Differentiability of Lipschitz Maps from Metric Measure Spaces to Banach Spaces with the Radon–Nikodym Property
Differentiability of Lipschitz Maps from Metric Measure Spaces to Banach Spaces with the Radon–Nikodym Property
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具有 Radon-Nikodym 性质的 Lipschitz 映射从度量空间到 Banach 空间的可微性
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
B. Kleiner
中科院分区：
文献类型：
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作者：
J. Cheeger;B. Kleiner
证明了Lipschitz映射X → V的可微性，其中X表示PI空间，即满足加倍条件和Poincaré不等式的完备度量测度空间，V表示具有Radon-Nikodym性质（RNP）的Banach空间.作为结果，我们得到了一个双Lipschitz非嵌入定理的RNP目标。微分定理依赖于一个新的规格的PI空间的可微结构，涉及方向导数的方向上的速度向量的可求长曲线。我们给出了两个不同的证明，其中第二个依赖于一个新的特征的最小上梯度。PI空间的无穷小结构有很强的意义，这将在其他地方讨论。
We prove the differentiability of Lipschitz maps X → V, where X denotes a PI space, i.e. a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon–Nikodym Property (RNP). As a consequence, we obtain a bi-Lipschitz nonembedding theorem for RNP targets. The differentiation theorem depends on a new specification of the differentiable structure for PI spaces involving directional derivatives in the direction of velocity vectors to rectifiable curves. We give two different proofs of this, the second of which relies on a new characterization of the minimal upper gradient. There are strong implications for the infinitesimal structure of PI spaces which will be discussed elsewhere.