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

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.


Introduction
Let denote the open unit disc in the complex plane and let ( ) denote the space of functions analytic in . Let denote the Banach space of complex-valued Borel measures on = { : | | = 1}, endowed with the total variation norm. For > 0, the space of fractional Cauchy transforms is the collection of functions of the form where ∈ . The principal branch of the logarithm is used here. The space is a Banach space, with norm given by where varies over all measures in for which (1) holds. The families have been studied extensively [1,2]. Let > 0. The Bloch-type space is the Banach space of functions analytic in such that sup ∈ (1 − | | 2 ) | ( )| < ∞, with norm The integral representation (1) implies that ⊂ +1 and there is a constant depending only on such that for ∈ .
It is known that any univalent ∈ ( ) belongs to for any > 2. MacGregor [3] constructed a univalent function such that ∉ 2 . Let denote the normalized function = ( − (0))/ (0). Then ∉ 2 . Since ∈ , the classical family of schlicht functions, the Distortion Theorem [4] yields ∈ 3 . Let Φ be an analytic self-map of and let ∈ ( ). The weighted composition operator ,Φ is defined for ∈ ( ) by If = 1, then the operator ,Φ reduces to the composition operator Φ defined by Φ ( ) = ∘ Φ. If Φ is the identity function, then ,Φ is the multiplication operator defined by ( ) = . This paper characterizes Φ and for which ,Φ : → is bounded or compact. Corollaries are obtained for the operators Φ and .

Boundedness
We follow the convention that denotes a positive constant, which may vary from one appearance to the next.
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]. ∈ and ‖ ‖ = 1.

Lemma 3.
Fix > 0 and let ∈ ( ). If ∈ +1 , then ∈ and there is a positive constant independent of such that The first statement in Lemma 4 is due to MacGregor [3]. The norm inequality is due to Hibschweiler and Nordgren [6].
Then ∈ and there is a constant independent of such that ‖ ‖ ≤ for all ∈ .
Proof. First assume = 1 and fix generic ∈ . A calculation shows that is in the Hardy space 1 and ‖ ‖ 1 ≤ 1. Since the inclusion 1 ⊂ 1 is bounded, this case is complete.
Fix > 1. Then By the case for = 1 and Lemma 2, is the product of a function in 1 and a function in −1 . By Lemma 4, ∈ and there is a constant independent of such that ‖ ‖ ≤ for all . Finally fix , 0 < < 1. By the previous case, and ‖ ‖ +1 ≤ for all . By Lemma 3, ∈ and ‖ ‖ ≤ . 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 +1 . Theorem 6. Fix > 0 and > 0. Let ∈ ( ) and let Φ be an analytic self-map of . Assume that ,Φ : → is bounded. Then Proof. Fix , , , and Φ as described. By assumption there is a constant independent of such that for all ∈ . The argument will first establish that 1 < ∞. Let ∈ and define By Lemmas 2 and 5, there is a constant such that ‖ ‖ ≤ for all ∈ . Therefore for all ∈ . Since it follows that In particular, (17) yields ∈ .
To obtain the second condition in the theorem, let ∈ and define Journal of Complex Analysis 3 By Lemma 5, there is a constant independent of such that ‖ℎ ‖ ≤ . Relation (13) yields it follows that for all ∈ . First consider ∈ with 1/2 < |Φ( )|. By the triangle inequality, relation (21) yields for such . By relation (17), it follows that Finally consider ∈ with |Φ( )| ≤ 1/2. Let ( ) = in relation (13). Thus Φ ∈ and for all ∈ . Therefore for all ∈ . Therefore Relations (23) and (26) yield and the proof is complete.
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 , > 0 and let ∈ ( ). The function is a multiplier of into if ( ) = ∈ for every ∈ . By the Closed Graph Theorem, it follows that : → is bounded. The collection of all such multipliers is denoted ( , ). In [5], Ohno et al. characterized ∈ ( , ). Let ( , ) denote the set of analytic functions for which : → is bounded. Corollary 9 follows from Corollary 7 and the characterization in [5] for the case = + 1 > 1.

Compactness
A characterization is given for functions , Φ for which ,Φ : → is compact.
Lemma 10. Fix > 0 and let ∈ . Define by Then ∈ and there is a constant such that ‖ ‖ ≤ for all ∈ .
Proof. First fix = 2 and let ∈ . A particular case of Lemma 5 provides a constant independent of ∈ such that for all ∈ . Since Lemma 4 now implies that ∈ 2 and ‖ ‖ 2 ≤ for all .
When > 2, By the previous case and Lemma 2, is the product of a function in 2 and a function in −2 . By Lemma 4, ∈ and ‖ ‖ ≤ for all ∈ .
Lemma 11 is the standard sequential criterion for compactness.
Let , > 0 and assume that Φ : → is bounded. In [7], Xiao provided additional conditions on Φ necessary and sufficient for Φ : → to be compact.

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