The Harnack inequality for a class of degenerate elliptic operators

TL;DR

This paper establishes a Harnack inequality for a class of hypoelliptic degenerate elliptic PDEs related to Kolmogorov operators, extending regularity results to these complex operators.

Contribution

It proves a Harnack inequality for distributional solutions to a new class of hypoelliptic degenerate elliptic PDEs involving variable coefficients.

Findings

Harnack inequality holds for the specified class of operators

Operators are hypoelliptic due to the non-vanishing derivatives of coefficients

Results extend regularity theory to more general degenerate elliptic equations

Abstract

We prove a Harnack inequality for distributional solutions to a type of degenerate elliptic PDEs in $N$ dimensions. The differential operators in question are related to the Kolmogorov operator, made up of the Laplacian in the last $N-1$ variables, a first-order term corresponding to a shear flow in the direction of the first variable, and a bounded measurable potential term. The first-order coefficient is a smooth function of the last $N-1$ variables and its derivatives up to certain order do not vanish simultaneously at any point, making the operators in question hypoelliptic.