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

Abstract

Let $M^{(u)}$, $H^{(u)}$ be the maximal operator and Hilbert transform along the parabola $(t,u{t}^2)$. 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.

14 pages, final version to appear in Proceedings of the Edinburgh Mathematical Society