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

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 $u\in C_0^{\mathrm{\infty }}({\mathbb{R}}_{k_{+}}^n)$. Using this decomposition and following the idea of Tertikas and Zographopoulos, we obtain the Filippas-Tertikas improvement of the Hardy inequality.