Duality Of Martingale Hardy-Lorentz Space And John-Nirenberg Inequality

The main contribution of this thesis is that by stopping time sequences we have defined a generalized BMO martingale space which enables us to characterize the dual space of martingale Hardy-Lorentz space when0<p<1. We prove the duality theorems by improving the previous atomic decomposition results of martingale Hardy-Lorentz spaces. Then under the condition of regularity, we prove the John-Nirenberg theorem of the generalized BMO martingale spaces. It is an extension of the John-Nirenberg theorem of the Lipschitz spaces. Finally, we extend the boundedness of fractional integrals from martingale Hardy spaces to martingale Hardy-Lorentz spaces.This thesis will be divided into the following five chapters. In Chapter1, we introduce some research background and the main point of this thesis briefly. In Chapter2, some preliminary knowledge is introduced. In Chapter3, we establish the atomic decomposition theorems of martingale Hardy-Lorentz spaces, which improve the previous atomic decomposition results of martingale Hardy-Lorentz spaces. In Chapter4, using the atomic decomposition theorems which are formulated in Chapter3, we prove some duality theorems of martingale Hardy-Lorentz spaces, and then by duality we prove the John-Nirenberg inequality of the generalized BMO martingale spaces. In Chapter5, we firstly introduce some previous results of the boundedness of fractional integrals and then extend the boundedness of fractional integrals to martingale Hardy-Lorentz spaces.