# Weighted Composition Operators between the Fractional Cauchy Spaces and the Bloch-Type Spaces

Authors: R. A. Hibschweiler
Journal: Journal of Complex Analysis
 (2017)

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## Abstract

We characterize boundedness and compactness of weighted composition operators mapping the families of fractional Cauchy transforms into the Bloch-type spaces. Corollaries are obtained about composition operators and multiplication operators.


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## Body

## 1. Introduction

LetD denote the open unit disc in the complex plane and let H(D) denote the space of functions analytic in D. Let M denote the Banach space of complex-valued Borel measures on T=ζ:ζ=1, endowed with the total variation norm. For α>0, the space Fα of fractional Cauchy transforms is the collection of functions of the form(1)fz=∫T11-ζ¯zαdμζz∈D, where μ∈M. The principal branch of the logarithm is used here. The space Fα is a Banach space, with norm given by(2)fFα=inf⁡μ, where μ varies over all measures in M for which (1) holds. The families Fα have been studied extensively [1, 2].Letβ>0. The Bloch-type space Bβ is the Banach space of functions analytic in D such that supz∈D⁡1-z2βf′z<∞, with norm(3)fBβ=f0+supz∈D⁡1-z2βf′z.The integral representation (1) implies that Fα⊂Bα+1 and there is a constant depending only on α such that(4)fBα+1≤CfFα for f∈Fα.It is known that any univalentf∈H(D) belongs to Fα for any α>2. MacGregor [3] constructed a univalent function f such that f∉F2. Let g denote the normalized function g=f-f0/f′(0). Then g∉F2. Since g∈S, the classical family of schlicht functions, the Distortion Theorem [4] yields g∈B3.LetΦ be an analytic self-map of D and let u∈H(D). The weighted composition operator Wu,Φ is defined for f∈H(D) by(5)Wu,Φfz=uzfΦzz∈D. If u=1, then the operator Wu,Φ reduces to the composition operator CΦ defined by CΦ(f)=f∘Φ. If Φ is the identity function, then Wu,Φ is the multiplication operator Mu defined by Mu(f)=uf.This paper characterizesΦ and u for which Wu,Φ:Fα→Bβ is bounded or compact. Corollaries are obtained for the operators CΦ and Mu.


## 2. Boundedness

We follow the convention thatC denotes a positive constant, which may vary from one appearance to the next.Theorem 1.
Fixα>0 and β>0. Let u∈H(D) and let Φ be an analytic self-map of D. If(6)supw∈Du′w1-w2β1-Φw2α<∞,supw∈D⁡uw1-w2β1-Φw2α+1Φ′w<∞,then Wu,Φ:Fα→Bβ is bounded.Proof.
Assume the hypotheses. By a theorem due to Ohno et al.,Wu,Φ:Bα+1→Bβ is bounded [5]. Thus there is a constant C such that Wu,ΦfBβ≤CfBα+1 for all f∈Bα+1. Relation (4) now yields Wu,ΦfBβ≤CfFα for all f∈Fα.The focus of this paper is to prove the converse of Theorem1. Several lemmas are needed. Proofs of Lemmas 2 and 3 appear in [2].Lemma 2.
Fixα>0 and let w∈D. Let fw(z)=1/1-w¯zα for z∈D. Then fw∈Fα and fwFα=1.Lemma 3.
Fixα>0 and let f∈H(D). If f′∈Fα+1, then f∈Fα and there is a positive constant C independent of f such that(7)fFα≤Cf′Fα+1+f0.The first statement in Lemma4 is due to MacGregor [3]. The norm inequality is due to Hibschweiler and Nordgren [6].Lemma 4.
Letα,β>0. If f∈Fα and g∈Fβ, then fg∈Fα+β and(8)fgFα+β≤fFαgFβ.Lemma5 will be used to develop test functions needed for the proof of the converse.Lemma 5.
Fixα>0. Let w∈D and define(9)kwz=1-w21-w¯zα+1z∈D. Then kw∈Fα and there is a constant C independent of w such that kwFα≤C for all w∈D.Proof.
First assumeα=1 and fix generic w∈D. A calculation shows that kw is in the Hardy space H1 and kwH1≤1. Since the inclusion H1⊂F1 is bounded, this case is complete.
Fixα>1. Then(10)kwz=1-w21-w¯z211-w¯zα-1z∈D. By the case for α=1 and Lemma 2, kw is the product of a function in F1 and a function in Fα-1. By Lemma 4, kw∈Fα and there is a constant C independent of w such that kwFα≤C for all w.
Finally fixα,0<α<1. By the previous case,(11)kw′z=α+1w¯1-w21-w¯zα+2∈Fα+1 and kw′Fα+1≤C for all w. By Lemma 3, kw∈Fα and kwFα≤C. The proof is complete.We now prove the converse of Theorem1. The test functions used in the proof first appeared in [5], in the context of the spaces Bα+1.Theorem 6.
Fixα>0 and β>0. Let u∈H(D) and let Φ be an analytic self-map of D. Assume that Wu,Φ:Fα→Bβ is bounded. Then(12)C1=supw∈D⁡u′w1-w2β1-Φw2α<∞,C2=supw∈D⁡uwΦ′w1-w2β1-Φw2α+1<∞.Proof.
Fixα, β, u, and Φ as described. By assumption there is a constant C independent of f such that(13)Wu,ΦfBβ≤CfFα for all f∈Fα.
The argument will first establish thatC1<∞. Let w∈D and define(14)gwz=α+11-Φw¯zα-α1-Φw21-Φw¯zα+1z∈D. By Lemmas 2 and 5, there is a constant C such that gwFα≤C for all w∈D. Therefore(15)supz∈D⁡1-z2βu′zgwΦz+uzgw′ΦzΦ′z≤Wu,ΦgwBβ≤C for all w∈D. Since(16)gwΦw=11-Φw2α,gw′Φw=0, it follows that(17)C1=supw∈D⁡u′w1-w2β1-Φw2α<∞. In particular, (17) yields u∈Bβ.
To obtain the second condition in the theorem, letw∈D and define(18)hwz=1-Φw21-Φw¯zα+1z∈D. By Lemma 5, there is a constant C independent of w such that hwFα≤C. Relation (13) yields(19)supz∈D⁡1-z2βu′zhwΦz+uzhw′ΦzΦ′z≤Wu,ΦhwBβ≤C for all w∈D. Since(20)hwΦw=11-Φw2α,hw′Φw=α+1Φw¯1-Φw2α+1, it follows that(21)1-w2βu′w1-Φw2α+α+1uwΦ′wΦw¯1-Φw2α+1≤C for all w∈D.
First considerw∈D with 1/2<Φw. By the triangle inequality, relation (21) yields(22)α+121-w2βuwΦ′w1-Φw2α+1≤α+11-w2βuwΦ′wΦw1-Φw2α+1≤C+1-w2βu′w1-Φw2αfor such w. By relation (17), it follows that(23)sup1/2<Φw⁡1-w2βuwΦ′w1-Φw2α+1<∞.
Finally considerw∈D with Φw≤1/2. Let f(z)=z in relation (13). Thus uΦ∈Bβ and(24)1-w2βuwΦ′w+u′wΦw≤Wu,ΦzBβ≤C for all w∈D. Therefore(25)1-w2βuwΦ′w≤C+1-w2βu′wΦw≤C+uBβ for all w∈D. Therefore(26)supΦw≤1/2⁡1-w2βuwΦ′w1-Φw2α+1<∞.
Relations (23) and (26) yield(27)C2=supw∈D⁡uwΦ′w1-w2β1-Φw2α+1<∞ and the proof is complete.Letγ,β>0. Ohno et al. [5] characterized u and Φ for which Wu,Φ:Bγ→Bβ is bounded. Theorems 1 and 6 and their result yield the following corollary.Corollary 7.
Fixα,β>0. Let u∈H(D) and let Φ be an analytic self-map.(28)Wu,Φ:Fα⟶Bβis  bounded⟺Wu,Φ:Bα+1⟶Bβis  bounded.Xiao [7] characterized the self-maps Φ for which CΦ:Bγ→Bβ is bounded for γ,β>0.Corollary 8.
Fixα,β, and Φ as above.(29)CΦ:Fα⟶Bβis  bounded⟺CΦ:Bα+1⟶Bβis  bounded⟺supw∈D⁡1-w2βΦ′w1-Φw2α+1<∞.Proof.
The equivalence of the first two conditions follows from Corollary7. The equivalence of the second and third conditions is due to Xiao.Letγ,β>0 and let u∈H(D). The function u is a multiplier of Bγ into Bβ if Mu(f)=uf∈Bβ for every f∈Bγ. By the Closed Graph Theorem, it follows that Mu:Bγ→Bβ is bounded. The collection of all such multipliers is denoted M(Bγ,Bβ). In [5], Ohno et al. characterized u∈M(Bγ,Bβ).LetM(Fα,Bβ) denote the set of analytic functions u for which Mu:Fα→Bβ is bounded. Corollary 9 follows from Corollary 7 and the characterization in [5] for the case γ=α+1>1.Corollary 9.
Fixα,β>0 and let u∈H(D).(30)MFα,Bβ=Bβ-α,ifβ>α+1,H∞,ifβ=α+1,0,ifβ<α+1.


## 3. Compactness

A characterization is given for functionsu,Φ for which Wu,Φ:Fα→Bβ is compact.Lemma 10.
Fixα>0 and let w∈D. Define Lw by(31)Lwz=1-w221-w¯zα+2z∈D. Then Lw∈Fα and there is a constant C such that LwFα≤C for all w∈D.Proof.
First fixα=2 and let w∈D. A particular case of Lemma 5 provides a constant C independent of w∈D such that(32)1-w21-w¯z2F1≤C for all w∈D. Since(33)Lwz=1-w21-w¯z22, Lemma 4 now implies that Lw∈F2 and LwF2≤C for all w.
Whenα>2,(34)Lwz=1-w221-w¯z411-w¯zα-2. By the previous case and Lemma 2, Lw is the product of a function in F2 and a function in Fα-2. By Lemma 4, Lw∈Fα and LwFα≤C for all w∈D.
Fixα,1≤α<2. By the previous cases Lw′∈Fα+1 and Lw′Fα+1≤C for all w∈D. Lemma 3 shows that Lw∈Fα and LwFα≤C. A similar argument applies when 0<α<1. The proof is complete.Lemma11 is the standard sequential criterion for compactness.Lemma 11.
Fixα,β>0. The operator Wu,Φ:Fα→Bβ is compact if and only if Wu,ΦfnBβ→0 as n→∞ for any sequence (fn) in Fα with fnFα≤C and fn→0 uniformly on compact subsets of D as n→∞.Theorem 12.
Fixα,β>0. Assume that Wu,Φ:Fα→Bβ is bounded. The operator Wu,Φ:Fα→Bβ is compact if and only if(35)limΦw→1⁡u′w1-w2β1-Φw2α=0,(36)limΦw→1⁡uwΦ′w1-w2β1-Φw2α+1=0.Proof.
Fixα,β>0 and assume that Wu,Φ:Fα→Bβ is bounded.
First assume the limit conditions (35) and (36). Corollary 7 implies that Wu,Φ:Bα+1→Bβ is bounded and it now follows as in [5] that Wu,Φ:Bα+1→Bβ is compact. Suppose that (fn) is a sequence in Fα such that fnFα≤C for all n and fn→0 uniformly on compact subsets. By relation (4), fnBα+1≤C and thus Wu,ΦfnBβ→0 as n→∞. By Lemma 11, Wu,Φ:Fα→Bβ is compact.
Now assume thatWu,Φ:Fα→Bβ is compact. We may assume that Φ∞=1. Let (wn) be any sequence in D with Φwn→1 as n→∞. For n=1,2,… define(37)hnz=1-Φwn21-Φwn¯zα+1z∈D. By Lemma 5, hnFα≤C for all n. Also hn→0 uniformly on compact subsets of D as n→∞. Thus Wu,ΦhnBβ→0 as n→∞ and(38)supw∈D⁡1-w2βu′whnΦw+uwhn′ΦwΦ′w⟶0 as n→∞. Calculations yield(39)1-wn2βu′wn1-Φwn2α+α+1uwnΦ′wnΦwn¯1-Φwn2α+1⟶0 as n→∞.
The argument will first establish that(40)1-wn2βu′wn1-Φwn2α⟶0 as n→∞. As in [5], define the test functions(41)fnz=α+21-Φwn21-Φwn¯zα+1-α+11-Φwn221-Φwn¯zα+2, where z∈D and n=1,2,…. Then fn→0 uniformly on compact subsets as n→∞. By Lemmas 10 and 5, there is a constant C with fnFα≤C for all n. It now follows that(42)supz∈D⁡1-z2βu′zfnΦz+uzfn′ΦzΦ′z≤Wu,ΦfnBβ⟶0asn⟶∞.In particular,(43)1-wn2βu′wnfnΦwn+uwnfn′ΦwnΦ′wn⟶0 as n→∞. (44)SincefnΦwn=11-Φwn2α,sincefn′Φwn=0, relation (40) is established. Since (wn) is a generic sequence with Φwn→1 as n→∞, relation (35) holds.
To complete the proof note that relations (39) and (40) yield(45)1-wn2βuwnΦ′wnΦwn1-Φwn2α+1⟶0 as n→∞. Since Φwn→0,(46)1-wn2βuwnΦ′wn1-Φwn2α+1⟶0 as n→∞. Condition (36) follows and the proof is complete.Corollary 13.
Fixα,β>0 and assume that Wu,Φ:Fα→Bβ is bounded.(47)Wu,Φ:Fα→Bβis  compact⟺Wu,Φ:Bα+1⟶Bβis  compact.Proof.
The hypothesis and Corollary7 yield that Wu,Φ:Bα+1→Bβ is bounded.
Assume thatWu,Φ:Bα+1→Bβ is compact. Since the inclusion Fα⊂Bα+1 is bounded, it follows that Wu,Φ:Fα→Bβ is compact.
Assume thatWu,Φ:Fα→Bβ is compact. By Theorem 12, conditions (35) and (36) hold. These conditions are sufficient to imply that the bounded operator Wu,Φ:Bα+1→Bβ is compact [5].Letγ,β>0 and assume that CΦ:Bγ→Bβ is bounded. In [7], Xiao provided additional conditions on Φ necessary and sufficient for CΦ:Bγ→Bβ to be compact.Corollary 14.
Fixα,β>0 and assume that CΦ:Fα→Bβ is bounded. The following are equivalent: (1)
CΦ:Fα→Bβ is compact.(2)
CΦ:Bα+1→Bβ is compact.(3)
limΦw→11-w2βΦ′w/1-Φw2α+1=0.Proof.
Corollary13 yields the equivalence of the first and second conditions.
SinceCΦ:Fα→Bβ is bounded, Corollary 8 yields that CΦ:Bα+1→Bβ is bounded. Under this hypothesis, Xiao [7] proved the equivalence of the second and third conditions.Fixγ,β>0. In [5], Ohno et al. characterized u for which the bounded operator Mu:Bγ→Bβ is compact.Letu∈H(D) and let β>0. Recall that u is in the little Bloch space B0β if(48)limz→1⁡1-z2βu′z=0.Corollary13 and the characterization in [5] for γ=α+1>1 yield the following result.Corollary 15.
Fixα,β>0 and assume Mu:Fα→Bβ is bounded. (1)
Assumeβ>α+1. Mu:Fα→Bβis compact⇔u∈B0β-α.(2)
Assumeβ≤α+1. Mu:Fα→Bβis compact⇔u≡0.



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