*Accepted Paper*

**Inserted:** 12 dec 2019

**Last Updated:** 13 oct 2023

**Journal:** J. Funct. Anal.

**Year:** 2019

**Abstract:**

We show that geometric integrals of the type $\int_\Omega f\,d g^1\wedge \,d g^2$ can be defined over a two-dimensional domain $\Omega$ when the functions $f$, $g^1$, $g^2\colon \mathbb{R}^2\to \mathbb{R}$ are just Hölder continuous with sufficiently large Hölder exponents and the boundary of $\Omega$ has sufficiently small dimension, by summing over a refining sequence of partitions the discrete Stratonovich or Itô type terms. This leads to a two-dimensional extension of the classical Young integral that coincides with the integral introduced recently by R. Züst. We further show that the Stratonovich-type summation allows to weaken the requirements on Hölder exponents of the map $g=(g^1,g^2)$ 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.

**Keywords:**
Young integral, Rough paths, Sewing lemma, Itô integral, Stratonovich integral

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