Gradient estimates for the fractional $p$-Poisson equation

Abstract

We consider local weak solutions to the fractional $p$-Poisson equation of order $s$, i.e. ${\left(-\hat{I}{”}_p\right)}^{s}u=f$. In the range $p>1$ and $s\in (\frac{p-1}{p},1)$ we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any $q>1$ that \begin{equation*} f\in L^{\frac{qp}{p-1}}_{\rm loc} \quad\Longrightarrow\quad \nabla u\in L^{qp}_{\rm loc}. \end{equation*} The qualitative result is accompanied by a local quantitative estimate.