Free Hilbert Transforms

TL;DR

This paper investigates free Hilbert transforms as Fourier multipliers on free groups, establishing their boundedness on non-commutative L^p spaces and exploring implications for free group decompositions and von Neumann algebra structures.

Contribution

It proves the complete boundedness of free Hilbert transforms on non-commutative L^p spaces and applies this to various problems in free group von Neumann algebras.

Findings

Complete boundedness of free Hilbert transforms on L^p spaces.

Resolution of Ozawa's compactness problem.

Length independent estimates for free Rosenthal inequalities.

Abstract

We study analogues of classical Hilbert transforms as fourier multipliers on free groups. We prove their complete boundedness on non commutative $L^p$ spaces associated with the free group von Neumann algebras for all $1<p<\infty$. This implies that the decomposition of the free group $\F_\infty$ into reduced words starting with distinct free generators is completely unconditional in $L^p$. We study the case of Voiculescu's amalgamated free products of von Neumann algebras as well. As by-products, we obtain a positive answer to a compactness-problem posed by Ozawa, a length independent estimate for Junge-Parcet-Xu's free Rosenthal inequality, a Littlewood-Paley-Stein type inequality for geodesic paths of free groups, and a length reduction formula for $L^p$-norms of free group von Neumann algebras.