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 , we also study a number of inequalities for probability measures of Boltzmann type of the form . These are obtained using the method of U-bounds. Poincaré inequalities are obtained as consequences of the log-Sobolev inequality. The connection between Poincaré 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.
Funding Statement
Financial support from EPSRC is also gratefully acknowledged.
Acknowledgments
The author wishes to thank Boguslaw Zegarlinski for introducing him to the problem and for useful advice, and the anonymous referee for carefully reading the paper and for their constructive suggestions.
Citation
Andrei Velicu. "Logarithmic Sobolev inequalities for Dunkl operators with applications to functional inequalities for singular Boltzmann-Gibbs measures." Electron. J. Probab. 27 1 - 25, 2022. https://doi.org/10.1214/22-EJP810
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