Weighted Hardy inequalities beyond Lipschitz domains

It is a well-known fact that in a Lipschitz domain Ω\subset R^n a p-Hardy inequality, with weight d(x,\partialΩ)^β, holds for all u\in C_0^\infty(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.

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