On the boundary value of Besov-Bergman spaces
type: Item
instance of: scholarly article
title: On the boundary value of Besov-Bergman spaces
zbMATH Open document ID: 0611.46042
author: Geraldo De Souza
published in: Real Analysis Exchange
publication date: 1985-01-01T00:00:00Z
review text: The author of this paper has previously, in 9 papers, introduced and studied some function spaces, denoted by \(B^ p\). These spaces are: \[ B^ p=\{f| f: [-\pi,\pi)\to {\mathbb{R}},\quad f(t)=\sum c_ nb_ n(t),\quad \sum | c_ n| <\infty \}, \] where \((c_ n)\) are sequences of real numbers and \((b_ n)\) are special atoms, that is functions \(b:[-\pi,\pi)\to {\mathbb{R}}\), defined by \(b(t)=\pi\), or for each interval \(I\subseteq [-\pi,\pi)\), \(b(t)=-| I|^{-1/p}\chi_ R(t)+| I|^{-1/p}\chi_ L(t)\), \(\chi_ E\) is the characteristic function of E. These space \(B^ p\) are endowed with a norm: \(\| f\|_{B^ p}=\inf \sum | c_ n|\), where the infimum is taken over all possible representations of f. In this paper, the author proves an embedding theorem of the space \(B^ p\) in a Besov-Bergman space and that \(B^ p\) is dense in this space. With these results he proves that \(f\in B^ p\), for \(1<p<\infty\), if and only if f belongs to a Besov-Bergman space. Well, this characterization is for a better understanding of the Besov- Bergman spaces, but it is for a better understanding of the \(B^ p\) spaces, too.
reviewed by: A.Donescu
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 3987988
zbMATH Keywords: embedding theorem
zbMATH Keywords: Besov-Bergman space
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3987988