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Assume that the operator $$P_\\mu:=-\\Delta-\\frac{\\mu}{\\delta_\\Omega^2(x)}$$ is {\\em subcritical} in $\\Omega$, where $\\delta_\\Omega$ is the distance function to the boundary of $\\Omega$ and $\\mu \\leq 1/4$. We show that under some smoothness assumption on $\\Omega$, the following improved Hardy-type inequality \\begin{equation*}\n  \\int_{\\Omega}|\\nabla \\varphi|^2\\,\\mathrm{d}x -\\mu\\int_{\\Omega} \\frac{|\\varphi|^2}{\\delta_\\Omega^2}\\,\\mathrm{d}x \\geq \\lambda(\\mu)\\int_{\\Omega} \\frac{|\\varphi|^2}{|x|^2}\\,\\mathrm{d}x \\qquad"},"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":"1502.05205","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-02-18T12:53:54Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"73a83dc74ae4c7675d8f1a7de5c3fdcabfd2c130b77077b28a924e4192ce1331","abstract_canon_sha256":"038ce600923a510f026782ed100a9a3682ca35d76c30d3393a9cedf9aa4be62c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:52.181890Z","signature_b64":"lPHajUvcbpQ0eOiG/Aac1NeOs9yzM71QBYNlA/jAdH0l8q+kyeu+3OXX7hMceQpB/hoHcU1aNWYNFwvOX4fUDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfceaa3e22518fa1b3ca4732560a409f6ae68f7eb02366d241ff34e3030f41f4","last_reissued_at":"2026-05-18T02:26:52.181419Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:52.181419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal Hardy inequalities in cones","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.SP","authors_text":"Baptiste Devyver, Georgios Psaradakis, Yehuda Pinchover","submitted_at":"2015-02-18T12:53:54Z","abstract_excerpt":"Let $\\Omega$ be an open connected cone in $\\mathbb{R}^n$ with vertex at the origin. 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We show that under some smoothness assumption on $\\Omega$, the following improved Hardy-type inequality \\begin{equation*}\n  \\int_{\\Omega}|\\nabla \\varphi|^2\\,\\mathrm{d}x -\\mu\\int_{\\Omega} \\frac{|\\varphi|^2}{\\delta_\\Omega^2}\\,\\mathrm{d}x \\geq \\lambda(\\mu)\\int_{\\Omega} \\frac{|\\varphi|^2}{|x|^2}\\,\\mathrm{d}x \\qquad"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.05205","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":"1502.05205","created_at":"2026-05-18T02:26:52.181486+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.05205v1","created_at":"2026-05-18T02:26:52.181486+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.05205","created_at":"2026-05-18T02:26:52.181486+00:00"},{"alias_kind":"pith_short_12","alias_value":"37HKUPRCKGH2","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_16","alias_value":"37HKUPRCKGH2DM6K","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_8","alias_value":"37HKUPRC","created_at":"2026-05-18T12:29:02.477457+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/37HKUPRCKGH2DM6KI4ZFMCSAT5","json":"https://pith.science/pith/37HKUPRCKGH2DM6KI4ZFMCSAT5.json","graph_json":"https://pith.science/api/pith-number/37HKUPRCKGH2DM6KI4ZFMCSAT5/graph.json","events_json":"https://pith.science/api/pith-number/37HKUPRCKGH2DM6KI4ZFMCSAT5/events.json","paper":"https://pith.science/paper/37HKUPRC"},"agent_actions":{"view_html":"https://pith.science/pith/37HKUPRCKGH2DM6KI4ZFMCSAT5","download_json":"https://pith.science/pith/37HKUPRCKGH2DM6KI4ZFMCSAT5.json","view_paper":"https://pith.science/paper/37HKUPRC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.05205&json=true","fetch_graph":"https://pith.science/api/pith-number/37HKUPRCKGH2DM6KI4ZFMCSAT5/graph.json","fetch_events":"https://pith.science/api/pith-number/37HKUPRCKGH2DM6KI4ZFMCSAT5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/37HKUPRCKGH2DM6KI4ZFMCSAT5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/37HKUPRCKGH2DM6KI4ZFMCSAT5/action/storage_attestation","attest_author":"https://pith.science/pith/37HKUPRCKGH2DM6KI4ZFMCSAT5/action/author_attestation","sign_citation":"https://pith.science/pith/37HKUPRCKGH2DM6KI4ZFMCSAT5/action/citation_signature","submit_replication":"https://pith.science/pith/37HKUPRCKGH2DM6KI4ZFMCSAT5/action/replication_record"}},"created_at":"2026-05-18T02:26:52.181486+00:00","updated_at":"2026-05-18T02:26:52.181486+00:00"}