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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 | | 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
$h:[0,infty) o[0,infty)$ 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 | | Services: | Forum | Review | PDF | Favorites | |
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