  Let $L$ be a L\'evy-type generator whose L\'evy measure is controlled from
below by that of a non-degenerate $\alpha$-stable ($0<\alpha<2$) process. In
this paper, we study the martingale problem for the operator
$\mathcal{L}_{t}=L+K_{t}$, with $K_{t}$ being a time-dependent non-local
operator defined by \[
K_{t}f(x):=\int_{\mathbb{R}^{d}\backslash\{0\}}[f(x+y)-f(x)-\mathbf{1}_{\alpha>1}\mathbf{1}_{\{|y|\le1\}}y\cdot\nabla
f(x)]M(t,x,dy), \] where $M(t,x,\cdot)$ is a L\'evy measure on
$\mathbb{R}^{d}\backslash\{0\}$ for each $(t,x)\in \mathbb{R}_+ \times
\mathbb{R}^{d}$. We show that if \[
\sup_{t\geq0,x\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}\backslash\{0\}}1\wedge|y|^{\beta}M(t,x,dy)<\infty
\] for some $0<\beta<\alpha$, then the martingale problem for $\mathcal{L}_{t}$
is well-posed.
