On Polynomial Carleson operators along quadratic hypersurfaces

We prove that a maximally modulated singular oscillatory integral operator along a hypersurface defined by $(y,Q(y))\subseteq {\mathbb{R}}^{n+1}$, for an arbitrary non-degenerate quadratic form $Q$, admits an a priori bound on $L^p$ for all $1<p<\mathrm{\infty }$, for each $n\ge 2$. This operator takes the form of a polynomial Carleson operator of Radon-type, in which the maximally modulated phases lie in the real span of $\{p_2,\dots ,p_d\}$ for any set of fixed real-valued polynomials $p_j$ such that $p_j$ is homogeneous of degree $j$, and $p_2$ is not a multiple of $Q(y)$. The general method developed in this work applies to quadratic forms of arbitrary signature, while previous work considered only the special positive definite case $Q(y)=\mathrm{\mid }y{\mathrm{\mid }}^2$.