Maximal characterisation of local Hardy spaces on locally doubling manifolds

Alessio Martini; Stefano Meda; Maria Vallarino

doi:10.1007/s00209-021-02856-x

Maximal characterisation of local Hardy spaces on locally doubling manifolds

Alessio Martini, Stefano Meda, Maria Vallarino

Mathematics

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Abstract

We prove a radial maximal function characterisation of the local atomic Hardy space h¹(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 h¹(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.

Original language	English
Journal	Mathematische Zeitschrift
Early online date	30 Aug 2021
DOIs	https://doi.org/10.1007/s00209-021-02856-x
Publication status	E-pub ahead of print - 30 Aug 2021

Bibliographical note

Funding Information:
Work partially supported by PRIN 2015 “Real and complex manifolds: geometry, topology and harmonic analysis”. The first- and third-named authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Publisher Copyright:
© 2021, The Author(s).

Keywords

Hardy space
Maximal function
Riemannian manifold
Exponential growth
Locally doubling space

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Cite this

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abstract = "We prove a radial maximal function characterisation of the local atomic Hardy space h1(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 h1(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 H{\"o}lder 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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AU - Meda, Stefano

AU - Vallarino, Maria

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N2 - We prove a radial maximal function characterisation of the local atomic Hardy space h1(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 h1(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.

AB - We prove a radial maximal function characterisation of the local atomic Hardy space h1(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 h1(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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KW - Riemannian manifold

KW - Exponential growth

KW - Locally doubling space

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Maximal characterisation of local Hardy spaces on locally doubling manifolds

Abstract

Bibliographical note

Keywords

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