Lacunarity for amalgams
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 \(L^{\infty}({\mathbb{T}})\) one has \(\hat f\in \ell^ p( {\mathbb{Z}})\). In the paper under review this concept is transferred to non- discrete groups, especially to \({\mathbb{R}}\). Let, as previously, \(F=E+[0,1)\) with \(E\subset {\mathbb{Z}}\). The following equivalence is proved: E is p- Sidon in \({\mathbb{Z}}\) if and only if for every F-function \(f\in (L^{\infty},\ell^ 1)( {\mathbb{R}})\) one has \(\hat f\in L^ p({\mathbb{R}})\) (also this theorem extends to other LCA groups). The author is not satisfied with this result and he conjectures that if E is p-Sidon then one has even \(\hat f\in (L^{\infty},\ell^ p)( {\mathbb{R}})\). Finally, he deduces from the two theorems above the following interesting statement: given a subset E of \({\mathbb{Z}}\) let ``\(\alpha E\)'' \((\alpha >0)\) be the set obtained by replacing each number in \(\alpha E\) by the nearest integer. Then, if E is a \(\Lambda(p)\) set (a p-Sidon set) then so is ``\(\alpha E\)''. There is an unusual number of misprints in the paper.
scientific article; zbMATH DE number 4021845