Regularity for free multiplicative convolution on the unit circle

TL;DR

This paper proves that the free multiplicative convolution of two nondegenerate measures on the unit circle is absolutely continuous, extending known results from additive convolutions and positive-line multiplicative convolutions.

Contribution

It establishes the regularity (absence of singular parts) for free multiplicative convolution on the unit circle, a previously unresolved case.

Findings

No continuous singular part in the convolution measure

Extension of regularity results to the unit circle case

Analogous to known results on the line and positive half-line

Abstract

It is shown that the free multiplicative convolution of two nondegenerate probability measures on the unit circle has no continuous singular part relative to arclength measure. Analogous results have long been known for free additive convolutions on the line and free multiplicative convolution on the positive half-line.