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 $f\in L^r_{\mathrm{loc}}(Ω)$. Assuming $p>1$, $s\in(0,1)$, and $sp’>1$, solutions belong to $W^{1,q}_{\mathrm{loc}}(Ω)$ 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.

💡 Research Summary

The paper establishes sharp gradient integrability results for weak solutions of fractional p‑Laplacian type equations with a non‑local right‑hand side. The authors consider the (s,p)‑Poisson equation

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