Wavelet characterization of functions with conditions on the mean oscillation

  • Hugo Aimar
  • Ana Bernardis
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


The space BMO of those real functions defined on IR n for which the mean oscillation over cubes is bounded, appears in the pioneer works of J. Moser [13] and John-Nirenberg [9] in the early sixties as a tool for the study of regularity of weak solutions of elliptic and parabolic differential equations. Their main result, known today as John-Nirenberg Theorem, provides a characterization of BMO in terms of the exponential decay of the distribution function on each cube. Although the depth of this result, the space BMO only became well known in harmonic analysis after the celebrated Fefferman-Stein theorem of duality for the Hardy spaces: BMO is the dual of the Hardy space H 1. Since H 1 was already known to be the good substitute of L 1 for many questions in analysis, the space BMO was realized as the natural substitute of L in the scale of the Lebesgue spaces. Also due to Fefferman and Stein is the Littlewood-Paley type characterization of BMO in terms of the derivatives of the harmonic extension and Carleson measures (see for example [16]). This result has a discrete version: the Lemarié-Meyer characterization of BMO using wavelets, [10].


Hardy Space Wavelet Coefficient Wavelet Basis Carleson Measure Homogeneous Type 
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  1. [1]
    Aimar H. Rearrangement and Continuity Properties of BMO(Æ) Functions on Spaces of Homogeneous Type. Ann. Scuola Norm. Sup. Pisa, Serie IV, Vol XVIII 3, 353–362, 1991MathSciNetGoogle Scholar
  2. [2]
    Campanato S. Propietà di hölderianità di alcune classi di fun-zioni. Ann. Scuola Norm. Sup. Pisa 17, 175–188, 1963.MathSciNetMATHGoogle Scholar
  3. [3]
    Daubechies I. Ten lectures on wavelets. SIAM CBMS-NSF Regional Conf. Series in Applied Math, 1992.Google Scholar
  4. [4]
    Daubechies I. Orthonormal Bases of Compactly Supported Wavelets. Comm. in Pure and Applied Math, Vol. 4l, 909–996, 1988.CrossRefGoogle Scholar
  5. [5]
    Fefferman C. and Stein E. H p spaces of several variables. Acta Math. 129, 137–193, 1971.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Holschneider R. and Tchamitchian F. Pointwise analysis of Rie-mann’s nondifferentiable function. Centre de Physique Théorique. Marseille.,1989.Google Scholar
  7. [7]
    Janson S. Generalizations of Lipschitz spaces an application to Hardy spaces and bounded mean oscillation. Duke Math. Journal, Vol 474, 959–982, 1980.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Jaffard S. and Laurencot Ph. Orthonormal Wavelets, Analysis of Operators and Applications to Numerical Analysis. Wavelets-A Tutorial in Theory and Applications, C. K. Chui (ed.) Academic Press, Inc., 543–601, 1991.Google Scholar
  9. [9]
    John F. and Nirenberg L. On Functions of Bounded Mean Oscillation. Comm. on Pure and Applied Math., Vol XIV, 415–426, 1961.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Lemarié P. and Meyer Y. Ondelettes et bases hilbertiennes. Revista Matemática Iberoamericana, Vol. 2, Nos.1 y 2, 1–18, 1986.CrossRefGoogle Scholar
  11. [11]
    Meyer Y. Ondelettes et Opérateurs I. Hermann éditeurs des sciences et des arts. Paris, 1990.Google Scholar
  12. [12]
    Meyers G. Mean oscillation over cubes and Hölder continuity. Proc. Amer. Math. Soc. 15, 717–724, 1964.MathSciNetMATHGoogle Scholar
  13. [13]
    Moser J. On Harnack’s Theorem for elliptic differential equations. Comm. Pure and Appl. Math., Vol XIV, 577–591, 1961.CrossRefGoogle Scholar
  14. [14]
    Spanne S. Some function spaces defined using the mean oscillation over cubes. Ann. Scuola Norm. Sup. Pisa 19, 593–608, 1965.MathSciNetMATHGoogle Scholar
  15. [15]
    Sawyer E. and Wheeden R. Wheighted inequalities for fractional integrals on euclidean and homogeneous spaces. American Journal of Math. 114, 813–874, 1962.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Torchinsky A. Real-Variable Methods in Harmonic Analysis. Academic Press, Inc., Pure and Appl. Math., Vol 123, 198Google Scholar
  17. [17]
    Viviani B. An Atomic Decomposition of the Predual of BMO(p). Revista Matemática Iberoamericana, Vol 3, Nos. 3 y4, 401–425, 1987.MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media New York 1997

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  • Hugo Aimar
  • Ana Bernardis

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