Admissible function spaces for weighted Sobolev inequalities

TL;DR

This paper characterizes admissible weighted function spaces for Hardy-Sobolev inequalities in product domains, identifying optimal conditions on weights for the inequalities to hold and conditions for attaining the best constant.

Contribution

It provides a comprehensive classification of function spaces for weights in Hardy-Sobolev inequalities on product domains, including conditions for existence of extremals.

Findings

Identified pairs of Lorentz, Lorentz-Zygmund, and weighted Lebesgue spaces for weights.

Established conditions under which the best constant is attained in the Beppo-Levi space.

Derived various admissible weight conditions depending on domain and parameters.

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