发表机构
School of Mathematics, Harbin Institute of Technology; School of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院; 哈尔滨工业大学数学学院与高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了超临界参数范围内分数阶 $p$-Laplacian 方程在测度数据下的逐点梯度估计，通过比较原理和仿射衰减估计，建立了 Fréchet 可微性及梯度上界。
AI 中文摘要
我们证明了具有测度数据的分数阶 $p$-Laplace 方程的逐点梯度估计。设 $n\ge 2$，$p>1$ 且 $\max\{n/p,1/p'\}<s<1$，其中 $p'=p/(p-1)$。假设 $u\in W^{s,p}(\mathbb R^n)$ 是方程 $(-\Delta_p)^s u=\mu$ 在 $\Omega$ 中的弱解，其中 $\mu\in\mathcal M_{\mathrm{loc}}(\Omega)$。令 $\gamma=s-(p-1)/p$。若 $B_{2R}(x_0)\Subset\Omega$ 且 $\mathcal{W}_{\gamma,p}^{|\mu|}(x_0,2R)<\infty$，则 $u$ 在 $x_0$ 处是 Fréchet 可微的，并且 $|\nabla u(x_0)|\le C[\mathcal A(u;x_0,2R)+\mathcal{W}_{\gamma,p}^{|\mu|}(x_0,2R)]$，其中 $\mathcal{W}_{\gamma,p}^{|\mu|}$ 表示截断的 Wolff 势，$\mathcal A$ 依赖于 $u$ 的局部振荡及其非局部尾部。证明使用了与分数阶 $p$-调和替换的比较，以及齐次解的仿射衰减估计，其中常数与仿射斜率无关。在大斜率区域，仿射衰减估计由仿射余项所满足的线性非局部方程的 Schauder 估计得出。
英文摘要
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 $n\ge2$, $p>2$, $0<s<1$, and $sp>p-1$, every SOLA belongs to $W^{1,p-1}_{\mathrm{loc}}$, 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.
CommentsRevised version with gradient potential estimates for SOLA in the range $p>2$ and $sp>p-1$. Title changed from "supercritical" to "superquadratic"
