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

Sato, Ryotaro

Sato, Ryotaro

Studia Mathematica, Tome 113 (1995), p. 227-236 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $n^{- 1} \sum_{i = 0}^{n - 1} f \circ τ^{i} (x)$ 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 $(r, s) : r = s = 1, o r 1 < r < \infty a n d 1 \leq s \leq \infty, o r r = s = \infty$ . Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized and unified

Publié le : 1995-01-01

Zbl 0835.47006

EUDML-ID : urn:eudml:doc:216189

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     author = {Ryotaro Sato},
     title = {Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations},
     journal = {Studia Mathematica},
     volume = {113},
     year = {1995},
     pages = {227-236},
     zbl = {0835.47006},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv114i3p227bwm}
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Sato, Ryotaro. Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations. Studia Mathematica, Tome 113 (1995) pp. 227-236. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv114i3p227bwm/

Bibliographie

[00000] [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. | Zbl 0558.60033

[00001] [2] I. Assani and J. Woś, An equivalent measure for some nonsingular transformations and application, Studia Math. 97 (1990), 1-12. | Zbl 0718.28006

[00002] [3] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience, New York, 1958. | Zbl 0084.10402

[00003] [4] S. Gładysz, Ergodische Funktionale und individueller ergodischer Satz, Studia Math. 19 (1960), 177-185. | Zbl 0097.11002

[00004] [5] R. A. Hunt, On L(p,q) spaces, Enseign. Math. 12 (1966), 177-185.

[00005] [6] Y. Ito, Uniform integrability and the pointwise ergodic theorem, Proc. Amer. Math. Soc. 16 (1965), 222-227. | Zbl 0135.36204

[00006] [7] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985.

[00007] [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. | Zbl 0802.28011

[00008] [9] H. L. Royden, Real Analysis, Macmillan, New York, 1988. | Zbl 0704.26006

[00009] [10] C. Ryll-Nardzewski, On the ergodic theorems. I. (Generalized ergodic theorems), Studia Math. 12 (1951), 65-73.

[00010] [11] R. Sato, Pointwise ergodic theorems for functions in Lorentz spaces $L_{p} q$ with p≠ ∞, Studia Math. 109 (1994), 209-216. | Zbl 0822.47011