Weighted composition operators preserving various Lipschitz constants
Let $\mathrm{Lip}(X)$, $\mathrm{Lip}^b(X)$, $\mathrm{Lip}^{\mathrm{loc}}(X)$ and $\mathrm{Lip}^\mathrm{pt}(X)$ 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 $Tf=h\cdot f\circ φ$ defines a bijection between such vector spaces preserving Lipschitz constants, local Lipschitz constants or pointwise Lipschitz constants, then $h= \pm1/α$ is a constant function for some scalar $α>0$ and $φ$ is an $α$-dilation. Let $U$ be open connected and $V$ be open, or both $U,V$ are convex bodies, in normed linear spaces $E, F$, respectively. Let $Tf=h\cdot f\circφ$ be a bijective weighed composition operator between the vector spaces $\mathrm{Lip}(U)$ and $\mathrm{Lip}(V)$, $\mathrm{Lip}^b(U)$ and $\mathrm{Lip}^b(V)$, $\mathrm{Lip}^\mathrm{loc}(U)$ and $\mathrm{Lip}^\mathrm{loc}(V)$, or $\mathrm{Lip}^\mathrm{pt}(U)$ and $\mathrm{Lip}^\mathrm{pt}(V)$, preserving the Lipschitz, locally Lipschitz, or pointwise Lipschitz constants, respectively. We show that there is a linear isometry $A: F\to E$, an $α>0$ and a vector $b\in E$ such that $φ(x)=αAx + b$, and $h$ is a constant function assuming value $\pm 1/α$. More concrete results are obtained for the special cases when $E=F=\mathbb{R}^n$, or when $U,V$ are $n$-dimensional flat manifolds.