Ji-Ye Yuan

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

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^{2\pi i...