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

We compute the exact John–Nirenberg constant of ${\mathrm{B}\mathrm{M}\mathrm{O}}^p((0,1))$ 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 ${\mathrm{B}\mathrm{M}\mathrm{O}}^p$ to $L^{\mathrm{\infty }}\mathrm{.}$ These results rely on sharp $L^p$- and weak-type estimates for logarithms of $A_{\mathrm{\infty }}$ weights, which in turn use the exact expressions for the corresponding Bellman functions.