Hardy spaces of differential forms and Riesz transforms on Riemannian manifolds

Pascal Auscher; Alan McIntosh; Emmanuel Russ

doi:10.1016/j.crma.2006.11.023

Harmonic Analysis

Hardy spaces of differential forms and Riesz transforms on Riemannian manifolds
[Espaces de Hardy de formes différentielles et transformées de Riesz sur des variétés riemanniennes]

Présenté par : Yves Meyer

Pascal Auscher ¹ ; Alan McIntosh ² ; Emmanuel Russ ³

¹ CNRS UMR 8628, université Paris-sud, 91405 Orsay cedex, France
² Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia
³ Université Paul-Cézanne LATP, avenue Escadrille Normandie-Niemen, 13397 Marseille cedex 20, France

Comptes Rendus. Mathématique, Volume 344 (2007) no. 2, pp. 103-108.

Abstract (VO)
Résumé

Let M be a complete Riemannian manifold. Assuming that the Riemannian measure is doubling, we define, for all $1 ⩽ p ⩽ + \infty$ , a Hardy space $H^{p} (Λ T^{*} M)$ of differential forms on M, and give two alternative characterizations of $H^{1} (Λ T^{*} M)$ . We also prove, for all $1 ⩽ p ⩽ + \infty$ , the $H^{p} (Λ T^{*} M)$ boundedness of Riesz transforms on M, and show that $H^{p} (Λ T^{*} M)$ has a bounded holomorphic functional calculus.

Soit M une variété riemannienne complète. Sous l'hypothèse que la mesure riemannienne est doublante, on définit, pour tout $1 ⩽ p ⩽ + \infty$ , un espace de Hardy $H^{p} (Λ T^{*} M)$ de formes différentielles sur M, et on donne deux autres caractérisations de $H^{1} (Λ T^{*} M)$ . On prouve également, pour tout $1 ⩽ p ⩽ + \infty$ , la continuité sur $H^{p} (Λ T^{*} M)$ des transformées de Riesz sur M, et on montre que $H^{p} (Λ T^{*} M)$ possède un calcul fonctionnel holomorphe borné.

Reçu le : 2006-08-22
Accepté le : 2006-11-14
Publié le : 2006-12-21

DOI : 10.1016/j.crma.2006.11.023

Affiliations des auteurs :

Pascal Auscher ¹ ; Alan McIntosh ² ; Emmanuel Russ ³

¹ CNRS UMR 8628, université Paris-sud, 91405 Orsay cedex, France
² Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia
³ Université Paul-Cézanne LATP, avenue Escadrille Normandie-Niemen, 13397 Marseille cedex 20, France

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     author = {Pascal Auscher and Alan McIntosh and Emmanuel Russ},
     title = {Hardy spaces of differential forms and {Riesz} transforms on {Riemannian} manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {103--108},
     publisher = {Elsevier},
     volume = {344},
     number = {2},
     year = {2007},
     doi = {10.1016/j.crma.2006.11.023},
     language = {en},
}

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Pascal Auscher; Alan McIntosh; Emmanuel Russ. Hardy spaces of differential forms and Riesz transforms on Riemannian manifolds. Comptes Rendus. Mathématique, Volume 344 (2007) no. 2, pp. 103-108. doi : 10.1016/j.crma.2006.11.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.11.023/

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Peter Sjögren; Maria Vallarino Hardy spaces and Riesz transforms on a Lie group of exponential growth, Proceedings of the Edinburgh Mathematical Society (2025), p. 1 | DOI:10.1017/s0013091524000920
Marcin Preisner; Adam Sikora; Lixin Yan Hardy spaces meet harmonic weights, Transactions of the American Mathematical Society, Volume 375 (2022) no. 9, pp. 6417-6451 | DOI:10.1090/tran/8695 | Zbl:1496.42030
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Pascal Auscher; Alan McIntosh; Emmanuel Russ Hardy spaces of differential forms on Riemannian manifolds, The Journal of Geometric Analysis, Volume 18 (2008) no. 1, pp. 192-248 | DOI:10.1007/s12220-007-9003-x | Zbl:1217.42043

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