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

Abstract

We prove local $W^{1,q}$-regularity for weak solutions to fractional $p$-Laplacian type equations with right-hand side $f\in L^r_{\mathrm{loc}}(\Omega)$. Assuming $p>1$, $s\in(0,1)$, and $sp'>1$, solutions belong to $W^{1,q}_{\mathrm{loc}}(\Omega)$ 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.