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 Hσ p to Lp, and the unique strong solvability of the corresponding non-local elliptic equations in Lp spaces. As a byproduct, we also obtain interior Lpestimates. 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. © 2011 Elsevier Inc. All rights reserved