Topics on Analysis in Metric Spaces
Analysis in metric spaces has considerably expanded in recent years. Experts in analysis on (weighted) Euclidean spaces, manifolds, Carnot groups, and fractals, who are interested in function spaces, harmonic analysis, geometry of curves or quasiconformal geometry, have observed that a ground for some of their methods can be reduced to metric structures and thus the theory can be developed once for all frameworks.
The book by Ambrosio and Tilli presents a representative part of the fundamentals of this development. Part of the text gives elements of measure and integration theory: Riesz representation theorem, weak convergence of measures, construction of measures and particularly Hausdorff measures, abstract integration (this for general increasing set functions by the method of De Giorgi and Choquet). Next, Lipschitz mappings are studied and results related to area and coarea formula are discussed. The part on the geodesic problem presents Busemann's existence result. This claims the existence of a geodesic (i.e., shortest) connection between two points x,y in a metric space E provided all bounded sets of E are compact and there exists a connection of finite length. It is also shown that it is equivalent to consider the connections as connected sets (measured by the one-dimensional Hausdorff measure) or as parameterized curves. This requires some nontrivial facts on rectifiability. The Gromov-Hausdorff convergence of metric spaces is introduced and compactness and embedding results are proved. The method of Gromov-Hausdorff convergence is used to prove an existence theorem for the Steiner problem, which is a generalization of the geodesic problem. Even in the geodesic case, the assumptions are further weakened. The theory of Sobolev spaces of Hajlasz type on metric spaces is developed. It is shown that this concept generalizes the ordinary Sobolev spaces on extension domains. Inequalities of Sobolev and Poincaré type are proved.
The book is based on lectures given by the authors at the Scuola Normale in Pisa and presents a well-chosen selection of what to say to students in the field, when time is limited, with suggestions for further studies. The enjoyable exposition of the subject is supplemented by exercises of various levels.