##### Department of Mathematics,

University of California San Diego

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### Math 243, functional analysis seminar

## Prof. Runlian Xia

#### University of Glasgow

## Cotlar identities for groups acting on tree-like structures

##### Abstract:

The Hilbert transform $H$ is a basic example of Fourier multipliers. Its behaviour on Fourier series is the following:

$$

\sum_{n\in \mathbb{Z}}a_n e^{inx} \longmapsto \sum_{n\in \mathbb{Z}}m(n)a_n e^{inx},

$$

with $m(n)=-i\,{\rm sgn} (n)$.

Riesz proved that $H$ is a bounded operator on $L_p(\mathbb{T})$ for all $1<p<\infty$.

We study Hilbert transform type Fourier multipliers on group algebras and their boundedness on corresponding non-commutative $L_p$ spaces.

The pioneering work in this direction is due to Mei and Ricard who proved $L_p$-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity. In this talk, we introduce a generalised Cotlar identity and a new geometric form of Hilbert transform for groups acting on tree-like structures. This class of groups includes amalgamated free products, HNN extensions, left orderable groups and many others.

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\noindent{\small

Joint work with Adri\'an Gonz\'alez and Javier Parcet.

Host: David Jekel and Priyanga Ganesan

### November 29, 2022

### 11:00 AM

Zoom (email djekel@ucsd.edu for Zoom info)

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