  The best constant and extremal functions are well known of the following
Caffarelli-Kohn-Nirenberg inequality \[ \int_{\mathbb{R}^N}|\nabla
u|^p\frac{\mathrm{d}x}{|x|^{\mu}}\geq \mathcal{S}
\left(\int_{\mathbb{R}^N}|u|^r\frac{\mathrm{d}x}{|x|^s} \right)^{\frac{p}{r}},
\quad \mbox{for all}\quad u\in C^\infty_c(\mathbb{R}^N), \] where
$1<p<p+\mu<N$, $\frac{\mu}{p}\leq \frac{s}{r}<\frac{\mu}{p}+1$,
$r=\frac{p(N-s)}{N-p-\mu}$. An important task is investigating the stability of
extremals for this inequality. Firstly, we give the classification to the
linearized problem related to the extremals which shows the extremals are
non-degenerate. Then we investigate the gradient type remainder term of
previous inequality by using spectral estimate combined with a compactness
argument which partially extends the work of Wei and Wu [Math. Ann., 2022] to a
general $p$-Laplace case, and also the work of Figalli and Zhang [Duke Math.
J., 2022] to a weighted case.
