Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 42, 2017, 105–118
UNIVERSAL TEICHMÜLLER SPACES AND F (p,q,s) SPACE
Xiaogao Feng, Shengjin Huo and Shuan Tang∗
China West Normal University, College of Mathematics and Information Nanchong, 637002, P. R. China; and Soochow University, Department of Mathematics Suzhou, 215006, P.R. China; [email protected] Tianjin Polytechnic University, Department of Mathematics Tianjin, 300387, P. R. China; [email protected] Guizhou Normal University, School of Mathematics Sciences Guiyang 550001, P.R. China; tsafl[email protected]
Abstract. In this paper, we introduce the F (p,s)-Teichmüller space and investigate its Schwarzian derivative model and pre-logarithmic derivative model. In particular, we prove that the pre-logarithmic derivative model is a disconnected subset of Besov type space F (p,s) and the Bers projection is holomorphic.
1. Introduction
Let ∆ = {z : |z| < 1} be the unit disk in the complex plane C, ∆∗ = C\∆ be the outside of the unit disk and S1 = {z ∈ C: |z| =1} be the unit circle. Let α> 0, the Bloch-type space Bα consists of all holomorphic functions f on ∆ such that 2 α ′ kfkB = sup(1 −|z| ) |f (z)| < ∞, z∈∆ α α and the subspace B0 consists of all functions f ∈ B such that lim (1 −|z|2)α|f ′(z)| =0. |z|→1 We denote by BMO(S1) the space of all integrable functions on S1 such that 1 (1) kukBMO = sup |u − uI | dθ < ∞, I |I| ˆI where I is any arc on S1, |I| denotes the Lebesgue measure of I, and 1 (2) uI = u dθ |I| ˆI is the average of u over I. A holomorphic function f on ∆ belongs to BMOA(∆) if and only if it is a Poisson integral of some function which belongs to BMO(S1). z−a For any a ∈ ∆, set ϕa(z) = 1−az , z ∈ ∆. For p > 1, q > −2 and s ≥ 0, the space F (p,q,s) consists of all holomorphic functions f on the unit disk ∆ with the
https://doi.org/10.5186/aasfm.2017.4208 2010 Mathematics Subject Classification: Primary 30C62; Secondary 30F60, 32G15. Key words: Quasiconformal mapping, Universal Teichmüller space, F (p,s) space, Bers projection. ∗The first and the corresponding author. 106 Xiaogao Feng, Shengjin Huo and Shuan Tang following finite norm
p ′ p 2 q 2 s (3) kfkFp,q,s = sup |f (z)| (1 −|z| ) (1 −|ϕa(z)| ) dxdy < ∞, a∈∆ ¨∆ and F0(p,q,s) consists of all functions f ∈ F (p,q,s) with
′ p 2 q 2 s (4) lim |f (z)| (1 −|z| ) (1 −|ϕa(z)| ) dxdy =0. |a|→1 ¨∆ The space F (p,q,s) was introduced by Zhao [23]. It is well known that F (p,q,s) is trivial if q + s ≤ −1. For p ≥ 1, F (p,q,s) is a Banach space contained in the Bloch- α α q+2 type space B and F0(p,q,s) ⊂ B0 with α = p . It is also known that F (2, 0, 1) is the BMOA space (see [8]) and F (2, 0,s) is the Qs space (see [19, 20]). In this paper, we shall mainly concentrate on the Besov type space F (p,s) = F (p,p − 2,s) and F0(p,s)= F0(p,p − 2,s). Let Ω = ∆ or Ω = ∆∗. Given an arc I of the unit circle S1, the Carleson box is defined by {z ∈ ∆: 1 −|I|≤|z| < 1,z/|z| ∈ I}, Ω = ∆, SΩ(I)= ∗ ∗ ({z ∈ ∆ : 1 ≤|z| < 1+ |I|,z/|z| ∈ I}, Ω = ∆ . Let s> 0. A positive measure λ on Ω is called an s-Carleson measure if
λ(SΩ(I)) kλkC,s = sup s < ∞, I⊂S1 |I| and a compact s-Carleson measure if λ(S (I)) lim Ω =0. |I|→0 |I|s
We denote by CMs(Ω) the set of all s-Carleson measures on Ω and CMs,0(Ω) the set of all compact s-Carleson measures on Ω. 1-Carleson measure is the classical Carleson measure. By [19], we know that a positive measure λ on ∆ is an s-Carleson measure if and only if 1 −|a|2 s (5) sup dλ(z) < ∞, ¨ | 1 − az |2 a∈∆ ∆ and is a compact s-Carleson measure if and only if 1 −|a|2 s (6) lim dλ(z)=0. |a|→1 ¨ | 1 − az |2 ∆ Let f be a quasiconformal mapping of the complex plane C onto itself. Then f is a homeomorphism with locally integral distributional derivatives, and satisfies the Beltrami equation fz = µfz with kµk∞ = supz∈C |µ(z)| < 1. Here we use the notations 1 ∂ ∂ 1 ∂ ∂ f = + i f, f = − i f. z 2 ∂x ∂y z 2 ∂x ∂y This function µ is called the complex dilatation of f. The measurable Riemann mapping theorem (see [1]) says that for each measurable function µ on the complex plane C with kµk∞ < 1, there is a quasiconformal mapping f on C with complex dilatation µ and f is unique up to a Möbius transformation of C onto itself. A homeomorphism h is said to be quasisymmetric if there is some M(h) > 0 such that |h(I∗)| ≤ M(h)|h(I)| for any interval I ⊂ S1 with |I| ≤ π, where I∗ is the interval with the same center as I but with double length. Denote by Universal Teichmüller spaces and F (p,q,s) space 107
QS(S1) the group of quasisymmetric homeomorphisms of the unit circle S1. It is well known that a sense preserving self-homeomorphism h is quasisymmetric if and only if it can be extended to a quasiconformal self-homeomorphism of the unit disk ∆ (see [3]). Douady and Earle [7] also gave a quasiconformal extension of h to the unit disk which is conformally invariant. Let Möb(S1) be the group of all Möbius transformations of ∆ onto itself. The universal Teichmüller space is the right coset space T = QS(S1)/Möb(S1). Let M(∆∗) denote the unit ball of the Banach space L∞(∆∗) of all bounded mea- surable functions on ∆∗. For any µ ∈ M(∆∗), there exists a unique quasiconformal ∗ mapping fµ of C whose complex dilatation is equal to µ in ∆ and is zero in ∆. We normalize fµ by ′ ′′ fµ(0) = fµ(0) − 1= fµ (0) = 0. ∗ We say that two Beltrami coefficients µ1 and µ2 in M(∆ ) are Teichmüller equivalent, which is denoted by µ1 ∼ µ2, if fµ1 (∆) = fµ2 (∆). The universal Teichmüller space T can be described as T = M(∆∗)/ ∼. We denote by [µ] the equivalent class containing µ ∈ M(∆∗). Let SQ be the class of all univalent analytic functions f in the unit disk ∆ with the normalized condition f(0) = f ′(0) − 1=0 that can be extended to a quasiconformal ′ mapping in the whole plane. Set T (1) = {log f : f belongs to SQ}. It is known that T (1) is an alternative model called pre-logarithmic derivative model of the universal Teichmüller space. T (1) is a disconnected subset of the Bloch space B1. Furthermore, ′ ′ iθ Tb = {log f ∈ T (1): f(∆) is bounded} and Tθ = {log f ∈ T (1): f(e ) = ∞}, θ ∈ [0, 2π), are the connected components of T (1) (see [24]). A quasisymmetric homeomorphism h is called strongly quasisymmetric homeo- morphism if for each ε > 0, there is a constant δ > 0 such that |E| ≤ δ|I| implies that |h(E)| ≤ ε|h(I)|, where I ⊂ S1 is an interval and E ⊂ I is a measurable subset. In other words, h is absolutely continuous and log h′ ∈ BMO(S1). It is equivalent to say that there exists a quasiconformal extension of h to ∆ such that its complex dilatation µ(z) satisfies |µ(z)|2 dxdy ∈ CM (∆) 1 −|z|2 1 (see [2]). The Teichmüller space called BMO-Teichmüller space with respect to the strongly quasisymmetric homeomorphisms has been much studied in recently years (see [6], [16]). In particular, Astala and Zinsmeister [2] proved that the pre- logarithmic derivative model BMOA ∩ T (1) of the BMO-Teichmüller space is dis- connected open subset of BMOA. Recently, Wulan and Ye [18] introduced the QK -Teichmüller space and showed that its pre-logarithmic derivative model is also disconnected subset of the QK space. We denote by N(p,s) the space of all holomorphic functions f on ∆ with the following finite norm (1 −|a|2)s (7) kfkp = sup |f(z)|p(1 −|z|2)s+2p−2 dx dy. Np,s 2s a∈∆ ¨∆ |1 − az|
We say an analytic function f belongs to N0(p,s) if f ∈ N(p,s) and
2 s p 2 s+2p−2 (1 −|a| ) (8) lim |f(z)| (1 −|z| ) 2s dxdy =0. |a|→1 ¨∆ |1 − az| 108 Xiaogao Feng, Shengjin Huo and Shuan Tang
Zorboska [25] obtained a characterization of the relationship between the pre- ′ logarithmic derivative log f in space F (p,s) and the Schwarzian derivative Sf in space N(p,s). Theorem A. [25] Let f be conformal on ∆, 0 ≤ s < ∞ and 1 ≤ p< ∞. Then ′ ′ log f ∈ F (p,s) if and only if Sf ∈ N(p,s), while log f ∈ F0(p,s) if and only if Sf ∈ N0(p,s). It should be pointed out that the F (2, 1) case was proved by Astala and Zins- meister in [2] and the F (2,s) case was proved by Pau and Peláez in [13]. In this paper, we introduce the F (p,s)-Teichmüller space and investigate its Schwarzian derivative model and pre-logarithmic derivative model. In what follows, we always assume that p ≥ 2 and s > 0. Denote by Mp,s(Ω) the Banach space of all essentially bounded measurable functions µ each of which induces an s-Carleson |µ(z)|p measure λµ(z) := |1−|z|2|2−s dxdy ∈ CMs(Ω). The norm of µ ∈ Mp,s(Ω) is defined as
1/p (9) k µ ks=k µ k∞ +kλµkC,s, where kλµkC,s is the s-Carleson norm of λµ on Ω. Mp,s,0(Ω) is the subspace of 1 Mp,s(Ω) which consists of all elements µ such that λµ(z) ∈ CMs,0(Ω). Set Mp,s(Ω) = 1 Mp,s(Ω) ∩ M(Ω) and Mp,s,0(Ω) = Mp,s,0(Ω) ∩ M(Ω), where M(Ω) denotes the unit ball of the Banach space L∞(Ω) of all bounded measurable functions on Ω. We 1 ∗ define the F (p,s)-Teichmüller space TF (p,s) as TF (p,s) = Mp,s(∆ )/ ∼ and the F0(p,s)- 1 ∗ Teichmüller space TF0(p,s) as TF0(p,s) = Mp,s,0(∆ )/ ∼. It is noted that F (2, 1)-Teichmüller space is the BMO-Teichmüller space and the limit case F (2, 0)-Teichmüller space is the Weil–Petersson Teichmüller space (see [5]) which has been much investigated in recently years and has wide applications to various areas such as mathematical physics, differential equation and computer vision. The limit case F (p, 0)-Teichmüller space is the p-integrable Teichmüller space (see [9, 17, 21]).
The pre-logarithmic derivative model TF (p,s) of F (p,s)-Teichmüller space is the set of functions log f ′, where f is conformal on ∆ and admits a quasiconformal extension to the whole plane C such that itse complex dilatation µ satisfies | µ(z) |p (10) dxdy ∈ CM (∆∗). (| z |2 −1)2−s s In this paper, we shall prove
Theorem 1.1. Let p ≥ 2 and 0 The Schwarzian derivative Sf of a conformal mapping f on ∆ is defined as f ′′ ′ 1 f ′′ 2 S = − . f f ′ 2 f ′ Universal Teichmüller spaces and F (p,q,s) space 109 ∗ The Bers projection Φ: M(∆ ) → B∞(∆) is defined by µ 7→ Sfµ . The holomorphy of the Bers projection is important in the theory of Teichmüller space. It is known ∗ that Φ: M(∆ ) → B∞(∆) is holomorphic and descends down to a mapping B : T → B∞(∆) known as the Bers embedding. Via the Bers embedding, T carries a natural complex Banach manifold structure so that B is a holomorphic split submersion. For 1 ∗ the Bers projection Φ on Mp,s(∆ ), we also obtain the following result. 1 ∗ Theorem 1.2. Let p ≥ 2 and 0 ∗ 1 ∗ z0 Fix z0 ∈ ∆ . For µ ∈ Mp,s(∆ ), let gµ (abbreviated to be gµ) be the quasi- conformal mapping on the complex plane C whose complex dilatation equals to µ ∗ ′ in ∆ and zero in ∆, normalized by gµ(0) = gµ(0) − 1=0, gµ(z0) = ∞. Consider 1 ∗ ′ the pre-Bers projection mapping Lz0 on Mp,s(∆ ) by setting Lz0 (µ) = log gµ. Then 1 ∗ 0 0 ∗ z0∈∆ Lz0 (Mp,s(∆ )) = TF (p,s) ∩ F (p,s) , where F (p,s) consists of all functions ϕ ∈ F (p,s) with ϕ(0) = 0. S e ∗ Theorem 1.3. Let p ≥ 2 and 0 (3) f can be extended to a quasiconformal mapping to the whole plane such that |µ(z)|p ∗ its complex dilatation µ satisfies (|z|2−1)2−s dxdy ∈ CMs,0(∆ ); p 2 p+s−2 (4) |U(f, z)| (1 −|z| ) dxdy ∈ CMs,0(∆); 2 2 p+s−2 (5) |φh(z)| (1 −|z| ) dxdy ∈ CMs,0(∆). In what follows, C(·) will denote constant that depends only on the elements put in the bracket. 2. Bers projection and pre-Bers projection In order to prove Theorem 1.2 and Theorem 1.3, we need some lemmas as follows. Lemma 2.1. [22] Suppose that k> −1, r,t > 0, and r+t−k > 2. If t ′ e kλkC,s ≤ C kλkC,s, while λ ∈ CMs,0(∆) if λ ∈ CMs,0(∆)e . Proof. Set k = α, r = α +β +2, t =2s and note that s< 1+α/2, it follows from Lemmae 2.1 that there exists a universal constant C > 0 such that for any a, w ∈ ∆, (1 −|z|2)α (1 −|w|2)−β (15) α+β+2 2s dxdy ≤ C 2s . ¨∆ |1 − zw| |1 − az| |1 − aw| By Lemma 4.1.1 in Xiao [19], there exist some constants C1 > 0 and C2 > 0 such that (1−| a |2)s (16) C1kλkC,s ≤ sup λ(z) 2s dxdy ≤ C2kλkC,s. a∈∆ ¨∆ | 1 − az | Consequently, we conclude from (15) that (1−| a |2)s λ(z) 2s dxdy ¨∆ | 1 − az | 2 α 2 β 2 s e (1 −|z| ) (1 −|w| ) (1−| a | ) = α+β+2 λ(w) dudv 2s dxdy ¨∆ ¨∆ |1 − zw| | 1 − az | (17) (1−| a |2)s (1 −|z|2)α = 2 −β λ(w) dudv α+β+2 2s dxdy ¨∆ (1 −|w| ) ¨∆ |1 − zw| |1 − az| (1−| a |2)s ≤ C λ(w) 2s du dv. ¨∆ | 1 − aw | Universal Teichmüller spaces and F (p,q,s) space 111 If λ ∈ CMs(∆), then from (16) and (17), we deduce that λ ∈ CMs(∆) and there is a constant C′ = CC2 such that kλk ≤ C′kλk . If λ ∈ CM (∆), then C1 C,s C,s s,0 e 1 −|a|2 s (18) lim λ(ew) dudv =0. |a|→1 ¨ | 1 − aw |2 ∆ We deduce from (17) that 1 −|a|2 s (19) lim λ(z) dxdy =0. |a|→1 ¨ | 1 − az |2 ∆ Therefore λ ∈ CMs,0(∆). e Let f be a conformal mapping on ∆∗. For any z ∈ ∆∗, set e 1+ wz (|z|2 − 1) (20) β (w)= and γ (w)= f ′(z). z w + z f w − f(z) ∗ Then βz is an automorphism of ∆ sending ∞ to z. We need a representation theorem of the Schwarzian derivative, which is proved by Astala and Zinsmeister [2]. Lemma 2.3. [2] Let f be a conformal mapping on ∆∗ and admits a quasicon- formal extension to the whole plane, then for any z ∈ ∆∗ and w = u + iv, 3(|z|2 − 1)−2 (21) Sf (z)= − ∂g(w) du dv, 2π ¨∆ where g = γf ◦ f ◦ βz. We first show that the Bers projection is well defined. 1 ∗ Proposition 2.4. Let p ≥ 2 and 0 b b Let g = γf ◦ f ◦ βz, where γf and βz are defined in (20). The area theorem of univalent functions yields b (23) Jg(ζ) dξ dη ≤ π, ¨∆ b where Jg is the Jacobian determinant of g. Noting that µg = µf◦βz and kµgk∞ = ||µ||∞, by (21), (23) and Hölder’s inequality, we get p p p 2 2p 3 |S b(z)| (|z| − 1) = (µ b ∂g)(ζ) dξ dη f 2π ¨ f◦βz ∆ p 3 π p ≤ p | µ b (ζ) | dξ dη (24) f ◦βz 2π (1 −kµ k2 ) 2 ¨ g ∞ ∆ p 2 2 b |µf (w)| (|z| − 1) = C1(||µ||∞) 4 du dv, ¨∆ | w − z | 112 Xiaogao Feng, Shengjin Huo and Shuan Tang 3 p π1−p p where C1(||µ||∞)= 2 . 2 (1−kµk∞) 2 ∗ Consequently, for b ∈ ∆ , set a = 1/b ∈ ∆, by (22), (24) and Lemma 2.2, we have 2 s p 2 2p+s−2 (|b| − 1) |S b(z)| (|z| − 1) dxdy f 2s ¨∆∗ |1 − bz| p 2 s b 2 2−s 2 s |µf (w)| (1 −|a| ) (1 −|w| ) (1 −|z| ) ≤ C1(||µ||∞) 2 2−s dudv 4 2s dxdy (25) ¨∆ (1 −|w| ) ¨∆ |1 − wz| |1 − az| p b 2 s |µf (w)| (1 −|a| ) ≤ C2(||µ||∞) 2 2−s 2s dudv ¨∆ (1 −|w| ) |1 − aw| ≤ C (||µ|| )kλ k = C (||µ|| )kλ k . 3 ∞ µfb C,s 3 ∞ µ C,s 1 1 b Noting that |Sf (ζ)| = |Sfµ ( ζ )| |ζ|4 , we get from (25) that 2 s p p 2 2p+s−2 (|a| − 1) (26) kSfµ k = sup |S b(z)| (|z| − 1) dxdy < ∞. Np,s ∗ f 2s a∈∆ ¨∆∗ |1 − az| 1 ∗ Which implies that Φ(µ)= Sfµ ∈ N(p,s) if µ ∈ Mp,s(∆ ). The proof follows. Before proving Theorem 1.2, we first recall some basic facts about the infinite dimensional holomorphy (see [11, p. 206], [12, p. 86–87]). Let E and F be two complex Banach space and U be an open subset in E, a mapping f : U → F is holomorphic if and only if it is continous (locally boundedness is also enough) and the complex Gateaux derivative dx(λ) defined as f(x + tλ) − f(x) dx(λ) = lim t→0 t exists for each (x, λ) ∈ U × E. Let F ∗ denote the dual space of F in the usual sense. For a subset A of F ∗, we define A⊥ = {y ∈ F : y∗(y)=0,y∗ ∈ A}. A subset A is called total if A⊥ = {0}. Proposition 2.5. [11, 12] f : U → F is holomorphic if and only if it satisfies one of the following conditions. (i) The mapping f is local bounded and for every (x, λ) ∈ U × E, the mapping t 7→ f(x + tλ) is holomorphic from an open neighborhood of zero in the complex plane C to F . (ii) The mapping f is continuous and there exists a total subset A of F ∗ such that for every y∗ ∈ A, the function y∗(f): U → C is holomorphic. We are now in a position to prove Theorem 1.2. Our proof is based on the proof of Theorem 3 in Cui [5]. 1 ∗ Proof of Theorem 1.2.. We first show that mapping Φ: Mp,s(∆ ) → N(p,s) is 1 ∗ 1 ∗ continuous. Let µ ∈ Mp,s(∆ ), ν ∈ Mp,s(∆ ). It is sufficient to show that there is a constant C(kµk∞, kνk∞) such that b b (27) kΦ(µ) − Φ(ν)kNp,s ≤ C(kµk∞, kνk∞)kµ − νks. b b Set f1 = fµb, f2 = fνb and fi(ζ) = s ◦ fi ◦ s(ζ), i = 1, 2, where s(ζ)=1/ζ. Then b b b b b b f1 is a quasiconformal mapping of C whose complex dilatation is equal to µ(ζ) = ′ b s (ζ) b ∗ b µ(s(ζ)) s′(ζ) in ∆ and is zero in ∆ , while f2 is a quasiconformal mapping of C whose ′ s (ζ) ∗ complex dilatation is equal to ν(ζ) = ν(s(ζ)) s′(ζ) in ∆ and is zero in ∆ . Thus the b b Universal Teichmüller spaces and F (p,q,s) space 113 correspondence between µ and µ is one-to-one and kµk∞ = kµk∞, kνk∞ = kνk∞. By a change of variable, we conclude that (28) kλ k =bkλ k , kλ k =bkλ k b µfb C,s µ C,s νfb C,s ν C,s and 2 s p p 2 2p+s−2 (|b| − 1) (29) kΦ(µ)−Φ(ν)k = sup |Sf1 (z)−Sf2 (z)| (|z| −1) dx dy. Np,s ∗ 2s b∈∆ ¨∆∗ |1 − bz| µ ν Let gb = γf1b◦ f1 ◦ βz and g = γf2 ◦ f2 ◦ βz. By Lemma 2.3, we have 2 −2 3(|z| − 1) µ ν (30) Sf1 (z) − Sf2 (z)= − ∂(g − g )(w) du dv. 2π ¨∆ ′ ′ βz(w) βz(w) Set µβz (w)= µ(βz(w)) ′ , νβz (w)= ν(βz(w)) ′ . Let H be the Beuring–Ahlfors βz(w) βz(w) operator defined as 1 φ(z) H(φ)(ζ)= − 2 dx dy, π ¨C (ζ − z) the integral is understood in the sense of Cauchy principal value. The representation µ µ ν theorem of quasiconformal mapping says that ∂g = µβz (I + H∂g ), ∂g = νβz (I + H∂gν) (see [1, Chapter V]). Consequently, we conclude that µ ν µ ν ∂(g − g )= µβz (I + H∂g ) − νβz (I + H∂g ) µ ν ν ν = µβz − νβz + µβz H∂g − µβz H∂g + µβz H∂g − νβz H∂g ν µ ν =(µβz − νβz )(H∂g + I)+ µβz H∂(g − g ). Since ∂gν = I + H∂gν, we have µ ν −1 ν −1 ν ∂(g − g )=(I − µβz H) (µβz − νβz )(I + H∂g )=(I − µβz H) ((µβz − νβz )∂g ). Thus it follows from (30) that 2 −2 3(|z| − 1) −1 ν (31) Sf2 (z) − Sf1 (z)= − (I − µβz H) ((µβz − νβz )∂g )(w) du dv. 2π ¨∆ −1 −1 Since (I − µβz H) = I + µβz H(I − µβz H) , we have Sf2 (z) − Sf1 (z) 2 −2 3(|z| − 1) −1 ν = − (I − µβz H) ((µβz − νβz )∂g )(ζ) dξ dη 2π ¨∆ 2 −2 3(|z| − 1) ν (32) = − ((µβz − νβz )∂g )(ζ) dξ dη 2π ¨∆ 2 −2 3(|z| − 1) −1 ν − µβz H(I − µβz H) ((µβz − νβz )∂g )(ζ) dξ dη 2π ¨∆ = L1 + L2, where 2 −2 3(|z| − 1) ν L1 = − ((µβz − νβz )∂g )(ζ) dξ dη 2π ¨∆ and 2 −2 3(|z| − 1) −1 ν L2 = − µβz H(I − µβz H) ((µβz − νβz )∂g )(ζ) dξ dη. 2π ¨∆ 114 Xiaogao Feng, Shengjin Huo and Shuan Tang By using the method similar to (24), we get |µ(w) − ν(w)|p(|z|2 − 1)2 (33) p 2 2p |L1| (|z| − 1) ≤ C(kµk∞) 4 du dv. ¨∆ | w − z | Consequently, similar to (25), by Lemma 2.2 and a change of variable, we obtain 2 s p 2 2p+s−2 (|b| − 1) sup |L1(z)| (|z| − 1) dxdy b∈∆∗ ¨ ∗ |1 − bz|2s (34) ∆ |µ(w) − ν(w)|p (1 −|a|2)s ≤ C(kµk∞) sup 2 2−s 2s du dv. a∈∆ ¨∆ (1 −|w| ) |1 − aw| We now estimate L2. It is noted that when kµk∞ < 1, the operator I − µH is 2 −1 invertible on L (∆) and the norm of its inverse (I − µH) is less than 1/(1 −kµk∞). Thus we have 3 2 2 | L |2(|z|2 − 1)4 = µ H(I − µ H)−1((µ − ν )∂gν)(ζ) dξ dη 2 2π ¨ βz βz βz βz ∆ 9 ≤ |µ |2 dξ dη |((µ − ν )∂gν)(ζ)| 2 dξ dη 2 2 βz βz βz 4π (1 −kµk∞) ¨∆ ¨∆ (35) 2 9kµ − νk∞ 2 ≤ 2 2 |µβz | dξ dη 4π(1 −kµk∞) (1 −kνk∞) ¨∆ 2 2 2 2 9kµ − νk∞ |µ(ζ)| (|z| − 1) = 2 2 4 dξ dη. 4π(1 −kµk∞) (1 −kνk∞) ¨∆ |ζ − z| Noting that p ≥ 2, by Hölder’s inequality, we get p p 2 2 −2 2 p 2 −1 p 9kµ − νk∞(|z| − 1) |µ(ζ)| 1 |L2| ≤ 2 2 4 dξ dη 4 dξ dη 4π(1 −kµk∞) (1 −kνk∞) ¨∆ |ζ − z| ¨∆ |ζ − z| p p |µ(ζ)| (36) ≤ C1(kµk∞, kνk∞)kµ − νk∞ 4 2 2p−2 dξ dη, ¨∆ |ζ − z| (|z| − 1) Similar to L1, we can deduce that 2 s p 2 2p+s−2 (|b| − 1) (37) sup |L2(z)| (|z| − 1) dxdy ∗ 2s b∈∆ ¨∆∗ |1 − bz| p 2 s p |µ(w)| (1 −|a| ) ≤ C2(kµk∞, kνk∞)kµ − νk∞ sup 2 2−s 2s du dv. a∈∆ ¨∆ (1 −|w| ) |1 − aw| Combining (28), (29), (32), (34) and (37), we deduce that (27) holds and thus the 1 ∗ mapping Φ: Mp,s(∆ ) → N(p,s) is continuous. 1 ∗ We now prove that the Bers projection Φ: Mp,s(∆ ) → N(p,s) is holomorphic. For each z ∈ ∆, we define a continuous linear functional lz on the Banach space N(p,s) by lz(ϕ) = ϕ(z) for ϕ ∈ N(p,s). Then the set A = {lz : z ∈ ∆} is a to- tal subset of the dual space of N(p,s). Now for each z ∈ ∆, each pair (µ, ν) ∈ 1 ∗ ∗ Mp,s(∆ ) × Mp,s(∆ ) and small t in the complex plane, by the well known holomor- phic dependence of quasiconformal mappings on parameters (see [11, Theorem 3.1 in Chapter II], [1, Chapter V]), we conclude that lz(Φ(µ+tν)) = Sfµ+tν (z) is a holomor- 1 ∗ phic function of t. From Proposition 2.5, the Bers projection Φ: Mp,s(∆ ) → N(p,s) is holomorphic. This completes the proof. Checking the proof of Theorem 1.2, we can show the following Universal Teichmüller spaces and F (p,q,s) space 115 1 ∗ Theorem 2.6. Let p ≥ 2 and 0 By Theorem 1.2, we conclude that (39) kSgµ (z) − Sgν (z)kNp,s ≤ C1(kµk∞, kνk∞)kµ − νks. It follows from Chapter 4 in [14] that there is a constant C2 > 0 which is independent of µ and ν such that g′′ g′′ p (1 −|z|2)p−2(1 −|a|2)s µ − ν dxdy ¨ g′ g′ |1 − az|2s ∆ µ ν (40) p g′′ g′′ p g′′ ′ g′′ ′ (1−|z|2)2p−2(1−|a|2)s µ ν µ ν ≤ C2 ′ (0)− ′ (0) +C2 ′ − ′ 2s dx dy. gµ gν ¨∆ gµ gν |1−az| By the definition of the Schwarzian derivative, we get p p ′′ ′ ′′ ′ ′′ 2 ′′ 2 gµ gν p p p gµ gν ′ − ′ ≤ 2 |Sgµ − Sgν | +2 ′ − ′ gµ gν gµ gν (41) g′′ g′′ p g′′ g ′′ p =2p|S − S |p +2p µ + ν µ − ν . gµ gν g′ g′ g′ g ′ µ ν µ ν Taking z =0 in (38), we get g′′ g′′ p (42) µ (0) − ν (0) ≤ Cp(kµk )kµ − νkp . g′ g′ ∞ ∞ µ ν It follows from (38), (39) and (41) that p ′′ ′ ′′ ′ 2 2p−2 2 s gµ gν (1 −|z| ) (1 −|a| ) ′ − ′ 2s dxdy ¨∆ gµ gν |1 − az| 2 2p−2 2 s p p (1 −|z| ) (1 −|a| ) ≤ 2 Sgµ − Sgν 2s dxdy ¨∆ |1 − az| g′′ g′′ p g′′ g′′ p (1 −|z|2)2p−2(1 −|a|2)s +2p µ + ν µ − ν dxdy (43) ¨ g′ g′ g′ g′ |1 − az|2s ∆ µ ν µ ν p p p ≤ 2 C1 (kµk ∞, kνk∞ )k µ − νks g′′ g′′ p g′′ g′′ p (1 −|z|2)p−2(1 −|a|2)s +2p sup(1 −|z|2)p µ − ν µ + ν dxdy g′ g′ ¨ g′ g′ |1 − az|2s z∈∆ µ ν ∆ µ ν p p p ≤ 2 C1 (kµk∞, kνk∞) kµ − νks p p ′ ′ p p +4 C (kµk∞)(k log gµkF (p,p−2,s) + k log gνkF (p,p−2,s)) kµ − νk∞. 116 Xiaogao Feng, Shengjin Huo and Shuan Tang Combining (40), (42) and (43), we get kLz0 (µ) − Lz0 (ν)kF (p,p−2,s) ′ ′ ≤ C3 kµk∞, kνk∞, k log gµkF (p,p−2,s), k log gνkF (p,p−2,s) kµ − νks. This completes the proof of Theorem 1.3. Similarly, we have the following ∗ Theorem 2.7. Let p ≥ 2 and 0 3. Proofs of Theorem 1.1 and Theorem 1.4 In this section, we prove Theorem 1.1 and Theorem 1.4. ′ Proof of Theorem 1.1. Let log f ∈ TF (p,s). Then f is a quasiconformal mapping |µ(z)|p of the complex plane C whose complex dilatation µ satisfies λµ = (|z|2−1)2−s dxdy ∈ ∗ et CMs(∆ ) and equals to zero in ∆. Let f be the quasiconformal mapping in C with f −1(∞)=(f t)−1(∞) and ∂f t = tµ∂f t. Consider the path t 7−→ log(f t)′, 0 ≤ t ≤ 1, in the space F (p,s). Set g = f t and h = f s. By Theorem 1.3, we conclude that ′ ′ ′ ′ k log g − log h kF (p,p−2,s) ≤ C kµks, k log g kF (p,p−2,s), k log h kF (p,p−2,s) |t − s|. This implies that the path t 7−→ log(f t)′, 0 ≤ t ≤ 1, is continuous in the space ′ F (p,s). Consequently, each log f ∈ TF (p,s) can be connected by a continuous path to an element log ϕ′ ∈ F (p,s), where ϕ is a Möbius transformation of C. If ϕ(∆) is unbounded, then f(ζ)= ϕ(ζ)= ∞ fore some ζ ∈ ∂∆. Otherwise ϕ(∆) is bounded, we ′ consider the path r 7−→ log ϕr, where ϕr = ϕ(rz), 0 ≤ r ≤ 1. It is easy to see that this is a path which connects the point log ϕ′ to the point 0 in F (p,s). It turns out that ′ ′ Tb = {log f ∈ TF (p,s) : f(∆) bounded } and Tθ = {log f ∈ TF (p,s) : limz→eiθ f(z) = ∞}, 0 ≤ θ ≤ 2π, are connected. By [24], elements in different classes can not bee connected evene in Bloch space. We concludee that Tb ande Tθ are the connected components of TF (p,s). Before proving Theorem 1.4, we need a lemma whiche wase proved by Shen and Wei in [16]. e Lemma 3.1. Let h be a quasisymmetric homeomorphism on S1 with normalized decomposition h = f −1 ◦ g and ν be the complex dilatation of a quasiconformal extension of h−1 to ∆. Then 2 1 2 2 2 2 2 1 |ν(w)| 1 (44) (1 −|z| ) |Sf (z)| ≤ U (f, z) ≤ φh(z) ≤ 2 4 du dv. 36 π ¨∆ 1 −|ν(w)| |1 − zw| Proof of Theorem 1.4. It follows from Theorem A that (1) ⇐⇒ (2). From Theorem 1.3, we conclude that (1) =⇒ (3). Lemma 3.1 gives (4) ⇐⇒ (5) and (4) =⇒ (2). Thus it remains to show that (3) =⇒ (1) and (3) =⇒ (5). We first prove that (3) =⇒ (5). Let h be a quasisymmetric homeomorphism on 1 the unit circle S . Then there exists a unique pair of conformal mappings f ∈ SQ on ∆ and g on ∆∗, such that f(0) = f ′(0) − 1=0, g(∞) = ∞ and h = f −1 ◦ g on S1 (see [11, Lemma 1.1 in Chapter III]). Suppose f can be extended to a quasiconformal mapping of the whole plane C, which is also denoted by f, such that its complex |µ(z)|p ∗ −1 dilatation µ satisfies (|z|2−1)2−s dxdy ∈ CMs,0(∆ ). It is noted that H = g ◦ f is a quasiconformal extension of h−1 to ∆∗ and has the same complex dilatation µ as f. b Universal Teichmüller spaces and F (p,q,s) space 117 Then H = j ◦ H ◦ j, where j(z)=1/z, is a quasiconformal extension of h−1 to ∆ with complex dilatation ν(z) satisfying |ν(z)| = |µ(1/z)|. A computation shows that |ν(z)|p (1−|z|2)2−s dxdy ∈b CMs,0(∆). By Lemma 2.2 and Lemma 3.1, we conclude that (5) holds. We now show that (1) =⇒ (3). Suppose that (1) holds. Noting that F0(p,s) is a 1 subspace of the little Bloch space B0 , we have lim (1 −|z|2)|f ′′(z)/f ′(z)| =0. |z|→1 Becker and Pommerenke (see [4]) constructed a quasiconformal extension of the con- formal mapping f to the whole plane C by the following formula f(z)= f(1/z)+ f ′(1/z)(z − 1/z), z ∈ ∆∗. By some computations we have |µ(z)| = |1/z|2(1 −|1/z|2)|f ′′(1/z)/f ′(1/z)|. For z, b ∈ ∆∗, we set w =1/z and b =1/a. A change of variable gives |µ(z)|p (|b|2 − 1)s dxdy 2 2−s 2s ¨∆∗ (|z| − 1) |1 − bz| (45) 2 s ′′ ′ p 2 p−2+s (1 −|a| ) ≤ |f (w)/f (w)| (1 −|w| ) 2s du dv. ¨∆ |1 − aw| ′ Noting that log f ∈ F0(p,s) and |b|→ 1 if and only if |a|→ 1, we conclude that |µ(z)|p dxdy ∈ CM (∆∗). (|z|2 − 1)2−s s,0 The proof follows. Acknowledgements. The authors would like to thank the referees for their careful reading and valuable suggestions, which improve the quality of this paper. 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