The Bers embedding plays an important role in Teichmüller theory. In terms of Schwarzian derivative, Teichmüller space is embedded onto an open subset of some complex Banach space of holomorphic functions. We emphasize that the Bers projection in Teichmüller 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 $ Q_K $-Teichmüller space. We begin with some notations and definitions.

Let $ \Delta = \{z:|z| < 1\} $ be the unit disk in the complex plane $ \mathbb{C} $, $ \Delta^{*} = \mathbb{C}\backslash \overline{\Delta} $ be the outside of the unit disk and $ S^1 = \{z\in \mathbb {C} : |z| = 1\} $ be the unit circle. For $ z, a\in\Delta $, we set $ \varphi_a(z) = \frac{z-a}{1-\overline{a}z} $ and denote by $ g(z, a) = \log\frac{1}{|\varphi_a(z)|} $ the Green function of with a pole at $ a $. For a nonnegative and nondecreasing function $ K $ on $ [0, \infty) $, the space $ Q_{K} $ consists of all analytic functions with the following finite norm

$ \begin{equation} \| f\|^2_{Q_{K}} = \sup\limits_{a\in\Delta} \iint_{\Delta} |f'(z)|^2 K(g(z, a))dxdy. \end{equation} $

(1.1)

We say $ f $ belongs to $ Q_{K, 0} $ space if $ f\in Q_{K} $ and

$ \begin{equation} \lim\limits_{|a|\rightarrow 1} \iint_{\Delta} |f'(z)|^2 K(g(z, a))dxdy = 0. \end{equation} $

(1.2)

Wulan and Wu introduced in [3] the space $ Q_{K} $ which was investigated in recent years (see [4-9]). Wulan and Wu [3] proved that $ Q_{K} $ spaces are always contained in the Bloch space $ \mathfrak{B} $, which consists of holomorphic functions $ f $ on $ \Delta $ such that

$ \| f\|_B = \sup\limits_{z\in\Delta}(1-|z|^2)|f'(z)|<\infty $

and $ Q_{K, 0}\subset \mathfrak{B}_0 $, which consists of all functions $ f\in \mathfrak{B} $ such that

$ \lim\limits_{|z|\rightarrow 1}(1-|z|^2)|f'(z)| = 0. $

If $ K(t) = t^p $ for $ 0 < p < \infty $, the space $ Q_{K} $ gives a $ Q_{p} $ space (see [10, 11]). In particular, If $ K(t) = t $, then $ Q_{K} = $BMOA. $ Q_{K} $ space is nontrivial if and only if

$ \begin{equation} \sup\limits_{t\in(0, 1)} \int_0^1 \frac{(1-t)^2}{(1-tr^2)^3} K\left( \log\frac{1}{r}\right) rdr<\infty, \end{equation} $

(1.3)

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

$ \begin{equation} \int_0^1 \frac{\varphi_K(s)}{s}ds<\infty \end{equation} $

(1.4)

and

$ \begin{equation} \int_1^{\infty} \frac{\varphi_K(s)}{s^{1+p}}ds<\infty, \; 0<p<2, \end{equation} $

(1.5)

where $ \varphi_K(s) = \sup\limits_{0\le t\le 1} K(st)/K(t) $, $ 0 < s < \infty $.

Let $ I $ be an arc of the unit circle $ S^1 $ with normalized arclength $ \ell(I) < 1 $, the Carleson box is defined by

$ {S_\Omega }(I) = \left\{ \begin{array}{l} \{ z \in \Delta :1 - \ell (I) \le |z| < 1,z/|z| \in I\} ,\Omega = \Delta ,\\ \{ z \in {\Delta ^*}:1 \le |z| < 1 + \ell (I),z/|z| \in I\} ,\Omega = {\Delta ^*}. \end{array} \right. $

A positive measure $ \lambda $ on $ { \Omega} $ is called a $ K $-Carleson measure if

$ \| \lambda\|_{C, K} = \sup\limits_{I\subset S^1} \iint_{S_\Omega(I)} K\left( \frac{1-|z|}{\ell(I)}\right) d\lambda(z)<\infty $

and a compact $ K $-Carleson measure if

$ \lim\limits_{|I|\rightarrow 0} \iint_{S_\Omega(I)} K\left( \frac{1-|z|}{\ell(I)}\right) d\lambda(z) = 0. $

We denote by $ CM_K({ \Omega}) $ the set of all $ K $-Carleson measures on $ { \Omega} $ and $ CM_{K, 0}({ \Omega}) $ the set of all compact $ K $-Carleson measures on $ { \Omega} $.

An orientation preserving homeomorphism $ f $ from domain $ \Omega $ onto $ f(\Omega) $ is quasiconformal if $ f $ has locally $ L^{2} $ integrable distributional derivative on $ \Omega $ and satisfies the following equation

$ f_{\overline{z}} = \mu(z) f_{z} $

for some measurable functions $ \mu $ with $ \|\mu \|_{\infty} $. We say a sense preserving self-homeomorphism $ h $ is quasisymmetric if there exists some quasiconformal homeomorphism of $ \Delta $ onto itself which has boundary value $ h $ (see [12]). Let $ QS(S^{1}) $ be the group of quasisymmetric homeomorphisms of the unit circle $ S^{1} $ and Möb($ S^{1} $) the group of Möbius transformations mapping $ \Delta $ onto itself. The universal Teichmüller space is defined as the right coset space $ T = QS(S^{1})/ $Möb($ S^{1} $).

The universal Teichmüller space $ T $ can also be described as $ T = M(\Delta^*)/\sim $, where $ M(\Delta^*) $ denotes the unit ball of the Banach space $ L^{\infty}(\Delta^*) $ of bounded measurable functions on $ \Delta^* $. Let $ f_{\mu} $ be the unique quasiconformal mapping whose complex dilatation is $ \mu $ in $ \Delta^{*} $ and is zero in $ \Delta $, normalized by

$ f_{\mu}(0) = f_{\mu}'(0)-1 = f_{\mu}''(0) = 0. $

We say that two Beltrami coefficients $ \mu_{1} $ and $ \mu_{2} $ in $ M(\Delta^*) $ are Teichmüller equivalent and denote by $ \mu_{1}\sim\mu_{2} $ if $ f_{\mu_{1}}(\Delta) = f_{\mu_{2}}(\Delta) $.

Let $ {\mathcal B}_{\infty}(\Delta) $ denote the Banach space of holomorphic functions on $ \Delta $ with norm

$ \|\varphi\|_{{\mathcal B}_{\infty}} = \sup\limits_{z\in\Delta}|\varphi(z)|(1-|z|^2)^2<\infty. $

For a conformal mapping $ f $ on $ \Delta $, its Schwarzian derivative $ S_{f} $ of is defined as

$ S_{f} = \left(\frac{f''}{f'}\right)'-\frac{1}{2}\left(\frac{f''}{f'}\right)^{2}. $

The Bers projection $ \Phi:M(\Delta^*) \rightarrow {\mathcal B}_\infty(\Delta) $ is defined by $ \mu\mapsto S_{f_{\mu}} $, which is a holomorphic split submersion onto its image and descends down to the Bers embedding $ B: T\rightarrow {\mathcal B}_\infty(\Delta) $. 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 $ \Delta $ such that its complex dilatation $ \mu(z) $ satisfies the condition

$ \frac{|\mu(z)|^2}{(1-|z|^2)}dxdy $

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

Recently, the author, Feng and Huo [16] introduced the $ F(p, q, s) $-Teichmüller 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 $ { \Omega} = \Delta $ or $ { \Omega} = \Delta^* $. We denote by $ M_{Q_K}({ \Omega}) $ the Banach space of all essentially bounded measurable functions $ \mu $ with

$ \lambda_\mu(z): = \frac{\mid \mu(z)\mid ^{2}}{||z|-1|^{2}}dxdy \in CM_K({\Omega}). $

The norm of $ M_{Q_K}({ \Omega}) $ is defined as

$ \begin{equation} \parallel \mu\parallel_{K} = \parallel \mu\parallel_{\infty}+ \| \lambda_\mu\|^{1/2}_{C, K}, \end{equation} $

(1.6)

where $ \| \lambda_\mu\|_{C, K} $ is the $ K $ Carleson norm of $ \lambda_\mu $ on $ { \Omega} $. Let $ M_{Q_{K, 0}}({ \Omega}) $ be the subspace of $ M_{Q_K}({ \Omega}) $ consists of all elements $ \mu $ such that $ \lambda_\mu(z)\in CM_{K, 0}({ \Omega}) $. Set $ M^1_{Q_K}(\Delta^*) = M_{Q_K}(\Delta^*)\cap M(\Delta^*) $ and $ M^1_{Q_{K, 0}}(\Delta^*) = M_{Q_{K, 0}}(\Delta^*)\cap M(\Delta^*) $. The $ Q_K $-Teichmüller space $ T_{Q_K} $ is defined as $ T_{Q_K} = M^1_{Q_K}(\Delta^*)/\sim $ and $ Q_{K, 0} $-Teichmüller space $ T_{Q_{K, 0}} = M^1_{Q_{K, 0}}(\Delta^*)/\sim $.

Let $ T_K $ be the set of all functions $ \log f' $, where $ f $ is conformal in $ \Delta $ with the normalized condition $ f(0) = f'(0)-1 = 0 $ and admits a quasiconformal extension to the whole plane such that its complex dilatation $ \mu $ satisfies $ \lambda_\mu(z)\in CM_{K}(\Delta^*) $. Then $ T_K $ gives another model of $ T_{Q_K} $. Wulan and Ye [6] proved the following.

Theorem 1.1 [6] Let $ K $ satisfy $ (1.4) $ and $ (1.5) $. Then $ T_K $ is a disconnected subset of $ Q_K $. Furthermore, $ T_{K, b} = \{ \log f'\in T_K: f(\Delta) \text{is bounded}\} $ and $ T_{K, \theta} = \{ \log f'\in T_K: f(e^{i\theta}) = \infty $, $ \theta\in [0, 2\pi] $, are the connected components of $ T_K $.

The authors also obtained some more characterizations of $ Q_K $-Teichmüller 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 $ Q_K $-Teichmüller space. In what follows, $ C $ will denote a positive universal constant which may vary from line to line and $ C(\cdot) $ will denote constant that depends only on the elements put in the brackets.

In this section, we prove the Bers projection is holomorphic in $ Q_K $-Teichmüller space. Noting that $ K $ satisfies condition (1.5), we conclude that there exists $ c > 0 $ such that $ K(t)/t^{p-c} $ is non-increasing and $ K(2t)\approx K(t) $ for $ 0 < t < \infty $. Wulan and Zhou proved the following.

Lemma 2.1 [8] Let $ K $ satisfy conditions $ (1.4) $ and $ (1.5) $. Set $ b+\alpha \ge 1+p $, $ b\ge p $, $ b-p+\beta+c > 1 $, and $ \alpha > 0 $. Then there exists a constant $ C $ (independent of the length $ \ell(I) $ of $ I $) such that

$ \begin{equation} \iint_{\Delta} \frac{K\left( \frac{1-|\zeta|}{\ell(I)}\right)(1-|w|^2)^{b-1+\beta}}{(1-|\zeta|)^{1-\alpha+\beta}|1-w\overline{\zeta}|^{b+\alpha}} dA(\zeta) \le C K\left( \frac{1-|w|}{\ell(I)}\right) \end{equation} $

(2.1)

for all $ w\in\Delta $ and arc $ I\subset S^1 $.

Lemma 2.2 Let $ K $ satisfy conditions $ (1.4) $ and $ (1.5) $. Let $ b+\alpha \ge 1+p $, $ b\ge p $, $ b-p+\alpha+c > 2 $ and $ \alpha > 0 $. Set

$ \begin{equation} \widetilde{\lambda}(\zeta) = \iint_{\Delta} \frac{(1-|w|^2)^{b+\alpha-2}}{|1-w\overline{\zeta}|^{b+\alpha}} \lambda(w)dA(w). \end{equation} $

(2.2)

Then $ \widetilde{\lambda}\in CM_K(\Delta) $ if $ \lambda\in CM_K(\Delta) $ and $ \|\widetilde{\lambda}\|_{C, K}\le C\|\lambda\|_{C, K} $, while $ \widetilde{\lambda}\in CM_{K, 0}(\Delta) $ if $ \lambda\in CM_{K, 0}(\Delta) $.

Proof For any arc $ I\subset S^1 $, 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

$ \iint_{S(I)} \iint_{\Delta}\frac{(1-|w|^2)^{b+\alpha-2}}{|1-w\overline{\zeta}|^{b+\alpha}} \lambda(w)dA(w) K\left( \frac{1-|\zeta|}{\ell(I)}\right) dA(\zeta) = L_1+L_2, $

where

$ L_1 = \iint_{S(I)} \iint_{S(2I)} \frac{(1-|w|^2)^{b+\alpha-2}}{|1-w\overline{\zeta}|^{b+\alpha}} \lambda(w)dA(w)K\left( \frac{1-|\zeta|}{\ell(I)}\right) dA(\zeta) $

and

$ L_2 = \iint_{S(I)} \iint_{\Delta\backslash S(2I)} \frac{(1-|w|^2)^{b+\alpha-2}}{|1-w\overline{\zeta}|^{b+\alpha}} \lambda(w)dA(w)K\left( \frac{1-|\zeta|}{\ell(I)}\right) dA(\zeta). $

It follows from lemma that

$ \begin{eqnarray} L_1 & = & \iint_{S(I)} \lambda(w)dA(w) \iint_{S(2I)} \frac{K\left( \frac{1-|\zeta|}{\ell(I)}\right)(1-|w|^2)^{b-2+\alpha}}{|1-w\overline{\zeta}|^{b+\alpha}} dA(\zeta) \\ &\le& C \sup\limits_{I}\iint_{S(I)} \lambda(w) K\left( \frac{1-|w|}{\ell(I)}\right)dA(w). \end{eqnarray} $

(2.3)

Now we estimate $ L_2 $. Set

$ S_n = S(2^nI) = \{r\zeta\in \Delta: 1-2^n\ell(I)<r<1, \; \zeta\in 2^nI\}, \quad n = 1, 2, \cdots. $

Let $ n_I $ be the minimum such that $ 2^nI\ge 1 $. Then $ S_n = \Delta $ when $ n\ge n_I $. Denote by $ w_I $ the center of $ I $ and $ w_1 = (1-\ell(I)/2)w_I $. If $ \zeta\in S(I) $ and $ w\in S_n\backslash S_{n-1} $, $ 1 < n < n_I $, then

$ \frac{2^{n-1}}{\pi} \ell(I)\le |\zeta-w_I|\le \frac{3}{\pi} 2^n\ell(I). $

This implies that

$ \begin{eqnarray*} |\zeta-w_1|&\le& |\zeta-w_I|+|w_I-z_1|\le 3\cdot 2^n\ell(I), \\ |\zeta-w_1|&\ge& |\zeta-w_I|-|w_I-z_1|\ge \frac{4-\pi}{8\pi} 2^n\ell(I). \end{eqnarray*} $

Consequently

$ 1-|\zeta|<\frac{8\pi}{4-\pi} 2^{-n} |\zeta-w_1| , \quad\quad 1-|w|<2^n\ell(I) \le\frac{8\pi}{4-\pi}|\zeta-w_1| $

and

$ \frac{1}{|1-w\overline{\zeta}|}\le \frac{C}{2^{n-1}\ell(I)}. $

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

$ \begin{eqnarray*} L_2 & = & \iint_{S(I)} \iint_{\Delta\backslash S(2I)} \frac{(1-|w|^2)^{b+\alpha-2}}{|1-w\overline{\zeta}|^{b+\alpha}} \lambda(w)dA(w)K\left( \frac{1-|\zeta|}{\ell(I)}\right) dA(\zeta) \\ & = & \iint_{S(I)} \sum\limits_{n = 2}^{n_I} \iint_{S_n\backslash S_{n-1}} \frac{(1-|w|^2)^{b+\alpha-2}}{|1-w\overline{\zeta}|^{b+\alpha}} \lambda(w)dA(w)K\left( \frac{1-|\zeta|}{\ell(I)}\right) dA(\zeta) \\ &\le& C \iint_{S(I)} \sum\limits_{n = 2}^{\infty} \iint_{S_n\backslash S_{n-1}} \frac{(1-|w|^2)^{b+\alpha-2}}{|1-w\overline{\zeta}|^{b+\alpha}} \lambda(w) K\left( \frac{|\zeta-w_1|}{2^n\ell(I)}\right) dA(w)dA(\zeta). \end{eqnarray*} $

Noting that $ K $ satisfies (1.4) and (1.5), we conclude that $ K(t)/t^{p-c} $ is non-increasing for some small $ c > 0 $. Consequently,

$ \begin{eqnarray} L_2 &\le& C \iint_{S(I)} \sum\limits_{n = 2}^{\infty} \iint_{S_n\backslash S_{n-1}} K\left( \frac{1-|w|}{2^n\ell(I)}\right) \\ && \times \frac{(1-|w|^2)^{b+\alpha-2+c-p}}{|\zeta-w_1|^{c-p}|1-w\overline{\zeta}|^{b+\alpha}} \lambda(w) dA(w)dA(\zeta) \\ &\le& C \iint_{S(I)} \sum\limits_{n = 2}^{\infty} \iint_{S_n\backslash S_{n-1}} K\left( \frac{1-|w|}{2^n\ell(I)}\right) \\ && \times \frac{(1-|w|^2)^{b+\alpha-2+c-p} (3\cdot 2^n \ell(I))^{p-c}}{|1-w\overline{\zeta}|^{b+\alpha}} \lambda(w) dA(w)dA(\zeta) \\ & = & C \iint_{S(I)} \sum\limits_{n = 2}^{\infty} \iint_{S_n\backslash S_{n-1}} \frac{4\cdot 6^{p-c}}{2^{2n}\ell(I)^2} \frac{(1-|w|^2)^{b+\alpha-2+c-p}}{|1-w\overline{\zeta}|^{b+\alpha-2+c-p}} \\ &&\times \lambda(w) K\left( \frac{1-|w|}{2^n\ell(I)}\right) dA(w)dA(\zeta)\\ &\le& C \sum\limits_{n = 2}^{\infty} \frac{4\cdot 6^{p-c}}{2^{2n}} \iint_{S_n}\lambda(w) K\left( \frac{1-|w|}{2^n\ell(I)}\right) dA(w) \\ &\le& C(p, c) \sup\limits_{I}\iint_{S(I)} \lambda(w) K\left( \frac{1-|w|}{\ell(I)}\right)dA(w). \end{eqnarray} $

(2.4)

Combing (2.3) and (2.4), we deduce that $ \|\widetilde{\lambda}\|_{C, K}\le C\|\lambda\|_{C, K} $, while $ \widetilde{\lambda}\in CM_{K, 0}(\Delta) $ if $ \lambda\in CM_{K, 0}(\Delta) $. The proof follows.

Let $ N_K(\Delta) $ denote the space which consists of all analytic functions $ f $ in $ \Delta $ with the following finite norm

$ \begin{equation} \|f\|^2_{N_K} = \sup\limits_{a\in\Delta} \iint_{\Delta} |f(z)|^2 (|z|^2-1)^{2} K(1-|\varphi_a(z)|^2) dxdy. \end{equation} $

(2.5)

We say a analytic function $ f $ belongs to $ N_{K, 0}(\Delta) $ if $ f\in N_K $ and

$ \begin{equation} \lim\limits_{|a|\rightarrow 1} \iint_{\Delta} |f(z)|^2 (|z|^2-1)^{2}K(1-|\varphi_a(z)|^2)dxdy = 0. \end{equation} $

(2.6)

The following is our main result.

Theorem 2.3 The Bers projection $ \Phi: M^1_{Q_K}(\Delta^*) \rightarrow N_K(\Delta) $ is holomorphic.

Proof We first show that the Bers projection $ \Phi: M^1_{Q_K}(\Delta^*) \rightarrow N_K(\Delta) $ is continuous. For any two elements $ \mu $ and $ \nu $ in $ M^1_{Q_K}(\Delta^*) $, set $ S_{f_\mu} = \Phi(\mu) $ and $ S_{f_\nu} = \Phi(\nu) $. By 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 constant $ C(\| \mu \|_{\infty}) $ such that

$ \begin{equation} |S_{f_\mu}-S_{f_\nu}|^2 \leq \frac{C(\| \mu \|_{\infty})}{(|z|^{2}-1)^{2}} \iint_{\Delta^*} \frac{|\mu(\zeta)-\nu(\zeta)|^{2}+ \| \mu -\nu\|^2_{\infty}|\mu(\zeta)|^2}{|\zeta-z|^{4}}d\xi d\eta. \end{equation} $

(2.7)

Let $ \widetilde{\mu}(z) = \mu(1/z) z^2/\overline{z}^2 $ and $ \widetilde{\nu}(z) = \nu(1/z) z^2/\overline{z}^2 $. Then a change of variable gives

$ \begin{equation} \| \lambda_{\mu-\nu}\|_{C, K} = \| \lambda_{\widetilde{\mu}-\widetilde{\nu}}\|_{C, K} \; \text{and}\; \| \lambda_\mu\|_{C, K} = \| \lambda_{\widetilde{\mu}} \|_{C, K}, \end{equation} $

(2.8)

where $ d\lambda_\mu(z) = \frac{\mid \mu(z)\mid ^{2}}{||z|^2-1|^{2}}dxdy $. Consequently, we have

$ \begin{equation} |S_{f_\mu}-S_{f_\nu}|^2 \leq \frac{C(\| \mu\|_{\infty})}{(|z|^{2}-1)^{2}} \iint_{\Delta} \frac{|\widetilde{\mu}(\zeta)-\widetilde{\nu}(\zeta)|^{2}+ \| \mu -\nu\|^2_{\infty}|\widetilde{\mu}(\zeta)|^2}{|1-\zeta z|^{4}}d\xi d\eta. \end{equation} $

(2.9)

From [5], the measure $ \lambda_{\mu}\in CM_K(\Delta) $ if and only if

$ \sup\limits_{a\in \Delta}\iint_{\Delta} K(1-|\varphi_a(z)|^2) d\lambda_{\mu}(z). $

Furthermore, there exist two positive constant $ C_1 $ and $ C_2 $ such that

$ C_1\| \lambda_{\mu}\|_{C, K} \le \sup\limits_{a\in \Delta}\iint_{\Delta} K(1-|\varphi_a(z)|^2) d\lambda_{\mu}(z)\le C_2\| \lambda_{\mu}\|_{C, K}. $

Thus, it follows from Lemma 2.2 that

$ \begin{eqnarray*} \|S_{f_\mu}-S_{f_\nu} \|^2_{N_K} &\le& C(\| \mu\|_{\infty}) \sup\limits_{a\in \Delta} \iint_{\Delta} K(1-|\varphi_a(z)|^2) \\ && \times \iint_{\Delta} \frac{|\widetilde{\mu}(\zeta)-\widetilde{\nu}(\zeta)|^{2}+ \| \mu -\nu\|^2_{\infty}|\widetilde{\mu}(\zeta)|^2}{|1-\zeta z|^{4}}d\xi d\eta dxdy\\ &\le& C(\| \mu \|_{\infty}) \sup\limits_{a\in \Delta} \iint_{\Delta} K(1-|\varphi_a(z)|^2) \iint_{\Delta} \frac{(1-|\zeta|^2)^2}{|1-\zeta z|^{4}} d\lambda_{\widetilde{\mu}-\widetilde{\nu}}(\zeta) dxdy\\ &&+ C(\| \mu \|_{\infty})\| \mu -\nu\|^2_{\infty} \sup\limits_{a\in \Delta} \iint_{\Delta} K(1-|\varphi_a(z)|^2) \iint_{\Delta} \frac{(1-|\zeta|^2)^2}{|1-\zeta z|^{4}} d\lambda_{\widetilde{\mu}}(\zeta) dxdy\\ &\le& C(\| \mu \|_{\infty}) (\| \lambda_{\widetilde{\mu}-\widetilde{\nu}}\|^2_{C, K} + \| \mu -\nu\|^2_{\infty} \|\lambda_{\widetilde{\mu}}\|^2_{C, K}) \\ &\le& C(\| \mu \|_{\infty})(1+ \|\lambda_{\mu}\|^2_{C, K})\|\mu-\nu\|^2_{K}. \end{eqnarray*} $

This shows the Bers projection $ \Phi: M^1_{Q_K}(\Delta^*) \rightarrow N_K(\Delta) $ is continuous.

We now prove that the mapping $ \Phi: M^1_{Q_K}(\Delta^*) \rightarrow N_{K}(\Delta) $ is holomorphic. For each $ z\in \Delta $, we define a continuous linear functional $ l_z $ on the Banach space $ N_{K}(\Delta) $ by $ l_z(\varphi) = \varphi(z) $ for $ \varphi\in N_{K}(\Delta) $. Then the set $ A = \{ l_z: z\in \Delta\} $ is a total subset of the dual space of $ N_{K}(\Delta) $. Now for each $ z\in\Delta $, each pair $ (\mu, \nu)\in M^1_{Q_K}(\Delta^*) \times M_{Q_K}(\Delta^*) $ and small $ t $ in the complex plane, it follows from the holomorphic dependence of quasiconformal mappings on parameters that $ l_z(\Phi(\mu+t\nu)) = S_{f_{\mu+t\nu}}(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 projection $ \Phi: M^1_{Q_K}(\Delta^*) \rightarrow N_K(\Delta) $ is holomrphic.

Similarly, we can prove the following

Theorem 2.4 The Bers projection $ \Phi: M^1_{Q_{K, 0}}(\Delta^*) \rightarrow N_{K, 0}(\Delta) $ is holomorphic.

Fix $ z_0\in \Delta^* $. For $ \mu\in M^1_{Q_K}(\Delta^*) $, let $ f^{z_0}_{\mu} $ be the quasiconformal mapping whose complex dilatation is $ \mu $ in $ \Delta^* $ and is zero in $ \Delta $, normalized by $ f_\mu(0) = f_\mu'(0)-1 = 0, f_\mu(z_0) = \infty $. The pre-Bers projection mapping $ L_{z_0} $ on $ M^1_{Q_K}(\Delta^*) $ is defined by the correspondence $ L_{z_0}(\mu) = \log f_\mu' $. Let $ Q^{0}_{K} $ be the space consisting of all functions $ \varphi\in Q_K $ with $ \varphi(0) = 0 $, then $ \cup_{z_0\in\Delta^*} L_{z_0}(M^1_{Q_K}(\Delta^*)) = T_{K, b}\cap Q^{0}_K $.

We prove the following.

Theorem 3.1 For $ z_0\in \Delta^* $, the pre-Bers projection mapping $ L_{z_0}: M^1_{Q_K}(\Delta^*) \rightarrow Q^{0}_{K} $ is holomorphic.

Proof Since we can prove $ L_{z_0}: M^1_{Q_K}(\Delta^*) \rightarrow Q^{0}_{K} $ is holomorphic by the same reasoning as the proof of the holomorphy of $ \Phi: M^1_{Q_K}(\Delta^*) \rightarrow N_K(\Delta) $, we need only to show that $ L_{z_0}: M^1_{Q_K}(\Delta^*) \rightarrow Q^{0}_{K} $ is continuous. For $ \mu, \nu\in M^1_{Q_K}(\Delta^*) $, for simplicity of notation, we let $ f $ and $ g $ stand for $ f_\mu^{z_0} $ and $ f_\nu^{z_0} $, respectively. It follows from Theorem 3.1 in Chapter Ⅱ in [2] that

$ \begin{eqnarray} \sup\limits_{z\in\Delta}(1-\mid z\mid^{2})\left|\frac{f''}{f'}-\frac{g''}{g'}\right| \leq C\|\mu-\nu \|_\infty. \end{eqnarray} $

(3.1)

By theorem, we conclude that

$ \begin{eqnarray} \sup\limits_{a\in \Delta}\iint_{\Delta}& (1-\mid z\mid^{2})^{2} \mid S_{f}(z)-S_{g}(z)\mid^{2} K(1-|\varphi_a(z)|^2)dxdy &\le C\|\mu-\nu\|^2_{K}. \end{eqnarray} $

(3.2)

It is known that a function of $ Q_K $ space can be characterized by its higher derivative [19]. Thus, for any $ a\in\Delta $, we get

$ \begin{equation} \begin{split} I(a):& = \iint_{\Delta} \left| \frac{f''}{f'}-\frac{g''}{g'}\right|^{2} K(1-|\varphi_a(z)|^2)dxdy\\ &\leq C \left|\frac{f''}{f'}(0)-\frac{g''}{g'}(0)\right|^{2} \\ &+ C\iint_{\Delta} (1-\mid z\mid^{2})^{2}\left|\left(\frac{f''}{f'}\right)'-\left(\frac{g''}{g'}\right)'\right|^{2} K(1-|\varphi_a(z)|^2)dxdy.\\ \end{split} \end{equation} $

(3.3)

Combing (3.2) with (3.1) gives

$ \begin{equation} \begin{split} I(a) \le& C \|\mu-\nu \|_\infty^2 + C\iint_{\Delta} (1-\mid z\mid^{2})^{2} \mid S_{f}-S_{g}\mid^{2} K(1-|\varphi_a(z)|^2)dxdy\\ &+ C\iint_{\Delta} \left| \left(\frac{f''}{f'}\right)^2-\left(\frac{g''}{g'}\right)^2\right|^{2} (1-\mid z\mid^{2})^{2} K(1-|\varphi_a(z)|^2)dxdy \\ \le& C \|\mu-\nu\|^2_{K}\\ &+ C\iint_{\Delta} (1-\mid z\mid^{2})^{2} \left| \frac{f''}{f'}+\frac{g''}{g'}\right|^{2} \left| \frac{f''}{f'}-\frac{g''}{g'}\right|^{2} K(1-|\varphi_a(z)|^2)dxdy \\ \le& C(\|\log g'_\mu\|^{2}_{Q_K}+\|\log g'_\nu\|^{2}_{Q_K})\|\mu-\nu\|^2_{K}. \end{split} \end{equation} $

(3.4)

This implies that $ L_{z_0}: M^1_{Q_K}(\Delta^*) \rightarrow Q^{0}_{K} $ is continuous. The proof follows.

Similar arguments apply to $ Q_{K, 0} $ space gives

Theorem 3.2 For $ z_0\in \Delta^* $, the pre-Bers projection mapping $ L_{z_0}: M^1_{Q_{K, 0}}(\Delta^*) \rightarrow Q^{0}_{K, 0} $ is holomorphic, where $ Q^{0}_{K, 0} $ consists of all functions $ \varphi\in Q_{K, 0} $ with $ \varphi(0) = 0 $.