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Journal d'Analyse Mathématique

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

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

  • Mariusz Mirek
Article

Abstract

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 N h = {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 N h .

Keywords

Maximal Function Ergodic Theorem Weak Type Analytic Number Theory Ergodic Average 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew University Magnes Press 2015

Authors and Affiliations

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

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