Hölder seminorm preserving linear bijections and isometries

Abstract

Let (X,d) be a compact metric and 0<α<1. The space Lip^α(X) of Hölder functions of order α is the Banach space of all functions f from X into K such that ||f||=max{||f||_∞, L(f )}<∞, where L(f)=sup{|f(x)−f(y)|/d^α(x, y): x,y ∈ X, x\neq y} is the Hölder seminorm of f. The closed subspace of functions f such that lim_{d(x,y)→0}|f(x)-f (y)|/d^α(x,y)=0 is denoted by lip^α(X). We determine the form of all bijective linear maps from lip^α(X) onto lip^α(Y ) that preserve the Hölder seminorm.