{
  "id": "2311.17356",
  "version": "v1",
  "published": "2023-11-29T05:08:28.000Z",
  "updated": "2023-11-29T05:08:28.000Z",
  "title": "Homogeneous Newton-Sobolev spaces in metric measure spaces and their Banach space properties",
  "authors": [
    "Nageswari Shanmugalingam"
  ],
  "comment": "10 pages",
  "categories": [
    "math.FA",
    "math.AP",
    "math.CV"
  ],
  "abstract": "In this note we prove the Banach space properties of the homogeneous Newton-Sobolev spaces $HN^{1,p}(X)$ of functions on an unbounded metric measure space $X$ equipped with a doubling measure supporting a $p$-Poincar\\'e inequality, and show that when $1<p<\\infty$, even with the lack of global $L^p$-integrability of functions in $HN^{1,p}(X)$, we have that every bounded sequence in $HN^{1,p}(X)$ has a strongly convergent convex-combination subsequence. The analogous properties for the inhomogeneous Newton-Sobolev classes $N^{1,p}(X)$ are proven elsewhere in existing literature",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-29T05:08:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E36",
      "31E05",
      "31L15"
    ],
    "keywords": [
      "banach space properties",
      "homogeneous newton-sobolev spaces",
      "unbounded metric measure space",
      "strongly convergent convex-combination subsequence",
      "inhomogeneous newton-sobolev classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}