Title: Dimension-free estimates for semigroup BMO and related classes
Speaker: Leonid Slavin (University of Cincinnati)
Abstract: 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 the same estimate holds for ${\rm BMO}_K(\mathbb{R}^n),$ with all Lebesgue 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 Zatitiskiy.