{
  "id": "2407.05187",
  "version": "v1",
  "published": "2024-07-06T21:19:59.000Z",
  "updated": "2024-07-06T21:19:59.000Z",
  "title": "Dimension dependence of factorization problems: Haar system Hardy spaces",
  "authors": [
    "Thomas Speckhofer"
  ],
  "comment": "20 pages, 2 figures",
  "categories": [
    "math.FA"
  ],
  "abstract": "For $n\\in \\mathbb{N}$, let $Y_n$ denote the linear span of the first $n+1$ levels of the Haar system in a Haar system Hardy space $Y$ (this class contains all separable rearrangement-invariant function spaces and also related spaces such as dyadic $H^1$). Let $I_{Y_n}$ denote the identity operator on $Y_n$. We prove the following quantitative factorization result: Fix $\\Gamma,\\delta,\\varepsilon > 0$, and let $n,N \\in \\mathbb{N}$ be chosen such that $N \\ge Cn^2$, where $C = C(\\Gamma,\\delta,\\varepsilon) > 0$ (this amounts to a quasi-polynomial dependence between $\\dim Y_N$ and $\\dim Y_n$). Then for every linear operator $T\\colon Y_N\\to Y_N$ with $\\|T\\|\\le \\Gamma$, there exist operators $A,B$ with $\\|A\\|\\|B\\|\\le 2(1+\\varepsilon)$ such that either $I_{Y_n} = ATB$ or $I_{Y_n} = A(I_{Y_N} - T)B$. Moreover, if $T$ has $\\delta$-large positive diagonal with respect to the Haar system, then we have $I_{Y_n} = ATB$ for some $A,B$ with $\\|A\\|\\|B\\|\\le (1+\\varepsilon)/\\delta$. If the Haar system is unconditional in $Y$, then an inequality of the form $N \\ge Cn$ is sufficient for the above statements to hold (hence, $\\dim Y_N$ depends polynomially on $\\dim Y_n$). Finally, we prove an analogous result in the case where $T$ has large but not necessarily positive diagonal entries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-06T21:19:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A68",
      "46B07",
      "46B25",
      "46E30",
      "30H10"
    ],
    "keywords": [
      "haar system hardy space",
      "factorization problems",
      "dimension dependence",
      "separable rearrangement-invariant function spaces",
      "positive diagonal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}