# On BMO and Carleson measures on Riemannian manifolds

## TL;DR

This paper characterizes BMO functions on certain Riemannian manifolds via Carleson measure conditions of their harmonic extensions, extending classical results and applying advanced harmonic analysis techniques.

## Contribution

It extends BMO and Carleson measure characterizations to Riemannian manifolds with specific curvature conditions, and applies these to Jacobian estimates for Sobolev maps.

## Key findings

- BMO functions characterized by Carleson measure conditions on manifolds.
- Extension of Coifman–Lions–Meyer–Semmes theorem to Riemannian manifolds.
- Use of harmonic extensions and trace space techniques in analysis.

## Abstract

Let $\mathcal{M}$ be a Riemannian $n$-manifold with a metric such that the manifold is Ahlfors-regular. We also assume either non-negative Ricci curvature, or that the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. We characterize BMO-functions $u: \mathcal{M} \to \mathbb{R}$ by a Carleson measure condition of their $\sigma$-harmonic extension $U: \mathcal{M} \times (0,\infty) \to \mathbb{R}$. We make crucial use of a $T(b)$ theorem proved by Hofmann, Mitrea, Mitrea, and Morris. As an application we show that the famous theorem of Coifman--Lions--Meyer--Semmes holds in this class of manifolds: Jacobians of $W^{1,n}$-maps from $\mathcal{M}$ to $\mathbb{R}^n$ can be estimated against BMO-functions, which now follows from the arguments for commutators recently proposed by Lenzmann and the second-named author using only harmonic extensions, integration by parts, and trace space characterizations.