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

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 ${\mu }_k$, we also study a number of inequalities for probability measures of Boltzmann type of the form $e^{-\mathrm{\mid }x{\mathrm{\mid }}^p}d{\mu }_k$. 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. Finally, we study application to exponential integrability and to functional inequalities for a class of singular Boltzmann-Gibbs measures.