Weighted Hardy inequalities beyond Lipschitz domains

Abstract:

It is a well-known fact that in a Lipschitz domain $\Omega \subset \mathbb {R}^n$, a $p$-Hardy inequality, with weight $\operatorname {dist}(x,\partial \Omega )^\beta$, holds for all $u\in C_0^\infty (\Omega )$ whenever $\beta <p-1$. 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 $\lambda =n-1$. Corresponding results also hold for smaller exponents, and, in fact, our methods work in general metric spaces satisfying standard structural assumptions.