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| Formato: | Preprint |
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2023
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| Acceso en línea: | https://arxiv.org/abs/2310.15452 |
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| _version_ | 1866918099366707200 |
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| author | Chen, Shaolin Huang, Manzi Wang, Xiantao Xiao, Jie |
| author_facet | Chen, Shaolin Huang, Manzi Wang, Xiantao Xiao, Jie |
| contents | One of the celebrated results by Riesz \cite{Rie} is the Riesz conjugate functions theorem for analytic functions in the complex plane $\mathbb{C}$. The study on the Riesz conjugate functions theorem for functions in higher dimensional spaces has attracted much attention. Fefferman and Stein \cite{FS-1972} established the Riesz conjugate functions theorem for the Cauchy-Riemann systems in the upper half real space $\mathbb{R}^{n+1}_{+}$. Astala and Koskela \cite{AS-2} investigated the Riesz conjugate functions theorem for quasiconformal mappings of the unit ball $\mathbf{B}^{n}$ in $\mathbb{R}^n$, and posed an open problem which is as follows: Does there exist a quasiconformal analog for the Riesz theorem on conjugate functions? The purpose of this paper is to develop some methods to study this topic further, in particular, Astala-Koskela's open problem. First, we prove a sharp Riesz conjugate functions theorem for a class of quasiregular mappings of $\mathbf{B}^{n}$ for all $n\geq 2$ which satisfy the so-called Heinz's nonlinear differential inequality. As a direct consequence of this result, we find that the answer to Astala-Koskela's open problem is affirmative for harmonic quasiregular mappings of $\mathbf{B}^{n}$ for all $n\geq 2$. Second, we obtain a sharp Riesz conjugate functions theorem for invariant harmonic $K$-quasiregular mappings of $\mathbf{B}^{n}$ for all $n\geq 2$ which shows that the answer to Astala-Koskela's open problem is affirmative for these mappings. At last, we introduce the family of $κ$-pluriharmonic mappings of the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$, and establish a sharp Riesz conjugate functions theorem for these mappings for all $n\geq 1$. Consequently, we generalize and improve all main results by Liu and Zhu \cite{L-Z}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_15452 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sharp Riesz conjugate functions theorems for quasiregular mappings Chen, Shaolin Huang, Manzi Wang, Xiantao Xiao, Jie Functional Analysis 30H10, 30C62 One of the celebrated results by Riesz \cite{Rie} is the Riesz conjugate functions theorem for analytic functions in the complex plane $\mathbb{C}$. The study on the Riesz conjugate functions theorem for functions in higher dimensional spaces has attracted much attention. Fefferman and Stein \cite{FS-1972} established the Riesz conjugate functions theorem for the Cauchy-Riemann systems in the upper half real space $\mathbb{R}^{n+1}_{+}$. Astala and Koskela \cite{AS-2} investigated the Riesz conjugate functions theorem for quasiconformal mappings of the unit ball $\mathbf{B}^{n}$ in $\mathbb{R}^n$, and posed an open problem which is as follows: Does there exist a quasiconformal analog for the Riesz theorem on conjugate functions? The purpose of this paper is to develop some methods to study this topic further, in particular, Astala-Koskela's open problem. First, we prove a sharp Riesz conjugate functions theorem for a class of quasiregular mappings of $\mathbf{B}^{n}$ for all $n\geq 2$ which satisfy the so-called Heinz's nonlinear differential inequality. As a direct consequence of this result, we find that the answer to Astala-Koskela's open problem is affirmative for harmonic quasiregular mappings of $\mathbf{B}^{n}$ for all $n\geq 2$. Second, we obtain a sharp Riesz conjugate functions theorem for invariant harmonic $K$-quasiregular mappings of $\mathbf{B}^{n}$ for all $n\geq 2$ which shows that the answer to Astala-Koskela's open problem is affirmative for these mappings. At last, we introduce the family of $κ$-pluriharmonic mappings of the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$, and establish a sharp Riesz conjugate functions theorem for these mappings for all $n\geq 1$. Consequently, we generalize and improve all main results by Liu and Zhu \cite{L-Z}. |
| title | Sharp Riesz conjugate functions theorems for quasiregular mappings |
| topic | Functional Analysis 30H10, 30C62 |
| url | https://arxiv.org/abs/2310.15452 |