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

TL;DR

This paper precisely computes the John--Nirenberg constant for ${ m BMO}^p$ spaces with $1\, extless p\, extless 2$, demonstrating its attainability and providing sharp estimates related to $A_$ weights.

Contribution

It provides the exact John--Nirenberg constant for ${ m BMO}^p$ spaces for all $p$ between 1 and 2, extending known results for $p=1$ and $p=2$.

Findings

Exact John--Nirenberg constant for ${ m BMO}^p$ with $1 extless p extless 2$.

Attainment of the constant in the weak-type inequality.

Sharp lower bounds for the distance from ${ m BMO}^p$ to $L^$.

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.