  Let $k,N \in \mathbb{N}$ with $1\le k\le N$ and let $\Omega=\Omega_1 \times
\Omega_2$ be an open set in $\mathbb{R}^k \times \mathbb{R}^{N-k}$. For $p\in
(1,\infty)$ and $q \in (0,\infty),$ we consider the following Hardy-Sobolev
type inequality:
  \begin{align}
  \int_{\Omega} |g_1(y)g_2(z)| |u(y,z)|^q \, dy \, dz \leq C \left(
\int_{\Omega} | \nabla u(y,z) |^p \, dy \, dz \right)^{\frac{q}{p}}, \quad
\forall \, u \in \mathcal{C}^1_c(\Omega),
  \end{align}
  for some $C>0$. Depending on the values of $N,k,p,q,$ we have identified
various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue
spaces for $(g_1, g_2)$ so that the above inequality holds. Furthermore, we
give a sufficient condition on $g_1,g_2$ so that the best constant in the above
inequality is attained in the Beppo-Levi space
$\mathcal{D}^{1,p}_0(\Omega)$-the completion of $\mathcal{C}^1_c(\Omega)$ with
respect to $\|\nabla u\|_{L^p(\Omega)}$.
