Carleson measure estimates for the Green function

Abstract: It is known that the oscillation of the Green function for the Laplacian in a domain is related to the flatness of the boundary of the domain. In a joint work with Guy David and Svitlana Mayboroda, we consider the Green function for a second-order elliptic operator in the half-space. We show that if the coefficients satisfy a quadratic Carleson condition, then the 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. Our results are optimal, in the sense that the class of the operators considered cannot be improved.

This work is motivated mainly by finding PDE characterizations of uniformly rectifiable sets with higher co-dimension, yet our result is new of this kind in the co-dimension one setting as well.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

OARS Online Analysis Research Seminar

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