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A General Approach to Littlewood-Paley Theorems for Orthogonal Families

Published online by Cambridge University Press:  20 November 2018

Kathryn E. Hare
Kathryn E. Hare*
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1
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Abstract

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A general lacunary Littlewood-Paley type theorem is proved, which applies in a variety of settings including Jacobi polynomials in [0, 1], SU(2), and the usual classical trigonometric series in [0, 2π). The theoremis used to derive new results for Lp multipliers on SU(2) and Jacobi Lp multipliers.

Keywords

Primary: 42B25secondary: 42C1043A80Littlewood-Paley theoremorthogonal decompositionsmultipliers
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 3 , 01 September 1997 , pp. 296 - 308
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Askey, R., A transplantation theorem for Jacobi series, Illinois J. Math. 13 (1969), 583–590.Google Scholar
2. Askey, R. and Hirschman, I., Mean summability for ultraspherical polynomials, Math. Scand. 12 (1963), 167–177.Google Scholar
3. Bavinck, H., A special class of Jacobi series and applications, J. Math. Anal. Appl. 37 (1972), 767–797.Google Scholar
4. Bonami, A. and Clerc, J.-L., Sommes de Cesaro et multiplicateurs des developpements en harmoniques spheriques, Trans. Amer.Math. Soc. 183 (1973), 223–263.Google Scholar
5. Clerc, J.-L., Sommes de Riesz et multiplicateurs sur un groupe de Lie compact, Ann. Inst. Fourier (Grenoble) 24 (1974), 149–172.Google Scholar
6. Coifman, R. and Weiss, G., Central multiplier theorems for compact Lie groups, Bull. Amer. Math. Soc. 80 (1974), 124–126.Google Scholar
7. Connett, W. and Schwartz, A., The theory of ultraspherical multipliers, Mem. Amer. Math. Soc. (183) 9 (1977).Google Scholar
8. Connett, W. and Schwartz, A., A multiplier theorem for Jacobi expansions, Studia Math. 52 (1975), 243–261.Google Scholar
9. Connett, W. and Schwartz, A., The Littlewood-Paley theory for Jacobi expansions, Trans.Amer.Math. Soc. 251 (1979), 219–234.Google Scholar
10. Edwards, R. and Gaudry, G., Littlewood-Paley and multiplier theory, Springer-Verlag, Berlin, Heidelberg, 1977.Google Scholar
11. Gaudry, G., Littlewood-Paley theorems for sum and difference sets, Math. Proc. Cambridge Philos. Soc. 83 (1978), 65–71.Google Scholar
12. Hare, K., Lp-improving measures on compact non-abelian groups, J. Austral. Math. Soc. 46 (1989), 402–414.Google Scholar
13. Hare, K., Properties and examples of (Lp, Lq) multipliers, Indiana Univ. Math. J. 38 (1989), 211–227.Google Scholar
14. Hare, K. and Klemes, I., A new type of Littlewood-Paley partition, Ark.Mat. 30 (1992), 297–309.Google Scholar
15. Hare, K. and Klemes, I., On permutations of lacunary intervals, Trans. Amer. Math. Soc. 347 (1995), 4105–4127.Google Scholar
16. Hewitt, E. and Ross, K., Abstract harmonic analysis, Vol.II, Springer-Verlag, Berlin, Heidelberg, New York, 1979.Google Scholar
17. Inglis, I., Central multipliers which vanish at infinity, J. London Math. Soc. 19 (1979), 102–106.Google Scholar
18. Muckenhoupt, B. and Stein, E., Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17–92.Google Scholar
19. Pollard, H., The mean convergence of orthogonal series. II,Trans. Amer.Math. Soc. 63 (1948), 355–367.Google Scholar
20. Price, J., Non ci sono insiemi infiniti di tipo Λ(p) per SU(2), Boll. Un. Mat. Ital. 4 (1971), 879–881.Google Scholar
21. Rubio de Francia, J., A Littlewood-Paley inequality for arbitrary intervals, Rev.Mat. Iberoamericana (2) 1 (1985), 1–14.Google Scholar
22. Sjogren, P. and Sjolin, P., Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets, Ann. Inst. Fourier (Grenoble) 31 (1981), 157–175.Google Scholar
23. Stein, E. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, New Jersey, 1971.Google Scholar
24. Strichartz, R., Multipliers for spherical harmonic expansions, Trans. Amer.Math. Soc. 167 (1972), 115–124.Google Scholar
25. Szego, G., Orthogonal polynomials, Amer.Math. Soc. 23, New York, 1975.Google Scholar
26. Weiss, N., Lp estimates for bi-invariant operators on compact Lie groups, Amer. J. Math. 94 (1972), 103–118.Google Scholar