Coercive Inequalities on Metric Measure Spaces

TL;DR

This paper establishes Log-Sobolev inequalities on non-doubling metric measure spaces, specifically on the Heisenberg group with certain measures, advancing understanding of functional inequalities in sub-Riemannian geometry.

Contribution

It proves Log-Sobolev inequalities on the Heisenberg group with heat kernel and Gaussian measures, introducing new U-bound techniques for non-doubling spaces.

Findings

Log-Sobolev inequality holds on the Heisenberg group with specified measures

Introduction of U-bound estimates as intermediate results

Extension of coercive inequalities to non-doubling metric measure spaces

Abstract

We study coercive inequalities on finite dimensional metric spaces with probability measures which do not have volume doubling property. This class of inequalities includes Poincar\'e and Log-Sobolev inequality. Our main result is proof of Log-Sobolev inequality on Heisenberg group equipped with either heat kernel measure or "gaussian" density build from optimal control distance. As intermediate results we prove so called U-bounds.