  In this paper, we investigate the following Caffarelli-Kohn-Nirenberg
inequality: \begin{equation*}
  \left(\int_{\mathbb{R}^n}|x|^{-pa}|\nabla u|^pdx\right)^{\frac{1}{p}}\geq
S(p,a,b)\left(\int_{\mathbb{R}^n}|x|^{-qb}|u|^qdx\right)^{\frac{1}{q}},\quad\forall\;
u\in D_a^p(\mathbb{R}^n), \end{equation*} where $S(p,a,b)$ is the sharp
constant and $a,b,p,q$ satisfy the relations: \begin{equation*}
  0\leq a<\frac{n-p}{p},\quad a\leq b<a+1,\quad 1<p<n,\quad
q=\frac{np}{n-p(1+a-b)}. \end{equation*} Our main results involve establishing
gradient stability within both the functional and the critical settings and
deriving some qualitative properties for the stability constant. In the
functional setting, the main method we apply is a simple but clever
transformation inspired by Horiuchi \cite{Hor}, which enables us to reduce the
inequality to some well-studied ones. Based on the gradient stability, we
establish several refined Sobolev-type embeddings involving weak Lebesgue norms
for functions supported in general domains. In the critical point setting, we
use some careful expansion techniques motivated by Figalli and Neumayer
\cite{Fig} to handle the nonlinearity appeared in the Euler-Lagrange equations.
We believe that the ideas presented in this paper can be applied to treat other
weighted Sobolev-type inequalities.
