Remarks about Hardy Inequalities on Metric Trees

where ψ > 0 is the so-called Hardy weight and C(Γ, ψ) is a positive constant which might depend on Γ and ψ, but which is independent of u. Evans, Harris and Pick [EHP] found a necessary and sufficient condition such that (1.1) holds for all functions u on Γ such that u(o) = 0 and such that the integral on the right hand side is finite. They consider even the case of non-symmetric Hardy weights. However, due to the existence of harmonic functions with finite Dirichlet integral, the Hardy weights have to decay rather fast. This led Naimark and Solomyak [NS1] to the study of (1.1) for functions in {u ∈ C 0 (Γ) : u(o) = 0} (and its closure with respect to the Dirichlet integral). For regular trees, see Subsection 2.1 for the definition, they gave a complete characterization of the validity of (1.1) on that class of functions. Note that in both papers the condition u(o) = 0 was imposed. It is clear that without this assumption inequality (1.1) cannot hold for all metric trees. To see this, it suffices to consider Γ = R+ as an example. Our first remark in this paper is that if the tree is regular and grows sufficiently fast, then there are weights ψ such that (1.1) hold for all u ∈ C 0 (Γ ∪ {o}) (without the condition u(o) = 0). Following the approach in [NS1] we can give a complete characterization of admissible weights and obtain two-sided estimates on the sharp constant C(ψ,Γ) in (1.1). As in the Euclidean case, Hardy weights typically decay like |x|−2 as |x| := dist(o, x) → ∞. The growth of a tree is reflected by its branching function g0(t) := # {x ∈ Γ : |x| = t}, t ∈ R+ , (1.2)