  Let $n \in \mathbb N$, let $\zeta_{n,1},...,\zeta_{n,n}$ be a sequence of
independent random variables with $\mathbb E \zeta_{n,i}=0$ and $\mathbb E
|\zeta_{n,i}|<\infty$ for each $i$, and let $\mu$ be an $\alpha$-stable
distribution having characteristic function $e^{-|\lambda|^{\alpha}}$ with
$\alpha\in (1,2)$. Denote $S_{n}=\zeta_{n,1}+...+\zeta_{n,n}$ and its
distribution by $\mathcal L(S_n)$, we bound the Wasserstein distance of
$\mathcal L(S_{n})$ and $\mu$ essentially by an $L^{1}$ discrepancy between two
kernels, this bound can be interpreted as a generalization of the Stein
discrepancy (in $L^{2}$ sense) introduced by Ledoux, Nourdin and Peccati. More
precisely, we prove the following inequality: \begin{equation} \begin{split}
d_W\left(\mathcal L (S_n), \mu\right) \ \le C \left[\sum_{i=1}^n\int_{-N}^N
\left|\frac{\mathcal K_\alpha(t,N)}n -\frac{ K_i(t,N)}{\alpha}\right| d t \ +\
\mathcal R_{N,n}\right], \end{split} \end{equation} where $d_{W}$ is the
Wasserstein distance of probability measures, $\mathcal K_\alpha(t,N)$ is the
kernel of a decomposition of the fractional Laplacian $\Delta^{\frac \alpha2}$,
$ K_i(t,N)$ is a kernel introduced by Chen, Goldstein and Shao with a
truncation which can be interpreted as an $L^1$ Stein kernel, and $\mathcal
R_{N,n}$ is a small remainder. The integral term $$\sum_{i=1}^n\int_{-N}^N
\left|\frac{\mathcal K_\alpha(t,N)}n -\frac{ K_i(t,N)}{\alpha}\right| d t$$ can
be interpreted as an $L^{1}$ Stein discrepancy.
  As an application, we prove a general theorem of stable law convergence rate
when $\zeta_{n,i}$ are i.i.d. and the distribution falls in the normal domain
of attraction of $\mu$. We also study four examples with comparing our
convergence rates and those known for these examples, among which the
distribution in the second example is not in the normal domain of attraction of
$\mu$.
