On permutations of lacunary intervals

Hare, Kathryn E.; Klemes, Ivo

doi:10.1090/S0002-9947-1995-1308014-7

On permutations of lacunary intervals
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by Kathryn E. Hare and Ivo Klemes PDF

Trans. Amer. Math. Soc. 347 (1995), 4105-4127 Request permission

Abstract:

Let $\{ {I_j}\}$ be an interval partition of the integers and consider the Littlewood-Paley type square function $S(f) = {(\sum {\left | {{f_j}} \right |^2})^{1/2}}$ 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 ${\left \| {S(f)} \right \|_p} \approx {\left \| f \right \|_p}$ 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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Sur les multiplicateurs des series de Fourier

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