## Inequalities for the ergodic maximal function and convergence of the averages in weighted $L^ p$-spaces

HTML articles powered by AMS MathViewer

- by F. J. Martín-Reyes PDF
- Trans. Amer. Math. Soc.
**296**(1986), 61-82 Request permission

## Abstract:

This paper is concerned with the characterization of those positive functions $w$ such that Hopf’s averages associated to an invertible measure preserving transformation $T$ and a positive function $g$ converge almost everywhere for every $f \in {L^p}(w d\mu )$. We also study mean convergence when $g$ satisfies a "doubling condition" over orbits. In order to do this, we first characterize the pairs of positive functions $(u,w)$ such that the ergodic maximal operator associated to $T$ and $g$ is of weak or strong type with respect to the measures $w d\mu$ and $u d\mu$.## References

- 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), no. 3, 231–244. MR**782660**, DOI 10.4064/sm-78-3-231-244 - E. Atencia and A. de la Torre,
*A dominated ergodic estimate for $L_{p}$ spaces with weights*, Studia Math.**74**(1982), no. 1, 35–47. MR**675431**, DOI 10.4064/sm-74-1-35-47 - R. Coifman, Peter W. Jones, and José L. Rubio de Francia,
*Constructive decomposition of BMO functions and factorization of $A_{p}$ weights*, Proc. Amer. Math. Soc.**87**(1983), no. 4, 675–676. MR**687639**, DOI 10.1090/S0002-9939-1983-0687639-3
N. Dunford and J. T. Schwartz, - John E. Gilbert,
*Nikišin-Stein theory and factorization with applications*, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 233–267. MR**545313** - R. A. Hunt, D. S. Kurtz, and C. J. Neugebauer,
*A note on the equivalence of $A_{p}$ and Sawyer’s condition for equal weights*, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 156–158. MR**730066**
F. J. Martíin-Reyes and A. de la Torre, - Benjamin Muckenhoupt,
*Weighted norm inequalities for the Hardy maximal function*, Trans. Amer. Math. Soc.**165**(1972), 207–226. MR**293384**, DOI 10.1090/S0002-9947-1972-0293384-6 - D. Revuz,
*Markov chains*, 2nd ed., North-Holland Mathematical Library, vol. 11, North-Holland Publishing Co., Amsterdam, 1984. MR**758799** - José L. Rubio de Francia,
*Boundedness of maximal functions and singular integrals in weighted $L^{p}$ spaces*, Proc. Amer. Math. Soc.**83**(1981), no. 4, 673–679. MR**630035**, DOI 10.1090/S0002-9939-1981-0630035-3 - Eric T. Sawyer,
*A characterization of a two-weight norm inequality for maximal operators*, Studia Math.**75**(1982), no. 1, 1–11. MR**676801**, DOI 10.4064/sm-75-1-1-11 - Eric T. Sawyer,
*Weighted norm inequalities for fractional maximal operators*, 1980 Seminar on Harmonic Analysis (Montreal, Que., 1980) CMS Conf. Proc., vol. 1, Amer. Math. Soc., Providence, R.I., 1981, pp. 283–309. MR**670111** - Janusz Woś,
*The filling scheme and the ergodic theorems of Kesten and Tanny*, Colloq. Math.**52**(1987), no. 2, 263–276. MR**893542**, DOI 10.4064/cm-52-2-263-276

*Linear operators*, Part I, Interscience, New York, 1957.

*A dominated ergodic theorem for invertible*${L_p}$-

*isometries in weighted*${L_p}$-

*spaces*(preprint).

## Additional Information

- © Copyright 1986 American Mathematical Society
- 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