{
  "id": "1505.06479",
  "version": "v1",
  "published": "2015-05-24T20:49:18.000Z",
  "updated": "2015-05-24T20:49:18.000Z",
  "title": "Cancellation for the multilinear Hilbert transform",
  "authors": [
    "Terence Tao"
  ],
  "comment": "16 pages, no figures",
  "categories": [
    "math.CA",
    "math.CO"
  ],
  "abstract": "For any natural number $k$, consider the $k$-linear Hilbert transform $$ H_k( f_1,\\dots,f_k )(x) := \\operatorname{p.v.} \\int_{\\bf R} f_1(x+t) \\dots f_k(x+kt) \\frac{dt}{t}$$ for test functions $f_1,\\dots,f_k: {\\bf R} \\to {\\bf C}$. It is conjectured that $H_k$ maps $L^{p_1}({\\bf R}) \\times \\dots \\times L^{p_k}({\\bf R}) \\to L^p({\\bf R})$ whenever $1 < p_1,\\dots,p_k,p < \\infty$ and $\\frac{1}{p} = \\frac{1}{p_1} + \\dots + \\frac{1}{p_k}$. This is proven for $k=1,2$, but remains open for larger $k$. In this paper, we consider the truncated operators $$ H_{k,r,R}( f_1,\\dots,f_k )(x) := \\int_{r \\leq |t| \\leq R} f_1(x+t) \\dots f_k(x+kt) \\frac{dt}{t}$$ for $R > r > 0$. The above conjecture is equivalent to the uniform boundedness of $\\| H_{k,r,R} \\|_{L^{p_1}({\\bf R}) \\times \\dots \\times L^{p_k}({\\bf R}) \\to L^p({\\bf R})}$ in $r,R$, whereas the Minkowski and H\\\"older inequalities give the trivial upper bound of $2 \\log \\frac{R}{r}$ for this quantity. By using the arithmetic regularity and counting lemmas of Green and the author, we improve the trivial upper bound on $\\| H_{k,r,R} \\|_{L^{p_1}({\\bf R}) \\times \\dots \\times L^{p_k}({\\bf R}) \\to L^p({\\bf R})}$ slightly to $o( \\log \\frac{R}{r} )$ in the limit $\\frac{R}{r} \\to \\infty$ for any admissible choice of $k$ and $p_1,\\dots,p_k,p$. This establishes some cancellation in the $k$-linear Hilbert transform $T$, but not enough to establish its boundedness in $L^p$ spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-24T20:49:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30",
      "42B20"
    ],
    "keywords": [
      "multilinear hilbert transform",
      "trivial upper bound",
      "cancellation",
      "arithmetic regularity",
      "natural number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150506479T"
    }
  }
}