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

TL;DR

This paper establishes log-Sobolev inequalities for Dunkl operators and related measures, deriving new functional inequalities and exploring their implications for exponential integrability and Boltzmann-Gibbs measures.

Contribution

It introduces novel log-Sobolev inequalities for Dunkl operators and probability measures of Boltzmann type, extending classical results and analyzing their applications.

Findings

Proved an equivalent of the classical log-Sobolev inequality for Dunkl measure μ_k

Derived Poincaré inequalities from log-Sobolev inequalities

Applied results to exponential integrability and singular Boltzmann-Gibbs measures

Abstract

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 $e^{-|x|^p} d\mu_k$. 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.