## Duke Mathematical Journal

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

### Free Hilbert transforms

Tao Mei and Éric Ricard

#### 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<p<\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**

Received: 5 July 2016

Revised: 10 February 2017

First available in Project Euclid: 28 April 2017

**Permanent link to this document**

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

**Subjects**

Primary: 46L07: Operator spaces and completely bounded maps [See also 47L25]

Secondary: 46L54: Free probability and free operator algebras 46L52: Noncommutative function spaces

**Keywords**

Hilbert transforms free group von Neumann algebra noncommutative $L^{p}$-spaces

#### 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