On sharp anisotropic Hardy inequalities

Abstract

Recently, Yanyan Li and Xukai Yan showed the following interesting Hardy inequalities with anisotropic weights: Let $n\ge 2$, $p\ge 1$, $p\hat{I}\pm >1-n$, $p(\hat{I}\pm +{\hat{I}}^2)>-n$, then there exists $C>0$ such that $$\Vert |x{|}^2|x^{\prime }{|}^{\hat{I}\pm +1}\nabla u{\Vert }_{L^p({\mathbb{R}}^n)}\ge C\Vert |x{|}^2|x^{\prime }{|}^{\hat{I}}\pm u{\Vert }_{L^p({\mathbb{R}}^n)},\forall u\in C_c^1({\mathbb{R}}^n).$$ Here $x^{\prime }=(x_1,\dots ,x_{n-1},0)$ 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 ${\hat{I}}^2\ge 0$. 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.

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