On the Hilbert transform and lacunary directions in the plane

Details On the Hilbert transform and lacunary directions in the plane On the Hilbert transform and lacunary directions in the plane

2000-10-20

arxiv.org/abs/math/0010196v2

Mathematics

Classical Analysis and ODEs

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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