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

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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 .

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Authors and Affiliations

  1. Mathematical Institute, Universität Bonn, Endenicher Allee 60, D-53115, Bonn, Germany

    Mariusz Mirek

  2. Mathematical Institute, Uniwersytet Wrocławski, Plac Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Mariusz Mirek

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  1. Mariusz Mirek
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Correspondence to Mariusz Mirek.

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The author was supported by NCN grant DEC-2012/05/D/ST1/00053.

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Mirek, M. Weak type (1, 1) inequalities for discrete rough maximal functions. JAMA 127, 247–281 (2015). https://doi.org/10.1007/s11854-015-0030-4

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  • DOI: https://doi.org/10.1007/s11854-015-0030-4

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