Logarithmic Sobolev inequalities for Dunkl operators

Andrei Velicu

Published 2019-10-15Version 1

In this paper we study several inequalities of log-Sobolev type for Dunkl operators. After proving an equivalent of the classical inequality for the usual Dunkl measure, we also study a number of inequalities for weighted probability Dunkl measures, with weights of the form $e^{-|x|^p}$. These are obtained using the method of $U$-bounds. Poincar\'e inequalities are obtained as consequences of the log-Sobolev inequality. The connection between Poincar\'e and log-Sobolev inequalities is further examined, obtaining in particular tight log-Sobolev inequalities.