Mutual Absolute Continuity of Harmonic and Surface Measures for Hormander Type Operators

TL;DR

This paper investigates the relationship between harmonic and surface measures for Hormander-type sub-Laplacians, establishing their mutual absolute continuity and providing a framework for solving the Dirichlet problem under geometric conditions.

Contribution

It proves mutual absolute continuity of harmonic and surface measures for Hormander operators and offers a representation for solutions to the Dirichlet problem.

Findings

Reversed Hölder inequality for the Poisson kernel

Mutual absolute continuity of harmonic and surface measures

Representation formula for Dirichlet problem solutions

Abstract

In this paper, we consider the Sub-Laplacian L which consists of sum of squares of smooth vector fields that satisfy Hormander's finite rank condition. We study the Dirichlet problem for this operator on domains that satisfy certain geometric conditions. For such domains, several key results are established. These results consist of 1) A reversed Holder inequality for the Poisson kernel 2) Harmonic measure (corresponding to L) and surface measure (as well as the H-Perimeter measure) are mutually absolutely continuous 3) A representation (hence solvability of the Dirichlet problem) for solutions to the Dirichlet problem.