# The Universal Teichmüller Space and Function Space

Authors: Yutong Liu; Yi Qi
Journal: Journal of Function Spaces
 (2020)

---

## Abstract

In this paper, a subspaceTF02,1−s,s of the universal Teichmüller space, which is related to the analytic function space F02,1−s,s, 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 T˜F02,1−s,s of TF02,1−s,s is a disconnected subset of the function space F02,1−s,s. Moreover, several equivalent descriptions of elements of TF02,1−s,s are obtained and the holomorphy of higher Bers maps is proved.


---

## Body

## 1. Introduction

LetΔ=z:z<1 be the unit disk of the extended complex plane ℂ^ and Δ∗=ℂ^\Δ¯. Denote by MΔ∗ the open unit ball of the Banach space L∞Δ∗ of essentially bounded measurable functions on Δ∗. For any μ∈MΔ∗, there exists a unique quasiconformal self-homeomorphism fμ of ℂ^ whose complex dilatation is µ in Δ∗ and zero in Δ, satisfying the following normalized condition:(1)fμ0=fμ′0−1=fμ″0=0.Two elementsμ,ν∈MΔ∗ are called Teichmüller equivalent and denoted by μ∼ν, if fμ=fν on Δ. The universal Teichmüller space T can be defined as(2)T=MΔ∗/∼=μ:μ∈MΔ∗,where μ is the Teichmüller equivalence class containing µ (see [1–3]).LetQΔ be the complex Banach space of all holomorphic quadratic differentials ϕ in Δ with the following norm:(3)ϕQ=supz∈Δ1−z22ϕz<∞.Then, there is a complex structure onT induced from QΔ via the Bers embedding:(4)β:T⟶QΔ;μ⟼Sfμ,where(5)Sfμ=Nfμ′−12Nfμ2,Nfμ=logfμ′′=fμ″fμ′.The imageβT is called the Schwarzian derivative model of T while Sfμ is the Schwarzian derivative of fμ.The complex structure ofT can be induced from MΔ∗ by the natural projection:(6)P:MΔ∗⟶T,μ⟼μ.Precisely, the Bers projection,(7)S:MΔ∗⟶QΔ,μ⟼Sfμ,is holomorphic. And the following diagram is commutative.(8)LetSQ be the class of all univalent holomorphic functions f on Δ which can be quasiconformally extended to ℂ and satisfy f0=f′0−1=0. Then,(9)T˜=logf′:f∈SQis sometimes called the prelogarithmic derivative model of T, since the image of T˜ under the map(10)Φ:T˜⟶QΔ;logf′⟼Sfis the Schwarzian derivative model βT. It is known [4] that T˜ is a disconnected subset of the Bloch space ℬ consisting all holomorphic functions on Δ satisfying(11)fℬ=supz∈Δ1−z2f′z<∞,with infinitely many connected components:(12)Tb=logf′∈T˜:fΔis bounded,Tθ=logf′∈T˜:feiθ=∞,θ∈0,2π.Fixz0∈Δ∗. For μ∈MΔ∗, let fμz0 (abbreviated to be fμ) be the quasiconformal mapping to ℂ^ whose complex dilation is µ in Δ∗ and zero in Δ, normalized by fμ0=fμ′0−1=0 and fμz0=∞. It is clear that there is an injective map as follows:(13)Lz0:MΔ∗⟶T˜;μ⟼logfμ′,which is called the pre-Bers map.Since there are many important subspaces ofℬ such as BMOA and VMOA, one obtains interesting subspaces of T via the map Φ. For example, BMO-Teichmüller space and VMO-Teichmüller space are subsets β−1ΦT˜∩BMOA and β−1ΦT˜∩VMOA of T, respectively [5–9].LetFp,q,s (p≥1,q>−2,s≥0) be the Banach space of all holomorphic functions f on Δ with the following norm:(14)fFp,q,sp=supa∈Δ∫∫Δf′zp1−z2q+s1−a2s1−a¯z2sdxdy<∞.It is introduced and studied by Zhao [10]. Fp,q,s is trivial if q+s≤−1 and Fp,q,s are contained in the Bloch-type space Bq+2/p, where Bα is the Banach space of all holomorphic functions f on Δ with the norm:(15)fBα=supz∈Δ1−z2αf′z<∞.LetF0p,q,s be the subspace of Fp,q,s with(16)lima⟶1∫∫Δf′zp1−z2q+s1−a2s1−a¯z2sdxdy=0,which plays an important role in this paper.Noticing that(17)Fp,s≔Fp,p−2,s⊂B1=ℬ.Feng et al. [11] introduced a subset TFp,s of T by the s-Carleson measure, and they prove that its corresponding prelogarithmic derivative model is(18)T˜Fp,s=T˜∩Fp,p−2,s,forp≥2,0<s<2.So they callTFp,s the Fp,s-Teichmüller space. Moreover, they prove that its prelogarithmic derivative model T˜Fp,s has infinitely many connected components in Fp,p−2,s and its Schwarzian derivative model is contained in Np,p−2,s. Here, Np,q,s is the Banach space of all holomorphic functions f on Δ with the norm:(19)fNp,q,sp=supa∈Δ∫∫Δfzp1−z2p+q+s1−a2s1−a¯z2sdxdy<∞.LetN0p,q,s be the subspace of Np,q,s with(20)lima⟶1∫∫Δfzp1−z2p+q+s1−a2s1−a¯z2sdxdy=0.The holomorphies of the corresponding Bers projection and pre-Bers map are proved, and several descriptions of the elements inT˜∩F0p,s are also given.Since(21)F0p,q,s⊂B0q+2/p,B0α⊂ℬ0,0<α≤1,it is clear that F02,1−s,s⊂ℬ0 as 1≤s<3/2. Enlightened by [11, 12], we will introduce a subset TF02,1−s,s of T by the compact s-Carleson measure and we prove that its corresponding prelogarithmic derivative model is T˜∩F02,1−s,s.Let(22)L2,s,0∞Δ∗=μ∈L∞Δ∗λμz=μz2z2−1∈CMs,0Δ∗,where CMs,0Δ∗ is the set of all compact s-Carlson measures on Δ∗ (see Section 2 in detail). Then, L2,s,0∞Δ∗ is a Banach space with the norm:(23)μs=μ∞+λμC,s1/2.Let(24)M2,s,0Δ∗≔L2,s,0∞Δ∗∩MΔ∗.It is clear thatM2,s,0Δ∗ is a subdomain of L2,s,0∞Δ∗. Then, we have Theorem 1.Theorem 1.
Let1≤s<3/2. The Bers projection S:M2,s,0Δ∗⟶N02,1−s,s is holomorphic.Let(25)TF02,1−s,s≔μ∈T:μ∈M2,s,0Δ∗,T˜F02,1−s,s=logf′:f∈SQandfhas an extension withμfΔ∗∈M2,s,0Δ∗.It is clear thatT˜F02,1−s,s the prelogarithmic derivative model of TF02,1−s,s and(26)∪z0∈Δ∗¯Lz0M2,s,0Δ∗=T˜F02,1−s,s.Furthermore, we have the following theorem.Theorem 2.
Let1≤s<3/2. Then, for given z0∈Δ∗¯, the pre-Bers map Lz0:M2,s,0Δ∗⟶F02,1−s,s is holomorphic.Theorem2 shows T˜F02,1−s,s is a subset of F02,1−s,s. Thus, we call TF02,1−s,s the F02,1−s,s-Teichmüller space. The connectivity of T˜F02,1−s,s in F02,1−s,s is also considered.Theorem 3.
Let1≤s<3/2. Then, T˜F02,1−s,s is a disconnected subset of F02,1−s,s, and its connected components are (i)
T˜b=logf′∈T˜F02,1−s,s:fΔis bounded and(ii)
T˜θ=logf′∈T˜F02,1−s,s:feiθ=∞,θ∈0,2π.Then, equivalent descriptions of elements ofTF02,1−s,s are studied. Precisely, we obtain the following theorem (for the relevant operators and kern function, see Section 3).Theorem 4.
Let1≤s<3/2. If f is conformal on Δ, then the following statements are equivalent:(1)
logf′∈F02,1−s,s.(2)
Sfz21−z23∈CMs,0Δ.(3)
f can be extended to a quasiconformal mapping to the whole plane such that its complex dilatation satisfies μz2/z2−1∈CMs,0Δ∗.(4)
Uf,z21−z2∈CMs,0Δ, where(27)Uf,z=1π∫∫Δf,ζ,z2dξdη1/2,z∈ΔUf,ζ,z=f′ζf′zfζ−fz2−1ζ−z2,ζ,z∈Δ×Δare the Grunsky kernel off.(5)
There exists a conformal mapg:Δ∗⟶ℂ\fΔ∗¯, such that h=f−1∘g is a quasisymmetric homeomorphism and ϕh2z¯1−z2∈CMs,0Δ, where(28)ϕhz=1π∫∫Δϕhζ,z2dξdη1/2,z∈Δ,ϕhζ,z=12πi∫S1hw1−ζw21−zhwdw,ζ,z∈Δ×Δ.The higher Bers maps are studied in [13], which are defined by the higher Schwarzian derivatives introduced in [14]. The higher Bers maps σnfn≥3 of a univalent function f is the generalization of the classical Schwarzian derivative Sf and pre-Schwarzian derivative Nf, which is defined inductively as σ3f=Sf and(29)σn+1fz=σn′fz−n−1Nfzσnfz,n≥3.Finally, the higher Bers maps ofM2,s,0Δ∗ are discussed. Let Np,q,s,n (n≥3) be the Banach space of all holomorphic functions f on Δ with the norm(30)fNp,q,s,np=supa∈Δ∫∫Δfzp1−z2n−2p+q+s1−a2s1−a¯z2sdxdy<∞.Denote byN0p,q,s,n the subspace of Np,q,s,n with(31)lima⟶1∫∫Δfzp1−z2n−2p+q+s1−a2s1−a¯z2sdxdy=0.Then, the following theorem is proved.Theorem 5.
If1≤s<3/2, n≥3. Then, the higher Bers maps σn:M2,s,0Δ∗⟶N02,1−s,s,n are holomorphic. Moreover, the differential D0σn at the origin is given by the following correspondence:(32)μ⟼−1nn!π∫∫Δ∗μwz−wn+1dudv.The paper is organized as follows. In Section2, we show the relevant notions and lemmas. We study the Schwarzian derivative model and the prelogarithmic derivative model of TF02,1−s,s by Theorems 1 and 2. Moreover, we consider the connected components by Theorem 3. In Section 3, we give the characterizations of TF02,1−s,s in Theorem 4. In Section 4, we consider the high Bers maps of TF02,1−s,s are holomorphic in Theorem 5.


## 2.F02,1−s,s-Teichmüller Space

### 2.1. Notions and Lemmas

LetΩ=Δ or Δ∗. Given an arc I on S1, the Carleson box on Ω is defined as(33)SΩI=z∈Δ:1−I≤z≤1,zz∈I,Ω=Δ;z∈Δ∗:1≤z≤1+I,zz∈I,Ω=Δ∗.A positive measureλ on Ω is called an s-Carleson measure (s>0) if(34)λC,s=supI⊂S1λSΩIIs<∞and a compact s-Carleson measure in addition if(35)limI⟶0λSΩIIs=0.Obviously, 1-Carleson measure is the classical Carleson measure (see [15]). Denoted by CMsΩ (or CMs,0Ω) is the set of all (compact) s-Carleson measures on Ω.Lemma 1 (see [11]).
Letα>0,β>0 and s<1+α/2. For a positive measure λ on Δ, set(36)λ˜z=∫∫Δ1−z2α1−w2β1−z¯wα+β+2λwdudv.Ifλ∈CMsΔ, then λ˜∈CMsΔ, and there exists a constant C>0 such that λ˜C,s≤CλC,s, while λ˜∈CMs,0Δ if λ∈CMs,0Δ.Lemma 2 (see [16]).
Lets>0; a positive measure λ on Δ is an s-Carleson measure if and only if(37)supa∈Δ∫∫Δ1−a21−a¯z2sλzdxdy<∞,and it is a compact s-Carleson measure if and only if(38)lima⟶1∫∫Δ1−a21−a¯z2sλzdxdy=0.Theorem 6 (see [12]).
Letf be conformal on Δ, 1≤p<∞, −2<q<∞, 0≤s<∞, and q+s>−1. If q+2≤p, then logf′∈F0p,q,s if and only if Sf∈N0p,q,s.Lemma 3.
Letφ be analytic in Δ, s>1,n∈N,andα>0. Then, the following statements hold:(1)
Ifφzn1−z2α∈CMsΔ, then supz∈Δφzn1−z2α+1<∞(2)
Ifφzn1−z2α∈CMs,0Δ, then limz⟶1φzn1−z2α+1=0.Proof.
Ifφzn1−z2α∈CMsΔ, then φzn1−z2α∈CMΔ. From Lemma 4.1 in [7], we obtain the required results immediately.Lemma 4 (see [17]).
Suppose thatk>−1, r,t>0, and r+t−k>2. If t<k+2<r, then there exists a universal constant C>0, such that for all z,ζ∈Δ,(39)∫∫Δ1−w2k1−w¯zr1−w¯ζtdudv≤C1−z22+k−r1−ζ¯zt.Lemma 5 (see [5]).
Letf be a conformal mapping on Δ∗ and admit a quasiconformal extension to the whole plane, if g=γf∘f∘βz; then,(40)Sfz=−3z2−1−22π∫∫Δ∂¯gwdudv,∀z∈Δ∗,whereβzw=1+wzw+z¯,γfw=z2−1w−fzf′z.Lemma 6 (see [2]).
f
:
U
⟶
F is holomorphic if and only if f is local bounded and every x,λ∈U×E; the map t⟼fx+tλ is holomorphic from an open neighborhood of zero in the complex plane ℂ to F, where U is an open subset of E.


### 2.2. Proof of Theorem 1

We prove Theorem1 by the following three steps:(1)
SM2,s,0Δ∗⊂N02,1−s,s. Set f^ζ=1/fμ1/ζ. Then, f^ζ is a quasiconformal self-homeomorphism of ℂ^ satisfying μf^ζ=μ1/ζ in Δ and μf^ζ=0 in Δ∗. If λμz∈CMs,0Δ∗, we have(41)λμf^=μf^ζ21−ζ2∈CMs,0Δ,λμf^C,s=λμC,s.Letg=γf^∘f^∘βz, where γf^ and βz are defined as (40). The area theorem of univalent functions yields(42)∫∫ΔJgζdξdη≤π,whereJg is the Jacobian determinant of g. Note that μg=μf^∘βz and μg∞=μ∞.By (40) and (42), we obtain(43)Sf^2z2−14≤32π2π1−μg∞2∫∫Δμf^∘βzζ2dξdη=32π2π1−μg∞2∫∫Δμf^w2z2−12w−z4dudv.Sinceζ4Sf^ζ=Sfμ1/ζ, we have(44)SfμN2,1−s,s2=supb∈Δ∗∫∫Δ∗Sf^z2z2−13b2−1sb¯z−12sz2s−2dxdy.From (41), (43), and Lemma 4, we obtain(45)SfμN2,1−s,s2≤32π2π1−μg∞2supa∈Δ∫∫Δμf^w21−a2s1−w2×∫∫Δ1−w21−z21−wz41−a¯z2sdxdydudv≤C1supa∈Δ∫∫Δμf^w21−a2s1−w21−a¯w2sdudv,whereC1 only depends on μ∞. Thus, Sμ=Sfμ∈N02,1−s,s by Lemma 5.(2)
S:M2,s,0Δ∗⟶N02,1−s,s is continuous. Indeed, we prove that there exists a constant C, such that(46)Sμ^−Sν^N2,1−s,s≤Cμ^−ν^s,for anyμ^,ν^∈M2,s,0Δ∗.Setf^1=fμ^, f^2=fν^, and fiζ=1/f^i1/ζ, i=1,2. Then, f1 is a quasiconformal mapping of ℂ whose complex dilation is μζ=μ^1/ζζ/ζ¯2 in Δ and is zero in Δ∗, while f2 is a quasiconformal mapping of ℂ whose complex dilation is νζ=ν^1/ζζ/ζ¯2 in Δ and is zero in Δ∗. Thus, the correspondence between µ and μ^ and ν and ν^ are one-to-one and μ^∞=μ∞ and ν^∞=ν∞. We have(47)Sμ^−Sν^N2,1−s,s2=supb∈Δ∗∫∫Δ∗Sf1z−Sf2z2z2−13b2−1s1−b¯z2sz2s−2dxdy.Letgμ=γf1∘f1∘βz and gν=γf2∘f2∘βz. By Lemma 6, we obtain(48)Sf1z−Sf2z=−3z2−1−22π∫∫Δ∂¯gμ−gνwdudv.Setμβzw=μβzwβz′w¯/βz′w and νβzw=νβzwβz′w¯/βz′w. Let H be the Beurling–Ahlfors operator defined as(49)Hϕζ=−1π∫∫ℂϕzζ−z2dxdy,where the integral is understood in the sense of Cauchy principal value. The representation theorem of quasiconformal mapping (see [18]) says that(50)∂¯gμ=μβzI+H∂¯gμ,∂¯gν=νβzI+H∂¯gν.Consequently, we conclude that(51)∂¯gμ−gν=μβz−νβzH∂¯gν+I+μβzH∂¯gμ−gν.Since∂gν=I+H∂¯gν, we have(52)∂¯gμ−gν=I−μβzH−1μβz−νβzI+H∂¯gν=I−μβzH−1μβz−νβz∂gν.Thus, it follows from (48):(53)Sf2z−Sf1z=−3z2−1−22π∫∫ΔI−μβzH−1μβz−νβz∂gνwdudv.SinceI−μβzH−1=I+μβzHI−μβzH−1, we have Sf2z−Sf1z≔L1+L2, where(54)L1=−3z2−1−22π∫∫Δμβz−νβz∂gνζdξdη,L2=−3z2−1−22π∫∫ΔμβzHI−μβzH−1μβz−νβz∂gνζdξdη.By using the method similar to (43) and (45), we have(55)supb∈Δ∗∫∫Δ∗L12z2−13b2−1s1−b¯z2sz2s−2dxdy≤C2supa∈Δ∫∫Δμw−νw21−w21−a2s1−a¯w2sdudv.Now, we estimateL2. When μ∞<1, the operator I−μH is invertible on L2Δ and the norm of I−μH−1 is less than 1/1−μ∞. Thus, we have(56)L22z2−14≤94π21−μ∞2∫∫Δμβzζ2dξdη∫∫Δμβz−νβz∂gνζ2dξdη≤9μ−ν∞24π1−μ∞21−ν∞2∫∫Δμβzζ2dξdη=9μ−ν∞24π1−μ∞21−ν∞2∫∫Δμζ2z2−12ζ−z4dξdη.Similar toL1, we have(57)supb∈Δ∗∫∫Δ∗L22z2−13b2−1s1−b¯z2sz2s−2dxdy≤C3supa∈Δ∫∫Δμw21−w21−a2s1−a¯w2sdudv,whereC3 depends on μ∞ and ν∞. From (47), (55), and (57), we obtain (46).(3)
S is holomorphic. For any z∈Δ, define a continuous linear function lzφ=φz for φ∈N02,1−s,s. For each pair μ,ν∈M2,s,0Δ∗×M2,s,0Δ∗ and small t in the complex plane, by the holomorphic dependence of quasiconformal mappings on parameters (see [2, 18]), we conclude that lzSμ+tν=Sfμ+tν is holomorphic of t. From Lemma 6, S is holomorphic.


### 2.3. Proof of Theorem 2

Theorem1 and Theorem 6 imply Lz0M2,s,0Δ∗⊂F02,1−s,s0 and the holomorphy of Lz0 is the same as Theorem 1. Thus, we only show that Lz0 is continuous. From Theorem 3.1 in Chapter 2 [2], we have(58)supz∈Δ1−z2gμ″gμ′−gν″gν′≤Cμ−ν∞,forμ,ν∈M2,s,0Δ∗,where C depends on μ∞ and ν∞. By Theorem 1, we conclude that(59)Sgμz−SgνzN2,1−s,s≤C1μ−νs,where C1 depends on μ∞ and ν∞.It follows from Chapter 4 in [19] that there is a constant C2>0 which is independent of µ and ν such that(60)Lz0μ−Lz0νF2,1−s,s2=supa∈Δ∫∫Δgμ″gμ′−gν″gν′21−z21−a2s1−a¯z2sdxdy≤C2gμ″gμ′0−gv″gv′02+C2supa∈Δ∫∫Δgμ″gμ′′−gν″gν′′21−z231−a2s1−a¯z2sdxdy≔C2I1+C2I2.From (58), I1≤Cμ−ν∞2. By the definition of Schwarzian derivative, we obtain(61)gμ″gμ′′−gν″gν′′2≤4Sgμ−Sgν2+4gμ″gμ′+gν″gν′2gμ″gμ′−gv″gν′2.It follows from (58), (59), and (61) that(62)I2≤4Sgμ−SgνN2,1−s,s2+4∫∫Δgμ″gμ′+gν″gν′2gμ″gμ′−gν″gν′21−z231−a2s1−a¯z2sdxdy≤4C1μ−νs2+4supz∈Δ1−z22gμ″gμ′−gν″gν′2∫∫Δgμ″gμ′+gν″gν′21−z21−a2s1−a¯z2sdxdy≤4C1μ−νs2+C3loggμ′F2,1−s,s+loggν′F2,1−s,s2μ−ν∞2.Combining (60) and (62), we obtain Lz0μ−Lz0νF2,1−s,s≤C4μ−νs.


### 2.4. Proof of Theorem 3

Iflogf′∈T˜F02,1−s,s, then f is a quasiconformal mapping of ℂ^ whose complex dilatation satisfies μz2/z2−1dxdy∈CMs,0Δ∗. Let ft be the quasiconformal map in ℂ^ with f−1∞=ft−1∞ and ∂¯ft=tμ∂ft.Consider the patht⟼logft′, 0≤t≤1, in F02,1−s,s. Set g=ft and h=fs. By Theorem 2, we conclude that(63)logg′−logh′F2,1−s,s≤Ct−ss.This implies that the patht⟼logft′ is continuous in F02,1−s,s.Consequently, eachlogf′∈T˜F02,1−s,s can be connected by a continuous path to an element logφ′∈T˜F02,1−s,s, where φ is a Möbius transformation of ℂ^. If φΔ is unbounded, then fζ=φζ=∞ for some ζ∈S1. Otherwise φΔ is bounded, we consider the path r⟼logφr′, where φr=φrz, 0≤r≤1. It is easy to see that this is a path which connects the point logφ′ to the point 0 in F02,1−s,s. It proves that T˜b = {logf′∈T˜F02,1−s,s:fΔ is bounded } and T˜θ = logf′∈T˜F02,1−s,s:feiθ=∞, θ∈0,2π are connected. By [4], elements in different classes cannot be connected in ℬ. We conclude the T˜b and T˜θ are the connected components of T˜F02,1−s,s.


## 2.1. Notions and Lemmas

LetΩ=Δ or Δ∗. Given an arc I on S1, the Carleson box on Ω is defined as(33)SΩI=z∈Δ:1−I≤z≤1,zz∈I,Ω=Δ;z∈Δ∗:1≤z≤1+I,zz∈I,Ω=Δ∗.A positive measureλ on Ω is called an s-Carleson measure (s>0) if(34)λC,s=supI⊂S1λSΩIIs<∞and a compact s-Carleson measure in addition if(35)limI⟶0λSΩIIs=0.Obviously, 1-Carleson measure is the classical Carleson measure (see [15]). Denoted by CMsΩ (or CMs,0Ω) is the set of all (compact) s-Carleson measures on Ω.Lemma 1 (see [11]).
Letα>0,β>0 and s<1+α/2. For a positive measure λ on Δ, set(36)λ˜z=∫∫Δ1−z2α1−w2β1−z¯wα+β+2λwdudv.Ifλ∈CMsΔ, then λ˜∈CMsΔ, and there exists a constant C>0 such that λ˜C,s≤CλC,s, while λ˜∈CMs,0Δ if λ∈CMs,0Δ.Lemma 2 (see [16]).
Lets>0; a positive measure λ on Δ is an s-Carleson measure if and only if(37)supa∈Δ∫∫Δ1−a21−a¯z2sλzdxdy<∞,and it is a compact s-Carleson measure if and only if(38)lima⟶1∫∫Δ1−a21−a¯z2sλzdxdy=0.Theorem 6 (see [12]).
Letf be conformal on Δ, 1≤p<∞, −2<q<∞, 0≤s<∞, and q+s>−1. If q+2≤p, then logf′∈F0p,q,s if and only if Sf∈N0p,q,s.Lemma 3.
Letφ be analytic in Δ, s>1,n∈N,andα>0. Then, the following statements hold:(1)
Ifφzn1−z2α∈CMsΔ, then supz∈Δφzn1−z2α+1<∞(2)
Ifφzn1−z2α∈CMs,0Δ, then limz⟶1φzn1−z2α+1=0.Proof.
Ifφzn1−z2α∈CMsΔ, then φzn1−z2α∈CMΔ. From Lemma 4.1 in [7], we obtain the required results immediately.Lemma 4 (see [17]).
Suppose thatk>−1, r,t>0, and r+t−k>2. If t<k+2<r, then there exists a universal constant C>0, such that for all z,ζ∈Δ,(39)∫∫Δ1−w2k1−w¯zr1−w¯ζtdudv≤C1−z22+k−r1−ζ¯zt.Lemma 5 (see [5]).
Letf be a conformal mapping on Δ∗ and admit a quasiconformal extension to the whole plane, if g=γf∘f∘βz; then,(40)Sfz=−3z2−1−22π∫∫Δ∂¯gwdudv,∀z∈Δ∗,whereβzw=1+wzw+z¯,γfw=z2−1w−fzf′z.Lemma 6 (see [2]).
f
:
U
⟶
F is holomorphic if and only if f is local bounded and every x,λ∈U×E; the map t⟼fx+tλ is holomorphic from an open neighborhood of zero in the complex plane ℂ to F, where U is an open subset of E.


## 2.2. Proof of Theorem 1

We prove Theorem1 by the following three steps:(1)
SM2,s,0Δ∗⊂N02,1−s,s. Set f^ζ=1/fμ1/ζ. Then, f^ζ is a quasiconformal self-homeomorphism of ℂ^ satisfying μf^ζ=μ1/ζ in Δ and μf^ζ=0 in Δ∗. If λμz∈CMs,0Δ∗, we have(41)λμf^=μf^ζ21−ζ2∈CMs,0Δ,λμf^C,s=λμC,s.Letg=γf^∘f^∘βz, where γf^ and βz are defined as (40). The area theorem of univalent functions yields(42)∫∫ΔJgζdξdη≤π,whereJg is the Jacobian determinant of g. Note that μg=μf^∘βz and μg∞=μ∞.By (40) and (42), we obtain(43)Sf^2z2−14≤32π2π1−μg∞2∫∫Δμf^∘βzζ2dξdη=32π2π1−μg∞2∫∫Δμf^w2z2−12w−z4dudv.Sinceζ4Sf^ζ=Sfμ1/ζ, we have(44)SfμN2,1−s,s2=supb∈Δ∗∫∫Δ∗Sf^z2z2−13b2−1sb¯z−12sz2s−2dxdy.From (41), (43), and Lemma 4, we obtain(45)SfμN2,1−s,s2≤32π2π1−μg∞2supa∈Δ∫∫Δμf^w21−a2s1−w2×∫∫Δ1−w21−z21−wz41−a¯z2sdxdydudv≤C1supa∈Δ∫∫Δμf^w21−a2s1−w21−a¯w2sdudv,whereC1 only depends on μ∞. Thus, Sμ=Sfμ∈N02,1−s,s by Lemma 5.(2)
S:M2,s,0Δ∗⟶N02,1−s,s is continuous. Indeed, we prove that there exists a constant C, such that(46)Sμ^−Sν^N2,1−s,s≤Cμ^−ν^s,for anyμ^,ν^∈M2,s,0Δ∗.Setf^1=fμ^, f^2=fν^, and fiζ=1/f^i1/ζ, i=1,2. Then, f1 is a quasiconformal mapping of ℂ whose complex dilation is μζ=μ^1/ζζ/ζ¯2 in Δ and is zero in Δ∗, while f2 is a quasiconformal mapping of ℂ whose complex dilation is νζ=ν^1/ζζ/ζ¯2 in Δ and is zero in Δ∗. Thus, the correspondence between µ and μ^ and ν and ν^ are one-to-one and μ^∞=μ∞ and ν^∞=ν∞. We have(47)Sμ^−Sν^N2,1−s,s2=supb∈Δ∗∫∫Δ∗Sf1z−Sf2z2z2−13b2−1s1−b¯z2sz2s−2dxdy.Letgμ=γf1∘f1∘βz and gν=γf2∘f2∘βz. By Lemma 6, we obtain(48)Sf1z−Sf2z=−3z2−1−22π∫∫Δ∂¯gμ−gνwdudv.Setμβzw=μβzwβz′w¯/βz′w and νβzw=νβzwβz′w¯/βz′w. Let H be the Beurling–Ahlfors operator defined as(49)Hϕζ=−1π∫∫ℂϕzζ−z2dxdy,where the integral is understood in the sense of Cauchy principal value. The representation theorem of quasiconformal mapping (see [18]) says that(50)∂¯gμ=μβzI+H∂¯gμ,∂¯gν=νβzI+H∂¯gν.Consequently, we conclude that(51)∂¯gμ−gν=μβz−νβzH∂¯gν+I+μβzH∂¯gμ−gν.Since∂gν=I+H∂¯gν, we have(52)∂¯gμ−gν=I−μβzH−1μβz−νβzI+H∂¯gν=I−μβzH−1μβz−νβz∂gν.Thus, it follows from (48):(53)Sf2z−Sf1z=−3z2−1−22π∫∫ΔI−μβzH−1μβz−νβz∂gνwdudv.SinceI−μβzH−1=I+μβzHI−μβzH−1, we have Sf2z−Sf1z≔L1+L2, where(54)L1=−3z2−1−22π∫∫Δμβz−νβz∂gνζdξdη,L2=−3z2−1−22π∫∫ΔμβzHI−μβzH−1μβz−νβz∂gνζdξdη.By using the method similar to (43) and (45), we have(55)supb∈Δ∗∫∫Δ∗L12z2−13b2−1s1−b¯z2sz2s−2dxdy≤C2supa∈Δ∫∫Δμw−νw21−w21−a2s1−a¯w2sdudv.Now, we estimateL2. When μ∞<1, the operator I−μH is invertible on L2Δ and the norm of I−μH−1 is less than 1/1−μ∞. Thus, we have(56)L22z2−14≤94π21−μ∞2∫∫Δμβzζ2dξdη∫∫Δμβz−νβz∂gνζ2dξdη≤9μ−ν∞24π1−μ∞21−ν∞2∫∫Δμβzζ2dξdη=9μ−ν∞24π1−μ∞21−ν∞2∫∫Δμζ2z2−12ζ−z4dξdη.Similar toL1, we have(57)supb∈Δ∗∫∫Δ∗L22z2−13b2−1s1−b¯z2sz2s−2dxdy≤C3supa∈Δ∫∫Δμw21−w21−a2s1−a¯w2sdudv,whereC3 depends on μ∞ and ν∞. From (47), (55), and (57), we obtain (46).(3)
S is holomorphic. For any z∈Δ, define a continuous linear function lzφ=φz for φ∈N02,1−s,s. For each pair μ,ν∈M2,s,0Δ∗×M2,s,0Δ∗ and small t in the complex plane, by the holomorphic dependence of quasiconformal mappings on parameters (see [2, 18]), we conclude that lzSμ+tν=Sfμ+tν is holomorphic of t. From Lemma 6, S is holomorphic.


## 2.3. Proof of Theorem 2

Theorem1 and Theorem 6 imply Lz0M2,s,0Δ∗⊂F02,1−s,s0 and the holomorphy of Lz0 is the same as Theorem 1. Thus, we only show that Lz0 is continuous. From Theorem 3.1 in Chapter 2 [2], we have(58)supz∈Δ1−z2gμ″gμ′−gν″gν′≤Cμ−ν∞,forμ,ν∈M2,s,0Δ∗,where C depends on μ∞ and ν∞. By Theorem 1, we conclude that(59)Sgμz−SgνzN2,1−s,s≤C1μ−νs,where C1 depends on μ∞ and ν∞.It follows from Chapter 4 in [19] that there is a constant C2>0 which is independent of µ and ν such that(60)Lz0μ−Lz0νF2,1−s,s2=supa∈Δ∫∫Δgμ″gμ′−gν″gν′21−z21−a2s1−a¯z2sdxdy≤C2gμ″gμ′0−gv″gv′02+C2supa∈Δ∫∫Δgμ″gμ′′−gν″gν′′21−z231−a2s1−a¯z2sdxdy≔C2I1+C2I2.From (58), I1≤Cμ−ν∞2. By the definition of Schwarzian derivative, we obtain(61)gμ″gμ′′−gν″gν′′2≤4Sgμ−Sgν2+4gμ″gμ′+gν″gν′2gμ″gμ′−gv″gν′2.It follows from (58), (59), and (61) that(62)I2≤4Sgμ−SgνN2,1−s,s2+4∫∫Δgμ″gμ′+gν″gν′2gμ″gμ′−gν″gν′21−z231−a2s1−a¯z2sdxdy≤4C1μ−νs2+4supz∈Δ1−z22gμ″gμ′−gν″gν′2∫∫Δgμ″gμ′+gν″gν′21−z21−a2s1−a¯z2sdxdy≤4C1μ−νs2+C3loggμ′F2,1−s,s+loggν′F2,1−s,s2μ−ν∞2.Combining (60) and (62), we obtain Lz0μ−Lz0νF2,1−s,s≤C4μ−νs.


## 2.4. Proof of Theorem 3

Iflogf′∈T˜F02,1−s,s, then f is a quasiconformal mapping of ℂ^ whose complex dilatation satisfies μz2/z2−1dxdy∈CMs,0Δ∗. Let ft be the quasiconformal map in ℂ^ with f−1∞=ft−1∞ and ∂¯ft=tμ∂ft.Consider the patht⟼logft′, 0≤t≤1, in F02,1−s,s. Set g=ft and h=fs. By Theorem 2, we conclude that(63)logg′−logh′F2,1−s,s≤Ct−ss.This implies that the patht⟼logft′ is continuous in F02,1−s,s.Consequently, eachlogf′∈T˜F02,1−s,s can be connected by a continuous path to an element logφ′∈T˜F02,1−s,s, where φ is a Möbius transformation of ℂ^. If φΔ is unbounded, then fζ=φζ=∞ for some ζ∈S1. Otherwise φΔ is bounded, we consider the path r⟼logφr′, where φr=φrz, 0≤r≤1. It is easy to see that this is a path which connects the point logφ′ to the point 0 in F02,1−s,s. It proves that T˜b = {logf′∈T˜F02,1−s,s:fΔ is bounded } and T˜θ = logf′∈T˜F02,1−s,s:feiθ=∞, θ∈0,2π are connected. By [4], elements in different classes cannot be connected in ℬ. We conclude the T˜b and T˜θ are the connected components of T˜F02,1−s,s.


## 3. Some Characterizations ofF02,1−s,s-Teichmüller Space

In this section, we give some characterizations ofF02,1−s,s-Teichnüller space.


### 3.1. Two Operators

For a quasisymmetric homeomorphismh, Hu [6] introduces two kernel functions as follows:(64)ϕhζ,z=12πi∫S1hw1−ζw21−zhwdw,ζ,z∈Δ×Δ;ψhζ,z=12πi∫S1hwζ−w21−zhwdw,ζ,z∈Δ×Δ.Clearly,ϕh and ψh are holomorphic. They define two operators on A2 induced by ϕh and ψh, respectively:(65)Th−ψζ=1π∫∫Δϕhζ,zψzdxdy,ψ∈A2,ζ∈Δ;Th+ψζ=1π∫∫Δψhζ,zψzdxdy,ψ∈A2,ζ∈Δ,where A2 is the complex Hilbert space that consists of all analytic functions ϕ on Δ with the inner product and norm(66)ϕ,ψ=1π∫∫Δϕwψw¯dudv,ϕ=ϕ,ϕ1/2<∞.We set(67)ϕhz=1π∫∫Δϕhζ,z2dξdη1/2,z∈Δ.


### 3.2. Grunsky Kernel Function

For an univalent functionf on Δ, define the Grunsky kernel (see [20]) as(68)Uf,ζ,z=f′ζf′zfζ−fz2−1ζ−z2,ζ,z∈Δ×Δ,Uf,z=1π∫∫ΔUf,ζ,z2dξdη1/2,z∈Δ.Lemma 7 (see [7]).
Letf be a univalent function on Δ, and h=f−1∘g be the corresponding quasisymmetric conformal welding. Denote ν by the Beltrami coefficient of a quasiconformal extension of h−1 to Δ. Then, we have(69)1−z2236Sfz2≤U2f,z≤ϕh2z¯≤1π∫∫Δνw21−νw211−z¯w4dudv.Lemma 8 (see [7]).
Letf be a univalent function on Δ, and h=f−1∘g be the corresponding quasisymmetric conformal welding. Then, we have(70)Uf,z≤ϕhζ,⋅≤Th+Uf,z,z∈Δ.


### 3.3. Proof of Theorem 4

(1) and (2) are equivalent in Theorem6.Now we prove,2⟹3. If Sfz21−z23∈CMs,0Δ, then we know logf′∈B0 by Theorem 6. Becker–Pommerenke [21] point out that f can be extended to a quasiconformal mapping f˜ to the whole plane ℂ, and its complex dilatation satisfies(71)μz=12Sf1z¯1−1z22,1<z<R.Hence, we haveμz2/z2−1∈CMs,0Δ∗.Because Lemmas7 and 8 imply that 5⟹2 and the equivalence of (4) and (5), we only need to consider 3⟹5.Leth be a quasisymmetric homeomorphism on S1. Then, there exists a unique pair of conformal mappings f∈SQ on Δ and g on Δ∗, such that f0=f′0−1=0, g∞=∞, and h=f−1∘g on S1. Suppose that f can be extended to a quasiconformal mapping f˜ to the whole plane ℂ, whose complex dilatation satisfies μz2/z2−1∈CMs,0Δ∗. Denote H˜=g−1∘f˜ is a quasiconformal extension of h−1 to Δ∗. We can notice that H˜ has the same complex dilation µ as f˜. Note that H^=j∘H˜∘j, where jz=1/z¯. By a simple computation, we find H^ is a quasiconformal extension of h−1 to Δ with its complex dilatation νz satisfying νz=μ1/z¯ and νz2/1−z2∈CMs,0Δ. By Lemma 7, we conclude that(72)ϕh2z¯1−z2≤1π∫∫Δνw21−w21−w21−z21−z¯w4dudv.Chooseα=1andβ=1 in Lemma 1, we have ϕh2z¯1−z2∈CMs,0Δ.


## 3.1. Two Operators

For a quasisymmetric homeomorphismh, Hu [6] introduces two kernel functions as follows:(64)ϕhζ,z=12πi∫S1hw1−ζw21−zhwdw,ζ,z∈Δ×Δ;ψhζ,z=12πi∫S1hwζ−w21−zhwdw,ζ,z∈Δ×Δ.Clearly,ϕh and ψh are holomorphic. They define two operators on A2 induced by ϕh and ψh, respectively:(65)Th−ψζ=1π∫∫Δϕhζ,zψzdxdy,ψ∈A2,ζ∈Δ;Th+ψζ=1π∫∫Δψhζ,zψzdxdy,ψ∈A2,ζ∈Δ,where A2 is the complex Hilbert space that consists of all analytic functions ϕ on Δ with the inner product and norm(66)ϕ,ψ=1π∫∫Δϕwψw¯dudv,ϕ=ϕ,ϕ1/2<∞.We set(67)ϕhz=1π∫∫Δϕhζ,z2dξdη1/2,z∈Δ.


## 3.2. Grunsky Kernel Function

For an univalent functionf on Δ, define the Grunsky kernel (see [20]) as(68)Uf,ζ,z=f′ζf′zfζ−fz2−1ζ−z2,ζ,z∈Δ×Δ,Uf,z=1π∫∫ΔUf,ζ,z2dξdη1/2,z∈Δ.Lemma 7 (see [7]).
Letf be a univalent function on Δ, and h=f−1∘g be the corresponding quasisymmetric conformal welding. Denote ν by the Beltrami coefficient of a quasiconformal extension of h−1 to Δ. Then, we have(69)1−z2236Sfz2≤U2f,z≤ϕh2z¯≤1π∫∫Δνw21−νw211−z¯w4dudv.Lemma 8 (see [7]).
Letf be a univalent function on Δ, and h=f−1∘g be the corresponding quasisymmetric conformal welding. Then, we have(70)Uf,z≤ϕhζ,⋅≤Th+Uf,z,z∈Δ.


## 3.3. Proof of Theorem 4

(1) and (2) are equivalent in Theorem6.Now we prove,2⟹3. If Sfz21−z23∈CMs,0Δ, then we know logf′∈B0 by Theorem 6. Becker–Pommerenke [21] point out that f can be extended to a quasiconformal mapping f˜ to the whole plane ℂ, and its complex dilatation satisfies(71)μz=12Sf1z¯1−1z22,1<z<R.Hence, we haveμz2/z2−1∈CMs,0Δ∗.Because Lemmas7 and 8 imply that 5⟹2 and the equivalence of (4) and (5), we only need to consider 3⟹5.Leth be a quasisymmetric homeomorphism on S1. Then, there exists a unique pair of conformal mappings f∈SQ on Δ and g on Δ∗, such that f0=f′0−1=0, g∞=∞, and h=f−1∘g on S1. Suppose that f can be extended to a quasiconformal mapping f˜ to the whole plane ℂ, whose complex dilatation satisfies μz2/z2−1∈CMs,0Δ∗. Denote H˜=g−1∘f˜ is a quasiconformal extension of h−1 to Δ∗. We can notice that H˜ has the same complex dilation µ as f˜. Note that H^=j∘H˜∘j, where jz=1/z¯. By a simple computation, we find H^ is a quasiconformal extension of h−1 to Δ with its complex dilatation νz satisfying νz=μ1/z¯ and νz2/1−z2∈CMs,0Δ. By Lemma 7, we conclude that(72)ϕh2z¯1−z2≤1π∫∫Δνw21−w21−w21−z21−z¯w4dudv.Chooseα=1andβ=1 in Lemma 1, we have ϕh2z¯1−z2∈CMs,0Δ.


## 4. Higher Bers Maps ofTF02,1−s,s

In this section, we consider the higher Bers maps ofTF02,1−s,s.Lemma 9.
Let1≤s<3/2 and n≥3. If μ∈M2,s,0Δ∗, then σnfμ∈N02,1−s,s,n.Proof.
From Theorem1, we obtain that σ3fμ∈N02,1−s,s,3.
Ifσnfμ∈N02,1−s,s,n, n≥3, we have σnz21−z22n−3∈CMs,0Δ from Lemma 2. Hence, we have σn′z21−z22n−1∈CMs,0Δ by lemma in [22].
Fromσn+1fμ≤σn′fμ+n−1Nfμσnfμ, we have(73)supa∈Δ∫∫Δσn+1fμ21−z22n−11−a21−a¯z2sdxdy≤supa∈Δ∫∫Δσn′fμ21−z22n−11−a21−a¯z2sdxdy+n−12supa∈Δ∫∫ΔNfμ2σnfμ21−z22n−11−a21−a¯z2sdxdy.
Sincefμ is a univalent analytic function in Δ, we have Nfz1−z2≤6. It is easy to obtain σn+1fμ∈N02,1−s,s,n+1 from Lemma 2.
By the mathematical induction, this lemma is proved completely.


### 4.1. Proof of Theorem 5

(1)
σ
n is continuous. For any μ,ν∈M2,s,0Δ∗, let f (or g) be the quasiconformal mapping whose complex dilatation is µ (or ν) in Δ∗ and is zero in Δ, normalized by f0=f′0−1=f″0=0 and g0=g′0−1=g″0=0. By the definition of the higher Schwarzian derivative,
(74)
σ
n
+
1
f
−
σ
n
+
1
g
N
2,1
−
s
,
s
,
n
+
1
2
≤
σ
n
′
f
−
σ
n
′
g
N
2,1
−
s
,
s
,
n
+
1
2
+
n
−
1
2
N
f
σ
n
f
−
N
g
σ
n
g
N
2,1
−
s
,
s
,
n
+
1
2
.
SinceNfQ≤6 and Nfσnf−Ngσng≤Nfσnf−σng+σngNf−Ng, we have
(75)
N
f
σ
n
f
−
N
g
σ
n
g
N
2,1
−
s
,
s
,
n
+
1
2
≤
N
f
Q
2
σ
n
f
−
σ
n
g
N
2,1
−
s
,
s
,
n
2
+
σ
n
g
N
2,1
−
s
,
s
,
n
2
N
f
−
N
g
Q
2
.
By one theorem in Chapter 2 in [2], there is a constant C such that
(76)
N
f
−
N
g
Q
≤
C
μ
−
ν
∞
.
From Lemma9 and σn′f−σn′gN2,1−s,s,n+1≈σnf−σngN2,1−s,s,n, we have
(77)
σ
n
+
1
f
−
σ
n
+
1
g
N
2,1
−
s
,
s
,
n
+
1
2
≤
C
1
σ
n
f
−
σ
n
g
N
2,1
−
s
,
s
,
n
2
+
C
2
σ
n
g
N
2,1
−
s
,
s
,
n
2
μ
−
ν
∞
2
.
Repeating this processn−3 times, it has
(78)
σ
n
+
1
f
−
σ
n
+
1
g
N
2,1
−
s
,
s
,
n
+
1
≤
C
3
σ
3
f
−
σ
3
g
N
2,1
−
s
,
s
,
3
+
C
4
μ
−
ν
∞
.
By Theorem1, we obtain σn+1f−σn+1gN2,1−s,s,n+1≤C5μ−νs.
(2)
σ
n is holomorphic. It is sufficient to prove that for any μ,ν∈M2,s,0Δ∗, σnμ+tν is holomorphic in a small neighborhood of t=0.Ifμ∈M2,s,0Δ∗, there exists a positive constant ε such that for any t with t<2ε, μ+tν∞<1, and μ+tνs<∞.Here, abbreviateσnμ+tν by ψt. For fixed z∈Δ, ψt is holomorphic in t<2ε. For t<ε and t0<ε, it follows from the Cauchy integral formula that(79)ψtz−ψt0zt−t0−ddtt=t0ψtz=t−t02π∫s=2εψszs−ts−t02ds≤t−t02πε3∫s=2εψszds.Consequently, by Fubini theorem, we have(80)∫∫Δψtz−ψt0zt−t0−ddtt=t0ψtz21−z22n−31−a21−a¯z2sdxdy≤t−t02πε32∫∫Δ∫s=2εψsz2ds1−z22n−31−a21−a¯z2sdxdy=t−t02πε32∫s=2ε∫∫Δψsz21−z22n−31−a21−a¯z2sdxdyds≤C6t−t02.Thus, it deduces thatlimt⟶t0ψt−ψt0/t−t0=d/dtt=t0ψt exists in N02,1−s,s,n. This implies that σn:M2,s,0Δ∗⟶N02,1−s,s,n is holomorphic.A theorem is shown in [13] that d/dtt=0ψtz=−1nn!/π∫∫Δ∗μw/z−wn+1dudv.


## 4.1. Proof of Theorem 5

(1)
σ
n is continuous. For any μ,ν∈M2,s,0Δ∗, let f (or g) be the quasiconformal mapping whose complex dilatation is µ (or ν) in Δ∗ and is zero in Δ, normalized by f0=f′0−1=f″0=0 and g0=g′0−1=g″0=0. By the definition of the higher Schwarzian derivative,
(74)
σ
n
+
1
f
−
σ
n
+
1
g
N
2,1
−
s
,
s
,
n
+
1
2
≤
σ
n
′
f
−
σ
n
′
g
N
2,1
−
s
,
s
,
n
+
1
2
+
n
−
1
2
N
f
σ
n
f
−
N
g
σ
n
g
N
2,1
−
s
,
s
,
n
+
1
2
.
SinceNfQ≤6 and Nfσnf−Ngσng≤Nfσnf−σng+σngNf−Ng, we have
(75)
N
f
σ
n
f
−
N
g
σ
n
g
N
2,1
−
s
,
s
,
n
+
1
2
≤
N
f
Q
2
σ
n
f
−
σ
n
g
N
2,1
−
s
,
s
,
n
2
+
σ
n
g
N
2,1
−
s
,
s
,
n
2
N
f
−
N
g
Q
2
.
By one theorem in Chapter 2 in [2], there is a constant C such that
(76)
N
f
−
N
g
Q
≤
C
μ
−
ν
∞
.
From Lemma9 and σn′f−σn′gN2,1−s,s,n+1≈σnf−σngN2,1−s,s,n, we have
(77)
σ
n
+
1
f
−
σ
n
+
1
g
N
2,1
−
s
,
s
,
n
+
1
2
≤
C
1
σ
n
f
−
σ
n
g
N
2,1
−
s
,
s
,
n
2
+
C
2
σ
n
g
N
2,1
−
s
,
s
,
n
2
μ
−
ν
∞
2
.
Repeating this processn−3 times, it has
(78)
σ
n
+
1
f
−
σ
n
+
1
g
N
2,1
−
s
,
s
,
n
+
1
≤
C
3
σ
3
f
−
σ
3
g
N
2,1
−
s
,
s
,
3
+
C
4
μ
−
ν
∞
.
By Theorem1, we obtain σn+1f−σn+1gN2,1−s,s,n+1≤C5μ−νs.
(2)
σ
n is holomorphic. It is sufficient to prove that for any μ,ν∈M2,s,0Δ∗, σnμ+tν is holomorphic in a small neighborhood of t=0.Ifμ∈M2,s,0Δ∗, there exists a positive constant ε such that for any t with t<2ε, μ+tν∞<1, and μ+tνs<∞.Here, abbreviateσnμ+tν by ψt. For fixed z∈Δ, ψt is holomorphic in t<2ε. For t<ε and t0<ε, it follows from the Cauchy integral formula that(79)ψtz−ψt0zt−t0−ddtt=t0ψtz=t−t02π∫s=2εψszs−ts−t02ds≤t−t02πε3∫s=2εψszds.Consequently, by Fubini theorem, we have(80)∫∫Δψtz−ψt0zt−t0−ddtt=t0ψtz21−z22n−31−a21−a¯z2sdxdy≤t−t02πε32∫∫Δ∫s=2εψsz2ds1−z22n−31−a21−a¯z2sdxdy=t−t02πε32∫s=2ε∫∫Δψsz21−z22n−31−a21−a¯z2sdxdyds≤C6t−t02.Thus, it deduces thatlimt⟶t0ψt−ψt0/t−t0=d/dtt=t0ψt exists in N02,1−s,s,n. This implies that σn:M2,s,0Δ∗⟶N02,1−s,s,n is holomorphic.A theorem is shown in [13] that d/dtt=0ψtz=−1nn!/π∫∫Δ∗μw/z−wn+1dudv.



---

*Source: 8451832-2020-01-21.xml*