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

The fractional Caffarelli-Kohn-Nirenberg inequality states that integral(Rn)integral(Rn)(u(x)-u(y))(2)|x|(alpha)|x-y|(n+2s)|y|(alpha)dxdy >= Lambda(n,s,p,alpha,beta)& Vert;u|x|(-beta)& Vert;(2)(Lp), for 0 < s= 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 & Ropf;(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.