Parabolic Poincaré inequalities and maximal function estimates for systems of partial differential equations

We study parabolic Poincaré inequalities for solutions to nonlinear systems of partial differential equations. Our main results show that these inequalities are self-improving. As applications, we establish reverse Hölder inequalities for the mean oscillation over parabolic cylinders and for the gradient. We also consider the corresponding Poincaré inequalities and self-improvement results on time intervals at fixed spatial points. These results are based on a pointwise maximal function estimate. As an application, we obtain regularity results in the time direction for solutions to nonlinear systems.