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