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

Feneuil, Joseph

doi:10.2140/apde.2023.16.545

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

Joseph Feneuil

Vol. 16 (2023), No. 2, 545–570

DOI: 10.2140/apde.2023.16.545

Abstract

In $ℝ_{+}^{d + 1}$ or in $ℝ^{n} ∖ ℝ^{d}$ ( $d < n - 1$ ), we study the Green function with pole at infinity defined for instance by David, Engelstein, and Mayboroda. In two cases, we deduce the equivalence between the elliptic measure and the Lebesgue measure on $ℝ^{d}$ . 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.

Keywords

Green function with pole at infinity, elliptic measure, $A_{\infty}$-absolute continuity

Mathematical Subject Classification

Primary: 35J25, 42B37

Milestones

Received: 22 October 2020

Revised: 26 May 2021

Accepted: 15 July 2021

Published: 3 May 2023

Authors

Joseph Feneuil
Laboratoire de mathématiques d’Orsay Université Paris-Saclay Orsay France

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Vol. 16, No. 2, 2023