On Kato-Sobolev spaces. The Wiener-L\'{e}vy theorem for Kato-Sobolev algebras $\mathcal{H}_{\mathtt{ul}}^{s}$

TL;DR

This paper studies the multiplication properties and develops an analytic functional calculus for Kato-Sobolev spaces, extending the Wiener-Levy theorem within these algebras using techniques from harmonic analysis.

Contribution

It introduces new multiplication properties and an analytic functional calculus for Kato-Sobolev algebras, adapting methods from Beurling algebras and Calderón's integral formulas.

Findings

Established multiplication properties of Kato-Sobolev spaces.

Developed an analytic functional calculus for these algebras.

Extended Wiener-Levy theorem to Kato-Sobolev algebras.

Abstract

We investigate some multiplication properties of Kato-Sobolev spaces by adapting the techniques used in the study of Beurling algebras by Coifman and Meyer. Also we develop an analytic functional calculus for Kato-Sobolev algebras based on an integral representation formula belonging A. P. Calder\'{o}n.