Equivalence of Hardy Spaces on Noncompact Manifolds

Definitions and mathematical statement

Let $M$ be a complete, connected, noncompact Riemannian manifold, and let $\mu$ denote its Riemannian volume measure. The supplied record asks about three Hardy spaces associated with the Riesz transform, heat maximal operator, and Poisson maximal operator, but does not specify their full definitions.

For clarity, the following describes a conventional global interpretation, conditional on choices that require confirmation from the source. If $M$ has no boundary, let $L=-\operatorname{div}\nabla$ be the nonnegative self-adjoint Laplace--Beltrami operator on $L^2(M,\mu)$. The associated operators are formally \[ \mathcal R=\nabla L^{-1/2},\qquad \mathcal M_H f(x)=\sup_{t>0}|e^{-tL}f(x)|,\qquad \mathcal M_P f(x)=\sup_{t>0}|e^{-t\sqrt L}f(x)|. \] Here $\mathcal Rf$ is a tangent vector field, whose $L^1$ norm uses the Riemannian pointwise norm. The spectral treatment of $\ker L$, the initial domain of $\mathcal R$, and its extension to the functions under consideration must be specified. If the semigroups and Riesz transform are given suitable extensions, conventional candidate definitions are \[ \begin{aligned} H_R^1(M)&=\{f\in L^1(M,\mu):\mathcal Rf\in L^1(TM,\mu)\},\\ H_H^1(M)&=\{f\in L^1(M,\mu):\mathcal M_Hf\in L^1(M,\mu)\},\\ H_P^1(M)&=\{f\in L^1(M,\mu):\mathcal M_Pf\in L^1(M,\mu)\}. \end{aligned} \] These formulas describe a possible interpretation; the supplied record does not confirm these precise conventions, whether the full range $t>0$ is intended, or whether the spaces are instead defined by completion of an initial domain. If manifolds with boundary are allowed, an operator realization and boundary conditions are also missing.

The supplied mathematical question is: for every complete, connected, noncompact Riemannian manifold $M$, do the intended spaces satisfy \[ H_R^1(M)=H_H^1(M)=H_P^1(M), \] and do they admit atomic decompositions? No additional geometric hypotheses are stated. The record does not specify whether coincidence includes equivalence of the defining norms.

An atomic decomposition requires a specified class $\mathcal A(M)$ of atoms and a specified topology of convergence for representations \[ f=\sum_{j=1}^{\infty}\lambda_j a_j, \qquad a_j\in\mathcal A(M),\qquad \sum_{j=1}^{\infty}|\lambda_j|<\infty. \] The record supplies neither this atom class nor its support, size, cancellation, scale, convergence, or norm-characterization requirements. Consequently, the displayed question preserves the stated target, but a uniquely specified atomic-decomposition assertion and definitive Hardy-space definitions cannot be recovered from the supplied record alone.

Short English statement

The question asks whether, on every complete, connected, noncompact Riemannian manifold, the Hardy spaces defined using the Riesz transform, heat maximal operator, and Poisson maximal operator coincide and have atomic decompositions. The exact operator conventions and admissible atoms are unspecified. The supplied status is partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry: supplied source record for permanent ID $30003538$, entitled Equivalence of Hardy Spaces on Noncompact Manifolds, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2017$ and names no individual proposer.
• Supplied report citation: Real Analysis, Harmonic Analysis, and Applications (2018), Oberwolfach Reports, pp. 2101--2164, DOI 10.4171/owr/2017/34. https://doi.org/10.4171/owr/2017/34. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Meda, Martini, Vallarino, Hardy spaces report. https://doi.org/10.4171/owr/2017/34.
• Supplied literature source: Martini--Meda--Vallarino, nondoubling Hardy spaces. https://arxiv.org/abs/1908.10057.
• Supplied literature source: local Hardy spaces on manifolds. https://link.springer.com/article/10.1007/s00209-021-02856-x.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that heat and Poisson maximal Hardy spaces have atomic characterizations under standard doubling and bounded-geometry hypotheses, that special symmetric spaces are understood, and that the general equality with the Riesz-transform Hardy space and atomic decomposition remains unresolved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.