A Homomorphism Theorem for Bilinear Multipliers

TL;DR

This paper establishes an abstract homomorphism theorem for bilinear multipliers on locally compact Abelian groups, extending classical results and providing new boundedness conditions in various functional spaces.

Contribution

It introduces a general homomorphism theorem for bilinear multipliers on LCA groups and extends known Euclidean results to broader group settings.

Findings

Proves an abstract homomorphism theorem for bilinear multipliers.

Extends K. de Leeuw's theorem to bilinear multipliers on LCA groups.

Provides necessary conditions for boundedness of bilinear multipliers in quasi Banach spaces.

Abstract

In this paper we prove an abstract homomorphism theorem for bilinear multipliers in the setting of locally compact Abelian (LCA) groups. We also provide some applications. In particular, we obtain a bilinear abstract version of K. de Leeuw's theorem for bilinear multipliers of strong and weak type. We also obtain necessary conditions on bilinear multipliers on non-compact LCA groups, yielding boundedness for the corresponding operators on products of rearrangement invariant spaces. Our investigations extend some existing results in Euclidean spaces to the framework of general LCA groups, and yield new boundedness results for bilinear multipliers in quasi Banach spaces.