A degree one Carleson operator along the paraboloid

We prove $L^p$ bounds, $\frac{d^2+4d+2}{(d+1{)}^2}<p<2(d+1)$, for maximal linear modulations of singular integrals along paraboloids with frequencies in certain subspaces of ${\mathbb{R}}^{d+1}$, for $d\ge 2$. This generalizes Carleson’s theorem on convergence of Fourier series, and complements a corresponding result by Pierce and Yung with polynomial modulations without linear terms.