The Kato Square Root Problem on Vector Bundles with Generalised Bounded Geometry

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• 作者：Lashi Bandara ; Alan McIntosh
• 关键词：Kato square root problem ; Square roots of elliptic operators ; Quadratic estimates ; Holomorphic functional calculi ; Dirac type operators ; Generalised bounded geometry ; 58J05 ; 47F05 ; 47B44 ; 47A60
• 刊名：Journal of Geometric Analysis
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：26
• 期：1
• 页码：428-462
• 全文大小：641 KB
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  15.Morris, A.J.: Local Quadratic Estimates and Holomorphic Functional Calculi. The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis. Proceedings of the Centre for Mathematics and its Applications. Australian National University, vol. 44, pp. pp. 211–231. Australian National University, Canberra (2010)
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• 作者单位：Lashi Bandara (1)
  Alan McIntosh (1)
  
  1. Centre for Mathematics and its Applications, Australian National University, Canberra, ACT, 0200, Australia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Differential Geometry
  Convex and Discrete Geometry
  Fourier Analysis
  Abstract Harmonic Analysis
  Dynamical Systems and Ergodic Theory
  Global Analysis and Analysis on Manifolds
  
• 出版者：Springer New York
• ISSN：1559-002X

文摘

We consider smooth, complete Riemannian manifolds which are exponentially locally doubling. Under a uniform Ricci curvature bound and a uniform lower bound on injectivity radius, we prove a Kato square root estimate for certain coercive operators over the bundle of finite rank tensors. These results are obtained as a special case of similar estimates on smooth vector bundles satisfying a criterion which we call generalised bounded geometry. We prove this by establishing quadratic estimates for perturbations of Dirac type operators on such bundles under an appropriate set of assumptions. Keywords Kato square root problem Square roots of elliptic operators Quadratic estimates Holomorphic functional calculi Dirac type operators Generalised bounded geometry