A family of Hardy-type spaces on nondoubling manifolds

Martini, Alessio; Meda, Stefano; Vallarino, Maria

doi:10.1007/s10231-020-00956-9

Cited by 5 publications

(16 citation statements)

References 44 publications

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“…It is straightforward to check that the function within square brackets is in H ∞ 0 (S ϕ ) and (τ + λ) −1/2 is in E (S ϕ ) by [Haa,Lemma 2.2.3]. The operator L is sectorial of angle π/2 on h 1 (N ) [MaMV1,Theorem 3.1]. Therefore the natural functional calculus [Haa,Theorem 2.3.3] implies that D − D τ is bounded on h 1 (N ).…”

Section: Writementioning

confidence: 99%

“…We observe in passing that on certain examples of nondoubling measure spaces, such as the hyperbolic disc, a perhaps surprising phenomenon occurs: the Hardy-type spaces defined in terms of the Riesz transform, the Poisson maximal operator and the heat maximal operator are different spaces [MaMVV]. For more on the attempts to define an effective Hardy-type space on noncompact symmetric spaces and generalisations thereof, see [An,Io,Lo,CMM1,MMV2,MMV3,MaMV1] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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Local Riesz transform and local Hardy spaces on Riemannian manifolds with bounded geometry

Meda¹,

Veronelli²

2020

Preprint

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We prove that if τ is a large positive number, then the atomic Goldbergtype space h 1 (N ) and the space h 1 Rτ (N ) of all integrable functions on N whose local Riesz transform Rτ is integrable are the same space on any complete noncompact Riemannian manifold N with Ricci curvature bounded from below and positive injectivity radius. We also relate h 1 (N ) to a space of harmonic functions on the slice N × (0, δ) for δ > 0 small enough.

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Section: Writementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Local Riesz transform and local Hardy spaces on Riemannian manifolds with bounded geometry

Meda¹,

Veronelli²

2020

Preprint

Self Cite

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“…It is virtually impossible in a research paper to give appropriate credit to all the mathematicians who have contributed to develop the theory of Hardy spaces in an impressive variety of settings besides R n . Some pointers on the existing literature may be found in the introduction of [MaMV1], to which we refer the interested reader.…”

Section: Introductionmentioning

confidence: 99%

“…It is natural to speculate whether a similar equality holds in wider generality. A discussion of this interesting problem may be found in the introduction of [MaMV1], to which the reader is referred for further information. Here we content ourselves to mention that, as a consequence of the efforts of various authors [AMR, DKKP, HLMMY],…”

Section: Introductionmentioning

confidence: 99%

Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds

Martini¹,

Meda²,

Vallarino³

et al. 2022

Preprint

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In this paper we establish inclusions and noninclusions between various Hardy type spaces on noncompact Riemannian manifolds M with Ricci curvature bounded from below, positive injectivity radius and spectral gap.Our first main result states that, if L is the positive Laplace-Beltrami operator on M , then the Riesz-Hardy space H 1 R (M ) is the isomorphic image of the Goldberg type space h 1 (M ) via the map L 1/2 (I + L ) −1/2 , a fact that is false in R n . Specifically, H 1 R (M ) agrees with the Hardy type space X 1/2 (M ) recently introduced by the the first three authors; as a consequence, we prove that H 1 R (M ) does not admit an atomic characterisation. Noninclusions are mostly proved in the special case where the manifold is a Damek-Ricci space S. Our second main result states that H 1 R (S), the heat Hardy space H 1 H (S) and the Poisson-Hardy space H 1 P (S) are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree.

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“…in the context of a nondoubling manifold M. Nevertheless, the results in the present paper turn out to be instrumental in the analysis of such spaces and their relation to the Hardy-type spaces X γ (M) introduced in [13,20], which we plan to develop in a future work [14].…”

Section: Introductionmentioning

confidence: 99%

Maximal characterisation of local Hardy spaces on locally doubling manifolds

2021

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We prove a radial maximal function characterisation of the local atomic Hardy space $${{\mathfrak {h}}}^1(M)$$ h 1 ( M ) on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to $${{\mathfrak {h}}}^1(M)$$ h 1 ( M ) if and only if either its local heat maximal function or its local Poisson maximal function is integrable. A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. Uchiyama.

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A family of Hardy-type spaces on nondoubling manifolds

Cited by 5 publications

References 44 publications

Local Riesz transform and local Hardy spaces on Riemannian manifolds with bounded geometry

Local Riesz transform and local Hardy spaces on Riemannian manifolds with bounded geometry

Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds

Maximal characterisation of local Hardy spaces on locally doubling manifolds

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