$L^q$ estimates for nonlocal p-Laplacian type equations with BMO kernel coefficients in divergence form

We study $s$-fractional $p$-Laplacian type equations with discontinuous kernel coefficients in divergence form to establish $W^{s+\sigma ,q}$ estimates for any choice of pairs $(\sigma ,q)$ with $q\in (p,\mathrm{\infty })$ and $\sigma \in (0,\min \{\frac{s}{p-1},1-s\})$ under the assumption that the associated kernel coefficients have small BMO seminorms near the diagonal. As a consequence, we find in the literature an optimal fractional Sobolev regularity of such a non-homogeneous nonlocal equation when the right-hand side is presented by a suitable fractional operator. Our results are new even in the linear case.