Weak integral conditions for BMO

A. A. Logunov, L. Slavin, D. M. Stolyarov, V. Vasyunin, P. B. Zatitskiy

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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 languageEnglish (US)
Pages (from-to)2913-2926
Number of pages14
JournalProceedings of the American Mathematical Society
Issue number7
StatePublished - 2015

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


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