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|^{-α} \ast u^{2_α^{\ast}}\big)u^{2_α^{\ast}} dx\right)^{\frac{1}{2_α^{\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 $\mathrm{\parallel }u{\mathrm{\parallel }}_{{\mathcal{D}}^{1,2}({\mathbb{R}}^N)}=\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2({\mathbb{R}}^N)}$. 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 $0<\alpha <N$.