{
  "annotation_id": "eb40dd80-8fab-44d0-8df5-1142550b6fdf",
  "date_created": "2026-02-17T05:53:20.237000Z",
  "date_modified": "2026-02-17T05:53:20.237000Z",
  "file_hash": "ee54b19ca002e2ae40e72a7f98701aec84f55a300793b88d69617a361b6675fc",
  "private": false,
  "record": {
    "abstract": "In this article, we establish a nearly sharp localized weighted inequality related to Gagliardo and Sobolev seminorms, respectively, with the sharp $A_1$-weight constant or with the specific $A_p$-weight constant when $p\\in (1,\\infty)$. As applications, we further obtain a new characterization of Muckenhoupt weights and, in the framework of ball Banach function spaces, an inequality related to Gagliardo and Sobolev seminorms on cubes, a Gagliardo--Nirenberg interpolation inequality, and a Bourgain--Brezis--Mironescu formula. All these obtained results have wide generality and are proved to be (nearly) sharp.\n  The original version of this article was published in [Adv. Math. 481 (2025), Paper No. 110537]. In this revised version, we correct an error appeared in Theorem 1.1 in the case where $p=1$, which was pointed out to us by Emiel Lorist.",
    "arxiv_id": "2601.09094",
    "authors": [
      "Pingxu Hu",
      "Yinqin Li",
      "Dachun Yang",
      "Wen Yuan"
    ],
    "categories": [
      "math.FA",
      "math.AP",
      "math.CA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "A Sharp Localized Weighted Inequality Related to Gagliardo and Sobolev Seminorms and Its Applications",
    "url": "https://arxiv.org/abs/2601.09094",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "5ad36ec0-5662-4a19-9dab-4040d4576ab9",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}