Admissible function spaces for weighted Sobolev inequalities

Let \begin{document}$ k,N\in \mathbb{N} $\end{document} with \begin{document}$ 1\le k\le N $\end{document} and let \begin{document}$ \Omega = \Omega_1 \times \Omega_2 $\end{document} be an open set in \begin{document}$ \mathbb{R}^k \times \mathbb{R}^{N-k} $\end{document} . For \begin{document}$ p\in (1,\infty) $\end{document} and \begin{document}$ q \in (0,\infty), $\end{document} we consider the following weighted Sobolev type inequality: \begin{document}$\begin{align} \int_{\Omega} |g_1(y)||g_2(z)| |u(y,z)|^q \, {\rm d}y {\rm d}z \leq C \left( \int_{\Omega} | \nabla u(y,z) |^p \, {\rm d}y {\rm d}z \right)^{\frac{q}{p}}, \quad \forall \, u \in \mathcal{C}^1_c(\Omega), \\(0.1)\end{align}$\end{document} for some \begin{document}$ C>0 $\end{document} . Depending on the values of \begin{document}$ N,k,p,q $\end{document} we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for \begin{document}$ (g_1, g_2) $\end{document} so that (0.1) holds. Furthermore, we give a sufficient condition on \begin{document}$ g_1,g_2 $\end{document} so that the best constant in (0.1) is attained in the Beppo-Levi space \begin{document}$ \mathcal{D}^{1,p}_0(\Omega) $\end{document} -the completion of \begin{document}$ \mathcal{C}^1_c(\Omega) $\end{document} with respect to \begin{document}$\|\nabla u\|_{L p(\Omega)}$\end{document} .