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Furthermore, let H(0) denote the Friedrichs realization of $\\nabla^*\\nabla$ and let $V$ be a potential. We prove that $V^-$ is H(0)-form bounded with bound $<1$, if the function $\\max\\sigma(V^-)$ is in the Kato class of $(M,g)$. In particular, this gives a sufficient condition under which one can define the form sum $H(V):=H(0)\\dotplus V$ on arbitrary Riemannian manifolds."},"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":"1105.0532","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-05-03T10:11:56Z","cross_cats_sorted":["math.AP","math.DG","math.MP","math.SP"],"title_canon_sha256":"032f984b59f346a20dae217eb434ca0d457c0540a3083ea5f0b158b3fd06b885","abstract_canon_sha256":"fc4313053d27651419c62fe287e64319af13a1464a979677c434d2ce1bf24ddf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:13:06.956943Z","signature_b64":"cE5QYbxUjYsx1OyLoBC54O57VqMk9xlM9VZxdCl5BoGpxiO+9zg5DLBA7JruuGDN6f8r9hnheowvC0vIRQOmCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"004a9558c1157686b17f5d4e61d6c8b34c13591d840f43b4916add5b4af80a1c","last_reissued_at":"2026-05-18T04:13:06.956501Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:13:06.956501Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Kato's inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DG","math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Batu G\\\"uneysu","submitted_at":"2011-05-03T10:11:56Z","abstract_excerpt":"Let $(M,g)$ be a Riemannian manifold with Laplace-Beltrami operator $-\\Delta$ and let $E\\to M$ be a Hermitian vector bundle with a Hermitian covariant derivative $\\nabla$. Furthermore, let H(0) denote the Friedrichs realization of $\\nabla^*\\nabla$ and let $V$ be a potential. We prove that $V^-$ is H(0)-form bounded with bound $<1$, if the function $\\max\\sigma(V^-)$ is in the Kato class of $(M,g)$. In particular, this gives a sufficient condition under which one can define the form sum $H(V):=H(0)\\dotplus V$ on arbitrary Riemannian manifolds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0532","kind":"arxiv","version":3},"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":"1105.0532","created_at":"2026-05-18T04:13:06.956563+00:00"},{"alias_kind":"arxiv_version","alias_value":"1105.0532v3","created_at":"2026-05-18T04:13:06.956563+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.0532","created_at":"2026-05-18T04:13:06.956563+00:00"},{"alias_kind":"pith_short_12","alias_value":"ABFJKWGBCV3I","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_16","alias_value":"ABFJKWGBCV3INML7","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_8","alias_value":"ABFJKWGB","created_at":"2026-05-18T12:26:24.575870+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/ABFJKWGBCV3INML7LVHGDVWIWN","json":"https://pith.science/pith/ABFJKWGBCV3INML7LVHGDVWIWN.json","graph_json":"https://pith.science/api/pith-number/ABFJKWGBCV3INML7LVHGDVWIWN/graph.json","events_json":"https://pith.science/api/pith-number/ABFJKWGBCV3INML7LVHGDVWIWN/events.json","paper":"https://pith.science/paper/ABFJKWGB"},"agent_actions":{"view_html":"https://pith.science/pith/ABFJKWGBCV3INML7LVHGDVWIWN","download_json":"https://pith.science/pith/ABFJKWGBCV3INML7LVHGDVWIWN.json","view_paper":"https://pith.science/paper/ABFJKWGB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1105.0532&json=true","fetch_graph":"https://pith.science/api/pith-number/ABFJKWGBCV3INML7LVHGDVWIWN/graph.json","fetch_events":"https://pith.science/api/pith-number/ABFJKWGBCV3INML7LVHGDVWIWN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ABFJKWGBCV3INML7LVHGDVWIWN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ABFJKWGBCV3INML7LVHGDVWIWN/action/storage_attestation","attest_author":"https://pith.science/pith/ABFJKWGBCV3INML7LVHGDVWIWN/action/author_attestation","sign_citation":"https://pith.science/pith/ABFJKWGBCV3INML7LVHGDVWIWN/action/citation_signature","submit_replication":"https://pith.science/pith/ABFJKWGBCV3INML7LVHGDVWIWN/action/replication_record"}},"created_at":"2026-05-18T04:13:06.956563+00:00","updated_at":"2026-05-18T04:13:06.956563+00:00"}