论文标题
加权欧几里得空间的无限希尔伯特式的简短证明
A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space
论文作者
论文摘要
我们提供了以下已知结果的快速证明：与欧几里得距离和任意的ra量相关的sobolev空间是希尔伯特。我们的新方法依赖于艾伯利 - 梅尔省（Alberti-Marches）的可分解性捆绑包的特性。由于我们的论点，我们还证明，如果Sobolev Norm在紧凑型的平滑函数上可以接近，那么相对于Lebesgue度量，参考度量绝对是连续的。
We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.
