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

The fractional Caffarelli-Kohn-Nirenberg inequality states that $$ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(u(x)-u(y))^2}{|x|^\alpha |x-y|^{n+2s} |y|^\alpha} \mathrm{d} x \, \mathrm{d} y \geq \Lambda_{n, s, p, \alpha,\beta} \|u |x|^{-\beta}\|_{L^p}^2, $$ for $0<s<\min\{1, n/2\}$, $2<p<2^*_s$, and $\alpha,\beta\in\mathbb R$ so that $\beta-\alpha = s - n\big(\frac12 - \frac1p\big)$ 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.