Boundedness from H 1 to L 1 of Riesz transforms on a Lie group of exponential growth
Annales de l'Institut Fourier, Volume 58 (2008) no. 4, pp. 1117-1151.

Let G be the Lie group 2 + endowed with the Riemannian symmetric space structure. Let X 0 ,X 1 ,X 2 be a distinguished basis of left-invariant vector fields of the Lie algebra of G and define the Laplacian Δ=-(X 0 2 +X 1 2 +X 2 2 ). In this paper we consider the first order Riesz transforms R i =X i Δ -1/2 and S i =Δ -1/2 X i , for i=0,1,2. We prove that the operators R i , but not the S i , are bounded from the Hardy space H 1 to L 1 . We also show that the second-order Riesz transforms T ij =X i Δ -1 X j are bounded from H 1 to L 1 , while the transforms S ij =Δ -1 X i X j and R ij =X i X j Δ -1 , for i,j=0,1,2, are not.

On considère le groupe de Lie G= 2 + muni de la structure Riemannienne d’espace symétrique. On choisit une base X 0 , X 1 , X 2 de champs vectoriels invariants à gauche de l’algèbre de Lie de G et on définit le Laplacien Δ=-(X 0 2 +X 1 2 +X 2 2 ). Dans cet article nous considérons les transformées de Riesz du premier ordre R i =X i Δ -1/2 et S i =Δ -1/2 X i , avec i=0,1,2. Nous prouvons que les opérateurs R i , mais non pas les S i , sont bornés de l’espace de Hardy H 1 à L 1 . Nous démontrons aussi que les transformées de Riesz du deuxième ordre T ij =X i Δ -1 X j sont bornées de H 1 à L 1 , tandis que les transformées S ij =Δ -1 X i X j et R ij =X i X j Δ -1 , i,j=0,1,2, ne sont pas bornées.

DOI: 10.5802/aif.2380
Classification: 43A80,  42B20,  42B30,  22E30
Keywords: Singular integrals, Riesz transforms, Hardy space, Lie groups, exponential growth
Sjögren, Peter 1; Vallarino, Maria 2

1 Göteborg University and Chalmers University of Technology Department of Mathematical Sciences 412 96 Göteborg (Sweden)
2 Università di Milano-Bicocca Dipartimento di Matematica e Applicazioni Via R. Cozzi 53 20125 Milano (Italy)
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Sjögren, Peter; Vallarino, Maria. Boundedness from $H^1$ to $L^1$ of Riesz transforms on a Lie group of exponential growth. Annales de l'Institut Fourier, Volume 58 (2008) no. 4, pp. 1117-1151. doi : 10.5802/aif.2380. http://www.numdam.org/articles/10.5802/aif.2380/

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