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A brief look at the statement of Gagliardo-Nirenberg interpolation inequality would suggest that there should exist a proof by a clever use of Jensen's inequality. In other words, there should be a function of the form $$F(a-b/n)=\|D^b f\|_a$$ such that $\log F(a-b/n)$ is convex. That would provide a short proof of the inequality. The existing proofs that I could find on the internet involved complicated double inductions, etc. Is anyone aware of a proof along these lines? Thanks!

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    $\begingroup$ The shortest proof I know of the Gagliardo-Nirenberg interpolation inequalities in full generality uses Besov spaces and their characterization in terms of the Littlewood-Paley (LP) decomposition. No induction argument in the indices is needed there, and the LP decompostion makes the interpolation parameter $\theta$ quite evident in the course of the proof. See e.g. Theorem 2.44, pp. 84-85 of the book Fourier Analysis and Nonilinear Partial Differential Equations by H. Bahouri, J.-Y. Chemin and R. Danchin (Springer-Verlag, 2011). $\endgroup$ Jun 22, 2023 at 16:12

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