no. 19-20
Harmonic Analysis
An extension of the Córdoba–Fefferman theorem on the equivalence between the boundedness of certain classes of maximal and multiplier operators
[Généralisation du théorème de Córdoba–Fefferman sur l'équivalence du caractère borné de certains opérateurs maximaux et de multiplicateurs]

Présenté par : Jean-Pierre Kahane

Hagelstein, Paul1 ; Stokolos, Alexander2
1 Department of Mathematics, Baylor University, Waco, TX 76798, USA
2 Department of Mathematics, DePaul University, Chicago, IL 60614, USA
Comptes Rendus. Mathématique, Tome 346 (2008) no. 19-20, pp. 1063-1065.

Les travaux récents de Bateman sur les opérateurs maximaux relatifs à des directions, et ceux des auteurs sur les opérateurs maximaux associés à des bases d'ensembles convexes invariantes par homothétie et vérifiant des conditions tauberiennes permettent d'étendre le théorème de Fefferman et Córdoba sur l'équivalence du caractère borné de certains opérateurs maximaux et de multiplicateurs.

The Córdoba–Fefferman theorem involving the equivalence between boundedness properties of certain classes of maximal and multiplier operators is extended utilizing the recent work of Bateman on directional maximal operators as well as the work of Hagelstein and Stokolos on geometric maximal operators associated to homothecy invariant bases of convex sets satisfying Tauberian conditions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.09.010
Hagelstein, Paul 1 ; Stokolos, Alexander 2

1 Department of Mathematics, Baylor University, Waco, TX 76798, USA
2 Department of Mathematics, DePaul University, Chicago, IL 60614, USA 
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Hagelstein, Paul; Stokolos, Alexander. An extension of the Córdoba–Fefferman theorem on the equivalence between the boundedness of certain classes of maximal and multiplier operators. Comptes Rendus. Mathématique, Tome 346 (2008) no. 19-20, pp. 1063-1065. doi : 10.1016/j.crma.2008.09.010. http://archive.numdam.org/articles/10.1016/j.crma.2008.09.010/

[1] Bateman, M. Kakeya sets and directional maximal operators in the plane | arXiv

[2] Bateman, M.; Katz, N.H. Kakeya sets in Cantor directions, Math. Res. Lett., Volume 15 (2008) no. 1, pp. 73-81

[3] Córdoba, A.; Fefferman, R. On the equivalence between the boundedness of certain classes of maximal and multiplier operators in Fourier analysis, Proc. Natl. Acad. Sci. USA, Volume 74 (1977) no. 2, pp. 423-425

[4] P. Hagelstein, A. Stokolos, Tauberian conditions for geometric maximal operators, Trans. Amer. Math. Soc., in press

[5] Stein, E.M. Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970

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