# 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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topSato, 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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- [2] I. Assani and J. Woś, An equivalent measure for some nonsingular transformations and application, Studia Math. 97 (1990), 1-12. Zbl0718.28006
- [3] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience, New York, 1958. Zbl0084.10402
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- [6] Y. Ito, Uniform integrability and the pointwise ergodic theorem, Proc. Amer. Math. Soc. 16 (1965), 222-227. Zbl0135.36204
- [7] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985.
- [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] H. L. Royden, Real Analysis, Macmillan, New York, 1988. Zbl0704.26006
- [10] C. Ryll-Nardzewski, On the ergodic theorems. I. (Generalized ergodic theorems), Studia Math. 12 (1951), 65-73.
- [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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