The Maximum Locus of the Bloch Norm
For a Bloch function f in the unit ball in ℂn, we study the maximal locus of the Bloch norm of f; namely, the set Lf where the Bergman length of the gradient vector field of f attains its maximum. We prove that for n ≥, the set Lf consists of a finite union of real analytic sets with dimensions at m...
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| Format: | Article |
| Language: | English |
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Sciendo
2023-05-01
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| Series: | Moroccan Journal of Pure and Applied Analysis |
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| Online Access: | https://doi.org/10.2478/mjpaa-2023-0019 |
| _version_ | 1797806218465509376 |
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| author | El Hassan Youssfi |
| author_facet | El Hassan Youssfi |
| author_sort | El Hassan Youssfi |
| collection | DOAJ |
| description | For a Bloch function f in the unit ball in ℂn, we study the maximal locus of the Bloch norm of f; namely, the set Lf where the Bergman length of the gradient vector field of f attains its maximum. We prove that for n ≥, the set Lf consists of a finite union of real analytic sets with dimensions at most 2n − 2. This is not the case for n = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set Lf. In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball. |
| first_indexed | 2024-03-13T06:03:57Z |
| format | Article |
| id | doaj.art-2c870c19103f423f8e4af4c074aa9ad9 |
| institution | Directory Open Access Journal |
| issn | 2351-8227 |
| language | English |
| last_indexed | 2024-03-13T06:03:57Z |
| publishDate | 2023-05-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Moroccan Journal of Pure and Applied Analysis |
| spelling | doaj.art-2c870c19103f423f8e4af4c074aa9ad92023-06-12T06:33:05ZengSciendoMoroccan Journal of Pure and Applied Analysis2351-82272023-05-019229130310.2478/mjpaa-2023-0019The Maximum Locus of the Bloch NormEl Hassan Youssfi01I2M UMR 7373, Université d’Aix-Marseille, 39 Rue Joliot-Curie 13453 Marseille Cedex 13, France.For a Bloch function f in the unit ball in ℂn, we study the maximal locus of the Bloch norm of f; namely, the set Lf where the Bergman length of the gradient vector field of f attains its maximum. We prove that for n ≥, the set Lf consists of a finite union of real analytic sets with dimensions at most 2n − 2. This is not the case for n = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set Lf. In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.https://doi.org/10.2478/mjpaa-2023-0019bergman kernelbergman metricbloch functionbloch32a18 |
| spellingShingle | El Hassan Youssfi The Maximum Locus of the Bloch Norm Moroccan Journal of Pure and Applied Analysis bergman kernel bergman metric bloch function bloch 32a18 |
| title | The Maximum Locus of the Bloch Norm |
| title_full | The Maximum Locus of the Bloch Norm |
| title_fullStr | The Maximum Locus of the Bloch Norm |
| title_full_unstemmed | The Maximum Locus of the Bloch Norm |
| title_short | The Maximum Locus of the Bloch Norm |
| title_sort | maximum locus of the bloch norm |
| topic | bergman kernel bergman metric bloch function bloch 32a18 |
| url | https://doi.org/10.2478/mjpaa-2023-0019 |
| work_keys_str_mv | AT elhassanyoussfi themaximumlocusoftheblochnorm AT elhassanyoussfi maximumlocusoftheblochnorm |