Comptes Rendus
no. 17-18
Harmonic Analysis/Functional Analysis
A simple real-variable proof that the Hilbert transform is an L2-isometry
[Une démonstration simple en variables réelles de la propriété d'isométrie L2 de la transformation de Hilbert H]

Présenté par : Yves Meyer

Enrico Laeng1
1 Politecnico di Milano, Dipartimento di Matematica “F. Brioschi”, Via Bonardi 9, 20133 Milano, Italy
Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 977-980.

La transformation de Hilbert H peut être étendue à une isometrie dans L2. On demontre cette propriété en utilsant directement la valeur principale de l'intégrale, sans utiliser la transformation de Fourier, ni des systèmes de fonctions orthogonales. L'approche proposée est liée à nos tentative de comprendre le proprietés de réarrangement de H.

The Hilbert transform H can be extended to an isometry of L2. We prove this fact working directly on the principal value integral, completely avoiding the use of the Fourier transform and the use of orthogonal systems of functions. Our approach here is a byproduct of our attempts to understand the rearrangement properties of H.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.07.002
Enrico Laeng 1

1 Politecnico di Milano, Dipartimento di Matematica “F. Brioschi”, Via Bonardi 9, 20133 Milano, Italy 
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Enrico Laeng. A simple real-variable proof that the Hilbert transform is an $ {L}^{2}$-isometry. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 977-980. doi : 10.1016/j.crma.2010.07.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.002/

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[2] L. Colzani; E. Laeng; L. Monzón Variations on a theme of Boole and Stein–Weiss, Journal of Mathematical Analysis and Applications, Volume 363 (2010), pp. 225-229

[3] J. Duoandikoetxea The Hilbert transform and Hermite functions: A real variable proof of the L2-isometry, Journal of Mathematical Analysis and Applications, Volume 347 (2008), pp. 592-596

[4] L. de Carli; E. Laeng Sharp Lp-estimates for the segment multiplier, Collectanea Mathematica, Volume 51 (2000) no. 3, pp. 309-326

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