Atomic Decompositions And John-Nirenberg Inequalities For Hardy Martingale Spaces

In this dissertation, by using the methods and techniques of analysis and prob-ability theory, we study the atomic decompositions and John-Nirenberg inequalities of Martingale Hardy-Lorentz spaces, Martingale Hardy-Orlicz spaces and Martingale rearrangement invariant spaces. This dissertation consists of five chapters.In chapter1, we sketch the historical background, status, the up-to-date progress for the discussed problems, and preliminary facts on Martingale spaces, atomic de-compositions and John-Nirenberg inequalities.In chapter2, we study the predual and John-Nirenberg inequalities on general-ized BMO martingale spaces. Firstly, we introduce the generalized BMO martingale spaces by stopping time sequences which enable us to characterize the dual spaces of martingale Hardy-Lorentz spaces Hp,qs q for0<p≤1,1<q<oo. Secondly, using atomic decompositions which are formulated in Section2, we prove some duality the-orems of martingale Hardy-Lorentz spaces. And our results improve and generalize the existing results. By duality, the new John-Nirenberg theorem for the generalized Lipschitz space is proved in Section4. Finally, the boundedness of fractional integrals on martingale Hardy-Lorentz spaces are investigated..In chapter3, we construct the atomic decomposition of Hardy spaces by atoms associated with rearrangement invariant Banach function spaces and prove the new John-Nirenberg inequalities. We first define new bounded mean oscillation martingale spaces associated with rearrangement invariant Banach function space. Then, we prove the atomic decomposition theorem in Hardy spaces by atoms associated with rearrangement invariant Banach function spaces. In the final Section, we prove the new John-Nirenberg inequalities by constructing the atomic decomposition of Hardy spaces Hp by atoms associated with rearrangement invariant Banach function space.In chapter4, we study some new atomic decomposition theorems for martingale Hardy-Orlicz spaces associated with concave functions are proved. As an application, we deduce a new type of John-Nirenberg for BMO^. We introduce the small index notation of martingale Hardy-Orlicz spaces and give some preliminaries necessary to the whole paper. After that, we prove the new John-Nirenberg inequalities by constructing atomic decomposition of Hardy-Orlicz spaces HΦs via atoms associated with r.i. spaces. At last, we introduce variable exponent spaces and establish John-Nirenberg inequalities on variable exponent spaces.