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.

1. Introduction

Let D 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=infμ, where μ varies over all measures in M for which (1) holds. The families Fα have been studied extensively [1, 2].

Let β>0. The Bloch-type space Bβ is the Banach space of functions analytic in D such that supz∈D1-z2βf′z<∞, with norm(3)fBβ=f0+supz∈D1-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 univalent f∈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.

Let Φ 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 characterizes Φ 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 that C denotes a positive constant, which may vary from one appearance to the next.

Theorem 1.

Fix α>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∈Duw1-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 Theorem 1. Several lemmas are needed. Proofs of Lemmas 2 and 3 appear in [2].

Lemma 2.

Fix α>0 and let w∈D. Let fw(z)=1/1-w¯zα for z∈D. Then fw∈Fα and fwFα=1.

Lemma 3.

Fix α>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 Lemma 4 is due to MacGregor [3]. The norm inequality is due to Hibschweiler and Nordgren [6].

Lemma 4.

Let α,β>0. If f∈Fα and g∈Fβ, then fg∈Fα+β and(8)fgFα+β≤fFαgFβ.

Lemma 5 will be used to develop test functions needed for the proof of the converse.

Lemma 5.

Fix α>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 assume α=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.

Fix α>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 fix α,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 Theorem 1. The test functions used in the proof first appeared in [5], in the context of the spaces Bα+1.

Theorem 6.

Fix α>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∈Du′w1-w2β1-Φw2α<∞,C2=supw∈DuwΦ′w1-w2β1-Φw2α+1<∞.

Proof.

Fix α, 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 that C1<∞. 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∈D1-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∈Du′w1-w2β1-Φw2α<∞. In particular, (17) yields u∈Bβ.

To obtain the second condition in the theorem, let w∈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∈D1-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 consider w∈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<Φw1-w2βuwΦ′w1-Φw2α+1<∞.

Finally consider w∈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≤1/21-w2βuwΦ′w1-Φw2α+1<∞.

Relations (23) and (26) yield(27)C2=supw∈DuwΦ′w1-w2β1-Φw2α+1<∞ and the proof is complete.

Let γ,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.

Fix α,β>0. Let u∈H(D) and let Φ be an analytic self-map.(28)Wu,Φ:Fα⟶Bβ is bounded ⟺Wu,Φ:Bα+1⟶Bβ is bounded.

Xiao [7] characterized the self-maps Φ for which CΦ:Bγ→Bβ is bounded for γ,β>0.

Corollary 8.

Fix α,β, and Φ as above.(29)CΦ:Fα⟶Bβ is bounded⟺CΦ:Bα+1⟶Bβ is bounded⟺supw∈D1-w2βΦ′w1-Φw2α+1<∞.

Proof.

The equivalence of the first two conditions follows from Corollary 7. The equivalence of the second and third conditions is due to Xiao.

Let γ,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).

Let M(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.

Fix α,β>0 and let u∈H(D).(30)MFα,Bβ=Bβ-α,if β>α+1,H∞,if β=α+1,0,if β<α+1.

3. Compactness

A characterization is given for functions u,Φ for which Wu,Φ:Fα→Bβ is compact.

Lemma 10.

Fix α>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 fix α=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.

When α>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.

Fix α,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.

Lemma 11 is the standard sequential criterion for compactness.

Lemma 11.

Fix α,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.

Fix α,β>0. Assume that Wu,Φ:Fα→Bβ is bounded. The operator Wu,Φ:Fα→Bβ is compact if and only if(35)limΦw→1u′w1-w2β1-Φw2α=0,(36)limΦw→1uwΦ′w1-w2β1-Φw2α+1=0.

Proof.

Fix α,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 that Wu,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∈D1-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∈D1-z2βu′zfnΦz+uzfn′ΦzΦ′z≤Wu,ΦfnBβ⟶0as n⟶∞.In particular,(43)1-wn2βu′wnfnΦwn+uwnfn′ΦwnΦ′wn⟶0 as n→∞. (44)Since fnΦwn=11-Φwn2α,since fn′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.

Fix α,β>0 and assume that Wu,Φ:Fα→Bβ is bounded.(47)Wu,Φ:Fα→Bβ is compact⟺Wu,Φ:Bα+1⟶Bβ is compact.

Proof.

The hypothesis and Corollary 7 yield that Wu,Φ:Bα+1→Bβ is bounded.

Assume that Wu,Φ:Bα+1→Bβ is compact. Since the inclusion Fα⊂Bα+1 is bounded, it follows that Wu,Φ:Fα→Bβ is compact.

Assume that Wu,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].

Let γ,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.

Fix α,β>0 and assume that CΦ:Fα→Bβ is bounded. The following are equivalent:

CΦ:Fα→Bβ is compact.

CΦ:Bα+1→Bβ is compact.

limΦw→11-w2βΦ′w/1-Φw2α+1=0.

Proof.

Corollary 13 yields the equivalence of the first and second conditions.

Since CΦ: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.

Fix γ,β>0. In [5], Ohno et al. characterized u for which the bounded operator Mu:Bγ→Bβ is compact.

Let u∈H(D) and let β>0. Recall that u is in the little Bloch space B0β if(48)limz→11-z2βu′z=0.

Corollary 13 and the characterization in [5] for γ=α+1>1 yield the following result.

Corollary 15.

Fix α,β>0 and assume Mu:Fα→Bβ is bounded.

Assume β>α+1. Mu:Fα→Bβ is compact ⇔u∈B0β-α.

Assume β≤α+1. Mu:Fα→Bβ is compact ⇔u≡0.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

CimaJ. A.MathesonA. L.RossW. T.HibschweilerR. A.MacGregorT. H.MacGregorT. H.Analytic and univalent functions with integral representations involving complex measuresDurenP. L.OhnoS.StroethoffK.ZhaoR.Weighted composition operators between Bloch-type spacesHibschweilerR.NordgrenE.Cauchy transforms of measures and weighted shift operators on the disc algebraXiaoJ.Composition operators associated with Bloch-type spaces