Molecules in Coorbit Spaces and Boundedness of Operators

TL;DR

This paper investigates molecules in coorbit spaces and establishes conditions under which operators extend to bounded operators on these spaces, with applications to modulation and Besov spaces.

Contribution

It introduces the concept of molecules in coorbit spaces and proves boundedness of operators mapping atoms to molecules, extending their applicability.

Findings

Operators mapping atoms to molecules are bounded on coorbit spaces.

Boundedness results are recovered for modulation spaces.

Boundedness on homogeneous Besov spaces is established.

Abstract

We study the notion of molecules in coorbit spaces. The main result states that if an operator, originally defined on an appropriate space of test functions, maps atoms to molecules, then it can be extended to a bounded operator on coorbit spaces. For time-frequency molecules we recover some boundedness results on modulation spaces, for time-scale molecules we obtain the boundedness on homogenous Besov spaces.