Let $K_t$ be either the heat or the Poisson kernel on $R^n$. Let $\mathcal{A}$ stand either for $BMO$ equipped with the quadratic seminorm or for $A_p$, $1< p\leq +\infty$. 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 $R^n$: If a given integral functional admits an estimate on $\mathcal{A}(I)$, then the same estimate holds for $\mathcal{A}_K(R^n)$, 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 ball-based $BMO(R^n)$. In particular, we show that the constant in the John-Nirenberg inequality for this space decays with dimension no faster than $n^{-1/2}$.

The talk is based on joint work with Leonid Slavin.