Structural transition between $L^{p}(G)$ and $L^{p}(G/H)$

Abstract

Let $H$ be a compact subgroup of a locally compact group $G$. We consider the homogeneous space $G/H$ equipped with a strongly quasi-invariant Radon measure $\mu$. For $1\leq p\leq +\infty$, we introduce a norm decreasing linear map from $L^{p}(G)$ onto $L^{p}(G/H,\mu)$ and show that $L^{p}(G/H,\mu)$ may be identified with a quotient space of $L^{p}(G)$. Also, we prove that $L^{p}(G/H,\mu)$ is isometrically isomorphic to a closed subspace of $L^{p}(G)$. These help us study the structure of the classical Banach spaces constructed on a homogeneous space via those created on topological groups.