2025/03/18 by Chen, Shaolin, Hamada, Hidetaka
#31A30 #35J40 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.13853
Let φ, ψ∈ C(\mathbbT), g∈ C(\mathbbD), where \mathbbD and \mathbbT denote the unit disk and the unit circle, respectively. Suppose that f∈ C4(\mathbbD) satisfies the following: (1) the inhomogeneous biharmonic equation Δ(Δf(z))=g(z) for z∈\mathbbD, (2) the Dirichlet boundary conditions ∂_zf(ζ)=φ(ζ) and f(ζ)=ψ(ζ) for ζ∈\mathbbT. Recently, the authors in [J. Geom. Anal. 29: 2469-2491, 2019] showed that if ω is a majorant with \limsupt→0+(ω(t)/t)<∞, ψ=0 and φ1 ∈\mathscrLω(\mathbbT), where φ1(eit)=φ(eit)e-it for t∈[0,2π], then f∈\mathscrLω(\mathbbD). The purpose of this paper is to improve and generalize this result. We not only prove that the condition "\limsupt→0+(ω(t)/t)<∞" is redundant, but also demonstrate that conditions "ψ=0" and "φ1∈\mathscrLω(\mathbbT)" can be replaced by weaker conditions.