A solution to the multidimensional additive homological equation

Journal article (2023)

Authors

Aleksei F. Ber National University of Uzbekistan named after M. Ulugbek
M.J. Borst Analysis - EEMCS
Sander J. Borst Centrum Wiskunde & Informatica (CWI)
F. Sukochev Analysis - EEMCS , University of New South Wales
Research Group
Analysis (EEMCS) (TU Delft)
Copyright: © 2023 Aleksei F. Ber, M.J. Borst, Sander J. Borst, F. Sukochev
DOI: https://doi.org/10.4213/im9319e
More Info
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Published Date

2023

Language

English

Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

Abstract

We prove that, for a finite-dimensional real normed space V, every bounded mean zero function f ∈ L∞([0, 1]; V) can be written in the form f = g ◦ T − g for some g ∈ L∞([0, 1]; V) and some ergodic invertible measure preserving transformation T of [0, 1]. Our method moreover allows us to choose g, for any given ε > 0, to be such that ∥g∥∞ ⩽ (SV + ε)∥f∥∞, where SV is the Steinitz constant corresponding to V.