Abstract
In this paper, a subspace of the universal Teichmüller space, which is related to the analytic function space , is introduced and the holomorphy of the Bers map is shown. It is also proved that the pre-Bers map is holomorphic and the prelogarithmic derivative model of is a disconnected subset of the function space . Moreover, several equivalent descriptions of elements of are obtained and the holomorphy of higher Bers maps is proved.
1. Introduction
Let be the unit disk of the extended complex plane and . Denote by the open unit ball of the Banach space of essentially bounded measurable functions on . For any , there exists a unique quasiconformal self-homeomorphism of whose complex dilatation is µ in and zero in , satisfying the following normalized condition:
Two elements are called Teichmüller equivalent and denoted by , if on . The universal Teichmüller space T can be defined aswhere is the Teichmüller equivalence class containing µ (see [1–3]).
Let be the complex Banach space of all holomorphic quadratic differentials ϕ in with the following norm:
Then, there is a complex structure on T induced from via the Bers embedding:where
The image is called the Schwarzian derivative model of T while is the Schwarzian derivative of .
The complex structure of T can be induced from by the natural projection:
Precisely, the Bers projection,is holomorphic. And the following diagram is commutative.
Let be the class of all univalent holomorphic functions f on which can be quasiconformally extended to and satisfy . Then,is sometimes called the prelogarithmic derivative model of T, since the image of under the mapis the Schwarzian derivative model . It is known [4] that is a disconnected subset of the Bloch space consisting all holomorphic functions on satisfyingwith infinitely many connected components:
Fix . For , let (abbreviated to be ) be the quasiconformal mapping to whose complex dilation is µ in and zero in , normalized by and . It is clear that there is an injective map as follows:which is called the pre-Bers map.
Since there are many important subspaces of such as BMOA and VMOA, one obtains interesting subspaces of T via the map . For example, BMO-Teichmüller space and VMO-Teichmüller space are subsets and of T, respectively [5–9].
Let () be the Banach space of all holomorphic functions f on with the following norm:
It is introduced and studied by Zhao [10]. is trivial if and are contained in the Bloch-type space , where is the Banach space of all holomorphic functions f on with the norm:
Let be the subspace of withwhich plays an important role in this paper.
Noticing that
Feng et al. [11] introduced a subset of T by the s-Carleson measure, and they prove that its corresponding prelogarithmic derivative model is
So they call the -Teichmüller space. Moreover, they prove that its prelogarithmic derivative model has infinitely many connected components in and its Schwarzian derivative model is contained in . Here, is the Banach space of all holomorphic functions f on with the norm:
Let be the subspace of with
The holomorphies of the corresponding Bers projection and pre-Bers map are proved, and several descriptions of the elements in are also given.
Sinceit is clear that as . Enlightened by [11, 12], we will introduce a subset of T by the compact s-Carleson measure and we prove that its corresponding prelogarithmic derivative model is .
Letwhere is the set of all compact s-Carlson measures on (see Section 2 in detail). Then, is a Banach space with the norm:
Let
It is clear that is a subdomain of . Then, we have Theorem 1.
Theorem 1. Let . The Bers projection is holomorphic.
Let
It is clear that the prelogarithmic derivative model of and
Furthermore, we have the following theorem.
Theorem 2. Let . Then, for given , the pre-Bers map is holomorphic.
Theorem 2 shows is a subset of . Thus, we call the -Teichmüller space. The connectivity of in is also considered.
Theorem 3. Let . Then, is a disconnected subset of , and its connected components are (i) and(ii)
Then, equivalent descriptions of elements of are studied. Precisely, we obtain the following theorem (for the relevant operators and kern function, see Section 3).
Theorem 4. Let . If f is conformal on , then the following statements are equivalent:(1).(2).(3)f can be extended to a quasiconformal mapping to the whole plane such that its complex dilatation satisfies .(4), where are the Grunsky kernel of f.(5)There exists a conformal map , such that is a quasisymmetric homeomorphism and , where
The higher Bers maps are studied in [13], which are defined by the higher Schwarzian derivatives introduced in [14]. The higher Bers maps of a univalent function f is the generalization of the classical Schwarzian derivative and pre-Schwarzian derivative , which is defined inductively as and
Finally, the higher Bers maps of are discussed. Let () be the Banach space of all holomorphic functions f on with the norm
Denote by the subspace of with
Then, the following theorem is proved.
Theorem 5. If , . Then, the higher Bers maps are holomorphic. Moreover, the differential at the origin is given by the following correspondence:
The paper is organized as follows. In Section 2, we show the relevant notions and lemmas. We study the Schwarzian derivative model and the prelogarithmic derivative model of by Theorems 1 and 2. Moreover, we consider the connected components by Theorem 3. In Section 3, we give the characterizations of in Theorem 4. In Section 4, we consider the high Bers maps of are holomorphic in Theorem 5.
2. F0-Teichmüller Space
2.1. Notions and Lemmas
Let or . Given an arc I on , the Carleson box on is defined as
A positive measure λ on is called an s-Carleson measure () ifand a compact s-Carleson measure in addition if
Obviously, 1-Carleson measure is the classical Carleson measure (see [15]). Denoted by (or ) is the set of all (compact) s-Carleson measures on .
Lemma 1 (see [11]). Let and . For a positive measure λ on , set
If , then , and there exists a constant such that , while if .
Lemma 2 (see [16]). Let ; a positive measure λ on is an s-Carleson measure if and only ifand it is a compact s-Carleson measure if and only if
Theorem 6 (see [12]). Let f be conformal on , , , , and . If , then if and only if .
Lemma 3. Let φ be analytic in , . Then, the following statements hold:(1)If , then (2)If , then
Proof. If , then . From Lemma 4.1 in [7], we obtain the required results immediately.
Lemma 4 (see [17]). Suppose that , , and . If , then there exists a universal constant , such that for all ,
Lemma 5 (see [5]). Let f be a conformal mapping on and admit a quasiconformal extension to the whole plane, if ; then,
Lemma 6 (see [2]). is holomorphic if and only if f is local bounded and every ; the map is holomorphic from an open neighborhood of zero in the complex plane to F, where U is an open subset of E.
2.2. Proof of Theorem 1
We prove Theorem 1 by the following three steps:(1). Set . Then, is a quasiconformal self-homeomorphism of satisfying in and in . If , we have Let , where and are defined as (40). The area theorem of univalent functions yields where is the Jacobian determinant of . Note that and . By (40) and (42), we obtain Since , we have From (41), (43), and Lemma 4, we obtain where only depends on . Thus, by Lemma 5.(2) is continuous. Indeed, we prove that there exists a constant C, such that Set , , and , . Then, is a quasiconformal mapping of whose complex dilation is in and is zero in , while is a quasiconformal mapping of whose complex dilation is in and is zero in . Thus, the correspondence between µ and and ν and are one-to-one and and . We have Let and . By Lemma 6, we obtain Set and . Let H be the Beurling–Ahlfors operator defined as where the integral is understood in the sense of Cauchy principal value. The representation theorem of quasiconformal mapping (see [18]) says that Consequently, we conclude that Since , we have Thus, it follows from (48): Since , we have , where By using the method similar to (43) and (45), we have Now, we estimate . When , the operator is invertible on and the norm of is less than . Thus, we have Similar to , we have where depends on and . From (47), (55), and (57), we obtain (46).(3)S is holomorphic. For any , define a continuous linear function for . For each pair and small t in the complex plane, by the holomorphic dependence of quasiconformal mappings on parameters (see [2, 18]), we conclude that is holomorphic of t. From Lemma 6, S is holomorphic.
2.3. Proof of Theorem 2
Theorem 1 and Theorem 6 imply and the holomorphy of is the same as Theorem 1. Thus, we only show that is continuous. From Theorem 3.1 in Chapter 2 [2], we havewhere C depends on and . By Theorem 1, we conclude thatwhere depends on and .
It follows from Chapter 4 in [19] that there is a constant which is independent of µ and ν such that
From (58), . By the definition of Schwarzian derivative, we obtain
It follows from (58), (59), and (61) that
Combining (60) and (62), we obtain .
2.4. Proof of Theorem 3
If , then f is a quasiconformal mapping of whose complex dilatation satisfies . Let be the quasiconformal map in with and .
Consider the path , , in . Set and . By Theorem 2, we conclude that
This implies that the path is continuous in .
Consequently, each can be connected by a continuous path to an element , where φ is a Möbius transformation of . If is unbounded, then for some . Otherwise is bounded, we consider the path , where , . It is easy to see that this is a path which connects the point to the point 0 in . It proves that = { is bounded } and = , are connected. By [4], elements in different classes cannot be connected in . We conclude the and are the connected components of .
3. Some Characterizations of -Teichmüller Space
In this section, we give some characterizations of -Teichnüller space.
3.1. Two Operators
For a quasisymmetric homeomorphism h, Hu [6] introduces two kernel functions as follows:
Clearly, and are holomorphic. They define two operators on induced by and , respectively:where is the complex Hilbert space that consists of all analytic functions ϕ on with the inner product and norm
We set
3.2. Grunsky Kernel Function
For an univalent function f on , define the Grunsky kernel (see [20]) as
Lemma 7 (see [7]). Let f be a univalent function on , and be the corresponding quasisymmetric conformal welding. Denote ν by the Beltrami coefficient of a quasiconformal extension of to . Then, we have
Lemma 8 (see [7]). Let f be a univalent function on , and be the corresponding quasisymmetric conformal welding. Then, we have
3.3. Proof of Theorem 4
(1) and (2) are equivalent in Theorem 6.
Now we prove, . If then we know by Theorem 6. Becker–Pommerenke [21] point out that f can be extended to a quasiconformal mapping to the whole plane , and its complex dilatation satisfies
Hence, we have .
Because Lemmas 7 and 8 imply that and the equivalence of (4) and (5), we only need to consider .
Let h be a quasisymmetric homeomorphism on . Then, there exists a unique pair of conformal mappings on and on , such that , , and on . Suppose that f can be extended to a quasiconformal mapping to the whole plane , whose complex dilatation satisfies . Denote is a quasiconformal extension of to . We can notice that has the same complex dilation µ as . Note that , where . By a simple computation, we find is a quasiconformal extension of to with its complex dilatation satisfying and . By Lemma 7, we conclude that
Choose in Lemma 1, we have .
4. Higher Bers Maps of
In this section, we consider the higher Bers maps of .
Lemma 9. Let and . If , then .
Proof. From Theorem 1, we obtain that .
If , , we have from Lemma 2. Hence, we have by lemma in [22].
From , we haveSince is a univalent analytic function in , we have . It is easy to obtain from Lemma 2.
By the mathematical induction, this lemma is proved completely.
4.1. Proof of Theorem 5
(1) is continuous. For any , let f (or ) be the quasiconformal mapping whose complex dilatation is µ (or ν) in and is zero in , normalized by and . By the definition of the higher Schwarzian derivative, Since and , we have By one theorem in Chapter 2 in [2], there is a constant C such that From Lemma 9 and , we have Repeating this process times, it has By Theorem 1, we obtain .(2) is holomorphic. It is sufficient to prove that for any , is holomorphic in a small neighborhood of .
If , there exists a positive constant such that for any t with , , and .
Here, abbreviate by . For fixed , is holomorphic in . For and , it follows from the Cauchy integral formula that
Consequently, by Fubini theorem, we have
Thus, it deduces that exists in . This implies that is holomorphic.
A theorem is shown in [13] that .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Nature Science Foundation of China (Grant no. 11871085).