  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 $p=1$.
