Muramatu’s integral formula is a very useful tool for the study of Sobolev spaces, although this does not seem to be widely recognized. Most theorems in Sobolev spaces can be proved by this formula combined with basic inequalities in analysis, and it is possible to directly treat not only the whole space but also a special Lipschitz domain. In this paper, we present an introduction to L _p -based Sobolev spaces of integer order by making Muramatu’s integral formula play a central role, as Cauchy’s integral formula does in complex analysis. The topics we take up are approximation by smooth functions, the interpolation inequality, the Sobolev embedding theorems, the trace theorem, construction of an extension operator, complex interpolation of Sobolev spaces and real interpolation of Sobolev spaces.

中文翻译：

---

通过Muramatu的积分公式介绍L Sobolev空间

穆拉玛图的积分公式是研究Sobolev空间的非常有用的工具，尽管这似乎尚未得到广泛认可。该公式结合分析中的基本不等式可以证明Sobolev空间中的大多数定理，并且不仅可以直接处理整个空间，还可以直接处理特殊的Lipschitz域。在本文中，我们通过使Muramatu的积分公式起中心作用，如Cauchy积分公式在复杂分析中的作用，介绍基于L _p的整数阶Sobolev空间。我们讨论的主题是光滑函数的近似值，插值不等式，Sobolev嵌入定理，迹定理，扩展算子的构造，Sobolev空间的复数插值和Sobolev空间的实数插值。