Noncommutative Riesz transforms -- Dimension free bounds and Fourier multipliers

TL;DR

This paper establishes dimension-free bounds for noncommutative Riesz transforms and Fourier multipliers on group von Neumann algebras, extending classical results to noncommutative and more general settings with new techniques.

Contribution

It introduces new dimension-free estimates for noncommutative Riesz transforms, including fractional Laplacians and free group word lengths, and links Fourier multipliers to Riesz transforms in noncommutative contexts.

Findings

Dimension free bounds for noncommutative Riesz transforms on group von Neumann algebras.

H"ormander-Mihlin multipliers as Littlewood-Paley averages of Riesz transforms.

New Sobolev/Besov smoothness conditions with dimension free constants.

Abstract

We obtain dimension free estimates for noncommutative Riesz transforms associated to conditionally negative length functions on group von Neumann algebras. This includes Poisson semigroups, beyond Bakry's results in the commutative setting. Our proof is inspired by Pisier's method and a new Khintchine inequality for crossed products. New estimates include Riesz transforms associated to fractional laplacians in $\mathbb{R}^n$ (where Meyer's conjecture fails) or to the word length of free groups. Lust-Piquard's work for discrete laplacians on LCA groups is also generalized in several ways. In the context of Fourier multipliers, we will prove that H\"ormander-Mihlin multipliers are Littlewood-Paley averages of our Riesz transforms. This is highly surprising in the Euclidean and (most notably) noncommutative settings. As application we provide new Sobolev/Besov type smoothness conditions. The Sobolev type condition we give refines the classical one and yields dimension free constants. Our results hold for arbitrary unimodular groups.