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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2312.15879 |
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| _version_ | 1866914073342377984 |
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| author | Zhong, Deguang Cai, Fangming Wei, Dongping |
| author_facet | Zhong, Deguang Cai, Fangming Wei, Dongping |
| contents | Suppose that $1<p\leq\infty$ and $φ\in L^{p}(\mathbb{B}^{n},\mathbb{R}^{n}).$ In this note, we use Hölder inequality and some basic properties of hypergeometric functions to establish the sharp constant $C_{p}$ and function $C_{p}(x)$ in the following inequalities $$|u(x)|\leq \frac{C_{p}}{(1-|x|^{2})^{(n-1)/p}}\cdot||φ||_{L^{p}}$$
and
$$|u(x)|\leq \frac{C_{p}(x)}{(1-|x|^{2})^{(n-1)/p}}\cdot||φ||_{L^{p}},$$
where $u$ are those mapping from the unit ball $\mathbb{B}^{n}$ into $\mathbb{R}^{n}$ admitting general Poisson representations. The obtained results generalize and extend some known results from harmonic mappings (\cite[Proposition 6.16]{ABR92} and \cite[Theorems 1.1 and 1.2]{DM12}) and hyperbolic harmonic mappings (\cite[Theorems 1.1 and 1.2]{CJLK20}). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_15879 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal estimates for mappings admitting general Poisson representations in the unit ball Zhong, Deguang Cai, Fangming Wei, Dongping Complex Variables 31B05, 31B10, 42B30 Suppose that $1<p\leq\infty$ and $φ\in L^{p}(\mathbb{B}^{n},\mathbb{R}^{n}).$ In this note, we use Hölder inequality and some basic properties of hypergeometric functions to establish the sharp constant $C_{p}$ and function $C_{p}(x)$ in the following inequalities $$|u(x)|\leq \frac{C_{p}}{(1-|x|^{2})^{(n-1)/p}}\cdot||φ||_{L^{p}}$$ and $$|u(x)|\leq \frac{C_{p}(x)}{(1-|x|^{2})^{(n-1)/p}}\cdot||φ||_{L^{p}},$$ where $u$ are those mapping from the unit ball $\mathbb{B}^{n}$ into $\mathbb{R}^{n}$ admitting general Poisson representations. The obtained results generalize and extend some known results from harmonic mappings (\cite[Proposition 6.16]{ABR92} and \cite[Theorems 1.1 and 1.2]{DM12}) and hyperbolic harmonic mappings (\cite[Theorems 1.1 and 1.2]{CJLK20}). |
| title | Optimal estimates for mappings admitting general Poisson representations in the unit ball |
| topic | Complex Variables 31B05, 31B10, 42B30 |
| url | https://arxiv.org/abs/2312.15879 |