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

We study the integro-differential operators $L$ with kernels $K(y) = a(y) J(y)$,where$J(y)dy$isaL\acute{e}vymeasureon$\bR^d$(i\mathrm{.}e\mathrm{.}$\int_{\bR^d}(1\wedge |y|^2)J(y)dy<\infty$)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 $Lu-\lambda u=f$ in $L_p$-spaces and present some $L_p$-estimates of the solutions.