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arXiv · 2501.01627

Zygmund's theorem for harmonic quasiregular mappings

Suman Das·Jie Huang·Antti Rasila
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Abstract

Given an analytic function $f=u+iv$ in the unit disk $\mathbb{D}$, Zygmund's theorem gives the minimal growth restriction on $u$ which ensures that $v$ is in the Hardy space $h^1$. This need not be true if $f$ is a complex-valued harmonic function. However, we prove that Zygmund's theorem holds if $f$ is a harmonic $K$-quasiregular mapping in $\ID$. Our work makes further progress on the recent Riesz-type theorem of Liu and Zhu (Adv. Math., 2023), and the Kolmogorov-type theorem of Kalaj (J. Math. Anal. Appl., 2025), for harmonic quasiregular mappings. We also obtain a partial converse, thus showing that the proposed growth condition is the best possible. Furthermore, as an application of the classical conjugate function theorems, we establish a harmonic analogue of a well-known result of Hardy and Littlewood.

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Suman Das, Jie Huang, Antti Rasila. 2025-01-03. Zygmund's theorem for harmonic quasiregular mappings. https://arxiv.org/abs/2501.01627

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