A certain critical density property for invariant Harnack inequalities in H-type groups

TL;DR

This paper establishes a critical density property and an invariant Harnack inequality for certain degenerate-elliptic operators in H-type groups, extending previous results from the Heisenberg group to a broader class.

Contribution

It extends the critical density estimate and Harnack inequality to second order degenerate-elliptic operators in H-type groups under a Cordes-Landis type condition.

Findings

Proves a critical density estimate for these operators.

Derives an invariant Harnack inequality for non-negative solutions.

Extends known results from the Heisenberg group to H-type groups.

Abstract

We consider second order linear degenerate-elliptic operators which are elliptic with respect to horizontal directions generating a stratified algebra of H-type. Extending a result by Guti\'errez and Tournier for the Heisenberg group, we prove a critical density estimate by assuming a condition of Cordes-Landis type. We then deduce an invariant Harnack inequality for the non-negative solutions from a result by Di Fazio, Guti\'errez, and Lanconelli.