We provide a draft of a theory of geometric integration of “rough differential forms” which are generalizations of classical (smooth) differential forms to similar objects with very low regularity, for instance, involving Hölder continuous functions that may be nowhere differentiable. Borrowing ideas from the theory of rough paths, we show that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions. This can be seen as the basis of geometric integration similar to that used in geometric measure theory, but without any underlying differentiable structure, thus allowing Lipschitz functions and rectifiable sets to be substituted by far less regular objects (e.g. Hölder functions and their images which may be purely unrectifiable). Our construction includes both the one-dimensional Young integral and multidimensional integrals introduced recently by Züst, and provides also an alternative (and more geometric) view on the standard construction of rough paths. To simplify the exposition, we limit ourselves to integration of rough k-forms with k≤ 2.

Original languageEnglish
Pages (from-to)2766 - 2828
Number of pages63
JournalJournal of Geometric Analysis
Volume31
Issue number3
Early online date2 Mar 2020
DOIs
StatePublished - Mar 2021
Externally publishedYes

    Research areas

  • Exterior differential calculus, Stokes theorem, Young integral

    Scopus subject areas

  • Geometry and Topology

ID: 53713030