  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 $0<\alpha<N$.
