Hölder Seminorm Preserving Linear Bijections and Isometries

Proceedings Mathematical Sciences

A Jiménez-Vargas²  M A Navarro¹ 

[1] $$;Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 00, Almería, Spain$$

关键词: Lipschitz function;    isometry;    linear preserver problem;    Banach-Stone theorem.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let (𝑋, 𝑑) be a compact metric and $0 < 𝛼 < 1$. The space $mathrm{Lip}^𝛼(X)$ of Hölder functions of order 𝛼 is the Banach space of all functions 𝑓 from 𝑋 into $mathbb{K}$ such that $| f|=max {| f|_∞,L(f)} <∞$, where$$L(f)=sup{|f(x)-f(y)|/d^𝛼(x,y):x,yin X, x≠ y}$$is the Hölder seminorm of 𝑓. The closed subspace of functions 𝑓 such that$$limlimits_{d(x,y)→ 0}|f(x)-f(y)|/d^𝛼(x,y)=0$$is denoted by $mathrm{lip}^𝛼(X)$. We determine the form of all bijective linear maps from $mathrm{lip}^𝛼(X)$ onto $mathrm{lip}^𝛼(Y)$ that preserve the Hölder seminorm.

【 授权许可】

Unknown