A reverse log-Sobolev inequality in the Segal-Bargmann space

This article is based on recent results of the author on the properties of the reproducing kernel of the Segal-Bargmann space. Those results are used here to demonstrate a family of energy-entropy inequalities in the Segal-Bargmann space. In some cases this is a log-Sobolev inequality while in other cases this is actually a reverse log-Sobolev inequality, which means that the energy term is bounded above by the entropy term, plus a norm term. This implies that in the Segal-Bargmann space the entropy is finite if and only if the energy is finite. Applications of this result to the Segal-Bargmann transform are given as well as a discussion of its possible relation with reverse hypercontractivity. It should be noted that all of the results of this article are proved without using hypercontractivity estimates.