Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.13853 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917959444725760 |
|---|---|
| author | Chen, Shaolin Hamada, Hidetaka |
| author_facet | Chen, Shaolin Hamada, Hidetaka |
| contents | Let $φ$, $ψ\in C(\mathbb{T})$, $g\in C(\overline{\mathbb{D}})$, where $\mathbb{D}$ and $\mathbb{T}$ denote the unit disk and the unit circle, respectively. Suppose that $f\in C^{4}(\mathbb{D})$ satisfies the following: (1) the inhomogeneous biharmonic equation $ Δ(Δf(z))=g(z)$ for $z\in\mathbb{D}$, (2) the Dirichlet boundary conditions $\partial_{\overline{z}}f(ζ)=φ(ζ)$ and $f(ζ)=ψ(ζ)$ for $ζ\in\mathbb{T}$. Recently, the authors in [J. Geom. Anal. 29: 2469-2491, 2019] showed that if $ω$ is a majorant with $\limsup_{t\rightarrow0^{+}}\left(ω(t)/t\right)<\infty$, $ψ=0$ and $φ_1 \in\mathscr{L}_ω(\mathbb{T})$, where $φ_1(e^{it})=φ(e^{it})e^{-it}$ for $t\in[0,2π]$, then $f\in\mathscr{L}_ω(\mathbb{D})$. The purpose of this paper is to improve and generalize this result. We not only prove that the condition "$\limsup_{t\rightarrow0^{+}}\left(ω(t)/t\right)<\infty$" is redundant, but also demonstrate that conditions "$ψ=0$" and "$φ_1\in\mathscr{L}_ω(\mathbb{T})$" can be replaced by weaker conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13853 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Remarks on "Schwarz-type lemma, Landau-type theorem, and Lipschitz-type space of solutions to inhomogeneous biharmonic equations" Chen, Shaolin Hamada, Hidetaka Complex Variables 31A30, 35J40 Let $φ$, $ψ\in C(\mathbb{T})$, $g\in C(\overline{\mathbb{D}})$, where $\mathbb{D}$ and $\mathbb{T}$ denote the unit disk and the unit circle, respectively. Suppose that $f\in C^{4}(\mathbb{D})$ satisfies the following: (1) the inhomogeneous biharmonic equation $ Δ(Δf(z))=g(z)$ for $z\in\mathbb{D}$, (2) the Dirichlet boundary conditions $\partial_{\overline{z}}f(ζ)=φ(ζ)$ and $f(ζ)=ψ(ζ)$ for $ζ\in\mathbb{T}$. Recently, the authors in [J. Geom. Anal. 29: 2469-2491, 2019] showed that if $ω$ is a majorant with $\limsup_{t\rightarrow0^{+}}\left(ω(t)/t\right)<\infty$, $ψ=0$ and $φ_1 \in\mathscr{L}_ω(\mathbb{T})$, where $φ_1(e^{it})=φ(e^{it})e^{-it}$ for $t\in[0,2π]$, then $f\in\mathscr{L}_ω(\mathbb{D})$. The purpose of this paper is to improve and generalize this result. We not only prove that the condition "$\limsup_{t\rightarrow0^{+}}\left(ω(t)/t\right)<\infty$" is redundant, but also demonstrate that conditions "$ψ=0$" and "$φ_1\in\mathscr{L}_ω(\mathbb{T})$" can be replaced by weaker conditions. |
| title | Remarks on "Schwarz-type lemma, Landau-type theorem, and Lipschitz-type space of solutions to inhomogeneous biharmonic equations" |
| topic | Complex Variables 31A30, 35J40 |
| url | https://arxiv.org/abs/2503.13853 |