论文标题
使用单个测试计划测试Sobolev属性
Testing the Sobolev property with a single test plan
论文作者
论文摘要
我们证明，在大量的度量度量空间（即，与之相关的Sobolev空间是可以分离的那些）的属性：可以使用单个测试计划来恢复任何Sobolev函数的最小弱上梯度。这意味着，为了确定哪些是弱上梯度不平等中的特殊曲线，就可以在曲线上考虑一组合适的borel措施，而不是$ p $ -Modulus的曲线。此外，在$ \ sf rcd $空间上，我们可以改善结果，表明还可以选择测试计划集中在Equi-Lipschitz曲线家族上。
We prove that in a vast class of metric measure spaces (namely, those whose associated Sobolev space is separable) the following property holds: a single test plan can be used to recover the minimal weak upper gradient of any Sobolev function. This means that, in order to identify which are the exceptional curves in the weak upper gradient inequality, it suffices to consider the negligible sets of a suitable Borel measure on curves, rather than the ones of the $p$-modulus. Moreover, on $\sf RCD$ spaces we can improve our result, showing that the test plan can be also chosen to be concentrated on an equi-Lipschitz family of curves.
