Rearrangement And The Weighted Logarithmic Sobolev Inequality

In this paper,we study some weighted logarithmic Sobolev inequalities which can come from the analysis on the singular Riemannian manifolds.We give the necessary and sufficient conditions such that the 1-dimension weighted logarithmic Sobolev inequalities are true and obtain some new estimates on the entropy.The paper is divided into three chapters.In the first chapter,we mainly introduce the development of the weighted logarithmic Sobolev inequalities and the main result of this thesis.In the second chapter,we recall the definitions and properties of the rearrangement and then give some propositions and lemmas that will be used in the proof.In the last chapter,the proof of the main theorem is given.By virtue of the rearrangement and the weighted Sobolev inequalities,we complete the proof.