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 ${\sup }_k|H_{k}f|$ 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.

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