Non-commutative Kolmogorov Inequality And Zygmund Inequality

This paper contains two chapters,the outline of the paper is arranged as follows:Chapter 1 contains three sections.Firstly,we give definitions of some operators and C~* algebra.Then,considering a von Neumann algebra with a normal faithful semi-finite trace (?),we introduce(?)—measurable operator and non-commutative L~p space.Last,we mainly introduce the distribution function and singular value of(?)-measurable operator.In chapter 2,considering the singular value of(?)—measurable operator,we get the distribution function of(?)—measurable operator,according to the singular value of(?)—measurable operator,and give some properties of the distribution function of(?)—measurable operatot. Then,we give definitions of(p,q)-type operator and weak(p,q)-type operator in non-commutative L~p space,and we generalize the Kolmogorov inequality and Zygmund inequality to the case of the non-commutative L~p space.Lastly,we prove the Marcinkiewicz interpolation theorem of diagonal form by some properties of the distribution function of (?)-measurable operator and get the results.