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arXiv · 0707.3227

Direct and reverse log-Sobolev inequalities in $μ$-deformed Segal-Bargmann analysis

Carlos Ernesto Angulo Aguila·Stephen Bruce Sontz
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Abstract

Both direct and reverse log-Sobolev inequalities, relating the Shannon entropy with a $μ$-deformed energy, are shown to hold in a family of $μ$-deformed Segal-Bargmann spaces. This shows that the $μ$-deformed energy of a state is finite if and only if its Shannon entropy is finite. The direct inequality is a new result, while the reverse inequality has already been shown by the authors but using different methods. Next the $μ$-deformed energy of a state is shown to be finite if and only if its Dirichlet form energy is finite. This leads to both direct and reverse log-Sobolev inequalities that relate the Shannon entropy with the Dirichlet energy. We obtain that the Dirichlet energy of a state is finite if and only if its Shannon entropy is finite. The main method used here is based on a study of the reproducing kernel function of these spaces and the associated integral kernel transform.

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Carlos Ernesto Angulo Aguila, Stephen Bruce Sontz. 2007-07-21. Direct and reverse log-Sobolev inequalities in $μ$-deformed Segal-Bargmann analysis. https://arxiv.org/abs/0707.3227

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