Fourier multiplier theorem for atomic Hardy spaces on unbounded Vilenkin groups
type: Item
instance of: scholarly article
title: Fourier multiplier theorem for atomic Hardy spaces on unbounded Vilenkin groups
zbMATH Open document ID: 1183.43003
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-12-22T00:00:00Z
review text: Let \(G\) be a locally compact Abelian group that contains a strictly decreasing sequence of open compact subgroups \((G_n)_{n=-\infty}^\infty\) such that \(\bigcup_{n=-\infty}^\infty G_n=G\), and \(\bigcap_{n=-\infty}^\infty G_n=\{0\}\). \(\Gamma\) denotes the dual group of \(G\). It is the union of the increasing sequence of its subgroups \(\Gamma_n= \{\gamma\in\Gamma: \gamma(x)=1\), \(\forall x\in G_n\}\). A complex function \(a\) is called an atom on \(G\) if \(\text{supp}(a)\subset y+G_n\), \(\|a\|_\infty\leq 1/\mu(G_n)\), \(\int_G {a(x)\,dx=0}\). The atomic Hardy space \(H^1\) consists of integrable functions \(f\) which can be represented as \(f=\sum_{j=1}^\infty \lambda_j a_j\), where each \(a_j\) is an atom and \(\sum_{j=1}^\infty|\lambda_j|<+\infty\). The norm in \(H^1\) is given by \[ \|f\|_{H^1}=\inf\sum_{j=1}^\infty |\lambda_j|, \] where the infimum is taken over all such decompositions of \(f\). The authors provide some characterization of atomic Hardy spaces on unbounded locally compact Vilenkin groups by means of a modified maximal function. The results are of independent interest, and make possible to reduce the investigation to the spaces defined by these maximal functions. The main results of the article are certain sufficient conditions on the kernel of the multiplier operator. Let us cite one of the results obtained. Theorem. Let \(\varphi\in L^\infty (\Gamma)\) and \(\sup_N\int_{G_N^c}|(\varphi-\varphi_{N+1})^\vee (y)|\,dy=O(1)\), where \(\varphi_{N+1}=\varphi{\mathbf{1}}_{\Gamma_{N+1}}\) and \(\vee\) denotes the inverse Fourier transform. Then, \(\varphi\) is a multiplier in \(H^1\), i.e., \(\varphi\in m(H^1)\). The article also contains an interesting comparison with preceding known results. However, when we have some sufficient condition for a multiplier, the question arises of how sharp it is, i.e., about some necessary conditions. It seems to be interesting and is a good question for further investigations. A bibliography reflects several …
Mathematics Subject Classification ID: 43A70
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 43A17
zbMATH DE Number: 5651214
zbMATH Keywords: Vilenkin group
zbMATH Keywords: Hardy space
zbMATH Keywords: multiplier
author: Muharem Avdispahić
author: N. Memić
reviewed by: Alexander Tovstolis
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.09.037
OpenAlex ID: W2075920080
Wikidata QID: Q57621857
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DOI: 10.1016/J.JMAA.2009.09.037
description: scientific article; zbMATH DE number 5651214