A maximal function for families of Hilbert transforms along homogeneous curves

Abstract

Let $H^{(u)}$ be the Hilbert transform along the parabola $(t,u{t}^2)$ where $u\in \mathbb{R}$. For a set $U$ of positive numbers consider the maximal function ${\mathcal{H}}^{U}f=\sup \{|H^{(u)}f|:u\in U\}$. We obtain an (essentially) optimal result for the $L^p$ operator norm of ${\mathcal{H}}^U$ when $2<p<\infty$. The results are proved for families of Hilbert transforms along more general nonflat homogeneous curves.

44 pages, fixed typos. A shortened version of this paper will appear in Mathematische Annalen