Fabes-Stroock approach to higher integrability of Green's functions and ABP estimates with $L_d$ drift

TL;DR

This paper uses the Fabes-Stroock approach to prove higher integrability of Green's functions and an improved ABP estimate for elliptic equations with $L_d$ drift, providing an alternative proof of Krylov's recent result.

Contribution

It introduces an analytic method employing the Fabes-Stroock approach to establish higher integrability and ABP estimates for elliptic equations with $L_d$ drift, offering a new proof of Krylov's theorem.

Findings

Derived a Gehring-type inequality for Green's functions.

Established higher integrability of Green's functions with $L_d$ drift.

Provided an alternative proof of Krylov's $L_d$ drift result.

Abstract

We explore the higher integrability of Green's functions associated with the second-order elliptic equation $a^{ij}D_{ij}u + b^i D_iu = f$ in a bounded domain $\Omega \subset \mathbb{R}^d$, and establish an enhanced version of Aleksandrov's maximum principle. In particular, we consider the drift term $b=(b^1, \ldots, b^d)$ in $L_d$ and the source term $f \in L_p$ for some $p < d$. This provides an alternative and analytic proof of a result by N. V. Krylov (\textit{Ann. Probab.}, 2021) concerning $L_d$ drifts. The key step involves deriving a Gehring-type inequality for Green's functions by using the Fabes-Stroock approach (\textit{Duke Math. J.}, 1984).