Superconvergence and regularity of densities in free probability

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