Hölder Gradient Estimates for $(s,p)$-Poisson Equations with measurable and Hölder Data

Abstract: We prove interior $C<sup>{1,α}$ estimates for weak solutions of the fractional $p$-Poisson equation, with $p\geq2$. For right-hand sides in $L<sup>q$, we obtain the estimate when $p&lt;(1-d/q)/(1-s)$. For $γ$-Hölder continuous right-hand sides, the range is $p&lt;(1+γ)/(1-s)$. Both ranges of parameters are optimal.