Title: Maximal polynomial modulations of singular integrals,
Abstract: Let $K$ be a standard Hölder continuous Calderón--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 Sjölin's multidimensional Carleson theorem and Lie's
polynomial Carleson theorem.