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

The operators DCΦ and CΦD are defined by DCΦð f Þ = ð f ∘ΦÞ′ and CΦDð f Þ = 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 B , where α > 0 and β > 0. In the case β < 2, the operator DCΦ : Fα ⟶ B is compact ⇔DCΦ : Fα ⟶ B is bounded ⇔Φ′ ∈ Bβ,ΦΦ′ ∈ B and kΦk∞ < 1. For β < 1, CΦD : Fα ⟶ B is compact ⇔CΦD : Fα ⟶ B is bounded ⇔Φ ∈ B and kΦk∞ < 1.


Introduction
Let U = fz ∈ C : |z|<1g and let HðUÞ denote the family of functions analytic on U. Let M denote the Banach space of complex Borel measures on T = fx ∈ C : jxj = 1g, endowed with the total variation norm. For α > 0, the space F α of fractional Cauchy transforms is the family of functions of the form where μ ∈ M. The principal branch of the logarithm is used here. The space F α is a Banach space, with norm where μ varies over all measures in M for which (1) holds. The families F α 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 ⊆ F α for any α > 2 [2]. The reference [2] also includes Mac-Gregor's construction of a function f ∈ S with f ∉ F 2 .
Let β > 0. The Bloch-type space B β is the Banach space of functions analytic in U such that sup z∈U ð1 − jzj 2 Þ β jf ′ ðzÞj The relation (1) implies that F α ⊂ B α+1 , and there is a con- Conditions on Φ are given, necessary and sufficient to imply boundedness or compactness of C Φ D : Products of composition and differentiation on the Bloch space were studied by Ohno in [3]. In [4], Li and Stević studied C Φ D and DC Φ acting between the weighted Bergman spaces and the Bloch-type spaces. In [5], Hibschweiler and Portnoy studied these operators between Bergman and Hardy spaces.

Preliminary Results
Fix α > 0. For fixed z ∈ U and for n = 0, 1, … , the relation (1) yields a constant C depending only on n such that jf ðnÞ ðzÞj [2].
For each w ∈ U, k1/ð1 − wzÞ α k F α = 1 [2]. We follow the convention that C denotes a positive constant, the precise value of which will differ from one appearance to the next. Lemma 1 and Lemma 2 will be used to develop test functions for F α . Proofs appear in [6].
Then, h w ∈ F α , and there is a constant C such that kh w k F α ≤ C for all w ∈ U.
Then, k w ∈ F α , and there is a constant C such that kk w k F α ≤ C for all w ∈ U.

The Operator DC
In [7], Shapiro proved that the condition kΦk ∞ < 1 is necessary for C Φ : 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 B γ when γ < 1 [8]. and for all z ∈ U. It follows that and thus, Φ ∈ B β/2 . Let w ∈ U and define By Lemma 1 and the preliminary results, there is a constant C independent of w such that kg w k F α ≤ C, and thus, for all w ∈ U. Calculations yield g w ′ðΦðwÞÞ = 0 and The substitution z = w in (11) now yields and thus, By the relation (9), Thus, It follows that By Xiao's result [9], Journal of Function Spaces as jΦðwÞj ⟶ 1. Thus, C Φ : B β/2 ⟶ B β/2 is compact [9], and it follows as in [7] that kΦk ∞ < 1. It has been established that the conditions Φ ′ ∈ B β , ΦΦ ′ ∈ B β , and kΦk ∞ < 1 are neces- Since f n ′ ⟶ 0 and f n ′′ ⟶ 0 uniformly on compact subsets as n ⟶ ∞, the argument shows that The remaining implication is clear, and the proof is complete.
First, assume (21) and (22). Let f ∈ F α . By (21) and the introductory remarks in Section 2, A similar argument using (22) yields for all z ∈ U. Thus, Since The argument leading to (16) remains valid for β ≥ 2. Thus, (22) holds. It remains to prove (21). First, note that For w ∈ U, define for z ∈ U. By Lemma 1 and Lemma 2, there is a constant C independent of w such that kH w k F α ≤ C. Thus, for all w ∈ U. An argument using H w ′ðΦðwÞÞ = ðα + 1Þ ΦðwÞ/ð1 − jΦðwÞj 2 Þ α+1 and H w ′′ðΦðwÞÞ = 0 yields The relations (25) and (28) establish relation (21), and the proof is complete. and

Journal of Function Spaces
Proof. First, assume that DC Φ : F α ⟶ B β is bounded and relations (30) and (31) hold. Let ð f n Þ be a bounded sequence in F α such that f n ⟶ 0 uniformly on compact subsets of U. As previously noted, there is a constant C depending only on α such that for n = 1, 2, … and z ∈ U. Relation (31) now implies that given ε > 0, there exists r 0 , 0 < r 0 < 1, such that for all n.
For the converse, assume that DC Φ : F α ⟶ B β is compact. We may assume that kΦk ∞ = 1. Let ðz n Þ be any sequence in U with jΦðz n Þj ⟶ 1 as n ⟶ ∞. For z ∈ U, define By the lemmas above, k f n k F α ≤ C. Also, f n ⟶ 0 uniformly on compact subsets. Therefore, kDC Φ ðf n Þk B β ⟶ 0 and as n ⟶ ∞. Calculations yield f n ′′ðΦðz n ÞÞ = 0 and Substitution into (40) yields as n ⟶ ∞. Since ðz n Þ is a generic sequence with |Φðz n Þ | ⟶1 as n ⟶ ∞, this yields the relation (30). A similar argument using the functions yields the relation (31). The details are omitted.
Proof. By assumption, there is a constant C such that kDC Φ ðf Þk B β ≤ Ck f k F α for all f ∈ F α . Fix γ with 0 < γ < α and let f ∈ F γ . Then, f ∈ F α and k f k F α ≤ k f k F γ [2]. Therefore, DC Φ : F γ ⟶ B β is bounded and Theorem 5 applies.