A polynomial Carleson operator along the paraboloid

TL;DR

This paper extends the polynomial Carleson operator to the setting of the paraboloid in higher dimensions, developing new techniques involving oscillatory integral estimates and Littlewood-Paley theory.

Contribution

It introduces a novel approach to analyze polynomial Carleson operators along the paraboloid using van der Corput estimates and smoothing approximations.

Findings

Boundedness results for polynomial Carleson operators along the paraboloid.

Development of a method to approximate oscillatory integrals with smoother operators.

Extension of Carleson operator theory to higher-dimensional paraboloid settings.

Abstract

In this work we extend consideration of the polynomial Carleson operator to the setting of a Radon transform acting along the paraboloid in $\mathbb{R}^{n+1}$ for $n \geq 2$. Inspired by work of Stein and Wainger on the original polynomial Carleson operator, we develop a method to treat polynomial Carleson operators along the paraboloid via van der Corput estimates. A key new step in the approach of this paper is to approximate a related maximal oscillatory integral operator along the paraboloid by a smoother operator, which we accomplish via a Littlewood-Paley decomposition and the use of a square function. The most technical aspect then arises in the derivation of bounds for oscillatory integrals involving integration over lower-dimensional sets. The final theorem applies to polynomial Carleson operators with phase belonging to a certain restricted class of polynomials with no linear terms and whose homogeneous quadratic part is not a constant multiple of the defining function $|y|^2$ of the paraboloid in $\mathbb{R}^{n+1}$.