{
  "id": "2110.15774",
  "version": "v2",
  "published": "2021-10-28T10:13:59.000Z",
  "updated": "2024-01-06T14:44:29.000Z",
  "title": "A note on the generalized Hausdorff and packing measures of product sets in metric space",
  "authors": [
    "Rihab Guedri",
    "Najmeddine Attia"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $\\mu$ and $\\nu$ be two Borel probability measures on two separable metric spaces $\\X$ and $\\Y$ respectively. For $h, g$ be two Hausdorff functions and $q\\in \\R$, we introduce and investigate the generalized pseudo-packing measure ${\\RRR}_{\\mu}^{q, h}$ and the weighted generalized packing measure ${\\QQQ}_{\\mu}^{q, h}$ to give some product inequalities : $${\\HHH}_{\\mu\\times \\nu}^{q, hg}(E\\times F) \\le {\\HHH}_{\\mu}^{q, h}(E) \\; {\\RRR}_{\\nu}^{q, g}(F) \\le {\\RRR}_{\\mu\\times \\nu}^{q, hg}(E\\times F)$$ and $${\\PPP}_{\\mu\\times \\nu}^{q, hg}(E\\times F) \\le {\\QQQ}_{\\mu}^{q, h}(E) \\; {\\PPP}_{\\nu}^{q, g}(F) $$ for all $E\\subseteq \\X$ and $F\\subseteq \\Y$, where ${\\HHH}_{\\mu}^{q, h}$ and ${\\PPP}_{\\mu}^{q, h}$ is the generalized Hausdorff and packing measures respectively. As an application, we prove that under appropriate geometric conditions, there exists a constant $c$ such that $${\\HHH}_{\\mu\\times \\nu}^{q, hg}(E\\times F) \\le c\\, {\\HHH}_{\\mu}^{q, h}(E) \\; {\\PPP}_{\\nu}^{q, g}(F) $$ $${\\HHH}_{\\mu}^{q, h}(E) \\; {\\PPP}_{\\nu}^{q, g}(F) \\le c\\, {\\PPP}_{\\mu}^{q, hg}(E \\times F) $$ $${\\PPP}_{\\mu\\times \\nu}^{q, hg}(E\\times F) \\le c\\, {\\PPP}_{\\mu}^{q, h}(E) \\; {\\PPP}_{\\nu}^{q, g}(F). $$ These appropriate inequalities are more refined than well know results since we do no assumptions on $\\mu, \\nu, h$ and $g$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-01-06T14:44:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "packing measure",
      "generalized hausdorff",
      "product sets",
      "borel probability measures",
      "appropriate geometric conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}