Characterizations and Extensions of Lipschitz-α Operators |
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作者姓名: | Huai Xin CAO Jian Hua ZHANG Zong Ben XU |
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作者单位: | [1]College of Mathematics and Information Science, Shaanxi Normal University,Xi'an 710062, P. R. China [2]Faculty of Science, Xi'an Jiaotong University, Xi'an 710049, P. R. China |
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基金项目: | This work is partly supported by NNSF of China (No. 19771056, No. 69975016, No. 10561113),Acknowledgments We would like to thank the referees for their valuable suggestions. |
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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.
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关 键 词: | Lipschitz-α算子 延伸性 度量空间 Banach空间 |
收稿时间: | 2004-03-23 |
修稿时间: | 2004-03-232004-09-06 |
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