  In this paper, we consider the Caffarelli-Kohn-Nirenberg (CKN) inequality:
\begin{eqnarray*} \bigg(\int_{{\mathbb
R}^N}|x|^{-b(p+1)}|u|^{p+1}dx\bigg)^{\frac{2}{p+1}}\leq C_{a,b,N}\int_{{\mathbb
R}^N}|x|^{-2a}|\nabla u|^2dx \end{eqnarray*} where $N\geq3$, $a<\frac{N-2}{2}$,
$a\leq b\leq a+1$ and $p=\frac{N+2(1+a-b)}{N-2(1+a-b)}$. It is well-known that
up to dilations $\tau^{\frac{N-2}{2}-a}u(\tau x)$ and scalar multiplications
$Cu(x)$, the CKN inequality has a unique extremal function $W(x)$ which is
positive and radially symmetric in the parameter region $b_{FS}(a)\leq b<a+1$
with $a<0$ and $a\leq b<a+1$ with $a\geq0$ and $a+b>0$, where $b_{FS}(a)$ is
the Felli-Schneider curve. We prove that in the above parameter region the
following stabilities hold: \begin{enumerate} \item[$(1)$] \quad stability of
CKN inequality in the functional inequality setting $$dist_{D^{1,2}_{a}}^2(u,
\mathcal{Z})\lesssim\|u\|^2_{D^{1,2}_a({\mathbb
R}^N)}-C_{a,b,N}^{-1}\|u\|^2_{L^{p+1}(|x|^{-b(p+1)},{\mathbb R}^N)}$$ where
$\mathcal{Z}= \{ c W_\tau\mid c\in\bbr\backslash\{0\}, \tau>0\}$;
\item[$(2)$]\quad stability of CKN inequality in the critical point setting (in
the class of nonnegative functions) \begin{eqnarray*} dist_{D_a^{1,2}}(u,
\mathcal{Z}_0^\nu)\lesssim\left\{\aligned &\Gamma(u),\quad p>2\text{ or
}\nu=1,\\ &\Gamma(u)|\log\Gamma(u)|^{\frac12},\quad p=2\text{ and }\nu\geq2,\\
&\Gamma(u)^{\frac{p}{2}},\quad 1<p<2\text{ and }\nu\geq2, \endaligned\right.
\end{eqnarray*} where $\Gamma (u)=\|div(|x|^{-a}\nabla
u)+|x|^{-b(p+1)}|u|^{p-1}u\|_{(D^{1,2}_a)^{'}}$ and
$$\mathcal{Z}_0^\nu=\{(W_{\tau_1},W_{\tau_2},\cdots,W_{\tau_\nu})\mid
\tau_i>0\}.$$
