Pointwise ergodic theorems for functions in Lorentz spaces L p q with p ≠ ∞

Ryotaro Sato

Studia Mathematica (1994)

  • Volume: 109, Issue: 2, page 209-216
  • ISSN: 0039-3223

Abstract

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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 L p q 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 ≠ ∞.

How to cite

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Sato, Ryotaro. "Pointwise ergodic theorems for functions in Lorentz spaces $L_{pq}$ with p ≠ ∞." Studia Mathematica 109.2 (1994): 209-216. <http://eudml.org/doc/216070>.

@article{Sato1994,
abstract = {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 $L_\{pq\}$ 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 ≠ ∞.},
author = {Sato, Ryotaro},
journal = {Studia Mathematica},
keywords = {pointwise ergodic theorems; $L_\{pq\}$ spaces; null preserving transformations; measure preserving transformations; positive contractions on $L_1$ spaces; spaces; positive contractions on spaces; null preserving point transformation on a finite measure space; almost everywhere convergence of the ergodic averages},
language = {eng},
number = {2},
pages = {209-216},
title = {Pointwise ergodic theorems for functions in Lorentz spaces $L_\{pq\}$ with p ≠ ∞},
url = {http://eudml.org/doc/216070},
volume = {109},
year = {1994},
}

TY - JOUR
AU - Sato, Ryotaro
TI - Pointwise ergodic theorems for functions in Lorentz spaces $L_{pq}$ with p ≠ ∞
JO - Studia Mathematica
PY - 1994
VL - 109
IS - 2
SP - 209
EP - 216
AB - 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 $L_{pq}$ 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 ≠ ∞.
LA - eng
KW - pointwise ergodic theorems; $L_{pq}$ spaces; null preserving transformations; measure preserving transformations; positive contractions on $L_1$ spaces; spaces; positive contractions on spaces; null preserving point transformation on a finite measure space; almost everywhere convergence of the ergodic averages
UR - http://eudml.org/doc/216070
ER -

References

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  1. [1] R. V. Chacon, A class of linear transformations, Proc. Amer. Math. Soc. 15 (1964), 560-564. Zbl0168.11702
  2. [2] N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Interscience, New York, 1958. Zbl0084.10402
  3. [3] A. M. Garsia, Topics in Almost Everywhere Convergence, Markham, Chicago, 1970. Zbl0198.38401
  4. [4] R. A. Hunt, On L(p,q) spaces, Enseign. Math. 12 (1966), 249-276. Zbl0181.40301
  5. [5] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985. 
  6. [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. Zbl0802.28011
  7. [7] C. Ryll-Nardzewski, On the ergodic theorems. I. (Generalized ergodic theorems), Studia Math. 12 (1951), 65-73. 
  8. [8] R. Sato, On pointwise ergodic theorems for positive operators, ibid. 97 (1990), 71-84. Zbl0755.47005

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