Ching-Jou Liao, Chih-Neng Liu, Jung-Hui Liu, N. Wong
{"title":"Weighted composition operators preserving various Lipschitz constants","authors":"Ching-Jou Liao, Chih-Neng Liu, Jung-Hui Liu, N. Wong","doi":"10.4310/amsa.2023.v8.n2.a4","DOIUrl":null,"url":null,"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 $Tf=h\\cdot f\\circ \\varphi$ defines a bijection between such vector spaces preserving Lipschitz constants, local Lipschitz constants or pointwise Lipschitz constants, then $h= \\pm1/\\alpha$ is a constant function for some scalar $\\alpha>0$ and $\\varphi$ 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 $Tf=h\\cdot f\\circ\\varphi$ 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 $\\varphi(x)=\\alpha Ax + b$, 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.","PeriodicalId":42896,"journal":{"name":"Annals of Mathematical Sciences and Applications","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2023-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Mathematical Sciences and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4310/amsa.2023.v8.n2.a4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
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 $Tf=h\cdot f\circ \varphi$ defines a bijection between such vector spaces preserving Lipschitz constants, local Lipschitz constants or pointwise Lipschitz constants, then $h= \pm1/\alpha$ is a constant function for some scalar $\alpha>0$ and $\varphi$ 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 $Tf=h\cdot f\circ\varphi$ 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 $\varphi(x)=\alpha Ax + b$, 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.