Gradient stability of Caffarelli-Kohn-Nirenberg inequality involving weighted p-Laplace

The best constant and extremal functions are well known of the following Caffarelli-Kohn-Nirenberg inequality $${\int }_{{\mathbb{R}}^N}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p\frac{\mathrm{d}x}{\mathrm{\mid }x{\mathrm{\mid }}^{\mu }}\ge \mathcal{S}{\left({\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^r\frac{\mathrm{d}x}{\mathrm{\mid }x{\mathrm{\mid }}^s}\right)}^{\frac{p}{r}},forallu\in C_c^{\mathrm{\infty }}({\mathbb{R}}^N),$$ where $1<p<p+\mu <N$, $\frac{\mu }{p}\le \frac{s}{r}<\frac{\mu }{p}+1$, $r=\frac{p(N-s)}{N-p-\mu }$. 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.