Doubling Metric Space And Lipschitz Approximation And Extension

The modulus of continuity of a map f:X →Y between two metric spaces is the function wf:[0,∞)→ R+U {∞} given byThe map f is said to be uniformly continuous,if there is a εo>0 such that wf(ε)<oo for ε<ε0,and limε→0+ wf(ε)= 0.In Mathematical analysis,the modulus of continuity of a map is used to measure quantitatively its uniform continuity.In general,it is difficult to determine explicitly the modulus of continuity wf of a map f,one is mainly interested in maps whose modulus of continuity can be dominated by a special class of functions.For instance,the inequality wf(ε)≤ cε for some constant c describes the Lipschitz continuity of f,and wf(ε)≤ cεα(0<α<1)describes the Holder continuity of f.In the first part of this thesis,our aim is to find a sufficient and necessary conditions for a subadditive function w dominating the modulus of continuity of a uniformly continuous map f.When the source space X is a doubling space and the target space V is a normed linear space,we get such a characterization.As the main part of this characterization,we give a constructive method of approximating a uniformly continuous map from X to V by Lipschitz maps,the corresponding approximation operation is linear.In the second part of the thesis,we study the classical Lipschitz extension problem in the case where the source space is a doubling space and the target space is a Banach space.We give a short and constructive proof of the famous Lee and Naor’s Lipschitz extension theorem,and get a uniform bound on the cardinality of relevant convex combinations.