Lectures on Lipschitz Analysis

(1.1) |f(a)− f(b)| ≤ L |a− b| for every pair of points a, b ∈ A. We also say that a function is Lipschitz if it is L-Lipschitz for some L. The Lipschitz condition as given in (1.1) is a purely metric condition; it makes sense for functions from one metric space to another. In these lectures, we concentrate on the theory of Lipschitz functions in Euclidean spaces. In Section 2, we study extension problems and Lipschitz retracts. In Section 3, we prove the classical differentiability theorems of Rademacher and Stepanov. In Section 4, we briefly discuss Sobolev spaces and Lipschitz behavior; another proof of Rademacher’s theorem is given there based on the Sobolev embedding. Section 5 is the most substantial. Therein we carefully develop the basic theory of flat differential forms of Whitney. In particular, we give a proof of the fundamental duality between flat chains and flat forms. The Lipschitz invariance of flat forms is also discussed. In the last section, Section 6, we discuss some recent developments in geometric analysis, where flat forms are used in the search for Lipschitz variants of the measurable Riemann mapping theorem. Despite the Euclidean framework, the material in these lectures should be of interest to students of general metric geometry. Many basic results about Lipschitz functions defined on subsets of R are valid in great generality, with similar proofs. Moreover, fluency in the classical theory is imperative in analysis and geometry at large. Lipschitz functions appear nearly everywhere in mathematics. Typically, the Lipschitz condition is first encountered in the elementary theory of ordinary differential equations, where it is used in existence theorems. In the basic courses on real analysis, Lipschitz functions appear as examples of functions of bounded variation, and it is proved