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

Abstract

We explore the higher integrability of Green's functions associated with the second-order elliptic equation $a^{ij}{D}_{ij}u+b^i{D}_{i}u=f$ in a bounded domain $\Omega \subset R^d$, and establish an enhanced version of Aleksandrov's maximum principle. In particular, we consider the drift term $b=(b^1,\dots ,b^d)$ 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 (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 (Duke Math. J., 1984).