Sharp Maximal Function Estimates for Hilbert Transforms Along Monomial Curves in Higher Dimensions

For any nonempty set \(U\subset {\mathbb {R}}^+\) $U\subset {\mathbb{R}}^{+}$ , we consider the maximal operator \({\mathcal {H}}^U\) ${\mathcal{H}}^U$ defined as \({\mathcal {H}}^Uf=\sup _{u\in U}|H^{(u)} f|\) ${\mathcal{H}}^{U}f={sup}_{u\in U}|H^{(u)}f|$ , where \(H^{(u)}\) $H^{(u)}$ represents the Hilbert transform along the monomial curve \(u\gamma (s)\) $u\gamma (s)$ . We focus on the \(L^p({\mathbb {R}}^d)\) $L^p({\mathbb{R}}^d)$ operator norm of \({\mathcal {H}}^U\) ${\mathcal{H}}^U$ for \(p\in (p_\circ (d),\infty )\) $p\in (p_{\circ }(d),\infty )$ , where \(p_\circ (d)\) $p_{\circ }(d)$ is the optimal exponent known for the \(L^p\) $L^p$ boundedness of the maximal averaging operator obtained by Ko–Lee–Oh (Invent Math 228:991–1035, 2022, Forum Math Pi 11:Paper No. e4, 33, 2023) and Beltran–Guo–Hickman–Seeger (Am J Math, https://arXiv.org/abs/2102.08272). 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 Guo–Roos–Seeger–Yung (Math Ann 377:69–114, 2020).