# Analysis Seminar

## Dimension-free estimates for semigroup BMO and related classes

Leonid Slavin, University of Cincinnati

Friday, December 1st, 2017
1:00 PM - 2:00 PM
Hylan 101

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

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.

Event contact: xchen84 at ur dot rochester dot edu