Carleson measure estimates for the Green function

TL;DR

This paper establishes sharp Carleson measure estimates for the Green function of elliptic operators in the half-space, showing near-affineness under quadratic Carleson coefficient oscillation conditions.

Contribution

It proves that the Green function is almost affine with sharp estimates under quadratic Carleson conditions on coefficients, extending understanding of elliptic PDEs in half-spaces.

Findings

Green function satisfies Carleson measure estimates

Results are sharp and optimal for the class of operators

Green function is almost affine at every scale

Abstract

In the present paper, we consider an elliptic divergence form operator in the half-space and prove that its Green function is almost affine, or more precisely, that the normalized difference between the Green function and a suitable affine function at every scale satisfies a Carleson measure estimate, provided that the oscillations of the coefficients satisfy the traditional quadratic Carleson condition. The results are sharp, and in particular, it is demonstrated that the class of the operators considered in the paper cannot be improved.