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Weighted composition operators preserving various Lipschitz constants
- Publication Year :
- 2023
-
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.<br />Comment: to appear in "Annals of Mathematical Sciences and Applications"
- Subjects :
- Mathematics - Functional Analysis
46B04, 51F30, 26A16
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2306.12824
- Document Type :
- Working Paper