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On the Hilbert transform and lacunary directions in the plane

Michael T. Lacey
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Abstract

Let $H_k$ be the one dimensional Hilbert transform computed in the direction $(1,2^k)$ in the plane. We show that the maximal operator $\sup_k |H_kf|$ maps $L^p$ of the plane into itself for $1<p<\infty$. The same result with the Hilbert transform replaced by the one dimensional maximal function was proved by Nagel, Stein and Wainger in 1978.

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Michael T. Lacey. 2001-12-21. On the Hilbert transform and lacunary directions in the plane. https://arxiv.org/abs/math/0010196

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