Weak Poincaré inequalities for convolution probabilities measures

In this paper, weak Poincaré inequalities are obtained for convolution probabilities with explicit rate functions by constructing suitable Lyapunov functions. Here, one of these Lyapunov functions is introduced for the first time and can be used to improve parts of results on the estimate of rate functions for super Poincaré inequalities 13 as well. In addition, these weak Poincaré inequalities are applied to some concrete examples, which can refine some of results even for the case without convolution.

On the Hausdorff Dimension of Bernoulli Convolutions

We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution $\nu _{\beta }$ to arbitrary given accuracy whenever $\beta $ is algebraic. In particular, if the Garsia entropy $H(\beta )$ is not equal to $\log (\beta )$ then we have a finite time algorithm to determine whether or not $\operatorname{dim_H} (\nu _{\beta })=1$.