A note on regularity for free convolutions

Abstract

Let μ ⊞ ν and μ ⊠ ν denote the free additive convolution and the free multiplicative convolution, respectively, of the Borel probability measures μ and ν. We analyze the boundary behavior of the functions G μ ⊞ ν ( z ) = ∫ 1 z − t d ( μ ⊞ ν ) ( t ) and ψ μ ⊠ ν ( z ) = ∫ z t 1 − z t d ( μ ⊠ ν ) ( t ) . We prove that, under certain conditions, these functions extend continuously to the boundary of their natural domains as functions with values in the extended complex plane C ∪ { ∞ } . As a consequence, we obtain that μ ⊞ ν (respectively μ ⊠ ν ) can never be purely singular, unless μ or ν is concentrated in one point.