Atomic Decompositions Of Weak Orlicz-lorentz Martingale Spaces

The atomic decomposition is a simple, quick and practical research tool, which hasbeen widely used in many aspects, especially in the harmonic analysis and martingaletheory etc. By using the atomic decomposition we can simplify our research problem,we also can easily handle the intractable problem in general case. More importantly, it isnot only widely used in the research of martingale spaces with small index, and it hassome advantages in the single index and multi index martingale theory. In this paper wewill discuss several kinds of commonly used scalar valued martingale spaces. Based onthese types of spaces we will research the atomic decomposition theory and itsapplication on weak martingale Orlicz-Lorentz space, for example we can researchsome martingale spaces bounded operate. It will be discussed in chapter five.Specifically, it includes the followings.(1) Weak Hardy martingale spaces. Hardy space not only plays an important role inthe function space, but also is one of the main research object in martingale space theory.In the second chapter, the first part discussed the atomic decomposition of this space,the second part is to discuss the application of the atomic decomposition in some waysbased on the front half part.(2) Weak Orlicz martingale spaces. It is a widely martingale spaces, as we know itis the special case of the classical Lp space. This paper mainly discusses atomicdecomposition theory in this space, including strong atomic decomposition and weakatomic decomposition, and using the atomic decomposition to obtain the boundednessof some operators.(3) Lorentz martingale spaces. With the help of rearrangement function, theLorentz space establish the relation between strong space and weak space. Moreover, itis the extension of the classical Lp spaces. By this space, it can helps us understand theclassical Lp spaces clearly.(4) Weak Orlicz-Lorentz martingale spaces. Based on Orlicz space and Lorentz space, we consider Orlicz-Lorentz space which combines these two types of space. Thispaper focuses on the research atomic decomposition of this space and the application ofatomic decomposition.