Smooth Fourier multipliers in group algebras via Sobolev dimension

TL;DR

This paper develops new criteria for Fourier multipliers on group algebras using Sobolev dimension, broadening the understanding of smoothness conditions and maximal operator control in noncommutative harmonic analysis.

Contribution

It introduces Sobolev dimension as a flexible measure for multiplier smoothness and extends maximal operator techniques to noncommutative $L_p$ spaces for spectral and non-spectral multipliers.

Findings

Established new Hörmander-Mikhlin criteria for multipliers.

Replaced cocycle dimension with Sobolev dimension for better flexibility.

Derived new $L_p$ estimates for smooth Fourier multipliers.

Abstract

We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new H\"ormander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative $L_p$ spaces. This general principle ---exploited in Euclidean harmonic analysis during the last 40 years--- is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimension by the Sobolev dimension. This is based on a noncommutative form of the Sobolev embedding theory for Markov semigroups initiated by Varopoulos, and yields more flexibility to measure the smoothness of the symbol. Third, we introduce a dual notion of polynomial growth to further exploit our maximal principle for non-spectral Fourier multipliers. The combination of these ingredients yields new $L_p$ estimates for smooth Fourier multipliers in group algebras.