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THE BEST-CONSTANT PROBLEM FOR A FAMILY OF GAGLIARDO–NIRENBERG INEQUALITIES ON A COMPACT RIEMANNIAN MANIFOLD

Published online by Cambridge University Press:  27 January 2003

Christophe Brouttelande
Christophe Brouttelande
Affiliation:
Université Paul Sabatier, Département de Mathématiques, 118 route de Narbonne, 31062 Toulouse Cedex 4, France (Christophe.Brouttelande@math.ups-tlse.fr)
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Abstract

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The best-constant problem for Nash and Sobolev inequalities on Riemannian manifolds has been intensively studied in the last few decades, especially in the compact case. We treat this problem here for a more general family of Gagliardo–Nirenberg inequalities including the Nash inequality and the limiting case of a particular logarithmic Sobolev inequality. From the latter, we deduce a sharp heat-kernel upper bound.

AMS 2000 Mathematics subject classification: Primary 58J05

Keywords

Sobolev logarithmic inequalityGagliardo–Nirenberg inequalitiesbest-constant problemoptimal inequalities
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 46 , Issue 1 , February 2003 , pp. 117 - 146
Copyright
Copyright © Edinburgh Mathematical Society 2003