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

Zhongwei Shen[1]

  • [1] University of Kentucky, Department of Mathematics, Lexington, KY 40506 (USA)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 1, page 173-197
  • ISSN: 0373-0956

Abstract

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Let ℒ = -div ( A ( x ) ∇ ) 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 ∇ ( ℒ ) - 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 V M O coefficients. The range of p is sharp. The closely related boundedness of ∇ ( ℒ ) - 1 / 2 on weighted L 2 spaces is also studied.

How to cite

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Shen, Zhongwei. "Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators." Annales de l’institut Fourier 55.1 (2005): 173-197. <http://eudml.org/doc/116183>.

@article{Shen2005,
abstract = {Let $\{\mathcal \{L\}\} =$ -div $(A(x)\nabla )$ be a second order elliptic operator with real, symmetric, bounded measurable coefficients on $\{\mathbb \{R\}\} ^n$ or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed $p&gt;2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform $\nabla (\{\mathcal \{L\}\})^\{-1/2\}$ on the $L^p$ space. As an application, for $1&lt;p&lt;3+\epsilon $, 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 $\nabla (\{\mathcal \{L\}\})^\{-1/2\}$ on weighted $L^2$ spaces is also studied.},
affiliation = {University of Kentucky, Department of Mathematics, Lexington, KY 40506 (USA)},
author = {Shen, Zhongwei},
journal = {Annales de l’institut Fourier},
keywords = {Riesz transform; elliptic operator; Lipschitz domain},
language = {eng},
number = {1},
pages = {173-197},
publisher = {Association des Annales de l'Institut Fourier},
title = {Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators},
url = {http://eudml.org/doc/116183},
volume = {55},
year = {2005},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 1
SP - 173
EP - 197
AB - Let ${\mathcal {L}} =$ -div $(A(x)\nabla )$ be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ${\mathbb {R}} ^n$ or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed $p&gt;2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform $\nabla ({\mathcal {L}})^{-1/2}$ on the $L^p$ space. As an application, for $1&lt;p&lt;3+\epsilon $, 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 $\nabla ({\mathcal {L}})^{-1/2}$ on weighted $L^2$ spaces is also studied.
LA - eng
KW - Riesz transform; elliptic operator; Lipschitz domain
UR - http://eudml.org/doc/116183
ER -

References

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