Study On A Class Of Zygmund Type Function Classes And Their Smoothness

In the smooth theory of Hardy space, if f(z)is analyzed in the unit disk, it appears that there is a very close relation between the mean growth of the derivative f'(z) and the smoothness of the boundary function f[ei?).Hardy G.H.,Littlewood J.E. and Zygmund A. are the mathematicians who study Zygmund type function and also gave classical conclusions. Soon after that, the Zygmund function class was further developed in analogy study, which opened a new field for the research on the smooth function of the Zygmund type. In recent years,many mathematics researchers at home and abroad became more and more interested in studying the Zygmund type function in different fields; Many profound results have been accomplished by them. Based on the previous studies,further research on the problem of the smoothness of the Zygmund type function is making the research more complete.This article gives the definition of Zygmund type function on the base of Lipschitz functions defined by GH hardy and JE little wood as well as Zygmund which is similar to those of Lipschitz functions. At the same time, this paper gets sufficient and necessary condition of the function for Zygmund type functions.This paper is divided into three chapters:first chapter introduces the purpose and significance of the topic. Secondly, it gives a brief description of the Zygmund theory about its research background and nowadays conditions at home and abroad. And main result in this paper is given at las. In the second chapter, based on 3 different classes of functions to define the ?*p functions, the paper mainly introduces 3 kinds of different functions defined by Hardy G.H, Littlewood J.E and A.Zygmund by combinaing the Lipschitz properties of the 3 kinds of different functions, and making use of the monotonous of the Lp function class.When p< q, the inclusion relations between the A*p and the ?*q function classes and the ?*p and the ?ap functions are given. Also, a specific function is given to verify that it is an ?* class functions rather than an ?* class functions. Besides, f(z) function is confirmed to be a sufficient conditions of Aa functions on the base of the definition of the 3 difference functions, that is:Let f(z)is analytic, f(ei?)is continuous and|f(ei(?+h))-2f(ei?)+f(ei(?-h)]? Aha,0<a<1, then f(z)???The third chapter mainly introduces the smooth characterization of 3 kinds functions in the unit disk which defined by Hardy G.H.,Littlewood J.E. and Zygmund A. It proves that the sufficient and necessary conditions between the smooth characterization of the ?*p function and the change of its derivative which based on 3 different f.unction classes in the unit disk.Secondly,after the study of the property of the self-conjugate of the ??p functions and the ?*p functions on the basis of the self-conjugate of the ?? functions and the ?* functions,this paper gets its main result of this paper:1.Let f(z)be analytic in |z|<1. Then f(z)?Hp and f(ei?)??*p if and only if Mp(r,f")=O(1-r/1).2.Let f(z)=u(z)+iv(z)be analytic in|z|<1.When u ?hp(1<p<?)and u(ei?) ???p(0<?<1),then v(ei?)???p.That means ??p is self-conjugate.3.Let f(z)=u(z)+iv(z)be analytic in |z|<1.When u ?hp(1<p<?)and u(ei?)??*p(0<?<1),then v(ei?)??*p. That means ?*p is self-conjugate.