Boundary Hardy inequality on functions of bounded variation

Classical boundary Hardy inequality, that goes back to 1988, states that if 1 < p <, \ ~ is bounded Lipschitz domain, then for all u C^₂ (), _ |u (x) |^p^{p_ (x) } dx C_ | u (x) |^pdx, where _ (x) is the distance function from 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 () 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.