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Wavelet decomposition of Calderón-Zygmund operators on function spaces

  • Ka-Sing Lau (a1) and Lixin Yan (a2)
Abstract
Abstract

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.

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[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.
[2]Beylkin G., Coifman R. and Rokhlin V., ‘Fast wavelet transforms and numerical algorithms’, Comm. Pure Appl. Math. 44 (1991), 141183.
[3]Daubechies I., Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Appl. Math. 61 (SIAM, Philadelphia, 1992).
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[5]Deng D. G., Yan L. X. and Yang Q. X., ‘Blocking analysis and T(1) theorem’, Science in China 41 (1998), 800808.
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[7]Frazier M., Torres R. and Weiss G., ‘The boundedness of Calderón-Zygmund operator on the spaces inline-graphic’, Rev. Mat. Iberoamericana 4 (1998), 4172.
[8]Han Y. and Hofmann S., ‘T(l) theorem for Besov and Triebel-Lizorkin spaces’, Trans. Amer. Math. Soc. 237 (1993), 839853.
[9]Han Y., Paluszynski M. and Weiss G., ‘A new atomic decomposition for the Triebel-Lizorkin spaces’, Contemporary Math. 189 (1995), 235249.
[10]Lemarié P. G., ‘Continuité sur les espaces de Besov and operatéurs definis par des intégrales singulières’, Ann. Inst. Fourier (Grenoble) 35 (1985), 175187.
[11]Meyer Y., La minimalité de l'espace de Besov inline-graphicet la continuité des opérateurs definis par des integrales singulières, Monografias de Matematicas, 4 (Univ. Autonoma de Madrid, 1986).
[12]Meyer Y., Ondelettes et opérateurs, Vols I, II (Hermann, Paris, 1990).
[13]Triebel H., Theory of function spaces (Birkhäuser, Basel, 1983).
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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