Note on the $q$-Logarithmic Sobolev and $p$-Talagrand Inequalities on Carnot Groups

TL;DR

This paper establishes $q$-Logarithmic Sobolev and $p$-Talagrand inequalities on Carnot groups, advancing the understanding of functional inequalities in sub-Riemannian geometry.

Contribution

It proves the $q$-Logarithmic Sobolev inequality and derives the $p$-Talagrand inequality using Hamilton-Jacobi equations in Carnot groups, a novel approach in this setting.

Findings

Proved $q$-Logarithmic Sobolev inequality for probability measures.

Derived $p$-Talagrand inequality and hypercontractivity results.

Connected inequalities with the Carnot-Carathéodory distance.

Abstract

In the setting of Carnot groups, we prove the $q-$Logarithmic Sobolev inequality for probability measures as a function of the Carnot-Carath\'eodory distance. As an application, we use the Hamilton-Jacobi equation in the setting of Carnot groups to prove the $p-$Talagrand inequality and hypercontractivity.