Sharp defective log-Sobolev inequalities on H-type groups

TL;DR

This paper establishes sharp defective log-Sobolev inequalities on H-type groups and applies them to demonstrate exponential integrability of Lipschitz functions under heat kernel measures, extending to Gaussian-like measures in sub-Riemannian settings.

Contribution

It introduces the first sharp defective log-Sobolev inequalities on H-type groups and applies these to analyze measure concentration and integrability properties.

Findings

Proved sharp defective log-Sobolev inequalities on H-type groups.

Established exponential integrability of Lipschitz functions under heat kernel measures.

Extended inequalities to Gaussian-like measures in sub-Riemannian geometry.

Abstract

In this paper we prove a sharp defective log-Sobolev inequality on H-type groups. Then we use such an inequality to show exponential integrability of Lipschitz functions with respect to the heat kernel measure. A defective log-Sobolev-type inequality for the Gaussian-like measure with respect to the sub-Riemannian distance is also proved on arbitrary H-type groups.