Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T00:23:45.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Littlewood–Paley theorems for sum and difference sets

Published online by Cambridge University Press:  24 October 2008

Garth I. Gaudry
Garth I. Gaudry
Affiliation:
Flinders University of South Australia
Get access Rights & Permissions [Opens in a new window]

Summary

Let α be a positive integer, and El, …, Eα Hadamard sets of positive integers. It is shown that E = E1 + … + Eα determines a Littlewood–Paley decomposition of Z.

Suppose that is a Hadamard set of positive integers such that nj+1/nj ≥ 2 for all j. Let α be a positive integer, and

We show that F(α) also determines a Littlewood-Paley decomposition of Z.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 1 , January 1978 , pp. 65 - 71
Copyright
Copyright © Cambridge Philosophical Society 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1) Bonami, A. Etude des coefficients de Fourier des fonctions de Lp(G). Ann. Inst. Fourier (Grenoble) 20 (1970), 335–402. MR 44, No. 727.CrossRefGoogle Scholar
(2) Edwards, R. E. and Gaudry, G. I. Littlewood-Paley and multiplier theory. Ergebnisse der Mathematik und Ihrer Grenzgebiete, vol. 90 (Berlin, Heidelberg, New York: Springer, 1977).CrossRefGoogle Scholar
(3) Gaudry, G. L. The Littlewood-Paley theorem and Fourier multipliers for certain locally compact groups. Proc. Convegno su analisi armonica e spazi di funzioni su gruppi localmente compatti, INAM, Roma, 1976 (to appear).Google Scholar
(4) Kahane, J. -P. Séries de Fourier absolument convergentes. Ergebnisse der Mathematik und Ihrer Grenzgebiete, vol. 50 (Berlin, Heidelberg, New York: Springer, 1970). MR 40, No. 8095.CrossRefGoogle Scholar
(5) Lizorkin, P. I. On a theorem of Marcinkiewicz type for H-valued functions. A continual form of the Paley–Littlewood theorem. Math. USSR Sbornik 16 (1972), 237–243.CrossRefGoogle Scholar
(6) Marcinkiewicz, J. and Zygmund, A. Quelques inégalités pour les opérations linéaires. Fund. Math. 32 (1939), 115–121. MR 52, No. 1162.CrossRefGoogle Scholar
(7) Stein, E. M. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 (Princeton: Princeton Univ. Press, 1970). MR 44, No. 7280.Google Scholar