Bloch-Type Spaces of Minimal Surfaces

It is readily seen that the set of all α-Bloch maps of h(D) is a Banach spaceB with the norm ‖f‖Bα = |f(0)| + ‖f‖α. Let ω : [0, +∞) 󳨀→ [0, +∞) be an increasing function with ω(0) = 0; we say that ω is a majorant if ω(t)/t is nonincreasing for t > 0 (cf. [2]). Given a majorant ω and α > 0, the ω-α-Bloch spaceBαω consists of all maps f ∈ h(D) such that 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩ω,α = sup z∈D ω ((1 − |z|2)) 󵄨󵄨󵄨󵄨fx (z)󵄨󵄨󵄨󵄨 < ∞ (5) and the little ω− α-Bloch spaceBαω,0 consists of the functions f ∈Bαω such that


Introduction and Main Results
Let D = { ∈ C : || < 1} be the unit disk of the complex plane C and  = ( 1 , . . .,   ) be a harmonic map from D into R  ( ≥ 2).The set of all pairs (, ()),  =  +  ∈ D, or simply the map  itself, is called a minimal surface if where   = ( 1 , . . .,   ) , are partial derivatives and the products are inner (cf.[1]).The set of all minimal surfaces in D is denoted by ℎ(D).
For each  ∈ D, the Möbius transformation   : D → D is defined by If ,  ∈ D and  ∈ (0, 1), we define the pseudohyperbolic disk with center  and radius  as It is known that (, ) is a Euclidean disk with center at Recall that, for 0 <  ≤ 1, the weighted hyperbolic metric   of D, introduced in [4], is defined as Suppose that ()(0 ≤  ≤ 1) is a continuous and piecewise smooth curve in D.Then, the length of () with respect to the weighted hyperbolic metric   is given by Consequently, the associated distance between  and  in D is where  is a continuous and piecewise smooth curve in D.
Let ,  ≥ 0 and  be a continuous function in D. If there exists a constant  such that for any ,  ∈ D, then we say that  is a weighted Euclidian (resp., hyperbolic) Lipschitz function of indices (, ).In particular, when  =  = 0, 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 D in terms of weighted Euclidian Lipschitz functions of indices (1/2, 1/2).In [8][9][10], the authors extended it to the holomorphic Bloch-type spaces in the unit ball of C  .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][13][14][15][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  ∈ ℎ(D) and  > 0. en,  ∈ B  if and only if there is a constant  > 0 such that Moreover, for all  ∈ B  .
Let  be a holomorphic self-mapping of D. The composition operator   , induced by , is defined by   () = ∘ for  ∈ ℎ(D).During the past few years, composition operators have been studied extensively on spaces of holomorphic functions on various domains in C and C  (see, e.g., [17][18][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 ,  > 0 and  be a holomorphic self-mapping of D. en,   : 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 / ≤  ≤ .

Proofs of Theorems 1 and 2
In order to prove the main results, we need some lemmas.The following lemma is proved in [11].
Proof of eorem .First, we show the "if" part.For any ,  ∈ D, from the definition of ℎ  (, ), we assume that () is the geodesic between  and  (parametrized by arc-length) with respect to ℎ  , that is, ( From the minimal length property of geodesics, for all ,  ∈ D. This completes the proof.
By adding a restriction  ∈ (, ), we characterize the space The proof of Theorem 8 is completed.
A similar result is also true for the little Bloch-type spaces B  ,0 .( The proof is almost the same as the one of Theorem 3.2 in [9].Thus, we omit it here. Remark .When () = , Li and Wulan [8] obtained the analogues of Theorems 3.7 and 3.8 for holomorphic Bloch space on the unit ball of C  .
From (3)  Recall that the classical Schwarz-Pick Lemma in the unit disk gives the notion that, for a holomorphic self-mapping  of D, (1 − || 2 )|  ()| ≤ 1 − |()| 2 holds for all  ∈ D. As an application of this result, it is easy to derive the following corollary.
Corollary 12. Let  be a holomorphic self-mapping of D. en,   : B 1 → B 1 is bounded.

.
Lemma 11.Let ,  > 0 and  be a holomorphic self-mapping of D. en, the composition operator   : B  ℎ → B   : B  → B  is bounded.For the converse, assume that   : B  → B  is a bounded composition operator with        ()     B  ≤          B ℎ are holomorphic -Bloch space and holomorphic -Bloch space, respectively.