Bounds of Riesz Transforms on L p Spaces for Second Order Elliptic Operators
Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 173-197.

Let = -div (A(x)) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on 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 () -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 () -1/2 on weighted L 2 spaces is also studied.

Soit = -div (A(x)) un opérateur elliptique du second ordre à coefficients réels mesurables bornés symétriques sur 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 () -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 () -1/2 est borné dans les espaces L 2 à poids.

DOI: 10.5802/aif.2094
Classification: 32J15,  35J25,  42B20
Keywords: Riesz transform, elliptic operator, Lipschitz domain
Shen, Zhongwei 1

1 University of Kentucky, Department of Mathematics, Lexington, KY 40506 (USA)
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Shen, Zhongwei. Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators. Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 173-197. doi : 10.5802/aif.2094. http://www.numdam.org/articles/10.5802/aif.2094/

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