Direct and reverse log-Sobolev inequalities in μ-deformed segal-bargmai analysis

Both direct and reverse log-Sobolev inequalities, relating the Shannon entropy with a mu-deformed energy, are shown to hold in a family of mu-deformed Segal-Bargmarm spaces. This shows that the p-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 P-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.