Carleson measure estimates for the Green function

Abstract: We are interested in the relations between an elliptic operator on a domain, the geometry of the domain, and the boundary behavior of the Green function. In joint work with Guy David and Svitlana Mayboroda, we show that if the coefficients of the operator satisfy a quadratic Carleson condition, then the Green function on the half-space 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. We demonstrate with counterexamples that our results are optimal, in the sense that the class of the operators considered are essentially the best possible.

This work is motivated mainly by finding PDE characterizations of uniform rectifiable sets with higher co-dimension. I’ll talk about this motivation and backgrounds, our recent results, as well as possible directions in the future.

analysis of PDEsclassical analysis and ODEsfunctional analysismetric geometry

Audience: researchers in the topic

HA-GMT-PDE Seminar

Series comments: Description: A senior graduate student/postdoc series in harmonic analysis, geometric measure theory, and partial differential equations.

Meetings will be weekly, and usually will occur on Monday, with some exceptions. For each talk, the Zoom link is made available in the website too. Contact Bruno Poggi at poggi008@umn.edu if you would like to subscribe to the seminar mailing list.