Sobolev Inequalities For Boltzmann Measures On Circles

In this paper,we study Sobolev inequalities for Boltzmann measures over rings.For the purpose of this paper,we first introduce several important inequalities: Poincaréinequality,log-Sobolev inequality and transportation inequality,and the relationship between these three inequalities and Sobolev inequality.In the second chapter,we introduce the main research methods of this paper: the dimension reduction theorem and the equivalent characterization of one-dimensional measure satisfying Sobolev inequality.In the third chapter,we give the main results and proof of this paper.The main results show an interesting phenomenon: the optimal Poincaré constant has an order of1/h when h tends to infinity,while the optimal log-Sobolev constant is a constant order with respect to h.Combined with an equivalent characterization of Poincaré inequality and Talagrand transportation inequality on compact manifolds,the results of this paper can prove that the log-Sobolev inequality is strictly stronger than the Talagrand transportation inequality to some extent.