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