Weighted composition operators preserving various Lipschitz constants

Let Lip(X), Lip^b (X), Lip^loc (X) and Lip^pt (X) be the vector spaces of Lipschitz, bounded Lipschitz, locally Lipschitz and pointwise Lipschitz (real-valued) functions defined on a metric space (X, d_X), respectively. We show that if a weighted composition operator T f = h · f ◦ ϕ defines a bijection between such vector spaces preserving Lipschitz constants, local Lipschitz constants or pointwise Lipschitz constants, then h = ±1/α is a constant function for some scalar α > 0 and ϕ is an α-dilation. Let V be open connected and U be open, or both U, V are convex bodies, in normed linear spaces E, F,FgYlUjdKvEL2idcCLp5asBNPpdRrkrespectively. Let T f = h · f ◦ ϕ be a bijective weighed composition operator between the vector spaces Lip(U) and Lip(V), Lip^b (U) and Lip^b (V), Lip^loc (U) and Lip^loc (V), or Lip^pt (U) and Lip^pt (V), preserving the Lipschitz, locally Lipschitz, or pointwise Lipschitz constants, respectively. We show that there is a linear isometry A: F → E, an α > 0 and a vector b ∈ E such that ϕ(x) = αAx+b, and h is a constant function assuming value ±1/α. More concrete results are obtained for the special cases when E = F = Rⁿ, or when U, V are n-dimensional flat manifolds.