Abstract approach to non homogeneous Harnack inequality in doubling quasi metric spaces

TL;DR

This paper develops an abstract framework to establish Harnack inequalities for non homogeneous PDEs in quasi metric spaces, extending classical results to more general settings involving subelliptic and X-elliptic operators.

Contribution

It introduces a novel abstract approach that adapts the double ball and critical density concepts to non homogeneous PDEs in quasi metric spaces, applicable to subelliptic and X-elliptic operators.

Findings

Established Harnack inequalities for subelliptic equations with Grushin vector fields.

Extended the theory to X-elliptic operators in divergence form.

Provided a unified abstract method for non homogeneous PDEs in quasi metric spaces.

Abstract

We develop an abstract theory to obtain Harnack inequality for non homogeneous PDEs in the setting of quasi metric spaces. The main idea is to adapt the notion of double ball and critical density property given by Di Fazio, Guti\'errez, Lanconelli, taking into account the right hand side of the equation. Then we apply the abstract procedure to the case of subelliptic equations in non divergence form involving Grushin vector fields and to the case of X-elliptic operators in divergence form.