# Fourier analysis associated to a vector measure on a compact group

## TL;DR

This paper extends Fourier analysis to functions and measures on compact groups using vector measures, establishing foundational properties and inequalities akin to classical results.

## Contribution

It introduces the Fourier transform for vector measure integrable functions on compact groups and explores convolution and Young's inequalities in this context.

## Key findings

- Fourier transform defined for vector measure integrable functions
- Established analogues of Young's inequalities for these transforms
- Analyzed convolution of scalar and vector measures

## Abstract

In this paper, we introduce and study the Fourier transform of functions which are integrable with respect to a vector measure on a compact group (not necessarily abelian). We also study the Fourier transform of vector measures. We also introduce and study the convolution of functions from $L^p$-spaces associated to a vector measure. We prove some analogues of the classical Young's inequalities. Similarly, we also study convolution of a scalar measure and a vector measure.