Weak-$ {L}^{p}$ bounds for the Carleson and Walsh–Carleson operators

Di Plinio, Francesco

doi:10.1016/j.crma.2014.02.005

Harmonic analysis

Weak-

L^{p}

bounds for the Carleson and Walsh–Carleson operators
[Estimation

L^{p, \infty}

pour les opérateurs de Carleson et de Walsh–Carleson]

Présenté par : Jean-Pierre Kahane

Di Plinio, Francesco ^1,²

¹ INdAM – Cofund Marie Curie Fellow at Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
² The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, IN 47405, USA

Comptes Rendus. Mathématique, Tome 352 (2014) no. 4, pp. 327-331.

Résumé
Abstract

Nous prouvons une estimation $L^{p, \infty}$ pour l'opérateur de Walsh–Carleson, pour p proche de 1, qui constitue une amélioration d'un théorème de Sjölin. Nous interprétons nos résultats par rapport à la conjecture selon laquelle la série de Fourier d'une fonction $f \in L \log L (T)$ est convergente presque partout.

We prove a weak- $L^{p}$ bound for the Walsh–Carleson operator for p near 1, improving on a theorem of Sjölin. We relate our result to the conjectures that the Walsh–Fourier and Fourier series of a function $f \in L \log L (T)$ converge for almost every $x \in T$ .

Reçu le : 2013-12-02
Accepté le : 2014-02-06
Publié le : 2014-02-21

DOI : 10.1016/j.crma.2014.02.005

Affiliations des auteurs :

Di Plinio, Francesco ^{1, 2}

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     title = {Weak-$ {L}^{p}$ bounds for the {Carleson} and {Walsh{\textendash}Carleson} operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {327--331},
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Di Plinio, Francesco. Weak-$ {L}^{p}$ bounds for the Carleson and Walsh–Carleson operators. Comptes Rendus. Mathématique, Tome 352 (2014) no. 4, pp. 327-331. doi : 10.1016/j.crma.2014.02.005. http://www.numdam.org/articles/10.1016/j.crma.2014.02.005/

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