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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Weak integral conditions for BMO
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by A. A. Logunov, L. Slavin, D. M. Stolyarov, V. Vasyunin and P. B. Zatitskiy PDF
Proc. Amer. Math. Soc. 143 (2015), 2913-2926 Request permission

Abstract:

We study the question of how much one can weaken the defining condition of BMO. Specifically, we show that if $Q$ is a cube in $\mathbb {R}^n$ and $h:[0,\infty )\to [0,\infty )$ is such that $h(t)\to \infty$ as $t\to \infty ,$ then \[ \sup _{J~\text {subcube}~Q}\frac 1{|J|}\int _J h\big (\big |\varphi -\textstyle {\frac 1{|J|} \int _J\varphi } \big |\big )<\infty \quad \Longrightarrow \quad \varphi \in \mathrm {BMO}(Q). \] Under some additional assumptions on $h$ we obtain estimates on $\|\varphi \|_{\mathrm {BMO}}$ in terms of the supremum above. We also show that even though the limit condition on $h$ is not necessary for this implication to hold, it becomes necessary if one considers the dyadic $\mathrm {BMO}$.
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Additional Information
  • A. A. Logunov
  • Affiliation: Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia
  • L. Slavin
  • Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45211-0025
  • MR Author ID: 121075
  • ORCID: 0000-0002-9502-8852
  • D. M. Stolyarov
  • Affiliation: Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia – and – Steklov Institute of Mathematics, St. Petersburg Branch, St. Petersburg, Russia
  • MR Author ID: 895114
  • V. Vasyunin
  • Affiliation: Steklov Institute of Mathematics, St. Petersburg Branch, St. Petersburg, Russia
  • P. B. Zatitskiy
  • Affiliation: Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia – and – Steklov Institute of Mathematics, St. Petersburg Branch, St. Petersburg, Russia
  • MR Author ID: 895184
  • Received by editor(s): September 26, 2013
  • Published electronically: March 6, 2015
  • Additional Notes: The second author was supported by the NSF (DMS-1041763)
    The third author was supported by a Rokhlin grant and the RFBR (grant 11-01-00526)
    The fourth author was supported by the RFBR (grant 11-01-00584-a)
    The fifth author was supported by the President of Russia grant for young researchers MK-6133.2013.1 and by the RFBR (grant 13-01-12422 ofi_m2)
    This research was supported in part by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under the RF Government grant 11.G34.31.0026, and by JSC “Gazprom Neft”
  • Communicated by: Alexander Iosevich
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 2913-2926
  • MSC (2010): Primary 42B35, 49K20
  • DOI: https://doi.org/10.1090/S0002-9939-2015-12424-0
  • MathSciNet review: 3336616