Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces. (10th May 2021) - Hibschweiler, R. A.
Abstract : The operators D C Φ and C Φ D are defined by D C Φ f = f ∘ Φ ′ and C Φ D f = 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 F α to the Bloch-type spaces B β, where α > 0 and β > 0 . In the case β < 2, the operator D C Φ : F α ⟶ B β is compact ⇔ D C Φ : F α ⟶ B β is bounded ⇔ Φ ′ ∈ B β, Φ Φ ′ ∈ B β and Φ ∞ < 1 . For β < 1, C Φ D : F α ⟶ B β is compact ⇔ C Φ D : F α ⟶ B β is bounded ⇔ Φ ∈ B β and Φ ∞ < 1 .