# Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces

Authors: R. A. Hibschweiler
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The operatorsDCΦ and CΦD are defined by DCΦf=f∘Φ′ and CΦDf=f′∘Φ where Φ is an analytic self-map of the unit disc and f is analytic in the disc. A characterization is provided for boundedness and compactness of the products of composition and differentiation from the spaces of fractional Cauchy transforms Fα to the Bloch-type spaces Bβ, where α>0 and β>0. In the case β<2, the operator DCΦ:Fα⟶Bβ is compact ⇔DCΦ:Fα⟶Bβ is bounded ⇔Φ′∈Bβ,ΦΦ′∈Bβ and Φ∞<1. For β<1, CΦD:Fα⟶Bβ is compact ⇔CΦD:Fα⟶Bβ is bounded ⇔Φ∈Bβ and Φ∞<1.


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## Body

## 1. Introduction

LetU=z∈C:∣z∣<1 and let HU denote the family of functions analytic on U. Let M denote the Banach space of complex Borel measures on T=x∈C:x=1, endowed with the total variation norm. For α>0, the space Fα of fractional Cauchy transforms is the family of functions of the form
(1)fz=∫T11−x¯zαdμxz<1,where μ∈M. The principal branch of the logarithm is used here. The space Fα is a Banach space, with norm
(2)fFα=infμ,where μ varies over all measures in M for which (1) holds. The families Fα have been studied extensively [1, 2]. Interest in these spaces was first established in connection with the classical family S of normalized univalent functions. It is known that S⊆Fα for any α>2 [2]. The reference [2] also includes MacGregor’s construction of a function f∈S with f∉F2.Letβ>0. The Bloch-type space Bβ is the Banach space of functions analytic in U such that supz∈U1−z2βf′z<∞, with norm
(3)fBβ=f0+supz∈U1−z2βf′z.The relation (1) implies that Fα⊂Bα+1, and there is a constant C depending only on α such that fBα+1≤CfFα for all f∈Fα.LetΦ be an analytic self-map of U. The composition operator CΦ is defined by CΦf=f∘Φ for f∈HU. The differentiation operator D is defined by Df=f′. In this paper, the products CΦDf=f′∘Φ and DCΦf=Φ′f′∘Φ are studied. Conditions on Φ are given, necessary and sufficient to imply boundedness or compactness of CΦD:Fα⟶Bβ and DCΦ:Fα⟶Bβ.Products of composition and differentiation on the Bloch space were studied by Ohno in [3]. In [4], Li and Stević studied CΦD and DCΦ acting between the weighted Bergman spaces and the Bloch-type spaces. In [5], Hibschweiler and Portnoy studied these operators between Bergman and Hardy spaces.


## 2. Preliminary Results

Fixα>0. For fixed z∈U and for n=0,1,…, the relation (1) yields a constant C depending only on n such that fnz≤CfFα/1−z2α+n [2].For eachw∈U, 1/1−w¯zαFα=1 [2].We follow the convention thatC denotes a positive constant, the precise value of which will differ from one appearance to the next.Lemma1 and Lemma 2 will be used to develop test functions for Fα. Proofs appear in [6].Lemma 1.
Fixα>0. For w∈U, define
(4)hwz=1−w21−w¯zα+1z∈U.
Then,hw∈Fα, and there is a constant C such that hwFα≤C for all w∈U.Lemma 2.
Fixα>0. For w∈U, define
(5)kwz=1−w221−w¯zα+2z∈U.
Then,kw∈Fα, and there is a constant C such that kwFα≤C for all w∈U.


## 3. The OperatorDCΦ:Fα⟶Bβ

In [7], Shapiro proved that the condition Φ∞<1 is necessary for CΦ:X⟶X to be compact, for Banach spaces X obeying boundary regularity and Möbius invariance. In particular, Shapiro’s result applies to the Lipschitz spaces and thus, to the space Bγ when γ<1 [8].Theorem 3.
Fixα>0 and 0<β<2. Let Φ be an analytic self-map of U.
(6)DCΦ:Fα⟶Bβisbounded⇔Φ′∈Bβ,ΦΦ′∈BβandΦ∞<1⇔DCΦ:Fα⟶Bβiscompact.Proof.
First, assume thatDCΦ:Fα⟶Bβ is bounded, that is, there is a constant C such that DCΦfBβ≤CfFα for all f∈Fα. It is clear that Φ′=DCΦz∈Bβ and ΦΦ′=DCΦz2/2∈Bβ. Thus,
(7)1−z2βΦ′′z≤C,and
(8)1−z2βΦzΦ′′z+Φ′z2≤C,for all z∈U. It follows that
(9)supz∈U1−z2βΦ′z2<∞,and thus, Φ∈Bβ/2.
Letw∈U and define
(10)gwz=α+11−Φw¯zα−α1−Φw21−Φw¯zα+1z∈U.
By Lemma1 and the preliminary results, there is a constant C independent of w such that gwFα≤C, and thus, DCΦgwBβ=gw′∘ΦΦ′Bβ≤C. It follows that
(11)supz∈U1−z2βgw′′ΦzΦ′z2+gw′ΦzΦ′′z≤C,for all w∈U. Calculations yield gw′Φw=0 and
(12)gw′′Φw=−αα+1Φw¯21−Φw2α+2.
The substitutionz=w in (11) now yields
(13)supw∈U1−w2βαα+1Φw2Φ′w21−Φw2α+2≤C,and thus,
(14)supΦw>1/21−w2βΦ′w21−Φw2α+2<∞.
By the relation (9),
(15)supΦz≤1/21−w2βΦ′w21−Φw2α+2<∞.
Thus,(16)C=supw∈U1−w2βΦ′w21−Φw2α+2<∞.
It follows that(17)supw∈U1−w2β/2Φ′w1−Φw2β/2≤supw∈U1−w2β/2Φ′w1−Φw2α+2/2<∞.
By Xiao’s result [9], CΦ:Bβ/2⟶Bβ/2 is bounded. Furthermore, (16) yields
(18)1−w2β/2Φ′w1−Φw2β/2≤C1−Φw2α−β+2/2⟶0,as Φw⟶1. Thus, CΦ:Bβ/2⟶Bβ/2 is compact [9], and it follows as in [7] that Φ∞<1. It has been established that the conditions Φ′∈Bβ,ΦΦ′∈Bβ, and Φ∞<1 are necessary if DCΦ:Fα⟶Bβ is bounded.
Next, assume thatΦ′∈Bβ, ΦΦ′∈Bβ, and Φ∞<1. To show that DCΦ:Fα⟶Bβ is compact, let fn be a bounded sequence in Fα with fn⟶0 uniformly on compact subsets of U as n⟶∞. It is enough to prove that DCΦfnBβ⟶0 as n⟶∞. First, note that fn′Φ0Φ′0⟶0 as n⟶∞. For z∈U, (9) yields
(19)1−z2βDCΦfn′z=1−z2βfn′′ΦzΦ′z2+fn′ΦzΦ′′z≤Cmaxw≤Φ∞fn′′w+Φ′Bβmaxw≤Φ∞fn′w.
Sincefn′⟶0 and fn′′⟶0 uniformly on compact subsets as n⟶∞, the argument shows that supz∈U1−z2βDCΦfn′z⟶0 as n⟶∞. Thus, DCΦfnBβ⟶0 as n⟶∞, and DCΦ:Fα⟶Bβ is compact, as required.
The remaining implication is clear, and the proof is complete.Theorem 4.
Fixα>0 and β≥2. Let Φ be an analytic self-map of U. Then,
(20)DCΦ:Fα⟶Bβisbounded⇔(21)supz∈U1−z2βΦ′′z1−Φz2α+1<∞,and
(22)supz∈U1−z2βΦ′z21−Φz2α+2<∞.Proof.
Fixα,β and Φ as described.
First, assume (21) and (22). Let f∈Fα. By (21) and the introductory remarks in Section 2,
(23)1−z2βf′ΦzΦ′′z≤1−z2βΦ′′zCfFα1−Φz2α+1≤CfFα.
A similar argument using (22) yields
(24)1−z2βf′′ΦzΦ′z2≤CfFα,for all z∈U. Thus, supz∈U1−z2βDCΦf′z≤CfFα. Since DCΦf0≤CfFα, it now follows that DCΦfBβ≤CfFα, as required.
For the converse, assume thatDCΦfBβ≤CfFα for a constant C independent of f∈Fα. In particular, Φ′∈Bβ.
The argument leading to (16) remains valid for β≥2. Thus, (22) holds. It remains to prove (21). First, note that
(25)supΦw≤1/21−w2βΦ′′w1−Φw2α+1≤43α+1Φ′Bβ<∞.
Forw∈U, define
(26)Hwz=α+31−Φw21−Φw¯zα+1−α+11−Φw221−Φw¯zα+2,for z∈U. By Lemma 1 and Lemma 2, there is a constant C independent of w such that HwFα≤C. Thus, DCΦHwBβ≤C for all w. It follows that
(27)supz∈U1−z2βHw′ΦzΦ′′z+Φ′z2Hw′′Φz<C,for all w∈U. An argument using Hw′Φw=α+1Φw¯/1−Φw2α+1 and Hw′′Φw=0 yields
(28)sup1/2<Φw1−w2βΦ′′w1−Φw2α+1<∞.
The relations (25) and (28) establish relation (21), and the proof is complete.Theorem 5.
Fixα>0 and assume β≥2. Let Φ be a self-map of U for which DCΦ:Fα⟶Bβ is bounded.
(29)DCΦ:Fα⟶Bβiscompact⇔(30)limΦz⟶11−z2βΦ′′z1−Φz2α+1=0,and
(31)lim∣Φz∣⟶11−z2βΦ′z21−Φz2α+2=0.Proof.
First, assume thatDCΦ:Fα⟶Bβ is bounded and relations (30) and (31) hold. Let fn be a bounded sequence in Fα such that fn⟶0 uniformly on compact subsets of U. As previously noted, there is a constant C depending only on α such that
(32)1−z2βfn′′ΦzΦ′z2≤C1−z2βΦ′z21−Φz2α+2,for n=1,2,… and z∈U. Relation (31) now implies that given ε>0, there exists r0,0<r0<1, such that
(33)supΦz>r01−z2βfn′′ΦzΦ′z2<ε,for all n.
SinceDCΦ:Fα⟶Bβ is bounded, relation (9) holds, and thus,
(34)1−z2βfn′′ΦzΦ′z2<Cfn′′Φz,for all z∈U. Since fn′′⟶0 uniformly on w:w≤r0, there exists N>0 such that
(35)supΦz≤r01−z2βfn′′ΦzΦ′z2<ε,for all n>N. The relations (33) and (35) yield
(36)supz∈U1−z2βfn′′ΦzΦ′z2<ε,for n>N.
A similar argument usingΦ′∈Bβ and (30) yields N1>0 such that
(37)supz∈U1−z2βfn′ΦzΦ′′z<ε,for n>N1. The relations (36) and (37) yield
(38)supz∈U1−z2βDCΦfn′z⟶0,as n⟶∞.
SinceDCΦfn0⟶0 as n⟶∞, the argument yields DCΦfnBβ⟶0 as n⟶∞ for any sequence fn as described, and therefore, DCΦ:Fα⟶Bβ is compact.
For the converse, assume thatDCΦ:Fα⟶Bβ is compact. We may assume that Φ∞=1. Let zn be any sequence in U with Φzn⟶1 as n⟶∞. For z∈U, define
(39)fnz=α+31−Φzn21−Φzn¯zα+1−α+11−Φzn221−Φzn¯zα+2.
By the lemmas above,fnFα≤C. Also, fn⟶0 uniformly on compact subsets. Therefore, DCΦfnBβ⟶0 and
(40)supz∈U1−z2βfn′ΦzΦ′′z+fn′′ΦzΦ′z2⟶0,as n⟶∞. Calculations yield fn′′Φzn=0 and
(41)fn′Φzn=α+1Φzn¯1−Φzn2α+1.
Substitution into (40) yields
(42)1−zn2βΦznΦ′′zn1−Φzn2α+1⟶0,as n⟶∞. Since zn is a generic sequence with ∣Φzn∣⟶1 as n⟶∞, this yields the relation (30).
A similar argument using the functions(43)gnz=α+21−Φzn21−Φzn¯zα+1−α+11−Φzn221−Φzn¯zα+2yields the relation (31). The details are omitted.Theorem3 implies that if DCΦ:Fα⟶Bβ is bounded for fixed α,β with β<2, then DCΦ:Fγ⟶Bβ is compact for all γ>0. The next corollary gives a related result when β≥2.Corollary 6.
Fixα>0 and β≥2. Let Φ be a self-map of U and assume that DCΦ:Fα⟶Bβ is bounded. Then, DCΦ:Fγ⟶Bβ is compact for any γ,0<γ<α.Proof.
By assumption, there is a constantC such that DCΦfBβ≤CfFα for all f∈Fα. Fix γ with 0<γ<α and let f∈Fγ. Then, f∈Fα and fFα≤fFγ [2]. Therefore, DCΦ:Fγ⟶Bβ is bounded and Theorem 5 applies.
SinceDCΦ:Fα⟶Bβ is bounded, (21) yields
(44)1−z2βΦ′′z1−Φz2γ+1≤C1−Φz2α−γ,and therefore,
(45)limΦz⟶11−z2βΦ′′z1−Φz2γ+1=0.
A similar argument using (22) yields
(46)limΦz⟶11−z2βΦ′z21−Φz2γ+2=0.
Theorem5 now yields DCΦ:Fγ⟶Bβ is compact.


## 4. The OperatorCΦD

In this section, characterizations are given for self-mapsΦ for which CΦD:Fα⟶Bβ is bounded or compact. The proofs are similar to those in Section 3, so details are kept to a minimum.Theorem 7.
Fixα>0 and 0<β<1.
(47)CΦD:Fα⟶Bβisbounded⇔Φ∈BβandΦ∞<1⇔CΦD:Fα⟶Bβiscompact.Proof.
First, assume that there is a constantC independent of f∈Fα such that CΦDfBβ≤CfFα. In particular, Φ∈Bβ. For w∈U, define
(48)gwz=11−Φw¯zαz∈U.
There is a constantC independent of w∈U such that gwFα≤C, and it follows that
(49)supz∈U1−z2βgw′′ΦzΦ′z<C,for all w∈U. The substitution z=w yields
(50)1−w2βαα+1Φw2Φ′w1−Φw2α+2<C,for all w∈U. Therefore,
(51)sup1/2<Φw1−w2βΦ′w1−Φw2α+2<∞.
SinceΦ∈Bβ,
(52)supΦw≤1/21−w2βΦ′w1−Φw2α+2<∞.
It follows that(53)supw∈U1−w2βΦ′w1−Φw2α+2<∞and therefore
(54)supw∈U1−w2βΦ′w1−Φw2β<∞.
By [9], CΦ:Bβ⟶Bβ is bounded. A further argument as in the proof of Theorem 3 yields that CΦ:Bβ⟶Bβ is compact. Since β<1, Shapiro’s result [7] applies and yields Φ∞<1. Thus, the conditions Φ∈Bβ and Φ∞<1 are necessary in order for CΦD:Fα⟶Bβ to be bounded.
Next, assumeΦ∈Bβ and Φ∞<1. Let fn be a bounded sequence in Fα with fn⟶0 uniformly on compact subsets of U. First, note that fn′Φ0⟶0 as n⟶∞. For z∈U,
(55)1−z2βfn′∘Φ′z≤ΦBβmaxw≤Φ∞fn′′w.
Sincefn′′⟶0 uniformly on compact subsets, the argument yields CΦDfnBβ⟶0 and CΦD:Fα⟶Bβ is compact.
The remaining implication is trivial, and the proof is complete.Theorem 8.
Fixα>0 and β≥1. Let Φ be a self-map of U.
(56)CΦD:Fα⟶Bβisbounded⇔supz∈U1−z2βΦ′z1−Φz2α+2<∞.Proof.
First, assume that the supremum is finite.
Letf∈Fα. By previous remarks, f′Φ0≤CfFα. By an argument as in the proof of Theorem 4,
(57)1−z2βf′∘Φ′z=1−z2βf′′ΦzΦ′z≤1−z2βCfFαΦ′z1−Φz2α+2≤CfFα,and thus, CΦDfBβ≤CfFα as required.
To complete the proof, assume thatCΦDfBβ≤CfFα for a constant C independent of f. The argument leading to (53) remains valid for β≥1. This proves the opposite implication, and the proof is complete.Theorem 9.
Fixα>0 and β≥1. Let Φ be a self-map of U and assume that CΦD:Fα⟶Bβ is bounded.
(58)CΦD:Fα⟶Bβiscompact⇔limΦz⟶11−z2βΦ′z1−Φz2α+2=0.Proof.
First, assume thatCΦD:Fα⟶Bβ is bounded and the limit condition holds. Let fn be a bounded sequence in Fα with fn⟶0 uniformly on compact subsets as n⟶∞. Clearly, fn′Φ0⟶0 as n⟶∞. As in previous arguments,
(59)1−z2βfn′′ΦzΦ′z≤C1−z2βΦ′z1−Φz2α+2,for all z∈U. The hypothesis now implies that, given ε>0, there exists r0,0<r0<1, such that
(60)supΦz>r01−z2βfn′′ΦzΦ′z<ε,for all n. Since Φ∈Bβ and since fn′′⟶0 uniformly on compact subsets,
(61)supΦz≤r01−z2βfn′′ΦzΦ′z⟶0,as n⟶∞. By (60) and (61),
(62)supz∈U1−z2βfn′∘Φ′z⟶0,as n⟶∞. The argument yields fn′∘ΦBβ⟶0 as n⟶∞ for any sequence fn as described above. Thus, CΦD:Fα⟶Bβ is compact.
Now, assume thatCΦD:Fα⟶Bβ is compact. We may assume that ∥Φ∥∞=1. Let zn be any sequence in U with ∣Φzn∣⟶1 as n⟶∞. For n=1,2,⋯, define
(63)fnz=1−Φzn21−Φzn¯zα+1,for z∈U. By Lemma 1, fnFα≤C for all n. Also, fn⟶0 uniformly on compact subsets. Therefore, CΦDfnBβ⟶0 as n⟶∞. Given ε>0, there exists N>0 such that
(64)supz∈U1−z2βfn′′ΦzΦ′z<ε,for all n>N. In particular, 1−zn2βfn′′ΦznΦ′zn<ε for n>N. Calculations yield
(65)1−zn2βα+1α+2Φzn2Φ′zn1−Φzn2α+2<ε,for n>N. Since zn is a generic sequence with Φzn⟶1, it follows that
(66)limΦz⟶11−z2βΦ′z1−Φz2α+2=0.
The proof is complete.Assume thatCΦD:Fα⟶Bβ is bounded for fixed α>0 and β<1. By Theorem 7, Φ∈Bβ and Φ∞<1. It follows that CΦD:Fγ⟶Bβ is compact for any γ>0. Corollary 10 gives a related result in the case β≥1.Corollary 10.
Fixα>0,β≥1 and assume that CΦD:Fα⟶Bβ is bounded. Then, CΦD:Fγ⟶Bβ is compact for any γ,0<γ<α.Proof.
Fix0<γ<α and let f∈Fγ. Then, f∈Fα and fFα≤fFγ [2]. Therefore, CΦD:Fγ⟶Bβ is bounded and Theorem 9 applies.
By Theorem8, there is a constant C with
(67)1−z2βΦ′z1−Φz2α+2≤C,for all z∈U. Therefore,
(68)1−z2βΦ′z1−Φz2γ+2≤C1−Φz2α−γ⟶0,as∣Φz∣⟶1. By Theorem 9, CΦD:Fγ⟶Bβ is compact.



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