关于度量值Sobolev映射的后复合方法的一个松弛化
A relaxation to the post-composition approach to metric-valued Sobolev maps
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中文总结 AI 辅助
本文证明了度量值Sobolev映射的后复合定义在特定空间条件下可仅由定性测试判定，显著推广了既有结果。
中文摘要 AI 辅助
我们证明了，从度量测度空间到度量空间的Sobolev映射的定义，若依赖于Ambrosio-Reshetnyak的后复合方法，则在空间满足某些透明假设下，承认一个实质上弱于主要版本的等价重述。具体而言，相应的定义要求此类映射被称为Sobolev时，需先通过一个“定性”测试，再通过一个“定量”测试。同时，我们的结果允许仅基于“定性”部分即可得出相关结论。值得注意的是，足以使此成立的空间条件完全契合该主题的奇异性质。更明确地说，当源空间具有有限Hausdorff维数并满足Sobolev到Luzin-Lipschitz性质，或目标空间具有有限Hausdorff维数时，这种现象就会出现。所有这些为该主题的若干早期成果提供了显著推广。
英文摘要
We establish that the definition of Sobolev mappings from metric measure spaces to metric spaces relying on the post-composition approach of Ambrosio-Reshetnyak admits, under certain transparent assumptions on the spaces, an equivalent reformulation that is substantially weaker than the principal version. In detail, the corresponding definition requires maps of this kind to meet two successive tests in order to be called Sobolev: first a ''qualitative'' one and then a ''quantitative'' one. At the same time, our result allows the relevant conclusion to be drawn based solely on the ''qualitative'' part. Remarkably, the conditions on the spaces sufficient for this to hold fit entirely into the singular flavor of the subject. More explicitly, such a phenomenon emerges if either the source space is of finite Hausdorff dimension and satisfies the Sobolev-to-Luzin-Lipschitz property or the target space is of finite Hausdorff dimension. All this provides a significant generalization of several earlier achievements on the topic.
发表机构
- International School for Advanced Studies（高等研究所）
- Moscow Institute of Physics and Technology（莫斯科物理技术学院）
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