Multiparameter pointwise ergodic theorems for Markov operators on L∞.
Sato, Ryotaro
Publicacions Matemàtiques, Tome 38 (1994), p. 395-410 / Harvested from Biblioteca Digital de Matemáticas
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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 < ∞ the averages

Anf = (n + 1)-d Σ0≤ni≤n P1 n1 P2 n2 ... Pd nd 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.

Publié le : 1994-01-01
DMLE-ID : 3727 
@article{urn:eudml:doc:41181,
     title = {Multiparameter pointwise ergodic theorems for Markov operators on L$\infty$.},
     journal = {Publicacions Matem\`atiques},
     volume = {38},
     year = {1994},
     pages = {395-410},
     mrnumber = {MR1316635},
     zbl = {0826.60026},
     language = {en},
     url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41181}
}

Sato, Ryotaro. Multiparameter pointwise ergodic theorems for Markov operators on L∞.. Publicacions Matemàtiques, Tome 38 (1994) pp. 395-410. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41181/