{
  "id": "2406.11298",
  "version": "v1",
  "published": "2024-06-17T08:05:12.000Z",
  "updated": "2024-06-17T08:05:12.000Z",
  "title": "New characterization of weighted inequalities involving superposition of Hardy integral operators",
  "authors": [
    "Amiran Gogatishvili",
    "Tuğuçe Ünver"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:2201.11437",
  "categories": [
    "math.AP",
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $1\\leq p <\\infty$ and $0 < q,r < \\infty$. We characterize the validity of the inequality for the composition of the Hardy operator, \\begin{equation*} \\bigg(\\int_a^b \\bigg(\\int_a^x \\bigg(\\int_a^t f(s)ds \\bigg)^q u(t) dt \\bigg)^{\\frac{r}{q}} w(x) dx \\bigg)^{\\frac{1}{r}} \\leq C \\bigg(\\int_a^b f(x)^p v(x) dx \\bigg)^{\\frac{1}{p}} \\end{equation*} for all non-negative measurable functions $f$ on $(a,b)$, $-\\infty \\leq a < b \\leq \\infty$. We construct a more straightforward discretization method than those previously presented in the literature, and we provide some new scales of weight characterizations of this inequality in both discrete and continuous forms and we obtain previous characterizations as the special case of the parameter.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-17T08:05:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26D10",
      "26D15"
    ],
    "keywords": [
      "hardy integral operators",
      "weighted inequalities",
      "inequality",
      "superposition",
      "special case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}