  Dolbeault, Esteban and Loss [Invent. Math., 2016] obtained an optimal
rigidity result, that is, when $a<0$ and $b_{\mathrm{FS}}(a)\leq b<a+1$ the
extremal function for best constant $\mathcal{S}_{a,b}>0$ of the following
Caffarelli-Kohn-Nirenberg inequality is symmetry, \[
\mathcal{S}_{a,b}\left(\int_{\mathbb{R}^2}|x|^{-qb}|u|^q
\mathrm{d}x\right)^{\frac{2}{q}}
  \leq \int_{\mathbb{R}^2}|x|^{-2a}|\nabla u|^2 \mathrm{d}x, \quad \mbox{for
all}\quad u\in C^\infty_0(\mathbb{R}^2), \] where
$b_{\mathrm{FS}}(a):=a-\frac{a}{\sqrt{a^2+1}}$, $q=\frac{2}{b-a}$. An important
task is investigating the stability of extremal functions set $\mathcal{M}$ for
this inequality. Firstly, we classify all solutions of the linearized problem
related to the extremals which fills the work of Felli and Schneider [J. Diff.
Equ., 2003]. When $b_{\mathrm{FS}}(a)< b<a+1$, we investigate the stability of
previous inequality by using spectral estimate combined with a compactness
argument that
  \begin{align*}
  \int_{\mathbb{R}^2}|x|^{-2a}|\nabla u|^2 \mathrm{d}x
  -\mathcal{S}_{a,b}\left(\int_{\mathbb{R}^2}|x|^{-qb}|u|^q
\mathrm{d}x\right)^{\frac{2}{q}}
  \geq \mathcal{B}
  \mathrm{dist}(u,\mathcal{M})^2,\quad \mbox{for all}\quad u\in
C^\infty_0(\mathbb{R}^2),
  \end{align*}
  for some $\mathcal{B}>0$, however it is false when $b=b_{\mathrm{FS}}(a)$,
which extends the work of Wei and Wu [Math. Ann., 2022] to $\mathbb{R}^2$.
Furthermore, we obtain the existence of minimizers for $\mathcal{B}$ which
extends the recent work of K\"{o}nig [J. Eur. Math. Soc., to appear].
