# Revista Matemática Iberoamericana

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**Volume 35, Issue 2, 2019, pp. 339–422**

**DOI: 10.4171/rmi/1057**

Published online: 2019-03-05

A polynomial Carleson operator along the paraboloid

Lillian B. Pierce^{[1]}and Po-Lam Yung

^{[2]}(1) Duke University, Durham, USA

(2) The Chinese University of Hong Kong, Hong Kong

In this work we extend consideration of the well-known 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 $\mathbb R^{n+1}$.

*Keywords: *Carleson operator, Radon transform, oscillatory integrals, square function, van der Corput estimates

Pierce Lillian, Yung Po-Lam: A polynomial Carleson operator along the paraboloid. *Rev. Mat. Iberoam.* 35 (2019), 339-422. doi: 10.4171/rmi/1057