Maximal polynomial modulations of singular integrals

Let $K$ be a standard H"older continuous Calder'on–Zygmund kernel on ${\mathbb{R}}^{\mathbf{d}}$ whose truncations define $L^2$ bounded operators. We show that the maximal operator obtained by modulating $K$ by polynomial phases of a fixed degree is bounded on $L^p({\mathbb{R}}^{\mathbf{d}})$ for $1 < p < \infty$. This extends Sj"olin’s multidimensional Carleson theorem and Lie’s polynomial Carleson theorem.