# A maximal function for families of Hilbert transforms along homogeneous curves

## TL;DR

This paper establishes nearly optimal bounds for the $L^p$ norms of maximal functions formed by Hilbert transforms along families of homogeneous curves, extending results to more general nonflat curves.

## Contribution

It provides the first essentially sharp $L^p$ bounds for maximal Hilbert transforms along families of homogeneous curves, including nonflat cases.

## Key findings

- Established sharp $L^p$ bounds for maximal Hilbert transforms along parabola families.
- Extended results to more general nonflat homogeneous curves.
- Achieved bounds for $2<p<
finite$ that are essentially optimal.

## Abstract

Let $H^{(u)}$ be the Hilbert transform along the parabola $(t, ut^2)$ 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.