Hardy-Littlewood Averages Operators On F_p,q^s,r(Rⁿ) Spaces

Harmonic Analysis originated in the research of Euler, Fourier and other famous scientists, starting with the application of heat conduction equation in the study. After 200 years of development, Harmonic Analysis closely links with many branches of mathematics, such as Wavelet Analysis, and nowdays has become one of the core disciplines of mathematics.Function space and related operator theory has been an important part of modern Harmonic Analysis, and has already accumulated and formed many rich research meth-ods and research results. This dissertation will concentrate on the boundness of the Hardy-Littlewood average operator in the space Fp,qs,t (R"), describe some properties of the space Fp,qs,t(Rn), and consider the boundness of the commutators of the weighted Hardy-Littlewood average operator and Lipschitz function in the space Fp,qs,t (Rn).Chapter one focuses on the development of function space and the latest research results, and then explain the relevant research background, knoledge for preparation and the status of the study for the next two chapters prepare the research work.In the second chapter, we discuss the properties of the weighted Hardy-Littlewood average operator and commutators generated by Lipschitz functions, and obtain the sufficient condition of the boundness of the commutators in the Morrey space。In the last chapter, we will describe the nature of the space Fp,qs,t(Rn), necessary and sufficient conditions of Hardy average operators and its adjoint operator bounded in the space Fp,qs,t (Rn) and the operator norm on this space.