When a function belonging to a fractional-order Sobolev space is supported in a proper subset of the Lipschitz domain on which the Sobolev space is defined, how is its Sobolev norm as a function on the smaller set compared to its norm on the whole domain? On what do the comparison constants depend on? Do different norms behave differently? This article addresses these issues. We prove some inequalities and disprove some misconceptions by counter-examples.

中文翻译：

---

关于分数阶Sobolev范数的注记

当在定义Sobolev空间的Lipschitz域的适当子集中支持属于分数阶Sobolev空间的函数时，与整个域的范数相比，它在较小集合上的Sobolev范数如何作为函数？比较常数取决于什么？不同的规范表现不同吗？本文解决了这些问题。我们通过反例证明了一些不平等，并驳斥了一些误解。