A Bourgain-Brezis-Mironescu -type characterization for Sobolev differential forms

Given a bounded domain Ω ⊂ Rⁿ, a result by Bourgain, Brezis, and Mironescu characterizes when a function f ∈ Lᵖ(Ω) is in the Sobolev space W1,p(Ω) based on the limiting behavior of its Besov seminorms. We prove a direct analogue of this result which characterizes when a differential k-form ω ∈ Lᵖ(ᵏ T^* Ω) has a weak exterior derivative dω ∈ Lᵖ(ᵏ⁺¹ T^* Ω), where the analogue of the Besov seminorm that our result uses is based on integration over simplices.