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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 for . 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 of the paraboloid in .
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Lillian B. Pierce, Po-Lam Yung, A polynomial Carleson operator along the paraboloid. Rev. Mat. Iberoam. 35 (2019), no. 2, pp. 339–422DOI 10.4171/RMI/1057