A polynomial Carleson operator along the paraboloid

• Lillian B. Pierce ^[1] ; Po-Lam Yung ^[2]
  1. [1] Duke University

     Duke University

     Township of Durham, Estados Unidos

     
  2. [2] Chinese University of Hong Kong

     Chinese University of Hong Kong

     RAE de Hong Kong (China)

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 35, Nº 2, 2019, págs. 339-422
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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 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.