On permutations of lacunary intervals
HTML articles powered by AMS MathViewer
- 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.References
- J. Bourgain, On square functions on the trigonometric system, Bull. Soc. Math. Belg. Sér. B 37 (1985), no. 1, 20–26. MR 847119
- R. E. Edwards and G. I. Gaudry, Littlewood-Paley and multiplier theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 90, Springer-Verlag, Berlin-New York, 1977. MR 0618663
- Garth I. Gaudry, Littlewood-Paley theorems for sum and difference sets, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 65–71. MR 470576, DOI 10.1017/S0305004100054293
- Kathryn E. Hare and Ivo Klemes, Properties of Littlewood-Paley sets, Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 3, 485–494. MR 985685, DOI 10.1017/S0305004100077860
- Kathryn E. Hare and Ivo Klemes, A new type of Littlewood-Paley partition, Ark. Mat. 30 (1992), no. 2, 297–309. MR 1289757, DOI 10.1007/BF02384876 J. Marcinkiewicz, Sur les multiplicateurs des series de Fourier, Studia Math. 8 (1939), 78-91.
- S. K. Pichorides, A note on the Littlewood-Paley square function inequality, Colloq. Math. 60/61 (1990), no. 2, 687–691. MR 1096408, DOI 10.4064/cm-60-61-2-687-691
- S. K. Pichorides, A remark on the constants of the Littlewood-Paley inequality, Proc. Amer. Math. Soc. 114 (1992), no. 3, 787–789. MR 1088445, DOI 10.1090/S0002-9939-1992-1088445-6
- José L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Rev. Mat. Iberoamericana 1 (1985), no. 2, 1–14. MR 850681, DOI 10.4171/RMI/7
- P. Sjögren and P. Sjölin, Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 1, vii, 157–175 (English, with French summary). MR 613033
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 4105-4127
- MSC: Primary 42B25; Secondary 42A45
- DOI: https://doi.org/10.1090/S0002-9947-1995-1308014-7
- MathSciNet review: 1308014