## Duke Mathematical Journal

### Free Hilbert transforms

#### Abstract

We study Fourier multipliers of Hilbert transform type on free groups. We prove that they are completely bounded on noncommutative $L^{p}$-spaces associated with the free group von Neumann algebras for all $1\lt p\lt \infty$. This implies that the decomposition of the free group $\mathbf{F}_{\infty}$ 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’s 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.

#### Article information

Source
Duke Math. J., Volume 166, Number 11 (2017), 2153-2182.

Dates
Revised: 10 February 2017
First available in Project Euclid: 28 April 2017

https://projecteuclid.org/euclid.dmj/1493344841

Digital Object Identifier
doi:10.1215/00127094-2017-0007

Mathematical Reviews number (MathSciNet)
MR3694567

Zentralblatt MATH identifier
1385.46040

#### Citation

Mei, Tao; Ricard, Éric. Free Hilbert transforms. Duke Math. J. 166 (2017), no. 11, 2153--2182. doi:10.1215/00127094-2017-0007. https://projecteuclid.org/euclid.dmj/1493344841

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