Let M be a smooth compact manifold of dimension n≥1\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 1$$\end{document} without boundary endowed with a volume form ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} and a fibrewise norm N:T∗M→R\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {N}:T^*M \rightarrow \mathbb {R}$$\end{document}. For any p>q≥1\documentclass[12pt]{minimal}
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\begin{document}$$p > q \ge 1$$\end{document} and corresponding interpolation parameter θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document}, we prove that the optimal normed Nash inequality holds for any smooth function u on M, ∫M|u|pω1/pθ≤Nopt∫MNp(du)ω1/p+B∫M|u|pω1/p∫M|u|qω(1-θ)/θq\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left( \int _M |u|^p\; \omega \right) ^{1/p \theta }\le & {} \left( N_{\mathrm{{opt}}} \left( \int _M \mathcal {N}^p(\mathrm{{d}}u)\; \omega \right) ^{1/p} \right. \\&+ \left. B\left( \int _M |u|^p\; \omega \right) ^{1/p} \right) \left( \int _M |u|^q\; \omega \right) ^{(1 - \theta )/\theta q} \end{aligned}$$\end{document}for some constant B, where Nopt\documentclass[12pt]{minimal}
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\begin{document}$$N_{\mathrm{{opt}}}$$\end{document} 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.
