Hs,p estimates for weak solutions of fractional nonlocal equations in divergence form

This paper establishes interior Hs,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∈ Lᵖloc, then the solution u∈ Hs,ploc. In contrast to Byun and Kim {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 W1,p regularity to fractional nonlocal settings under minimal kernel assumptions.