Optimal Normed Sobolev Type Inequalities on Manifolds I: The Nash Prototype

Abstract

Let M be a smooth compact manifold of dimension $$n \ge 1$$ without boundary endowed with a volume form $$\omega $$ and a fibrewise norm $$\mathcal {N}:T^*M \rightarrow \mathbb {R}$$ . For any $$p > q \ge 1$$ and corresponding interpolation parameter $$\theta $$ , we prove that the optimal normed Nash inequality holds for any smooth function u on M, $$\begin{aligned} \left( \int _M |u|^p\; \omega \right) ^{1/p \theta }\le & {} \left( N_{\mathrm{{opt}}} \left( \int _M \mathcal {N}^p(\mathrm{{d}}u)\; \omega \right) ^{1/p} \right. \\&+ \left. B\left( \int _M |u|^p\; \omega \right) ^{1/p} \right) \left( \int _M |u|^q\; \omega \right) ^{(1 - \theta )/\theta q} \end{aligned}$$ for some constant B, where $$N_{\mathrm{{opt}}}$$ is the best possible constant. Its importance can be viewed from two perspectives. Firstly, this inequality is a powerful tool in the study of normed entropy and isoperimetrical inequalities on manifolds which have been established in the flat context by Gentil (J Funct Anal 202:591–599, 2003) and Cordero-Erausquin, Nazaret and Villani (Adv Math 182:307–332, 2004), , respectively. Secondly, this work introduces an appropriate framework to study Sobolev type inequalities on manifolds endowed with a very general way of measuring the involved quantities, instead of using the restricted Riemannian context.