2 results
Borderline gradient continuity for fractional heat type operators
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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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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].
Borderline gradient estimates at the boundary in Carnot groups
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 6 / December 2021
- Published online by Cambridge University Press:
- 02 December 2020, pp. 1920-1953
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- December 2021
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In this article, we prove the continuity of the horizontal gradient near a C1,Dini non-characteristic portion of the boundary for solutions to
$\Gamma ^{0,{\rm Dini}}$ perturbations of horizontal Laplaceans as in (1.1) below, where the scalar term is in scaling critical Lorentz space L(Q, 1) with Q being the homogeneous dimension of the group. This result can be thought of both as a sharpening of the 
$\Gamma ^{1,\alpha }$ boundary regularity result in [4] as well as a subelliptic analogue of the main result in [1] restricted to linear equations.
