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 $\left(M,g\right)$ 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 $\mathsc{ℛ}\left(d\right)\cap {L}^{p}\left({\Lambda }^{1}{T}^{\ast }M\right)$ when $\nu ∕\left(\nu -1\right), where $\nu >2$ is related to the volume growth. Throughout, $\mathsc{ℛ}\left(d\right)$ denotes the range of $d$ as an unbounded operator from ${L}^{2}$ to ${L}^{2}\left({\Lambda }^{1}{T}^{\ast }M\right)$. 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