Admissible function spaces for weighted Sobolev inequalities

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 \(\|\nabla u\|_{L^p(\Omega)}\).