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

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 $1\le k\le n$. Furthermore, the spherical harmonic decomposition is given for a function $u\in C_0^{\infty }({\mathbb{R}}_{k_{+}}^n)$. Using this decomposition and following the idea of Tertikas and Zographopoulos, we obtain the Filippas-Tertikas improvement of the Hardy inequality.

7 pages

References in corpus (3)

• Hardy inequalities with boundary singularities
• Rellich inequalities with weights
• On the Hardy-Poincaré inequality with boundary singularities

Cited by in corpus (1)

• The Hardy inequality and large time behaviour of the heat equation on ${\mathbb{R}}^{N-k}\times (0,\infty {)}^k$