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

Abstract

We compute the exact John--Nirenberg constant of ${\mathrm{BMO}}^p((0,1))$ 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 ${\mathrm{BMO}}^p$ 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.

arXiv comment27 pages, 3 figures

References in corpus (1)

• Uchiyama's lemma and the John-Nirenberg inequality

Cited by in corpus (2)

• Unions of cubes in ${\mathbb{R}}^n$, combinatorics in ${\mathbb{Z}}^n$ and the John-Nirenberg and John-Strömberg inequalities
• The John--Nirenberg constant of ${\mathrm{BMO}}^p$, $p>2$