JCA Journal of Complex Analysis 2314-4971 2314-4963 Hindawi 10.1155/2017/9486907 9486907 Research Article Weighted Composition Operators between the Fractional Cauchy Spaces and the Bloch-Type Spaces http://orcid.org/0000-0002-8389-3718 Hibschweiler R. A. 1 Grudsky Sergei Department of Mathematics and Statistics University of New Hampshire Durham NH 03824 USA unh.edu 2017 12112017 2017 31 08 2017 16 10 2017 12112017 2017 Copyright © 2017 R. A. Hibschweiler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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¯zαdμζzD, where μM. 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 supzD1-z2βfz<, with norm(3)fBβ=f0+supzD1-z2βfz.

The integral representation (1) implies that FαBα+1 and there is a constant depending only on α such that(4)fBα+1CfFα for fFα.

It is known that any univalent fH(D) belongs to Fα for any α>2. MacGregor  constructed a univalent function f such that fF2. Let g denote the normalized function g=f-f0/f(0). Then gF2. Since gS, the classical family of schlicht functions, the Distortion Theorem  yields gB3.

Let Φ be an analytic self-map of D and let uH(D). The weighted composition operator Wu,Φ is defined for fH(D) by(5)Wu,Φfz=uzfΦzzD. If u=1, then the operator Wu,Φ reduces to the composition operator CΦ defined by CΦ(f)=fΦ. 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:FαBβ 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 uH(D) and let Φ be an analytic self-map of D. If(6)supwDuw1-w2β1-Φw2α<,supwDuw1-w2β1-Φw2α+1Φw<,then Wu,Φ:FαBβ is bounded.

Proof.

Assume the hypotheses. By a theorem due to Ohno et al., Wu,Φ:Bα+1Bβ is bounded . Thus there is a constant C such that Wu,ΦfBβCfBα+1 for all fBα+1. Relation (4) now yields Wu,ΦfBβCfFα for all fFα.

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 .

Lemma 2.

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

Lemma 3.

Fix α>0 and let fH(D). If fFα+1, then fFα and there is a positive constant C independent of f such that(7)fFαCfFα+1+f0.

The first statement in Lemma 4 is due to MacGregor . The norm inequality is due to Hibschweiler and Nordgren .

Lemma 4.

Let α,β>0. If fFα and gFβ, then fgFα+β and(8)fgFα+βfFαgFβ.

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

Lemma 5.

Fix α>0. Let wD and define(9)kwz=1-w21-w¯zα+1zD. Then kwFα and there is a constant C independent of w such that kwFαC for all wD.

Proof.

First assume α=1 and fix generic wD. A calculation shows that kw is in the Hardy space H1 and kwH11. Since the inclusion H1F1 is bounded, this case is complete.

Fix α>1. Then(10)kwz=1-w21-w¯z211-w¯zα-1zD. 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, kwFα and there is a constant C independent of w such that kwFαC for all w.

Finally fix α,0<α<1. By the previous case,(11)kwz=α+1w¯1-w21-w¯zα+2Fα+1 and kwFα+1C for all w. By Lemma 3, kwFα and kwFαC. The proof is complete.

We now prove the converse of Theorem 1. The test functions used in the proof first appeared in , in the context of the spaces Bα+1.

Theorem 6.

Fix α>0 and β>0. Let uH(D) and let Φ be an analytic self-map of D. Assume that Wu,Φ:FαBβ is bounded. Then(12)C1=supwDuw1-w2β1-Φw2α<,C2=supwDuwΦw1-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,ΦfBβCfFα for all fFα.

The argument will first establish that C1<. Let wD and define(14)gwz=α+11-Φw¯zα-α1-Φw21-Φw¯zα+1zD. By Lemmas 2 and 5, there is a constant C such that gwFαC for all wD. Therefore(15)supzD1-z2βuzgwΦz+uzgwΦzΦzWu,ΦgwBβC for all wD. Since(16)gwΦw=11-Φw2α,gwΦw=0, it follows that(17)C1=supwDuw1-w2β1-Φw2α<. In particular, (17) yields uBβ.

To obtain the second condition in the theorem, let wD and define(18)hwz=1-Φw21-Φw¯zα+1zD. By Lemma 5, there is a constant C independent of w such that hwFαC. Relation (13) yields(19)supzD1-z2βuzhwΦz+uzhwΦzΦzWu,ΦhwBβC for all wD. Since(20)hwΦw=11-Φw2α,hwΦw=α+1Φw¯1-Φw2α+1, it follows that(21)1-w2βuw1-Φw2α+α+1uwΦwΦw¯1-Φw2α+1C for all wD.

First consider wD with 1/2<Φw. By the triangle inequality, relation (21) yields(22)α+121-w2βuwΦw1-Φw2α+1α+11-w2βuwΦwΦw1-Φw2α+1C+1-w2βuw1-Φw2αfor such w. By relation (17), it follows that(23)sup1/2<Φw1-w2βuwΦw1-Φw2α+1<.

Finally consider wD with Φw1/2. Let f(z)=z in relation (13). Thus uΦBβ and(24)1-w2βuwΦw+uwΦwWu,ΦzBβC for all wD. Therefore(25)1-w2βuwΦwC+1-w2βuwΦwC+uBβ for all wD. Therefore(26)supΦw1/21-w2βuwΦw1-Φw2α+1<.

Relations (23) and (26) yield(27)C2=supwDuwΦw1-w2β1-Φw2α+1< and the proof is complete.

Let γ,β>0. Ohno et al.  characterized u and Φ for which Wu,Φ:BγBβ is bounded. Theorems 1 and 6 and their result yield the following corollary.

Corollary 7.

Fix α,β>0. Let uH(D) and let Φ be an analytic self-map.(28)Wu,Φ:FαBβ  is  bounded  Wu,Φ:Bα+1Bβ  is  bounded.

Xiao  characterized the self-maps Φ for which CΦ:BγBβ is bounded for γ,β>0.

Corollary 8.

Fix α,β, and Φ as above.(29)CΦ:FαBβ  is  boundedCΦ:Bα+1Bβ  is  boundedsupwD1-w2βΦw1-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 uH(D). The function u is a multiplier of Bγ into Bβ if Mu(f)=ufBβ for every fBγ. By the Closed Graph Theorem, it follows that Mu:BγBβ is bounded. The collection of all such multipliers is denoted M(Bγ,Bβ). In , Ohno et al. characterized uM(Bγ,Bβ).

Let M(Fα,Bβ) denote the set of analytic functions u for which Mu:FαBβ is bounded. Corollary 9 follows from Corollary 7 and the characterization in  for the case γ=α+1>1.

Corollary 9.

Fix α,β>0 and let uH(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:FαBβ is compact.

Lemma 10.

Fix α>0 and let wD. Define Lw by(31)Lwz=1-w221-w¯zα+2zD. Then LwFα and there is a constant C such that LwFαC for all wD.

Proof.

First fix α=2 and let wD. A particular case of Lemma 5 provides a constant C independent of wD such that(32)1-w21-w¯z2F1C for all wD. Since(33)Lwz=1-w21-w¯z22, Lemma 4 now implies that LwF2 and LwF2C 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, LwFα and LwFαC for all wD.

Fix α,1α<2. By the previous cases LwFα+1 and LwFα+1C for all wD. Lemma 3 shows that LwFα and LwFαC. 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:FαBβ is compact if and only if Wu,ΦfnBβ0 as n for any sequence (fn) in Fα with fnFαC and fn0 uniformly on compact subsets of D as n.

Theorem 12.

Fix α,β>0. Assume that Wu,Φ:FαBβ is bounded. The operator Wu,Φ:FαBβ is compact if and only if(35)limΦw1uw1-w2β1-Φw2α=0,(36)limΦw1uwΦw1-w2β1-Φw2α+1=0.

Proof.

Fix α,β>0 and assume that Wu,Φ:FαBβ is bounded.

First assume the limit conditions (35) and (36). Corollary 7 implies that Wu,Φ:Bα+1Bβ is bounded and it now follows as in  that Wu,Φ:Bα+1Bβ is compact. Suppose that (fn) is a sequence in Fα such that fnFαC for all n and fn0 uniformly on compact subsets. By relation (4), fnBα+1C and thus Wu,ΦfnBβ0 as n. By Lemma 11, Wu,Φ:FαBβ is compact.

Now assume that Wu,Φ:FαBβ is compact. We may assume that Φ=1. Let (wn) be any sequence in D with Φwn1 as n. For n=1,2, define(37)hnz=1-Φwn21-Φwn¯zα+1zD. By Lemma 5, hnFαC for all n. Also hn0 uniformly on compact subsets of D as n. Thus Wu,ΦhnBβ0 as n and(38)supwD1-w2βuwhnΦw+uwhnΦwΦw0 as n. Calculations yield(39)1-wn2βuwn1-Φwn2α+α+1uwnΦwnΦwn¯1-Φwn2α+10 as n.

The argument will first establish that(40)1-wn2βuwn1-Φwn2α0 as n. As in , define the test functions(41)fnz=α+21-Φwn21-Φwn¯zα+1-α+11-Φwn221-Φwn¯zα+2, where zD and n=1,2,. Then fn0 uniformly on compact subsets as n. By Lemmas 10 and 5, there is a constant C with fnFαC for all n. It now follows that(42)supzD1-z2βuzfnΦz+uzfnΦzΦzWu,ΦfnBβ0as  n.In particular,(43)1-wn2βuwnfnΦwn+uwnfnΦwnΦwn0 as n. (44)Since  fnΦwn=11-Φwn2α,since  fnΦwn=0, relation (40) is established. Since (wn) is a generic sequence with Φwn1 as n, relation (35) holds.

To complete the proof note that relations (39) and (40) yield(45)1-wn2βuwnΦwnΦwn1-Φwn2α+10 as n. Since Φwn0,(46)1-wn2βuwnΦwn1-Φwn2α+10 as n. Condition (36) follows and the proof is complete.

Corollary 13.

Fix α,β>0 and assume that Wu,Φ:FαBβ is bounded.(47)Wu,Φ:FαBβ  is  compactWu,Φ:Bα+1Bβ  is  compact.

Proof.

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

Assume that Wu,Φ:Bα+1Bβ is compact. Since the inclusion FαBα+1 is bounded, it follows that Wu,Φ:FαBβ is compact.

Assume that Wu,Φ:FαBβ is compact. By Theorem 12, conditions (35) and (36) hold. These conditions are sufficient to imply that the bounded operator Wu,Φ:Bα+1Bβ is compact .

Let γ,β>0 and assume that CΦ:BγBβ is bounded. In , Xiao provided additional conditions on Φ necessary and sufficient for CΦ:BγBβ to be compact.

Corollary 14.

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

CΦ:FαBβ is compact.

CΦ:Bα+1Bβ is compact.

limΦw11-w2βΦw/1-Φw2α+1=0.

Proof.

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

Since CΦ:FαBβ is bounded, Corollary 8 yields that CΦ:Bα+1Bβ is bounded. Under this hypothesis, Xiao  proved the equivalence of the second and third conditions.

Fix γ,β>0. In , Ohno et al. characterized u for which the bounded operator Mu:BγBβ is compact.

Let uH(D) and let β>0. Recall that u is in the little Bloch space B0β if(48)limz11-z2βuz=0.

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

Corollary 15.

Fix α,β>0 and assume Mu:FαBβ is bounded.

Assume β>α+1. Mu:FαBβ is compact uB0β-α.

Assume βα+1. Mu:FαBβ is compact u0.

Conflicts of Interest

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

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