Comptes Rendus
no. 4
Harmonic analysis
Weak-Lp bounds for the Carleson and Walsh–Carleson operators
[Estimation Lp,∞ pour les opérateurs de Carleson et de Walsh–Carleson]

Présenté par : Jean-Pierre Kahane

Francesco Di Plinio1, 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, Volume 352 (2014) no. 4, pp. 327-331.

Nous prouvons une estimation Lp,∞ 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∈LlogL(T) est convergente presque partout.

We prove a weak-Lp 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∈LlogL(T) converge for almost every x∈T.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.02.005
Francesco Di Plinio 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 
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Francesco Di Plinio. Weak-$ {L}^{p}$ bounds for the Carleson and Walsh–Carleson operators. Comptes Rendus. Mathématique, Volume 352 (2014) no. 4, pp. 327-331. doi : 10.1016/j.crma.2014.02.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.02.005/

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