### 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 Rn and h: [0,∞) → [0,∞) is such that h(t)→∞ as t→∞, then sup J subcube Q 1 |J|_ J h ϕ – 1 |J| J ϕ _< ∞ =⇒ ϕ ∈ BMO(Q). Under some additional assumptions on h we obtain estimates on _ϕ_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 BMO.

Original language | English (US) |
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Pages (from-to) | 2913-2926 |

Number of pages | 14 |

Journal | Proceedings of the American Mathematical Society |

Volume | 143 |

Issue number | 7 |

DOIs | |

State | Published - Jan 1 2015 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

Logunov, A. A., Slavin, L., Stolyarov, D. M., Vasyunin, V., & Zatitskiy, P. B. (2015). Weak integral conditions for BMO.

*Proceedings of the American Mathematical Society*,*143*(7), 2913-2926. https://doi.org/10.1090/S0002-9939-2015-12424-0