The paper deals with some regularity issues for weak solutions of the Cauchy-Dirichlet problem \[ \begin{cases} u_t-D_\alpha(a^{\alpha\beta}(x,t)D_\beta u)= D_\alpha f^\alpha(x,t) & \text{in}\;Q=\Omega\times(0,T),\\ u(x,t)=0 & \text{on}\;\partial_PQ, \end{cases} \] where \(\Omega\subset \mathbb{R}^n\) is a bounded domain and \(\partial_PQ\) is the parabolic boundary of the cylinder \(Q.\)
Under the assumptions of partial BMO smallness of the coefficients \(a^{\alpha\beta}\) and Reifenberg flatness of \(\partial\Omega,\) the authors obtain global regularity in generalized Morrey spaces for the spatial gradient of the weak solutions, developing in this way a Calderón-Zygmund-type theory for the operators considered. Problems as those considered in the paper arise in the modeling of composite materials and in the mechanics of membranes and films of simple nonhomogeneous materials which form a linear laminated medium.