On Petermichl's dyadic shift and the Hilbert transform

Hytönen, Tuomas

Comptes Rendus. Mathématique, Tome 346 (2008) no. 21-22, pp. 1133-1136.

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 $L^p(\mathbb{R})$, $p\in (1,\infty )$, ce qui améliore le résultat antérieur que affirme la convergence faible dans $L^2(\mathbb{R})$.

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 $L^p(\mathbb{R})$, $p\in (1,\infty )$, which improves on the earlier result of weak convergence in $L^2(\mathbb{R})$.

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