Regularity properties of free multiplicative convolution on the positive line

Given two nondegenerate Borel probability measures $\mu$ and $\nu$ on ${\mathbb{R}}_{+}=[0,\mathrm{\infty })$, we prove that their free multiplicative convolution $\mu ⊠\nu$ 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 $(0,\mathrm{\infty })$ having power law behavior with exponents in $(-1,1)$, we prove that $\mu ⊠\nu$ is another Jacobi measure whose density has square root decay at the edges of its support.