PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 131, Number 1, Pages 231–233
S 0002-9939(02)06756-4
Article electronically published on June 27, 2002
A REMARK ON REAL COBOUNDARY
COCYCLES IN L∞ -SPACE
RYOTARO SATO
(Communicated by Michael Handel)
Abstract. Let T be an ergodic automorphism of a probability measure space
(Ω, A, m) and let f be a real-valued measurable function on Ω. We deduce
a necessary and sufficient condition for the existence of L∞ -solutions of the
cohomology equation f = h ◦ T − h, by using the recent result of Alonso, Hong
and Obaya.
Let (Ω, A, m) be a probability measure space and let T be an ergodic automorphism of (Ω, A, m). Every real-valued measurable function f on Ω defines an
additive real T -cocycle
(1)
Sn f =
n−1
X
f ◦ T j,
n > 0.
j=0
Recently Alonso, Hong and Obaya proved in [1] that a necessary and sufficient
condition for the existence of Lp -solutions of the cohomology equation f = h◦T −h,
with 0 < p < ∞, is that f ∈ Lp (Ω, m) and there exists a measurable subset A ⊂ Ω
with m(A) > 0 and an increasing sequence {nk } of positive integers with
nk Z
1 X
(2)
|Sj f |p dm < ∞.
sup
k≥1 nk j=1 A
For related results we refer the reader to [2] and [3]. In this note we would like to
consider the case p = ∞ and prove the following theorem.
Theorem. Let f ∈ Lp (Ω, m) for some p with 0 < p ≤ ∞. Then there exists a
function h in L∞ (Ω, m) with f = h ◦ T − h if and only if there exists a measurable
subset A ⊂ Ω with m(A) > 0 and an increasing sequence {nk } of positive integers
with
nk
1 X
(3)
kSj f · χA k∞ < ∞.
sup
k≥1 nk j=1
Proof. Suppose f = h ◦ T − h for some h ∈ L∞ (Ω, m). Then, since Sj f = h ◦ T j − h
for j ≥ 1, we have kSj f k∞ ≤ 2khk∞, whence (3) holds for A = Ω and nk = k.
Received by the editors August 31, 2001.
2000 Mathematics Subject Classification. Primary 37A20, 28D05, 47A35.
Key words and phrases. Ergodic automorphism, additive real cocycle, cohomology equation,
coboundary.
c
2002
American Mathematical Society
231
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232
RYOTARO SATO
Conversely, suppose the second condition of the Theorem holds. Since 0 < r < p
implies Lp (Ω, m) ⊂ Lr (Ω, m), we may assume from the beginning that 0 < p ≤ 1.
Then we have
Z
|Sj f |p dm ≤ 1 + kSj f · χA k∞ for j ≥ 1,
A
which implies
nk Z
1 X
|Sj f |p dm < ∞.
sup
k≥1 nk j=1 A
Thus, by Theorem 3.2 of [1], there exists a function h in Lp (Ω, m) with f = h◦T −h.
It suffices to show that h is a function in L∞ (Ω, m).
To do so, we notice that there exists a positive number δ with
m(A ∩ {ω : |h(ω)| ≤ δ}) > 0.
Therefore, we may assume from the beginning that
|h| ≤ δ
(4)
on A.
Then, using the relations
kh ◦ T j · χA k∞ = k(Sj f + h) · χA k∞ ≤ kSj f · χA k∞ + δ,
we see from (3) that
(5)
nk
1 X
kh ◦ T j · χA k∞ < ∞.
M := sup
k≥1 nk j=1
By this inequality, we will prove below that h ∈ L∞ (Ω, m).
For this purpose, let α be a positive number such that there exists a measurable
subset B ⊂ Ω with
(6)
m(B) > 0
and |h| ≥ α
on B.
Since T is an ergodic automorphism by hypothesis, the Birkhoff Ergodic Theorem
implies that
nk
1 X
χA (T −j ω) = m(A) > 0
lim
k→∞ nk
j=1
for almost all ω ∈ Ω. Hence, we can choose a measurable subset B1 ⊂ B with
m(B1 ) > 0 and a positive integer K with
(7)
nK
1 X
χA (T −j ω) > 2−1 m(A)
nK j=1
for all ω ∈ B1 .
Thus, taking into account (5) and (6), we get
nK
nK
1 X
1 X
j
kh ◦ T · χA k∞ =
kh · χA ◦ T −j k∞
∞>M ≥
nK j=1
nK j=1


nK
nK
X
1 X
1
−j 

≥
kαχB · (χA ◦ T −j )k∞ ≥ α · ·
χ
◦
T
χ
B1
A
nK j=1
nK j=1
≥ α · 2−1 m(A).
∞
This shows that α cannot be so large, and thus h ∈ L∞ (Ω, m) must follow. This
completes the proof.
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A REMARK ON REAL COBOUNDARY COCYCLES IN L∞ -SPACE
233
References
[1] A. I. Alonso, J. Hong and R. Obaya, Absolutely continuous dynamics and real coboundary
cocycles in Lp -spaces, 0 < p < ∞, Studia Math. 138 (2000), 121–134. MR 2001i:37010
[2] I. Assani, A note on the equation Y = (I − T )X in L1 , Illinois J. Math. 43 (1999), 540–541.
MR 2000j:47016
[3] M. Lin and R. Sine, Ergodic theory and the functional equation (I − T )x = y, J. Operator
Theory 10 (1983), 153–166. MR 84m:47015
Department of Mathematics, Okayama University, Okayama, 700-8530 Japan
E-mail address: [email protected]
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