{
  "id": "2212.07236",
  "version": "v1",
  "published": "2022-12-14T14:17:07.000Z",
  "updated": "2022-12-14T14:17:07.000Z",
  "title": "Hardy inequalities on metric measure spaces, IV: The case $p=1$",
  "authors": [
    "Michael Ruzhansky",
    "Anjali Shriwastawa",
    "Bankteshwar Tiwari"
  ],
  "comment": "11 pages, comments and suggestions are welcome",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case $p=1$ and $1 \\leq q <\\infty.$ This result complements the Hardy inequalities obtained in \\cite{RV} in the case $1< p\\le q<\\infty.$ The case $p=1$ requires a different argument and does not follow as the limit of known inequalities for $p>1.$ As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan-Hadamard manifolds for the case $p=1$ and $1\\le q<\\infty.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-14T14:17:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "inequality",
      "measure space possessing polar decompositions",
      "metric measure space possessing polar",
      "two-weight hardy inequalities",
      "best constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}