Elaboration on Kwapien's theorem: Representing bounded mean zero functions f as coboundary f = g ◦ T − g
M.J. Borst (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Mark Veraar – Mentor (TU Delft - Analysis)
B. van den Dries – Graduation committee member (TU Delft - Analysis)
C. Kraaikamp – Graduation committee member (TU Delft - Applied Probability)
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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