{
  "id": "2004.08988",
  "version": "v1",
  "published": "2020-04-19T23:16:55.000Z",
  "updated": "2020-04-19T23:16:55.000Z",
  "title": "Neccessary conditions for two weight inequalities for singular integral operators",
  "authors": [
    "David Cruz-Uribe",
    "John-Oliver MacLellan"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We prove necessary conditions on pairs of measures $(\\mu,\\nu)$ for a singular integral operator $T$ to satisfy weak $(p,p)$ inequalities, $1\\leq p<\\infty$, provided the kernel of $T$ satisfies a weak non-degeneracy condition first introduced by Stein, and the measure $\\mu$ satisfies a weak doubling condition related to the non-degeneracy of the kernel. We also show similar results for pairs of measures $(\\mu,\\sigma)$ for the operator $T_\\sigma f = T(f\\,d\\sigma)$, which has come to play an important role in the study of weighted norm inequalities. Our major tool is a careful analysis of the strong type inequalities for averaging operators; these results are of interest in their own right. Finally, as an application of our techniques, we show that in general a singular operator does not satisfy the endpoint strong type inequality $T : L^1(\\nu) \\rightarrow L^1(\\mu)$. Our results unify and extend a number of known results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-19T23:16:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B35"
    ],
    "keywords": [
      "singular integral operator",
      "weight inequalities",
      "neccessary conditions",
      "endpoint strong type inequality",
      "weak non-degeneracy condition first"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}