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11 August 2020
 
  » arxiv » 1309.6780

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Weak integral conditions for BMO
Alexander A. Logunov ; Leonid Slavin ; Dmitriy M. Stolyarov ; Vasily Vasyunin ; Pavel B. Zatitskiy ;
Date 26 Sep 2013
AbstractWe 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 $h:[0,infty) o[0,infty)$ is such that $h(t)underset{t oinfty}{longrightarrow}infty,$ then
$$ sup_{J ext{subcube} Q} frac1{|J|}int_J h(|varphi-frac1{|J|} int_Jvarphi |)<infty Longrightarrow varphiin BMO(Q). $$
Under some additional assumptions on $h$ we obtain estimates on $|varphi|_{BMO}$ in terms of the supremum above. We also show that even though the condition $h(t)underset{t oinfty}{longrightarrow}infty$ is not necessary for this implication to hold, it becomes necessary if one considers the dyadic BMO.
Source arXiv, 1309.6780
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