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

The fractional Caffarelli-Kohn-Nirenberg inequality states that ∫Rⁿ∫Rⁿ (u(x)-u(y))²|x|^α |x-y|ⁿ⁺²ˢ |y|^α d x \, d y ≥ Λn, s, p, α,β \|u |x|-β\|Lᵖ², for 0<s<min\1, n/2\, 2<p<2^*ₛ, and α,β∈ R so that β-α = s - n(12 - 1p) and -2s < α < n-2s/2. Continuing the program started in Ao et al. (2022), we establish the non-degeneracy and sharp quantitative stability of minimizers for α≥ 0. Furthermore, we show that minimizers remain symmetric when α<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.