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On permutations of lacunary intervals. (English) Zbl 0863.42016

Let \(\{I_j\}\) be an interval partition of the integers and consider the Littlewood-Paley type square function \(S(f)=(\sum|f_j|^2)^{1/2}\), where \(\widehat f_j=\widehat f\chi_{I_j}\). The authors showed that if the lengths \(l(I_j)\) of the intervals \(I_j\) satisfy \(l(I_{j+1})/l(I_j)\to\infty\), then \(|S(f)|_p\sim|f|_p\) for \(1<p<\infty\). Since these intervals need not be adjacent, such partitions can be thought as permutation of lacunary intervals. This naturally generalizes the Littlewood-Paley theorem: The above conclusion holds when \(\{I_j\}\) consists of \((-\infty,0)\) and intervals of the form \([n_j,n_{j+1})\) with \(n_{j+1}\geq\lambda n_j\geq 0\) for a fixed \(\lambda>1\).
Reviewer: K.Yabuta (Nara)

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42A45 Multipliers in one variable harmonic analysis
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