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

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 $\mathrm{\Omega }\subset {\mathbb{R}}^d$, and establish an enhanced version of Aleksandrov’s maximum principle. In particular, we consider the drift term $b=(b^1,\dots ,b^d)$ 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 ($\mathit{\text{Ann.}}\mathit{\text{Probab.}}$, 2021) concerning $L_d$ drifts. The key step involves deriving a Gehring-type inequality for Green’s functions by using the Fabes-Stroock approach ($\mathit{\text{Duke}}\mathit{\text{Math.}}\mathit{\text{J.}}$, 1984).