Weighted composition operators preserving various Lipschitz constants
Volume 8, Issue 2 (2023), pp. 269–285
Pub. online: 26 July 2023
Type: Article
Received
8 June 2023
8 June 2023
Accepted
21 June 2023
21 June 2023
Published
26 July 2023
26 July 2023
Notes
Dedicated to Anthony To-Ming Lau on the occasion of his 80th birthday
Abstract
$\def\Lip{\mathrm{Lip}}\def\Lipb{\mathrm{Lip^b}}\def\Liploc{\mathrm{Lip^{loc}}}\def\Lippt{\mathrm{Lip^{pt}}}$Let $\Lip(X)$, $\Lipb(X)$, $\Liploc(X)$ and $\Lippt(X)$ 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 $Tf = h \cdot f \circ \varphi$ 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 $\alpha \gt 0$ and $\varphi$ is an $\alpha$-dilation.
Let $V$ be open connected and $U$ be open, or both $U,V$ are convex bodies, in normed linear spaces $E,F$, respectively. Let $Tf = h \cdot f \circ \varphi$ be a bijective weighed composition operator between the vector spaces $\Lip(U)$ and $\Lip(V)$, $\Lipb(U)$ and $\Lipb(V),$ $\Liploc(U)$ and $\Liploc(V)$, or $\Lippt(U)$ and $\Lippt(V)$, preserving the Lipschitz, locally Lipschitz, or pointwise Lipschitz constants, respectively.We show that there is a linear isometry $A : F \to E$, an $\alpha \gt 0$ and a vector $b \in E$ such that $\varphi(x) = \alpha Ax+b$, 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.