# Optimal extension of the Fourier transform and convolution operator on compact groups

## TL;DR

This paper characterizes the optimal domain and extension of Fourier and convolution operators on Orlicz spaces over compact groups, broadening the understanding of harmonic analysis in non-abelian settings.

## Contribution

It establishes the precise conditions for extending Fourier and convolution operators on Orlicz spaces over compact groups, including non-abelian cases.

## Key findings

- Identifies the optimal domain for Fourier transform on Orlicz spaces.
- Determines the extended convolution operator on these spaces.
- Provides a framework for harmonic analysis on non-abelian compact groups.

## Abstract

Let $G$ be a compact group (not necessarily abelian) and let $\Phi$ be a Young function satisfying the $\Delta_2$-condition. We determine the optimal domain and the associated extended operator for both Fourier transform and the convolution operator defined on the Orlicz spaces $L^\Phi(G).$