Into linear isometries between spaces of Lipschitz functions

In this paper we state a Lipschitz version of a known Holsztynski's theorem on linear isometries of C(X)-spaces. Let Lip(X) be the Banach space of all scalar-valued Lipschitz functions f on a compact metric space X endowed with the norm ||f|| = max{sup{|f(x)|: x in X}, L(f)}, where L(f) is the Lipschitz constant of f. We prove that any linear isometry T from Lip(X) into Lip(Y) satisfying that L(T1X)< 1 is essentially a weighted composition operator Tf(y) = a(y)f(b(y)) for all f in Lip(X) and all y in Y0, where Y0 is a closed subset of Y, b is a Lipschitz map from Y0 onto X with L(b) less or equal than max{1,diam(X)}, and a is a function in Lip(Y) with ||a|| = 1 and |a(y)| = 1 for all y in Y0. We improve this representation in the case of onto linear isometries and we classify codimension 1 linear isometries in two types