One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: remarks on duality and flows (Q2922850)

The authors study a family of one dimensional Gagliardo-Nirenberg-Sobolev inequalities which can be written as NEWLINE\[NEWLINE \begin{aligned} \|f\|_{L^p(\mathbb R)}\leq\mathbf C_{\text{GN}}(p)\|f'\|_{L^2(\mathbb R)}^\theta \|f\|_{L^2(\mathbb R)}^{1-\theta}\quad\text{if }\, p\in(2,\infty), \\ \|f\|_{L^2(\mathbb R)}\leq\mathbf C_{\text{GN}}(p)\|f'\|_{L^2(\mathbb R)}^\eta \|f\|_{L^p(\mathbb R)}^{1-\eta}\quad\text{if }\, p\in(1,2), \end{aligned} NEWLINE\]NEWLINE with \(\theta=\frac{p-2}{2p}\) and \(\eta=\frac{2-p}{2+p}\), \(\mathbf C_{\text{GN}}(p)\) denotes the best constant.NEWLINENEWLINEIn a few particular cases (e.g. Nash's inequality and some interpolation inequalities on the sphere), the best constants are explicit and optimal functions can be simply characterized. Then there is a non-linear flow associated with the inequalities. This flow can be considered as a gradient flow of an entropy functional with respect to Wasserstein's distance (see [\textit{F. Otto}, Commun. Partial Differ. Equations 26, No. 1--2, 101--174 (2001; Zbl 0984.35089)]). Let us also mention a duality argument based on the mass transportation method which allows these inequalities to be related to much simpler ones (cf. [\textit{M. Agueh} et al., Geom. Funct. Anal. 14, No. 1, 215--244 (2004; Zbl 1122.82022); \textit{D. Cordero-Erausquin} et al., Adv. Math. 182, No. 2, 307--332 (2004; Zbl 1048.26010)]). The main purpose of the paper is to study analogues of these properties in connection with the inequalities mentioned above.