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 $a^{ij}D_{ij}u + b^i D_iu = f$ 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).