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

Let $H$ be a compact subgroup of a locally compact group $G$ and let $m$ be the normalized $G$-invariant measure on homogeneous space $G\mathrm{/}H$ associated with Weil’s formula. Let $\phi$ be a Young function satisfying ${\mathrm{\Delta }}_2$-condition. We introduce the notion of left module action of $L^1(G\mathrm{/}H,m)$ on the Orlicz spaces $L^{\phi }(G\mathrm{/}H,m)\mathrm{.}$ We also introduce a Banach left $L^1(G\mathrm{/}H,m)$-submodule of $L^{\phi }(G\mathrm{/}H,m)\mathrm{.}$