On the best constants of Hardy inequality in Rn−k×k(R+) and related improvements

We compute the explicit sharp constants of Hardy inequalities in the cone with 1⩽k⩽n. Furthermore, the spherical harmonic decomposition is given for a function . Using this decomposition and following the idea of Tertikas and Zographopoulos, we obtain the Filippas–Tertikas improvement of the Hardy inequality.