2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/136495Let (M,g) be a compact Riemannian manifold of dimension n \geq 2. In this work we prove the validity of the optimal L^p-Riemannian Gagliardo-Nirenberg inequality for 1 < p \leq 2. Our proof relies strongly on a new distance lemma which. In particular, we extend L^p-Euclidean Gagliardo-Nirenberg inequalities due to Del Pino and Dolbeault and the optimal L^2-Riemannian Gagliardo-Nirenberg inequality due to Broutteland in a unified framework.23 pages. To appear in Mathematische ZeitschriftDifferential GeometryAnalysis of PDEstext