Wavelet transforms and operators in various function spaces

  • Shinya Moritoh


We define a class of wavelet transforms as a continuous and micro-local version of the Littlewood-Paley decompositions. Hörmander’s wave front sets (see [3]) as well as the Besov and Triebel-Lizorkin spaces (see [6] and [7]) may be characterized in terms of our wavelet transforms. By using the results obtained above (see [4]), we characterize the wave front sets in the sense of the Besov-Triebel-Lizorkin regularity in terms of our wavelet transforms. Finally, Päivärinta’s results on the continuity of pseudodifferential operators in the Besov-Triebel-Lizorkin spaces (see [9]) may be microlocalized. In other words, we show the pseudo-microlocal properties in the sense of the Besov-Triebel-Lizorkin regularity. We remark that the components of our decompositions are not linearly independent but can be treated as if they were.


Function Space Besov Space Pseudodifferential Operator Inverse Fourier Transform Inversion Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. Daubechies, Ten Lectures on Wavelets, SIAM CBMS-NSF regional conference series in applied mathematics-61, Philadelphia, 1992.Google Scholar
  2. 2.
    C. Fefferman and E.M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    L. Hörmander, Fourier integral operators.I, Acta.Math. 127 (1971), 79–183.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    S. Moritoh, Wavelet transforms in Euclidean spaces — their relation with wave front sets and Besov, Triebel-Lizorkin spaces —, Tôhoku Mathematical Journal 47 (1995), 555–565.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    S. Moritoh, Wave front sets in the sense of Besov-Triebel-Lizorkin regularity and pseudo-microlocal property, (preprint).Google Scholar
  6. 6.
    J. Peetre, New Thoughts on Besov Spaces, Duke Univ., Durham, 1976.Google Scholar
  7. 7.
    H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.CrossRefGoogle Scholar
  8. 8.
    R. Murenzi, Wavelet transforms associated to the n-dimensional Euclidean group with dilations: signals in more than one dimension, Wavelets (J.M. Combes, A. Grossmann and Ph. Tchamitchian, eds.), Springer-Verlag, Berlin, Heidelberg, New York, 1989.Google Scholar
  9. 9.
    L. Päivärinta, Pseudodifferential operators in Hardy-Triebel spaces, Zeitschrift für Analysis und ihre Anwendungen 2(3) (1983), 235–242.MATHGoogle Scholar

Copyright information

© Springer-Verlag Tokyo 1997

Authors and Affiliations

  • Shinya Moritoh
    • 1
  1. 1.Department of MathematicsNara Women’s UniversityKita-Uoya Nishimachi, Nara 630Japan

Personalised recommendations