LAUSR

For a compact metric space (X, d) and $$\alpha \in (0,1)$$α∈(0,1), let $$\mathrm{Lip}^\alpha (X)$$Lipα(X) be the linear space of all complex-valued functions f on X satisfying and $$\mathrm{lip}^\alpha (X)$$lipα(X) be the subspace of $$\mathrm{Lip}^\alpha (X)$$Lipα(X) consisting of functions f with $$\lim \frac{f(x)-f(y)}{d^\alpha (x,y)} =0$$limf(x)-f(y)dα(x,y)=0 as $$d(x,y) \rightarrow 0$$d(x,y)→0. In this paper, we give a characterization of a bijective map $$T:\mathrm{lip}^\alpha (X)\longrightarrow \mathrm{lip}^\alpha (Y)$$T:lipα(X)⟶lipα(Y), not necessarily linear, which is an isometry with respect to the Hölder seminorm $$L(\cdot )$$L(·). It is shown that there exist $$K_0>0$$K0>0, a surjective map $$\Psi : Y \longrightarrow X$$Ψ:Y⟶X with $$d^\alpha (y,z)= K_0 \, d^\alpha (\Psi (y),\Psi (z))$$dα(y,z)=K0dα(Ψ(y),Ψ(z)) for all $$y,z\in Y$$y,z∈Y, and a function $$\Lambda : \mathrm{lip}^\alpha (X) \longrightarrow {\mathbb {C}}$$Λ:lipα(X)⟶C (which is linear or real-linear if T is so) such that either $$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0\, f(\Psi (y))+\Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y) \end{aligned}$$Tf(y)=T0(y)+τ¯K0f(Ψ(y))+Λ(f)(f∈lipα(X),y∈Y)or $$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0 \,\overline{f(\Psi (y))}+ \Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y), \end{aligned}$$Tf(y)=T0(y)+τ¯K0f(Ψ(y))¯+Λ(f)(f∈lipα(X),y∈Y),where $$\tau =e^{i\theta }$$τ=eiθ for some $$\theta \in [0,\pi )$$θ∈[0,π).