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

Zhongwei Shen

Type: Article

Publication Date: 2005-01-01

Citations: 161

DOI: https://doi.org/10.5802/aif.2094

Abstract

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 VMO coefficients. The range of p is sharp. The closely related boundedness of ∇(ℒ) -1/2 on weighted L 2 spaces is also studied.

Locations

arXiv (Cornell University) - View - PDF
French digital mathematics library (Numdam) - View - PDF
Annales de l’institut Fourier - View - PDF

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+	An $L^p$ -estimate for the gradient of solutions of second order elliptic divergence equations	1963	Norman G. Meyers
+	Singular Integrals and Differentiability Properties of Functions.	1971	Elias M. Stein
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+	Heat Kernels and Spectral Theory	1989	E. B. Davies