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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 $H_\\delta$ is a generalized Gru\\v{s}in operator, \\[ H_\\delta=-\\nabla_{x_1}\\,|x_1|^{(2\\delta_1,2\\delta_1')}\\,\\nabla_{x_1}-|x_1|^{(2\\delta_2,2\\delta_2')}\\,\\nabla_{x_2}^2 \\;. \\] Here $x_1\\in\\Ri^n$, $x_2\\in\\Ri^m$, $\\delta_1,\\delta_1'\\in[0,1\\rangle$,\n  $\\delta_2,\\delta_2'\\geq0$ and $|x_1|^{(2\\delta,2\\delta')}=|x_1|^{2\\delta}$ if $|x_"},"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":"1305.6998","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-05-30T04:06:43Z","cross_cats_sorted":[],"title_canon_sha256":"e1cb84971ef0ba7938c1b4ab4978ef28c985962d3c640d3dc668e813581eb7cb","abstract_canon_sha256":"de84a1a08b7f852ca4a9134a9e48745e7d2b0d27224c67051ffb07a9d5daf832"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:00.375282Z","signature_b64":"DeNjgQnHvx1ed5jFQAVDkaq1YCPkfx0RHKXmU34FHaUNSkLIexrcqWTBDh7Z1wHQTiQdb63ykPb2eZCBzP7EBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46c5c4fac873b6bd2766fdbc3be0b802a21c8b7dc50bf113ce2d9079d3609732","last_reissued_at":"2026-05-18T02:32:00.374689Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:00.374689Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The limitations of the Poincar{\\'e} inequality","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":"2013-05-30T04:06:43Z","abstract_excerpt":"We examine the validity of the Poincar\\'e inequality for 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 $H_\\delta$ is a generalized Gru\\v{s}in operator, \\[ H_\\delta=-\\nabla_{x_1}\\,|x_1|^{(2\\delta_1,2\\delta_1')}\\,\\nabla_{x_1}-|x_1|^{(2\\delta_2,2\\delta_2')}\\,\\nabla_{x_2}^2 \\;. \\] Here $x_1\\in\\Ri^n$, $x_2\\in\\Ri^m$, $\\delta_1,\\delta_1'\\in[0,1\\rangle$,\n  $\\delta_2,\\delta_2'\\geq0$ and $|x_1|^{(2\\delta,2\\delta')}=|x_1|^{2\\delta}$ if $|x_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6998","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":"1305.6998","created_at":"2026-05-18T02:32:00.374786+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.6998v1","created_at":"2026-05-18T02:32:00.374786+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.6998","created_at":"2026-05-18T02:32:00.374786+00:00"},{"alias_kind":"pith_short_12","alias_value":"I3C4J6WIOO3L","created_at":"2026-05-18T12:27:46.883200+00:00"},{"alias_kind":"pith_short_16","alias_value":"I3C4J6WIOO3L2J3G","created_at":"2026-05-18T12:27:46.883200+00:00"},{"alias_kind":"pith_short_8","alias_value":"I3C4J6WI","created_at":"2026-05-18T12:27:46.883200+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/I3C4J6WIOO3L2J3G7W6DXYFYAK","json":"https://pith.science/pith/I3C4J6WIOO3L2J3G7W6DXYFYAK.json","graph_json":"https://pith.science/api/pith-number/I3C4J6WIOO3L2J3G7W6DXYFYAK/graph.json","events_json":"https://pith.science/api/pith-number/I3C4J6WIOO3L2J3G7W6DXYFYAK/events.json","paper":"https://pith.science/paper/I3C4J6WI"},"agent_actions":{"view_html":"https://pith.science/pith/I3C4J6WIOO3L2J3G7W6DXYFYAK","download_json":"https://pith.science/pith/I3C4J6WIOO3L2J3G7W6DXYFYAK.json","view_paper":"https://pith.science/paper/I3C4J6WI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.6998&json=true","fetch_graph":"https://pith.science/api/pith-number/I3C4J6WIOO3L2J3G7W6DXYFYAK/graph.json","fetch_events":"https://pith.science/api/pith-number/I3C4J6WIOO3L2J3G7W6DXYFYAK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I3C4J6WIOO3L2J3G7W6DXYFYAK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I3C4J6WIOO3L2J3G7W6DXYFYAK/action/storage_attestation","attest_author":"https://pith.science/pith/I3C4J6WIOO3L2J3G7W6DXYFYAK/action/author_attestation","sign_citation":"https://pith.science/pith/I3C4J6WIOO3L2J3G7W6DXYFYAK/action/citation_signature","submit_replication":"https://pith.science/pith/I3C4J6WIOO3L2J3G7W6DXYFYAK/action/replication_record"}},"created_at":"2026-05-18T02:32:00.374786+00:00","updated_at":"2026-05-18T02:32:00.374786+00:00"}