Zygmund's theorem for harmonic quasiregular mappings

TL;DR

The paper addresses extending Zygmund's minimal growth condition from analytic to harmonic $K$-quasiregular mappings $f=u+iv$, proving that $u\in{{\bf h}\log^+{\bf h}}$ (under $u\ge1$ or $u\le-1$ and $v(0)=0$) forces $f\in h^1$ with explicit bounds, and hence $v\in h^1$. The authors derive these results via Green's theorem and a key differential inequality $\Delta\{|f|\} \le K^2 \Delta\{u\log u\}$, and they also establish a partial converse showing the optimality of the growth condition. Building on Riesz and Kolmogorov-type theorems, they derive Riesz-like and Kolmogorov-like consequences for harmonic Hardy spaces and provide a harmonic Hardy-Littlewood-type application. Overall, the work sharpens the understanding of how minimal growth of the real part governs the Hardy-class membership of the harmonic conjugate, within the harmonic quasiregular setting, with precise, asymptotically sharp constants.

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.

Theorems & Definitions (17)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Theorem 1
• Remark 1
• Theorem 2
• Lemma 1
• proof