Weighted $L^p-$spaces on nilpotent, locally compact groups

TL;DR

This paper proves the $L^p_{ ext{omega}}$-conjecture for nilpotent, locally compact groups by revising spectral theory for commutative Banach algebras and avoiding structural theorems, extending results from abelian groups.

Contribution

It introduces an alternative spectral theory approach to prove the $L^p_{ ext{omega}}$-conjecture for nilpotent, locally compact groups, without relying on structural theorems.

Findings

Proved the $L^p_{ ext{omega}}$-conjecture for abelian groups.

Extended the proof to nilpotent, locally compact groups.

Provided a new spectral theory framework for these groups.

Abstract

Our paper begins with a revision of spectral theory for commutative Banach algebras, which enables us to prove the $L^p_{\omega}-$conjecture for locally compact abelian groups. We follow an alternative approach to the one known in the literature. In particular, we do not resort to any structural theorems for locally compact groups. Subsequently, we discuss nilpotent, locally compact groups. The climax of the paper is the proof of the $L^p_{\omega}-$conjecture for these groups.