Hardy spaces of differential forms and Riesz transforms on Riemannian manifolds

Auscher, Pascal; McIntosh, Alan; Russ, Emmanuel

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

Auscher, Pascal ¹ ; McIntosh, Alan ² ; Russ, Emmanuel 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, Tome 344 (2007) no. 2, pp. 103-108.

Résumé
Abstract

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.

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.

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 :

Auscher, Pascal ¹ ; McIntosh, Alan ² ; Russ, Emmanuel ³

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Auscher, Pascal; McIntosh, Alan; Russ, Emmanuel. Hardy spaces of differential forms and Riesz transforms on Riemannian manifolds. Comptes Rendus. Mathématique, Tome 344 (2007) no. 2, pp. 103-108. doi : 10.1016/j.crma.2006.11.023. http://www.numdam.org/articles/10.1016/j.crma.2006.11.023/

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