Inequalities for the ergodic maximal function and convergence of the averages in weighted -spaces

Author:
F. J. Martín-Reyes

Journal:
Trans. Amer. Math. Soc. **296** (1986), 61-82

MSC:
Primary 28D05; Secondary 47A35

DOI:
https://doi.org/10.1090/S0002-9947-1986-0837798-1

MathSciNet review:
837798

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Abstract: This paper is concerned with the characterization of those positive functions such that Hopf's averages associated to an invertible measure preserving transformation and a positive function converge almost everywhere for every . We also study mean convergence when satisfies a "doubling condition" over orbits. In order to do this, we first characterize the pairs of positive functions such that the ergodic maximal operator associated to and is of weak or strong type with respect to the measures and .

**[1]**E. Atencia and F. J. Martín-Reyes,*Weak type inequalities for the maximal ergodic function and the maximal ergodic Hilbert transform in weighted spaces*, Studia Math.**78**(1984), 231-244. MR**782660 (86f:28020)****[2]**E. Atencia and A. de la Torre,*A dominated ergodic estimate for**spaces with weights*, Studia Math.**74**(1982), 35-47. MR**675431 (84f:47005)****[3]**R. Coifman, Peter W. Jones and José L. Rubio de Francia,*On a constructive decomposition of**functions and factorizations of**weights*, Proc. Amer. Math. Soc.**87**(1983), 675-676. MR**687639 (84c:42031)****[4]**N. Dunford and J. T. Schwartz,*Linear operators*, Part I, Interscience, New York, 1957.**[5]**J. E. Gilbert,*Nikishin-Stein theory and factorization with applications*, Proc. Sympos. Pure Math., vol. 35, Amer. Math. Soc., Providence, R. I., 1979, pp. 233-267. MR**545313 (81g:42023)****[6]**R. Hunt, D. Kurtz and C. Neugebauer,*A note on the equivalence of**and Sawyer's condition for equal weights*, Conference on Harmonic Analysis in honor of Antoni Zygmund, Vols. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, Calif., 1983, pp. 156-158. MR**730066 (85f:42032)****[7]**F. J. Martíin-Reyes and A. de la Torre,*A dominated ergodic theorem for invertible*-*isometries in weighted*-*spaces*(preprint).**[8]**B. Muckenhoupt,*Weighted norm inequalities for the Hardy maximal function*, Trans. Amer. Math. Soc.**165**(1972), 207-226. MR**0293384 (45:2461)****[9]**D. Revuz,*Markov chains*, North-Holland Mathematical Library, 1975. MR**758799 (86a:60097)****[10]**J. L. Rubio de Francia,*Boundedness of maximal functions and singular integrals in weighted**spaces*, Proc. Amer. Math. Soc.**83**(1981), 673-679. MR**630035 (83a:42018)****[11]**E. T. Sawyer,*A characterization of a two-weight norm inequality for maximal operators*, Studia Math.**75**(1982), 1-11. MR**676801 (84i:42032)****[12]**-,*Two weight norm inequalities for certain maximal and integral operators*, Harmonic Analysis, Proceedings, Minneapolis, 1981, Springer-Verlag, Berlin and New York, 1982, pp. 102-127. MR**654182 (83k:42020b)****[13]**J. Woś,*The filling scheme and the ergodic theorems of Kesten and Tanny*, Colloq. Math. (to appear). MR**893542 (88k:28023)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0837798-1

Keywords:
Ergodic maximal function,
mean convergence,
almost everywhere convergence,
weighted inequalities,
Hopf's averages

Article copyright:
© Copyright 1986
American Mathematical Society