Decompositions of \(L^p\) and Hardy spaces of polyharmonic functions
type: Item
instance of: scholarly article
title: Decompositions of \(L^p\) and Hardy spaces of polyharmonic functions
zbMATH Open document ID: 0912.46032
author: Miroslav Pavlović
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-02-09T00:00:00Z
review text: Denote by \(B\) the open Euclidean unit ball in \(\mathbb{R}^n\). For \(0< p, q\leq\infty\), \(\alpha\in \mathbb{R}\) and \(k\in\mathbb{N}\) denote by \(H^{p,q,\alpha}_k\) the space of all \(f\in C^\infty(B)\) satisfying \(\Delta^k f=0\) for which \[ \| f\|_{p,q,\alpha}:= \Biggl(\int_B (M_p(f,| x|))^q(1-| x|^2)^{q\alpha- 1}dx\Biggr)^{1/q}< \infty, \] where \[ M_p(f, r):= \Biggl(\int_{\partial B}| f(ry)|^p d\sigma(y)\Biggr)^{1/p},\quad 0\leq r< 1, \] and \(d\sigma\) the normalized surface measure of \(\partial B\). The author shows that for \(k\geq 2\) and \(\alpha> -1\) the map \[ Q: H^{p,q,\alpha}_k\to H^{p,q,\alpha}_1\oplus H^{p,q,\alpha+ 2}_{k- 1},\quad Q(f)= (Pf,\Delta f) \] is an isomorphism, for \(Pf\) being the Poisson integral of \(f\). From this it follows that \(H^{p,q,\alpha}_k\) is isomorphic to the direct sum of the spaces \(H^{p,q,\alpha+ 2j}_1\), \(j= 0,\dots,k- 1\), of harmonic functions. Furthermore, another decomposition result of this type is given which uses the Almansi representation theorem.
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 31B30
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 1115583
zbMATH Keywords: Hardy spaces
zbMATH Keywords: polyharmonic functions
zbMATH Keywords: normalized surface measure
zbMATH Keywords: isomorphism
zbMATH Keywords: Poisson integral
zbMATH Keywords: direct sum
zbMATH Keywords: harmonic functions
zbMATH Keywords: Almansi representation theorem
reviewed by: Reinhold Meise
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5675
OpenAlex ID: W2034231550
cites work: Q3660069
cites work: \(H^p\) spaces of several variables
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cites work: Q4834554
cites work: Inequalities for the Integral Means of Holomorphic Functions and their Derivatives in the Unit Ball of C n
DOI: 10.1006/JMAA.1997.5675
description: scientific article; zbMATH DE number 1115583