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 $J(y)dy$ is a Lévy measure on $\bR^d$ (i.e. $\int_{\bR^d}(1\wedge |y|^2)J(y)dy<\infty$) 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 $Lu-\hat{I}»u=f$ in $L_p$-spaces and present some $L_p$-estimates of the solutions.