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We assume the coefficients are real symmetric and $a_1H_\\delta\\geq H\\geq a_2H_\\delta$ for some $a_1,a_2>0$ where \\[ H_\\delta=-{\\nabla}_{x_1}\\cdot(c_{\\delta_1, \\delta'_1}(x_1)\\,\\nabla_{x_1})-c_{\\delta_2, \\delta'_2}(x_1)\\,\\nabla_{x_2}^2 \\;. \\] Here $x_1\\in\\Ri^n$, $x_2\\in\\Ri^m$ and $c_{\\delta_i, \\delta'_i}$ are positive measurable functions such that $c_{\\delta_i, \\delta'_i}(x)$ behaves like $|x|^{\\delta_i}$ as $x\\to0$ and $|x|^{\\delta_i'}$ as $x\\to\\infty$ with $\\delta_1,\\delta_1'\\in[0,1"},"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":"1305.7002","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-05-30T04:44:11Z","cross_cats_sorted":[],"title_canon_sha256":"23ca1afde64ab5193ec2ad3c6be0aa4e041db3e05539a90b5919c91168d4e361","abstract_canon_sha256":"26ae3d59adbd477b599996150bb5449d5b4fd5cb459f9458105e9b44a766abba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:32.155564Z","signature_b64":"pGiiAS7gXFoXa8yg6R7KGEt2v+ubsAD7ZztGPELRJKa7OGKyfZ3ZBvZ9cf0Rgp8wQ2bSVcKSc5T2DtDAyn+qDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e20e2f240d5dc4c137ff28a15a3dceb79742a3f5aaf060a2a072a10cfa0b20da","last_reissued_at":"2026-05-18T03:03:32.154951Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:32.154951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Analysis of degenerate elliptic operators of Gru\\v{s}in type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Adam Sikora, Derek W. Robinson","submitted_at":"2013-05-30T04:44:11Z","abstract_excerpt":"We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\\Ri^{n}\\times\\Ri^{m})$. We assume the coefficients are real symmetric and $a_1H_\\delta\\geq H\\geq a_2H_\\delta$ for some $a_1,a_2>0$ where \\[ H_\\delta=-{\\nabla}_{x_1}\\cdot(c_{\\delta_1, \\delta'_1}(x_1)\\,\\nabla_{x_1})-c_{\\delta_2, \\delta'_2}(x_1)\\,\\nabla_{x_2}^2 \\;. \\] Here $x_1\\in\\Ri^n$, $x_2\\in\\Ri^m$ and $c_{\\delta_i, \\delta'_i}$ are positive measurable functions such that $c_{\\delta_i, \\delta'_i}(x)$ behaves like $|x|^{\\delta_i}$ as $x\\to0$ and $|x|^{\\delta_i'}$ as $x\\to\\infty$ with $\\delta_1,\\delta_1'\\in[0,1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.7002","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":"1305.7002","created_at":"2026-05-18T03:03:32.155078+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.7002v1","created_at":"2026-05-18T03:03:32.155078+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.7002","created_at":"2026-05-18T03:03:32.155078+00:00"},{"alias_kind":"pith_short_12","alias_value":"4IHC6JANLXCM","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_16","alias_value":"4IHC6JANLXCMCN77","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_8","alias_value":"4IHC6JAN","created_at":"2026-05-18T12:27:34.582898+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/4IHC6JANLXCMCN77FCQVUPOOW6","json":"https://pith.science/pith/4IHC6JANLXCMCN77FCQVUPOOW6.json","graph_json":"https://pith.science/api/pith-number/4IHC6JANLXCMCN77FCQVUPOOW6/graph.json","events_json":"https://pith.science/api/pith-number/4IHC6JANLXCMCN77FCQVUPOOW6/events.json","paper":"https://pith.science/paper/4IHC6JAN"},"agent_actions":{"view_html":"https://pith.science/pith/4IHC6JANLXCMCN77FCQVUPOOW6","download_json":"https://pith.science/pith/4IHC6JANLXCMCN77FCQVUPOOW6.json","view_paper":"https://pith.science/paper/4IHC6JAN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.7002&json=true","fetch_graph":"https://pith.science/api/pith-number/4IHC6JANLXCMCN77FCQVUPOOW6/graph.json","fetch_events":"https://pith.science/api/pith-number/4IHC6JANLXCMCN77FCQVUPOOW6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4IHC6JANLXCMCN77FCQVUPOOW6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4IHC6JANLXCMCN77FCQVUPOOW6/action/storage_attestation","attest_author":"https://pith.science/pith/4IHC6JANLXCMCN77FCQVUPOOW6/action/author_attestation","sign_citation":"https://pith.science/pith/4IHC6JANLXCMCN77FCQVUPOOW6/action/citation_signature","submit_replication":"https://pith.science/pith/4IHC6JANLXCMCN77FCQVUPOOW6/action/replication_record"}},"created_at":"2026-05-18T03:03:32.155078+00:00","updated_at":"2026-05-18T03:03:32.155078+00:00"}