# Harnack's inequality for a class of non-divergent equations in the Heisenberg group

## TL;DR

This paper establishes an invariant Harnack's inequality for a class of non-divergent operators structured on Heisenberg vector fields, using barrier constructions and an axiomatic approach.

## Contribution

It introduces a novel method for proving Harnack's inequality for non-divergent equations in the Heisenberg group with continuous, symplectic, positive definite coefficients.

## Key findings

- Proves Harnack's inequality in the Heisenberg group setting.
- Develops barrier functions for supersolutions.
- Utilizes an axiomatic framework to derive the inequality.

## Abstract

We prove an invariant Harnack's inequality for operators in non-divergence form structured on Heisenberg vector fields when the coefficient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach from [DGL08] to obtain Harnack's inequality.