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))⊆Rⁿ⁺¹, for an arbitrary non-degenerate quadratic form Q, admits an a priori bound on L^p for all 1

2,…,pd} for any set of fixed real-valued polynomials pj such that pj is homogeneous of degree j, and p2 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)=|y|².