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.
Bibliographical noteFunding Information:
G. David was partially supported by the European Community H2020 Grant GHAIA 777822, and the Simons Foundation Grant 601941, GD. S. Mayboroda was partly supported by the NSF RAISE-TAQS Grant DMS-1839077 and the Simons foundation Grant 563916, SM.
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature.