Let -div be a second order elliptic operator with real, symmetric, bounded measurable coefficients on or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed , a necessary and sufficient condition is obtained for the boundedness of the Riesz transform on the space. As an application, for , we establish the boundedness of Riesz transforms on Lipschitz domains for operators with coefficients. The range of is sharp. The closely related boundedness of on weighted spaces is also studied.
Soit -div un opérateur elliptique du second ordre à coefficients réels mesurables bornés symétriques sur ou sur un domaine à bord Lipschitzien, soumis à une condition au bord de type Dirichlet. Pour tout , nous obtenons une condition nécessaire et suffisante pour que la transformée de soit bornée sur l’espace . A titre d’application, nous établissons pour , le caractère borné en norme des transformées de Riez d’opérateurs à coefficients sur les domaines à bord Lipschitzien. L’intervalle obtenu pour est optimal. Nous étudions également si est borné dans les espaces à poids.
Keywords: Riesz transform, elliptic operator, Lipschitz domain
Mot clés : transformées de Riesz, opérateur elliptique, domaine à bord Lipschitzien
@article{AIF_2005__55_1_173_0, author = {Shen, Zhongwei}, title = {Bounds of {Riesz} {Transforms} on $L^p$ {Spaces} for {Second} {Order} {Elliptic} {Operators}}, journal = {Annales de l'Institut Fourier}, pages = {173--197}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {55}, number = {1}, year = {2005}, doi = {10.5802/aif.2094}, mrnumber = {2141694}, zbl = {1068.47058}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2094/} }
TY - JOUR AU - Shen, Zhongwei TI - Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators JO - Annales de l'Institut Fourier PY - 2005 SP - 173 EP - 197 VL - 55 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2094/ DO - 10.5802/aif.2094 LA - en ID - AIF_2005__55_1_173_0 ER -
%0 Journal Article %A Shen, Zhongwei %T Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators %J Annales de l'Institut Fourier %D 2005 %P 173-197 %V 55 %N 1 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2094/ %R 10.5802/aif.2094 %G en %F AIF_2005__55_1_173_0
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://archive.numdam.org/articles/10.5802/aif.2094/
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