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Moreover, the optimality of the constant $c_{0,\\mu}$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential \\[ Lu+Vu=\\left(\\Delta u+\\frac{\\nabla \\mu}{\\mu}\\cdot \\nabla u\\right)+\\frac{c}{|x|^2}u \\] for which the existence of positive solu"},"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":"1703.10567","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-30T17:04:43Z","cross_cats_sorted":[],"title_canon_sha256":"5081a6fc0eed431a4ee3f09dc2917787db677c866ba1f1d84fb02f26ba59394d","abstract_canon_sha256":"f38c393317c63a8a067cb45a28b15495dcd46fe773ace79b4ccee4cb8acef7eb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:12.468198Z","signature_b64":"usyf8ukY+7b7HKHZ3GVDISMiiDnWXhMacGI6hUtWCIqh+G/Ky5YzH4zjKLqM8gYJKYH+FS4dQgyVSkxgyRpBCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d605667b3b7c36c6a0bd6bb11f40259b005461dd1dc22a19033bf08d9fbcc68f","last_reissued_at":"2026-05-18T00:39:12.467667Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:12.467667Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weighted Hardy's inequalities and Kolmogorov-type operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Abdelaziz Rhandi, Anna Canale, Cristian Tacelli, Federica Gregorio","submitted_at":"2017-03-30T17:04:43Z","abstract_excerpt":"We give general conditions to state the weighted Hardy inequality \\[ c\\int_{\\mathbb{R}^N}\\frac{\\varphi^2} {|x|^2}d\\mu\\leq\\int_{\\mathbb{R}^N}|\\nabla \\varphi |^2 d\\mu+C\\int_{\\mathbb{R}^N} \\varphi^2d\\mu,\\quad \\varphi\\in C_c^{\\infty}(\\mathbb{R}^N),\\,c\\leq c_{0,\\mu}, \\] with respect to a probability measure $d\\mu$. Moreover, the optimality of the constant $c_{0,\\mu}$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential \\[ Lu+Vu=\\left(\\Delta u+\\frac{\\nabla \\mu}{\\mu}\\cdot \\nabla u\\right)+\\frac{c}{|x|^2}u \\] for which the existence of positive solu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.10567","kind":"arxiv","version":2},"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":"1703.10567","created_at":"2026-05-18T00:39:12.467764+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.10567v2","created_at":"2026-05-18T00:39:12.467764+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.10567","created_at":"2026-05-18T00:39:12.467764+00:00"},{"alias_kind":"pith_short_12","alias_value":"2YCWM6Z3PQ3M","created_at":"2026-05-18T12:30:55.937587+00:00"},{"alias_kind":"pith_short_16","alias_value":"2YCWM6Z3PQ3MNIF5","created_at":"2026-05-18T12:30:55.937587+00:00"},{"alias_kind":"pith_short_8","alias_value":"2YCWM6Z3","created_at":"2026-05-18T12:30:55.937587+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/2YCWM6Z3PQ3MNIF5NOYR6QBFTM","json":"https://pith.science/pith/2YCWM6Z3PQ3MNIF5NOYR6QBFTM.json","graph_json":"https://pith.science/api/pith-number/2YCWM6Z3PQ3MNIF5NOYR6QBFTM/graph.json","events_json":"https://pith.science/api/pith-number/2YCWM6Z3PQ3MNIF5NOYR6QBFTM/events.json","paper":"https://pith.science/paper/2YCWM6Z3"},"agent_actions":{"view_html":"https://pith.science/pith/2YCWM6Z3PQ3MNIF5NOYR6QBFTM","download_json":"https://pith.science/pith/2YCWM6Z3PQ3MNIF5NOYR6QBFTM.json","view_paper":"https://pith.science/paper/2YCWM6Z3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.10567&json=true","fetch_graph":"https://pith.science/api/pith-number/2YCWM6Z3PQ3MNIF5NOYR6QBFTM/graph.json","fetch_events":"https://pith.science/api/pith-number/2YCWM6Z3PQ3MNIF5NOYR6QBFTM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2YCWM6Z3PQ3MNIF5NOYR6QBFTM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2YCWM6Z3PQ3MNIF5NOYR6QBFTM/action/storage_attestation","attest_author":"https://pith.science/pith/2YCWM6Z3PQ3MNIF5NOYR6QBFTM/action/author_attestation","sign_citation":"https://pith.science/pith/2YCWM6Z3PQ3MNIF5NOYR6QBFTM/action/citation_signature","submit_replication":"https://pith.science/pith/2YCWM6Z3PQ3MNIF5NOYR6QBFTM/action/replication_record"}},"created_at":"2026-05-18T00:39:12.467764+00:00","updated_at":"2026-05-18T00:39:12.467764+00:00"}