Let τ be a null preserving point transformation on a finite measure space. Assuming τ is invertible, P. Ortega Salvador has recently obtained sufficient conditions for the almost everywhere convergence of the ergodic averages in with 1 < p < ∞, 1 < q < ∞. In this paper we obtain necessary and sufficient conditions for the almost everywhere convergence, without assuming that τ is invertible and only assuming that p ≠ ∞.
@article{bwmeta1.element.bwnjournal-article-smv109i2p209bwm,
author = {Ryotaro Sato},
title = {Pointwise ergodic theorems for functions in Lorentz spaces $L\_{pq}$ with p $\neq$ $\infty$},
journal = {Studia Mathematica},
volume = {108},
year = {1994},
pages = {209-216},
zbl = {0822.47011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv109i2p209bwm}
}
Sato, Ryotaro. Pointwise ergodic theorems for functions in Lorentz spaces $L_{pq}$ with p ≠ ∞. Studia Mathematica, Tome 108 (1994) pp. 209-216. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv109i2p209bwm/
[00000] [1] R. V. Chacon, A class of linear transformations, Proc. Amer. Math. Soc. 15 (1964), 560-564. | Zbl 0168.11702
[00001] [2] N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Interscience, New York, 1958. | Zbl 0084.10402
[00002] [3] A. M. Garsia, Topics in Almost Everywhere Convergence, Markham, Chicago, 1970. | Zbl 0198.38401
[00003] [4] R. A. Hunt, On L(p,q) spaces, Enseign. Math. 12 (1966), 249-276. | Zbl 0181.40301
[00004] [5] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985.
[00005] [6] P. Ortega Salvador, Weights for the ergodic maximal operator and a.e. convergence of the ergodic averages for functions in Lorentz spaces, Tôhoku Math. J. 45 (1993), 437-446. | Zbl 0802.28011
[00006] [7] C. Ryll-Nardzewski, On the ergodic theorems. I. (Generalized ergodic theorems), Studia Math. 12 (1951), 65-73.
[00007] [8] R. Sato, On pointwise ergodic theorems for positive operators, ibid. 97 (1990), 71-84. | Zbl 0755.47005