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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.

Regularity Properties of Free Multiplicative Convolution on the Positive Line