{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:VBMBV6LYQ7DO3QVITYMJ5ICXBB","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":"b0ca9786fb65c9a1a96c5ccee7f87571e21b2079991d8a566d7f50c803fca18e","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-12-15T11:13:35Z","title_canon_sha256":"10359197aa8cca42e87a70e146a3b9de716f428a9922312fd21c91dd8ad75437"},"schema_version":"1.0","source":{"id":"1612.07144","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.07144","created_at":"2026-05-18T00:54:14Z"},{"alias_kind":"arxiv_version","alias_value":"1612.07144v1","created_at":"2026-05-18T00:54:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.07144","created_at":"2026-05-18T00:54:14Z"},{"alias_kind":"pith_short_12","alias_value":"VBMBV6LYQ7DO","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VBMBV6LYQ7DO3QVI","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VBMBV6LY","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:b899c2f3a78ca99c44029f12edc2774cc1def2810466a614c5d6e5e831739d05","target":"graph","created_at":"2026-05-18T00:54:14Z","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 paper, we consider nonlocal Schr\\\"odinger equations with certain potentials $V$ given by an integro-differential operator $L_K$ as follows; \\begin{equation*}L_K u+V u=f\\,\\,\\text{ in $\\BR^n$ }\\end{equation*} where $V\\in\\rh^q$ for $q>\\f{n}{2s}$ and $0<s<1$. We denote the solution of the above equation by $\\cS_V f:=u$, which is called {\\it the inverse of the nonlocal Schr\\\"odinger operator $L_K+V$ with potential $V$}; that is, $\\cS_V=(L_K+V)^{-1}$. Then we obtain a weak Harnack inequality of weak subsolutions of the nonlocal equation \\begin{equation}\\begin{cases}L_K u+V u=0\\,\\,&\\text{ in ","authors_text":"Woocheol Choi, Yong-Cheol Kim","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-12-15T11:13:35Z","title":"$L^p$ mapping properties for nonlocal Schr\\\"odinger operators with certain potential"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.07144","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:f183e51fbc76cbf180f46cd073003fc80215900a86eb1a4a23025646c04e6460","target":"record","created_at":"2026-05-18T00:54:14Z","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":"b0ca9786fb65c9a1a96c5ccee7f87571e21b2079991d8a566d7f50c803fca18e","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-12-15T11:13:35Z","title_canon_sha256":"10359197aa8cca42e87a70e146a3b9de716f428a9922312fd21c91dd8ad75437"},"schema_version":"1.0","source":{"id":"1612.07144","kind":"arxiv","version":1}},"canonical_sha256":"a8581af97887c6edc2a89e189ea057084af26bf5621589a7a6c7715d7b67a819","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a8581af97887c6edc2a89e189ea057084af26bf5621589a7a6c7715d7b67a819","first_computed_at":"2026-05-18T00:54:14.584673Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:54:14.584673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"V0JxHgSQPaYk6BiK6Ui3/BJJAPfm7IyngGzB/ado0f/pKq3Um1SbNHnG+pEuiv3wa5W/fC6im7SmOLNYpIpvAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:54:14.585108Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.07144","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f183e51fbc76cbf180f46cd073003fc80215900a86eb1a4a23025646c04e6460","sha256:b899c2f3a78ca99c44029f12edc2774cc1def2810466a614c5d6e5e831739d05"],"state_sha256":"19cd69709d7e2920e84367aa1aa19457b62d57ba9f073b5da1abfdec0c8b2b54"}