Harmonic K-quasiconformal Koebe functions: construction and application to Pavlovic's problem
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911549153607680 |
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| author | Wang, Zhi-Gang Wang, Xiao-Yuan Rasila, Antti Qiu, Jia-Le |
| author_facet | Wang, Zhi-Gang Wang, Xiao-Yuan Rasila, Antti Qiu, Jia-Le |
| contents | We first construct the harmonic K-quasiconformal Koebe functions, filling a long-standing foundational gap in geometric function theory. This construction provides a unified parametric candidate extremal function framework for conformal mappings, quasiconformal mappings, and harmonic mappings, and we formulate related conjectures for the extremal theory of harmonic K-quasiconformal mappings. By combining this construction with Astala and Koskela's Hp-theory for quasiconformal mappings, we establish a sharp result concerning the optimal order of harmonic K-quasiconformal mappings with bounded Schwarzian norm in harmonic Hardy spaces. Motivated by the work of Chuaqui, Hernandez, and Martin [Math. Ann. 367, 1099-1122, 2017], this result gives a partial solution to Pavlovic's 2014 open problem on the embeddings of harmonic quasiconformal mappings into Hardy spaces, and outlines a path toward its complete solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_19852 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Harmonic K-quasiconformal Koebe functions: construction and application to Pavlovic's problem Wang, Zhi-Gang Wang, Xiao-Yuan Rasila, Antti Qiu, Jia-Le Complex Variables 30C55, 30C62, 30H10, 31A05 We first construct the harmonic K-quasiconformal Koebe functions, filling a long-standing foundational gap in geometric function theory. This construction provides a unified parametric candidate extremal function framework for conformal mappings, quasiconformal mappings, and harmonic mappings, and we formulate related conjectures for the extremal theory of harmonic K-quasiconformal mappings. By combining this construction with Astala and Koskela's Hp-theory for quasiconformal mappings, we establish a sharp result concerning the optimal order of harmonic K-quasiconformal mappings with bounded Schwarzian norm in harmonic Hardy spaces. Motivated by the work of Chuaqui, Hernandez, and Martin [Math. Ann. 367, 1099-1122, 2017], this result gives a partial solution to Pavlovic's 2014 open problem on the embeddings of harmonic quasiconformal mappings into Hardy spaces, and outlines a path toward its complete solution. |
| title | Harmonic K-quasiconformal Koebe functions: construction and application to Pavlovic's problem |
| topic | Complex Variables 30C55, 30C62, 30H10, 31A05 |
| url | https://arxiv.org/abs/2405.19852 |