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Zeitschrift für Analysis und ihre Anwendungen

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Volume 40, Issue 2, 2021, pp. 217–236
DOI: 10.4171/ZAA/1681

Published online: 2021-03-30

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

Alberto Fiorenza[1], Maria Rosaria Formica[2], Tomáš G. Roskovec[3] and Filip Soudský[4]

(1) Università di Napoli Federico II, Italy
(2) Università di Napoli Parthenope, Italy
(3) University of South Bohemia, České Budějovice and Czech Technical University, Prague, Czechia
(4) University of South Bohemia, České Budějovice, Czechia

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.

Keywords: Gagliardo–Nirenberg inequality, interpolation inequality, intermediate derivatives, Sobolev spaces, Sobolev embedding theorem, inequalities for derivatives

Fiorenza Alberto, Formica Maria Rosaria, Roskovec Tomáš, Soudský Filip: Detailed Proof of Classical Gagliardo–Nirenberg Interpolation Inequality with Historical Remarks. Z. Anal. Anwend. 40 (2021), 217-236. doi: 10.4171/ZAA/1681