Towards geometric integration of rough differential forms

Abstract: 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 H\"older 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. H\"older functions and their images which may be purely unrectifiable). Our construction includes both the one-dimensional Young integral and multidimensional integrals introduced recently by R. Z\"ust. To simplify the exposition, we limit ourselves to integration of forms of dimensions not exceeding two.