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Weak integral conditions for BMO


Authors: A. A. Logunov, L. Slavin, D. M. Stolyarov, V. Vasyunin and P. B. Zatitskiy
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
Published electronically: March 6, 2015
MathSciNet review: 3336616
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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

$\displaystyle \sup _{J~\text {subcube}~Q}\frac 1{\vert J\vert}\int _J h\big (\b... ...g \vert\big )<\infty \quad \Longrightarrow \quad \varphi \in \mathrm {BMO}(Q). $

Under some additional assumptions on $ h$ we obtain estimates on $ \Vert\varphi \Vert _{\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

D. M. Stolyarov
Affiliation: Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia – and – Steklov Institute of Mathematics, St. Petersburg Branch, St. Petersburg, Russia

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

DOI: https://doi.org/10.1090/S0002-9939-2015-12424-0
Keywords: BMO, different norms, Bellman function
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
Article copyright: © Copyright 2015 American Mathematical Society