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

Abstract

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

33 pages; updated version with the introduction significantly revised in order to state a version of the main result earlier and with less non-standard notation