2019/12/18 by Giovanni Alberti, Alberti, Giovanni, Eugene Stepanov +3
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #History and Theory of Mathematics
paper · pdf · doi:10.48550/arxiv.1912.08796
openalex publication_date 2019/12/18 · openalex created_date 2019/12/26 · openalex updated_date 2026/07/28
We show that geometric integrals of the type ∫Ωf d g1\wedge d g2 can be defined over a two-dimensional domain Ω when the functions f, g1, g2\colon ℝ2→ ℝ are just Hölder continuous with sufficiently large Hölder exponents and the boundary of Ω 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=(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.