Sharp gradient integrability for $(s,p)$-Poisson type equations

We prove local $W^{1,q}$-regularity for weak solutions to fractional $p$-Laplacian type equations with right-hand side $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^r(\mathrm{\Omega })$. Assuming $p>1$, $s\in (0,1)$, and $s{p}^{\mathrm{\prime }}>1$, solutions belong to $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,q}(\mathrm{\Omega })$ for the optimal exponent $q=q(n,p,s,r)$. We obtain quantitative local gradient estimates involving nonlocal tail terms. The optimality of $q$ is confirmed by a counterexample.