Abstract
Based on a new martingale representation formula, we prove some quantitative upper bound estimates of the L p-norm of some singular integral operators on complete Riemannian manifolds. This leads us to establish the Weak L p-Hodge decomposition theorem and to prove the L p-boundedness of the Beurling–Ahlfors transforms on complete non-compact Riemannian manifolds with non-negative Weitzenböck curvature operator.
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X.-D. Li research was partially supported by Hundred Talents Project of the Chinese Academy of Sciences, NSFC No. 10971032, Shanghai Pujiang Talent Project No. 09PJ1401600 and Key Laboratory RCSDS, CAS, No. 2008DP173182.
An erratum to this article is available at http://dx.doi.org/10.1007/s00440-014-0561-0.
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Li, XD. On the weak L p-Hodge decomposition and Beurling–Ahlfors transforms on complete Riemannian manifolds. Probab. Theory Relat. Fields 150, 111–144 (2011). https://doi.org/10.1007/s00440-010-0270-2
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DOI: https://doi.org/10.1007/s00440-010-0270-2
Keywords
- Beurling–Ahlfors transforms
- Hodge decomposition
- Martingale representation formula
- Weitzenböck curvature
Mathematics Subject Classification (2000)
- Primary 58A14
- 58J35
- Secondary 58J40
- 58J65