Hardy-Sobolev inequalities involving mixed radially and cylindrically symmetric weights

TL;DR

This work develops sharp weighted Hardy–Sobolev inequalities on $\mathbb{R}^d$ with anisotropic weights $|y|^a|z|^{-b}$ and analyzes the associated best constants $S_{a,b,\gamma}(q)$. The authors establish necessary and sufficient conditions for positivity, use variational methods and concentration-compactness (including Ekeland’s principle and Sacks–Uhlenbeck-type arguments) to prove existence of extremals in broad regimes, and perform a detailed study of critical ($q=p^*$) and bottom ($\gamma=b$) limiting cases. They introduce a cylindrical-to-mixed-weights reduction via the parameter $t=a-b+\gamma$ and connect the results to Maz'ya constants, with a Hilbertian treatment of the bottom case $p=2$ clarifying existence/nonexistence thresholds. The paper thereby clarifies how mixed cylindrical and spherical weights influence sharp constants, extremals, and symmetry aspects, enriching the classical Il’in–Caffarelli–Kohn–Nirenberg framework and Maz'ya-type inequalities.

Abstract

We deal with weighted Hardy-Sobolev type inequalities for functions on $\mathbb{R}^d$, $d\geq 2$. The weights involved are anisotropic, given by products of powers of the distance to the origin and to a nontrivial subspace. We establish necessary and sufficient conditions for validity of these inequalities, and investigate the existence/nonexistence of extremal functions.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Bottom case
• Theorem 1.4
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Theorem 4.1