Gradient estimates for the fractional p-Laplacian in the superquadratic regime
Abstract
We establish gradient potential estimates for solutions obtained as limits of approximations (SOLA) to the fractional p-Laplace equation with finite signed Radon measure data. Under the assumptions n≥2, p>2, 0<s<1, and sp>p−1, every SOLA belongs to W1,p−1loc, and its weak gradient satisfies a Wolff potential estimate at every Lebesgue point of the weak gradient. Under the additional condition sp>n, the solution has a continuous representative that is Fréchet differentiable at every point where the potential is finite. These results give a partial answer to questions raised by Diening and Nowak [Ann. PDE, \textbf{11}(2025)] and by Diening, Kim, Lee and Nowak~[J. Eur. Math. Soc. (JEMS), 2025] concerning gradient potential estimates for the fractional p-Laplacian. The proof combines homogeneous affine decay with constants independent of the affine slope and comparison estimates to obtain an affine excess recurrence. Iteration then yields the Wolff potential bound.
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BibTeX
@misc{li2026gradient,
title = {{Gradient estimates for the fractional $p$-Laplacian in the superquadratic regime}},
author = {Ying Li and Chao Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06924},
eprint = {2609.06924},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06924},
}
RIS
TY - UNPB
TI - Gradient estimates for the fractional p-Laplacian in the superquadratic regime
AU - Ying Li
AU - Chao Zhang
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.06924
UR - https://arxiv.org/abs/2609.06924
AB - We establish gradient potential estimates for solutions obtained as limits of approximations (SOLA) to the fractional p-Laplace equation with finite signed Radon measure data. Under the assumptions n≥2, p>2, 0<s<1, and sp>p−1, every SOLA belongs to W^1,p−1_loc, and its weak gradient satisfies a Wolff potential estimate at every Lebesgue point of the weak gradient. Under the additional condition sp>n, the solution has a continuous representative that is Fréchet differentiable at every point where the potential is finite. These results give a partial answer to questions raised by Diening and Nowak [Ann. PDE, \textbf{11}(2025)] and by Diening, Kim, Lee and Nowak~[J. Eur. Math. Soc. (JEMS), 2025] concerning gradient potential estimates for the fractional p-Laplacian. The proof combines homogeneous affine decay with constants independent of the affine slope and comparison estimates to obtain an affine excess recurrence. Iteration then yields the Wolff potential bound.
ER -