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<p<∞, for each n≥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 {p₂,…,p_d} for any set of fixed real-valued polynomials p_j such that p_j 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.