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

The fractional Caffarelli-Kohn-Nirenberg inequality states that ₑ䂞ₑ䂞 (u (x) -u (y) ) ²|x|^ |x-y|^{n+2s |y|^} d x \, d y ₍, ₒ, ₏, , \|u |x|^-\|₋㵵², for 0<s<\1, n/2\, 2<p<2^*ₛ, and, R so that - = s - n (12 - 1p) and -2s < < n-2s2. Continuing the program started in Ao et al. (2022), we establish the non-degeneracy and sharp quantitative stability of minimizers for 0. Furthermore, we show that minimizers remain symmetric when <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 Rⁿ, 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.