Boundedness from $H^1$ to $L^1$ of Riesz transforms on a Lie group of exponential growth

Sjögren, Peter; Vallarino, Maria

doi:10.5802/aif.2380

Boundedness from

H^{1}

L^{1}

of Riesz transforms on a Lie group of exponential growth
[Transformées de Riesz bornées de

H^{1}

L^{1}

sur un groupe de Lie à croissance exponentielle]

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)

Annales de l'Institut Fourier, Tome 58 (2008) no. 4, pp. 1117-1151.

Résumé
Abstract

On considère le groupe de Lie $G = ℝ^{2} ⋉ ℝ^{+}$ 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 $T_{i j} = X_{i} Δ^{- 1} X_{j}$ sont bornées de $H^{1}$ à $L^{1}$ , tandis que les transformées $S_{i j} = Δ^{- 1} X_{i} X_{j}$ et $R_{i j} = X_{i} X_{j} Δ^{- 1}$ , $i, j = 0, 1, 2$ , ne sont pas bornées.

Let $G$ be the Lie group $ℝ^{2} ⋉ ℝ^{+}$ 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 $T_{i j} = X_{i} Δ^{- 1} X_{j}$ are bounded from $H^{1}$ to $L^{1}$ , while the transforms $S_{i j} = Δ^{- 1} X_{i} X_{j}$ and $R_{i j} = X_{i} X_{j} Δ^{- 1}$ , for $i, j = 0, 1, 2$ , are not.

DOI : 10.5802/aif.2380

Classification : 43A80, 42B20, 42B30, 22E30
Keywords: Singular integrals, Riesz transforms, Hardy space, Lie groups, exponential growth
Mot clés : intégrales singulières, transformées de Riesz, espaces de Hardy, groupes de Lie, croissance exponentielle

Affiliations des auteurs :

Sjögren, Peter ¹ ; Vallarino, Maria ²

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     title = {Boundedness from $H^1$ to $L^1$ of {Riesz} transforms on a {Lie} group of exponential growth},
     journal = {Annales de l'Institut Fourier},
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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, Tome 58 (2008) no. 4, pp. 1117-1151. doi : 10.5802/aif.2380. https://aif.centre-mersenne.org/articles/10.5802/aif.2380/

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