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

Sjögren, Peter; Vallarino, Maria

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

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 $\Delta =-(X_0^2+X_1^2+X_2^2)$. Dans cet article nous considérons les transformées de Riesz du premier ordre $R_i=X_i{\Delta }^{-1/2}$ et $S_i={\Delta }^{-1/2}{X}_i$, 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 $T_{ij}=X_i{\Delta }^{-1}{X}_j$ sont bornées de $H^1$ à $L^1$, tandis que les transformées $S_{ij}={\Delta }^{-1}{X}_i{X}_j$ et $R_{ij}=X_i{X}_j{\Delta }^{-1}$, $i,j=0,1,2$, 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 $\Delta =-(X_0^2+X_1^2+X_2^2)$. In this paper we consider the first order Riesz transforms $R_i=X_i{\Delta }^{-1/2}$ and $S_i={\Delta }^{-1/2}{X}_i$, 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 $T_{ij}=X_i{\Delta }^{-1}{X}_j$ are bounded from $H^1$ to $L^1$, while the transforms $S_{ij}={\Delta }^{-1}{X}_i{X}_j$ and $R_{ij}=X_i{X}_j{\Delta }^{-1}$, for $i,j=0,1,2$, are not.

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