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

TL;DR

This paper establishes new and existing log-Sobolev inequalities in $$-deformed Segal-Bargmann spaces, linking Shannon entropy with $$-deformed energy and Dirichlet form energy, using kernel analysis.

Contribution

It introduces a new direct log-Sobolev inequality and connects $$-deformed energy with Dirichlet energy in Segal-Bargmann spaces.

Findings

The $$-deformed energy is finite iff Shannon entropy is finite.

The $$-deformed energy is finite iff Dirichlet energy is finite.

Established both direct and reverse log-Sobolev inequalities relating these quantities.

Abstract

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-Bargmann spaces. This shows that the $\mu$-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 $\mu$-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.