# zbMATH — the first resource for mathematics

The authors study, whether regularity properties of probability measures $$\mu$$, $$\nu$$ on $$\mathbb R$$ are inherited by the free additive convolution $$\mu\boxplus\nu$$. Having established in [Indiana Univ. Math. J. 42, No. 3, 733-773 (1993; Zbl 0806.46070)] that the operation $$\boxplus$$ introduced by D. Voiculescu [Lect. Notes Math. 1132, 556-588 (1985; Zbl 0618.46048)] extends to arbitrary probability measures, they generalize some results known in the case of compactly supported $$\mu$$ and $$\nu$$. In terms of the Cauchy transform defined by $$G_\mu(z)=\int_{-\infty}^{+\infty}(z-t)^{-1} d\mu(t)$$, $$z=x+iy$$, $$y>0$$, $$G_{\mu\boxplus\nu}$$ is subordinated to $$G_\mu$$ in $$\mathbb C^+$$, i.e. there exists an analytic function $$\omega :\mathbb C^+\to\mathbb C^+$$ such that 1) $$G_{\mu\boxplus\nu}=G_\mu\circ \omega$$, 2) $$\mathop{\text{Im}}\omega(z)\geqslant\mathop{\text{Im}}z$$, $$z\in\mathbb C^+$$ and 3) $$\omega(z)=z(1+o(1))$$, $$z\to\infty$$ nontangentially to $$\mathbb R$$. Then, for any fixed $$y>0$$ and $$1<p\leqslant\infty$$, $$\| (G_{\mu\boxplus\nu})_y\| _p\leqslant\| (G_\mu)_y\| _p$$ in $$L^p(\mathbb R)$$, and similar relations hold for the real (resp. imaginary) parts of the transforms. If $$\mu$$ is absolutely continuous with density $$f\in L^p(\mathbb R)$$, so is $$\mu\boxplus\nu$$ and its density $$g$$ satisfies $$\| g\| _p\leqslant\| f\| _p$$. Some inequalities involving the Riesz energies are obtained and all the above are symmetrical in $$\mu$$ and $$\nu$$. If the distribution function $${\mathcal F}_\mu$$ of $$\mu$$ is $$\alpha$$-Hölder, $$\alpha\in(0,1]$$, (with constant $$c$$), the same is true of $${\mathcal F}_{\mu\boxplus\nu}$$ (with constant $$\leqslant c$$). But there exist compactly supported $$\mu$$, $$\nu$$ having densities of class $$C^\infty$$ and $$C^1$$, respectively, such that the density of $$\mu\boxplus\nu$$ is not of class $$C^1$$. A description of atoms is given: $$\gamma\in\mathbb R$$ is an atom of $$\mu\boxplus\nu\iff$$ there exist atoms $$\alpha,\beta$$ for $$\mu,\nu$$, respectively, with $$\gamma=\alpha+\beta$$ and $$(\mu\boxplus\nu)(\{\gamma\})=\mu(\{\alpha\})+\nu(\{\beta\})-1>0$$. The background of noncommutative free random variables is mentioned occasionally.