On Polynomial Carleson Operators Along Quadratic Hypersurfaces

Abstract

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 Lp 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,…,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.