Boundedness Of Pseudo-differential Operators On Matrix Weighted Triebel-Lizorkin-Besov Spaces

Pseudo-differential operators play an important role in study of linear partial differential operators with variable coefficients.They are not only applied to many branches of mathematics,such as partial differential equations,number theory,complex analysis,probability theory,image processing,but also used in physics,finance and other fields.Besides,matrix-weighted function spaces have been widely used in data analysis,and many important spaces in analysis can be included in the scale of Triebel-Lizorkin and Besov spaces.In the dissertation we study the boundedness of pseudo-differential operators on the matrix-weighted Triebel-Lizorkin spaces and Besov spaces.Firstly,we characterize the matrix-weighted inhomogenous Triebel-Lizorkin spaces and Besov spaces by Peetre maximal function and approximation.Then using these characterizations and the equivalence between matrix weighted Triebel-Lizorkin-Besov spaces and the averaging Triebel-Lizorkin-Besov spaces,we obtain the boundedness of pseudo-differential operators with symbols in the H(?)rmander’s class on matrix weighted Besov and Triebel-Lizorkin spaces.We also obtain the boundedness of non-regular pseudo-differential operators with symbols in Besov spaces on matrix weighted BesovTriebel-Lizorkin spaces.These symbols include the classical H(?)rmander classes.