Hostname: page-component-7c8c6479df-hgkh8 Total loading time: 0 Render date: 2024-03-27T13:41:02.427Z Has data issue: false hasContentIssue false

Wavelet decomposition of Calderón-Zygmund operators on function spaces

Published online by Cambridge University Press:  09 April 2009

Ka-Sing Lau
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong e-mail: kslau@math.cuhk.edu.hk
Lixin Yan
Affiliation:
Department of Mathematics, Zhongshan University, Guangzhou, 510275, P. R. China e-mail: lixin@ics.mq.edu.au
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.

We make use of the Beylkin-Coifman-Rokhlin wavelet decomposition algorithm on the Calderón-Zygmund kernel to obtain some fine estimates on the operator and prove the T(l) theorem on Besov and Triebel-Lizorkin spaces. This extends previous results of Frazier et al., and Han and Hofmann.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Aguirre, J., Escobedo, M., Perel, J. C. and Tchamitchian, Ph., ‘Basis of wavelets and atomic decompositions of H 1 (Rn) and H 1 (Rn × Rn),’, Proc. Amer. Math. Soc. 111 (1991), 683693.Google Scholar
[2]Beylkin, G., Coifman, R. and Rokhlin, V., ‘Fast wavelet transforms and numerical algorithms’, Comm. Pure Appl. Math. 44 (1991), 141183.CrossRefGoogle Scholar
[3]Daubechies, I., Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Appl. Math. 61 (SIAM, Philadelphia, 1992).CrossRefGoogle Scholar
[4]David, G. and Journé, J. L., ‘A boundedness criterion for generalized Calderón-Zygmund operators’, Ann. of Math. 120 (1984), 371397.CrossRefGoogle Scholar
[5]Deng, D. G., Yan, L. X. and Yang, Q. X., ‘Blocking analysis and T(1) theorem’, Science in China 41 (1998), 800808.Google Scholar
[6]Frazier, M., Jawerth, B. and Weiss, G., Littlewood-Paley theory and the study of functions, CBMS– Regional Conference Series in Mathematics 79 (AMS, Providence, RI, 1991).CrossRefGoogle Scholar
[7]Frazier, M., Torres, R. and Weiss, G., ‘The boundedness of Calderón-Zygmund operator on the spaces ’, Rev. Mat. Iberoamericana 4 (1998), 4172.CrossRefGoogle Scholar
[8]Han, Y. and Hofmann, S., ‘T(l) theorem for Besov and Triebel-Lizorkin spaces’, Trans. Amer. Math. Soc. 237 (1993), 839853.Google Scholar
[9]Han, Y., Paluszynski, M. and Weiss, G., ‘A new atomic decomposition for the Triebel-Lizorkin spaces’, Contemporary Math. 189 (1995), 235249.CrossRefGoogle Scholar
[10]Lemarié, P. G., ‘Continuité sur les espaces de Besov and operatéurs definis par des intégrales singulières’, Ann. Inst. Fourier (Grenoble) 35 (1985), 175187.Google Scholar
[11]Meyer, Y., La minimalité de l'espace de Besov et la continuité des opérateurs definis par des integrales singulières, Monografias de Matematicas, 4 (Univ. Autonoma de Madrid, 1986).Google Scholar
[12]Meyer, Y., Ondelettes et opérateurs, Vols I, II (Hermann, Paris, 1990).Google Scholar
[13]Triebel, H., Theory of function spaces (Birkhäuser, Basel, 1983).CrossRefGoogle Scholar