  Let $K_t$ be either the heat or the Poisson kernel on $\mathbb{R}^n.$ Let
$\mathcal{A}$ stand either for BMO equipped with the quadratic seminorm or for
$A_p,$ $1< p\le\infty.$ We establish the following transference between the
class $\mathcal{A}$ on an interval $I\subset\mathbb{R}$ and its $K$-version,
$\mathcal{A}^K,$ on $\mathbb{R}^n$: If a given integral functional admits an
estimate on $\mathcal{A}(I),$ then the same estimate holds for
$\mathcal{A}^K(\mathbb{R}^n),$ with all Lebesgue averages replaced by
$K$-averages. In particular, all such estimates are dimension-free. As an
application, via the heat kernel, we obtain a weakly-dimensional theory for
${\rm BMO}(\mathbb{R}^n)$ on balls. In particular, we show that the
John--Nirenberg constant of this space decays with dimension no faster than
$n^{-1/2}.$
