**Memoirs of the American Mathematical Society**

2006;
119 pp;
Softcover

MSC: Primary 43; 46;
Secondary 35

Print ISBN: 978-0-8218-3911-9

Product Code: MEMO/182/857

List Price: $66.00

AMS Member Price: $39.60

MAA Member Price: $59.40

**Electronic ISBN: 978-1-4704-0461-1
Product Code: MEMO/182/857.E**

List Price: $66.00

AMS Member Price: $39.60

MAA Member Price: $59.40

# Non-Doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-Carathéodory Spaces

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*Donatella Danielli; Nicola Garofalo; Duy-Minh Nhieu*

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

# Table of Contents

## Non-Doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-Caratheodory Spaces

- Contents vii8 free
- Chapter 1. Introduction 112 free
- Chapter 2. Carnot groups 1526
- Chapter 3. The characteristic set 2334
- Chapter 4. X-variation, X-perimeter and surface measure 3344
- Chapter 5. Geometric estimates from above on CC balls for the perimeter measure 3748
- Chapter 6. Geometric estimates from below on CC balls for the perimeter measure 4152
- Chapter 7. Fine differentiability properties of Sobolev functions 5768
- Chapter 8. Embedding a Sobolev space into a Besov space with respect to an upper Ahlfors measure 6576
- Chapter 9. The extension theorem for a Besov space with respect to a lower Ahlfors measure 7990
- Chapter 10. Traces on the boundary of (ε, δ) domains 8596
- Chapter 11. The embedding of B[sup(p)][sub(β)](Ω, dμ) into L[sup(q)](Ω, dμ) 93104
- Chapter 12. Returning to Carnot groups 99110
- Chapter 13. The Neumann problem 103114
- Chapter 14. The case of Lipschitz vector fields 109120
- Bibliography 111122