On the exterior product of Hölder differential forms

Abstract

We introduce a complex of cochains, $\alpha$-fractional charges ($0<\alpha \le 1$), whose regularity is between that of De Pauw-Moonens-Pfeffer's charges and that of Whitney's flat cochains. We show that $\alpha$-Hölder differential forms and their exterior derivative can be realized as $\alpha$-fractional charges, and that it is possible to define the exterior product between an $\alpha$-fractional and a $\beta$-fractional charge, under the condition that $\alpha +\beta >1$. This construction extends the Young integral in arbitrary dimension and codimension.

This preprint will not be submitted for publication is kept on arXiv for reference. More complete and updated results are to be found in arXiv:2602.09151 and arXiv:2503.09298