A carefully written Nirenberg's proof of the famous Gagliardo–Nirenberg interpolation inequality for intermediate derivatives in $\mathbb R^n$ seems, surprisingly, to be missing in literature. In our paper, we shall first introduce this fundamental result and provide information about its historical background. Afterwards, we present a complete, student-friendly proof. In our proof, we use the architecture of Nirenberg's argument, the explanation is, however, much more detailed, also containing some differences. The reader can find a short comparison of differences and similarities in the final chapter.

中文翻译：

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经典加利亚多-尼伦贝格插值不等式的详细证明，带有历史性论述

令人吃惊的是，精心编写的Nirenberg证明了$ \ mathbb R ^ n $中中间导数的著名Gagliardo–Nirenberg插值不等式的证明在文学上似乎是令人惊讶的。在本文中，我们将首先介绍这一基本结果，并提供有关其历史背景的信息。之后，我们会提供一个完整的，对学生友好的证明。在我们的证明中，我们使用Nirenberg的论证体系，但是解释更加详细，也包含一些差异。读者可以在最后一章中找到差异和相似之处的简短比较。