Optimal Hardy inequalities in cones

Abstract

Let Ω be an open connected cone in ℝn with vertex at the origin. Assume that the Operatoris subcritical in Ω, where δΩ is the distance function to the boundary of Ω and μ ⩽ 1/4. We show that under some smoothness assumption on Ω the improved Hardy-type inequalityholds true, and the Hardy-weight λ(μ)|x|–2 is optimal in a certain definite sense. The constant λ(μ) > 0 is given explicitly.