  We study a class of non-divergence form elliptic and parabolic equations with
singular first-order coefficients in an upper half space with the homogeneous
Dirichlet boundary condition. In the simplest setting, the operators in the
equations under consideration appear in the study of fractional heat and
fractional Laplace equations. Intrinsic weighted Sobolev spaces are found in
which the existence and uniqueness of strong solutions are proved under certain
smallness conditions on the weighted mean oscillations of the coefficients in
small parabolic cylinders. Our results are new even when the coefficients are
constants and they cover the case where the weights may not be in the
$A_p$-Muckenhoupt class.
