Abstract
We study Bloch-type spaces of minimal surfaces from the unit disk into and characterize them in terms of weighted Lipschitz functions. In addition, the boundedness of a composition operator acting between two Bloch-type spaces is discussed.
1. Introduction and Main Results
Let be the unit disk of the complex plane and be a harmonic map from into (). The set of all pairs , or simply the map itself, is called a minimal surface ifwhereare partial derivatives and the products are inner (cf. [1]). The set of all minimal surfaces in is denoted by .
If is a minimal surface, then there exist holomorphic functions in such that , . We say that is an admissible system of (cf. [1]). Thus, by a simple computation, the formula in (1) can be rewritten as
For , a map is called -Bloch if It is readily seen that the set of all -Bloch maps of is a Banach space with the norm .
Let be an increasing function with ; we say that is a majorant if is nonincreasing for (cf. [2]). Given a majorant and , the --Bloch space consists of all maps such that and the little -Bloch space consists of the functions such thatIn particular, when , , we remark that the space can be regarded as the holomorphic -Bloch space (cf. [3]).
For each , the Möbius transformation is defined byIf and , we define the pseudohyperbolic disk with center and radius asIt is known that is a Euclidean disk with center at and the radius .
Recall that, for , the weighted hyperbolic metric of , introduced in [4], is defined asSuppose that is a continuous and piecewise smooth curve in . Then, the length of with respect to the weighted hyperbolic metric is given byConsequently, the associated distance between and in iswhere is a continuous and piecewise smooth curve in . Note that () is the hyperbolic distance on . For more information on weighted hyperbolic metric , see [4, 5].
Let and be a continuous function in . If there exists a constant such thatfor any , then we say that is a weighted Euclidian (resp., hyperbolic) Lipschitz function of indices . In particular, when , we say that is a Euclidian (resp., hyperbolic) Lipschitz function (cf. [6]).
The relationship between Bloch spaces and (weighted) Lipschitz functions has attracted much attention in recent years. Holland and Walsh [7] characterized holomorphic Bloch space in in terms of weighted Euclidian Lipschitz functions of indices . In [8–10], the authors extended it to the holomorphic Bloch-type spaces in the unit ball of . In [11], Zhu proved that a holomorphic function belongs to Bloch space if and only if it is hyperbolic Lipschitz. For the related results of harmonic functions, we refer to [6, 12–16] and the references therein. Recently, Huang and Wulan [3] considered the corresponding problems in the setting of Bloch minimal surface and established some analogous characterizations for Bloch minimal surfaces in terms of weighted Lipschitz functions. As the first aim of this paper, we consider the similar results of the abovementioned type for Bloch-type spaces of minimal surfaces. Our results in this line read as follows.
Theorem 1. Let and . Then, if and only if there is a constant such thatMoreover,for all .
Theorem 2. Let , , and . Then, the following are equivalent:(i);(ii)There exists a constant such that(iii)There exists a constant such that, for all ,where denotes the area of and denotes the normalized Lebesgue area measure in .
Remark 3. If we take , and , then Theorems 1 and 2 coincide with Theorems 1 and 5 in [3], respectively.
Let be a holomorphic self-mapping of . The composition operator , induced by , is defined by for . During the past few years, composition operators have been studied extensively on spaces of holomorphic functions on various domains in and (see, e.g., [17–19]). As the second aim of this paper, we also discuss the boundedness of composition operators between Bloch-type spaces of minimal surfaces.
Theorem 4. Let and be a holomorphic self-mapping of . Then, is bounded if and only if
This paper is organized as follows. In Section 2, we shall prove Theorems 1 and 2. The proof of Theorem 4 will be presented in Section 3.
Throughout this paper, constants are denoted by ; they are positive and may differ from one occurrence to the other. The notation means that there is a positive constant such that .
2. Proofs of Theorems 1 and 2
In order to prove the main results, we need some lemmas. The following lemma is proved in [11].
Lemma 5. Let , . Then, we have
Lemma 6 (see [15]). Let be a majorant and , . Then, for ,
As applications of Lemmas 5 and 6, we have the following.
Lemma 7. Let , , and be a majorant. Then,
In this section, we come to prove Theorems 1 and 2.
Proof of Theorem 1. First, we show the “if” part. For any , from the definition of , we assume that is the geodesic between and (parametrized by arc-length) with respect to , that is, . Since , we haveDividing both sides by and then letting in the abovementioned inequality giveFrom the minimal length property of geodesics,we obtain thatIt follows that and hence withNext, we prove the “only if” part. Assume that . Let and be a smooth curve from to . Then, bytaking the infimum over all piecewise continuous curves connecting and , we conclude thatfor all . This completes the proof.
Proof of Theorem 2 (). Following [1], for and , we haveSince for and it follows from Lemma 6 thatwhere the last integral converges since . Thus,This implies that holds.
. Suppose that andBy Lemma 7,which leads to follows the following inequality (see [3]):This completes the proof.
By adding a restriction , we characterize the space in terms of as follows.
Theorem 8. Let and . Then, if and only if
Proof (sufficiency). For , by Lemma 5 and (35), we haveThis givesfrom which we see that .
Conversely, let and for any , from the proof of Theorem 2, we haveThus,The proof of Theorem 8 is completed.
A similar result is also true for the little Bloch-type spaces .
Theorem 9. Let and . Then, if and only if
The proof is almost the same as the one of Theorem 3.2 in [9]. Thus, we omit it here.
Remark 10. When , Li and Wulan [8] obtained the analogues of Theorems 3.7 and 3.8 for holomorphic Bloch space on the unit ball of .
3. Proof of Theorem 4
In order to prove Theorem 4, we need the following lemma which comes from [17].
Lemma 11. Let and be a holomorphic self-mapping of . Then, the composition operator is bounded if and only ifwhere and are holomorphic -Bloch space and holomorphic -Bloch space, respectively.
Proof of Theorem 4. Suppose thatFor and , we haveandHence, is bounded.
For the converse, assume that is a bounded composition operator withfor all . Let and . Then, it is easy to check that with an admissible system .
From (3) and assumption (47), we haveSince is an arbitrary element in , we conclude that the composition operator is bounded. It follows from Lemma 11 that (17) holds.
Recall that the classical Schwarz-Pick Lemma in the unit disk gives the notion that, for a holomorphic self-mapping of , holds for all . As an application of this result, it is easy to derive the following corollary.
Corollary 12. Let be a holomorphic self-mapping of . Then, is bounded.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All the authors read and approved the final manuscript.
Acknowledgments
This work was partly supported by National Natural Science Foundation of China (no. 11501374).