Arkiv för Matematik

  • Ark. Mat.
  • Volume 38, Number 2 (2000), 209-221.

Densité de l'intégrale d'aire et intégrales singulières

Lucien Chevalier and Alain Dufresnoy

Full-text: Open access

Abstract

In his 1983 paper [3], R. F. Gundy introduced a new functional related to the Littlewood-Paley theory, called the density of the area integral. In this paper, we prove that this functional (although highly non-linear) can be expressed as the principal value of an explicit singular integral. This result provides us with a new and precise connection between the density of the area integral and the theory of Calderón-Zygmund operators. It does not seem to be a consequence of the standard Calderón-Zygmund-Cotlar theory, because the sign of a harmonic function in the half-space fails to have, in some appropriate sense, boundary limits.

Article information

Source
Ark. Mat., Volume 38, Number 2 (2000), 209-221.

Dates
Received: 12 October 1998
Revised: 12 January 1999
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898683

Digital Object Identifier
doi:10.1007/BF02384317

Mathematical Reviews number (MathSciNet)
MR1785399

Zentralblatt MATH identifier
1028.42012

Rights
2000 © Institut Mittag-Leffler

Citation

Chevalier, Lucien; Dufresnoy, Alain. Densité de l'intégrale d'aire et intégrales singulières. Ark. Mat. 38 (2000), no. 2, 209--221. doi:10.1007/BF02384317. https://projecteuclid.org/euclid.afm/1485898683


Export citation

Bibliographie

  • Chevalier, L., Une «formule de Tanaka» en analyse harmonique et quelques applications, Adv. Math. 138 (1998), 182–210.
  • Chevalier, L. et Dufresnoy, A., Sur les changements de signe d'une fonction harmonique dans le demi-plan, Prépublication, 1999.
  • Gundy, R. F., The density of the area integral, dans Conference on Harmonic Analysis on Honor of Antoni Zygmund (Beckman, W., Calderón, A. P., Fefferman, R. et Jones, P. W., éds.), p. 138–149, Wadsworth, Belmont, Calif., 1983.
  • Meyer, Y., Opérateurs de Calderón-Zygmund, Hermann, Paris, 1990.