H^{s,p} estimates for weak solutions of fractional nonlocal equations in divergence form

Abstract

This paper establishes interior $H^{s,p}$ regularity for weak solutions to fractional nonlocal equations in divergence form with general kernels. Under the $(δ, R)$-vanishing condition on the coefficients, we prove that for $p>2$, if the source term $f\in L^p_{loc}$, then the solution $u\in H^{s,p}_{loc}$. In contrast to Byun and Kim \cite{BK2025}, our result holds in the borderline case for $f$ without requiring differentiability assumptions. Our approach simplifies previous methods by using the stability properties of the fractional gradient to derive the good-inequality directly, thereby avoiding reliance on maximal functions. The proof integrates Vitali's covering lemma, Hardy-Littlewood maximal theory, self-improving estimates, and compactness arguments. This work extends classical $W^{1,p}$ regularity to fractional nonlocal settings under minimal kernel assumptions.