On a pointwise ergodic theorem for multiparameter semigroups.
Sato, Ryotaro
Publicacions Matemàtiques, Tome 38 (1994), p. 81-87 / Harvested from Biblioteca Digital de Matemáticas
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Let Ti (i = 1, 2, ..., d) be commuting null preserving transformations on a finite measure space (X, F, μ) and let 1 ≤ p < ∞. In this paper we prove that for every f ∈ Lp(μ) the averages

Anf(x) = (n + 1)-d Σ0≤ni≤n f(T1 n1 T2 n2 ... Td nd x)

converge a.e. on X if and only if there exists a finite invariant measure ν (under the transformations Ti) absolutely continuous with respect to μ and a sequence {XN} of invariant sets with XN ↑ X such that νB > 0 for all nonnull invariant sets B and such that the Radon-Nykodim derivative v = dν/dμ satisfies v ∈ Lq(xN,μ), 1/p + 1/q = 1, for each N ≥ 1.

Publié le : 1994-01-01
DMLE-ID : 3747 
@article{urn:eudml:doc:41203,
     title = {On a pointwise ergodic theorem for multiparameter semigroups.},
     journal = {Publicacions Matem\`atiques},
     volume = {38},
     year = {1994},
     pages = {81-87},
     mrnumber = {MR1291955},
     zbl = {0817.28010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41203}
}

Sato, Ryotaro. On a pointwise ergodic theorem for multiparameter semigroups.. Publicacions Matemàtiques, Tome 38 (1994) pp. 81-87. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41203/