Bourgain-Brezis-Mironescu domains. (October 2020) - Bal, Kaushik; Mohanta, Kaushik; Roy, Prosenjit
Abstract: Bourgain et al. (2001) proved that for p > 1 and smooth bounded domain Ω ⊆ R N, lim s → 1 ( 1 − s ) ∬ Ω × Ω | f ( x ) − f ( y ) | p | x − y | N + s p d x d y = κ ∫ Ω | ∇ f ( x ) | p d x for all f ∈ L p ( Ω ) . This gives a characterization of W 1, p ( Ω ) by means of W s, p ( Ω ) seminorms only. For the case p = 1, Dávila (2002) proved that when Ω is a bounded domain with Lipschitz boundary, lim s → 1 ( 1 − s ) ∬ Ω × Ω | f ( x ) − f ( y ) | | x − y | N + s d x d y = κ [ f ] B V ( Ω ) for all f ∈ L 1 ( Ω ) . This characterizes B V ( Ω ) in terms of W s, 1 ( Ω ) seminorm. In this paper we extend the first result and partially extend the second result to extension domains.