Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces. (10th May 2021)

• Record Type: Journal Article
• Authors: Hibschweiler, R. A.
• Other Names: Murugusundaramoorthy Gangadharan Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2021(2021)
• Publication Date: 2021-05-10
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2021/9991716
• Languages: English
• ISSNs: 2314-8896

Abstract

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 .