2015/05/14 by Lillian B. Pierce, Po‐Lam Yung, Pierce, L. B. +1 · 1 citation
Mathematics · #42B20 #42B25 #43A50 #44A12 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.1505.03882
openalex publication_date 2015/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we extend consideration of the polynomial Carleson operator to the setting of a Radon transform acting along the paraboloid in ℝn+1 for n ≥ 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 ℝn+1.