{
  "id": "2205.09851",
  "version": "v1",
  "published": "2022-05-19T20:51:05.000Z",
  "updated": "2022-05-19T20:51:05.000Z",
  "title": "The full range of uniform bounds for the bilinear Hilbert transform",
  "authors": [
    "Gennady Uraltsev",
    "Michał Warchalski"
  ],
  "comment": "107 pages, 1 figure. The authors welcome comments, corrections, and questions",
  "categories": [
    "math.CA"
  ],
  "abstract": "We prove uniform uniform $L^{p}$ bounds for the family of bilinear Hilbert transforms $\\mathrm{BHT}_{\\beta} [f_1, f_2] (x) := \\mathrm{p.v.} \\int_{\\mathbb{R}} f_1 (x - t) f_2 (x + \\beta t) \\frac{\\mathrm{d} t}{t}$. We show that the operator $\\mathrm{BHT}_{\\beta}$ maps $L^{p_{1}}\\times L^{p_{2}}$ into $L^{p}$ as long as $p_1 \\in (1, \\infty)$, $p_2 \\in (1, \\infty)$, and $p > \\frac{2}{3}$ with a bound independent of $\\beta\\in(0,1]$. This is the full open range of exponents where the modulation invariant class of bilinear operators containing $\\mathrm{BHT}_{\\beta}$ can be bounded uniformly. This is done by proving boundedness of certain affine transformations of the frequency-time-scale space $\\mathbb{R}^{3}_{+}$ in terms of iterated outer Lebesgue spaces. This results in new linear and bilinear wave packet embedding bounds well suited to study uniform bounds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-19T20:51:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B35",
      "46B70",
      "47A07",
      "47B38"
    ],
    "keywords": [
      "bilinear hilbert transform",
      "full range",
      "bilinear wave packet embedding bounds",
      "full open range",
      "iterated outer lebesgue spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 107,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}