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Geometric Maximal Operators and \(\mathrm {{BMO}}{}{}{}\) on Product Bases

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Abstract

We consider the problem of the boundedness of maximal operators on \(\mathrm {BMO}_{}^{}\) on shapes in \({\mathbb {R}}^n\). We prove that for bases of shapes with an engulfing property, the corresponding maximal function is bounded from \(\mathrm {BMO}_{}^{}\) to \(\mathrm {BLO}_{}^{}\), generalising a known result of Bennett for the basis of cubes. When the basis of shapes does not possess an engulfing property but exhibits a product structure with respect to lower-dimensional shapes coming from bases that do possess an engulfing property, we show that the corresponding maximal function is bounded from \(\mathrm {BMO}_{}^{}\) to a space we define and call rectangular \(\mathrm {BLO}_{}^{}\).

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Acknowledgements

The authors would like to thank Alex Stokolos for bringing to their attention the open problem of the boundedness of the strong maximal function on strong \(\mathrm {BMO}_{}^{}\) and for fruitful discussions.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Concordia University, Montréal, QC, H3G 1M8, Canada

    Galia Dafni

  2. Département de mathématiques et de statistique, Université Laval, Québec, QC, G1V 0A6, Canada

    Ryan Gibara

  3. Department of Mathematics, Georgia College and State University, Milledgeville, GA, 31061, USA

    Hong Yue

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  1. Galia Dafni
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  2. Ryan Gibara
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  3. Hong Yue
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Correspondence to Ryan Gibara.

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G.D. was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, the Centre de recherches mathématiques (CRM), and the Fonds de recherche du Québec – Nature et technologies (FRQNT). R.G. was partially supported by the Centre de recherches mathématiques (CRM), the Institut des sciences mathématiques (ISM), and the Fonds de recherche du Québec – Nature et technologies (FRQNT). H.Y. was partially supported by GCSU Faculty Development Funds.

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Dafni, G., Gibara, R. & Yue, H. Geometric Maximal Operators and \(\mathrm {{BMO}}{}{}{}\) on Product Bases. J Geom Anal 31, 5740–5765 (2021). https://doi.org/10.1007/s12220-020-00501-3

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