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.

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.

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author = {Carbery, Anthony},
title = {Differentiation in lacunary directions and an extension of the {Marcinkiewicz} multiplier theorem},
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Carbery, Anthony. 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. http://www.numdam.org/articles/10.5802/aif.1127/

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