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 $u\in C_c^{\infty }(\Omega )$, $${\int }_{\Omega }\frac{|u(x){|}^p}{{\delta }_{\Omega }^p(x)}dx\le C{\int }_{\Omega }|\nabla u(x){|}^{p}dx,$$ 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 $s\to 1^{-}$ plays an important role in the proof. Moreover, we also derive an infinite series Hardy inequality for the case $p=1$.

25 pages