Regularity for free multiplicative convolution on the unit circle

Serban T. Belinschi, Hari Bercovici and Ching-Wei Ho

Vol. 322 (2023), No. 2, 243–250

DOI: 10.2140/pjm.2023.322.243

Abstract

Suppose that ${\mu }_1$ and ${\mu }_2$ are Borel probability measures on the unit circle, both different from unit point masses, and let $\mu$ denote their free multiplicative convolution. We show that $\mu$ has no continuous singular part (relative to arclength measure) and that its density can only be locally unbounded at a finite number of points, entirely determined by the point masses of ${\mu }_1$ and ${\mu }_2$. Analogous results were proved earlier for the free additive convolution on $ℝ$ and for the free multiplicative convolution of Borel probability measures on the positive half-line.

Keywords

free probability, free multiplicative convolution, regularity, analytic subordination

Mathematical Subject Classification

Primary: 46L35

Secondary: 30D05

Milestones

Received: 26 May 2022

Revised: 27 January 2023

Accepted: 5 February 2023

Published: 23 May 2023