2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60168Let $H_k$ 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 1978.11 pages, 1 fiure. This is version to appear in Illinois J. MathClassical Analysis and ODEs42b20text