{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:7HJ3AYMZKIIRTU2KIC2CHN2F7N","short_pith_number":"pith:7HJ3AYMZ","schema_version":"1.0","canonical_sha256":"f9d3b06199521119d34a40b423b745fb5be11972b75d80fe71030939837baef2","source":{"kind":"arxiv","id":"1803.04613","version":2},"attestation_state":"computed","paper":{"title":"Application BMO type space to parabolic equations of Navier-Stokes type with the Neumann boundary condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chao Zhang, Minghua Yang","submitted_at":"2018-03-13T04:31:00Z","abstract_excerpt":"Let $L$ be a Neumann operator of the form $L=-\\Delta_{N}$ acting on $L^2(\\mathbb R^n)$. Let ${BMO}_{\\Delta_{N}}(\\mathbb R^n)$ denote the BMO space on $\\mathbb R^n$ associated to the Neumann operator $\\L$. In this article we will show that a function $f\\in { BMO}_{\\Delta_{N}}(\\mathbb R^n)$ is the trace of the solution of $${\\mathbb L}u=u_{t}+L u=0, u(x,0)= f(x),$$\n  where $u$ satisfies a Carleson-type condition \\begin{eqnarray*}\n  \\sup_{x_B, r_B} r_B^{-n}\\int_0^{r_B^2}\\int_{B(x_B, r_B)} |\\nabla u(x,t)|^2 {dx dt } \\leq C <\\infty, \\end{eqnarray*} for some constant $C>0$. Conversely, this Carleson"},"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":"1803.04613","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-03-13T04:31:00Z","cross_cats_sorted":[],"title_canon_sha256":"56560658b063e358c1503cf1169e631ae1e5948f36d2a4d15bf2a442911287b2","abstract_canon_sha256":"d0f378817ad2249b72157a0a7a52e6bd34a7e4b9bc00512a623b2fe7e8806398"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:37.517570Z","signature_b64":"Nx2cf83Clf8D8Ms3anyDGbk/VNZy6krfwM5+ZenSKkyyQMGEKX+GUcR5lqakYAMkIL2v9L1aVvKmE1A0yGnRBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9d3b06199521119d34a40b423b745fb5be11972b75d80fe71030939837baef2","last_reissued_at":"2026-05-18T00:00:37.517021Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:37.517021Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Application BMO type space to parabolic equations of Navier-Stokes type with the Neumann boundary condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chao Zhang, Minghua Yang","submitted_at":"2018-03-13T04:31:00Z","abstract_excerpt":"Let $L$ be a Neumann operator of the form $L=-\\Delta_{N}$ acting on $L^2(\\mathbb R^n)$. Let ${BMO}_{\\Delta_{N}}(\\mathbb R^n)$ denote the BMO space on $\\mathbb R^n$ associated to the Neumann operator $\\L$. In this article we will show that a function $f\\in { BMO}_{\\Delta_{N}}(\\mathbb R^n)$ is the trace of the solution of $${\\mathbb L}u=u_{t}+L u=0, u(x,0)= f(x),$$\n  where $u$ satisfies a Carleson-type condition \\begin{eqnarray*}\n  \\sup_{x_B, r_B} r_B^{-n}\\int_0^{r_B^2}\\int_{B(x_B, r_B)} |\\nabla u(x,t)|^2 {dx dt } \\leq C <\\infty, \\end{eqnarray*} for some constant $C>0$. Conversely, this Carleson"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.04613","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":"1803.04613","created_at":"2026-05-18T00:00:37.517111+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.04613v2","created_at":"2026-05-18T00:00:37.517111+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.04613","created_at":"2026-05-18T00:00:37.517111+00:00"},{"alias_kind":"pith_short_12","alias_value":"7HJ3AYMZKIIR","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_16","alias_value":"7HJ3AYMZKIIRTU2K","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_8","alias_value":"7HJ3AYMZ","created_at":"2026-05-18T12:32:11.075285+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/7HJ3AYMZKIIRTU2KIC2CHN2F7N","json":"https://pith.science/pith/7HJ3AYMZKIIRTU2KIC2CHN2F7N.json","graph_json":"https://pith.science/api/pith-number/7HJ3AYMZKIIRTU2KIC2CHN2F7N/graph.json","events_json":"https://pith.science/api/pith-number/7HJ3AYMZKIIRTU2KIC2CHN2F7N/events.json","paper":"https://pith.science/paper/7HJ3AYMZ"},"agent_actions":{"view_html":"https://pith.science/pith/7HJ3AYMZKIIRTU2KIC2CHN2F7N","download_json":"https://pith.science/pith/7HJ3AYMZKIIRTU2KIC2CHN2F7N.json","view_paper":"https://pith.science/paper/7HJ3AYMZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.04613&json=true","fetch_graph":"https://pith.science/api/pith-number/7HJ3AYMZKIIRTU2KIC2CHN2F7N/graph.json","fetch_events":"https://pith.science/api/pith-number/7HJ3AYMZKIIRTU2KIC2CHN2F7N/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7HJ3AYMZKIIRTU2KIC2CHN2F7N/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7HJ3AYMZKIIRTU2KIC2CHN2F7N/action/storage_attestation","attest_author":"https://pith.science/pith/7HJ3AYMZKIIRTU2KIC2CHN2F7N/action/author_attestation","sign_citation":"https://pith.science/pith/7HJ3AYMZKIIRTU2KIC2CHN2F7N/action/citation_signature","submit_replication":"https://pith.science/pith/7HJ3AYMZKIIRTU2KIC2CHN2F7N/action/replication_record"}},"created_at":"2026-05-18T00:00:37.517111+00:00","updated_at":"2026-05-18T00:00:37.517111+00:00"}