Carleson estimates for the Green function on domains with lower dimensional boundaries

TL;DR

This paper establishes Carleson estimates for the Green function of elliptic operators in domains with lower-dimensional boundaries, showing near-affine behavior despite oscillatory coefficients.

Contribution

It proves that the Green function is almost affine with Carleson measure estimates in domains with lower-dimensional boundaries, extending previous results to more oscillatory coefficients.

Findings

Green function exhibits near-affine behavior at all scales

Carleson measure estimates hold despite highly oscillatory coefficients

Results apply to elliptic operators in domains with lower-dimensional boundaries

Abstract

In the present paper, we consider an elliptic divergence form operator in $\mathbb{R}^n\setminus\mathbb{R}^d$ with $d<n-1$ and prove that its Green function is almost affine, in the sense that the normalized difference between the Green function with a sufficiently far away pole and a suitable affine function at every scale satisfies a Carleson measure estimate. The coefficients of the operator can be very oscillatory, and only need to satisfy some condition similar to the traditional quadratic Carleson condition.