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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On permutations of lacunary intervals

Authors: Kathryn E. Hare and Ivo Klemes
Journal: Trans. Amer. Math. Soc. 347 (1995), 4105-4127
MSC: Primary 42B25; Secondary 42A45
MathSciNet review: 1308014
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Abstract: Let $ \{ {I_j}\} $ be an interval partition of the integers and consider the Littlewood-Paley type square function $ S(f) = {(\sum {\left\vert {{f_j}} \right\vert^2})^{1/2}}$ where $ {\hat f_j} = \hat f\chi {I_j}$. We prove that if the lengths $ \ell ({I_j})$ of the intervals $ {I_j}$ satisfy $ \ell ({I_{j + 1}})/\ell ({I_j}) \to \infty $, then $ {\left\Vert {S(f)} \right\Vert _p} \approx {\left\Vert f \right\Vert _p}$ for $ 1 < p < \infty $. As these intervals need not be adjacent, such partitions can be thought of as permutations of lacunary intervals. This work generalizes the specific partition considered in a previous paper [H2]. We conjecture that it suffices to assume $ \ell ({I_{j + 1}})/\ell ({I_j}) \geqslant \lambda > 1$, and we also conjecture a necessary and sufficient condition.

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