Supports, regularity, and $\boxplus$-infinite divisibility for measures of the form $(\mu^{\boxplus p})^{\uplus q}$

Abstract

Let $\mathcal{M}$ be the set of Borel probability measures on $\mathbb{R}$. We denote by $\mu^{\mathrm{ac}}$ the absolutely continuous part of $\mu\in\mathcal{M}$. The purpose of this paper is to investigate the supports and regularity for measures of the form $(\mu^{\boxplus p})^{\uplus q}$, $\mu\in\mathcal{M}$, where $\boxplus$ and $\uplus$ are the operations of free additive and Boolean convolution on $\mathcal{M}$, respectively, and $p\geq1$, $q>0$. We show that for any $q$ the supports of $((\mu^{\boxplus p})^{\uplus q})^{\mathrm{ac}}$ and $(\mu^{\boxplus p})^{\mathrm{ac}}$ contain the same number of components and this number is a decreasing function of $p$. Explicit formulas for the densities of $((\mu^{\boxplus p})^{\uplus q})^{\mathrm{ac}}$ and criteria for determining the atoms of $(\mu^{\boxplus p})^{\uplus q}$ are given. Based on the subordination functions of free convolution powers, we give another point of view to analyze the set of $\boxplus$-infinitely divisible measures and provide explicit expressions for their Voiculescu transforms in terms of free and Boolean convolutions.