# Dimension-free estimates for semigroup BMO and $A_p$

## TL;DR

This paper establishes dimension-free estimates for semigroup BMO and A_p spaces, showing that certain integral estimates transfer from intervals to Euclidean spaces with heat or Poisson kernels, and analyzes the decay of the John--Nirenberg constant.

## Contribution

It introduces a transference principle linking BMO and A_p estimates on intervals to their K-versions on al{R}^n, ensuring dimension-free bounds and analyzing the John--Nirenberg constant decay.

## Key findings

- Estimates on intervals transfer to al{R}^n with K-averages.
- All such estimates are dimension-free.
- The John--Nirenberg constant decays at least as fast as n^{-1/2}.

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