Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

Let (X, d) be a pointed compact metric space, let 0 < α < 1, and let φ : X → X be a base point preserving Lipschitz map. We prove that the essential norm of the composition operator Cφ induced by the symbol φ on the spaces lip0(X, d α) and Lip0(X, d α) is given by the formula ‖Cφ‖e = lim t→0 sup 0<d(x,y)<t d(φ(x), φ(y))α d(x, y)α whenever the dual space lip0(X, d α)∗ has the approximation property. This happens in particular when X is an infinite compact subset of a finite-dimensional normed linear space.