First and second sharp constants in Riemannian Gagliardo–Nirenberg inequalities

Let be a smooth compact Riemannian manifold of dimension , , and be real parameters. Consider the Gagliardo–Nirenberg inequality which follows directly from the Euclidean inequality for some constants . The optimal first constant is defined as the infimum of all such that the inequality above is valid for some . The validity of the optimal inequality was established in [Ceccon‐Durán, J. Math. Anal. Appl. 433 (2016), 260–281]. In this paper, we complete the study of the optimal constants by using global concentration methods in order to determine the value of the first optimal constant : it coincides with the optimal constant for the Euclidean case; this result holds independently of the validity of the optimal inequality. Given that the optimal inequality is valid, the optimal second constant is defined in a similar fashion. Then, we also provide a lower bound for the second optimal constant depending on the scalar curvature of and data lifted from the extremal function of the corresponding Euclidean inequality.