Hostname: page-component-54dcc4c588-r5qjk Total loading time: 0 Render date: 2025-09-12T16:40:04.918Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on Littlewood–Paley Analysis

Published online by Cambridge University Press:  20 November 2018

Kwok-Pun Ho
Kwok-Pun Ho*
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China e-mail: makho@ust.hk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Littlewood–Paley analysis is generalized in this article. We show that the compactness of the Fourier support imposed on the analyzing function can be removed. We also prove that the Littlewood–Paley decomposition of tempered distributions converges under a topology stronger than the weak-star topology, namely, the inductive limit topology. Finally, we construct a multiparameter Littlewood–Paley analysis and obtain the corresponding “renormalization” for the convergence of this multiparameter Littlewood–Paley analysis.

Keywords

42B25Littlewood–Paley analysisdistributions

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 6 , 01 December 2008 , pp. 1283 - 1305
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Bownik, M. and Ho, K. P., Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces. Trans. Amer. Math. Soc. 358(2006), no. 4, 1469–1510.Google Scholar
[2] Chang, S.-Y. and Fefferman, R., A continuous version of duality of H1 with BMO on the bidisc. Ann. of Math. 112(1980), no. 1, 179–201.Google Scholar
[3] Chang, S.-Y. and Fefferman, R., The Calderón-Zygmund decomposition on product domains. Amer. J. Math. 104(1982), no. 3, 455–468.Google Scholar
[4] Chang, S.-Y. and Fefferman, R., Some recent developments in Fourier analysis and Hp-theory on product domains. Bull. Amer. Math. Soc. 12(1985), no. 1, 1–43.Google Scholar
[5] Fefferman, R., The atomic decomposition of H1 in product spaces. Adv. in Math. 55(1985), no. 1, 90–100.Google Scholar
[6] Fefferman, R., Singular integrals on product Hp spaces. Rev. Mat. Iberoamericana 1(1985), no. 2, 25–31.Google Scholar
[7] Fefferman, R., Fourier analysis in several parameters. Rev. Mat. Iberoamericana 2(1986), no. 1-2, 89–98.Google Scholar
[8] Fefferman, R., Harmonic analysis on product spaces. Ann. of Math. 126(1987), no. 1, 109–130.Google Scholar
[9] Fefferman, R., Multiparameter Calderón-Zygmund theory. In: Harmonic Analysis and Partial Differential Equations, 207–221, University of Chicago Press, Chicago, IL, 1999, pp. 207–221.Google Scholar
[10] Frazier, M., and Jawerth, B., A discrete transform and decomposition of distribution spaces. J. Funct. Anal. 93(1990), no. 1, 34–170.Google Scholar
[11] Frazier, M., Jawerth, B., and G.Weiss, Littlewood-Paley Theory and the Study of Function Spaces. CBMS Regional Conference Series inMathematics 79, AmericanMathematical Society, Providence, RI, 1991.Google Scholar
[12] Gundy, R. F., and Stein, E. M., Hp theory for the poly-disc. Proc. Nat. Acad. Sci. U.S.A. 76(1979), no. 3, 1026–1029.Google Scholar
[13] Ho, K.-P., Frames associated with expansive matrix dilations. Collect. Math. 54(2003), no. 3, 217–254.Google Scholar
[14] Littlewood, J. E. and Paley, R. E. A. C., Theorems on Fourier series and power series. I. J. London Math. Soc. 6(1931), 230–233.Google Scholar
[15] Meyer, Y.,Wavelets and Operators. Cambridge Studies in Advanced Mathematics 37, Cambridge University Press, Cambridge, 1992.Google Scholar
[16] Meyer, Y.,Wavelets, Vibrations and Scalings. CRM Monograph Series 9, AmericanMathematical Society, Providence, RI, 1998.Google Scholar
[17] Peetre, J., New Thoughts on Besov Spaces. Duke University Mathematics Series 1, Mathematics Department, Duke University, Durham, NC. 1976.Google Scholar
[18] Rudin, W., Functional Analysis. Second edition. McGraw-Hill, New York, 1991.Google Scholar