# Composition Operators and the Closure of Morrey Space in the Bloch Space

Authors: Nanhui Hu; Xiangling Zhu
Journal: Journal of Function Spaces
 (2019)

---

## Abstract

In this paper, we characterize the closure of the Morrey space in the Bloch space. Furthermore, the boundedness and compactness of composition operators from the Bloch space to the closure of the Morrey space in the Bloch space are investigated.


---

## Body

## 1. Introduction

LetD={z:z<1} be the open unit disk in the complex plane C and H(D) be the space of analytic functions on D. For a fixed point a∈D, let σa(z)=a-z/1-a-z denote the Mo¨bius transformation on D. Recall that the Bloch space, denoted by B=B(D), is the space of all f∈H(D) for which (1)fB=f0+supz∈D⁡1-z2f′z<∞.It is a Banach space with the above norm ·B. The little Bloch space B0 consists of all f∈H(D) such that (2)limz→1⁡1-z2f′z=0.It is easy to see that the little Bloch space B0 is the subspace of B. It is well known that B0 is the closure of polynomials in B.For0<p<∞, the Hardy space Hp(D) consists of all functions f∈H(D) with (3)fHpp=sup0<r<1⁡12π∫02πfreiθpdθ<∞.Denote by H∞=H∞(D) the space of bounded analytic functions on D.For an arcI⊂∂D, let |I|=1/2π∫Idζ be the normalized length of I and S(I) be the corresponding Carleson box; i.e., (4)SI=z∈D:1-I≤z<1andzz∈I.Clearly, if I=∂D, then S(I)=D. Let s>0. A nonnegative measure μ on D is said to be an s-Carleson measure (see [1]) if (5)supI⊂∂D⁡μSIIs<∞.If s=1, a bounded s-Carleson measure is the classical Carleson measure.For0<λ≤1, the Morrey space L2,λ(D) is the set of all f∈H2(D) such that (6)supI⊂∂D⁡1Iλ∫Ifζ-fI2dζ2π<∞.Here fI=1/|I|∫If(ζ)|dζ|/2π. Clearly, L2,1(D)=BMOA, the space of analytic functions whose boundary functions have bounded mean oscillation. From [2] or [3], the norm of functions f∈L2,λ(D) can be defined as follows. (7)fL2,λ=f0+supI⊂∂D⁡1Iλ∫SIf′z21-z2dAz.We remark that B⊈L2,λ and L2,λ⊈B. It is well known that the function (8)fz=∑n=0∞z2n∈B,butf∉L2,λ.After a calculation, we see that g(z)=(log⁡1/1-z)2∉B, but g∈L2,λ. See [2–6] for the study of Morrey space and related operators.For every self-mapφ on D, the composition operator Cφ is defined on H(D) by (9)Cφfz=fφz,z∈D.It is a simple consequence of the Schwartz-Pick lemma that any analytic self-mapping φ of D induces a bounded composition operator Cφ on the Bloch space. Madigan and Matheson in [7] proved that Cφ:B→B is compact if and only if limφz→1⁡φ♯z=0. Here and henceforth (10)φ♯z=1-z21-φz2φ′z.See, for example, [7–14] for more characterizations of the boundedness and compactness of composition operators on the Bloch space.In 1974, Anderson, Clunie, and Pommerenke posed the problem of how to describe the closure ofH∞ in the Bloch norm (see [15]). This is still an open problem. Anderson in [16] mentioned that Jones gave a characterization of CB(BMOA), the closure of BMOA in the Bloch norm (an unpublished result). A complete proof was provided by Ghatage and Zheng in [17]. Zhao in [18] studied the closures of some Möbius invariant spaces in the Bloch space. Monreal Galán and Nicolau in [19] characterized the closure in the Bloch norm of the space Hp for 1<p<∞. Later, Galanopoulos et al. in [20] studied the closure in the Bloch norm of Hp on the unit ball in Cn. Moreover, they have extended this result to the whole range 0<p<∞. Bao and Göğüş [21] studied the closure of Dirichlet type spaces Dα(-1<α≤1) in the Bloch space. See [22–26] for some related results.It is well known that (when0<λ<1) (11)H∞⊊BMOA⊊L2,λ⊊H2andH∞⊊BMOA⊊B.Hence, (12)CBBMOA⊆CBL2,λ∩B⊆CBH2∩B.From [19], we see that a Bloch function f is in CB(H2∩B) if and only if, for every ϵ>0, (13)∫∂D∫Γaξ∩ΩϵfdAz1-z22dξ<∞.Here (14)Γaξ=z∈D:z-ξ<a1-z,a>1,ξ∈∂D,and (15)Ωϵf=z∈D:1-z2f′z≥ϵ.From [16, 17], we see that a Bloch function f is in CB(BMOA) if and only if, for every ϵ>0, (16)supa∈D⁡∫Ωϵf1-σaz21-z2dAz1-z2<∞.It is natural to ask what is CB(L2,λ∩B), the closure of the Morrey type space L2,λ (0<λ<1) in the Bloch norm?The purpose of this paper is to characterizeCB(L2,λ∩B). Moreover, we study the boundedness and compactness of composition operators Cφ:B(B0)→CB(L2,λ∩B) and Cφ:CB(L2,λ∩B)→CB(L2,λ∩B).Throughout this paper, we say thatA≲B if there exists a constant C such that A≤CB. The symbol A≈B means that A≲B≲A.


## 2. Main Results and Proofs

In this section we give our main results and proofs. For this purpose, we need the following well-known estimate which can be found in [27] or [18].Lemma 1.
Lets>-1,r,t>0 and r+t-s>2. If t<s+2<r, then (17)∫D1-η2s1-η-zr1-η-ξtdAη≤C1-z2r-s-21-ξ-zt.The following lemma is Lemma 3.1.1 in [1].Lemma 2.
Lets,t∈(0,∞) and a nonnegative measure μ on D. Then μ is a s-Carleson measure if and only if (18)supa∈D⁡∫D1-a2t1-a-zs+tdμz<∞Lemma 3.
Let0<λ<1. Then f∈L2,λ if and only if (19)supa∈D⁡∫Df′z21-z21-λ1-σaz2λdAz<∞.Moreover, (20)fL2,λ≈f0+supa∈D⁡∫Df′z21-z21-λ1-σaz2λdAz1/2<∞.Proof.
Denote(21)dμz=f′z21-z2dAz.Then, from (7) we have that f∈L2,λ if and only if dμ is a λ-Carleson measure. Hence, by Lemma 2, we get that f∈L2,λ if and only if (22)supa∈D⁡∫D1-a2t1-a-zλ+tdμz=supa∈D⁡∫D1-a2t1-a-zλ+tf′z21-z2dAz<∞.Let t=λ. We get the desired result.Now we present and prove our main results in this paper.Theorem 4.
Let0<λ<1 and f∈B. Then f∈CB(L2,λ∩B) if and only if, for any ϵ>0, (23)supa∈D⁡∫Ωϵf1-σaz21-z2λdAz1-z2<∞.Proof.
Takef∈CB(L2,λ∩B) and ϵ>0. Then there exists a g∈L2,λ∩B such that f-gB≤ϵ/2. Since (24)1-z2f′z≤supw∈D⁡1-w2f′w-g′w+1-z2g′z≤ϵ2+1-z2g′z,z∈D,we see that Ωϵ(f)⊆Ωϵ/2(g). Then by Lemma 3 we get (25)supa∈D⁡∫Ωϵf1-σaz21-z2λdAz1-z2≤supa∈D⁡∫Ωϵ/2g1-σaz2λ1-z22-λg′z21-z22g′z2dAz1-z2≤4ϵ2supa∈D⁡∫Dg′z21-z21-λ1-σaz2λdAz<∞,as desired.
Conversely, suppose that (23) holds. Fix ϵ>0 and let f satisfy (23). Without loss of generality, we may assume that f(0)=0. For any z∈D, by Proposition 4.27 in [28], (26)fz=1β+1∫Df′w1-w21+β1-zw¯2+βw¯dAw,where β>0. Following [18], we decompose f as f(z)=f1(z)+f2(z), where (27)f1z=1β+1∫Ωϵff′w1-w21+β1-zw¯2+βw¯dAwand (28)f2z=1β+1∫D∖Ωϵff′w1-w21+β1-zw¯2+βw¯dAw.After a calculation, we get (29)f1′z=β+2β+1∫Ωϵff′w1-w21+β1-zw¯3+βdAwand (30)f2′z=β+2β+1∫D∖Ωϵff′w1-w21+β1-zw¯3+βdAw.Let g=f1-f1(0). Then g(0)=0; we obtain (31)f-gB=supz∈D⁡1-z2f2′z≲supz∈D⁡1-z2∫D∖Ωϵff′w1-w21+β1-zw¯3+βdAw≲ϵsupz∈D⁡1-z2∫D1-w2β1-zw¯3+βdAw.Then by Lemma 3.10 of [28] we get (32)f-gB≲ϵ.Hence g∈B. Applying Fubini’s theorem and Lemma 1, we deduce that (33)supa∈D⁡∫Dg′z21-z21-λ1-σaz2λdAz=supa∈D⁡∫Df1′z21-z21-λ1-σaz2λdAz≲f1Bsupa∈D⁡∫Ωϵff′w1-w21+βdAw∫D1-z2-λ1-σaz2λ1-zw-3+βdAz≲supa∈D⁡∫Ωϵf1-σaw2λ1-w2β1-a-w2λ1-w2λdAw∫DdAz1-z-w3+β1-z-a2λ≲supa∈D⁡∫Ωϵf1-σaw2λ1-w2λdAw1-w2<∞;that is, g∈L2,λ. Thus, for any ϵ>0, there exists a function g∈L2,λ∩B such that f-gB≲ϵ; i.e., f∈CB(L2,λ∩B). The proof is complete.Next, we consider the boundedness and compactness of composition operators fromB to CB(L2,λ∩B).Theorem 5.
Let0<λ<1 and let φ be an analytic self-map of D. Then Cφ:B→CB(L2,λ∩B) is bounded if and only if, for any ϵ>0, (34)supa∈D⁡∫φ♯z≥ϵ1-σaz21-z2λdAz1-z2<∞.Proof.
Assume thatCφ:B→CB(L2,λ∩B) is bounded. From [29], we see that there exists two functions f1,f2∈B such that (35)f1′z+f2′z≥11-z2.By the boundedness of Cφ, we get f1∘φ,f2∘φ∈CB(L2,λ∩B). Hence, Theorem 4 implies that, for any ϵ>0, (36)supa∈D⁡∫Ωϵ/2f1∘φ1-σaz21-z2λdAz1-z2<∞and (37)supa∈D⁡∫Ωϵ/2f2∘φ1-σaz21-z2λdAz1-z2<∞.When φ♯z≥ϵ, we get (38)f1∘φ′z+f2∘φ′z1-z2=f1′φz+f2′φzφ′z1-z2=f1′φz+f2′φz1-φ′z2φ′z1-z21-φ′z2≥φ♯z≥ϵ,which implies that either (39)f1∘φ′z1-z2≥ϵ2or (40)f2∘φ′z1-z2≥ϵ2.Hence, (41)supa∈D⁡∫φ♯z≥ϵ1-σaz21-z2λdAz1-z2≤supa∈D⁡∫Ωϵ/2f1∘φ∪Ωϵ/2f2∘φ1-σaz21-z2λdAz1-z2≤supa∈D⁡∫Ωϵ/2f1∘φ1-σaz21-z2λdAz1-z2+supa∈D⁡∫Ωϵ/2f2∘φ1-σaz21-z2λdAz1-z2<∞.
Conversely, suppose that (34) holds. Let f∈B. Then (42)f∘φ′z1-z2=f′φz1-φz2φ′z1-z21-φz2≤fBφ♯z.Therefore, for any δ>0, we obtain(43)supa∈D⁡∫Ωδf∘φ1-σaz21-z2λdAz1-z2≤supa∈D⁡∫φ♯z≥ϵ1-σaz21-z2λdAz1-z2<∞.From Theorem 4, we have f∘φ∈CB(L2,λ∩B), i.e., Cφ:B→CB(L2,λ∩B) is bounded. The proof is complete.Theorem 6.
Let0<λ<1 and let φ be an analytic self-map of D. Then Cφ:B0→CB(L2,λ∩B) is automatically bounded.Proof.
Sinceφ∈H∞⊊BMOA⊊L2,λ and φ∈H∞⊊B, we see that φ∈CB(L2,λ∩B). Let f∈B0. For any ϵ>0, there is a constant r (0<r<1) such that (44)f′z1-z2<ϵ2;whenever z>r. Let z∈Ωϵ(f∘φ). Then, by the assumption and Schwarz-Pick Lemma, we have (45)ϵ≤f′φzφ′z1-z2≤f′φz1-φz2φ′z1-z21-φz2≤f′φz1-φz2,which implies that φz<r. Thus, (46)ϵ≤f′φzφ′z1-z2≤fBφ′z1-z21-φz2≤fB1-r2φ′z1-z2.Let δ=ϵ(1-r2)/fB. Then φ′z(1-|z|2)≥δ. Hence, Ωϵ(f∘φ)⊆Ωδ(φ). Since φ∈CB(L2,λ∩B), by Theorem 4 we get (47)supa∈D⁡∫Ωϵf∘φ1-σaz21-z2λdAz1-z2≤supa∈D⁡∫Ωδφ1-σaz21-z2λdAz1-z2<∞.By Theorem 4 again, we see that f∘φ∈CB(L2,λ∩B). Hence Cφ:B0→CB(L2,λ∩B) is bounded. The proof is complete.Theorem 7.
Let0<λ<1 and let φ be an analytic self-map of D. Then the following statements are equivalent. (i)
Cφ:B→CB(L2,λ∩B) is compact;(ii)
Cφ:B0→CB(L2,λ∩B) is compact;(iii)
(48)limφz→1⁡φ′z1-z21-φz2=0.Proof.
( i ) ⇒ ( i i ). It is clear.
( i i ) ⇒ ( i i i ) . Since CB(L2,λ∩B)⊆B, we see that Cφ:B0→B is compact. Using [9, Theorem 1], (48) follows.
( i i i ) ⇒ ( i ) . By the assumption, we see that there exists r(0<r<1), such that (49)φ♯w<ϵ2,whenφw≥r.Let z∈D such that |φ♯(z)|≥ϵ. Then, |φ(z)|<r. Therefore, (50)ϵ≤φ♯z≤φ′z1-z21-r2,which implies that (51)ϵ1-r2≤φ′z1-z2.Let δ=ϵ(1-r2). Then z∈Ωδ(φ). Hence (52)supa∈D⁡∫φ♯z≥ϵ1-σaz21-z2λdAz1-z2≤supa∈D⁡∫Ωδφ1-σaz21-z2λdAz1-z2.Since φ∈CB(L2,λ∩B), by Theorem 4 we have (53)supa∈D⁡∫Ωδφ1-σaz21-z2λdAz1-z2<∞.By (52), (53), and Theorem 5, Cφ:B→CB(L2,λ∩B) is bounded.
Since (48) holds, from [7, Theorem 2], we see that Cφ:B→B is compact. Therefore Cφ:B→CB(L2,λ∩B) is compact. The proof is complete.Theorem 8.
Let0<λ<1 and let φ be an analytic self-map of D. Then Cφ:CB(L2,λ∩B)→CB(L2,λ∩B) is compact if and only if (54)limφz→1⁡φ′z1-z21-φz2=0.Proof.
Suppose that (54) holds. By Theorem 7, Cφ:B→CB(L2,λ∩B) is compact. Since CB(L2,λ∩B)⊆B, we get that Cφ:CB(L2,λ∩B)→CB(L2,λ∩B) is compact, as desired.
Conversely, assume thatCφ:CB(L2,λ∩B)→CB(L2,λ∩B) is compact. It is clear that φ∈CB(L2,λ∩B) since z∈CB(L2,λ∩B). Since B0 is closure of all polynomials in B and the space L2,λ contains all polynomials, hence, Cφ:B0→CB(L2,λ∩B) is compact. By Theorem 7 we see that (54) holds. The proof is complete.From Theorems7 and 8 and [13, Theorem 3], we immediately get the following corollary.Corollary 9.
Let0<λ<1 and let φ be an analytic self-map of D. Then the following statements are equivalent. (i)
Cφ:B→CB(L2,λ∩B) is compact;(ii)
Cφ:B0→CB(L2,λ∩B) is compact;(iii)
Cφ:B→B is compact;(iv)
Cφ:B0→B is compact;(v)
Cφ:CB(L2,λ∩B)→CB(L2,λ∩B) is compact;(vi)
limn→∞φnB=0;(vii)
lim|φ(z)|→1φ′z(1-|z|2)/1-|φ(z)|2=0.



---

*Source: 2834865-2019-04-01.xml*