Weighted pointwise Hardy inequalities

Abstract

We introduce the concept of a visual boundary of a domain Ω ⊂ ℝn and show that the weighted Hardy inequality ∫Ω|u|p dΩβ−p ⩽ C ∫Ω |∇u|pdΩβ, where dΩ(x) = dist(x, ∂Ω), holds for all u ∈ C0∞ (Ω) with exponents β < β0 when the visual boundary of Ω is sufficiently large. Here β0 = β0(p, n, Ω) is explicit, essentially sharp, and may even be greater than p – 1, which is the known bound for smooth domains. For instance, in the case of the usual von Koch snowflake domain the sharp bound is shown to be β0 = p − 2 + λ, with λ = log 4/log 3. These results are based on new pointwise Hardy inequalities.