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

  • Jurandir Ceccon
  • Carlos DuránEmail author
  • Marcos Montenegro


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.


Optimal Sobolev inequalities Blow-up analysis Pointwise estimates Finsler geometry 

Mathematics Subject Classification

58C35 47J05 46E35 53C15 



The first author was supported by CNPq [PQ 308244/2017-6] and the third author was supported by CNPq (PQ 306855/2016-0) and Fapemig (APQ 02574-16). The authors are indebted to the anonymous referee, whose careful reading and invaluable suggestions, comments and remarks helped us improve substantially the presentation of this work.


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© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal do ParanáCuritibaBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil

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