Bounds of riesz transforms on Lp spaces for second order elliptic operators

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Abstract

Let ℒ = -div (A(x) ∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ℝn 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 Lp 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 L2 spaces is also studied.

Original languageEnglish
Pages (from-to)173-197+VII+X-XI
JournalAnnales de l'Institut Fourier
Volume55
Issue number1
DOIs
StatePublished - 2005

Keywords

  • Elliptic operator
  • Lipschitz domain
  • Riesz transform

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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