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The Moser-Trudinger-Onofri inequality

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Abstract

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.

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  1. Ceremade, CNRS UMR 7534 and Université Paris-Dauphine, Place de Lattre de Tassigny, 75775, Paris Cédex 16, France

    Jean Dolbeault, Maria J. Esteban & Gaspard Jankowiak

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  1. Jean Dolbeault
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  2. Maria J. Esteban
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  3. Gaspard Jankowiak
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Correspondence to Jean Dolbeault.

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In Honor of the Scientific Contributions of Professor Luc Tartar

This work was supported by the Projects STAB and Kibord of the French National Research Agency (ANR), the Project NoNAP of the French National Research Agency (ANR) and the ECOS Project (No. C11E07).

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Dolbeault, J., Esteban, M.J. & Jankowiak, G. The Moser-Trudinger-Onofri inequality. Chin. Ann. Math. Ser. B 36, 777–802 (2015). https://doi.org/10.1007/s11401-015-0976-7

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