On Polynomial Carleson Operators Along Quadratic Hypersurfaces

Abstract

We prove that a maximally modulated singular oscillatory integral operator along a hypersurface defined by \((y,Q(y))\subseteq \mathbb {R}^{n+1}\), for an arbitrary non-degenerate quadratic form Q, admits an a priori bound on \(L^p\) for all \(1<p<\infty \), 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 \(\{p_2,\ldots ,p_d\}\) 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 \(Q(y)=|y|^2\).