Boundary Hardy inequality on functions of bounded variation

Abstract

Classical boundary Hardy inequality, that goes back to 1988, states that if $1 < p < \infty, \ ~\Omega$ is bounded Lipschitz domain, then for all $u \in C^{\infty}_{c}(\Omega)$, $$\int_{\Omega} \frac{|u(x)|^{p}}{\delta^{p}_{\Omega}(x)} dx \leq C\int_{\Omega} |\nabla u(x) |^{p}dx,$$ where $\delta_\Omega(x)$ is the distance function from $\Omega^c$. 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 $W^{s,1}(\Omega)$ 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 $p=1$.