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We study $s$-fractional $p$-Laplacian type equations with discontinuouskernel coefficients in divergence form to establish $W^{s+\sigma,q}$ estimatesfor any choice of pairs $( \sigma,q)$ with $q\in(p,\infty)$ and$\sigma\in\left(0,\min\left\{\frac{s}{p-1},1-s\right\}\right)$ under theassumption that the associated kernel coefficients have small BMO seminormsnear the diagonal. As a consequence, we find in the literature an optimalfractional Sobolev regularity of such a non-homogeneous nonlocal equation whenthe right-hand side is presented by a suitable fractional operator. Our resultsare new even in the linear case.

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