Weighted Hardy inequalities beyond Lipschitz domains

Abstract

It is a well-known fact that in a Lipschitz domain Ω ⊂ R n a p-Hardy inequality, with weight dist(x, ∂Ω)β , holds for all u ∈ C ∞0 (Ω) whenever β < p − 1. We show that actually the same is true under the sole assumption that the boundary of the domain satisfies a uniform density condition with the exponent λ = n − 1. Corresponding results also hold for smaller exponents, and, in fact, our methods work in general metric spaces satisfying standard structural assumptions.