On the best constants of Hardy inequality in R{double-struck} ~(n-k)×(R{double-struck} _+) ~k and related improvements

摘要

We compute the explicit sharp constants of Hardy inequalities in the cone R{double-struck} _(k+) ~n:=R{double-struck} ~(n-k)×(R{double-struck}+) ~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{double-struck} _(k+) ~n). Using this decomposition and following the idea of Tertikas and Zographopoulos, we obtain the Filippas-Tertikas improvement of the Hardy inequality.