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

We explore the higher integrability of Green's functions associated with the second-order elliptic equation aⁱʲDᵢⱼu + bⁱ Dᵢu = f in a bounded domain Ω ⊂ Rᵈ, and establish a version of Aleksandrov's maximum principle. In particular, we consider the drift term b=(b¹, …, bᵈ) in Ld and the source term f ∈ Lₚ for some $p < d$. This provides an alternative and analytic proof of a result by N. V. Krylov (2021) concerning Ld drifts. The key step involves deriving a Gehring type inequality for Green's functions by using the Fabes-Stroock approach (1984).