Hölder–Zygmund estimates for degenerate parabolic systems

We consider energy solutions of the inhomogeneous parabolic p  -Laplacian system ∂tu−div(|Du|p−2Du)=−divg∂tu−div(|Du|p−2Du)=−divg. We show in the case p⩾2p⩾2 that if the right hand side g   is locally in L∞(BMO)L∞(BMO), then u   is locally in L∞(C1)L∞(C1), where C1C1 is the 1-Hölder–Zygmund space. This is the borderline case of the Calderón–Zygmund theory. We provide local quantitative estimates. We also show that finer properties of g are conserved by Du, e.g. Hölder continuity. Moreover, we prove a new decay for gradients of p  -caloric solutions for all 2nn+2