Accepted Paper

Inserted: 24 apr 2026
Last Updated: 24 apr 2026

Journal: ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
Year: 2024
Doi: 10.2422/2036-2145.202407_011

Abstract:

Classical boundary Hardy inequality, that goes back to 1988, states that if $1 < p < \infty, \ ~Ω$ is bounded Lipschitz domain, then for all $u \in C^{\infty}_{c}(Ω)$, $$\int_Ω \frac{
u(x)
^{p}}{δ{p}_Ω(x)} dx \leq C\int_Ω
\nabla u(x)
^{p}dx,$$ where $δ_Ω(x)$ is the distance function from $Ω^c$. In this article, we address the long standing open question on the case $p=1$ by establishing appropriate boundary Hardy inequalities in the space of functions of bounded variation. We first establish appropriate inequalities on fractional Sobolev spaces $W^{s,1}(Ω)$ and then Brezis, Bourgain and Mironescu's result on limiting behavior of fractional Sobolev spaces as $s\rightarrow 1^{-}$ plays an important role in the proof. Moreover, we also derive an infinite series Hardy inequality for the case $p=1$.