# Kernel Theorems in Coorbit Theory

## TL;DR

This paper establishes general kernel theorems for operators between coorbit spaces, unifying and extending previous results for various function spaces like modulation and Besov spaces.

## Contribution

It provides a unified framework for kernel theorems in coorbit theory, encompassing known results and introducing new theorems for Besov spaces.

## Key findings

- Unified kernel theorems for coorbit spaces
- Recovery of Feichtinger's kernel theorem as a special case
- New kernel theorem for Besov space operators

## Abstract

We prove general kernel theorems for operators acting between coorbit spaces. These are Banach spaces associated to an integrable representation of a locally compact group and contain most of the usual function spaces (Besov spaces, modulation spaces, etc.). A kernel theorem describes the form of every bounded operator between a coorbit space of test functions and distributions by means of a kernel in a coorbit space associated to the tensor product representation. As special cases we recover Feichtinger's kernel theorem for modulation spaces and the recent generalizations by Cordero and Nicola. We also obtain a kernel theorem for operators between the Besov spaces $\dot{B}^0_{1,1}$ and $\dot{B}^{0}_{\infty, \infty }$.