Stability of Hardy Littlewood Sobolev Inequality under Bubbling

In this note we will generalize the results deduced in arXiv:1905.08203 and arXiv:2103.15360 to fractional Sobolev spaces. In particular we will show that for $s\in (0,1)$, $n>2s$ and $\nu\in \mathbb{N}$ there exists constants $\delta = \delta(n,s,\nu)>0$ and $C=C(n,s,\nu)>0$ such that for any function $u\in \dot{H}^s(\mathbb{R}^n)$ satisfying, \begin{align*} \left\| u-\sum_{i=1}^{\nu} \tilde{U}_{i}\right\|_{\dot{H}^s} \leq \delta \end{align*} where $\tilde{U}_{1}, \tilde{U}_{2},\cdots \tilde{U}_{\nu}$ is a $\delta-$interacting family of Talenti bubbles, there exists a family of Talenti bubbles $U_{1}, U_{2},\cdots U_{\nu}$ such that \begin{align*} \left\| u-\sum_{i=1}^{\nu} U_{i}\right\|_{\dot{H}^s} \leq C\left\{\begin{array}{ll} \Gamma & \text { if } 2s < n < 6s,\\ \Gamma|\log \Gamma|^{\frac{1}{2}} & \text { if } n=6s, \\ \Gamma^{\frac{p}{2}} & \text { if } n > 6s \end{array}\right. \end{align*} for $\Gamma=\left\|\Delta u+u|u|^{p-1}\right\|_{H^{-s}}$ and $p=2^*-1=\frac{n+2s}{n-2s}.$