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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Optimal constants in the exceptional case of Sobolev inequalities on Riemannian manifolds
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by Zoé Faget PDF
Trans. Amer. Math. Soc. 360 (2008), 2303-2325 Request permission

Abstract:

Let $(M,g)$ be a Riemannian compact $n$-manifold. We know that for any $\varepsilon >0$, there exists $C_\varepsilon >0$ such that for any $u\in H_1^n(M)$, $\int _Me^u\mathrm {dv}_g\le C_\varepsilon \exp [(\mu _n+\varepsilon )\int _M|\nabla u|^n\mathrm {dv}_g+\frac {1}{\mathrm {vol}(M)}\int _Mu\mathrm {dv}_g]$, $\mu _n$ being the smallest constant possible such that the inequality remains true for any $u\in H_1^n(M)$. We call $\mu _n$ the “first best constant”. We prove in this paper that it is possible to choose $\varepsilon =0$ and keep $C_\varepsilon$ a finite constant. In other words we prove the existence of a “second best constant” in the exceptional case of Sobolev inequalities on compact Riemannian manifolds.
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Additional Information
  • Zoé Faget
  • Affiliation: Departement Mathematik, ETH-Zentrum, CH-8092, Zurich, Switzerland
  • Address at time of publication: Equipe Géométrie et Dynamique, Institut Mathématiques, 173 rue de Chevaleret, 75013 Paris, France
  • Email: zoe.faget@math.ethz.ch, fagetzoe@math.jussieu.fr
  • Received by editor(s): December 8, 2005
  • Published electronically: December 11, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 2303-2325
  • MSC (2000): Primary 46E35
  • DOI: https://doi.org/10.1090/S0002-9947-07-04308-5
  • MathSciNet review: 2373315