Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces
The operators DCΦ 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 tran...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2021/9991716 |
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| author | R. A. Hibschweiler |
| author_facet | R. A. Hibschweiler |
| author_sort | R. A. Hibschweiler |
| collection | DOAJ |
| description | The operators DCΦ 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. |
| format | Article |
| id | doaj-art-afb852858fe346b0a6776310ef5f0b0e |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-afb852858fe346b0a6776310ef5f0b0e2025-08-20T03:39:05ZengWileyJournal of Function Spaces2314-88962314-88882021-01-01202110.1155/2021/99917169991716Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type SpacesR. A. Hibschweiler0Department of Mathematics and Statistics, University of New Hampshire, 03824 Durham, New Hampshire, USAThe operators DCΦ 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.http://dx.doi.org/10.1155/2021/9991716 |
| spellingShingle | R. A. Hibschweiler Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces Journal of Function Spaces |
| title | Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces |
| title_full | Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces |
| title_fullStr | Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces |
| title_full_unstemmed | Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces |
| title_short | Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces |
| title_sort | products of composition and differentiation between the fractional cauchy spaces and the bloch type spaces |
| url | http://dx.doi.org/10.1155/2021/9991716 |
| work_keys_str_mv | AT rahibschweiler productsofcompositionanddifferentiationbetweenthefractionalcauchyspacesandtheblochtypespaces |