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## Studia Mathematica

1995 | 114 | 3 | 227-236
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
227-236
Opis fizyczny
Daty
wydano
1995
otrzymano
1993-04-19
poprawiono
1993-12-10
Twórcy
autor
• Department of Mathematics, Okayama University, Okayama, 700 Japan
Bibliografia
• [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.
• [2] I. Assani and J. Woś, An equivalent measure for some nonsingular transformations and application, Studia Math. 97 (1990), 1-12.
• [3] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience, New York, 1958.
• [4] S. Gładysz, Ergodische Funktionale und individueller ergodischer Satz, Studia Math. 19 (1960), 177-185.
• [5] R. A. Hunt, On L(p,q) spaces, Enseign. Math. 12 (1966), 177-185.
• [6] Y. Ito, Uniform integrability and the pointwise ergodic theorem, Proc. Amer. Math. Soc. 16 (1965), 222-227.
• [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.
• [9] H. L. Royden, Real Analysis, Macmillan, New York, 1988.
• [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 $L_pq$ with p≠ ∞, Studia Math. 109 (1994), 209-216.
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Bibliografia
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