On 𝑑-parameter pointwise ergodic theorems in 𝐿₁

Hasegawa, Shigeru; Sato, Ryotaro

doi:10.1090/S0002-9939-1995-1249881-0

On $d$-parameter pointwise ergodic theorems in $L_ 1$
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by Shigeru Hasegawa and Ryotaro Sato PDF

Proc. Amer. Math. Soc. 123 (1995), 3455-3465 Request permission

Abstract:

Let ${P_1}, \ldots ,{P_d}$ be commuting positive linear contractions on ${L_1}$ and let ${T_1}, \ldots ,{T_d}$ be (not necessarily commuting) linear contractions on ${L_1}$ such that $|{T_i}f| \leq {P_i}|f|$ for $1 \leq i \leq d$ and $f \in {L_1}$. In this paper we prove that if each ${P_i},1 \leq i \leq d$, satisfies the mean ergodic theorem, then the averages ${A_n}({T_1}, \ldots ,{T_d})f = {A_n}({T_1}) \cdots {A_n}({T_d})f$, where ${A_n}({T_i}) = {n^{ - 1}}\sum \nolimits _{k = 0}^{n - 1} {T_i^k}$, converge a.e. for every $f \in {L_1}$. When ${T_1}, \ldots ,{T_d}$ commute, we further prove that the ${L_1}$-norm convergence of the averages ${A_n}({P_1}, \ldots ,{P_d})f$ for every $f \in {L_1}$ implies the a.e. convergence of the averages ${A_n}({T_1}, \ldots ,{T_d})f$ for every $f \in {L_1}$. These improve Çömez and Lin’s ergodic theorem.

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contractions and pointwise ergodic theorems

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