Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators

Shen, Zhongwei

Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 173-197.

Soit $ℒ=$ -div $\left(A\right(x\left)\nabla \right)$ un opérateur elliptique du second ordre à coefficients réels mesurables bornés symétriques sur ${ℝ}^n$ ou sur un domaine à bord Lipschitzien, soumis à une condition au bord de type Dirichlet. Pour tout $p>2$, nous obtenons une condition nécessaire et suffisante pour que la transformée de $\nabla {\left(ℒ\right)}^{-1/2}$ soit bornée sur l’espace $L^p$. A titre d’application, nous établissons pour $1<p<3+ϵ$, le caractère borné en norme $L^p$ des transformées de Riez d’opérateurs à coefficients $VMO$ sur les domaines à bord Lipschitzien. L’intervalle obtenu pour $p$ est optimal. Nous étudions également si $\nabla {\left(ℒ\right)}^{-1/2}$ est borné dans les espaces $L^2$ à poids.

Let $ℒ=$ -div $\left(A\right(x\left)\nabla \right)$ be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ${ℝ}^n$ or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed $p>2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform $\nabla {\left(ℒ\right)}^{-1/2}$ on the $L^p$ space. As an application, for $1<p<3+ϵ$, we establish the $L^p$ boundedness of Riesz transforms on Lipschitz domains for operators with $VMO$ coefficients. The range of $p$ is sharp. The closely related boundedness of $\nabla {\left(ℒ\right)}^{-1/2}$ on weighted $L^2$ spaces is also studied.

Classification : 32J15, 35J25, 42B20
Keywords: Riesz transform, elliptic operator, Lipschitz domain
Mot clés : transformées de Riesz, opérateur elliptique, domaine à bord Lipschitzien

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