The Gleason's problem for some polyharmonic and hyperbolic harmonic function spaces
type: Item
instance of: scholarly article
title: The Gleason's problem for some polyharmonic and hyperbolic harmonic function spaces
zbMATH Open document ID: 1113.46020
author: Zhangjian Hu
author: Xiaomin Tang
published in: Science in China. Series A
publication date: 2007-01-24T00:00:00Z
review text: Let \(\Omega\subseteq{\mathbb R}^n\) be a bounded convex domain with \({\mathcal C}^2\) boundary and let \(\lambda\) be the defining function of \(\Omega,\) that is, \(\Omega=\{x: \lambda(x)<0\}.\) A \({\mathcal C}^\infty\) function \(f\) is said to be polyharmonic of order \(k\) if \(\Delta^k f\equiv 0\) for the \(k\)-th formal power of the Laplacian \(\Delta.\) For \(0<p,q \leq \infty \) and a normal weight \(\varphi,\) the mixed norm \(\| f\| _{p,q,\varphi}\) is the \(L_q (0,\varepsilon)\) norm of the weighted function \(M_p(f,r){\varphi(r) \over r^{1/q}},\) where \(M_p(f,r)=\| f\| _{L_p(\partial\Omega_r,d\sigma_r)}\) for the surface measure \(\sigma_r\) of the boundary \(\partial\Omega_r=\{x: \lambda(x)=-r \}.\) The mixed norm space \(H^{p,q,\varphi}_k(\Omega)\) consists of all polyharmonic functions \(f\) of order \(k\) for which the mixed norm is finite. The authors prove that the Gleason problem \((\Omega,a,H^{p,q,\varphi}_k)\) is always solvable for any reference point \(a \in \Omega.\) Also, the Gleason problem for the polyharmonic \(\varphi\)-Bloch (little \(\varphi\)-Bloch) space is solvable. Parallel results for the hyperbolic harmonic mixed norm space are also obtained.
reviewed by: Pablo Galindo
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 31C05
zbMATH DE Number: 5118318
zbMATH Keywords: Gleason's problem
zbMATH Keywords: mixed norm space
zbMATH Keywords: Bloch-type space
MaRDI profile type: Publication
full work available at URL: s11425-006-1128-z
OpenAlex ID: W2217744858
cites work: Boundary Behavior of Holomorphic Functions of Several Complex Variables. (MN-11)
cites work: Some properties of fractional integrals. II
cites work: $L^{p}$-behaviour of the integral means of analytic functions
cites work: On the rate of growth of the means \(M_ p\) of holomorphic and pluriharmonic functions on bounded symmetric domains of \({\mathbb{C}}^ n\)
cites work: The Bergman Spaces, The Bloch Space, and Gleason's Problem
cites work: Estimates for the integral mean of harmonic functions on bounded domains in \(\mathbb{R}^ n\)
cites work: Derivatives of harmonic Bergman and Bloch functions on the ball
cites work: Projections for harmonic Bergman spaces and applications
cites work: Decompositions of \(L^p\) and Hardy spaces of polyharmonic functions
cites work: A note on polyharmonic functions
cites work: Q3958740
cites work: The gleason problem in bergman–sobolev spaces
cites work: Boundedness of the Bergman type operators on mixed norm spaces
cites work: Finitely generated ideals in certain function algebras
cites work: Holomorphic Lipschitz Functions in Pseudoconvex Domains
cites work: Gleason's problem in weighted Bergman space on egg domains
cites work: Weighted Harmonic Bloch Spaces and Gleason's Problem
cites work: Boundary behavior of Gleason's problem in hyperbolic harmonic Bergman spaces
cites work: Q4051162
cites work: The Gleason's problem on mixed norm spaces in convex domains
cites work: Q3924489
cites work: Harmonic functions on the real hyperbolic ball II Hardy-Sobolev and Lipschitz spaces
cites work: Q5187597
DOI: 10.1007/S11425-006-1128-Z
description: scientific article; zbMATH DE number 5118318