On Hirschman and log-Sobolev inequalities in μ-deformed Segal–Bargmann analysis

We consider a μ-deformation of the Segal–Bargmann transform, which is a unitary map from a μ-deformed ground state representation onto a μ-deformed Segal–Bargmann space. We study the μ-deformed Segal–Bargmann transform as an operator between Lp spaces and then we obtain sufficient conditions on the Lebesgue indices for this operator to be bounded. A family of Hirschman inequalities involving the Shannon entropies of a function and of its μ-deformed Segal–Bargmann transform are proved. We also prove a parametrized family of log-Sobolev inequalities, in which a new quantity that we call 'dilation energy' appears. This quantity generalizes the 'energy term' that has appeared in a previous work.