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

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