Integration of nonsmooth 2-forms: From Young to Itô and Stratonovich

We show that geometric integrals of the type 'omega f dg1 perpendicular to dg2 can be defined over a two-dimensional domain 12 when the functions f, g1, g2: R2 -> R are just Holder continuous with sufficiently large Holder exponents and the boundary of 12 has sufficiently small dimension, by summing over a refining sequence of partitions the discrete Stratonovich or Ito type terms. This leads to a two-dimensional extension of the classical Young integral that coincides with the integral introduced recently by R. Zust. We further show that the Stratonovich-type summation allows to weaken the requirements on Holder exponents of the map g = (g1, g2) when f(x) = F(x, g(x)) with F sufficiently regular. The technique relies upon an extension of the sewing lemma from Rough paths theory to alternating functions of two-dimensional oriented simplices, also proven in the paper.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).