$L^T_p-$ functions on locally compact groups

TL;DR

This paper introduces and studies the structure of $L_p^T$-functions on locally compact groups, characterizing positive functions for amenable groups and applying Plancherel's theorem in abelian cases.

Contribution

It defines $L_p^T$-functions, analyzes their structure, characterizes positive functions for amenable groups, and applies Plancherel's theorem to abelian groups.

Findings

Complete characterization of $L^{T}_p(G)^+$ for amenable groups

Structural insights into $L_p^T$-functions

Applications of Plancherel Theorem on $L^{T}_2(G)$ for abelian groups

Abstract

Let $G$ be a locally compact group and $1\leq p<\infty$. Based on some important earlier works, in this paper the concept of $L_p^T-$function is introduced. Then the structure of the space $L^{T}_p(G)$, which is consisting of all $L_p^T-$functions, is investigated. As an important result, $L^{T}_p(G)^+$ is completely characterized, for the class of amenable locally compact groups. Furthermore, it is verified some applications of Plancherel Theorem on $L^{T}_2(G)$, in the class of abelian locally compact groups.