Rough metrics on manifolds and quadratic estimates

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• 作者：Lashi Bandara
• 刊名：Mathematische Zeitschrift
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：283
• 期：3-4
• 页码：1245-1281
• 全文大小：732 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823
• 卷排序：283

文摘

We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of “rough” Riemannian-like metrics that are permitted to be of low regularity and degenerate on sets of measure zero. We also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi-Lipschitz. As a consequence, we demonstrate the invariance of the Kato square root problem under Lipschitz transformations and obtain solutions to this problem on functions and forms on compact manifolds with a rough metric. Furthermore, we show that a lower bound on the injectivity radius is not a necessary condition to solve the Kato square root problem.KeywordsRough metricsQuadratic estimatesKato square root problem