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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 thatthis constant is attained in the weak-type John--Nirenberg inequality andobtain 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 thecorresponding Bellman functions.

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