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

TL;DR

This paper investigates the support and regularity properties of measures formed by free and Boolean convolutions, providing explicit formulas, criteria for atoms, and insights into $oxplus$-infinite divisibility.

Contribution

It offers new explicit formulas and criteria for measures of the form $(oxplus p)^{isq}(nd$ analyzes their support structure and infinite divisibility using subordination functions.

Findings

Supports of measures have a decreasing number of components with increasing p.

Explicit density formulas for the absolutely continuous parts are derived.

Criteria for atoms in the measures are established.

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.