Continuity of extensions of Lipschitz maps and of monotone maps

IF 1 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2024-11-02 DOI:10.1112/jlms.70014

Krzysztof J. Ciosmak

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引用次数: 0

Abstract

Let $X$ be a subset of a Hilbert space. We prove that if $v : X \to R^{m}$ is such that

Moreover, if either $m \in {1, 2, 3}$ or $X$ is convex, we prove the converse: We show that a map $v : X \to R^{m}$ that allows for a 1-Lipschitz, uniform distance preserving extension of any 1-Lipschitz map on a subset of $X$ also satisfies the above set of inequalities. We also prove a similar continuity result concerning extensions of monotone maps. Our results hold true also for maps taking values in infinite-dimensional spaces.

Abstract Image

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Lipschitz 地图和单调地图扩展的连续性

让 X $X$ 是一个希尔伯特空间的子集。我们证明，如果 v : X → R m $v\colon X\rightarrow \mathbb {R}^m$ 是这样的，而且，如果 m ∈ { 1 , 2 , 3 } 或者 X $X$ 是凸的，我们证明反过来：如果 v : X → R m $v\colon X\rightarrow \mathbb {R}^m$ 是这样的。或 X $X$ 是凸的，我们证明反过来：我们证明了一个映射 v : X → R m $v\colon X\rightarrow \mathbb {R}^m$ 可以在 X $X$ 的子集上对任何 1-Lipschitz 映射进行 1-Lipschitz、均匀距离保持的扩展，这个映射也满足上述不等式集。我们还证明了关于单调映射扩展的类似连续性结果。我们的结果也适用于在无限维空间取值的映射。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.