Zygmund's theorem for harmonic quasiregular mappings

TL;DR

This paper extends Zygmund's theorem to harmonic quasiregular mappings, establishing minimal growth conditions for harmonic functions to belong to Hardy spaces, and advances related theorems in harmonic analysis.

Contribution

It proves Zygmund's theorem for harmonic K-quasiregular mappings, providing the best possible growth condition and linking it to recent progress in harmonic analysis.

Findings

Zygmund's theorem holds for harmonic K-quasiregular mappings.

Established a partial converse showing optimality of the growth condition.

Derived a harmonic analogue of a classical Hardy-Littlewood result.

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.