Vector-valued properties of the Weyl transform

TL;DR

This paper extends the Weyl transform to vector-valued functions and measures on phase space, establishing foundational properties and inequalities, including a vector-valued Hausdorff-Young inequality, for functions associated with locally compact abelian groups.

Contribution

It introduces the Weyl transform for vector measures and functions, and develops a vector-valued Hausdorff-Young inequality, advancing harmonic analysis in this context.

Findings

Weyl transform defined for vector measures and functions

Established convolution properties for vector-valued functions

Proved a vector-valued Hausdorff-Young inequality

Abstract

In this paper, we introduce and study the Weyl transform of functions which are integrable with respect to a vector measure on a phase space associated to a locally compact abelian group. We also study the Weyl transform of vector measures. Later, we also introduce and study the convolution of functions from $L^p$-spaces associated to a vector measure. We also study the Weyl transform of vector-valued functions and prove a vector-valued analogue of the Hausdorff-Young inequality.