# Pointwise ergodic theorems for functions in Lorentz spaces ${L}_{pq}$ with p ≠ ∞

Studia Mathematica (1994)

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

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topSato, 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

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

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