Parabolic equations with variably partially VMO coefficients

The $W^{1,2}_p$-solvability of second-order parabolic equations in nondivergence form in the whole space is proved for $p\in(1,\infty)$. The leading coefficients are assumed to be measurable in one spatial direction and have vanishing mean oscillation (VMO) in the orthogonal directions and the time variable in each small parabolic cylinder with direction allowed to depend on the cylinder. This extends a recent result by Krylov for elliptic equations. The novelty in the current paper is that the restriction $p>2$ is removed.