On the best constants of Hardy inequality in and related improvements

We compute the explicit sharp constants of Hardy inequalities in the cone R k + n : = R n − k × ( R + ) k = { ( x 1 , … , x n ) | x n − k + 1 > 0 , … , x n > 0 } with 1 ⩽ k ⩽ n . Furthermore, the spherical harmonic decomposition is given for a function u ∈ C 0 ∞ ( R k + n ) . Using this decomposition and following the idea of Tertikas and Zographopoulos, we obtain the Filippas–Tertikas improvement of the Hardy inequality.