Poincaré Inequality And Log-Sobolev Inequality Of Boltzmann Measure On The Unit Sphere

In the study of functional problems,Poincaré inequality and logarithm Sobolev inequality are effective tools for discussing measurement concentration and ergodic the-ory,and they are the key points and difficulties in current research.This paper focuses on the functional inequalities of the Boltzmann measure ?h on the unit sphere S2 in three-dimensional space.Using the conclusions from the document[13],the problem in the three-dimensional space is reduced to the one-dimensional space,and we can get the image measure vh of ?h.By estimating the two best constants Cp(vh)and CLS(vh),we can then derive the consistent Poincaré inequality and Log-Sobolev inequality for the parameter h.Among them,for the case of one-dimensional space,this paper can give a more detailed conclusion:When h is large enough,CP(vh)has a rank of 1/h,and CLS(vh)has a constant order.Simultaneously:(1).When h is O,the Poincaré constant Cp(vh)is 1/2.When h is infinite,Cp(vh)is O.This result shows that our conclusions are precise to a certain extent?(2).For the Log-Sobolev constant,when H is O,log2/4?CLS(vh)8e2/e2-1;htends to infinity,1/4log2?CLS(vh)?1+2/log2 This conclusion also supports the conclusion that the Poincaré inequality is weaker than the logarithm Sobolev inequality.