This paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory. More precisely, we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non-microstates free Fisher information. It is shown that their analytic distributions have H\"older continuous cumulative distribution functions with an explicit H\"older exponent that depends only on the degree of the considered polynomial. This, in particular, guarantees that such polynomial evaluations have finite logarithmic energy and thus finite (non-microstates) free entropy. We further provide a general criterion that gives for weak approximations of measures having H\"older continuous cumulative distribution functions explicit rates of convergence in terms of the Kolmogorov distance. Finally, we apply these results to study the asymptotic eigenvalue distributions of polynomials in GUEs or matrices with more general 查看全文>>