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 $n\ge 2$, $p>2$, $0<s<1$, and $sp>p-1$, every SOLA belongs to $W_{\mathrm{loc}}^{1,p-1}$, 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.

arXiv commentRevised version with gradient potential estimates for SOLA in the range $p>2$ and $sp>p-1$. Title changed from "supercritical" to "superquadratic"