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

Pita-Ruiz, Claudio; Sontz, Stephen Bruce

doi:10.1088/0305-4470/39/27/006

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

Journal

Journal of Physics A: Mathematical and General

ISSN

0305-4470

1361-6447

Date Issued

2006

Author(s)

Pita-Ruiz, Claudio

Facultad de Ingeniería - CampCM

Sontz, Stephen Bruce

Type

text::journal::journal article

DOI

10.1088/0305-4470/39/27/006

URL

https://scripta.up.edu.mx/handle/20.500.12552/3723

Abstract

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.