Quantitative stability of a nonlocal Sobolev inequality

In this paper, we study the quantitative stability of the nonlocal Soblev inequality \begin{equation*} S_{HL}\left(\int_{\mathbb{R}^N}\big(|x|^{-μ} \ast |u|^{2_μ^{\ast}}\big)|u|^{2_μ^{\ast}} dx\right)^{\frac{1}{2_μ^{\ast}}}\leq\int_{\mathbb{R}^N}|\nabla u|^2 dx , \quad \forall~u\in \mathcal{D}^{1,2}(\mathbb{R}^N), \end{equation*} where $2_μ^{\ast}=\frac{2N-μ}{N-2}$ and $S_{HL}$ is a positive constant depending only on $N$ and $μ$. For $N\geq3$, and $0<μ<N$, it is well-known that, up to tra...