Free Hilbert Transforms

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.

Added two remarks (4.12, 4.13). Corrected a few misprints

Cited by in corpus (8)

• Asymptotic freeness in tracial ultraproducts
• Sequential commutation in tracial von Neumann algebras
• Multilinear transference of Fourier and Schur multipliers acting on non-commutative $L_p$-spaces
• Free Fourier Multipliers associated with the firstSegment
• Fourier multipliers in ${\mathrm{SL}}_n(\mathbf{R})$
• Trigonometric chaos and ${\mathrm{X}}_p$ inequalities I -- Balanced Fourier truncations over discrete groups
• The local geometry of idempotent Schur multipliers
• A Hörmander-Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras