Coercive Inequalities in Higher-Dimensional Anisotropic Heisenberg Group

TL;DR

This paper investigates inequalities in a higher-dimensional anisotropic Heisenberg group by computing the fundamental solution of the sub-Laplacian and establishing related Poincaré and Logarithmic Sobolev inequalities.

Contribution

It provides explicit fundamental solutions and proves key inequalities in the anisotropic Heisenberg group setting, advancing understanding of geometric analysis in this context.

Findings

Computed the fundamental solution for the sub-Laplacian.

Proved Poincaré inequalities for measures related to the fundamental solution.

Established β-Logarithmic Sobolev inequalities in the anisotropic setting.

Abstract

In the setting of higher-dimensional anisotropic Heisenberg group, we compute the fundamental solution for the sub-Laplacian, and we prove Poincar\'e and $\beta-$Logarithmic Sobolev inequalities for measures as a function of this fundamental solution.