Sharp maximal function estimates for Hilbert transforms along monomial curves in higher dimensions (Q6632430)

This paper is concerned with maximal function estimates for Hilbert transforms along curves in higher dimensions.\N\NFor an integer \(d \geq 2\), let \(\{\alpha_l\}_{l=1}^d\) be a sequence of distinct positive constants. Define the curve \(\gamma : \mathbb{R} \rightarrow \mathbb{R}^d\) by \(\gamma(s) = (s^{\alpha_1}, s^{\alpha_2}, \ldots, s^{\alpha_d})\). The Hilbert transform along the curve \(u\gamma(s)\) for \(u > 0\) is defined by \N\[\NH^{(u)}f(x) = p. v. \int_\mathbb{R} f(x + u\gamma(s))\frac{ds}{s}\;.\N\]\NFor an arbitrary nonempty set \(U \subset \mathbb{R}^+\), we define the related maximal function \N\[\N\mathcal{H}^U f(x) = \sup_{u \in U}\left|H^{(u)}f(x)\right|\;.\N\]\NAlso associated to \(U\) is the constant \N\[\N\mathfrak{R}(U) =\#\{n \in \mathbb{Z} : [2^n, 2^{n+1}) \cap U \neq \emptyset\}\;.\N\]\NWe define \(p_0(d)\) by \N\[\Np_0(d) = \begin{cases} d & if \;\; d = 2,3 \\\N2d - 2 & if \;\; d \geq 4\;. \end{cases} \N\]\NThe main theorem of the paper is the following. \N\NLet \(d \geq 3\). For each \(p \in (p_0(d), \infty)\), the operator \(\mathcal{H}^U\) is \(L^p\)-bounded if and only if \(\mathfrak{R}(U) < \infty\). Moreover, we have \N\[\N\left\|\mathcal{H}^U\right\|_{L^p \rightarrow L^p} \sim \sqrt{\log ( e + \mathfrak{R}(U))}\;.\N\]