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

We consider non-local elliptic operators with kernel K (y) = a (y)/vertical bar y vertical bar(d+sigma), where 0 < sigma < 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 Hp to L. and the unique strong solvability of the corresponding non-local elliptic equations in L-p spaces. As a byproduct, we also obtain interior L p 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. (C) 2011 Elsevier Inc. All rights reserved.