Regularity properties of free multiplicative convolution on the positive line

Abstract

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