The Green function with pole at infinity applied to the study of the elliptic measure

TL;DR

This paper investigates the Green function with pole at infinity in certain domains and establishes the equivalence and absolute continuity of elliptic measure with Lebesgue measure, extending known results to broader classes of operators.

Contribution

It introduces new applications of the Green function with pole at infinity to prove measure equivalences and absolute continuity for elliptic measures in complex domains.

Findings

Elliptic measure is equivalent to Lebesgue measure in specific domains.

Proves $A_ abla$-absolute continuity of elliptic measure for a wider class of operators.

Extends previous results on elliptic measure and Green functions.

Abstract

In $\mathbb R^{d+1}_+$ or in $\mathbb R^n\setminus \mathbb R^d$ ($d<n-1$), we study the Green function with pole at infinity introduced by David, Engelstein, and Mayboroda. In two cases, we deduce the equivalence between the elliptic measure and the Lebesgue measure on $\mathbb R^d$; and we further prove the $A_\infty$-absolute continuity of the elliptic measure for operators that can be related to the two previous cases via Carleson measures, extending the range of operators for which the $A_\infty$-absolute continuity of the elliptic measure is known.