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Borderline gradient continuity for fractional heat type operators
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 5 / October 2023
- Published online by Cambridge University Press:
- 14 October 2022, pp. 1651-1682
- Print publication:
- October 2023
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- Article
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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].
