no. 21-22
Harmonic Analysis
On Petermichl's dyadic shift and the Hilbert transform
[La translation dyadique de Petermichl et la transformée d'Hilbert]

Présenté par : Gilles Pisier

Hytönen, Tuomas1
1 Department of Mathematics and Statistics, University of Helsinki, Gustaf Hällströmin katu 2b, 00014 Helsinki, Finland
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 Lp(R), p∈(1,∞), ce qui améliore le résultat antérieur que affirme la convergence faible dans L2(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 Lp(R), p∈(1,∞), which improves on the earlier result of weak convergence in L2(R).

Reçu le :
Accepté le :
Publié le :
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 
@article{CRMATH_2008__346_21-22_1133_0,
     author = {Hyt\"onen, Tuomas},
     title = {On {Petermichl's} dyadic shift and the {Hilbert} transform},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1133--1136},
     publisher = {Elsevier},
     volume = {346},
     number = {21-22},
     year = {2008},
     doi = {10.1016/j.crma.2008.09.021},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2008.09.021/}
}
TY  - JOUR
AU  - Hytönen, Tuomas
TI  - On Petermichl's dyadic shift and the Hilbert transform
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 1133
EP  - 1136
VL  - 346
IS  - 21-22
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2008.09.021/
DO  - 10.1016/j.crma.2008.09.021
LA  - en
ID  - CRMATH_2008__346_21-22_1133_0
ER  - 
%0 Journal Article
%A Hytönen, Tuomas
%T On Petermichl's dyadic shift and the Hilbert transform
%J Comptes Rendus. Mathématique
%D 2008
%P 1133-1136
%V 346
%N 21-22
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2008.09.021/
%R 10.1016/j.crma.2008.09.021
%G en
%F CRMATH_2008__346_21-22_1133_0
Hytönen, Tuomas. On Petermichl's dyadic shift and the Hilbert transform. Comptes Rendus. Mathématique, Tome 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/

[1] Figiel, T. Singular integral operators: a martingale approach, Strobl, 1989 (London Math. Soc. Lecture Note Ser.), Volume vol. 158, Cambridge Univ. Press, Cambridge (1990), pp. 95-110

[2] Nazarov, F.; Treil, S.; Volberg, A. The Tb-theorem on non-homogeneous spaces, Acta Math., Volume 190 (2003) no. 2, pp. 151-239

[3] Petermichl, S. Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, C. R. Acad. Sci. Paris Sér. I Math., Volume 330 (2000) no. 6, pp. 455-460

[4] Petermichl, S. The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical Ap characteristic, Amer. J. Math., Volume 129 (2007) no. 5, pp. 1355-1375

[5] Petermichl, S.; Pott, S. A version of Burkholder's theorem for operator-weighted spaces, Proc. Amer. Math. Soc., Volume 131 (2003) no. 11, pp. 3457-3461 (electronic)

Cité par Sources :