Sharp Gagliardo–Nirenberg Trace Inequalities via Mass Transportation Method and Their Affine Versions

Abstract

Exploiting the mass transportation method, we prove a dual principle which implies directly the sharp Gagliardo–Nirenberg trace inequalities which was recently proved by Bolley et al. (Int Math Res Not, 2017, https://doi.org/10.1093/imrn/rny111). Moreover, we determine all optimal functions for these obtained sharp Gagliardo–Nirenberg trace inequalities. This settles a question left open in Bolley et al. (Int Math Res Not, 2017, https://doi.org/10.1093/imrn/rny111). Finally, we use the sharp Gagliardo–Nirenberg trace inequality to establish their affine versions (i.e. the sharp affine Gagliardo–Nirenberg trace inequalities) which generalize a recent result of De Napoli et al. (Math Ann 370:287–308, 2018). It was shown that the affine versions are stronger and imply the sharp Gagliardo–Nirenberg trace inequalities. We also determine all extremal functions for the sharp affine Gagliardo–Nirenberg trace inequalities.