Journal d'Analyse Mathématique

, Volume 127, Issue 1, pp 247–281 | Cite as

Weak type (1, 1) inequalities for discrete rough maximal functions



The aim of this paper is to show that the discrete maximal function
$${M_h}f(x) = \mathop {\sup }\limits_{N \in {\Bbb N}} \frac{1}{{\left| {{N_h} \cap \left[ {1,N} \right]} \right|}}\left| {\sum\limits_{n \in {N_h} \cap \left[ {1,N} \right]} {f(x - n)} } \right|,{\text{ for }}x \in {\Bbb Z}$$
, where Nh = {n ∈ ℕ: there exists m ∈ ℕ such that n = ⌊h(m)⌋} for an appropriate function h, is of weak type (1, 1). As a consequence, we also obtain a pointwise ergodic theorem along the set Nh.


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  1. [1]
    M. Boshernitzan, G. Kolesnik, A. Quas, and M. Wierdl, Ergodic averaging sequences, J. Anal. Math. 95 (2005), 63–103.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), 39–72.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    J. Bourgain, On the pointwise ergodic theorem on L p for arithmetic sets, Israel J. Math. 61 (1988), 73–84.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Etudes Sci. Publ. Math. 69 (1989), 5–45.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Z. Buczolich, Universally L 1 good sequences with gaps tending to infinity, Acta Math. Hungar. 117 (2007), 91–140.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Z. Buczolich and R. D. Mauldin, Divergent square averages, Ann. of Math. (2) 171 (2010), 1479–1530.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    M. Christ, Weak type (1, 1) bounds for rough operators, Ann. of Math. (2) 128 (1988), 19–42.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    M. Christ, A weak type (1, 1) inequality for maximal averages over certain sparse sequences, preprint, (2011).Google Scholar
  9. [9]
    C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    D. R. Heath-Brown, The Pjateckiĭ-Šapiro prime number theorem, J. Number Theory 16 (1983), 242–266.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society, Providence, RI, 2004.MATHGoogle Scholar
  12. [12]
    P. LaVictoire, An L 1 ergodic theorem for sparse random subsequences, Math. Res. Lett. 16 (2009), 849–859.MathSciNetCrossRefGoogle Scholar
  13. [13]
    P. LaVictoire, Universally L 1-bad arithmetic sequences, J. Anal. Math. 113 (2011), 241–263.MATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    M. Mirek, p(ℤ)-boundedness of discrete maximal functions along thin subsets of primes and pointwise ergodic theorems, Math. Z. 279 (2015), 27–59.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    M. Mirek, Roth’s Theorem in the Piatetski-Shapiro primes, Rev. Math. Iberoam. 31 (2015), 617–656.MathSciNetCrossRefGoogle Scholar
  16. [16]
    M. B. Nathanson, Additive Number Theory. The Classical Bases, Springer-Verlag, 1996.Google Scholar
  17. [17]
    I. Piatetski-Shapiro, On the distribution of prime numbers in sequences of the formf (n)⌋, Math. Sbornik 33 (1953), 559–566.Google Scholar
  18. [18]
    L. B. Pierce, Discrete analogues in harmonic analysis, Ph.D. Thesis, Princeton University, 2009.Google Scholar
  19. [19]
    J. Rosenblatt and M. Wierdl, Pointwise ergodic theorems via harmonic analysis, Ergodic Theory and its Connections with Harmonic Analysis, Cambridge University Press, Cambridge, 1995, pp. 3–151.Google Scholar
  20. [20]
    R. Urban and J. Zienkiewicz, Weak Type (1, 1) estimates for a class of discrete rough maximal functions, Math. Res. Lett. 14 (2007), 227–237.MATHMathSciNetCrossRefGoogle Scholar

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© Hebrew University Magnes Press 2015

Authors and Affiliations

  1. 1.Mathematical InstituteUniversität BonnBonnGermany
  2. 2.Mathematical InstituteUniwersytet WrocławskiWrocławPoland

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