Optimal improved LpLp-Riemannian Gagliardo–Nirenberg inequalities

Let (M,g)(M,g) be a compact Riemannian manifold of dimension n≥3n≥3. This article is devoted to proving the validity of the optimal improved Gagliardo–Nirenberg type inequality for p>2p>2: (∫M|u|rdvg)2rθ≤(A(p,q,r)2p(∫M|∇gu|pdvg)2p+B(∫M|u|pdvg)2p)(∫M|u|qdvg)2(1−θ)θq, which supplements the results in Ceccon and Mentenegro (2008) [13].