Comptes Rendus
Mathematical Analysis/Harmonic Analysis
Pointwise regularity criteria
Comptes Rendus. Mathématique, Volume 339 (2004) no. 11, pp. 757-762.

A wavelet characterization of the pointwise regularity condition Tup(x0) of Calderón and Zygmund is obtained. The extremal case (a pointwise BMO condition) yields the sharpest wavelet condition which is implied by pointwise Hölder regularity; in particular, this criterium is sharper than the usual two-microlocal condition.

On obtient une caractérisation par ondelettes de la condition de régularité ponctuelle Tup(x0) de Calderón et Zygmund. Le cas extrème (une condition de type BMO local) fournit la condition la plus précise sur les modules des coefficients d'ondelette impliquée par la régularité Hölderienne ponctuelle ; en particulier elle est plus fine que le critère deux-microlocal usuel.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.10.011
Stéphane Jaffard 1, 2

1 Laboratoire d'analyse et de mathématiques appliquées, université Paris XII, 61, avenue du Général de Gaulle, 94010 Créteil cedex, France
2 Institut universitaire de France, 103, boulevard Saint-Michel 75005 Paris, France
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Stéphane Jaffard. Pointwise regularity criteria. Comptes Rendus. Mathématique, Volume 339 (2004) no. 11, pp. 757-762. doi : 10.1016/j.crma.2004.10.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.10.011/

[1] A. Arneodo; B. Audit; N. Decoster; J.-F. Muzy; C. Vaillant Wavelet-based multifractal formalism: applications to DNA sequences, satellite images of the cloud structure and stock market data (A. Bunde; J. Kropp; H.J. Schellnhuber, eds.), The Science of Disasters, Springer, 2002, pp. 27-102

[2] A.P. Calderon; A. Zygmund Local properties of solutions of elliptic partial differential equations, Studia Math., Volume 20 (1961), pp. 171-227

[3] S. Jaffard Pointwise smoothness, two-microlocalization and wavelet coefficients, Publ. Mat., Volume 35 (1991), pp. 155-168

[4] S. Jaffard, Wavelet techniques in multifractal analysis, in: M. Lapidus, M. van Frankenhuysen, Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Proc. Sympos. Pure Math., American Mathematical Society, 2004, in press

[5] S. Jaffard, C. Melot, Wavelet analysis of fractal boundaries, Preprint, 2004

[6] Y. Meyer Ondelettes et opérateurs, Hermann, 1990

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