On BMO and Carleson Measures on Riemannian Manifolds. (11th June 2020)

• Record Type: Journal Article
• Authors: Brazke, Denis; Schikorra, Armin; Sire, Yannick
• Is Part Of: International mathematics research notices. Volume 2022:Number 2(2022)
• Page Start: 1245
• Page End: 1269
• Publication Date: 2020-06-11
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa140
• Languages: English
• ISSNs: 1073-7928

Abstract

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 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 2nd-named author using only harmonic extensions, integration by parts, and trace space characterizations.