{
  "id": "2407.17096",
  "version": "v1",
  "published": "2024-07-24T08:49:11.000Z",
  "updated": "2024-07-24T08:49:11.000Z",
  "title": "Gaussian Poincaré inequalities on the half-space with singular weights",
  "authors": [
    "Luigi Negro",
    "Chiara Spina"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove Rellich-Kondrachov type theorems and weighted Poincar\\'{e} inequalities on the half-space $\\mathbb{R}^{N+1}_+=\\{z=(x,y): x \\in \\mathbb{R}^N, y>0\\}$ endowed with the weighted Gaussian measure $\\mu :=y^ce^{-a|z|^2}dz$ where $c+1>0$ and $a>0$. We prove that for some positive constant $C>0$ one has \\begin{align*} \\left\\|u-\\overline u\\right\\|_{L^2_\\mu(\\mathbb{R}^{N+1}_+)}\\leq C \\|\\nabla u\\|_{L^2_\\mu (\\mathbb{R}^{N+1}_+)},\\qquad \\forall u\\in H^1_\\mu(\\mathbb{R}^{N+1}_+) \\end{align*} where $\\overline u=\\frac 1{\\mu(\\mathbb{R}^{N+1}_+)}\\int_{\\mathbb{R}^{N+1}_+} u\\,d\\mu(z)$. Besides this we also consider the local case of bounded domains of $\\mathbb{R}^{N+1}_+$ where the measure $\\mu$ is $y^cdz$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-24T08:49:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K08",
      "35K67",
      "47D07",
      "35J70",
      "35J75",
      "35B65"
    ],
    "keywords": [
      "singular weights",
      "inequalities",
      "half-space",
      "rellich-kondrachov type theorems",
      "weighted gaussian measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}