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Dedicated to David Adams, on his 60th birthday Contents Chapter 1. Introduction 1.1. Carnot-Carathéodory spaces 1.2. The Chow-Rashevsky's accessibility theorem and CC metrics 1.3. The Nagel-Stein-Wainger polynomial and the size of the CC balls Chapter 2. Carnot groups 2.1. Carnot groups of step 2 2.2. The Kaplan mapping 2.3. Groups of Heisenberg type Chapter 3. The characteristic set 3.1. A result of Derridj on the size of the characteristic set 3.2. Some geometric examples 3.3. Non-characteristic manifolds 3.4. Manifolds with controlled characteristic set Chapter 4. X-variation, X-perimeter and surface measure 4.1. The structure of functions in BV X,loc 4.2. X-Caccioppoli sets 4.3. X-perimeter and the perimeter measure Chapter 5. Geometric estimates from above on CC balls for the perimeter measure 5.1. A fundamental estimate 5.2. The X-perimeter of a C 1,1 domain is an upper 1-Ahlfors measure Chapter 6. Geometric estimates from below on CC balls for the perimeter measure 6.1. The relative isoperimetric inequality and Theorem 6.1 6.2. A basic geometric lemma 6.3. Further analysis for Hörmander vector fields of step 2 6.4. Second proof of Theorem 6.1 6.5. Failure of the 1-Ahlfors condition for the X-perimeter of C 1,α domains Chapter 7. Fine differentiability properties of Sobolev functions 7.1. Poincaré inequality, fractional integrals and improved representation formulas 7.2. Fine mapping properties of fractional integration on metric spaces 7.3. Differentiation with respect to an upper Ahlfors measure 7.4. Upper Ahlfors measures and Hausdorff measure vii viii CONTENTS Chapter 8. Embedding a Sobolev space into a Besov space with respect to an upper Ahlfors measure 8.1. Some results from harmonic analysis 8.2. Two simple growth-estimates 8.3. A key continuity estimate for a singular integral 8.4. The main theorem Chapter 9. The extension theorem for a Besov space with respect to a lower Ahlfors measure 9.1. Some auxiliary results 9.2. Proof of Theorem 9.1 Chapter 10. Traces on the boundary of (, δ) domains 10.1. The (, δ) condition is optimal for the existence of traces 10.2. Characterization of the traces on the boundary Chapter 11. The embedding of B p β (Ω, dµ) into L q (Ω, dµ) Chapter 12. Returning to Carnot groups Chapter 13. The Neumann problem

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Abstract. One of the main objectives of this paper is to unravel a new interesting phenomenon of the sub-Riemannian Bernstein problem with respect to its Euclidean ancestor, with the purpose of also indicating a possible line of attack toward its solution. We show that the global intrinsic graphs (1.2) are unstable critical points of the horizontal perimeter. As a consequence of this fact, the study of the stability acquires a central position in the problem itself.

In groups of Heisenberg type we introduce a large class of domains, which we call ADP, admissible for the Dirichlet problem , and we prove that on the boundary of such domains, harmonic measure, ordinary surface measure, and the perimeter measure, are mutually absolutely continuous. We also establish the solvability of the Dirichlet problem when the boundary datum belongs to L p , 1 < p ≤ ∞, with respect to the ordinary surface measure. Here, the harmonic measure is that relative to a sub-Laplacian associated with a basis of the first layer of the Lie algebra. A domain is called ADP if it is a nontangentially accessible domain and it satisfies an intrinsic outer ball condition.