Gradient estimates for the fractional $p$-Laplacian in the supercritical regime

Abstract

We prove a pointwise gradient estimate for the fractional $p$-Laplace equation with measure data. Let $n\ge 2$, $p>1$ and $\max\{n/p,1/p'\}<s<1$, where $p'=p/(p-1)$. Suppose $u\in W^{s,p}(\mathbb R^n)$ is a weak solution of \[ (-\Delta_p)^s u=\mu \quad\text{in }\Omega, \] with $\mu\in\mathcal M_{\mathrm{loc}}(\Omega)$. Set $\gamma=s-(p-1)/p$. If $B_{2R}(x_0)\Subset\Omega$ and $\mathcal{W}_{\gamma,p}^{|\mu|}(x_0,2R)<\infty$, then $u$ is Fr\'echet differentiable at $x_0$, and \[|\nabla u(x_0)|\le C\bigl[\mathcal A(u;x_0,2R)+\mathcal{W}_{\gamma,p}^{|\mu|}(x_0,2R)\bigr], \] where $\mathcal{W}_{\gamma,p}^{|\mu|}$ denotes the truncated Wolff potential, and $\mathcal A$ depends on the local oscillation of $u$ and its nonlocal tail. The proof uses a comparison with fractional $p$-harmonic replacements and an affine decay estimate for homogeneous solutions, with constants independent of the affine slope. In the large slope regime, the affine decay estimate follows from a Schauder estimate for the linear nonlocal equation satisfied by the affine remainder.