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 ) ) ⊆ R n + 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 { p 2 , … , 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 .