eBook ISBN:  9781470404611 
Product Code:  MEMO/182/857.E 
List Price:  $66.00 
MAA Member Price:  $59.40 
AMS Member Price:  $39.60 
eBook ISBN:  9781470404611 
Product Code:  MEMO/182/857.E 
List Price:  $66.00 
MAA Member Price:  $59.40 
AMS Member Price:  $39.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 182; 2006; 119 ppMSC: Primary 43; 46; Secondary 35
The object of the present study is to characterize the traces of the Sobolev functions in a subRiemannian, or CarnotCarathéodory 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

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


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The object of the present study is to characterize the traces of the Sobolev functions in a subRiemannian, or CarnotCarathéodory 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.

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