Wolfe's theorem for weakly differentiable cochains

A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension m   in RnRn with the space of flat m-cochains, that is, the dual space of flat chains of dimension m   in RnRn. The main purpose of the present paper is to generalize Wolfe's theorem to the setting of Sobolev differential forms and Sobolev cochains in RnRn. A suitable theory of Sobolev cochains has recently been initiated by the second and third author. It is based on the concept of upper norm and upper gradient of a cochain, introduced in analogy with Heinonen–Koskela's concept of upper gradient of a function.