Let \(K_t\) be either the heat or the Poisson kernel on \(\mathbb{R}^n\) and consider \({\rm BMO}_K(\mathbb{R}^n)\) equipped with the norm

\[\|\varphi\|^{}_K:=\sup_{z\in\mathbb{R}^{n+1}_+} \big(\varphi^2(z)-\varphi(z)^2\big)^{1/2},\]

where \(g(z)\) denotes the \(K\)-extension of a function \(g\) on \(\mathbb R^n\) into the upper half-space: \(g(x,t)=(K_t*g)(x).\)

We establish the following transference principle between the classical \({\rm BMO}(Q)\) on an interval and \({\rm BMO}_K(\mathbb{R}^n):\) If an integral functional admits an estimate on \({\rm BMO}(Q),\) then exactly the same estimate holds for \({\rm BMO}_K(\mathbb{R}^n),\) with all Euclidean averages replaced by \(K\)-averages. In particular, all such estimates are dimension-free. The proof uses Bellman functions for \({\rm BMO}(Q)\) as locally concave majorants for their \(K\)-analogs, in conjunction with the probabilistic representation of the kernel \(K_t.\) Analogous results hold for related function classes, such as \(A_p.\) This is joint work with Pavel Zatitiskii.