Chinese Annals of Mathematics, Series B

, Volume 36, Issue 5, pp 777–802 | Cite as

The Moser-Trudinger-Onofri inequality

  • Jean DolbeaultEmail author
  • Maria J. Esteban
  • Gaspard Jankowiak


This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary remarks.

Various new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods (in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality. In the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally, a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.


Moser-Trudinger-Onofri inequality Duality Mass transportation Fast diffusion equation Rigidity 

2000 MR Subject Classification

26D10 46E35 35K55 58J60 


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Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Jean Dolbeault
    • 1
    Email author
  • Maria J. Esteban
    • 1
  • Gaspard Jankowiak
    • 1
  1. 1.Ceremade, CNRS UMR 7534 and Université Paris-DauphineParis Cédex 16France

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