An inequality for the distance between densities of free convolutions

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This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures μᵢ and νᵢ, i=1,2, are close to each other in terms of the Lévy metric and if the free convolution μ₁μ₂ is sufficiently smooth, then ν₁ν₂ is absolutely continuous, and the densities of measures ν₁ν₂ and μ₁μ₂ are close to each other. In particular, convergence in distribution μ₁⁽ⁿ⁾→ μ₁, μ₂⁽ⁿ⁾→μ₂ implies that the density of μ₁⁽ⁿ⁾μ₂⁽ⁿ⁾ is defined for all sufficiently large n and converges to the density of μ₁μ₂. Some applications are provided, including: (i) a new proof of the local version of the free central limit theorem, and (ii) new local limit theorems for sums of free projections, for sums of -stable random variables and for eigenvalues of a sum of two N-by-N random matrices.