# Regularity properties of free multiplicative convolution on the positive line

## TL;DR

This paper studies the regularity of free multiplicative convolution of probability measures on the positive real line, showing it has no singular continuous part and describing the density behavior for specific classes of measures.

## Contribution

It establishes the absence of singular continuous parts and characterizes the density properties of free multiplicative convolutions for measures with power law behavior.

## Key findings

- The free multiplicative convolution has zero singular continuous part.
- The density of the convolution is bounded by x^{-1}.
- For Jacobi measures, the convolution results in a measure with square root decay at the edges.

## Abstract

Given two nondegenerate Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}_{+}=[0,\infty)$, we prove that their free multiplicative convolution $\mu\boxtimes\nu$ has zero singular continuous part and its absolutely continuous part has a density bounded by $x^{-1}$. When $\mu$ and $\nu$ are compactly supported Jacobi measures on $(0,\infty)$ having power law behavior with exponents in $(-1,1)$, we prove that $\mu\boxtimes\nu$ is another Jacobi measure whose density has square root decay at the edges of its support.