Covariant Functions of Characters of Compact Subgroups

Abstract

This paper presents a systematic study for abstract harmonic analysis on classical Banach spaces of covariant functions of characters of compact subgroups. Let $G$ be a locally compact group and $H$ be a compact subgroup of $G$. Suppose that $\xi:H\to\mathbb{T}$ is a continuous character, $1\le p<\infty$ and $L_\xi^p(G,H)$ is the set of all covariant functions of $\xi$ in $L^p(G)$. It is shown that $L^p_\xi(G,H)$ is isometrically isomorphic to a quotient space of $L^p(G)$. It is also proven that $L^q_\xi(G,H)$ is isometrically isomorphic to the dual space $L^p_\xi(G,H)^*$, where $q$ is the conjugate exponent of $p$. The paper is concluded by some results for the case that $G$ is compact.