{
  "annotation_id": "20911677-e748-4f1d-8b64-4abcb037c96f",
  "date_created": "2026-02-17T05:53:03.789000Z",
  "date_modified": "2026-02-17T05:53:03.789000Z",
  "file_hash": "44c67a165a048632604c420409bc105c21d5f2d13c089af5da42af65b769ab2e",
  "private": false,
  "record": {
    "abstract": "We establish a new $T1$ theorem for the compactness of bi-parameter Calder\\\u0027on-Zygmund singular integral operators. Namely, we show that if a bi-parameter CZO $T$ satisfies the product weak compactness property, the mixed weak compactness/CMO property, and $T1, T^t1,$ $T_t1, T_t^t1 \\in \\text{CMO}(\\mathbb{R}^{n_1}\\times\\mathbb{R}^{n_2})$, then $T$ is compact on $L^2(\\mathbb{R}^{n_1}\\times\\mathbb{R}^{n_2})$. We also obtain endpoint compactness results for these operators and use them to deduce the necessity of most of our hypotheses. In particular, our conditions characterize the simultaneous $L^2(\\mathbb{R}^{n_1}\\times\\mathbb{R}^{n_2})$-compactness of a bi-parameter CZO and its partial transpose. Our assumptions improve upon previously known sufficient conditions, and our proof, which is shorter and simpler than earlier arguments, utilizes a new abstract compactness criterion for partially localized operators on tensor products of Hilbert spaces.",
    "arxiv_id": "2601.05454",
    "authors": [
      "Cody B. Stockdale",
      "Cody Waters"
    ],
    "categories": [
      "math.CA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "On the compactness of bi-parameter singular integrals",
    "url": "https://arxiv.org/abs/2601.05454",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "c7d4a66d-2bf7-450d-b437-6da281d623d9",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}