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arXiv · 1506.04969

The John--Nirenberg constant of ${\rm BMO}^p,$ $1\le p\le 2$

Leonid Slavin
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Abstract

We compute the exact John--Nirenberg constant of ${\rm BMO}^p((0,1))$ for $1\le p\le 2,$ which has been known only for $p=1$ and $p=2.$ We also show that this constant is attained in the weak-type John--Nirenberg inequality and obtain a sharp lower estimate for the distance in ${\rm BMO}^p$ to $L^\infty.$ These results rely on sharp $L^p$- and weak-type estimates for logarithms of $A_\infty$ weights, which in turn use the exact expressions for the corresponding Bellman functions.

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Leonid Slavin. 2015-06-16. The John--Nirenberg constant of ${\rm BMO}^p,$ $1\le p\le 2$. https://arxiv.org/abs/1506.04969

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