# Orlicz Modules over Coset Spaces of Compact Subgroups in Locally compact Groups

## TL;DR

This paper introduces a new framework for analyzing Orlicz spaces over coset spaces of compact subgroups in locally compact groups, focusing on module actions and submodules with respect to invariant measures.

## Contribution

It develops the concept of left module actions of $L^1(G/H, m)$ on Orlicz spaces and defines Banach submodules within this context.

## Key findings

- Defined left module actions of $L^1(G/H, m)$ on Orlicz spaces.
- Established the existence of Banach left submodules of these Orlicz spaces.
- Extended the theory of modules to Orlicz spaces over coset spaces.

## Abstract

Let $H$ be a compact subgroup of a locally compact group $G$ and let $m$ be the normalized $G$-invariant measure on homogeneous space $G/H$ associated with Weil's formula. Let $\varphi$ be a Young function satisfying $\Delta_2$-condition. We introduce the notion of left module action of $L^1(G/H, m)$ on the Orlicz spaces $L^\varphi(G/H, m).$ We also introduce a Banach left $L^1(G/H, m)$-submodule of $L^\varphi(G/H, m).$