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

Let $ℒ=$ -div $\left(A\left(x\right)\nabla \right)$ 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, 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.

Soit $ℒ=$ -div $\left(A\left(x\right)\nabla \right)$ 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, 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.

DOI : https://doi.org/10.5802/aif.2094
Classification:  32J15,  35J25,  42B20
Keywords: Riesz transform, elliptic operator, Lipschitz domain
@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},
publisher = {Association des Annales de l'institut Fourier},
volume = {55},
number = {1},
year = {2005},
pages = {173-197},
doi = {10.5802/aif.2094},
zbl = {1068.47058},
mrnumber = {2141694},
language = {en},
url = {http://www.numdam.org/item/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://www.numdam.org/item/AIF_2005__55_1_173_0/

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