Dimension-free estimates for semigroup BMO and $A_p$

Abstract

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 $\mathrm{BMO}({\mathbb{R}}^n)$ on balls. In particular, we show that the John--Nirenberg constant of this space decays with dimension no faster than $n^{-1/2}.$