Hardy spaces on Riemannian manifolds with quadratic curvature decay

Devyver, Baptiste; Russ, Emmanuel

doi:10.2140/apde.2022.15.1169

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Hardy spaces on Riemannian manifolds with quadratic curvature decay

Baptiste Devyver and Emmanuel Russ

Vol. 15 (2022), No. 5, 1169–1213

DOI: 10.2140/apde.2022.15.1169

Abstract

Let $(M, g)$ be a complete Riemannian manifold. Assume that the Ricci curvature of $M$ has quadratic decay and that the volume growth is strictly faster than quadratic. We establish that the Hardy spaces of exact $1$ -differential forms on $M$ , introduced by Auscher et al. (J. Geom. Anal. 18:1 (2008), 192–248), coincide with the closure in $L^{p}$ of $ℛ (d) \cap L^{p} (Λ^{1} T^{*} M)$ when $ν ∕ (ν - 1) < p < ν$ , where $ν > 2$ is related to the volume growth. Throughout, $ℛ (d)$ denotes the range of $d$ as an unbounded operator from $L^{2}$ to $L^{2} (Λ^{1} T^{*} M)$ . This result applies, in particular, when $M$ has a finite number of Euclidean ends.

Keywords

Hardy spaces, Riesz transforms, heat kernel

Mathematical Subject Classification

Primary: 42B30, 58J35

Milestones

Received: 18 October 2019

Revised: 9 November 2020

Accepted: 18 January 2021

Published: 29 September 2022

Authors

Baptiste Devyver
Institut Fourier Université Grenoble Alpes Saint-Martin d’Hères France	Department of Mathematics Technion, Israel Institute of Technology Haifa Israel

Emmanuel Russ
Institut Fourier Université Grenoble Alpes Saint-Martin d’Hères France

Vol. 15, No. 5, 2022