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

We prove interior $C^{1,\alpha }$ estimates for weak solutions of the fractional $p$-Poisson equation, with $p\ge 2$. For right-hand sides in $L^q$, we obtain the estimate when $p<(1-d\mathrm{/}q)\mathrm{/}(1-s)$. For $\gamma$-Hölder continuous right-hand sides, the range is $p<(1+\gamma )\mathrm{/}(1-s)$. Both ranges of parameters are optimal.