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Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces

About this Title

Donatella Danielli, Nicola Garofalo and Duy-Minh Nhieu

Publication: Memoirs of the American Mathematical Society
Publication Year: 2006; Volume 182, Number 857
ISBNs: 978-0-8218-3911-9 (print); 978-1-4704-0461-1 (online)
DOI: https://doi.org/10.1090/memo/0857
MathSciNet review: 2229731
MSC: Primary 43A85; Secondary 35H20, 46E35

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Table of Contents

Chapters

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