Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-06-08T03:49:19.537Z Has data issue: false hasContentIssue false
  • English
  • Français

Calderón-type reproducing formulae on Lipschitz curves and surfaces

Part of: Nontrigonometric harmonic analysis Generalized function theory Abstract harmonic analysis Harmonic analysis in several variables

Published online by Cambridge University Press:  09 April 2009

Tao Qian
Tao Qian
Affiliation:
Faculty of Science and Technology, The University of Macau, P.O. Box 3001, Macau (via Hong Kong) e-mail: fsttq@umac.mo
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.

Calderón type reproducing formulae with applications have been studied on one- and higher-dimensional Lipschitz graphs. In this note we study higher order Calderón reproducing formulae on star-shaped and non-star-shaped closed Lipschitz curves and surfaces.

Keywords

reproducing kernelLittlewood-Paley theorycontinuous and discrete wavelet decompositionClifford analysis

MSC classification

Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 30G35: Functions of hypercomplex variables and generalized variables 42C15: General harmonic expansions, frames 43A99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 1 , February 2002 , pp. 33 - 46
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Brackx, F., Delanghe, R. and Sommen, F., Clifford analysis, Research Notes Math. 76 (Pitman (Advanced Publishing Program), Boston, 1982).Google Scholar
[2]Coifman, R., McIntosh, A. and Meyer, Y., ‘L'intégrale de Cauchy définit un opérateur borné sur L2 pour les courbes lipschitziennes’, Ann. of Math. (2) 116 (1982), 361–387.Google Scholar
[3]David, G., Journé, J.-L. and Semmes, S., ‘Opérateurs de Calderón-Zygmund fonctions paraaccrétives et interpolation’, Rev. Mat. Iberoamericana 1 (1985), 1–56.Google Scholar
[4]Delanghe, R., Sommen, F. and Soucek, V., Clifford algebras and spinor valued functions: A function theory for Dirac operator, Mathematics and its Application 53 (Kluwer Acad. Publ., Dordrecht, 1992).CrossRefGoogle Scholar
[5]Deng, D. G. and Han, Y. S., ‘The Besov and Triebel-Lizorkin spaces on Lipschitz curves (I)’, Acta Math. Sinica 32 (1992), 608–619;Google Scholar
‘(II)’, Acta Math. Sinica 36 (1992), 122–135.Google Scholar
[6]Deng, D. G. and Han, Y. S., ‘The Besov and Triebel-Lizorkin spaces of higher order on Lipschitz curves’, Approx. Theory Appl. 9 (1993), 89–106.CrossRefGoogle Scholar
[7]Jerison, D. and Kenig, C. E., ‘Hardy spaces, A∞, and singular integrals on chord-arc domains’, Math. Scand. 50 (1982), 221–247.Google Scholar
[8]Jones, P., Constructions with functions of bounded mean oscillation (Ph.D. Thesis, University of California, 1978).Google Scholar
[9]Kenig, C., Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conf. Ser. in Math. 83 (Amer. Math. Soc., Providence, 1994).Google Scholar
[10]Qian, T., ‘Singular integrals with holomorphic kernels and H∞-Fourier multipliers on star-shaped Lipschitz curves’, Studia Math. 123 (1997), 195–216.Google Scholar
[11]Qian, T., ‘Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space’, Math. Ann. 310 (1998), 601–630.CrossRefGoogle Scholar
[12]Qian, T., ‘Fourier analysis on star-shaped Lipschitz surfaces’, J. Funct. Anal. to appear.Google Scholar
[13]Qian, T. and Yan, L.-X., ‘BMO decomposition on Lipschitz graphs’, preprint, University of New England, 1998.Google Scholar
[14]Verchota, G., ‘Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains’, J. Funct. Anal. 59 (1984), 572–611.CrossRefGoogle Scholar
[15]Yan, L.-X., ‘Wavelet frames on Lipschitz curves’, J. Fourier Anal. Appl. 6 (2000), 559–582.CrossRefGoogle Scholar