Boundedness Of Pseudodifferential Operators On Besov-Triebel-Lizorkin Spaces With Variable Exponents

Pseudodifferential operators are a kind of operators including constant coefficient partial differential operators.They are a main research content of the theory of partial differential equations.They are closely related to harmonic analysis,serval complex variable functions,differential geometry and number theory.They are not only applied to many branches of mathematics,but also widely used in physics,finance and other fields.Because variable exponent function spaces have been widely used in differential equations,electrorheological variable fluid,elasticity,image restoration and so on in recent decades,the theory of variable exponent function spaces has developed rapidly.Motivated by the development of the theory of variable exponent function spaces,the dissertation is devoted to estimates of pseudodifferential operators in Triebel-Lizorkin and Besov spaces with variable exponents.The plan is as follows.Firstly,we give an approximation characterization of Triebel-Lizorkin spaces with variable exponent.Then we obtain estimates of bilinear pseudodifferential operators associated to bilinear H¨ormander classes on Euclidean spaces in Besov and Triebel-Lizorkin spaces with variable exponents.Finally,we obtain the boundedness of non regular psedudodifferential operators on variable Triebel-Lizorkin spaces.