On Polynomial Carleson operators along quadratic hypersurfaces

TL;DR

This paper establishes $L^p$ bounds for a class of polynomial Carleson operators along quadratic hypersurfaces, extending previous results to arbitrary quadratic forms of any signature.

Contribution

It introduces a general method to prove boundedness of polynomial Carleson operators along quadratic hypersurfaces for all signatures, broadening the scope beyond positive definite forms.

Findings

Proved $L^p$ bounds for operators along quadratic hypersurfaces.

Extended previous positive definite results to arbitrary quadratic forms.

Developed a versatile method applicable to forms of any signature.

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 $n \geq 2$. 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 $Q(y)=|y|^2$.