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The solvability in Sobolev spaces $W^{1,2}_p$ is proved for nondivergenceform second order parabolic equations for $p>2$ close to 2. The leadingcoefficients are assumed to be measurable in the time variable and twocoordinates of space variables, and almost VMO (vanishing mean oscillation)with respect to the other coordinates. This implies the $W^{2}_p$-solvabilityfor the same $p$ of nondivergence form elliptic equations with leadingcoefficients measurable in two coordinates and VMO in the others. Underslightly different assumptions, we also obtain the solvability results when$p=2$.

Second-order elliptic and parabolic equations with $B(\mathbb R^{2}, VMO)$ coefficients