Vol. 15, No. 5, 2022

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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
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 Lp of (d) Lp(Λ1TM) when ν(ν 1) < p < ν, where ν > 2 is related to the volume growth. Throughout, (d) denotes the range of d as an unbounded operator from L2 to L2(Λ1TM). 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