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 dimens...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.展开更多
We derive the sharp Moser-Trudinger-Onofri inequalities on the standard n-sphere and CR(2n+1)-sphere as the limit of the sharp fractional Sobolev inequalities for all n≥1.On the 2-sphere and 4-sphere,this was establi...We derive the sharp Moser-Trudinger-Onofri inequalities on the standard n-sphere and CR(2n+1)-sphere as the limit of the sharp fractional Sobolev inequalities for all n≥1.On the 2-sphere and 4-sphere,this was established recently by Chang and Wang.Our proof uses an alternative and elementary argument.展开更多
In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below.Our proof does not rely on the uniformization theorem and the Onofri inequality,thu...In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below.Our proof does not rely on the uniformization theorem and the Onofri inequality,thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow.Such an analytic approach also sheds light on how to obtain the boundedness for E1 energy in the study of general Kähler manifolds.展开更多
基金supported by the Projects STAB and Kibord of the French National Research Agency(ANR)the Project No NAP of the French National Research Agency(ANR)the ECOS Project(No.C11E07)
文摘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.
基金supported in part by NSFC 11501034,NSFC 11571019 and the key project NSFC 11631002.
文摘We derive the sharp Moser-Trudinger-Onofri inequalities on the standard n-sphere and CR(2n+1)-sphere as the limit of the sharp fractional Sobolev inequalities for all n≥1.On the 2-sphere and 4-sphere,this was established recently by Chang and Wang.Our proof uses an alternative and elementary argument.
基金We thank Yuxiang Li for pointing out the proof of Lemma 3.5 in an early version is incomplete.We also thank the referee for the careful reviewing and comments.
文摘In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below.Our proof does not rely on the uniformization theorem and the Onofri inequality,thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow.Such an analytic approach also sheds light on how to obtain the boundedness for E1 energy in the study of general Kähler manifolds.
