  Suppose that $d\ge 1$ and $0<\beta<\alpha<2$. We establish the existence and
uniqueness of the fundamental solution $q^b(t, x, y)$ to a class of (possibly
nonsymmetric) non-local operators $L^b=\Delta^{\alpha/2}+S^b$, where
  $$
  S^bf(x):=A(d, -\beta) \int_{R^d} ( f(x+z)-f(x)- \nabla f(x) \cdot z
1_{\{|z|\leq 1\}} ) \frac{b(x, z)}{|z|^{d+\beta}}dz $$ and $b(x, z)$ is a
bounded measurable function on $R^d\times R^d$ with $b(x, z)=b(x, -z)$ for $x,
z\in R^d$. Here $A(d, -\beta)$ is a normalizing constant so that
$S^b=\Delta^{\beta/2}$ when $b(x, z)\equiv 1$. We show that if $b(x, z) \geq
-\frac{{\cal A}(d, -\alpha)}{A(d, -\beta)}\, |z|^{\beta -\alpha}$, then $q^b(t,
x, y)$ is a strictly positive continuous function and it uniquely determines a
conservative Feller process $X^b$, which has strong Feller property. The Feller
process $X^b$ is the unique solution to the martingale problem of $(L^b, {\cal
S} (R^d))$, where ${\cal S}(R^d)$ denotes the space of tempered functions on
$R^d$. Furthermore, sharp two-sided estimates on $q^b(t, x, y)$ are derived. In
stark contrast with the gradient perturbations, these estimates exhibit
different behaviors for different types of $b(x, z)$. The model considered in
this paper contains the following as a special case. Let $Y$ and $Z$ be
(rotationally) symmetric $\alpha$-stable process and symmetric $\beta$-stable
processes on $R^d$, respectively, that are independent to each other. Solution
to stochastic differential equations $dX_t=dY_t + c(X_{t-})dZ_t$ has
infinitesimal generator $L^b$ with $b(x, z)=| c(x)|^\beta$.
