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We show that weak solutions to parabolic equations in divergence form withzero Dirichlet boundary conditions are continuously differentiable up to theboundary when the leading coefficients have Dini mean oscillation and the lowerorder coefficients verify certain conditions. Similar results are obtained fornon-divergence form parabolic operators and their adjoint operators. Undersimilar conditions, we also establish a Harnack inequality for nonnegativeadjoint solutions, together with upper and lower Gaussian bounds for the globalfundamental solution.

On $C^{1/2,1}$, $C^{1,2}$, and $C^{0,0}$ estimates for linear parabolic operators