Illinois Journal of Mathematics

Moving averages in the plane

Laurent Moonens and Joseph M. Rosenblatt

Full-text: Open access


We study the almost everywhere behavior of the maximal operator associated to moving averages in the plane, both for Lebesgue derivatives and ergodic averages. We show that the almost everywhere behavior of the maximal operator associated to a sequence of moving rectangles $v_{i}+Q_{i}$, with $(0,0)\in Q_{i}$, depends both on the way the rectangles are moved by $v_{i}$ and the structure of the rectangles ($Q_{i}$) as a partially ordered set.

Article information

Illinois J. Math., Volume 56, Number 3 (2012), 759-793.

First available in Project Euclid: 31 January 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 28D05: Measure-preserving transformations


Moonens, Laurent; Rosenblatt, Joseph M. Moving averages in the plane. Illinois J. Math. 56 (2012), no. 3, 759--793. doi:10.1215/ijm/1391178547.

Export citation


  • P. Avramidou, On certain weighted moving averages and their differentiation analogues, New York J. Math. 12 (2006), 19–37.
  • A. Bellow, R. Jones and J. Rosenblatt, Convergence for moving averages, Ergodic Theory Dynam. Systems 10 (1990), no. 1, 43–62.
  • A. Bellow and R. L. Jones, A Banach principle for $L^\infty$, Adv. Math. 120 (1996), no. 1, 155–172.
  • A.-P. Calderón, Ergodic theory and translation-invariant operators, Proc. Natl. Acad. Sci. USA 59 (1968), 349–353.
  • R. P. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math. (2) 51 (1950), 161–166.
  • P. Hagelstein and A. Stokolos, Weak type inequalities for ergodic strong maximal operators, Acta Sci. Math. (Szeged) 76 (2010), 365–379.
  • P. Hagelstein and A. Stokolos, Weak type inequalities for maximal operators associated to double ergodic sums, New York J. Math. 17 (2011), 233–250.
  • B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Fund. Math. 25 (1935), 217–234.
  • A. Nagel and E. M. Stein, On certain maximal functions and approach regions, Adv. in Math. 54 (1984), no. 1, 83–106.
  • D. S. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 161–164.
  • M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker Inc., New York, 1991.
  • J. M. Rosenblatt and M. Wierdl, A new maximal inequality and its applications, Ergodic Theory Dynam. Systems 12 (1992), no. 3, 509–558.
  • S. Sawyer, Maximal inequalities of weak type, Ann. of Math. (2) 84 (1966), 157–174.
  • E. M. Stein, On limits of seqences of operators, Ann. of Math. (2) 74 (1961), 140–170.
  • A. M. Stokolos, On the differentiation of integrals of functions from $L\varphi (L)$, Studia Math. 88 (1988), no. 2, 103–120.
  • A. Zygmund, An individual ergodic theorem for non-commutative transformations, Acta Sci. Math. Szeged 14 (1951), 103–110.
  • A. Zygmund, Trigonometric series, 2nd ed. vols. I, II, Cambridge Univ. Press, New York, 1959.