Journal d'Analyse Mathématique

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

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

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 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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Copyright information

© Hebrew University Magnes Press 2015

Authors and Affiliations

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

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