{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:ZYJRDJMV7WFYPEPZ4QDISGFCL6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8cd408f4ed1393af99d285afe8de535f4d9885985cb17bbda17c6ad04d0101cb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-04-17T14:47:34Z","title_canon_sha256":"885a4c590ced2e9a5b03a5d613bf674aac260230630072d84c0b34f6d9f8527a"},"schema_version":"1.0","source":{"id":"1804.06296","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.06296","created_at":"2026-05-18T00:18:21Z"},{"alias_kind":"arxiv_version","alias_value":"1804.06296v1","created_at":"2026-05-18T00:18:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.06296","created_at":"2026-05-18T00:18:21Z"},{"alias_kind":"pith_short_12","alias_value":"ZYJRDJMV7WFY","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"ZYJRDJMV7WFYPEPZ","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"ZYJRDJMV","created_at":"2026-05-18T12:33:07Z"}],"graph_snapshots":[{"event_id":"sha256:18352a5ecc5569cc87423df15c67b24978b84bf3f9bb48ef7eaa37f04478974c","target":"graph","created_at":"2026-05-18T00:18:21Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study the boundedness properties of commutators formed by $b$ and $T$, where $T$ is a bilinear bi-parameter singular integral satisfying natural $T1$ type conditions and $b$ is a little BMO function. For paraproduct free bilinear bi-parameter singular integrals $T$ we prove that $[b, T]_1 \\colon L^p(\\mathbb{R}^{n+m}) \\times L^q(\\mathbb{R}^{n+m}) \\to L^r(\\mathbb{R}^{n+m})$ in the full range $1 < p, q \\le \\infty$, $1/2 < r < \\infty$ satisfying $1/p+1/q = 1/r$. A special case is when $T$ is a bilinear bi-parameter multiplier. We also prove the corresponding Banach range result for all singular","authors_text":"Emil Vuorinen, Henri Martikainen, Kangwei Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-04-17T14:47:34Z","title":"Commutators of bilinear bi-parameter singular integrals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.06296","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:11382300e9a719b8ab8d9d7082ce52801e533049ead5b8e9b145858ad662d151","target":"record","created_at":"2026-05-18T00:18:21Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8cd408f4ed1393af99d285afe8de535f4d9885985cb17bbda17c6ad04d0101cb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-04-17T14:47:34Z","title_canon_sha256":"885a4c590ced2e9a5b03a5d613bf674aac260230630072d84c0b34f6d9f8527a"},"schema_version":"1.0","source":{"id":"1804.06296","kind":"arxiv","version":1}},"canonical_sha256":"ce1311a595fd8b8791f9e4068918a25f8286b3e1062aa8540257bed97cca954d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ce1311a595fd8b8791f9e4068918a25f8286b3e1062aa8540257bed97cca954d","first_computed_at":"2026-05-18T00:18:21.092102Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:18:21.092102Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OuxW4lMMUyrvgP5tBQSRqpWkDVs88sPspJyTmdkry0eHzpXLMxV31xlQLeYU2FT+NrUV6yn0fPU+R0TlMtQkAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:18:21.093051Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.06296","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:11382300e9a719b8ab8d9d7082ce52801e533049ead5b8e9b145858ad662d151","sha256:18352a5ecc5569cc87423df15c67b24978b84bf3f9bb48ef7eaa37f04478974c"],"state_sha256":"28f42b3e50f5c13264bdd56a30152c13d388e2fe2aa395fea068e742c19f39a7"}