The Minimal Operator And Weighted Inequalities For Martingales

Since the early 1970's, martingale has undergone a flourishing development, and become more and more important in theory and application. Martingale ideas and methods have given much more simplified proofs for some important conclusions, at the same time brought out new problems and solved it easily, furthermore, martingale has many applications to the real lives, such as lottery tickets' collection, estimation of infectious disease and some problems on economical mathematics, etc.In the course of the development of martingale theory, martingale space and inequalities have been a concerned research hot spot. People always study the properties of operators via the corresponding martingale inequalities, whereby we obtain the relationship between two operators, further, the inclusions of many martingale spaces are established. Weight theory plays an important role in martingale setting. Only if we give a proper weight, then the inequalities can be extended to more general measure space, not only the research field enlarged but also operator properties and associated space structure changed correspondingly. Hence we say the study on weighted inequalities is worthwhile to be investigated.Prof. Long-Ruilin mentioned that if we want to study weighted inequalities for martingales, such as the maximal inequalities, the inequalities for the (conditional) square operator and so on, the condition imposed on weights are mainly Av and S, but we don't know whether this condition is superfluous in all problems we consider. Since the condition can not be weakened for the known operators, now we consider a new operator to discuss its Lp(wdu) - boundedness, then another problem arises, under what condition can we get it?We note that b-condition is Ap-weight's extension, and b-condition has been discussed only in the case of < 0 and >1 respectively, in the case of 0 < < 1, it's non-trivial but not studied, why? it's because that by the known operators we can not characterize it. Therefore, the minimal operator is born, then we would ask that wether the same operator has meaning, can solve and apply what problems and so on, these are mainly contents we would devote to in the below.In this dissertation, we extend the index including the condition of 0 < p < 1 and systematically investigate the properties of certain class of weights, such as Wp,p weight, Wp,P weight, W weight, W* weight etc. and the relationship between Wp,p weight and Ap,p weight. Concretely we introduce the minimal operator and the geometrical maximal operator in martingale setting, and establish the relative weighted inequalities. A series of equivalent relationships have been found between the properties of the weights and the relative weighted inequalities, then the class of RHs is developed via the minimal operator. We discuss the structure of RHs class, that is, we present the factorization of the RHs weights which satisfy the reverse Holder inequality in the case of s > 1. Also we deeply studies the extrapolation of Ap-weight. In the end, some open questions are given.The framework of this thesis consists of the following five parts:Chapter 1 is an introduction on the background, motivation and the principle results of the dissertation.In chapter 2, we define a new operator- the minimal operator on martingale space, it's well known that the maximal operator controls where given martingale / is large, intuitively, we say that the minimal operator controls where a martingale is small. Therefore, it's natural to estimate those values of operators such as (mf)-p , log mf and so on in order to investigate the weighted inequalities for the minimal operator. We discuss them under two conditions respectively: In one-weight case, we establish the relative weighted inequalities under the condition of Ap-weight and ^-weight respectively, and the equivalent condition has been found between the strong (p, p)-type weighted inequality and the property of Wp-weight; while in the case of two-weight condition, we characterize the equivalent relations between the wea...