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Let $p(\\cdot):\\ \\mathbb{R}^n\\to(0,\\,\\infty)$ be a variable exponent function satisfying the globally $\\log$-H\\\"{o}lder continuous condition. When $p(\\cdot):\\ \\mathbb{R}^n\\to(1,\\,\\infty)$, the authors prove that the operators $VL^{-1}$, $V^{1/2}\\nabla L^{-1}$ and $\\nabla^2L^{-1}$ are b"},"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":"1811.10768","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-11-27T01:53:00Z","cross_cats_sorted":[],"title_canon_sha256":"53e7772f804be047819968690de390202f047a1cdf42cedf17d5f0758775171a","abstract_canon_sha256":"393c854fe3ee1c3195b48cd4a43f4e4e478dccbc16a50d7dd89d40111f832c30"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:47.592180Z","signature_b64":"WDb2qXIrCdhJXFB61e1+WM67qNXeADGvhTnbEH7mCKTlD+fXL7zI4KDYAXkfcTnZOFwVchSQSyStaym3/JwnCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e2e15746a2ba1ae2a5d108ff453cf6ea78729cf3681acf900abb4e658e66938c","last_reissued_at":"2026-05-17T23:59:47.591794Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:47.591794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Estimates of Schr\\\"{o}dinger Type Operators on Variable Lebesgue and Hardy Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Junqiang Zhang, Zongguang Liu","submitted_at":"2018-11-27T01:53:00Z","abstract_excerpt":"In this article, the authors consider the Schr\\\"{o}dinger type operator $L:=-{\\rm div}(A\\nabla)+V$ on $\\mathbb{R}^n$ with $n\\geq 3$, where the matrix $A$ satisfies uniformly elliptic condition and the nonnegative potential $V$ belongs to the reverse H\\\"{o}lder class $RH_q(\\mathbb{R}^n)$ with $q\\in(n/2,\\,\\infty)$. 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When $p(\\cdot):\\ \\mathbb{R}^n\\to(1,\\,\\infty)$, the authors prove that the operators $VL^{-1}$, $V^{1/2}\\nabla L^{-1}$ and $\\nabla^2L^{-1}$ are b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.10768","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":"1811.10768","created_at":"2026-05-17T23:59:47.591853+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.10768v1","created_at":"2026-05-17T23:59:47.591853+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.10768","created_at":"2026-05-17T23:59:47.591853+00:00"},{"alias_kind":"pith_short_12","alias_value":"4LQVORVCXINO","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_16","alias_value":"4LQVORVCXINOFJOR","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_8","alias_value":"4LQVORVC","created_at":"2026-05-18T12:32:05.422762+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/4LQVORVCXINOFJORBD7UKPHW5J","json":"https://pith.science/pith/4LQVORVCXINOFJORBD7UKPHW5J.json","graph_json":"https://pith.science/api/pith-number/4LQVORVCXINOFJORBD7UKPHW5J/graph.json","events_json":"https://pith.science/api/pith-number/4LQVORVCXINOFJORBD7UKPHW5J/events.json","paper":"https://pith.science/paper/4LQVORVC"},"agent_actions":{"view_html":"https://pith.science/pith/4LQVORVCXINOFJORBD7UKPHW5J","download_json":"https://pith.science/pith/4LQVORVCXINOFJORBD7UKPHW5J.json","view_paper":"https://pith.science/paper/4LQVORVC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.10768&json=true","fetch_graph":"https://pith.science/api/pith-number/4LQVORVCXINOFJORBD7UKPHW5J/graph.json","fetch_events":"https://pith.science/api/pith-number/4LQVORVCXINOFJORBD7UKPHW5J/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4LQVORVCXINOFJORBD7UKPHW5J/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4LQVORVCXINOFJORBD7UKPHW5J/action/storage_attestation","attest_author":"https://pith.science/pith/4LQVORVCXINOFJORBD7UKPHW5J/action/author_attestation","sign_citation":"https://pith.science/pith/4LQVORVCXINOFJORBD7UKPHW5J/action/citation_signature","submit_replication":"https://pith.science/pith/4LQVORVCXINOFJORBD7UKPHW5J/action/replication_record"}},"created_at":"2026-05-17T23:59:47.591853+00:00","updated_at":"2026-05-17T23:59:47.591853+00:00"}