{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:4LQVORVCXINOFJORBD7UKPHW5J","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"393c854fe3ee1c3195b48cd4a43f4e4e478dccbc16a50d7dd89d40111f832c30","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-11-27T01:53:00Z","title_canon_sha256":"53e7772f804be047819968690de390202f047a1cdf42cedf17d5f0758775171a"},"schema_version":"1.0","source":{"id":"1811.10768","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.10768","created_at":"2026-05-17T23:59:47Z"},{"alias_kind":"arxiv_version","alias_value":"1811.10768v1","created_at":"2026-05-17T23:59:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.10768","created_at":"2026-05-17T23:59:47Z"},{"alias_kind":"pith_short_12","alias_value":"4LQVORVCXINO","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_16","alias_value":"4LQVORVCXINOFJOR","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_8","alias_value":"4LQVORVC","created_at":"2026-05-18T12:32:05Z"}],"graph_snapshots":[{"event_id":"sha256:037cf82c895f76f844cb086fc553c8fbe31585dab50cd64acf35dfc64b69a85b","target":"graph","created_at":"2026-05-17T23:59:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $q\\in(n/2,\\,\\infty)$. 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","authors_text":"Junqiang Zhang, Zongguang Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-11-27T01:53:00Z","title":"Some Estimates of Schr\\\"{o}dinger Type Operators on Variable Lebesgue and Hardy Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.10768","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6280308d1db148c023b48b293829bef5017230bf0fe37b547ecdc1235f98951a","target":"record","created_at":"2026-05-17T23:59:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"393c854fe3ee1c3195b48cd4a43f4e4e478dccbc16a50d7dd89d40111f832c30","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-11-27T01:53:00Z","title_canon_sha256":"53e7772f804be047819968690de390202f047a1cdf42cedf17d5f0758775171a"},"schema_version":"1.0","source":{"id":"1811.10768","kind":"arxiv","version":1}},"canonical_sha256":"e2e15746a2ba1ae2a5d108ff453cf6ea78729cf3681acf900abb4e658e66938c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e2e15746a2ba1ae2a5d108ff453cf6ea78729cf3681acf900abb4e658e66938c","first_computed_at":"2026-05-17T23:59:47.591794Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:47.591794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WDb2qXIrCdhJXFB61e1+WM67qNXeADGvhTnbEH7mCKTlD+fXL7zI4KDYAXkfcTnZOFwVchSQSyStaym3/JwnCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:47.592180Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.10768","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6280308d1db148c023b48b293829bef5017230bf0fe37b547ecdc1235f98951a","sha256:037cf82c895f76f844cb086fc553c8fbe31585dab50cd64acf35dfc64b69a85b"],"state_sha256":"729efe6b4c3e0d616e2c0fd618020aad92cda62d5fdc541e802e251dcee02299"}