{
  "id": "1505.03882",
  "version": "v1",
  "published": "2015-05-14T20:37:33.000Z",
  "updated": "2015-05-14T20:37:33.000Z",
  "title": "A polynomial Carleson operator along the paraboloid",
  "authors": [
    "L. B. Pierce",
    "P. -L. Yung"
  ],
  "comment": "76 pages",
  "categories": [
    "math.CA"
  ],
  "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 $\\mathbb{R}^{n+1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-14T20:37:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "43A50",
      "42B25",
      "44A12"
    ],
    "keywords": [
      "paraboloid",
      "related maximal oscillatory integral operator",
      "van der corput estimates",
      "treat polynomial carleson operators",
      "original polynomial carleson operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 76,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150503882P"
    }
  }
}