BMO averages on nested cubes grow at most logarithmically

If $Q=R$ the claim is trivial, so assume $Q⊊R$. Let $m\ge 0$ be the least integer such that $2^m\mathrm{\ell }(Q)\ge 2\mathrm{\ell }(R)$, and let $R^{\mathrm{\prime }}$ be the cube concentric with $Q$ of side length $2^m\mathrm{\ell }(Q)$. The centres of $Q$ and $R$ both lie in $R$, so their coordinatewise distance is at most $\mathrm{\ell }(R)\mathrm{/}2$; every point of $R$ is therefore within coordinate distance $\mathrm{\ell }(R)$ of the centre of $Q$, and $R\subseteq R^{\mathrm{\prime }}$. Since $Q⊊R$, $\mathrm{\ell }(Q)<\mathrm{\ell }(R)$, so $m\ge 1$. Minimality gives $2^{m-1}\mathrm{\ell }(Q)<2\mathrm{\ell }(R)$ and hence $2^m\mathrm{\ell }(Q)<4\mathrm{\ell }(R)$. Therefore $\mathrm{\mid }{R}^{\mathrm{\prime }}\mathrm{\mid }=2^{mn}\mathrm{\mid }Q\mathrm{\mid }<4^n\mathrm{\mid }R\mathrm{\mid }$.

Let $Q=Q_0\subseteq Q_1\subseteq \cdots \subseteq Q_m=R^{\mathrm{\prime }}$ be the concentric cubes of side lengths $2^k\mathrm{\ell }(Q)$. Since $\mathrm{\mid }{Q}_{k+1}\mathrm{\mid }=2^n\mathrm{\mid }{Q}_k\mathrm{\mid }$, [F1] gives $\mathrm{\mid }{b}_{Q_k}-b_{Q_{k+1}}\mathrm{\mid }\le \mathrm{\mid }{Q}_k{\mathrm{\mid }}^{-1}{\int }_{Q_k}\mathrm{\mid }b-b_{Q_{k+1}}\mathrm{\mid }\le 2^n\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$ for each $k<m$, and the triangle inequality over the $m$ steps gives $\mathrm{\mid }{b}_Q-b_{R^{\mathrm{\prime }}}\mathrm{\mid }\le m{2}^n\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$.

Since $R\subseteq R^{\mathrm{\prime }}$, [F1] and step 1.1 give $\mathrm{\mid }{b}_{R^{\mathrm{\prime }}}-b_R\mathrm{\mid }\le \mathrm{\mid }R{\mathrm{\mid }}^{-1}{\int }_R\mathrm{\mid }b-b_{R^{\mathrm{\prime }}}\mathrm{\mid }\le (\mathrm{\mid }{R}^{\mathrm{\prime }}\mathrm{\mid }\mathrm{/}\mathrm{\mid }R\mathrm{\mid })\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le 4^n\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$.

Combining steps 2.1 and 2.2, $\mathrm{\mid }{b}_Q-b_R\mathrm{\mid }\le (2^{n}m+4^n)\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$. From $2^m\mathrm{\ell }(Q)<4\mathrm{\ell }(R)$ we get $m<2+{\log }_2(\mathrm{\ell }(R)\mathrm{/}\mathrm{\ell }(Q))$, hence $m\le 2+{\log }_2(\mathrm{\ell }(R)\mathrm{/}\mathrm{\ell }(Q))\le 2(1+{\log }_2(\mathrm{\ell }(R)\mathrm{/}\mathrm{\ell }(Q)))$. Therefore $\mathrm{\mid }{b}_Q-b_R\mathrm{\mid }\le (2^{n+1}+4^n)(1+{\log }_2(\mathrm{\ell }(R)\mathrm{/}\mathrm{\ell }(Q)))\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$, which is the claim with $C_n=2^{n+1}+4^n$.