A degree one Carleson operator along the paraboloid

Abstract

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 $d \geq 2$. 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.