TANG Shu-an

(School of Mathematics Sciences, Guizhou Normal University, Guiyang 550001, China)

Abstract: This paper study the Bers projection and pre-projection of QK-Teichmller space.By means of quasiconformal mapping theory, we prove that the Bers projection and pre-projection of QK-Teichmller space are holomorphic.

Keywords: quasisymmetric homeomorphism; universal Teichmller space; Qk-Teichmller space; Bers projection

1 Introduction

The Bers embedding plays an important role in Teichmller theory.In terms of Schwarzian derivative,Teichmller space is embedded onto an open subset of some complex Banach space of holomorphic functions.We emphasize that the Bers projection in Teichmller space is a holomorphic split submersion (see [1], [2]for more details).The main purpose of this paper is to investigate the Bers projection and pre-projection of QK-Teichmller space.We begin with some notations and definitions.

Let ∆ = {z : |z| < 1} be the unit disk in the complex plane C,be the outside of the unit disk and S1= {z ∈ C : |z| = 1} be the unit circle.For z,a ∈ ∆, we setand denote by g(z,a)=the Green function of with a pole at a.For a nonnegative and nondecreasing function K on[0,∞), the space QKconsists of all analytic functions with the following finite norm

We say f belongs to QK,0space if f ∈QKand

Wulan and Wu introduced in [3]the space QKwhich was investigated in recent years(see [4–9]).Wulan and Wu [3]proved that QKspaces are always contained in the Bloch space B, which consists of holomorphic functions f on ∆such that

and QK,0⊂ B0, which consists of all functions f ∈ B such that

If K(t) = tpfor 0 < p < ∞, the space QKgives a Qpspace (see [10, 11]).In particular, If K(t)=t, then QK=BMOA. QKspace is nontrivial if and only if

here and in what follows we always assume that K(0) = 0 and condition (1.3) is satisfied.Furthermore, we require two more conditions on K as follows

Let I be an arc of the unit circle S1with normalized arclength< 1, the Carleson box is defined by

A positive measure λ on Ω is called a K-Carleson measure if

and a compact K-Carleson measure if

We denote by CMK(Ω) the set of all K-Carleson measures on Ω and CMK,0(Ω) the set of all compact K-Carleson measures on Ω.

An orientation preserving homeomorphism f from domain Ω onto f(Ω) is quasiconformal if f has locally L2integrable distributional derivative on Ω and satisfies the following equation

for some measurable functionsµ withWe say a sense preserving self-homeomorphism h is quasisymmetric if there exists some quasiconformal homeomorphism of ∆onto itself which has boundary value h (see [12]).Let QS(S1) be the group of quasisymmetric homeomorphisms of the unit circle S1and Mb(S1) the group of Mbius transformations mapping ∆onto itself.The universal Teichmller space is defined as the right coset space T =QS(S1)/Mb(S1).

We say that two Beltrami coefficients µ1and µ2in M(∆∗) are Teichmller equivalent and denote by µ1∼ µ2if fµ1(∆)=fµ2(∆).

Let B∞(∆) denote the Banach space of holomorphic functions on ∆ with norm

For a conformal mapping f on ∆, its Schwarzian derivative Sfof is defined as

The Bers projection Φ : M(∆∗) → B∞(∆) is defined by µ → Sfµ, which is a holomorphic split submersion onto its image and descends down to the Bers embedding B :T → B∞(∆).Thus T carries a natural complex Banach manifold structure so that B is a holomorphic split submersion.

A quasisymmetric homeomorphism h is strongly quasisymmetric if h has a quasiconformal extension to ∆ such that its complex dilatation µ(z) satisfies the condition

is a Carleson measure (see [13]).Strongly quasisymmetric homeomorphisms and its Teichmller theory were studied in recent years (see [14, 15]).

Recently, the author, Feng and Huo [16]introduced the F(p,q,s)-Teichmller space and showed that its pre-logarithmic derivative model is disconnected subset of the F(p,q,s)space.It was also shown that the Bers projection of this space is holomorphic.

Let Ω = ∆ or Ω = ∆∗.We denote by MQK(Ω) the Banach space of all essentially bounded measurable functions µ with

The norm of MQK(Ω) is defined as

Let TKbe the set of all functions logwhere f is conformal in ∆ with the normalized condition f(0) =− 1 = 0 and admits a quasiconformal extension to the whole plane such that its complex dilatation µ satisfies λµ(z)∈ CMK(∆∗).Then TKgives another model of TQK.Wulan and Ye [6]proved the following.

Theorem 1.1[6]Let K satisfy (1.4) and (1.5).Then TKis a disconnected subset of QK.Furthermore, TK,b= {log∈ TK: f(∆)is bounded} and TK,θ= {log∈ TK:f(eiθ)= ∞, θ ∈ [0,2π], are the connected components of TK.

The authors also obtained some more characterizations of QK-Teichmller space by using Grunsky kernel functions (see [17]).

The main purpose of this paper is to deal with the holomorphy of the Bers projection and the pre-Bers projection in QK-Teichmller space.In what follows, C will denote a positive universal constant which may vary from line to line and C(·) will denote constant that depends only on the elements put in the brackets.

2 Holomorphy of Bers Projection

In this section, we prove the Bers projection is holomorphic in QK-Teichmller space.Noting that K satisfies condition (1.5), we conclude that there exists c > 0 such that K(t)/tp−cis non-increasing and K(2t) ≈ K(t) for 0 < t < ∞.Wulan and Zhou proved the following.

Lemma 2.1[8]Let K satisfy conditions (1.4) and (1.5).Set b+ α ≥ 1+p, b ≥ p,b − p+ β +c > 1, and α > 0.Then there exists a constant C (independent of the lengthsuch that

for all w ∈ ∆ and arc I ⊂ S1.

Lemma 2.2Let K satisfy conditions (1.4) and (1.5).Let b+ α ≥ 1+p, b ≥ p,b − p+ α +c>2 and α >0.Set

ProofFor any arc I ⊂S1,let 2I is the arc with the same center as I but with double length.We divide the following integral into two part as follows

It follows from lemma that

Now we estimate L2.Set

Let nIbe the minimum such that 2nI ≥ 1.Then Sn= ∆ when n ≥ nI.Denote by wIthe center of I and w1=If ζ ∈ S(I) and w ∈ SnSn−1, 1

This implies that

Consequently

and

Since K(t) is a nondecreasing function, we have

Noting that K satisfies (1.4) and (1.5), we conclude that K(t)/tp−cis non-increasing for some small c>0.Consequently,

Combing (2.3) and (2.4), we deduce thatif λ ∈ CMK,0(∆).The proof follows.

Let NK(∆) denote the space which consists of all analytic functions f in ∆ with the following finite norm

We say a analytic function f belongs to NK,0(∆) if f ∈ NKand

The following is our main result.

Theorem 2.3The Bers projection Φ:is holomorphic.

ProofWe first show that the Bers projection Φis continuous.For any two elements µ and ν inBy an integral representation of the Schwarzian derivative by means of the representation theorem of quasiconformal mappings, Astala and Zinsmeister [13]proved that there exists some constantsuch that

where dλµ(z)=Consequently, we have

From [5], the measure λµ∈ CMK(∆) if and only if

Furthermore, there exist two positive constant C1and C2such that

Thus, it follows from Lemma 2.2 that

This shows the Bers projectionis continuous.

We now prove that the mappingis holomorphic.For each z ∈ ∆, we define a continuous linear functional lzon the Banach space NK(∆) by lz(ϕ) =ϕ(z) for ϕ ∈ NK(∆).Then the set A = {lz: z ∈ ∆} is a total subset of the dual space of NK(∆).Now for each z ∈ ∆, each pair (µ,ν) ∈and small t in the complex plane, it follows from the holomorphic dependence of quasiconformal mappings on parameters that lz(Φ(µ +tν))=Sfµ+tν(z) is a holomorphic function of t (see [18, 2, 1]).From the infinite dimensional holomorphy theory (see Proposition 1.6.2 in [1]), we deduce that the Bers projectionis holomrphic.

Similarly, we can prove the following

Theorem 2.4The Bers projectionis holomorphic.

3 Holomorphy of Pre-Bers Projection

Fix z0∈ ∆∗.Forbe the quasiconformal mapping whose complex dilatation is µ in ∆∗and is zero in ∆, normalized by fµ(0) =−1 = 0,fµ(z0) =∞.The pre-Bers projection mapping Lz0onis defined by the correspondencebe the space consisting of all functions ϕ ∈QKwith ϕ(0) = 0,then

We prove the following.

Theorem 3.1For z0∈ ∆∗, the pre-Bers projection mapping Lz0:is holomorphic.

ProofSince we can prove Lz0:is holomorphic by the same reasoning as the proof of the holomorphy of Φ :we need only to show thatis continuous.For µ,ν ∈for simplicity of notation, we let f and g stand forrespectively.It follows from Theorem 3.1 in Chapter II in [2]that

By theorem , we conclude that

It is known that a function of QKspace can be characterized by its higher derivative [19].Thus, for any a ∈ ∆, we get

Combing (3.2) with (3.1) gives

Similar arguments apply to QK,0space gives

Theorem 3.2For z0∈ ∆∗,the pre-Bers projection mappingis holomorphic, whereconsists of all functions ϕ ∈ QK,0with ϕ(0)=0.