shows equivalence of operator norm and kernel modulation space norm.

## Abstract

We deal with kernel theorems for modulation spaces. We completely characterize the continuity of a linear operator on the modulation spaces $M^p$ for every $1\leq p\leq\infty$, by the membership of its kernel to (mixed) modulation spaces. Whereas Feichtinger's kernel theorem (which we recapture as a special case) is the modulation space counterpart of Schwartz' kernel theorem for temperate distributions, our results do not have a couterpart in distribution theory. This reveals the superiority, in some respects, of the modulation space formalism upon distribution theory, as already emphasized in Feichtinger's manifesto for a post-modern harmonic analysis, tailored to the needs of mathematical signal processing. The proof uses in an essential way a discretization of the problem by means of Gabor frames. We also show the equivalence of the operator norm and the modulation space norm of the corresponding kernel. For operators acting on $M^{p,q}$ a similar characterization is not expected, but sufficient conditions for boundedness can be sated in the same spirit.