Discrete analogues of maximally modulated singular integrals of Stein–Wainger type

Consider the maximal operator $$\mathscr{C} f(x) = \sup_{\lambda\in\mathbb{R}}\Big|\sum_{\substack{y\in\mathbb{Z}^n\setminus\{0\}}} f(x-y) e(\lambda |y|^{2d}) K(y)\Big|,\quad (x\in\mathbb{Z}^n),$$ where $d$ is a positive integer, $K$ Calder\'on-Zygmund kernel and $n\ge 1$. This discrete analogue of real-variable studied by Stein Wainger. The nonlinearity phase introduces variety new difficulties that are not present in setting. We prove $\ell^2(\mathbb{Z}^n)$-bounds for $\mathscr{C}$, answering question posed Lillian Pierce.