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

Given a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, a result by Bourgain, Brezis, and Mironescu characterizes when a function $f\in L^p(\mathrm{\Omega })$ is in the Sobolev space $W^{1,p}(\mathrm{\Omega })$ based on the limiting behavior of its Besov seminorms. We prove a direct analogue of this result which characterizes when a differential $k$-form $\omega \in L^p({\wedge }^k{T}^{\ast }\mathrm{\Omega })$ has a weak exterior derivative $d\omega \in L^p({\wedge }^{k+1}{T}^{\ast }\mathrm{\Omega })$, where the analogue of the Besov seminorm that our result uses is based on integration over simplices.