A sharp representation of multiplicative isomorphisms of uniformly continuous functions

Let X, Y   be complete metric spaces, U(X)U(X), U(Y)U(Y) the spaces of uniformly continuous functions with real values defined on X and Y  . We will show the form that every multiplicative isomorphism T:U(Y)→U(X)T:U(Y)→U(X) has: for x   outside a uniformly isolated subset S⊂XS⊂X,Tf(x)=sign(f(τ(x)))|f(τ(x))|1+p(x),Tf(x)=sign(f(τ(x)))|f(τ(x))|1+p(x), where τ:X→Yτ:X→Y is a uniform homeomorphism and p:X∖S→Rp:X∖S→R is such that p⋅h⋅log⁡hp⋅h⋅log⁡h is uniformly continuous for every h∈U(X,(2,∞))h∈U(X,(2,∞)).