Harmonic Analysis of Covariant Functions of Characters of Normal Subgroups

Arash Ghaani Farashahi

Published 2021-02-17Version 1

This paper presents a systematic study for abstract harmonic analysis on classical Banach spaces of covariant functions of characters of closed normal subgroups. Let $G$ be a locally compact group and $N$ be a closed normal subgroup of $G$. Suppose that $\xi:N\to\mathbb{T}$ is a continuous character and $L_\xi^1(G,N)$ is the $L^1$-space of all covariant functions of $\xi$ on $G$. We showed that $L^1_\xi(G,N)$ is isometrically isomorphic to a quotient space of $L^1(G)$. It is also proved that the dual space $L^1_\xi(G,N)^*$ is isometrically isomorphic to $L^\infty_\xi(G,N)$. We then investigate some analytical aspects of the presented theory for the case of semi-direct product groups. The paper is concluded by realization of the theory in the case of some interesting examples.