# Multiparameter pointwise ergodic theorems for Markov operators on L∞.

• Volume: 38, Issue: 2, page 395-410
• ISSN: 0214-1493

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## Abstract

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Let P1, ..., Pd be commuting Markov operators on L∞(X,F,μ), where (X,F,μ) is a probability measure space. Assuming that each Pi is either conservative or invertible, we prove that for every f in Lp(X,F,μ) with 1 ≤ p &lt; ∞ the averagesAnf = (n + 1)-d Σ0≤ni≤n P1n1 P2n2 ... Pdnd f (n ≥ 0)converge almost everywhere if and only if there exists an invariant and equivalent finite measure λ for which the Radon-Nikodym derivative v = dλ/dμ is in the dual space Lp'(X,F,μ). Next we study the case in which exists p1, with 1 ≤ p1 ≤ ∞, such that for every f in Lp(X,F,μ) the limit function belongs to Lp1(X,F,μ). We give necessary and sufficient conditions for this problem.

## How to cite

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Sato, Ryotaro. "Multiparameter pointwise ergodic theorems for Markov operators on L∞.." Publicacions Matemàtiques 38.2 (1994): 395-410. <http://eudml.org/doc/41181>.

@article{Sato1994,
abstract = {Let P1, ..., Pd be commuting Markov operators on L∞(X,F,μ), where (X,F,μ) is a probability measure space. Assuming that each Pi is either conservative or invertible, we prove that for every f in Lp(X,F,μ) with 1 ≤ p &lt; ∞ the averagesAnf = (n + 1)-d Σ0≤ni≤n P1n1 P2n2 ... Pdnd f (n ≥ 0)converge almost everywhere if and only if there exists an invariant and equivalent finite measure λ for which the Radon-Nikodym derivative v = dλ/dμ is in the dual space Lp'(X,F,μ). Next we study the case in which exists p1, with 1 ≤ p1 ≤ ∞, such that for every f in Lp(X,F,μ) the limit function belongs to Lp1(X,F,μ). We give necessary and sufficient conditions for this problem.},
author = {Sato, Ryotaro},
journal = {Publicacions Matemàtiques},
keywords = {Teoría ergódica; Espacio de medida; Teorema de Radon-Nikodym; ergodic theorems; commuting Markov operators; Radon-Nikodym derivative},
language = {eng},
number = {2},
pages = {395-410},
title = {Multiparameter pointwise ergodic theorems for Markov operators on L∞.},
url = {http://eudml.org/doc/41181},
volume = {38},
year = {1994},
}

TY - JOUR
AU - Sato, Ryotaro
TI - Multiparameter pointwise ergodic theorems for Markov operators on L∞.
JO - Publicacions Matemàtiques
PY - 1994
VL - 38
IS - 2
SP - 395
EP - 410
AB - Let P1, ..., Pd be commuting Markov operators on L∞(X,F,μ), where (X,F,μ) is a probability measure space. Assuming that each Pi is either conservative or invertible, we prove that for every f in Lp(X,F,μ) with 1 ≤ p &lt; ∞ the averagesAnf = (n + 1)-d Σ0≤ni≤n P1n1 P2n2 ... Pdnd f (n ≥ 0)converge almost everywhere if and only if there exists an invariant and equivalent finite measure λ for which the Radon-Nikodym derivative v = dλ/dμ is in the dual space Lp'(X,F,μ). Next we study the case in which exists p1, with 1 ≤ p1 ≤ ∞, such that for every f in Lp(X,F,μ) the limit function belongs to Lp1(X,F,μ). We give necessary and sufficient conditions for this problem.
LA - eng
KW - Teoría ergódica; Espacio de medida; Teorema de Radon-Nikodym; ergodic theorems; commuting Markov operators; Radon-Nikodym derivative
UR - http://eudml.org/doc/41181
ER -

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