Orlicz-hardy Inequalities

We relate Orlicz-Hardy inequalities on a bounded Euclidean domain to certain fatness conditions on the complement. In the case of certain log-scale distortions of Ln, this relationship is necessary and sufficient, thus extending results of Ancona, Lewis, and Wannebo. 0. Introduction Suppose Ω ⊂ R is a bounded domain and let d(x) = dist(x, ∂Ω). We consider integral Hardy inequalities (0.1) ∀u ∈ C∞ 0 (Ω) : ∫ Ω Ψ ( |u(x)| d(x) ) dx ≤ C ∫ Ω Ψ(|∇u(x)|) dx, and norm Hardy inequalities (0.2) ∀u ∈ C∞ 0 (Ω) : ∥∥∥∥ |u(x)| d(x) ∥∥∥∥ LΨ(Ω) ≤ C‖ |∇u| ‖LΨ(Ω), where Ψ : [0,∞) → [0,∞) is any of a certain class of Orlicz functions with polynomial growth; see Section 1 for a definition of the Luxemburg norm ‖ · ‖LΨ(Ω). Extending results of Ancona [A], Lewis [L], and Wannebo [W], who considered the case of power functions Ψ(t) = t, we relate the validity of such inequalities to certain fatness conditions on the complement of Ω. For power functions, it is clear that (0.1) and (0.2) are mutually equivalent, but this is not so in general. However, (0.1) implies (0.2); see Section 1. In the classical case Ψ(t) = t, (0.1) holds for all 1 < p ≤ n on domains whose complement is locally uniformly p-fat, but it is only when p = n that we get the equivalence of Hardy and local uniform fatness of the complement Received July 28, 2003; received in final form August 6, 2004. 2000 Mathematics Subject Classification. 46E35. This paper was written while we were both at the University of Michigan for the Fred and Lois Gehring Special Year in Complex Analysis. We are both grateful for the hospitality of the Mathematics Department. The first author was partially supported by Enterprise Ireland. c ©2004 University of Illinois