Weighted mixed-norm $L_p$ estimates for equations in non-divergence form with singular coefficients: the Dirichlet problem

We study a class of elliptic and parabolic equations in non-divergence form with singular coefficients in an upper half space with the homogeneous Dirichlet boundary condition. Intrinsic weighted Sobolev spaces are found in which the existence and uniqueness of strong solutions are proved when the partial oscillations of coefficients in small parabolic cylinders are small. Our results are new even when the coefficients are constants