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Kogoj, Sergio Polidoro","submitted_at":"2015-09-17T13:18:08Z","abstract_excerpt":"We consider nonnegative solutions $u:\\Omega\\longrightarrow \\mathbb{R}$ of second order hypoelliptic equations \\begin{equation*} \\mathscr{L} u(x) =\\sum_{i,j=1}^n \\partial_{x_i} \\left(a_{ij}(x)\\partial_{x_j} u(x) \\right) + \\sum_{i=1}^n b_i(x) \\partial_{x_i} u(x) =0, \\end{equation*} where $\\Omega$ is a bounded open subset of $\\mathbb{R}^{n}$ and $x$ denotes the point of $\\Omega$. For any fixed $x_0 \\in \\Omega$, we prove a Harnack inequality of this type $$\\sup_K u \\le C_K u(x_0)\\qquad \\forall \\ u \\ \\mbox{ s.t. } \\ \\mathscr{L} u=0, u\\geq 0,$$ where $K$ is any compact subset of the interior of the "},"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":"1509.05245","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-09-17T13:18:08Z","cross_cats_sorted":[],"title_canon_sha256":"4584b249bdec419fe158307c11800796c0cdad275b0c887679ffceacc6821bd1","abstract_canon_sha256":"458e5c2195c5751e753750e5c6ee973a4bb07997298aab361c92b66075f6637f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:49.512535Z","signature_b64":"zfRWG8wK4NcqgCKEfRIbkIK+ur91kpMJ4yh7I4UczzPOG5hfNyyq+btz4NAUOGFAzPIzFc2O6jC9D60PxhwhCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"454f2bc02828f08da4e63e0fd7da5fcbd80e01feac3d7d33f39b1f94366ee228","last_reissued_at":"2026-05-18T01:32:49.511747Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:49.511747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Harnack inequality for hypoelliptic second order partial differential operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessia E. Kogoj, Sergio Polidoro","submitted_at":"2015-09-17T13:18:08Z","abstract_excerpt":"We consider nonnegative solutions $u:\\Omega\\longrightarrow \\mathbb{R}$ of second order hypoelliptic equations \\begin{equation*} \\mathscr{L} u(x) =\\sum_{i,j=1}^n \\partial_{x_i} \\left(a_{ij}(x)\\partial_{x_j} u(x) \\right) + \\sum_{i=1}^n b_i(x) \\partial_{x_i} u(x) =0, \\end{equation*} where $\\Omega$ is a bounded open subset of $\\mathbb{R}^{n}$ and $x$ denotes the point of $\\Omega$. For any fixed $x_0 \\in \\Omega$, we prove a Harnack inequality of this type $$\\sup_K u \\le C_K u(x_0)\\qquad \\forall \\ u \\ \\mbox{ s.t. } \\ \\mathscr{L} u=0, u\\geq 0,$$ where $K$ is any compact subset of the interior of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05245","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":"1509.05245","created_at":"2026-05-18T01:32:49.511880+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.05245v1","created_at":"2026-05-18T01:32:49.511880+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05245","created_at":"2026-05-18T01:32:49.511880+00:00"},{"alias_kind":"pith_short_12","alias_value":"IVHSXQBIFDYI","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"IVHSXQBIFDYI3JHG","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"IVHSXQBI","created_at":"2026-05-18T12:29:25.134429+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/IVHSXQBIFDYI3JHGHYH5PWS7ZP","json":"https://pith.science/pith/IVHSXQBIFDYI3JHGHYH5PWS7ZP.json","graph_json":"https://pith.science/api/pith-number/IVHSXQBIFDYI3JHGHYH5PWS7ZP/graph.json","events_json":"https://pith.science/api/pith-number/IVHSXQBIFDYI3JHGHYH5PWS7ZP/events.json","paper":"https://pith.science/paper/IVHSXQBI"},"agent_actions":{"view_html":"https://pith.science/pith/IVHSXQBIFDYI3JHGHYH5PWS7ZP","download_json":"https://pith.science/pith/IVHSXQBIFDYI3JHGHYH5PWS7ZP.json","view_paper":"https://pith.science/paper/IVHSXQBI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.05245&json=true","fetch_graph":"https://pith.science/api/pith-number/IVHSXQBIFDYI3JHGHYH5PWS7ZP/graph.json","fetch_events":"https://pith.science/api/pith-number/IVHSXQBIFDYI3JHGHYH5PWS7ZP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IVHSXQBIFDYI3JHGHYH5PWS7ZP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IVHSXQBIFDYI3JHGHYH5PWS7ZP/action/storage_attestation","attest_author":"https://pith.science/pith/IVHSXQBIFDYI3JHGHYH5PWS7ZP/action/author_attestation","sign_citation":"https://pith.science/pith/IVHSXQBIFDYI3JHGHYH5PWS7ZP/action/citation_signature","submit_replication":"https://pith.science/pith/IVHSXQBIFDYI3JHGHYH5PWS7ZP/action/replication_record"}},"created_at":"2026-05-18T01:32:49.511880+00:00","updated_at":"2026-05-18T01:32:49.511880+00:00"}