Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem

Carbery, Anthony

doi:10.5802/aif.1127

Carbery, Anthony

Annales de l'Institut Fourier, Tome 38 (1988) no. 1, pp. 157-168.

Résumé
Abstract

Nous montrons que l’opérateur maximal associé à la famille de rectangles en $R^{3}$ dont un des côtés est parallèle à $(1, 2^{j}, 2^{k})$ pour quelques $j, k \in Z$ est borné sur $L^{p}$ , $1 < p < \infty$ . Nous appliquons ce théorème pour obtenir une extension du théorème de multiplicateurs de Marcinkiewicz.

We show that the maximal operator associated to the family of rectangles in $R^{3}$ one of whose sides is parallel to $(1, 2^{j}, 2^{k})$ for some j,k $\in H Z$ is bounded on $L^{p}$ , $1 < p < \infty$ . We give an application of this theorem to obtain an extension of the Marcinkiewicz multiplier theorem.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1127

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     author = {Carbery, Anthony},
     title = {Differentiation in lacunary directions and an extension of the {Marcinkiewicz} multiplier theorem},
     journal = {Annales de l'Institut Fourier},
     pages = {157--168},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {1},
     year = {1988},
     doi = {10.5802/aif.1127},
     mrnumber = {89h:42026},
     zbl = {0607.42009},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1127/}
}

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Carbery, Anthony. Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem. Annales de l'Institut Fourier, Tome 38 (1988) no. 1, pp. 157-168. doi : 10.5802/aif.1127. http://www.numdam.org/articles/10.5802/aif.1127/

Bibliographie
Cité par

[1] A. Carbery. — An almost-orthogonality principle with applications to maximal functions associated to convex bodies, B.A.M.S., 14-2 (1986), 269-273. | MR | Zbl

[2] A. Carbery. — Variants of the Calderón-Zygmund theory for Lp-spaces, Revista Matemática Ibero Americana, 2-4 (1986), 381-396. | MR | Zbl

[3] M. Christ. — Personal communication.

[4] M. Christ, J. Duoandikoetxea AND J. L. Rubio De Francia. Maximal operators related to the Radon transform and the Calderón-Zygmund method of rotations, Duke Math. J., 53-1 (1986), 189-209. | MR | Zbl

[5] A. Nagel, E.M. Stein AND S. Wainger. — Differentiation in lacunary directions, P.N.A.S. (USA), 75-3 (1978), 1060-1062. | MR | Zbl

[6] E.M. Stein. — Singular integrals and differentiability properties of functions, Princeton University Press, Princeton N.J., 1970. | MR | Zbl

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