Borderline gradient continuity for fractional heat type operators

Abstract

In this paper, we establish gradient continuity for solutions to \[ (\partial_t - \operatorname{div}(A(x) \nabla ))^{s} u =f,\quad s \in (1/2, 1), \] when $f$ belongs to the scaling critical function space $L\left (\frac {n+2}{2s-1}, 1\right )$ . Our main results theorems 1.1 and 1.2 can be seen as a nonlocal generalization of a well-known result of Stein in the context of fractional heat type operators and sharpen some of the previous gradient continuity results which deal with $f$ in subcritical spaces. Our proof is based on an appropriate adaptation of compactness arguments, which has its roots in a fundamental work of Caffarelli in [13].