  We consider both divergence and non-divergence parabolic equations on a half
space in weighted Sobolev spaces. All the leading coefficients are assumed to
be only measurable in the time and one spatial variable except one coefficient,
which is assumed to be only measurable either in the time or the spatial
variable. As functions of the other variables the coefficients have small
bounded mean oscillation (BMO) semi-norms. The lower-order coefficients are
allowed to blow up near the boundary with a certain optimal growth condition.
As a corollary, we also obtain the corresponding results for elliptic
equations.
