  We investigate weighted Sobolev regularity of weak solutions of
non-homogeneous parabolic equations with singular divergence-free drifts.
Assuming that the drifts satisfy some mild regularity conditions, we establish
local weighted $L^p$-estimates for the gradients of weak solutions. Our results
improve the classical one to the borderline case by replacing the
$L^\infty$-assumption on solutions by solutions in the John-Nirenberg
\textup{BMO} space. The results are also generalized to parabolic equations in
divergence form with small oscillation elliptic symmetric coefficients and
therefore improve many known results.

Analysis of PDEs