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

Let Φ : exp Lip ( X 1 ) → exp Lip ( X 2 ) be a surjective mapping where X 1 and X 2 are compact metric spaces. We prove that if Φ satisfies the non-symmetric-quotient norm condition for the uniform norm: ‖ g f − 1 ‖ ∞ = ‖ Φ ( g ) Φ ( f ) − 1 ‖ ∞ ( f , g ∈ exp Lip ( X 1 ) ) , then Φ is of the form Φ ( f ) ( y ) = { Φ ( 1 ) ( y ) f ( ϕ ( y ) ) if y ∈ K , Φ ( 1 ) ( y ) f ( ϕ ( y ) ) ¯ if y ∈ X 2 \ K ( f ∈ exp Lip ( X 1 ) ) , where ϕ : X 2 → X 1 is a homeomorphism and K is a closed open subset of X 2 . On the other hand, if Φ satisfies the non-symmetric-quotient norm condition for the Lipschitz algebra norm: ‖ g f − 1 ‖ ∞ + ‖ g f − 1 ‖ L = ‖ Φ ( g ) Φ ( f ) − 1 ‖ ∞ + ‖ Φ ( g ) Φ ( f ) − 1 ‖ L ( f , g ∈ exp Lip ( X 1 ) ) , we show that Φ is of the form Φ ( f ) ( y ) = Φ ( 1 ) ( y ) f ( ϕ ( y ) ) ( y ∈ X 2 , f ∈ exp Lip ( X 1 ) ) , or Φ ( f ) ( y ) = Φ ( 1 ) ( y ) f ( ϕ ( y ) ) ¯ ( y ∈ X 2 , f ∈ exp Lip ( X 1 ) ) , where ϕ : X 2 → X 1 is a surjective isometry.