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

Abstract

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

1. Introduction

Throughout this paper, we adopt the conventional symbols, and , to denote the unit disk in the extended complex plane and the disk with center and radius , respectively. Moreover, use to denote the boundary of .

A complex-valued function is said to be complex-valued harmonic in if the real part and the imaginary part of are real harmonic in . Notice that every complex-valued harmonic function can be written as , where both and are analytic in . Moreover, a complex-valued harmonic function is called to be locally univalent if its Jacobian determinant does not vanish in . It follows that if is locally univalent, then is either sense-preserving or sense-reversing depending on the conditions and in , respectively.

Notice that if is sense-preserving, then is sense-reversing. Let be the (second complex) dilatation of . It follows from [1] that, if a locally univalent harmonic function is sense-preserving, then its analytic part is locally univalent and is analytic with . Moreover, a locally univalent harmonic function is univalent if is injective.

Recall that the harmonic pre-Schwarzian derivative of a locally univalent harmonic function with is defined by and the harmonic Schwarzian derivative [2–4] is defined as

Since , we can obtain that , where 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 .

Now, we shall introduce some basic concepts concerning the Fuchsian group. Firstly, a Möbius transformation of is defined by

A Fuchsian group is a discrete Möbius group acting on the unit disk . For a Fuchsian group , it is cocompact if is compact and is convex cocompact if is finitely generated without parabolic elements. Furthermore, a Fuchsian group is of the first kind if its limit set is ; 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 is called to be of divergence type if and to be of convergence type otherwise, where 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 compatible with a Fuchsian group if and only if , for any . Correspondingly, the Schwarzian derivative is called a -compatible Schwarzian derivative if is a -compatible locally univalent harmonic function. Since then a -compatible Schwarzian derivative should satisfy

A positive measure , defined on a simply connected domain , is called a Carleson measure if there exists a positive constant , independent of , such that for all and ,

The Carleson norm of is defined by

Denote by the set of all Carleson measures on . Correspondingly, is the set of all Carleson measures on . For more details, see [6].

Let be the Dirichlet fundamental domain of a Fuchsian group in centered at and be the boundary at infinity of . Huo [7] considered a Beltrami coefficient in compatible with a finitely generated Fuchsian group of the second kind and showed that if satisfies the Carleson condition on , then is a Carleson measure in . Naturally, one may ask whether it is right for the Schwarzian derivative of a -compatible locally univalent harmonic function or not. For the case of a -compatible univalent harmonic function, the following theorem will give an affirmation of the above problem.

Theorem 1. Let be any finitely generated Fuchsian group of the second kind and be the Dirichlet fundamental domain of centered at . For a -compatible univalent harmonic function , if there exists a constant such that, for any and any , then is in , where is the characteristic function of .

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.

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 that is a univalent harmonic function in . If then there exists a constant such that, for any and , where depends only on the Carleson norm of .

Proof. Choose and fix it. For any , if , it is obviously right. For , if the Euclidean distance (this case only happens when ), then, by Theorem 5 of [1], we have where and are universal positive constants.
In the case of , we can choose a point such that Then, we have and where is the Carleson norm of the measure .
Therefore, set and the proof of this lemma is complete.

By the above lemma, we know that for any simply connected domain , if is a Carleson measure on , then it is also a Carleson measure on .

Lemma 3. Let be a convergence-type Fuchsian group and be a -compatible univalent harmonic function. If there exists such that the support set of is contained in , then .

Proof. Suppose that the support set of , denoted by , is contained in . Recall that a sequence is called an interpolating sequence of , if it satisfies the following two conditions: where stands for the Dirac mass at .
Huo [7] has shown that the sequence is an interpolating sequence in . Hence, for any and , by Theorem 5 of [1], we have where is a universal positive constant.
Note that the hyperbolic radius of the Euclidean disk is . Therefore, for any , the disk is a hyperbolic disk with center and hyperbolic radius . By [8], we know that is contained in the Euclidean disk and , where is a constant depending on .
Combined with the above discussion, we have where depends on , and the Carleson norm of the measure .

Lemma 4 [9]. Let be a chord-arc domain. Then, the following statements are equivalent: (a) is a Carleson measure in (b)For , , we havewhere and only depends on the Carleson norm of .

3. Proof of Theorem 1

In order to prove Theorem 1, we divide two parts for the finitely generated Fuchsian group of the second kind as follows: the first case is to consider the finitely generated Fuchsian group of the second kind without any parabolic element; the second case is to discuss the finitely generated Fuchsian group of the second kind with some parabolic elements.

Theorem 5. Let be a finitely generated Fuchsian group of the second kind without any parabolic element and be the Dirichlet fundamental domain of centered at . For a -compatible univalent harmonic function , if there exists a constant such that, for any and any , then is in , where is the characteristic function of the Dirichlet fundamental domain .

Proof. Let be a finitely generated Fuchsian group of the second kind without any parabolic element and be the Dirichlet domain of with center Let be a -compatible univalent harmonic function. Then, the intersection of the closure of with contains finitely many intervals which are called free edges of , denoted by , ,
For any , let be the endpoints of . It is known that both do not belong to the limit set. Both sides of are free sides of Dirichlet fundamental domains with different centers.
By the statement of the theorem, there exists a constant such that, for , choose a ball such that contains no limit points of , and . Then, for any point , , we have Furthermore, the set is compact, where is the closure of .
By Lemma 2, we know that is a Carleson measure in .
We shall now divide into two parts , where . By Lemma 3, we know that the measure is a Carleson measure on . Next, we only need to show that is also a Carleson measure.
Let be an arbitrary point on and . In the following proof, we will find a positive constant which does not depend on and such that We first consider one special case: there exists such that . By Lemma 2, we know that is a Carleson measure on the domain . Then, we have The second above equality holds since Since is a Möbius transformation is a chord-arc domain, from Lemma 4, we have where depends only on the the Carleson norm. Hence, we have For any , since contains no limit points of and there are finitely many belonging to such that then we can get that the measure is a Carleson measure on .
Now, we consider the general case. Let be the set of all the elements in such that When , there are at most three possibilities as follows: (a)There exists , (b)There exists , and (c)There exists , and In Case (a), we have where the second above inequality holds by Lemma 4 and only depends on the Carleson norm of on .
For Case (b), we have For Case (c), notice that is a triangle with three circle arcs and the angle corresponding to the side is bigger than some constant. Thus, we have where depends on the Carleson norm of on and the angle between and .
Similar to Case (a), we have For any , the arc does not contain the limit points of . Therefore, for any , if and , then the images of under , do not overlap. Thus, we have where equals to the maximum value of the constants, appearing in the proof of this theorem, and . The proof of this theorem is complete.

Theorem 6. Let be a finitely generated Fuchsian group of the second kind with some parabolic elements and be the Dirichlet fundamental domain of centered at . For a -compatible univalent harmonic function , if there exists a constant such that, for any and any , then is in , where is the characteristic function of the Dirichlet fundamental domain .

Proof. Let be a finitely generated Fuchsian group of second kind with some parabolic elements. Without loss of generality, suppose that the generator of contains only one parabolic element and is one fixed point of .
Similar to the proof of Theorem 5, we divide into two parts as follows: where and is a sufficiently small disk with center and radius such that intersects with the sides of having as a common vertex. Let be the arc of between the sides of which have as a common vertex.
By Theorem 5, we know that . Then, we only need to show that is a Carleson measure in .
For any , we have where is any point in . The first above inequality holds by Theorem 5 of [1], and the second and final above equalities hold by In the second above inequality, the hyperbolic length of is finite, where depends on and the hyperbolic length of . In the final inequality, the finiteness of the hyperbolic area of is given by Theorem 1.2 in Huo [7].
Let be any point in and . By the proof of Lemma 2.2 in Huo [7], we know that the sequence is an interpolating sequence. Therefore, we obtain that where stands for the Dirac mass at . Thus, the images of under , contained in the disk , satisfy Thus, the proof of the theorem is complete.

Proof of Theorem 1. Combining Theorem 5 and Theorem 6, the proof of Theorem 1 is complete.

Data Availability

The (data type) data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported the National Natural Science Foundation of China Grant Number 11871085 and PhD research startup foundation of Jinling Institute of Technology Grant Numbers jit-b-202011, jit-b-202049, and jit-fhxm-2018.