Geodesic
A solution to the multidimensional additive homological equation
Izvestiya. Mathematics , Tome 87 (2023) no. 2, pp. 201-251

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We prove that, for a finite-dimensional real normed space $V$, every bounded mean zero function $f\in L_\infty([0,1];V)$ can be written in the form $f=g\circ T-g$ for some $g\in L_\infty([0,1];V)$ and some ergodic invertible measure preserving transformation $T$ of $[0,1]$. Our method moreover allows us to choose $g$, for any given $\varepsilon>0$, to be such that $\|g\|_\infty\leq (S_V+\varepsilon)\|f\|_\infty$, where $S_V$ is the Steinitz constant corresponding to $V$.
Keywords: additive homological equation, coboundary problem, measure preserving transformation.
Mots-clés : Kwapień's theorem, Steinitz constant 
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     title = {A solution to the multidimensional additive homological equation},
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A. F. Ber; M. Borst; S. Borst; F. A. Sukochev. A solution to the multidimensional additive homological equation. Izvestiya. Mathematics , Tome 87 (2023) no. 2, pp. 201-251. http://geodesic.mathdoc.fr/item/IM2_2023_87_2_a0/