On $L_p$-estimates for a class of non-local elliptic equations

We consider non-local elliptic operators with kernel $K(y)=a(y)\mathrm{/}\mathrm{\mid }y{\mathrm{\mid }}^{d+\sigma }$, 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 $H_p^{\sigma }$ 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 $L$.