Superconvergence and regularity of densities in free probability

Abstract

The phenomenon of superconvergence, first observed in the central limit theorem of free probability, was subsequently extended to arbitrary limit laws for free additive convolution. We show that the same phenomenon occurs for the multiplicative versions of free convolution on the positive line and on the unit circle. We also show that a certain Hölder regularity, first demonstrated by Biane for the density of a free additive convolution with a semicircular law, extends to free (additive and multiplicative) convolutions with arbitrary freely infinitely divisible distributions.