Estimates in Generalized Morrey Spaces for Weak Solutions to Divergence Degenerate Parabolic Systems

Let $\mathrm{X}=(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_q)$ be a family of real smooth vector fields satisfying H"{o}mander’s condition. The purpose of this paper is to establish gradient estimates in generalized Morrey spaces for weak solutions of the divergence degenerate parabolic system related to $X$ :%$$u_t^i+X_{\alpha }^{\ast }(a_{ij}^{\alpha \beta }(z)X_{\beta }{u}^j$$ where $\alpha ,\beta =1,2,\mathrm{.}\mathrm{.}\mathrm{.},q,$ $i,j=1,2,\mathrm{.}\mathrm{.}\mathrm{.},N$, $X_{\alpha }^{\ast }$ is the transposed vector field of $X_{\alpha }$, $z=(t,x)\in {\mathbb{R}}^{n+1}$, and coefficients $a_{ij}^{\alpha \beta }(z)$ belong to the space $VMO$ induced by the vector fields $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_q$.