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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 $u\in C^{\infty}_{0}(\mathbb{R}_{k_+}^{n})$. Using this decomposition and following the idea of Tertikas and Zographopoulos, we obtain the Filippas-Tertikas improvement of the Hardy inequality.