Embedding Theorem For Vector Fields Satisfying H(?)rmander’s Condition In Variable Exponential Spaces

With the development of the study of many nonlinear problems in natural science and engineering technology,many nonuniform materials have appeared,such as electrorheological fluid dynamics,elastic mechanics,image processing and etc.The classical Sobolev space is no longer satisfied,and the emergence of variable exponential function space provides a reliable theoretical basis for the research and development of the field.The characterization of the properties of variable exponent spaces plays an important role in promoting their development,embedding theorem is a basic tool in the study of the regularity of partial differential equations,and its research has important theoretical significance and application value.In this thesis,the embedding theorem of variable exponential space in H(?)rmander’s vector fields is studied based on the embedding theorem in thel Sobolev spaces in the variable exponential spaces and the constant index embedding theorem in H(?)rmander’s vector fields,by using the knowledge of density theory and closed image theorem in functional analysis.The main structure of this thesis is as follows:In Chapter 2,we will give the H(?)rmander’s vector fields and the basic properties,the variable exponential space in H(?)rmander’s vector fields and many preparation tools such as H(?)lder’s inequality,Fatou lemma,etc.;later,in chapter 3,the key lemma of this thesis and its detailed proof are given.The main conclusions of this thesis are as follows:1.The embedding theorem of the index p(x)satisfying Lipschitz condition is given and proved;2.The embedding theorem of the index p(x)under the condition of uniform continuity is obtained;3.The compact embedding theorem under the bounded condition of the region is proved.