  In this paper, we establish a new improved Sobolev inequality based on a
weighted Morrey space. To be precise, there exists $C=C(n,m,s,\alpha)>0$ such
that for any $u,v \in {\dot{H}}^s(\mathbb{R}^{n})$ and for any $\theta \in
(\bar{\theta},1)$, it holds that \begin{equation} \label{eq0.3} \Big( \int_{
\mathbb{R}^{n} } \frac{ |(uv)(y)|^{\frac{2^*_{s}(\alpha)}{2} } } {
|y'|^{\alpha} } dy \Big)^{ \frac{1}{ 2^*_{s} (\alpha) }} \leq C
||u||_{{\dot{H}}^s(\mathbb{R}^{n})}^{\frac{\theta}{2}}
  ||v||_{{\dot{H}}^s(\mathbb{R}^{n})}^{\frac{\theta}{2}}
||(uv)||^{\frac{1-\theta}{2}}_{ L^{1,n-2s+r}(\mathbb{R}^{n},|y'|^{-r}) },
\end{equation} where $s \!\in\! (0,1)$, $0\!<\!\alpha\!<\!2s\!<\!n$,
$2s\!<\!m\!<\!n$, $\bar{\theta}=\max \{ \frac{2}{2^*_{s}(\alpha)},
1-\frac{\alpha}{s}\cdot\frac{1}{2^*_{s}(\alpha)},
\frac{2^*_{s}(\alpha)-\frac{\alpha}{s}}{2^*_{s}(\alpha)-\frac{2\alpha}{m}} \}$,
$r=\frac{2\alpha}{ 2^*_{s}(\alpha) }$ and $y\!=\!(y',y'') \in \mathbb{R}^{m}
\times \mathbb{R}^{n-m}$.
  By using mountain pass lemma and (\ref{eq0.3}), we obtain a nontrivial weak
solution to a doubly critical system involving fractional Laplacian in
$\mathbb{R}^{n}$ with partial weight in a direct way. Furthermore, we extend
inequality (\ref{eq0.3}) to more general forms on purpose of studying some
general systems with partial weight, involving p-Laplacian especially.
