Maps which preserve norms of non-symmetrical quotients between groups of exponentials of Lipschitz functions

Let Φ:expLip(X1)→expLip(X2) be a surjective mapping where X1X1 and X2X2 are compact metric spaces. We prove that if Φ satisfies the non-symmetric-quotient norm condition for the uniform norm:‖gf−1‖∞=‖Φ(g)Φ(f)−1‖∞(f,g∈expLip(X1)), then Φ is of the formΦ(f)(y)={Φ(1)(y)f(ϕ(y))if y∈K,Φ(1)(y)f(ϕ(y))¯if y∈X2\K(f∈expLip(X1)), where ϕ:X2→X1ϕ:X2→X1 is a homeomorphism and K   is a closed open subset of X2X2. On the other hand, if Φ satisfies the non-symmetric-quotient norm condition for the Lipschitz algebra norm:‖gf−1‖∞+‖gf−1‖L=‖Φ(g)Φ(f)−1‖∞+‖Φ(g)Φ(f)−1‖L(f,g∈expLip(X1)), we show that Φ is of the formΦ(f)(y)=Φ(1)(y)f(ϕ(y))(y∈X2,f∈expLip(X1)), orΦ(f)(y)=Φ(1)(y)f(ϕ(y))¯(y∈X2,f∈expLip(X1)), where ϕ:X2→X1ϕ:X2→X1 is a surjective isometry.