Abstract
We study mapping properties of the centered Hardy–Littlewood maximal operator \(\mathcal {M}\) acting on Lorentz spaces \(L^{p,q}({\mathfrak {X}})\) in the context of certain non-doubling metric measure spaces \({\mathfrak {X}}\). The special class of spaces for which these properties are very peculiar is introduced and many examples are given. In particular, for each \(p_0, q_0, r_0 \in (1, \infty )\) with \(r_0 \ge q_0\) we construct a space \({\mathfrak {X}}\) for which the associated operator \(\mathcal {M}\) is bounded from \(L^{p_0,q_0}({\mathfrak {X}})\) to \(L^{p_0,r}({\mathfrak {X}})\) if and only if \(r \ge r_0\).
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Acknowledgements
I would like to express my deep gratitude to Professor Krzysztof Stempak for his suggestion to study the problem discussed in this article. I also thank him for insightful comments and continuous help during the preparation of the paper. Research was supported by the National Science Centre of Poland, Project No. 2016/21/N/ST1/01496.
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This paper constitutes a part of PhD Thesis written in the Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, under the supervision of Professor Krzysztof Stempak.
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Kosz, D. Maximal operators on Lorentz spaces in non-doubling setting. Math. Z. 298, 1523–1543 (2021). https://doi.org/10.1007/s00209-020-02650-1
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DOI: https://doi.org/10.1007/s00209-020-02650-1