Dominated ergodic theorems in rearrangement invariant spaces

• Autorzy: Michael Braverman , Ben-Zion Rubshtein , Alexander Veksler
• Języki publikacji: EN

Abstrakty

EN

We study conditions under which Dominated Ergodic Theorems hold in rearrangement invariant spaces. Consequences for Orlicz and Lorentz spaces are given. In particular, our results generalize the classical theorems for the spaces $L_p$ and the classes $L log^nL$.

Słowa kluczowe

EN

rearrangement invariant space; ergodic theorem; Hardy-Littlewood property

Kategorie tematyczne

• 47A35: Ergodic theory
• 46E10: Topological linear spaces of continuous, differentiable or analytic functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1998
• Tom: 128
• Numer: 2
• Strony: 145-157

Daty

wydano

1998

otrzymano

1997-01-30

poprawiono

1997-08-28

Twórcy

autor

Michael Braverman

• Department of Mathematics & Computer Sciences, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

autor

Ben-Zion Rubshtein

• Department of Mathematics & Computer Sciences, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

autor

Alexander Veksler

• Department of Mathematics, Tashkent State University, Tashkent, Uzbekistan

Bibliografia

• [AM90] M. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (1990), 727-735.
• [B93] M. Braverman, On a class of operators, J. London Math. Soc. 47 (1993), 119-128.
• [BM77] M. Braverman and A. Mekler, On the Hardy-Littlewood property of symmetric spaces, Siberian Math. J. 18 (1977), 371-385.
• [C66] A. P. Calderón, Spaces between $L^1$ and $L^∞$ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273-299.
• [Do53] J. L. Doob, Stochastic Processes, Wiley, New York, 1953.
• [De73] Y. Derriennic, On integrability of the supremum of ergodic ratios, Ann. Probab. 1 (1973), 338-340.
• [DS58] N. Dunford and J. Schwartz, Linear Operators, part I, Interscience, New York, 1958.
• [ES92] G. A. Edgar and L. Sucheston, Stopping Times and Directed Processes, Encyclopedia Math. Appl., Cambridge Univ. Press, 1992.
• [G86] D. Gilat, The best bound in the L logL inequality of Hardy and Littlewood and its martingale counterpart, Proc. Amer. Math. Soc. 97 (1986), 429-436.
• [KR61] M. A. Krasnosel'skiĭ and Ya. B. Rutitskiĭ, Convex Functions and Orlicz Spaces, Noordhoff, 1961.
• [KPS82] S. G. Krein, Yu. Petunin and E. Semenov, Interpolation of Linear Operators, Transl. Math. Monographs 54, Amer. Math. Soc., Providence, 1982.
• [K85] U. Krengel, Ergodic Theorems, de Gruyter Stud. Math., de Gruyter, Berlin, 1985.
• [LT79] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. Function Spaces, Springer, 1979.
• [M65] B. S. Mityagin, An interpolation theorem for modular spaces, Mat. Sb. 66 (1965), 473-482 (in Russian).
• [O71] D. S. Ornstein, A remark on the Birkhoff ergodic theorem, Illinois J. Math. 15 (1971), 77-79.
• [Sa90] E. T. Sawyer, Boundedness of classical operators in classical Lorentz spaces, Studia Math. 96 (1990), 145-158.
• [SW71] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971.
• [St93] V. D. Stepanov, The weighted Hardy's inequality for nonincreasing functions, Trans. Amer. Math. Soc. 338 (1993), 173-186.
• [V85] A. Veksler, An ergodic theorem in symmetric spaces, Sibirsk. Mat. Zh. 24 (1985), 189-191 (in Russian).