Characterizations and Extensions of Lipschitz-α Operators

作     者：Huai Xin CAO Jian Hua ZHANG Zong Ben XU 

作者机构：College of Mathematics and Information ScienceShaanxi Normal University Faculty of ScienceXi'an Jiaotong University 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2006年第22卷第3期

页      面：671-678页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：partly supported by NNSF of China(No.19771056 No.69975016 No.10561113) 

主　　题：Characterization Extension Lipschitz-α operator 

摘      要：In this work, we prove that a map F from a compact metric space K into a Banach space X over F is a Lipschitz-α operator if and only if for each σ in X^＊ the map σoF is a Lipschitz-α function on K. In the case that K = [a, b], we show that a map f from [a, b] into X is a Lipschitz-1 operator if and only if it is absolutely continuous and the map σ→ （σ o f）＇ is a bounded linear operator from X^＊ into L^∞（[a, b]）. When K is a compact subset of a finite interval （a, b） and 0 〈 α ≤ 1, we show that every Lipschitz-α operator f from K into X can be extended as a Lipschitz-α operator F from [a, b] into X with Lα（f） ≤ Lα（F） ≤ 3^1-α Lα（f）. A similar extension theorem for a little Lipschitz-α operator is also obtained.