The Cantor Boundary Behavior Of Analytic Function And Composition Type Operator Between Function Spaces

Let D={z:|z|< 1} denote the open unit disc and C00 the Riemann sphere. Suppose that f(z) is analytic in D and has a continuous extension to the boundary (?)D, then f((?)D) is a locally connected compact set. Let C∞\f((?)D)= Uj≥0Wj be the decomposition as connected components, then Wj is a simple connected domain with locally connected boundary. If are Cantor-type sets in (?)D, i.e., nowhere dense sets in (?)D, where Wj is any connected component with f(D)∩Wj≠(?), then we say that f(z) has the Cantor boundary behavior in D.This article consists of six chapters, among which we study the Cantor boundary behavior of analytic functions in D and problems concerning with it in the first half of this article, as well as the composition type operator on some holomorphic function spaces of Cn in the second half.In the first Chapter, the background and current situation of the Cantor boundary behavior of analytic functions and the boundedness and compactness of composition type operators are introduced. At the same time, the main results of this article are given.In Chapter 2, the measure of the Cantor type set C=f-1((?)f(D)) is studied, and a sufficient condition for the Lebesgue measure of C being positive is got, which is " If f(z) has the Cantor boundary behavior and f'(z)∈H1, then|C|> 0 ". As applications, some examples with positive Lebesgue measure are given.In Chapter 3, we mainly consider the Weierstrass function whereλ> 1,0<β< 1, which is analytic in D={z:|z|< 1}(Ifλ≠integer, we let D={z:|z|< 1}\[0,1) and zλn denote the principal branch in D.) and continuous on D. In this chapter, we first define the local Cantor boundary behavior, and then prove that the Weierstrass function f(z) has local bound-ary behavior on L={eiθ:<θ< 2π}. At the same time, we also prove that the Hadamard gap series has Cantor boundary behavior in D if inf andIn Chapter 4, according to the relationship between pre-Schwarzian deriva-tive and univalent functions, another sufficient condition of Cantor boundary behavior is given, and some applications are given.In Chapter 5, the necessary and sufficient conditions are given for the weighted composition operator Tψ,ψto be bounded or compact from little Bloch type spacesβ0p toβ0q for all 0< p<∞and 0< q<∞on the unit ball of Cn.In Chapter 6, the Careson measure and multipliers are studied. For 0< p≤q<∞, we first give the necessary and sufficient conditions for positive Borel measureμon the uint ball B such that Dαp (?) Lq(dμ), and then get the necessary and sufficient conditions for multipliers to be bounded from Dαp to Dβq.