## 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 $\mathrm {BMO}$.## References

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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