Poincaré Inequalities On Metric Measure Spaces

Since the introduction of Sobolev space,mathematicians have conducted indepth research and exploration on its theory and application.To this day,Sobolev space theory has become a central analytical tool in the study of partial differential equations and calculus of variations.In addition,Sobolev space theory is also widely used in differential geometry,algebraic topology,complex analysis,probability theory and other fields.In recent years,mathematicians try to extend Sobolev space theory to more general metric measure space.This is because the study of Sobolev space and Poincaréinequality on the metric measure space is helpful to the development of Carnot-Carathéodory metric geometry,quasiconformal mapping and fractal analysis.In this thesis,we mainly introduce the different definitions of Sobolev spaces on metric measure space and their relations,and introduce Poincaréinequalities and differentiability theo-rem related to them.Then,we take stratified Lie groups as the specific research object,and study polynomials,differential operators,differentiability theorems,Sobolev space and Poincaréin-equalities on them.This thesis consists of the following three chapters:In Chapter 1,we mainly introduce the definitions of three different Sobolev spaces on the metric measure space,and the relationship between them.At the same time,we introduce Poincaréinequality and doubling measures,and introduce the self-improvement property of Poincaréinequality and Cheeger differentiability theorem.Finally,we give the main research content of this thesis.In Chapter 2,we take stratified Lie groups as the specific research object.First,we intro-duce the definition and related content of stratified Lie groups,as well as the definition and rep-resentation formula of Sobolev spaces on stratified Lie groups.Then,we introduceA_pweight theory,Poincaréinequality,and Pansu differentiability theorem on stratified Lie groups.We also provide a new proof of Calderón’s theorem on stratified Lie groups by using first-order representation formula.In Chapter 3,a special class of Poincaréinequality on stratified Lie groups is proposed and proved.The conclusions of the Poincaréinequality of Chapter 2 in the special cases are supplemented,and the weighted Poincaréinequalities in the entire space are proved by using these conclusions.