Contents Chapter 1. Introduction 1 1.1. Carnot-Carat heodory spaces 9 1.2. The Chow-Rashevsky's accessibility theorem and CC metrics 9 1.3. The Nagel-Stein-Wainger polynomial and the size of the CC balls 11 15 17 Chapter 2. Carnot groups 2.1. Carnot groups of step 2 2.2. The Kaplan mapping 19 2.3. Groups of Heisenberg type 20 Chapter 3. The characteristic set 23 3.1. A result of Derridj on the size of the characteristic set 23 3.2. Some geometric examples 24 3.3. Non-characteristic manifolds 24 3.4. Manifolds with controlled characteristic set 27 Chapter 4. X-variation, X-perimeter and surface measure 4.1. The structure of functions in BVx,ioc 4.2. X-Caccioppoli sets 34 4.3. X-perimeter and the perimeter measure 33 33 36 Chapter 5. Geometric estimates from above on CC balls for the perimeter measure 37 5.1. A fundamental estimate 37 5.2. The X-perimeter of a C1,1 domain is an upper 1-Ahlfors measure 39 Chapter 6. Geometric estimates from below on CC balls for the perimeter measure 41 6.1. The relative isoperimetric inequality and Theorem 6.1 42 6.2. A basic geometric lemma 43 6.3. Further analysis for Hormander vector fields of step 2 46 6.4. Second proof of Theorem 6.1 52 6.5. Failure of the 1-Ahlfors condition for the X-perimeter of C1,oc domains 55 Chapter 7. Fine differentiability properties of Sobolev functions 57 7.1. Poincare inequality fractional integrals and improved representation formulas 57 7.2. Fine mapping properties of fractional integration on metric spaces 61 7.3. Differentiation with respect to an upper Ahlfors measure 62 7.4. Upper Ahlfors measures and Hausdorff measure 63

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