# Carleson Measure of Harmonic Schwarzian Derivatives Associated with a Finitely Generated Fuchsian Group of the Second Kind

Authors: Guangming Hu; Yutong Liu; Yu Sun; Xinjie Qian
Journal: Journal of Function Spaces
 (2021)

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## Abstract

LetSHf be the Schwarzian derivative of a univalent harmonic function f in the unit disk D, compatible with a finitely generated Fuchsian group G of the second kind. We show that if SHf21−z23dxdy satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain F of G, then SHf21−z23dxdy is a Carleson measure in D.


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## Body

## 1. Introduction

Throughout this paper, we adopt the conventional symbols,D=z:z<1 and Bz,r, to denote the unit disk in the extended complex plane ℂ^ and the disk with center z and radius r, respectively. Moreover, use S1=∂D to denote the boundary of D.A complex-valued functionf is said to be complex-valued harmonic in D if the real part and the imaginary part of f are real harmonic in D. Notice that every complex-valued harmonic function f can be written as f=h+g¯, where both g and h are analytic in D. Moreover, a complex-valued harmonic function f is called to be locally univalent if its Jacobian determinant
(1)Jf=fz2−fz¯2=h′2−g′2does not vanish in D. It follows that if f is locally univalent, then f is either sense-preserving or sense-reversing depending on the conditions Jf>0 and Jf<0 in D, respectively.Notice that iff is sense-preserving, then f¯ is sense-reversing. Let ω=g′/h′ be the (second complex) dilatation of f=h+g¯. It follows from [1] that, if a locally univalent harmonic function f=h+g¯ is sense-preserving, then its analytic part h is locally univalent and ω=g′/h′ is analytic with ω<1. Moreover, a locally univalent harmonic function f is univalent if f is injective.Recall that the harmonic pre-Schwarzian derivativePH of a locally univalent harmonic function f with Jf is defined by
(2)PHf=∂∂zlogJf=h″h′−ω¯ω′1−ω2,and the harmonic Schwarzian derivative SH [2–4] is defined as
(3)SHf=∂∂zPHf−12PH2f=Sh+ω¯1−ω2h″ω′h′−ω″−32ω¯ω′1−ω22.SincePHf=∂/∂zlogJf, we can obtain that PHf=PHf¯, where f is harmonic and sense-preserving. It implies that both the pre-Schwarzian derivative and the Schwarzian derivative are well-defined for the locally univalent harmonic function (sense-preserving or sense-reversing) in D.Now, we shall introduce some basic concepts concerning the Fuchsian group. Firstly, a Möbius transformation ofD is defined by
(4)Az=eiθz−a1−a¯z,a∈D,θ∈0,2π.A Fuchsian group is a discrete Möbius groupG acting on the unit disk D. For a Fuchsian group G, it is cocompact if D/G is compact and is convex cocompact if G is finitely generated without parabolic elements. Furthermore, a Fuchsian group G is of the first kind if its limit set is S1; otherwise, it is of the second kind. Notice that all cocompact groups are the first kind and convex cocompact groups minus cocompact groups are the second kind.A finitely generated Fuchsian groupG is called to be of divergence type if
(5)∑g∈G1−g0=∞or∑g∈Gexp−ρ0,g0=∞and to be of convergence type otherwise, where ρ·,· is the hyperbolic distance. We know that all finitely generated Fuchsian groups of the second kind are of convergence type. For more details about Fuchsian groups, see [5].On the basis of the above definitions, we call a locally univalent harmonic functionf compatible with a Fuchsian group G if and only if f∘A=f, for any A∈G. Correspondingly, the Schwarzian derivative SHf is called a G-compatible Schwarzian derivative if f is a G-compatible locally univalent harmonic function. Since
(6)SHf∘A=SHf∘A·A′2,then a G-compatible Schwarzian derivative SHf should satisfy
(7)SHf=SHf∘A·A′2,A∈G.A positive measureλ, defined on a simply connected domain Ω, is called a Carleson measure if there exists a positive constant C, independent of r, such that for all 0<r<diameter∂Ω and z∈∂Ω,
(8)λΩ∩Bz,r≤Cr.The Carleson normλΩ of λ is defined by
(9)λΩ=supλΩ∩Bz,rr:z∈∂Ω,0<r<diameterΩ<∞.Denote byCMΩ the set of all Carleson measures on Ω. Correspondingly, CMD is the set of all Carleson measures on D. For more details, see [6].LetF be the Dirichlet fundamental domain of a Fuchsian group G in D centered at z=0 and F∞ be the boundary at infinity of F. Huo [7] considered a Beltrami coefficient μ in D compatible with a finitely generated Fuchsian group G of the second kind and showed that if μ/1−z21−z2dxdy satisfies the Carleson condition on F∞, then μ/1−z21−z2dxdy is a Carleson measure in D. Naturally, one may ask whether it is right for the Schwarzian derivative SHf of a G-compatible locally univalent harmonic function or not. For the case of a G-compatible univalent harmonic function, the following theorem will give an affirmation of the above problem.Theorem 1.
LetG be any finitely generated Fuchsian group of the second kind and F be the Dirichlet fundamental domain of G centered at 0. For a G-compatible univalent harmonic function f, if there exists a constant C>0 such that, for any ξ∈F∞ and any 0<r<2,
(10)∬Bξ,rSHf21−z23χFdxdy≤Cr,then SHf21−z23dxdy is in CMD, where χF is the characteristic function of F.The rest of this paper is organized as follows. In Section2, we give some related lemmas. In Section 3, we divide two parts to give the proof of Theorem 1.


## 2. Some Lemmas

The following lemma is used several times in this paper, and we shall give a short proof in this section.Lemma 2.
Suppose thatf is a univalent harmonic function in D. If
(11)SHf21−z23dxdy∈CMD,then there exists a constant C>0 such that, for any ξ∈D¯ and 0<r<2,
(12)∬Bξ,r∩DSHf21−z23dxdy≤Cr,where C depends only on the Carleson norm of SHf21−z23dxdy.Proof.
Choose0<r<2 and fix it. For any ξ∈D¯, if ξ∈∂D, it is obviously right. For ξ∈D, if the Euclidean distance distξ,∂D≥2r (this case only happens when 0<r<1/2), then, by Theorem 5 of [1], we have
(13)∬Bξ,rSHf21−z23dxdy≤∬Bξ,rC11−z2dxdy≤C1πr21−1−r2=C1πr2−r≤C2r,where C1 and C2 are universal positive constants.
In the case ofdistξ,∂D<2r, we can choose a point η∈∂D such that distη,ξ<2r. Then, we have Bξ,r⊂Bη,4r and
(14)∬Bξ,r∩DSHf21−z23dxdy≤∬Bη,4r∩DSHf21−z23dxdy≤4C∗r,where C∗ is the Carleson norm of the measure SHf21−z23dxdy.
Therefore, setC=maxC2,4C∗ and the proof of this lemma is complete.By the above lemma, we know that for any simply connected domainΩ⊂D, if SHf21−z23dxdy is a Carleson measure on D, then it is also a Carleson measure on Ω.Lemma 3.
LetG be a convergence-type Fuchsian group and f be a G-compatible univalent harmonic function. If there exists 0<t<1 such that the support set of SHfχF is contained in B0,t, then SHf21−z23dxdy∈CMD.Proof.
Suppose that the support set ofSHfχF, denoted by SuppSHf, is contained in B0,t. Recall that a sequence zj is called an interpolating sequence of D, if it satisfies the following two conditions:
(15)∃η>0,ρzj,zk≥ηforj≠k,∑1−zi2δzi∈CMD,where δz stands for the Dirac mass at z.
Huo [7] has shown that the sequence g0g∈G is an interpolating sequence in D. Hence, for any ξ∈∂D and 0<r<2, by Theorem 5 of [1], we have
(16)L=∬D∩Bξ,rSHf21−z23dxdy≤∑g∈G∬gB0,t∩Bξ,rSHf21−z23dxdy=∑g∈G∬gB0,tSHf21−z23χBξ,rdxdy≤∑g∈G∬gB0,tC1−z2χBξ,rdxdy,
whereC is a universal positive constant.
Note that the hyperbolic radiustρ of the Euclidean disk B0,t is ln1+t1+t/1−t1−t. Therefore, for any g∈G, the disk gB0,t is a hyperbolic disk with center g0 and hyperbolic radius tρ. By [8], we know that gB0,t is contained in the Euclidean disk Bg0,Rg and Rg≤C31−g0, where C3 is a constant depending on t.
Combined with the above discussion, we have(17)L≤∑g0∈Bξ,rCπRg21−1−Rg2≤C4∑g0∈Bξ,r1−g0≤C∗r,where C∗ depends on C3,C4, and the Carleson norm of the measure ∑g∈G1−∣g0∣δg0.Lemma 4 [9].
LetΩ be a chord-arc domain. Then, the following statements are equivalent:
(a)
dμ is a Carleson measure in Ω(b)
For0<p<∞, f∈HpΩ, we have(18)∬Ωfpdμ≤C∫∂Ωfpds,where HpΩ=f:fisanalyticinΩand∫∂Ωfpds<∞ and C only depends on the Carleson norm of dμ.


## 3. Proof of Theorem1

In order to prove Theorem1, we divide two parts for the finitely generated Fuchsian group of the second kind G as follows: the first case is to consider the finitely generated Fuchsian group of the second kind G without any parabolic element; the second case is to discuss the finitely generated Fuchsian group of the second kind G with some parabolic elements.Theorem 5.
LetG be a finitely generated Fuchsian group of the second kind without any parabolic element and F be the Dirichlet fundamental domain of G centered at 0. For a G-compatible univalent harmonic function f, if there exists a constant C>0 such that, for any ξ∈F∞ and any 0<r<2,
(19)∬Bξ,rSHf21−z23χFdxdy≤Cr,then SHf21−z23dxdy is in CMD, where χF is the characteristic function of the Dirichlet fundamental domain F.Proof.
LetG be a finitely generated Fuchsian group of the second kind without any parabolic element and F be the Dirichlet domain of G with center 0. Let f be a G-compatible univalent harmonic function. Then, the intersection of the closure of F with ∂D contains finitely many intervals which are called free edges of F, denoted by I1, I2, ⋯,In.
For any1≤i≤n, let qi,1,qi,2 be the endpoints of Ii. It is known that both qi,1,qi,2 do not belong to the limit set. Both sides of qi,jj=1,or2 are free sides of Dirichlet fundamental domains with different centers.
By the statement of the theorem, there exists a constantC>0 such that, for 1≤i≤n, choose a ball Bi such that Bi∩∂D contains no limit points of G, and Ii⊂Bi∩∂D. Then, for any point ξ∈Ii, 0<r<2, we have
(20)∬Bξ,r∩DSHf21−z23χBi∩Ddxdy≤Cr.
Furthermore, the setFc=F¯−⋃i=1nBi∩F is compact, where F¯ is the closure of F.
By Lemma2, we know that SHf21−z23dxdy is a Carleson measure in Bi∩F.
We shall now dividef into two parts f1,f2,
(21)f1=∑g∈GfχgFc,f2=∑g∈GfχgB,where B=⋃i=1nBi∩F. By Lemma 3, we know that the measure SHf121−z23dxdy is a Carleson measure on D. Next, we only need to show that SHf221−z23dxdy is also a Carleson measure.
Letξ be an arbitrary point on ∂D and 0<r<2. In the following proof, we will find a positive constant C∗ which does not depend on ξ and r such that
(22)∬Bξ,r∩DSHf221−z23dxdy≤C∗r.
We first consider one special case: there existsg∈G such that gBξ,r∩D⊂F. By Lemma 2, we know that SHf221−z23dxdy is a Carleson measure on the domain gBξ,r∩D. Then, we have
(23)∬Bξ,r∩DSHf2w21−w23dudv=∬gBξ,r∩DSHf2∘g−1z21−g−1z23g−1′z2dxdy=∬gBξ,r∩DSHf2∘g−1z21−z23g−1′z5dxdy=∬gBξ,r∩DSHf2z21−z23g−1′z5dxdy=∬gBξ,r∩DSHf2∘g−1z21−z23g−1′zdxdy=∬gBξ,r∩DSHf2z21−z23g−1′zdxdy.
The second above equality holds since(24)g′z1−gz2=11−z2foranyg∈G.
Sinceg is a Möbius transformation gBξ,r∩D is a chord-arc domain, from Lemma 4, we have
(25)∬gBξ,r∩DSHf2z21−z23g−1′zdxdy≤C5∫∂gBξ,r∩Dg−1′zds=C5∫∂Bξ,r∩Dds≤2πC5r,where C5 depends only on the the Carleson norm. Hence, we have
(26)∬Bξ,r∩DSHf2w21−w23dudv≤2πC5r.
For any1≤i≤n, since Bi∩∂D contains no limit points of G and there are finitely many g1,⋯,gm belonging to G such that
(27)Bi∩D⊂⋃1mgjF,then we can get that the measure SHf2z21−z23dxdy is a Carleson measure on Bi∩D.
Now, we consider the general case. LetG∗ be the set of all the elements g in G such that gB∩Bξ,r≠∅. When g∈G∗, there are at most three possibilities as follows:
(a)
There exists1≤i≤n, gBi∩F⊂Bξ,r(b)
There exists1≤i≤n, gBi∩Bξ,r≠∅ and gIi⊂Bξ,r∩∂D(c)
There exists1≤i≤n, gBi∩Bξ,r≠∅ and gIi∩Bξ,r∩∂D≠∅
In Case (a), we have(28)∬gBi∩FSHf2w21−w23dudv≤∬gBi∩DSHf2w21−w23dudv=∬Bi∩DSHf2∘gz21−gz23g′z2dxdy=∬Bi∩DSHf2∘gz21−z23g′z5dxdy=∬Bi∩DSHf2z21−z23g′z5dxdy=∬Bi∩DSHf2∘gz21−z23g′zdxdy=∬Bi∩DSHf2z21−z23g′zdxdy≤C5∫∂Bi∩Dg′zds=C5∫∂gBi∩Dds≤πC5lengthgBi∩∂D,where the second above inequality holds by Lemma 4 and C5 only depends on the Carleson norm of SHf2z21−z23dxdy on Bi∩D.
For Case (b), we have(29)∬gBi∩F∩Bξ,rSHf2w21−w23dudv≤∬gBi∩D∩Bξ,rSHf2w21−w23dudv≤πC5lengthBi∩∂D.
For Case (c), notice thatgBi∩D∩Bξ,r is a triangle with three circle arcs and the angle corresponding to the side gBi∩∂D∩Bξ,r is bigger than some constant. Thus, we have
(30)length∂gBi∩D∩Bξ,r≤C6lengthgBi∩∂D∩Bξ,r,where C6 depends on the Carleson norm of SHf221−z23dxdy on Bi∩D and the angle between ∂Bi and ∂D.
Similar to Case (a), we have(31)∬gBi∩F∩Bξ,rSHf2w21−w23dudv≤πC6lengthgBi∩∂D∩Bξ,r.
For any1≤i≤n, the arc Bi∩∂D does not contain the limit points of G. Therefore, for any g1,g2∈G∗, if g1Bi∩Bξ,r≠∅ and g2Bi∩Bξ,r≠∅, then the images of Bi∩∂D under g1, g2 do not overlap. Thus, we have
(32)∬Bξ,r∩DSHf2w21−w23dudv≤πC∗∑g∈G∗lengthgB∩Bξ,r∩∂D≤πC∗lengthBξ,r∩∂D≤2π2C∗r,where C∗ equals to the maximum value of the constants, appearing in the proof of this theorem, and B=⋃inBi∩D. The proof of this theorem is complete.Theorem 6.
LetG be a finitely generated Fuchsian group of the second kind with some parabolic elements and F be the Dirichlet fundamental domain of G centered at 0. For a G-compatible univalent harmonic function f, if there exists a constant C>0 such that, for any ξ∈F∞ and any 0<r<2,
(33)∬Bξ,rSHf21−z23χFdxdy≤Cr,then SHf21−z23dxdy is in CMD, where χF is the characteristic function of the Dirichlet fundamental domain F.Proof.
LetG be a finitely generated Fuchsian group of second kind with some parabolic elements. Without loss of generality, suppose that the generator of G contains only one parabolic element γ and ξ∈F∞ is one fixed point of γ.
Similar to the proof of Theorem5, we divide f into two parts f1,f2 as follows:
(34)f1=∑g∈GfχgF∗,f2=∑g∈GfχgB∩F.where F∗=F−B∩F and B is a sufficiently small disk with center ξ and radius r0 such that ∂B intersects with the sides of F having ξ as a common vertex. Let γ0 be the arc of ∂B between the sides of F which have ξ as a common vertex.
By Theorem5, we know that SHf1z21−z23dxdy∈CMD. Then, we only need to show that SHf2z21−z23dxdy is a Carleson measure in D.
For anyg∈G, we have
(35)∬gB∩FSHf2w21−w23dudv≤∬gB∩FC11−w2dudv=∬B∩FC1g′z21−gz2dxdy=∬B∩FC1g′z1−z2dxdy=C1∬B∩F1−gz21−z22dxdy≤C21−gz02∬B∩F11−z22dxdy≤C1−gz02,where z0 is any point in γ0. The first above inequality holds by Theorem 5 of [1], and the second and final above equalities hold by
(36)g′z1−gz2=11−z2foranyg∈G.
In the second above inequality, the hyperbolic length ofγ0 is finite, where C2 depends on z0 and the hyperbolic length of γ0. In the final inequality, the finiteness of the hyperbolic area of B∩F is given by Theorem 1.2 in Huo [7].
Letη be any point in D and 0<r<1. By the proof of Lemma 2.2 in Huo [7], we know that the sequence gz0g∈G is an interpolating sequence. Therefore, we obtain that
(37)∑g∈G1−gz02δgz0∈CMD,where δz stands for the Dirac mass at z. Thus, the images of B∩F under g, contained in the disk Bη,r, satisfy
(38)∬B∩DSHf2w21−w23dudv≤∑g∈G,gB∩F∩Bη,r≠∅∬gB∩FSHf2w21−w23dudv≤Cr.
Thus, the proof of the theorem is complete.Proof of Theorem 1.
Combining Theorem5 and Theorem 6, the proof of Theorem 1 is complete.



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*Source: 5523454-2021-03-22.xml*