基于慢变函数的几类鞅空间理论研究及应用

本项目的研究属于泛函分析空间理论与概率论的交叉学科范畴，涉及到Banach空间几何学、调和分析及随机分析等。该项目集中对基于慢变函数的几类鞅空间理论及应用进行研究。更确切地说，首先本项目将改进现有关于联系慢变函数Orlicz型鞅空间理论的系列研究中的条件，释放对慢变函数的单调非降限制，扩展现有理论并开发新工具。另一方面，变指数鞅空间是一个既有理论意义又有实用价值的崭新研究领域。该项目也将对基于慢变函数的变指数Lorentz型鞅空间和Banach-值变指数Lorentz型鞅空间的原子分解、对偶、鞅不等式等进行深入研究，并应用到Banach空间几何属性分析中。本项目试图采用乘子、指数逼近、空间与算子的超积等方法进行研究，可以预见相关研究将为鞅空间理论提供新的工具和通道，推动其发展上升到一个新台阶。

This project is placed at the intersection between functional analysis and probability theory, involved in geometry of Banach spaces, harmonic analysis, stochastic analysis and other subject areas. We will focuses on the deep and systemic study of some martingale spaces associated with slowly varying functions. To be more specific, first we will release the non-monotonic decreasing limit of the slow varying function which is in the current series of studies on the martingale spaces associated with slowly varying functions, then extend existing theories and develop new tools. On the other hand, the theory of martingale spaces with variable exponents have increasingly becoming one of the new international topics. We shall study the atomic decompositions, dualities, martingale inequalities and others in the framework of variable Lorentz type martingale spaces and Banach-valued Lorentz-type martingale spaces associated with slow varying function. This project attempts to study several types of martingale space theory with slowly varying functions by multiplier, exponential function approximation, ultra-product of spaces and operators, etc.. The project will develop the new methods and techniques. Meanwhile, it plays an important role in promoting the development of martingale spaces theory to a new level.