DIRECT AND REVERSE LOG-SOBOLEV INEQUALITIES IN μ-DEFORMED SEGAL–BARGMANN ANALYSIS

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.