On Sharp Anisotropic Hardy Inequalities

Abstract Recently, Yanyan Li and Xukai Yan showed in [8, 9] 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 $$ \begin{align*} &\big \||x|^{\beta}|x^{\prime}|^{\alpha+1} \nabla u \big\|_{L^p(\mathbb{R}^n)} \geq C \big\||x|^\beta|x^{\prime}|^\alpha u \big\|_{L^p(\mathbb{R}^n)}, \quad \forall\; u\in C_c^1(\mathbb{R}^n).\end{align*} $$ Here $x^{\prime} = (x_{1},\ldots , x_{n-1}, 0)$ 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 [9], we provide explicit estimate for the anisotropic $L^{p}$-Caffarelli–Kohn–Nirenberg inequality.