Reverse Poincare inequalities, Isoperimetry, and Riesz transforms in Carnot groups

TL;DR

This paper establishes an optimal reverse Poincaré inequality for heat semigroups on Carnot groups, leading to new proofs of isoperimetric inequalities and Riesz transform boundedness, advancing analysis on these geometric structures.

Contribution

It introduces an optimal reverse Poincaré inequality for sub-Laplacian heat semigroups on Carnot groups, providing novel proofs for key geometric and harmonic analysis results.

Findings

Proved an optimal reverse Poincaré inequality for Carnot groups.

Derived new proofs of isoperimetric inequalities.

Established boundedness of Riesz transforms in Carnot groups.

Abstract

We prove an optimal reverse Poincar\'e inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.