Constructions of regular algebras $L_p^w(G)$

TL;DR

This paper extends the criterion for regularity of weighted algebras from the case p=1 to p>1 on locally compact abelian groups, constructing such algebras on any sigma-compact group.

Contribution

It generalizes the regularity criterion to translation invariant weighted algebras $L_p^w(G)$ for p>1 and constructs these algebras on all sigma-compact abelian groups.

Findings

Extended regularity criterion to $L_p^w(G)$ for p>1

Constructed regular algebras on any sigma-compact abelian group

Confirmed sigma-compactness as necessary for such algebras

Abstract

Criterion of (Shilov) regularity for weighted algebras $L_1^w(G)$ on a locally compact abelian group $G$ is known by works of Beurling (1949) and Domar (1956). In the present paper this criterion is extended to translation invariant weighted algebras $L_p^w(G)$ with $p>1$. Regular algebras $L_p^w(G)$ are constructed on any sigma-compact abelian group $G$. It was proved earlier by the author that sigma-compactness is necessary (in the abelian case) for the existence of weighted algebras $L_p^w(G)$ with $p>1$.