On [formula omitted]-estimates for a class of non-local elliptic equations

Abstract

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 H p σ to L p , 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.