An L-p-theory for a class of non-local elliptic equations related to nonsymmetric measurable kernels

Abstract

We study the integro-differential operators L with kernels K(y) = a(y) J(y), where J(y) is rotationally invariant and J(y)dy is a Levy measure on R-d (i.e. integral(Rd) (1 Lambda vertical bar y vertical bar(2)) J(y)dy < infinity) and a(y) is an only measurable function with positive lower and upper bounds. Under few additional conditions on J(y), we prove the unique solvability of the equation Lu - lambda u = f in L-p-spaces and present some L-p-estimates of the solutions. (C) 2015 Elsevier Inc. All rights reserved.