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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.13853 |
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Table of 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.