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

For each integer $k\ge 3$, we establish the following estimate for all $1\le p<\mathrm{\infty }$ and $\epsilon >0$: $${\parallel \underset{t\in \mathbb{T}}{\sup }\mid \sum _{n=1}^{N}e(nx+n^{k}t)\mid \parallel }_{L^p(\mathbb{T})}{\ll }_{k,p,\epsilon }{N}^{\max \{1-1\mathrm{/}{K}_k,1-1\mathrm{/}p\}+\epsilon },$$ where $K_k=\min \{2^{k-1},k(k+1)\mathrm{/}2\}$, $e(z)=e^{2\pi iz}$, and $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$. For $k\ge 5$, 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 $N^{\epsilon }$ 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}.