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In this article, the authors establish some weighted $L^p$ estimates of Kato square roots associated to the degenerate elliptic operators $L_w$. More precisely, the authors prove that, for $w\\in A_{p}(\\mathbb{R}^n)$, $p\\in(\\frac{2n}{n+1},\\,2]$ and any $f\\in C^\\infty_c(\\mathbb{R}^n)$, $\\|L_w^{1/2}(f)\\|_{L^p(w,\\,\\mathbb{R}^n)}\\sim \\|\\nabla f\\|_{L^p(w,\\,\\mathbb{R}^n)}$, where $C_c^\\infty(\\mathbb{R}^n)$ denotes the set"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1509.05478","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-09-18T00:19:58Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"1fc5ec45bf58e503fbc25e985e456f41b8cbf205c0e9ad4efa1ef0f32ad522bd","abstract_canon_sha256":"22710b297e7de1badcbbf08c0c26081a5d643a3a8b48e567c3776a04deb8302d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:43.069531Z","signature_b64":"j5c/kLr6IGCMoWeGeV/bcpOGZ+imjVemIEK9/aZ1aSnbf1Oym0FNhpcyINpMhKEkuKn+XSOHyQmPZ+pSRsQPBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e89974a677f88c4260893a6d52bf5aff07ff695872d0a968130b4cd3c2cd57c","last_reissued_at":"2026-05-18T01:32:43.069084Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:43.069084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weighted $L^p$ Estimates of Kato Square Roots Associated to Degenerate Elliptic Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Dachun Yang, Junqiang Zhang","submitted_at":"2015-09-18T00:19:58Z","abstract_excerpt":"Let $w$ be a Muckenhoupt $A_2(\\mathbb{R}^n)$ weight and $L_w:=-w^{-1}\\mathop\\mathrm{div}(A\\nabla)$ the degenerate elliptic operator on the Euclidean space $\\mathbb{R}^n$, $n\\geq 2$. In this article, the authors establish some weighted $L^p$ estimates of Kato square roots associated to the degenerate elliptic operators $L_w$. More precisely, the authors prove that, for $w\\in A_{p}(\\mathbb{R}^n)$, $p\\in(\\frac{2n}{n+1},\\,2]$ and any $f\\in C^\\infty_c(\\mathbb{R}^n)$, $\\|L_w^{1/2}(f)\\|_{L^p(w,\\,\\mathbb{R}^n)}\\sim \\|\\nabla f\\|_{L^p(w,\\,\\mathbb{R}^n)}$, where $C_c^\\infty(\\mathbb{R}^n)$ denotes the set"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05478","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.05478","created_at":"2026-05-18T01:32:43.069161+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.05478v1","created_at":"2026-05-18T01:32:43.069161+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05478","created_at":"2026-05-18T01:32:43.069161+00:00"},{"alias_kind":"pith_short_12","alias_value":"L2EZOSTHP6EM","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_16","alias_value":"L2EZOSTHP6EMIJQI","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_8","alias_value":"L2EZOSTH","created_at":"2026-05-18T12:29:29.992203+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/L2EZOSTHP6EMIJQISOTNKK7VV7","json":"https://pith.science/pith/L2EZOSTHP6EMIJQISOTNKK7VV7.json","graph_json":"https://pith.science/api/pith-number/L2EZOSTHP6EMIJQISOTNKK7VV7/graph.json","events_json":"https://pith.science/api/pith-number/L2EZOSTHP6EMIJQISOTNKK7VV7/events.json","paper":"https://pith.science/paper/L2EZOSTH"},"agent_actions":{"view_html":"https://pith.science/pith/L2EZOSTHP6EMIJQISOTNKK7VV7","download_json":"https://pith.science/pith/L2EZOSTHP6EMIJQISOTNKK7VV7.json","view_paper":"https://pith.science/paper/L2EZOSTH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.05478&json=true","fetch_graph":"https://pith.science/api/pith-number/L2EZOSTHP6EMIJQISOTNKK7VV7/graph.json","fetch_events":"https://pith.science/api/pith-number/L2EZOSTHP6EMIJQISOTNKK7VV7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L2EZOSTHP6EMIJQISOTNKK7VV7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L2EZOSTHP6EMIJQISOTNKK7VV7/action/storage_attestation","attest_author":"https://pith.science/pith/L2EZOSTHP6EMIJQISOTNKK7VV7/action/author_attestation","sign_citation":"https://pith.science/pith/L2EZOSTHP6EMIJQISOTNKK7VV7/action/citation_signature","submit_replication":"https://pith.science/pith/L2EZOSTHP6EMIJQISOTNKK7VV7/action/replication_record"}},"created_at":"2026-05-18T01:32:43.069161+00:00","updated_at":"2026-05-18T01:32:43.069161+00:00"}