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

The fractional Caffarelli-Kohn-Nirenberg inequality states that ˆRn ˆRn (u(x) − u(y)) 2 |x| α |x − y| n+2s |y| α dx dy ≥ n,s,p,α,β u|x| −β 2 L p , for 0 < s < min{1, n/2}, 2 < p < 2 * s , and α, β ∈ R so that β − α = s − n 1 2 − 1 p and −2s < α < n−2s 2. Continuing the program started in Ao et al. [1], 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 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.