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Local Hardy-Littlewood maximal operator. (English) Zbl 1211.42017
Authors’ 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 ${w \in A^p_{\text{loc}}}$, the class of local $A _{p }$ weights which is larger than the Muckenhoupt $A _{p }$ class. Also, the condition ${w \in A^p_{\text{loc}}}$ turns out to be necessary for the weighted weak type $(p,p), p \geq 1$, inequality to hold.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory
##### Keywords:
maximal operator; weights
Full Text:
##### References:
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