Sharp maximal function estimates for Hilbert transforms along monomial curves in higher dimensions

Abstract

For any nonempty set $U\subset {\mathbb{R}}^{+}$, we consider the maximal operator $\h^U$ defined as $\h^Uf=\sup_{u\in U}|H^{(u)} f|$, where $H^{(u)}$ represents the Hilbert transform along the monomial curve $u\gamma (s)$. We focus on the $L^p({\mathbb{R}}^d)$ operator norm of $\h^U$ for $p\in (p_{\circ }(d),\infty )$, where $p_{\circ }(d)$ is the optimal exponent known for the $L^p$ boundedness of the maximal averaging operator obtained by Ko-Lee-Oh \cite{KLO22,KLO23} and Beltran-Guo-Hickman-Seeger \cite{BGHS}. To achieve this goal, we employ a novel bootstrapping argument to establish a maximal estimate for the Mihlin-Hörmander-type multiplier, along with utilizing the local smoothing estimate for the averaging operator and its vector-valued extension to obtain crucial decay estimates. Furthermore, our approach offers an alternative means for deriving the upper bound established in \cite{Guo20}.

Final version, to appear in JFAA