Improved $L^p$ maximal estimates for higher-degree Weyl sums

Abstract

For each integer $k\ge3$, we establish the following estimate for all $1\le p<\infty$ and $\varepsilon>0$: \[ \left\|\sup_{t\in\mathbb{T}}\left|\sum_{n=1}^N e(nx+n^kt)\right|\right\|_{L^p(\mathbb{T})} \ll_{k,p,\varepsilon} N^{\max\{1-1/K_k,\,1-1/p\}+\varepsilon}, \] where $K_k=\min\{2^{k-1},k(k+1)/2\}$, $e(z)=e^{2πiz}$, and $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. For $k\ge5$, this improves our earlier bound \cite{chen-miao-yuan-zhao} for $1\le p<D_k$, where $D_k=\min\{2^{k-1},k(k-1)\}$, and, together with the corresponding lower bound, extends the sharp range up to $N^\varepsilon$ from $p\ge D_k$ to $p\ge K_k$. The main new ingredient is a mixed minor-arc moment estimate obtained before taking the time supremum. The minor-arc maximal estimate then follows from a smooth reproducing kernel argument, while the major arcs are handled directly by Poisson summation and classical complete exponential-sum estimates. The reduction from suprema to mean values also yields a short and direct proof of improved $L^p$ estimates for maximal operators on Weyl sums with general polynomial phases, strengthening \cite[Theorem 1.1]{baker-chen-shparlinski}.