Regularity Questions for Free Convolution

Abstract

Free additive convolution is a binary operation on the set M of all probability measures on the real line R. This operation was first defined in [7] for measures with finite moments of all orders (in particular for compactly supported measures). Maassen [5] extended this operation to measures with finite variance, and the extension to arbitrary measures was done in [1]. The free convolution of µ,v ∈ M is denoted µ ⊞ v. Unlike classical convolution, free convolution is a highly nonlinear operation, and therefore it is not obvious that various regularity properties of µ (like absolute continuity, differentiability, etc.) should be passed on to µ ⊞ v. In some respects however free convolution has a stronger regularizing effect than classical convolution. It is our purpose in this paper to examine a few instances in which regularity properties of µ ⊞ v can be inferred. Some of the earlier results in this direction were only proved for measures with compact support. We will extend these results to general probability measures, giving as much of the technical detail as necessary. Among the new results, we show that there may be a loss of smoothness under free convolution. We also give a complete description of the atoms of a free convolution of probability measures.KeywordsCompact SupportSelfadjoint OperatorAbsolute ContinuityFree ConvolutionUnbounded SupportThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.