Comptes Rendus
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

Presented by: Jean-Pierre Kahane

Paul Hagelstein1; Alexander Stokolos2
1 Department of Mathematics, Baylor University, Waco, TX 76798, USA
2 Department of Mathematics, DePaul University, Chicago, IL 60614, USA
Comptes Rendus. Mathématique, Volume 346 (2008) no. 19-20, pp. 1063-1065.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.09.010

Paul Hagelstein 1; Alexander Stokolos 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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Paul Hagelstein; Alexander Stokolos. An extension of the Córdoba–Fefferman theorem on the equivalence between the boundedness of certain classes of maximal and multiplier operators. Comptes Rendus. Mathématique, Volume 346 (2008) no. 19-20, pp. 1063-1065. doi : 10.1016/j.crma.2008.09.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.09.010/

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

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

[3] A. Córdoba; R. Fefferman 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] E.M. Stein Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970

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