On Lp-estimates for a class of non-local elliptic equations

We consider non-local elliptic operators with kernel K(y)=a(y)/|y|d+σ, where 0<σ<2 is a constant and a is a bounded measurable function. By using a purely analytic method, we prove the continuity of the non-local operator L from the Bessel potential space to Lp, and the unique strong solvability of the corresponding non-local elliptic equations in Lp spaces. As a byproduct, we also obtain interior Lp-estimates. The novelty of our results is that the function a is not necessarily to be homogeneous, regular, or symmetric. An application of our result is the uniqueness for the martingale problem associated to the operator L.