# Maximal functions associated with families of homogeneous curves: $L^p$ bounds for $p\le 2$

## TL;DR

This paper investigates $L^p$ bounds for maximal functions and Hilbert transforms along families of homogeneous curves, especially parabolas, for $p$ in the range (1,2), extending to more general curves.

## Contribution

It provides new $L^p$ estimates for maximal functions and Hilbert transforms along families of homogeneous curves, including non-flat cases, for $p
eq 2$.

## Key findings

- Established $L^p$ bounds for maximal functions along parabola families.
- Extended results to more general non-flat homogeneous curves.
- Focused on the range $1 < p extless 2$ for these operators.

## Abstract

Let $M^{(u)}$, $H^{(u)}$ be the maximal operator and Hilbert transform along the parabola $(t, ut^2) $. For $U\subset(0,\infty)$ we consider $L^p$ estimates for the maximal functions $\sup_{u\in U}|M^{(u)} f|$ and $\sup_{u\in U}|H^{(u)} f|$, when $1<p\le 2$. The parabolae can be replaced by more general non-flat homogeneous curves.