On sharp anisotropic Hardy inequalities

TL;DR

This paper determines the optimal constants in certain anisotropic Hardy inequalities for specific parameters and provides explicit estimates for related anisotropic Caffarelli-Kohn-Nirenberg inequalities, advancing understanding of anisotropic functional inequalities.

Contribution

It identifies the best constants for anisotropic Hardy inequalities when p=2 or 20, and offers explicit estimates for anisotropic Caffarelli-Kohn-Nirenberg inequalities, refining previous results.

Findings

Optimal constants determined for p=2 and 20 cases.

Explicit estimates provided for anisotropic Caffarelli-Kohn-Nirenberg inequalities.

Enhanced understanding of anisotropic Hardy inequalities and their sharpness.

Abstract

Recently, Yanyan Li and Xukai Yan showed the following interesting Hardy inequalities with anisotropic weights: Let $n\geq 2$, $p \geq 1$, $p\alpha > 1-n$, $p(\alpha + \beta)> -n$, then there exists $C > 0$ such that $$\||x|^{\beta}|x'|^{\alpha+1} \nabla u\|_{L^p(\mathbb{R}^n)} \geq C\||x|^\beta|x'|^\alpha u\|_{L^p(\mathbb{R}^n)}, \quad \forall\; u\in C_c^1(\mathbb{R}^n).$$ 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 $\beta \geq 0$. 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.