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In this paper we study several inequalities of log-Sobolev type for Dunkloperators. After proving an equivalent of the classical inequality for theusual Dunkl measure $\mu_k$, we also study a number of inequalities forprobability measures of Boltzmann type of the form $e^{-|x|^p} d\mu_k$. Theseare obtained using the method of $U$-bounds. Poincar\'e inequalities areobtained as consequences of the log-Sobolev inequality. The connection betweenPoincar\'e and log-Sobolev inequalities is further examined, obtaining inparticular tight log-Sobolev inequalities. Finally, we study application toexponential integrability and to functional inequalities for a class ofsingular Boltzmann-Gibbs measures.

Logarithmic Sobolev inequalities for Dunkl operators with applications to functional inequalities for singular Boltzmann-Gibbs measures