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. ${(-{\Delta }_p)}^{s}u=f$. 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$f\in L_{\mathrm{loc}}^{\frac{qp}{p-1}}⟹\nabla u\in L_{\mathrm{loc}}^{qp}.$ The qualitative result is accompanied by a local quantitative estimate.