On the Lp-Integrability of Green’s function for Elliptic Operators

• Main Author: Alharbi, Abdulrahman
• Other Authors: Gomes, Diogo A.
• Language: en
• Published: 2019
• Subjects: Green's Functions; non-divergence form; Fabes-Stroock; ABP Estimate; Muckenhoupt Weights
• Online Access: Alharbi, A. (2019). On the Lp-Integrability of Green’s function for Elliptic Operators. KAUST Research Repository. https://doi.org/10.25781/KAUST-71531; http://hdl.handle.net/10754/655516

In this thesis, we discuss some of the results that were proven by Fabes and Stroock in 1984. Our main purpose is to give a self-contained presentation of the proof of this results. The first result is on the existence of a “reverse H ̈older inequality” for the Green’s function. We utilize the work of Muckenhoupt on the reverse Ho ̈lder inequality and its connection to the A∞ class to establish a comparability property for the Green’s functions. Additionally, we discuss some of the underlying preliminaries. In that, we prove the Alexandrov-Bakelman-Pucci estimate, give a treatment to the Ap and A∞ classes of Muckenhoupt, and establish two intrinsic lemmas on the behavior of Green’s function.