On fractional Hardy-type inequalities in general open sets

TL;DR

This paper establishes sharp bounds for fractional Hardy inequalities in open sets, connecting the Hardy constant of punctured space to general domains, and explores limits and applications to eigenvalues and Cheeger inequalities.

Contribution

It provides a new lower bound for Hardy constants in open sets based on the punctured space case and analyzes their limits as parameters vary.

Findings

Sharp Hardy constant bounds for open sets when sp>N

Limit of Hardy constants as s approaches 1 and p approaches infinity

Improved Cheeger inequality with non-vanishing constant as p approaches infinity

Abstract

We show that, when $sp>N$, the sharp Hardy constant $\mathfrak{h}_{s,p}$ of the punctured space $\mathbb R^N\setminus\{0\}$ in the Sobolev-Slobodecki\u{\i} space provides an optimal lower bound for the Hardy constant $\mathfrak{h}_{s,p}(\Omega)$ of an open $\Omega\subsetneq \mathbb R^N$. The proof exploits the characterization of Hardy's inequality in the fractional setting in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation and relies on the construction of suitable supersolutions by means of the distance function from the boundary of $\Omega$. Moreover, we compute the limit of $\mathfrak{h}_{s,p}$ as $s\nearrow 1$, as well as the limit when $p \nearrow \infty$. Finally, we apply our results to establish a lower bound for the non-local eigenvalue $\lambda_{s,p}(\Omega)$ in terms of $\mathfrak{h}_{s,p}$ when $sp>N$, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as $p\nearrow \infty$.