Stability of Caffarelli-Kohn-Nirenberg inequality

In this paper, we consider the Caffarelli-Kohn-Nirenberg (CKN) inequality: begin{eqnarray*} bigg(int_{{mathbb R}^N}|x|^{-b(p+1)}|u|^{p+1}dxbigg)^{frac{2}{p+1}}leq C_{a,b,N}int_{{mathbb R}^N}|x|^{-2a}| abla u|^2dx end{eqnarray*} where $Ngeq3$, $a<frac{N-2}{2}$, $aleq bleq a+1$ and $p=frac{N+2(1+a-b)}{N-2(1+a-b)}$. It is well-known that up to dilations $tau^{frac{N-2}{2}-a}u(tau x)$ and scalar multiplications $Cu(x)$, the CKN inequality has a unique extremal function $W(x)$ which is positive and radially symmetric in the parameter region $b_{FS}(a)leq b<a+1$ with $a<0$ and $aleq b<a+1$ with $ageq0$ and $a+b>0$, where $b_{FS}(a)$ is the Felli-Schneider curve. We prove that in the above parameter region the following stabilities hold: begin{enumerate} item[$(1)$] quad stability of CKN inequality in the functional inequality setting $$dist_{D^{1,2}_{a}}^2(u, mathcal{Z})lesssim|u|^2_{D^{1,2}_a({mathbb R}^N)}-C_{a,b,N}^{-1}|u|^2_{L^{p+1}(|x|^{-b(p+1)},{mathbb R}^N)}$$ where $mathcal{Z}= { c W_taumid cinbbrbackslash{0}, tau>0}$; item[$(2)$]quad stability of CKN inequality in the critical point setting (in the class of nonnegative functions) begin{eqnarray*} dist_{D_a^{1,2}}(u, mathcal{Z}_0^ u)lesssimleft{aligned &Gamma(u),quad p>2text{ or } u=1, &Gamma(u)|logGamma(u)|^{frac12},quad p=2text{ and } ugeq2, &Gamma(u)^{frac{p}{2}},quad 1<p<2text{ and } ugeq2, endalignedright. end{eqnarray*} where $Gamma (u)=|div(|x|^{-a} abla u)+|x|^{-b(p+1)}|u|^{p-1}u|_{(D^{1,2}_a)^{}}$ and $$mathcal{Z}_0^ u={(W_{tau_1},W_{tau_2},cdots,W_{tau_ u})mid tau_i>0}.$$