On a planar Pierce--Yung operator

We show that the operator \begin{equation*} $\mathcal{C}$ f(x,y) := \sup_{v\in $\mathbb{R}$} \Big|$\mathrm{p}\mathrm{.}\mathrm{v}\mathrm{.}$ \int_{$\mathbb{R}$} f(x-t, y-t^2) e^{i v t^3} \frac{$\mathrm{d}$ t}{t} \Big| \end{equation*} is bounded on $L^p({\mathbb{R}}^2)$ for every $1<p<\mathrm{\infty }$. This gives an affirmative answer to a question of Pierce and Yung.