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

Let S H ð f Þ be the Schwarzian derivative of a univalent harmonic function f in the unit disk D , compatible with a ﬁ nitely generated Fuchsian group G of the second kind. We show that if j S H ð f Þj 2 ð 1 − j z j 2 Þ 3 dxdy satis ﬁ es the Carleson condition on the in ﬁ nite boundary of the Dirichlet fundamental domain F of G , then j S H ð f Þj 2 ð 1 − j z j 2 Þ 3 dxdy is a Carleson measure in D .


Introduction
Throughout this paper, we adopt the conventional symbols, D = fz : jzj < 1g and Bðz, rÞ, to denote the unit disk in the extended complex plane b ℂ and the disk with center z and radius r, respectively. Moreover, use S 1 = ∂D to denote the boundary of D.
A complex-valued function f is said to be complexvalued harmonic in D if the real part and the imaginary part of f are real harmonic in D. Notice that every complexvalued 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 does 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 J f > 0 and J f < 0 in D, respectively.
Notice that if f is sense-preserving, then f is sensereversing. Let ω = g ′ /h ′ be the (second complex) dilatation of f = h + g. It follows from [1] that, if a locally univalent har-monic function f = h + g is sense-preserving, then its analytic part h is locally univalent and ω = g ′ /h ′ is analytic with jωj < 1. Moreover, a locally univalent harmonic function f is univalent if f is injective.
Recall that the harmonic pre-Schwarzian derivative P H of a locally univalent harmonic function f with J f is defined by Since P H ð f Þ = ð∂/∂zÞ log J f , we can obtain that P H ð f Þ = P H ð f Þ, 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 of D is defined by A Fuchsian group is a discrete Möbius group G 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 S 1 ; 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 group G is called to be of divergence type if and to be of convergence type otherwise, where ρð·, · Þ 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 function f compatible with a Fuchsian group G if and only if f ∘ A = f , for any A ∈ G. Correspondingly, the Schwarzian derivative S H ð f Þ is called a G-compatible Schwarzian derivative if f is a G-compatible locally univalent harmonic function. Since then a G-compatible Schwarzian derivative S H ðf Þ should satisfy A positive measure λ, 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 < diameterð∂ΩÞ and z ∈ ∂Ω, The Carleson norm kλk Ω of λ is defined by Denote by CMðΩÞ the set of all Carleson measures on Ω. Correspondingly, CMðDÞ is the set of all Carleson measures on D. For more details, see [6].
Let F be the Dirichlet fundamental domain of a Fuchsian group G in D centered at z = 0 and Fð∞Þ 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 ðjμj/ð1 − jzj 2 Þ1 − jzj 2 Þdx dy satisfies the Carleson condition on Fð∞Þ, then ðjμj/ð1 − jzj 2 Þ1 − jzj 2 Þdxdy is a Carleson measure in D. Naturally, one may ask whether it is right for the Schwarzian derivative S H ð f Þ 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. Let G 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 ∈ Fð∞Þ and any 0 < r < 2, The rest of this paper is organized as follows. In Section 2, we give some related lemmas. In Section 3, we divide two parts to give the proof of Theorem 1.

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 that f is a univalent harmonic function in D.
If then there exists a constant C > 0 such that, for any ξ ∈ D and 0 < r < 2, where C depends only on the Carleson norm of jS H ð f Þj 2 ð1 − jzj 2 Þ 3 dxdy.
Proof. Choose 0 < r < 2 and fix it. For any ξ ∈ D, if ξ ∈ ∂D, it is obviously right. For ξ ∈ D, if the Euclidean distance distðξ, ∂ DÞ ≥ 2r (this case only happens when 0 < r < 1/2), then, by 2 Journal of Function Spaces where C 1 and C 2 are universal positive constants.
In the case of distðξ, ∂DÞ < 2r, we can choose a point η ∈ ∂D such that distðη, ξÞ < 2r: Then, we have Bðξ, rÞ ⊂ Bðη, 4rÞ and where C * is the Carleson norm of the measure Therefore, set C = max fC 2 , 4C * g and the proof of this lemma is complete.
By the above lemma, we know that for any simply con- where δ z stands for the Dirac mass at z. Huo [7] has shown that the sequence fgð0Þg g∈G is an interpolating sequence in D. Hence, for any ξ ∈ ∂D and 0 < r < 2, by Theorem 5 of [1], we have where C is a universal positive constant.
Note that the hyperbolic radius t ρ of the Euclidean disk Bð0, tÞ is ln ðð1 + tÞ1 + t/ð1 − tÞ1 − tÞ. Therefore, for any g ∈ G, the disk gðBð0, tÞÞ is a hyperbolic disk with center gð0Þ and hyperbolic radius t ρ . By [8], we know that gðBð0, tÞÞ is contained in the Euclidean disk Bðgð0Þ, R g Þ and R g ≤ C 3 ð1 − jgð0ÞjÞ, where C 3 is a constant depending on t.
Combined with the above discussion, we have where C * depends on C 3 , C 4 , and the Carleson norm of the measure ∑ g∈G ð1−|gð0Þ | Þδ gð0Þ .
Lemma 4 [9]. Let Ω be a chord-arc domain. Then, the following statements are equivalent: where H p ðΩÞ = f f : f is analytic in Ω and Ð ∂Ω jf j p ds<∞g and C only depends on the Carleson norm of dμ.

Proof of Theorem 1
In order to prove Theorem 1, 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. Let G 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 ∈ Fð∞Þ and any 0 < r < 2, then jS H ð f Þj 2 ð1 − jzj 2 Þ 3 dxdy is in CMðDÞ, where χ F is the characteristic function of the Dirichlet fundamental domain F.
Proof. Let G 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 I 1 , I 2 , ⋯, I n :

Journal of Function Spaces
For any 1 ≤ i ≤ n, let q i,1 , q i,2 be the endpoints of I i . It is known that both q i,1 , q i,2 do not belong to the limit set. Both sides of q i,j ðj = 1, or 2Þ are free sides of Dirichlet fundamental domains with different centers.
By the statement of the theorem, there exists a constant C > 0 such that, for 1 ≤ i ≤ n, choose a ball B i such that B i ∩ ∂D contains no limit points of G, and I i ⊂ B i ∩ ∂D. Then, for any point ξ ∈ I i , 0 < r < 2, we have Furthermore, the set F is the closure of F.
We shall now divide f into two parts f 1 , f 2 , where B = S n i=1 ðB i ∩ FÞ. By Lemma 3, we know that the measure jS H ð f 1 Þj 2 ð1 − jzj 2 Þ 3 dxdy is a Carleson measure on D.
Next, we only need to show that jS H ð f 2 Þj 2 ð1 − jzj 2 Þ 3 dxdy is also a Carleson measure. Let ξ 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 We first consider one special case: there exists g ∈ G such that gðBðξ, rÞ ∩ DÞ ⊂ F. By Lemma 2, we know that jS H ð f 2 Þj 2 ð1 − jzj 2 Þ 3 dxdy is a Carleson measure on the domain gðBðξ, rÞ ∩ DÞ. Then, we have The second above equality holds since Since g is a Möbius transformation gðBðξ, rÞ ∩ DÞ is a chord-arc domain, from Lemma 4, we have where C 5 depends only on the the Carleson norm. Hence, we have For any 1 ≤ i ≤ n, since B i ∩ ∂D contains no limit points of G and there are finitely many g 1 , ⋯, g m belonging to G such that then we can get that the measure jS H ð f 2 ðzÞÞj 2 ð1 − jzj 2 Þ 3 d xdy is a Carleson measure on B i ∩ D. Now, we consider the general case. Let G * be the set of all the elements g in G such that gðBÞ ∩ Bðξ, rÞ ≠ ∅: When g ∈ G * , there are at most three possibilities as follows: (a) There exists 1 ≤ i ≤ n, gðB i ∩ FÞ ⊂ Bðξ, rÞ where the second above inequality holds by Lemma 4 and C 5 only depends on the Carleson norm of jS H ð f 2 ðzÞÞj 2 ð1 − jzj 2 Þ 3 dxdy on B i ∩ D.

Journal of Function Spaces
For Case (b), we have For Case (c), notice that gðB i ∩ DÞ ∩ Bðξ, rÞ is a triangle with three circle arcs and the angle corresponding to the side gðB i ∩ ∂DÞ ∩ Bðξ, rÞ is bigger than some constant. Thus, we have where C 6 depends on the Carleson norm of jS H ð f 2 Þj 2 ð1 − jzj 2 Þ 3 dxdy on B i ∩ D and the angle between ∂B i and ∂D.
Similar to Case (a), we have For any 1 ≤ i ≤ n, the arc B i ∩ ∂D does not contain the limit points of G. Therefore, for any g 1 , g 2 ∈ G * , if g 1 ðB i Þ ∩ Bðξ, rÞ ≠ ∅ and g 2 ðB i Þ ∩ Bðξ, rÞ ≠ ∅, then the images of B i ∩ ∂D under g 1 , g 2 do not overlap. Thus, we have where C * equals to the maximum value of the constants, appearing in the proof of this theorem, and B = S n i ðB i ∩ DÞ. The proof of this theorem is complete.