Detailed proof of classical gagliardo–nirenberg interpolation inequality with historical remarks

A carefully written Nirenberg’s proof of the famous Gagliardo–Nirenberg interpolation inequality for intermediate derivatives in Rn 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.