Borderline gradient continuity for fractional heat type operators

Abstract

In this paper, we establish gradient continuity for solutions to \[ (\partial_t - \operatorname{div}(A(x) \nabla u))^s =f,\ s \in (1/2, 1), \] when $f$ belongs to the scaling critical function space $L(\frac{n+2}{2s-1}, 1)$. 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 sharpens some of the previous gradient continuity results which deals 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].