Covariant Functions of Characters of Compact Subgroups

TL;DR

This paper systematically analyzes covariant functions of characters of compact subgroups within classical Banach spaces, establishing isometric isomorphisms and duality relations in the context of harmonic analysis on locally compact groups.

Contribution

It introduces a framework for understanding covariant functions of characters as quotient spaces and explores their duality properties, extending harmonic analysis theory.

Findings

$L^p_\xi(G,H)$ is isometrically isomorphic to a quotient of $L^p(G)$

$L^q_\xi(G,H)$ is isometrically isomorphic to the dual of $L^p_\xi(G,H)$

Results specialized for the case when $G$ is compact

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.