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

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.

22 pages

References in corpus (7)

• Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem
• Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations
• Carleson measure estimates for the Green function
• Dirichlet problem in domains with lower dimensional boundaries
• The Dirichlet problem for elliptic operators having a BMO anti-symmetric part
• Square function estimates, BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets
• Square functions, non-tangential limits and harmonic measure in co-dimensions larger than one