Inversion problem in algebras of integrable functions with summable Fourier transforms

TL;DR

The paper addresses the norm-controlled inversion problem for Fourier-type Banach algebras $A_p(G)$ and their unitizations $L^{p}(G)_1$ when $G$ is a non-discrete locally compact abelian group. It develops explicit quantitative inversion bounds in terms of the spectral gap $\delta = \inf_{\gamma} |\widehat{x}(\gamma)|$, using a symmetrization technique with $x*x^{*}$ and, for $1\le p\le 2$, Hausdorff–Young, while for $p>2$ it employs a power-raising strategy to reduce to the $p\le 2$ case. The results extend to unitized algebras with parallel bounds and provide concrete formulas involving $n$, $\Delta_n$, and $c_n(\delta)$, thereby delivering complete positive answers to norm-controlled inversion in these settings. The work also discusses Bezout-type equations, highlights limitations for noncompact groups such as $\mathbb{R}$, and outlines open problems for further exploration in noncommutative or broader group contexts.

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.

Theorems & Definitions (19)

• Theorem 1: Theorem 3 in LarsenLiuWang1964
• Theorem 2: Theorem 4 in LarsenLiuWang1964
• Corollary 1
• Theorem 3
• Theorem 4: Hausdorff--Young inequalities
• Theorem 5
• proof
• Theorem 6
• proof
• Theorem 7