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

This paper is a continuation of earlier work by the first author who determined the John–Nirenberg constant of ${\mathrm{B}\mathrm{M}\mathrm{O}}^p((0,1))$ for the range $1\le p\le 2.$ Here, we compute that constant for $p>2.$ As before, the main results rely on Bellman functions for the $L^p$ norms of logarithms of $A_{\mathrm{\infty }}$ weights, but for $p>2$ these functions turn out to have a significantly more complicated structure than for $1\le p\le 2.$