  We show that weak solutions to parabolic equations in divergence form with
zero Dirichlet boundary conditions are continuously differentiable up to the
boundary when the leading coefficients have Dini mean oscillation and the lower
order coefficients verify certain conditions. Similar results are obtained for
non-divergence form parabolic operators and their adjoint operators. Under
similar conditions, we also establish a Harnack inequality for nonnegative
adjoint solutions, together with upper and lower Gaussian bounds for the global
fundamental solution.
