变指数鞅空间理论的新进展

Progress in the theory for martingale spaces with variable exponents

本文阐述近年发展起来的变指数鞅空间理论中的若干问题,分别就可数生成σ-代数序列和一般σ-代数序列两种情形介绍了此类鞅空间中的基本不等式,包括Doob极大不等式和Burkholder-Gundy-Davis不等式,以及各种类型的Hardy鞅空间和Lorentz-Hardy鞅空间.列举这些空间的相互连续嵌入关系以及原子分解、共轭空间、分数次积分及其在二进Fourier分析中的应用.同时还介绍Musielak-Orlicz鞅空间的有关情形.最后提出研究中的一些公开问题.

We summarize some results on variable martingale space theory which was developed in recent years.We separate two cases,countably generateσ-algebra sequences and generalσ-algebra sequences,to introduce some kinds of basic inequalities,including Doob’s maximal inequality and Burkholder-Gundy-Davis inequality,and some kinds of variable Hardy spaces,variable Lorentz-Hardy spaces.We enumerate continuous embeds between them and those results,including atomic decompositions,dual spaces,fractional integrals and their applications to dyadic Fourier analysis.Musielak-Orlicz Hardy spaces are also introduced.At last we isolate some open problems.