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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Mean oscillation bounds on rearrangements
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by Almut Burchard, Galia Dafni and Ryan Gibara PDF
Trans. Amer. Math. Soc. 375 (2022), 4429-4444 Request permission

Abstract:

We use geometric arguments to prove explicit bounds on the mean oscillation for two important rearrangements on ${\mathbb {R}^n}$. For the decreasing rearrangement $f^*$ of a rearrangeable function $f$ of bounded mean oscillation (BMO) on cubes, we improve a classical inequality of Bennett–DeVore–Sharpley, $\|f^*\|_{{\operatorname {BMO}}(\mathbb {R}_+)}\leq C_n \|f\|_{{\operatorname {BMO}}(\mathbb {R}^n)}$, by showing the growth of $C_n$ in the dimension $n$ is not exponential but at most of the order of $\sqrt {n}$. This is achieved by comparing cubes to a family of rectangles for which one can prove a dimension-free Calderón–Zygmund decomposition. By comparing cubes to a family of polar rectangles, we provide a first proof that an analogous inequality holds for the symmetric decreasing rearrangement, $Sf$.
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Additional Information
  • Almut Burchard
  • Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
  • MR Author ID: 319824
  • ORCID: 0000-0002-3542-1919
  • Email: almut@math.toronto.edu
  • Galia Dafni
  • Affiliation: Department of Mathematics and Statistics, Concordia University, Montréal, QC H3G 1M8, Canada
  • MR Author ID: 255789
  • ORCID: 0000-0002-5078-7724
  • Email: galia.dafni@concordia.ca
  • Ryan Gibara
  • Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025
  • MR Author ID: 1268389
  • Email: ryan.gibara@gmail.com
  • Received by editor(s): June 11, 2021
  • Received by editor(s) in revised form: November 11, 2021
  • Published electronically: March 17, 2022
  • Additional Notes: The first author was partially supported by Natural Sciences and Engineering Research Council (NSERC) of Canada. The second author was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Centre de recherches mathématiques (CRM). The third author 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)
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 4429-4444
  • MSC (2020): Primary 42B35, 46E30
  • DOI: https://doi.org/10.1090/tran/8629
  • MathSciNet review: 4419064