Triebel-lizorkin Spaces, Boundedness Of Operators,

In the classical and morden mathematics, function spaces play an important role. In harmonic analysis one soon encounters the Lebesgue space L^p, the Hardy spaces H^p(p ≤ 1), various forms of Lipschitz spaces and the space BMO. Similarly the homogeneous (or inhomogenous) Soblev spaces L_p^k (or L_p^k) are basic in the study of partial differential equations. From the original deinitions of these spaces, it may not appear that they are closely related. In 1970's, with the deep development of interpolation theory and " hard methods " (such as Fourier analysis, Maximal inequality and so on), homogenous (or inhomogenous) Triebel-Lizorkin spaces F_p^s,q(or F_p^s,q) and Besov spaces B_p^s,q (or B_p^s,q) which can unify these all function spaces mentioned above came up. The principal references for the theory of Besov and Triebel-Lizorkin spaces are [64, 76, 77]. On the other hand, sigular integrals, oscillatory sigular integrals, fractional integrals and commutators play a profound and extensive role in harmonic analysis and the partial differential equations. So it is nature of interest to extend the L^p, H^p, L_p^k boundedness on these operators to the more general spaces F_p^s,q, B_p^s,q. In this thesis, we will study the properties of these kinds of operators on Besov and Triebel-Lizorkin spaces.There are five chapters in this thesis.In Chapter 1 we first introduce the definitions of spaces of F_p^s,q ( F_p^s,q) and B_p^s,q ( B_p^s,q), the basic properties of these spaces and their atomic decompositions and molecule decompositions. Also, the definitions of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type are given.In Chapter 2 we mainly consider the weighted Triebel-Lizorkin boundedness of sin-...