Inversion problem in algebras of integrable functions with summable Fourier transforms

TL;DR

This paper investigates the norm-controlled inversion problem within specific algebras of integrable functions, demonstrating positive solutions without extra restrictions, unlike the classical $L^{1}(G)$ case.

Contribution

The paper extends the understanding of inversion in algebras of integrable functions by establishing positive results without additional constraints.

Findings

Positive solution to the inversion problem in new algebras

No extra restrictions needed for inversion

Contrasts with classical $L^{1}(G)$) case

Abstract

In this paper, we study the norm-controlled inversion problem in two classes of algebras of integrable functions. In contrast of the classical case of $L^{1}(G)$, we prove that this problem has a positive solution in our setting without any additional restrictions.