Admissible function spaces for weighted Sobolev inequalities

Abstract

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_Ω |g_1(y)g_2(z)| |u(y,z)|^q \, dy \, dz \leq C \left( \int_Ω | \nabla u(y,z) |^p \, dy \, dz \right)^{\frac{q}{p}}, \quad \forall \, u \in \mathcal{C}^1_c(Ω), \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 ${\mathcal{D}}_0^{1,p}(\Omega )$-the completion of ${\mathcal{C}}_c^1(\Omega )$ with respect to $\Vert \nabla u{\Vert }_{L^p(\Omega )}$.

38 pages

References in corpus (2)

• In between the inequalities of Sobolev and Hardy
• On the Generalized Hardy-Rellich Inequalities