ANNALES DE L'INSTITUT FOURIER

Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem
Annales de l'Institut Fourier, Volume 38 (1988) no. 1, p. 157-168
We show that the maximal operator associated to the family of rectangles in ${\mathbf{R}}^{3}$ one of whose sides is parallel to $\left(1,{2}^{j},{2}^{k}\right)$ for some j,k$\in \mathbf{H}Z$ is bounded on ${L}^{p}$, $1. We give an application of this theorem to obtain an extension of the Marcinkiewicz multiplier theorem.
Nous montrons que l’opérateur maximal associé à la famille de rectangles en ${\mathbf{R}}^{3}$ dont un des côtés est parallèle à $\left(1,{2}^{j},{2}^{k}\right)$ pour quelques $j,k\in \mathbf{Z}$ est borné sur ${L}^{p}$, $1. Nous appliquons ce théorème pour obtenir une extension du théorème de multiplicateurs de Marcinkiewicz.
@article{AIF_1988__38_1_157_0,
author = {Carbery, Anthony},
title = {Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
volume = {38},
number = {1},
year = {1988},
pages = {157-168},
doi = {10.5802/aif.1127},
mrnumber = {89h:42026},
zbl = {0607.42009},
language = {en},
url = {https://aif.centre-mersenne.org/item/AIF_1988__38_1_157_0}
}

Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem. Annales de l'Institut Fourier, Volume 38 (1988) no. 1, pp. 157-168. doi : 10.5802/aif.1127. https://aif.centre-mersenne.org/item/AIF_1988__38_1_157_0/

[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 87k:42015 | Zbl 0588.42012

[2] A. Carbery. — Variants of the Calderón-Zygmund theory for Lp-spaces, Revista Matemática Ibero Americana, 2-4 (1986), 381-396. | MR 89f:42011 | Zbl 0632.42013

[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 88d:42032 | Zbl 0656.42010

[5] A. Nagel, E.M. Stein AND S. Wainger. — Differentiation in lacunary directions, P.N.A.S. (USA), 75-3 (1978), 1060-1062. | MR 57 #6349 | Zbl 0391.42015

[6] E.M. Stein. — Singular integrals and differentiability properties of functions, Princeton University Press, Princeton N.J., 1970. | MR 44 #7280 | Zbl 0207.13501