Extensions of Lipschitz Maps

Abstract

The main part of this chapter deals with the theory of Lang and Schlichenmaier [LSchl-2005] devoted to Lipschitz extensions of maps acting between metric spaces. The theory gives a uni_ed approach to most previously established results of this kind which now become consequences of the main extension theorem established in [LSchl-2005]. All of these results are not sharp in the sense that the extensions do not preserve Lipschitz constants; in particular, the classical Kirszbraun and Valentine results cannot be proved in this way. Moreover, the estimates of Lipschitz extension constants in the main theorem contain unspeci_ed quantities whose dependence on the basic parameters are either implicit or too rough. More precise estimates require, for every special case, new approaches and methods. Sparse results of this kind are discussed in the _nal part of the chapter.