Optimal Hardy inequalities in cones

Let Omega be an open connected cone in R-n with vertex at the origin. Assume that the operator P mu := -Delta - mu/delta(2)(Omega) (x) is subcritical in Omega, where delta(Omega) is the distance function to the boundary of Omega and mu <= 1/ 4. We show that under some smoothness assumption on Omega the improved Hardy-type inequality integral(Omega)vertical bar del phi vertical bar(2) dx - mu integral(Omega) vertical bar phi vertical bar(2) /delta(2)(Omega) dx >= lambda(mu) integral(Omega) vertical bar phi vertical bar(2) /vertical bar x vertical bar(2) dx for all phi epsilon C-0(infinity) (Omega) holds true, and the Hardy-weight lambda( mu)vertical bar x vertical bar(-2) is optimal in a certain definite sense. The constant lambda(mu) > 0 is given explicitly.