  In this paper, we study some regularity issues concerning the gradient of
weak solutions of $u_t - {\rm div} \mathcal{A}(x,t,\nabla u) = g$, where
$\mathcal{A}(x,t,\nabla u)$ is modeled after the $p$-Laplace operator. The main
results we are interested in is to obtain optimal conditions on the datum $g$
(independent of time) such that borderline higher integrability of the gradient
and Lipschitz estimates for the weak solution holds. Moreover, we develop a
theory where we can obtain elliptic type estimates using parabolic theory,
which gives improved potential estimates for the elliptic systems.
