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Martingale transforms and L p-norm estimates of Riesz transforms on complete Riemannian manifolds

An Erratum to this article was published on 11 April 2014

Abstract

Under the condition that the Bakry–Emery Ricci curvature is bounded from below, we prove a probabilistic representation formula of the Riesz transforms associated with a symmetric diffusion operator on a complete Riemannian manifold. Using the Burkholder sharp L p-inequality for martingale transforms, we obtain an explicit and dimension-free upper bound of the L p-norm of the Riesz transforms on such complete Riemannian manifolds for all 1 < p < ∞. In the Euclidean and the Gaussian cases, our upper bound is asymptotically sharp when p→ 1 and when p→ ∞.

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Correspondence to Xiang-Dong Li.

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Dedicated to my daughter Yun-Xuan.

Research partially supported by a Delegation in CNRS at the University of Paris-Sud during the 2005–2006 academic year.

An erratum to this article is available at http://dx.doi.org/10.1007/s00440-014-0560-1.

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Li, XD. Martingale transforms and L p-norm estimates of Riesz transforms on complete Riemannian manifolds. Probab. Theory Relat. Fields 141, 247–281 (2008). https://doi.org/10.1007/s00440-007-0085-y

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Mathematics Subject Classification (2000)

  • Primary: 53C21
  • 58J65
  • Secondary: 58J40
  • 60J65

Keywords

  • Riesz transforms
  • Bakry–Emery Ricci curvature
  • Martingale transforms
  • Burkholder sharp L p-inequality for martingale subordination