Contents 1. Introduction 2. Classical Sobolev spaces 3. Curves in metric spaces 4. Borel and doubling measures 5. Modulus of the path family 6. Upper gradient 7. Sobolev spaces N 1,p 8. Sobolev spaces M 1,p 9. Sobolev spaces P 1,p 10. Abstract derivative and Sobolev spaces H 1,p 11. Spaces supporting Poincaré inequality 12. Historical notes References 1991 Mathematics Subject Classification. 46E35.

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We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. Inspired by Falconer's work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot-Carathéodory self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups provide a rich source of examples.

We describe a procedure for constructing "polar coordinates" in a certain class of Carnot groups. We show that our construction can be carried out in groups of Heisenberg type and we give explicit formulas for the polar coordinate decomposition in that setting. The construction makes use of nonlinear potential theory, specifically, fundamental solutions for the p-sub-Laplace operators. As applications of this result we obtain exact capacity estimates, representation formulas and an explicit sharp constant for the Moser-Trudinger inequality. We also obtain topological and measuretheoretic consequences for quasiregular mappings.

We use a Riemannnian approximation scheme to define a notion of sub-Riemannian Gaussian curvature for a Euclidean C 2 -smooth surface in the Heisenberg group H away from characteristic points, and a notion of sub-Riemannian signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss-Bonnet theorem. An application to Steiner's formula for the Carnot-Carathéodory distance in H is provided.