On sharp anisotropic Hardy inequalities

TL;DR

The paper advances sharp anisotropic Hardy-type inequalities with weights $|x'|^{\alpha}|x|^{\beta}$ by determining the best constant for the case $p=2$ (and $\beta\ge0$ for general $p$), using a divergence-identity framework with $f=|x'|^{\theta}|x|^{\lambda}$ and a strategic decomposition of $-\mathrm{div}(V\nabla f)/f$. It provides an explicit best constant $C_{n,\alpha,\beta}=\frac{(n-1+2\alpha)^2-[\sqrt{\max(K,1)}-1]^2}{4}$ where $K=-4\beta(n+2\alpha+\beta)$, and proves sharpness via singular test functions, with special cases yielding anisotropic Leray-type inequalities. The work extends to general $p>1$, giving exact constants $C_{n,\alpha,\beta}=((n-1+p\alpha)/p)^p$ for $\beta\ge0$ and a lower bound for $\beta<0$, all derived through the same $f$-based optimization and density arguments. Additionally, it derives an explicit anisotropic $L^p$-Caffarelli–Kohn–Nirenberg inequality and discusses sharpness and a DFLL-type generalization, thereby linking to Musina–Nazarov's framework and enriching the theory of weighted Sobolev embeddings in anisotropic settings.

Abstract

Recently, Yanyan Li and Xukai Yan showed the following interesting Hardy inequalities with anisotropic weights: Let $n\geq 2$, $p \geq 1$, $pα> 1-n$, $p(α+ β)> -n$, then there exists $C > 0$ such that $$\||x|^β|x'|^{α+1} \nabla u\|_{L^p(\mathbb{R}^n)} \geq C\||x|^β|x'|^αu\|_{L^p(\mathbb{R}^n)}, \quad \forall\; u\in C_c^1(\mathbb{R}^n).$$ Here $x' = (x_1,\ldots, x_{n-1}, 0)$ for $x = (x_i) \in \mathbb{R}^n$. In this note, we will determine the best constant for the above estimate when $p=2$ or $β\geq 0$. Moreover, as refinement for very special case of Li-Yan's result in Adv. Math. 2023, we provide explicit estimate for the anisotropic $L^p$-Caffarelli-Kohn-Nirenberg inequality.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Lemma 2.1
• Remark 2.2
• Remark 2.3
• Remark 4.1
• Theorem 5.1