Boundary Hardy inequality on functions of bounded variation

Classical boundary Hardy inequality, that goes back to 1988, states that if 1 < p < ∞, \ ~Ω is bounded Lipschitz domain, then for all u ∈ C∞c(Ω), ∫Ω |u(x)|ᵖδᵖΩ(x) dx ≤ C∫Ω |∇ u(x) |ᵖdx, where δ_Ω(x) is the distance function from Ωᶜ. 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 Ws,1(Ω) and then Brezis, Bourgain and Mironescu's result on limiting behavior of fractional Sobolev spaces as s→ 1⁻ plays an important role in the proof. Moreover, we also derive an infinite series Hardy inequality for the case $p=1$.