  Given $2\leq p<\infty$, $s\in (0, 1)$ and $t\in (1, 2s)$, we establish
interior $W^{t,p}$ Calderon-Zygmund estimates for solutions of nonlocal
equations of the form \[
  \int_{\Omega} \int_{\Omega} K\left (x,|x-y|,\frac{x-y}{|x-y|}\right )
\frac{(u(x)-u(y))(\varphi(x)-\varphi(y))}{|x-y|^{n+2s}} dx dy = g[\varphi],
\quad \forall \phi\in C_c^{\infty}(\Omega) \] where $\Omega\subset
\mathbb{R}^{n}$ is an open set.
  Here we assume $K$ is bounded, nonnegative and continuous in the first entry
-- and ellipticity is ensured by assuming that $K$ is strictly positive in a
cone.
  The setup is chosen so that it is applicable for nonlocal equations on
manifolds, but the structure of the equation is general enough that it also
applies to the certain fractional $p$-Laplace equations around points where $u
\in C^1$ and $|\nabla u| \neq 0$.
