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Hardy–Littlewood maximal operator on reflexive variable Lebesgue spaces over spaces of homogeneous type

Volume 254 / 2020

Alexei Karlovich Studia Mathematica 254 (2020), 149-178 MSC: Primary 43A85; Secondary 46E30. DOI: 10.4064/sm180816-16-9 Published online: 1 April 2020

Abstract

We show that the Hardy–Littlewood maximal operator is bounded on a reflexive variable Lebesgue space $L^{p(\cdot )}$ over a space of homogeneous type $(X,d,\mu )$ if and only if it is bounded on its dual space $L^{p’(\cdot )}$, where $1/p(x)+1/p’(x)=1$ for $x\in X$. This result extends the corresponding result of Lars Diening from the Euclidean setting of $\mathbb {R}^n$ to the setting of spaces $(X,d,\mu )$ of homogeneous type.

Authors

  • Alexei KarlovichCentro de Matemática e Aplicações
    Departamento de Matemática
    Faculdade de Ciências e Tecnologia
    Universidade Nova de Lisboa
    Quinta da Torre, 2829-516 Caparica, Portugal
    e-mail

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