On the Hilbert transform and lacunary directions in the plane

Mathematics – Classical Analysis and ODEs

Scientific paper

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11 pages, 1 fiure. This is version to appear in Illinois J. Math

Scientific paper

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 $1The 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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