Admissible function spaces for weighted Sobolev inequalities

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ k,N\in \mathbb{N} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ 1\le k\le N $\end{document}</tex-math></inline-formula> and let <inline-formula><tex-math id="M3">\begin{document}$ \Omega = \Omega_1 \times \Omega_2 $\end{document}</tex-math></inline-formula> be an open set in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^k \times \mathbb{R}^{N-k} $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M5">\begin{document}$ p\in (1,\infty) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ q \in (0,\infty), $\end{document}</tex-math></inline-formula> we consider the following weighted Sobolev type inequality:</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE1">\begin{document}$\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}$\end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>for some <inline-formula><tex-math id="M7">\begin{document}$ C&gt;0 $\end{document}</tex-math></inline-formula>. Depending on the values of <inline-formula><tex-math id="M8">\begin{document}$ N,k,p,q $\end{document}</tex-math></inline-formula> we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for <inline-formula><tex-math id="M9">\begin{document}$ (g_1, g_2) $\end{document}</tex-math></inline-formula> so that (0.1) holds. Furthermore, we give a sufficient condition on <inline-formula><tex-math id="M10">\begin{document}$ g_1,g_2 $\end{document}</tex-math></inline-formula> so that the best constant in (0.1) is attained in the Beppo-Levi space <inline-formula><tex-math id="M11">\begin{document}$ \mathcal{D}^{1,p}_0(\Omega) $\end{document}</tex-math></inline-formula>-the completion of <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{C}^1_c(\Omega) $\end{document}</tex-math></inline-formula> with respect to <inline-formula><tex-math id="M13">\begin{document}$\|\nabla u\|_{L p(\Omega)}$\end{document}</tex-math></inline-formula>.</p>