Zygmund theorem for harmonic quasiregular mappings
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909501426237440 |
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| author | Kalaj, David |
| author_facet | Kalaj, David |
| contents | Let $K\ge 1$. We prove Zygmund theorem for $K-$quasiregular harmonic mappings in the unit disk $\mathbb{D}$ in the complex plane by providing a constant $C(K)$ in the inequality $$\|f\|_{1}\le C(K)(1+\|\mathrm{Re}\,(f)\log^+ |\mathrm{Re}\, f|\|_1),$$ provided that $\mathrm{Im}\,f(0)=0$. Moreover for a quasiregular harmonic mapping $f=(f_1,\dots, f_n)$ defined in the unit ball $\mathbb{B}\subset \mathbb{R}^n$, we prove the asymptotically sharp inequality $$\|f\|_{1}-|f(0)|\le (n-1)K^2(\|f_1\log f_1\|_1- f_1(0)\log f_1(0)),$$ when $K\to 1$, provided that $f_1$ is positive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01814 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zygmund theorem for harmonic quasiregular mappings Kalaj, David Complex Variables Let $K\ge 1$. We prove Zygmund theorem for $K-$quasiregular harmonic mappings in the unit disk $\mathbb{D}$ in the complex plane by providing a constant $C(K)$ in the inequality $$\|f\|_{1}\le C(K)(1+\|\mathrm{Re}\,(f)\log^+ |\mathrm{Re}\, f|\|_1),$$ provided that $\mathrm{Im}\,f(0)=0$. Moreover for a quasiregular harmonic mapping $f=(f_1,\dots, f_n)$ defined in the unit ball $\mathbb{B}\subset \mathbb{R}^n$, we prove the asymptotically sharp inequality $$\|f\|_{1}-|f(0)|\le (n-1)K^2(\|f_1\log f_1\|_1- f_1(0)\log f_1(0)),$$ when $K\to 1$, provided that $f_1$ is positive. |
| title | Zygmund theorem for harmonic quasiregular mappings |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2501.01814 |