Optimal Riemannian [formula omitted]-Gagliardo–Nirenberg inequalities revisited

Abstract

Let (M,g) be a smooth compact Riemannian manifold of dimension n⩾2 and let 1<p<n and 1⩽q<r<p⁎=npn−p be real parameters. This paper concerns to the validity of the optimal Gagliardo–Nirenberg inequality(∫M|u|rdvg)prθ⩽(Aopt∫M|∇gu|pdvg+B∫M|u|pdvg)(∫M|u|qdvg)p(1−θ)θq on the range 1<p⩽2 for some constant B. Namely, in Ceccon and Montenegro (2008) [10], the authors established its validity when p<r. We here solve the remaining case p⩾r. This last one is technically more delicate since its proof requires a new distance lemma which relies on a global concentration analysis. One knows that the validity sometimes fails for p>2.