Weighted Hardy inequalities beyond Lipschitz domains

TL;DR

This paper extends weighted Hardy inequalities to broader classes of domains and metric spaces, showing they hold under minimal boundary density conditions rather than Lipschitz regularity.

Contribution

It proves Hardy inequalities with weights for domains with boundary density conditions, generalizing beyond Lipschitz domains and applying to metric spaces.

Findings

Hardy inequalities hold under boundary density conditions.

Results extend to metric spaces with structural assumptions.

Applicable for weights with exponents smaller than p-1.

Abstract

It is a well-known fact that in a Lipschitz domain \Omega\subset R^n a p-Hardy inequality, with weight d(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.