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Local Hardy–Littlewood maximal operator

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Abstract

In this article we define and investigate a local Hardy–Littlewood maximal operator in Euclidean spaces. It is proved that this operator satisfies weighted L p, p > 1, and weighted weak type (1,1) estimates with weight function \({w \in A^p_{\rm{loc}}}\), the class of local A p weights which is larger than the Muckenhoupt A p class. Also, the condition \({w \in A^p_{\rm{loc}}}\) turns out to be necessary for the weighted weak type (p,p), p ≥ 1, inequality to hold.

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Authors and Affiliations

  1. Department of Mathematics, National Central University, Chung-Li, 320, Taiwan, Republic of China

    Chin-Cheng Lin

  2. Instytut Matematyki i Informatyki, Politechnika Wrocławska, Wyb. Wyspiańskiego 27, 50-370, Wrocław, Poland

    Krzysztof Stempak

  3. Katedra Matematyki i Zastosowań Informatyki, Politechnika Opolska, ul. Mikołajczyka 5, 45-271, Opole, Poland

    Krzysztof Stempak

Authors
  1. Chin-Cheng Lin
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  2. Krzysztof Stempak
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Correspondence to Krzysztof Stempak.

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To the memory of Józef Marcinkiewicz on the centenary of his birth.

Research of C.-C. Lin supported by NSC and NCTS of Taiwan

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Lin, CC., Stempak, K. Local Hardy–Littlewood maximal operator. Math. Ann. 348, 797–813 (2010). https://doi.org/10.1007/s00208-010-0499-1

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  • DOI: https://doi.org/10.1007/s00208-010-0499-1

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