Weighted composition operators preserving various Lipschitz constants

Abstract

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 $Tf=h\cdot f\circ \phi$ defines a bijection between such vector spaces preserving Lipschitz constants, local Lipschitz constants or pointwise Lipschitz constants, then $h=\pm 1/\alpha$ is a constant function for some scalar $\alpha >0$ and $\phi$ is an $\alpha$-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 $Tf=h\cdot f\circ \phi$ 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 $\alpha >0$ and a vector $b\in E$ such that $\phi (x)=\alpha Ax+b$, and $h$ is a constant function assuming value $\pm 1/\alpha$. More concrete results are obtained for the special cases when $E=F={\mathbb{R}}^n$, or when $U,V$ are $n$-dimensional flat manifolds.

to appear in "Annals of Mathematical Sciences and Applications"