Generalized Morrey estimates for the gradient of divergence form parabolic operators with discontinuous coefficients

We consider the Cauchy-Dirichlet problem for linear divergence form parabolic operators in bounded Reifenberg-flat domains. The coefficients are supposed to be only measurable in one of the space variables and small BMO with respect to the others. We obtain boundedness of the Hardy-Littlewood maximal operator in the generalized Morrey spaces Wp,φ, p∈(1,∞) and weight φ satisfying certain supremum condition. This permits us to obtain Calderón-Zygmund type estimate for the gradient of the weak solution of the problem.