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Non-Doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-Carathéodory Spaces
Donatella Danielli, Purdue University, West Lafayette, IN, Nicola Garofalo, Purdue University, West Lafayette, IN, and Università di Padova, Italy, and Duy-Minh Nhieu, Georgetown University, Washington, DC

Memoirs of the American Mathematical Society
2006; 119 pp; softcover
Volume: 182
ISBN-10: 0-8218-3911-X
ISBN-13: 978-0-8218-3911-9
List Price: US$66
Individual Members: US$39.60
Institutional Members: US$52.80
Order Code: MEMO/182/857
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The object of the present study is to characterize the traces of the Sobolev functions in a sub-Riemannian, or Carnot-Carathéodory space. Such traces are defined in terms of suitable Besov spaces with respect to a measure which is concentrated on a lower dimensional manifold, and which satisfies an Ahlfors type condition with respect to the standard Lebesgue measure. We also study the extension problem for the relevant Besov spaces. Various concrete applications to the setting of Carnot groups are analyzed in detail and an application to the solvability of the subelliptic Neumann problem is presented.

Table of Contents

  • Introduction
  • Carnot groups
  • The characteristic set
  • \(X\)-variation, \(X\)-perimeter and surface measure
  • Geometric estimates from above on CC balls for the perimeter measure
  • Geometric estimates from below on CC balls for the perimeter measure
  • Fine differentiability properties of Sobolev functions
  • Embedding a Sobolev space into a Besov space with respect to an upper Ahlfors measure
  • The extension theorem for a Besov space with respect to a lower Ahlfors measure
  • Traces on the boundary of \((\epsilon,\delta)\) domains
  • The embedding of \(B^p_\beta(\Omega,d\mu)\) into \(L^q(\Omega,d\mu)\)
  • Returning to Carnot groups
  • The Neumann problem
  • The case of Lipschitz vector fields
  • Bibliography
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