Lipschitz可微空间的可嵌入性与可修正性
Embeddability and rectifiability of Lipschitz differentiability spaces
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中文总结 AI 辅助
本文通过可分解束和爆破分析，证明了双Lipschitz嵌入RNP空间的Lipschitz可微空间可数可修正，并给出新的刻画及解决Le Donne问题。
中文摘要 AI 辅助
我们证明，双Lipschitz嵌入到RNP空间中的Lipschitz可微空间是可数可修正的。与Cheeger和Kleiner的早期方法相比，我们的方法不依赖于对RNP目标的微分，而是利用可分解束和精细的爆破分析。我们还以避开Alberti表示的方式呈现可分解束，并将Alberti--Marchese的方法推广到RNP空间中的测度。此外，我们研究了到RNP目标的片段可微性，给出了RNP可微空间的一个新的${\rm Lip}-{\rm lip}$型刻画，并解决了Le Donne提出的一个问题，该问题要求刻画空间$(X,\mu)\subset\ell^2$，其Gromov--Hausdorff切空间是$\ell^2$中当$r\to 0$时$r^{-1}(X-x)$的Hausdorff极限。
英文摘要
We prove that Lipschitz differentiability spaces which bi-Lipschitz embed into an RNP-space are countably rectifiable. In contrast to earlier methods of Cheeger and Kleiner, our approach does not rely on differentiating RNP-targets, and uses instead decomposability bundles and a careful blow-up analysis. We also present decomposability bundles in a way which avoids the mention of Alberti representations and generalizes the approach of Alberti--Marchese to measures in RNP-spaces. We moreover study fragment-wise differentiability into RNP-targets, give a new ${\rm Lip}-{\rm lip}$-type characterization of RNP-differentiability spaces, and address a question of Le Donne asking for a characterization of spaces $(X,μ)\subset\ell^2$ whose Gromov--Hausdorff tangents are Hausdorff limits of $r^{-1} (X-x)$ in $\ell^2$ as $r\to 0$.
