# Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations

Studia Mathematica (1995)

• Volume: 114, Issue: 3, page 227-236
• ISSN: 0039-3223

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## Abstract

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Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average ${n}^{-1}{\sum }_{i=0}^{n-1}f\circ {\tau }^{i}\left(x\right)$ converges almost everywhere to a function f* in $L\left({p}_{1},{q}_{1}\right]$, where (pq) and $\left({p}_{1},{q}_{1}\right]$ are assumed to be in the set $\left(r,s\right):r=s=1,or1. Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized and unified

## How to cite

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Sato, Ryotaro. "Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations." Studia Mathematica 114.3 (1995): 227-236. <http://eudml.org/doc/216189>.

@article{Sato1995,
abstract = {Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^\{-1\} ∑^\{n-1\}_\{i=0\} f∘τ^\{i\}(x)$ converges almost everywhere to a function f* in $L(p_1,q_1]$, where (pq) and $(p_1,q_1]$ are assumed to be in the set $\{(r,s) : r=s=1, or 1 < r < ∞ and 1 ≤ s ≤ ∞, or r = s = ∞\}$. Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized and unified},
author = {Sato, Ryotaro},
journal = {Studia Mathematica},
keywords = {null preserving transformation; Lorentz spaces; pointwise ergodic theorems},
language = {eng},
number = {3},
pages = {227-236},
title = {Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations},
url = {http://eudml.org/doc/216189},
volume = {114},
year = {1995},
}

TY - JOUR
AU - Sato, Ryotaro
TI - Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations
JO - Studia Mathematica
PY - 1995
VL - 114
IS - 3
SP - 227
EP - 236
AB - Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^{-1} ∑^{n-1}_{i=0} f∘τ^{i}(x)$ converges almost everywhere to a function f* in $L(p_1,q_1]$, where (pq) and $(p_1,q_1]$ are assumed to be in the set ${(r,s) : r=s=1, or 1 < r < ∞ and 1 ≤ s ≤ ∞, or r = s = ∞}$. Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized and unified
LA - eng
KW - null preserving transformation; Lorentz spaces; pointwise ergodic theorems
UR - http://eudml.org/doc/216189
ER -

## References

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1. [1] I. Assani, Quelques résultats sur les opérateurs positifs à moyennes bornées dans ${L}_{p}$, Ann. Sci. Univ. Clermont-Ferrand II Probab. Appl. 3 (1985), 65-72. Zbl0558.60033
2. [2] I. Assani and J. Woś, An equivalent measure for some nonsingular transformations and application, Studia Math. 97 (1990), 1-12. Zbl0718.28006
3. [3] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience, New York, 1958. Zbl0084.10402
4. [4] S. Gładysz, Ergodische Funktionale und individueller ergodischer Satz, Studia Math. 19 (1960), 177-185. Zbl0097.11002
5. [5] R. A. Hunt, On L(p,q) spaces, Enseign. Math. 12 (1966), 177-185.
6. [6] Y. Ito, Uniform integrability and the pointwise ergodic theorem, Proc. Amer. Math. Soc. 16 (1965), 222-227. Zbl0135.36204
7. [7] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985.
8. [8] 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
9. [9] H. L. Royden, Real Analysis, Macmillan, New York, 1988. Zbl0704.26006
10. [10] C. Ryll-Nardzewski, On the ergodic theorems. I. (Generalized ergodic theorems), Studia Math. 12 (1951), 65-73.
11. [11] R. Sato, Pointwise ergodic theorems for functions in Lorentz spaces ${L}_{p}q$ with p≠ ∞, Studia Math. 109 (1994), 209-216. Zbl0822.47011

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