Sharp Gagliardo--Nirenberg trace inequalities via mass transportion method and their affine versions

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. [BCFGG17]. Moreover, we determine all optimal functions for these obtained sharp Gagliardo-Nirenberg trace inequalities. This settles a question left open in [BCFGG17]. 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 N\'apoli et al. [DeNapoli]. 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.