Martingale Transforms On The Hardy-orlicz Spaces Of Martingale And Their Applications

This thesis is devoted to studying martingale transforms on the Hardy spaces and the Hardy-Orlicz spaces of martingale and their applications. It mainly includes the following aspects: martingale transforms between Q1 and QΦof martingale spaces, martingale transforms between H 1s and H_Φ~s of martingale spaces, the further discussion about the above theorems and their applications. This thesis consists of six chapters as follows:In chapter 1, the background and motivation are introduced, and the main results are presented.Chapter 2 is the preliminary parts of the thesis. The basic concepts, the properties of martingale and the convexΦfunction and the related lemmas are introduced in the preliminary.The main contributions are investigated in chapter 3 and 4. Applying the techniques of martingale transform, the relationship between Hardy spaces and Hardy-Orlicz spaces of martingale is characterized .Given any Young functionΦwith finite power p_Φthe corresponding QΦspace is the martingale transform of Q1 and the converse result is also true. Given any Young functionΦwith finite power p_Φthe corresponding H_Φ~s space is the martingale transform of H 1s and the converse result is also true.From the compound martingale transforms aspect, the relationship between Hardy spaces and Hardy-Orlicz spaces of martingale characterized by martingale transforms is further discussed in chapter 5. A non-negative adapt multiplier is constructed in this chapter.The whole thesis in the form of some conclusions and some further researches are illustrated in chapter 6.