Non-degeneracy, stability and symmetry for the fractional Caffarelli-Kohn-Nirenberg inequality

The fractional Caffarelli-Kohn-Nirenberg inequality states that $${\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\frac{(u(x)-u(y){)}^2}{\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }\mathrm{\mid }x-y{\mathrm{\mid }}^{n+2s}\mathrm{\mid }y{\mathrm{\mid }}^{\alpha }}\mathrm{d}x\mathrm{d}y\ge {\mathrm{\Lambda }}_{n,s,p,\alpha ,\beta }\mathrm{\parallel }u\mathrm{\mid }x{\mathrm{\mid }}^{-\beta }{\mathrm{\parallel }}_{L^p}^2,$$ for $0<s<\min \{1,n\mathrm{/}2\}$, $2<p<2_s^{\ast }$, and $\alpha ,\beta \in \mathbb{R}$ so that $\beta -\alpha =s-n(\frac{1}{2}-\frac{1}{p})$ and $-2s<\alpha <\frac{n-2s}{2}$. Continuing the program started in Ao et al. (2022), we establish the non-degeneracy and sharp quantitative stability of minimizers for $\alpha \ge 0$. Furthermore, we show that minimizers remain symmetric when $\alpha <0$ for $p$ very close to $2$. Our results fit into the more ambitious goal of understanding the symmetry region of the minimizers of the fractional Caffarelli-Kohn-Nirenberg inequality. We develop a general framework to deal with fractional inequalities in ${\mathbb{R}}^n$, striving to provide statements with a minimal set of assumptions. Along the way, we discover a Hardy-type inequality for a general class of radial weights that might be of independent interest.