Harmonic Analysis
On Petermichl's dyadic shift and the Hilbert transform
Comptes Rendus. Mathématique, Volume 346 (2008) no. 21-22, pp. 1133-1136.

Petermichl's representation for the Hilbert transform as an average of dyadic shifts has important applications. Here it is shown that the integrals involved in (a variant of) this representation converge both almost everywhere and strongly in Lp(R), p(1,), which improves on the earlier result of weak convergence in L2(R).

La représentation, dû à Petermichl, pour la transformée d'Hilbert comme une moyenne des translations dyadiques a des applications importantes. Ici, on montre que les integrals dans (une forme de) cette représentation convergent à la fois presque partout et fortement dans Lp(R), p(1,), ce qui améliore le résultat antérieur que affirme la convergence faible dans L2(R).

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DOI: 10.1016/j.crma.2008.09.021
Hytönen, Tuomas 1

1 Department of Mathematics and Statistics, University of Helsinki, Gustaf Hällströmin katu 2b, 00014 Helsinki, Finland
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Hytönen, Tuomas. On Petermichl's dyadic shift and the Hilbert transform. Comptes Rendus. Mathématique, Volume 346 (2008) no. 21-22, pp. 1133-1136. doi : 10.1016/j.crma.2008.09.021. http://www.numdam.org/articles/10.1016/j.crma.2008.09.021/

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