On the best constants of Hardy inequality in $\mathbb{R}^{n-k}\times (\mathbb{R}_{+})^{k}$ and related improvements

TL;DR

This paper determines the exact sharp constants for Hardy inequalities in certain cones in Euclidean space and provides an improved version using spherical harmonic decomposition and previous methods.

Contribution

It explicitly computes the sharp constants for Hardy inequalities in cones and introduces an improved inequality via spherical harmonic decomposition.

Findings

Explicit sharp constants for Hardy inequalities in cones

Spherical harmonic decomposition for functions in the cone

An improved Hardy inequality following Tertikas and Zographopoulos

Abstract

We compute the explicit sharp constants of Hardy inequalities in the cone $\mathbb{R}_{k_+}^{n}:=\mathbb{R}^{n-k}\times (\mathbb{R}_{+})^{k}=\{(x_{1},...,x_{n})|x_{n-k+1}>0,...,x_{n}>0\}$ with $1\leq k\leq n$. Furthermore, the spherical harmonic decomposition is given for a function $u\in C^{\infty}_{0}(\mathbb{R}_{k_+}^{n})$. Using this decomposition and following the idea of Tertikas and Zographopoulos, we obtain the Filippas-Tertikas improvement of the Hardy inequality.