On Some Results Of Some Subspaces Of The Universal Teichmuller Space

Let 2≤p＜∞. A quasisymmetric homeomorphism of S1 is called a p-integrable asymptotic affine if it admits a quasiconformal extension into△so that its complex dilatationμ(z) is p-integrable in the Poincare metric on△, this means where p(z)|dz| is the Poincare metric on△. Let QSP(S1) denote the space of p-integrable asymptotic affine homeomorphisms of S1. Then one can denote by Tp= QSp(S1)/Mob(S1) the right coset space.The first part of this paper discuss some complex analytic characterizations of the p-integrable Teichmuller space. The followings are main results:(1) The complex dilatation of the Douady-Earle extension of the p-integrable asymp-totic affine homeomorphism is p-integrable in the Poincare metric on△;(2) The Bers projection of the p-integrable Teichmuller space is holomorphic;(3) We give a logarithmic derivative model Tp of the p-integrable Teichmuller space and discuss its topology structure, we prove that Tp is a connected open subset of the Besov space Bp;(4) Takhtajan and Teo asked that how to characterize the the 2-integrable asymptotic affine homeomorphism. In this paper, we give a necessary condition for the p-integrable asymptotic affine homeomorphism.The second part of this paper discuss the boundedness and compactness of com-position operators in the weighted Bergman space on the simple connected domain of the complex plane. LetΩbe a simple connected domain of the complex plane C.Let 0＜p＜∞and -1＜α＜∞, the weighted Bergman space Apαis consist of all the analytic functions onΩwith the followings hold where dA is the Lebesgue measure andδ(ω,(?)Ω)denote the distance fromωto the boundary (?)Ω.We obtain the following results:(1)LetΩbe a simple connected domain and letφ:Ω→Ωbe a finite valent analytic function.For-1≤α,β＜∞,we obtain a necessary condition for the composition operatorCφ:Aα2(Ω)→Aβ2(Ω)to be bounded or compacted.We also show that the condition is not suffcient by constructing an example.(2)LetQ be a simple connected domain and letφ:Ω→Ωbe a analytic function. For-1≤α,β＜∞,we obtain a sufficient condifion for the composition operator Cφ:Aα2(Ω)→Aβ2(Ω)to be bounded or compacted.We also show that the condition is not necessary by constructing an example.(3)WhenΩis a Lavrentiev domain,we obtain a sufficient and necessary condition for the composifion operator Cφ:Aα2(Ω)→Aβ2(Ω)to be bounded and compacted.