# Annals of Mathematical Sciences and Applications

## Volume 8 (2023)

### Special issue dedicated to Anthony To-Ming Lau on his 80th birthday

Guest Editors: Xiaolong Qin, Ngai-Ching Wong and Jen-Chih Yao

### Weighted composition operators preserving various Lipschitz constants

Pages: 269 – 285

DOI: https://dx.doi.org/10.4310/AMSA.2023.v8.n2.a4

#### Authors

Ching-Jou Liao (Department of Mathematics, Hong Kong Baptist University, Hong Kong)

Chih-Neng Liu (Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan)

Jung-Hui Liu (School of Information Engineering, Sanming University, Sanming City, Fujian, China)

Ngai-Ching Wong (Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan)

#### 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.

#### Keywords

(local/pointwise) Lipschitz functions, (local/pointwise) Lipschitz constants, weighted composition operators, flat manifolds

#### 2010 Mathematics Subject Classification

Primary 46B04, 51Fxx. Secondary 26A16.