Self-improvement of fractional Hardy inequalities in metric measure spaces via hyperbolic fillings

Abstract

In this paper, we prove a self-improvement result for $(\theta ,p)$-fractional Hardy inequalities, in both the exponent $1<p<\infty$ and the regularity parameter $0<\theta <1$, for bounded domains in doubling metric measure spaces. The key conceptual tool is a Caffarelli-Silvestre-type argument, which relates fractional Sobolev spaces on $Z$ to Newton-Sobolev spaces in the hyperbolic filling ${\overline{X}}_{\epsilon }$ of $Z$ via trace results. Using this insight, it is shown that a fractional Hardy inequality in an open subset of $Z$ is equivalent to a classical Hardy inequality in the filling ${\overline{X}}_{\epsilon }$. The main result is then obtained by applying a new weighted self-improvement result for $p$-Hardy inequalities. The exponent $p$ can be self-improved by a classical Koskela-Zhong argument, but a new theory of regularizable weights is developed to obtain the self-improvement in the regularity parameter $\theta$. This generalizes a result of Lehrbäck and Koskela on self-improvement of $d_{\Omega }^{\beta }$-weighted $p$-Hardy inequalities by allowing a much broader class of weights. Using the equivalence of fractional Hardy inequalities with Hardy inequalities in the fillings, we also give new examples of domains satisfying fractional Hardy inequalities.