Borderline gradient continuity for fractional heat type operators

TL;DR

This paper proves gradient continuity for solutions to fractional heat type operators with critical space data, extending classical results and sharpening previous regularity findings.

Contribution

It introduces a nonlocal generalization of Stein's classical result, establishing gradient continuity for fractional heat operators with critical space data.

Findings

Gradient continuity established for fractional heat operators

Extension of classical regularity results to nonlocal operators

Sharpens previous gradient regularity results in subcritical spaces

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].