Remainder terms of a nonlocal Sobolev inequality1

In this note we study a nonlocal version of the Sobolev inequality \begin{equation*} \int_{\mathbb{R}^N}|\nabla u|^2 dx \geq S_{HLS}\left(\int_{\mathbb{R}^N}\big(|x|^{-\alpha} \ast u^{2_\alpha^{\ast}}\big)u^{2_\alpha^{\ast}} dx\right)^{\frac{1}{2_\alpha^{\ast}}}, \quad \forall u\in \mathcal{D}^{1,2}(\mathbb{R}^N), \end{equation*} where $S_{HLS}$ is the best constant, $\ast$ denotes the standard convolution and $\mathcal{D}^{1,2}(\mathbb{R}^N)$ denotes the classical Sobolev space with respect to the norm $\|u\|_{\mathcal{D}^{1,2}(\mathbb{R}^N)}=\|\nabla u\|_{L^2(\mathbb{R}^N)}$. By using the nondegeneracy property of the extremal functions, we prove that the existence of the gradient type remainder term and a reminder term in the weak $L^{\frac{N}{N-2}}$-norm of above inequality for all $0<\alpha<N$.