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arXiv · 2405.19852

Harmonic K-quasiconformal Koebe functions: construction and application to Pavlovic's problem

Zhi-Gang Wang·Xiao-Yuan Wang·Antti Rasila·Jia-Le Qiu
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Abstract

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.

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Zhi-Gang Wang, Xiao-Yuan Wang, Antti Rasila, Jia-Le Qiu. 2024-05-30. Harmonic K-quasiconformal Koebe functions: construction and application to Pavlovic's problem. https://arxiv.org/abs/2405.19852

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