Lacunarity for amalgams
type: Item
instance of: scholarly article
title: Lacunarity for amalgams
zbMATH Open document ID: 0628.43007
DOI: 10.1216/RMJ-1987-17-2-277
author: John J. F. Fournier
published in: Rocky Mountain Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Given a measurable function on \({\mathbb{R}}\) and \(p,q\in [0,1]\) let \(\| f\|_{p,q}=[\sum_{n\in {\mathbb{Z}}}((\int_{I_ n}| f|^ p)^{1/p})^ q]^{1/q}\) where \(I_ n=[n,n+1)\). The space \((L^ p,\ell^ q)\) of functions for which \(\| f\|_{p,q}<\infty\) is called an amalgam. One has \((L^ p,\ell^ p)=L^ p\). The Fourier transform \(\hat f\) of an \(f\in (L^ p,\ell^ q)\) is viewed as a distribution. A measurable set F in a noncompact locally compact abelian group is called a \(\Lambda(p)\)-set \((1<p<\infty)\) iff \(f\in L^ p(G)\) whenever \(f\in L^ 1(G)\) and supp\(\hat f\subset F\). If f satisfies the last condition then it is said to be an F-function. \(\Lambda(p)\)-sets have been treated mostly in discrete abelian groups. Let now E be a subset of \({\mathbb{Z}}\) and \(F=E+[0,1)\). It is proved that ``(i) E is a \(\Lambda(p)\)-set in \({\mathbb{Z}}''\) and ``(ii) F is a \(\Lambda(p)\)-set in \({\mathbb{R}}''\) are equivalent. The notion of an amalgam is not involved in this statement but it is used in its proof. Namely, it is shown that each of the conditions (i) and (ii) is equivalent to (iii): For each \(q\in [1,\infty]\), if an F-function f belongs to the amalgam \((L^ 1,\ell^ q)( {\mathbb{R}})\) then it belongs to the amalgam \((L^ p,\ell^ q)( {\mathbb{R}})\). Since \(L^ r( {\mathbb{R}})\subset (L^ 1,\ell^ r)( {\mathbb{R}})\) (r\(\geq 1)\) the statement (iii) implies that any F-function belonging to \(L^ r( {\mathbb{R}})\) with \(r<p\) belongs also to \(L^ p( {\mathbb{R}})\)- this being a new result if \(p>r>2\). The assumption supp\(\hat f\subset F\) does not say anything about the global size of f, what means that the assertion in (iii) cannot be strengthened to \(f\in (L^ p,\ell^{q'})({\mathbb{R}})\) for any \(q'<q\). The concept of amalgam can be transferred to LCA groups other than \({\mathbb{R}}\) or \({\mathbb{R}}^ n\) and the above equivalences reappear. A set E in \({\mathbb{Z}}\) is called p-Sidon \((p\in [1,2))\) if for every E- function in \(L^{\in…
reviewed by: Stanisław Hartman
Mathematics Subject Classification ID: 43A46
zbMATH DE Number: 4021845
zbMATH Keywords: amalgam
zbMATH Keywords: \(\Lambda \) (p)-set
zbMATH Keywords: LCA groups
zbMATH Keywords: p-Sidon set
MaRDI profile type: Publication
full work available at URL: rmj-1987-17-2-277
OpenAlex ID: W2067988781
description: scientific article; zbMATH DE number 4021845