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

Abstract

We establish gradient potential estimates for SOLA to the fractional $p$-Laplace equation with finite signed measure data. Under the assumptions $n\ge2$, $p>2$, $0 p-1$, every SOLA belongs to $W^{1,p-1}_{\mathrm{loc}}$ and its weak gradient satisfies a Wolff potential estimate at every Lebesgue point. Under the additional condition $sp>n$, the solution has a continuous representative that is Fréchet differentiable at every point where the potential is finite. The proof combines homogeneous affine decay with constants independent of the affine slope and an $L^{p-1}$ comparison estimate. We establish weak differentiability by mollification and then identify the pointwise gradient. When $sp>n$, an additional $L^\infty$ comparison yields Fréchet differentiability.