Elaboration on Kwapien's theorem: Representing bounded mean zero functions f as coboundary f = g ◦ T − g

Abstract

In [8] Kwapien proved that every mean zero
function f ∈ L∞[0, 1] we can

write as f = g ◦ T − g for some g ∈ L∞[0, 1] and some measure preserving

transformation T of [0, 1]. However, as was
discovered in [4] there is a gap

in the proof for the case that f is not
continuous. The aim of this bachelor

thesis is filling in that gap in the proof. We
first extend Kwapien’s proof for continuous functions to certain other measure
spaces. Thereafter, we use the method of proof suggested by Kwapien, to proof the theorem for mean zero
function f ∈ L∞[0, 1] for which λ(f⁻¹({x})) = 0 for all x ∈ R.
Using this result we then proof that every mean zero function f ∈ L∞[0, 1] can be written as a sum f =(g₁ ◦ T₁ − g₁) +
(g₂ ◦ T₂ − g₂) where g₁, g₂
∈ L∞[0, 1]
and where T₁, T₂ are

measure preserving transformations of [0, 1]. We
finish this thesis with an

application of Kwapien’s theorem in the study to
singular traces