# Bloch-Type Spaces of Minimal Surfaces

Authors: Guanghua He; Xi Fu; Hancan Zhu
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We study Bloch-type spaces of minimal surfaces from the unit diskD into Rn and characterize them in terms of weighted Lipschitz functions. In addition, the boundedness of a composition operator Cϕ acting between two Bloch-type spaces is discussed.


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## Body

## 1. Introduction and Main Results

LetD={z∈C:|z|<1} be the unit disk of the complex plane C and f=(f1,…,fn) be a harmonic map from D into Rn (n≥2). The set of all pairs (z,f(z)),z=x+iy∈D, or simply the map f itself, is called aminimal surface if(1)fxfy=0,fxfx=fyfyinD,where(2)fx=f1x,…,fnx,fy=f1y,…,fnyare partial derivatives and the products are inner (cf. [1]). The set of all minimal surfaces in D is denoted by h(D).Iff=(f1,...,fn):D→Rn is a minimal surface, then there exist holomorphic functions gj in D such that fj=Regj, 1≤j≤n. We say that (g1,...,gn) is an admissible system of f (cf. [1]). Thus, by a simple computation, the formula in (1) can be rewritten as (3)∑j=1ngj′2=0,fxz2=12∑j=1ngj′2.Forα>0, a map f∈h(D) is called α-Bloch if (4)fα=supz∈D⁡1-z2αfxz<∞.It is readily seen that the set of all α-Bloch maps of h(D) is a Banach space Bα with the norm fBα=f0+fα.Letω:[0,+∞)→[0,+∞) be an increasing function with ω(0)=0; we say that ω is amajorant if ω(t)/t is nonincreasing for t>0 (cf. [2]). Given a majorant ω and α>0, the ω-α-Bloch spaceBωα consists of all maps f∈h(D) such that (5)fω,α=supz∈D⁡ω1-z2αfxz<∞and thelittleω-α-Bloch spaceBω,0α consists of the functions f∈Bωα such that(6)limz→1-⁡ω1-z2αfxz=0.In particular, when n=2, ω(t)=t, we remark that the space Bωα can be regarded as the holomorphic α-Bloch spaceBhα (cf. [3]).For eacha∈D, the Möbius transformation φa:D→D is defined by(7)φaz=a-z1-a¯z,z∈D.If a,z∈D and r∈(0,1), we define thepseudohyperbolic disk with center a and radius r as(8)Ea,r=z∈D:ρa,z=φaz<r.It is known that E(a,r) is a Euclidean disk with center at (1-r2)a/1-a2r2 and the radius 1-|a|2r/1-|a|2r2.Recall that, for0<α≤1, the weighted hyperbolic metric dsα of D, introduced in [4], is defined as(9)dsα2=dz21-z22α.Suppose that γ(t)(0≤t≤1) is a continuous and piecewise smooth curve in D. Then, the length of γ(t) with respect to the weighted hyperbolic metric dsα is given by(10)Lhαγ=∫γdsα=∫01γ′t1-γt2αdt.Consequently, the associated distance between z and w in D is(11)hαz,w=inf⁡Lhαγ:γ0=z,γ1=w,where γ is a continuous and piecewise smooth curve in D. Note that h1 (α=1) is the hyperbolic distance h on D. For more information on weighted hyperbolic metric dsα, see [4, 5].Lets,t≥0 and f be a continuous function in D. If there exists a constant C such that(12)1-z2s1-w2tfz-fw≤Cz-wresp.≤Chz,w,for any z,w∈D, then we say that f is a weighted Euclidian (resp., hyperbolic) Lipschitz function of indices (s,t). In particular, when s=t=0, we say that f 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 [8–10], the authors extended it to the holomorphic Bloch-type spaces in the unit ball of Cn. 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.
Letf∈h(D) and α>0. Then, f∈Bα if and only if there is a constant C>0 such that(13)fz-fw≤Chαz,w,z,w∈D.Moreover,(14)fα=supz,w∈D,z≠w⁡fz-fwhαz,wfor all f∈Bα.Theorem 2.
Letf∈h(D), 0<β<1,β≤α<1+β, and 0≤λ≤1. Then, the following are equivalent:(i)
f∈Bωα;(ii)
There exists a constantC>0 such that(15)supz,w∈D,z≠w⁡ω1-z2β1-w2α-βfz-fwz-wλ1-z¯w1-λ<∞;(iii)
There exists a constantC>0 such that, for all r∈(0,1),(16)1Ez,r∫Ez,rfz-fwdAw≤C1-z2ω1-z2α,where |E(z,r)| denotes the area of E(w,r) and dA(z) denotes the normalized Lebesgue area measure in D.Remark 3.
If we takeα=1,β=1/2,λ=1, and ω(t)=t, then Theorems 1 and 2 coincide with Theorems 1 and 5 in [3], respectively.Letϕ be a holomorphic self-mapping of D. The composition operator Cϕ, induced by ϕ, is defined by Cϕ(f)=f∘ϕ for f∈h(D). During the past few years, composition operators have been studied extensively on spaces of holomorphic functions on various domains in C and Cn (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.
Letα,β>0 and ϕ be a holomorphic self-mapping of D. Then, Cϕ:Bα→Bβ is bounded if and only if(17)supz∈D⁡1-z2βϕ′z1-ϕz2α<∞.This paper is organized as follows. In Section2, we shall prove Theorems 1 and 2. The proof of Theorem 4 will be presented in Section 3.Throughout this paper, constants are denoted byC; they are positive and may differ from one occurrence to the other. The notation A≍B means that there is a positive constant C such that B/C≤A≤CB.


## 2. Proofs of Theorems1 and 2

In order to prove the main results, we need some lemmas. The following lemma is proved in [11].Lemma 5.
Letr∈(0,1), w∈E(a,r). Then, we have(18)1-a2≍1-w2≍1-a¯w.Lemma 6 (see [15]).
Letω(t) be a majorant and p∈(0,1], q∈(1,∞). Then, for t∈(0,∞),(19)ωpt≥pωt,ωqt≤qωt.As applications of Lemmas5 and 6, we have the following.Lemma 7.
Letr∈(0,1), w∈E(a,r), and ω(t) be a majorant. Then,(20)ω1-a2≍ω1-w2.In this section, we come to prove Theorems1 and 2.Proof of Theorem1.
First, we show the “if” part. For anyz,w∈D, from the definition of hα(z,w), we assume that γ(s) is the geodesic between z and w (parametrized by arc-length) with respect to hα, that is, γ(0)=z,γ(s)=w. Since hα(γ(0),γ(s))=s, we have(21)fz-fw≤Cs.Dividing both sides by s and then letting s→0 in the abovementioned inequality give(22)fxz∥γ′0≤C.From the minimal length property of geodesics,(23)hαγ0,γs=∫0sγ′t1-γt2αdt=s,0<s<ϵ,we obtain that(24)lims→0⁡1s∫0sγ′t1-γt2αdt=γ′01-z2α=1.It follows that (1-|z|2)α|fx(z)|≤C and hence f∈Bα with(25)1-z2αfxz:z∈D≤sup⁡fz-fwhαz,w:z≠w.
Next, we prove the “only if” part. Assume thatf∈Bα. Let z,w∈D and γ(t)(0≤t≤1) be a smooth curve from z to w. Then, by(26)fz-fw=∫01dfdtγtdt≤∫01fxγtγ′tdt≤fα∫01γ′t1-γt2αdt≤fαhαγt,taking the infimum over all piecewise continuous curves connecting z and w, we conclude that(27)fz-fw≤fαhαz,wfor all z,w∈D. This completes the proof.Proof of Theorem2 (i⇒ii).
Following [1], for z,w∈D and γ(t)=tw+(1-t)z(0≤t≤1), we have(28)fz-fw≤∫01fxγtγ′tdt≤Cz-w∫01dsω1-sw+1-sz2α≤Cz-wλ1-z¯w1-λ∫01dsω1-sw+1-sz2α.Since for z,w∈D and s∈[0,1](29)1-sw+1-sz2α≥1-sw+1-szα≥s1-w22+1-s1-z22α≥s2α-β1-s2β1-w2α-β1-z2β,it follows from Lemma 6 that(30)fz-fwz-wλ1-z¯w1-λ≤C∫01dsω1-sz+1-sw2α≤C∫01dsωs/2α-β1-s/2β1-z2β1-w2α-β≤Cω1-z2β1-w2α-β∫01dssα-β1-sβ≤Cω1-z2β1-w2α-β,where the last integral converges since α<1+β. Thus,(31)ω1-z2β1-w2α-βfz-fwz-wλ1-z¯w1-λ<∞.This implies that (i)⇒(ii) holds.
( i i ) ⇒ ( i i i ). Suppose that f∈h(D) and(32)K=supz,w∈D,z≠w⁡ω1-z2β1-w2α-βfz-fwz-wλ1-z¯w1-λ<∞.By Lemma 7,(33)∫Ez,rfw-fzdAw≤K∫Ez,rz-wλ1-z¯w1-λω1-z2β1-w2α-βdAw≤C∫Ez,r1-z2ω1-z2αdAw,which leads to(34)1Ez,r∫Ez,rfz-fwdAw≤C1-z2ω1-z2α.
( i i i ) ⇒ ( i ) follows the following inequality (see [3]):(35)fxz1-z2≤CEz,r∫Ez,rfw-fzdAw.This completes the proof.By adding a restrictionw∈E(z,r), we characterize the space Bωα in terms of ω((1-|z|2)α)|f(z)-f(w)|/|z-w|λ|1-z¯w|1-λ as follows.Theorem 8.
Letr∈(0,1),0≤λ≤1 and f∈h(D). Then, f∈Bωα if and only if(36)M=supw∈Ez,r,z≠w⁡ω1-z2αfz-fwz-wλ1-z¯w1-λ<∞.Proof (sufficiency).
Forz∈D, by Lemma 5 and (35), we have(37)fxz≤CEz,r∫Ez,rfw-fz1-z2dAw(38)≤CEz,r∫Ez,rfw-fzz-wλ1-z¯w1-λdAw.This gives(39)fxz≤CMω1-z2α,from which we see that f∈Bωα.
Conversely, letf∈Bωα and for any w∈E(z,r),z≠w, from the proof of Theorem 2, we have(40)fz-fw≤Cz-w∫01dsω1-sw+1-sz2α≤Cz-wλ1-z¯w1-λω1-z2α.Thus,(41)supw∈Ez,r,z≠w⁡ω1-z2αfz-fwz-wλ1-z¯w1-λ<∞.The proof of Theorem 8 is completed.A similar result is also true for the little Bloch-type spacesBω,0α.Theorem 9.
Letr∈(0,1),0≤λ≤1 and f∈Bωα. Then, f∈Bω,0α if and only if(42)limz→1-⁡supw∈Ez,r,z≠w⁡ω1-z2αfz-fwz-wλ1-z¯w1-λ=0.The proof is almost the same as the one of Theorem 3.2 in [9]. Thus, we omit it here.Remark 10.
Whenω(t)=t, Li and Wulan [8] obtained the analogues of Theorems 3.7 and 3.8 for holomorphic Bloch space on the unit ball of Cn.


## 3. Proof of Theorem4

In order to prove Theorem4, we need the following lemma which comes from [17].Lemma 11.
Letα,β>0 and ϕ be a holomorphic self-mapping of D. Then, the composition operator Cϕ:Bhα→Bhβ is bounded if and only if(43)supz∈D⁡1-z2βϕ′z1-ϕz2α<∞,where Bhα and Bhβ are holomorphic α-Bloch space and holomorphic β-Bloch space, respectively.Proof of Theorem4.
Suppose that(44)L=supz∈D⁡1-z2βϕ′z1-ϕz2α.For f∈Bα and z∈D, we have(45)1-z2βCϕfx=1-z2βf∘ϕxz=limw→z⁡1-z2βf∘ϕz-f∘ϕwz-w=limw→z⁡1-z2βfϕz-fϕwϕz-ϕwϕz-ϕwz-w≤L1-ϕz2αfxϕz≤Cfα,and(46)fϕ0≤CfBα.Hence, Cϕ:Bα→Bβ is bounded.
For the converse, assume thatCϕ:Bα→Bβ is a bounded composition operator with(47)CϕfBβ≤CfBαfor all f∈Bα. Let g=u+iv∈Bhβ and h=(u,v,0…,0)=(Reg,Re(-ig),0,…,0). Then, it is easy to check that h∈Bα with an admissible system (g,-ig,0,…,0).
From (3) and assumption (47), we have(48)CϕgBhβ≤2CϕhBβ≤ChBα≤C1gBhα.Since g is an arbitrary element in Bhα, we conclude that the composition operator Cϕ:Bhα→Bhβ 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-mappingϕ of D, (1-|z|2)|ϕ′(z)|≤1-|ϕ(z)|2 holds for all z∈D. As an application of this result, it is easy to derive the following corollary.Corollary 12.
Letϕ be a holomorphic self-mapping of D. Then, Cϕ:B1→B1 is bounded.



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