Detailed Proof of Classical Gagliardo–Nirenberg Interpolation Inequality with Historical Remarks

Abstract

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.