弱型BMO鞅空间与John-Nirenberg定理

As the theory further develops, weak type spaces and their applications in martingale theory and harmonic analysis have acquired more and more attention over the past few years. This project is devoted to investigating several new weak type BMO martingale spaces associated with concave functions or slowly varying functions and establishing corresponding John-Nirenberg inequalities. To achieve this goal, a number of relevant classical problems, e.g. atomic decompositions, dualities, the boundedness of σ-sublinear operators and martingale inequalities, need to be discussed in new frames of martingale spaces. In order to overcome the difficulties caused by the involvement of concave functions or slowly varying functions, not only some traditional tools are needed, but also some new insights and techniques are required. The results to be obtained shall enrich the existing theory of BMO martingale spaces and lead to the emergence of some other new interesting problems.

随着理论的深入发展，弱型空间及其在鞅论和调和分析中的应用近年来获得人们越来越多的关注。本项目致力于研究几类新型的与凹函数或慢变函数相联系的弱BMO鞅空间，同时建立与之相对应的几类John-Nirenberg不等式。为了达成这一目标，我们需要在新的鞅空间框架中讨论若干相关经典问题，如原子分解定理、对偶、σ-次线性算子的有界性、鞅不等式等。该项目的研究不仅依赖于一些传统工具，同时为了克服凹函数、慢变函数等的卷入所引起的困难还需要引入新的思想和方法，其所得结论不仅可以扩展现有的BMO鞅空间理论，同时还会引发一些其他有趣的新问题。

本项目研究了几类新型的与凹函数或慢变函数相联系的弱BMO鞅空间，并且建立了与之相对应的John-Nirenberg不等式，解决了项目书中的各个问题，完成了既定目标。同时，项目组还考虑了变指标鞅Hardy-Morrey空间和非交换对称空间中2-缺项序列。以上述结果为基础，项目负责人及团队成员共完成SCI论文5篇，分别发表于Proc.Amer.Math.Soc.、J. Math. Anal. Appl.、Acta Math. Hungar等。