On the Hilbert transform and lacunary directions in the plane
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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2000
 DOI:
 10.48550/arXiv.math/0010196
 arXiv:
 arXiv:math/0010196
 Bibcode:
 2000math.....10196L
 Keywords:

 Classical Analysis and ODEs;
 42b20
 EPrint:
 11 pages, 1 fiure. This is version to appear in Illinois J. Math