Analysis of Banach function spaces by means of martingale method

我们项目的研究结果如下：让X成为一个超越非原子概率空间的Banach功能空间。给予一个martingale f=（f_n），拒绝由Sf的平方函数。我在伯克霍尔德方功能不平等的X上有一个必要和有效的条件c-f-∞-X [少于或等于] Sf-X [少于或等于] C-f-∞-X要持有，哪里f-∞拒绝给予f.的最明确限度给予一个统一的整数f=（f_n）， let Af=（af_n）拒绝由f_∞的绝对值生成的martingale。我对Sf和S（Af）的Banach功能空间X有一种必要和适当的条件，完全属于X。让X在一个非原子概率空间上成为一个真正的、可变的Banach功能空间。我在X、H_p（X）和K（X）等不同功能区中确定了各种新的结婚标准，其中H_p（X）和K（X）被定义为与X有一种深层和合适的关系。让X成为一个超越非原子概率空间的Banach功能空间我对戴维斯的不平等. Mf. X有一个必要和适当的条件，[小于或等于] C. Sf. X要持有，哪里Mf拒绝f.的最大函数。作为一个结果，如果这个不平等性持有，那么反向的不平等性Mf_X [小于或等于] C-Sf_X也持有。Give a martingale f=（f_n）， define a process θf=（θ f_n） by setting of_n = sup_<0 [小于或等于] n [小于或等于] m<∞>E[|f_m-f_<n-1>/-F_n]。我在Banach功能空间X中给出了一些必要的和适当的条件。

The research results of our project are as follows :・Let X be a Banach function space over a non-atomic probability space. Given a martingale f=(f_n), denote by Sf the square-function of f. I gave a necessary and sufficient condition on X for the Burkholder square-function inequality c‖f_∞‖_X【less than or equal】‖Sf‖_X【less than or equal】C‖f_∞‖_X to hold, where f_∞ denotes the almost sure limit of f.・Given a uniformly integrable martingale f=(f_n), let Af=(Af_n) denote the martingale generated by the absolute value of f_∞. I gave a necessary and sufficient condition on a Banach function space X for that Sf and S(Af) belong to X simultaneously.・Let X be a rearrangement-invariant Banach function space over a non-atomic probability space. I established various new martingale inequalities in the rearrangement-invariant function spaces X, H_p(X) and K(X), where H_p(X) and K(X) are defined so as to have a deep and suitable relation with X.・Let X be a Banach function space over a non-atomic probability space. I gave a necessary and sufficient condition on X for the Davis inequality ‖Mf‖_X【less than or equal】C‖Sf‖_X to hold, where Mf denote the maximal function of f. As a result, it follows that if this inequality holds, then the reversed inequality ‖Mf‖_X【less than or equal】C‖Sf‖_X also holds.・Give a martingale f=(f_n), define a process θf=(θf_n) by setting Of_n=sup_<0【less than or equal】n【less than or equal】m<∞>E[|f_m-f_<n-1>‖F_n]. I gave necessary and sufficient conditions for some inequalities involving the norm of θf in a Banach function space X.