{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:VH77CWDLUACJILAG2GTY733DCL","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":"a5708408cf36ffc06249cf3ac4ce96742127464063c13ed5de1acbf4362026c0","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-12-02T04:01:11Z","title_canon_sha256":"0ba19186ab4bd309406325b3572a1f52e50008577e72bb32511d881bf863dd02"},"schema_version":"1.0","source":{"id":"1512.00569","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.00569","created_at":"2026-05-18T01:25:17Z"},{"alias_kind":"arxiv_version","alias_value":"1512.00569v2","created_at":"2026-05-18T01:25:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.00569","created_at":"2026-05-18T01:25:17Z"},{"alias_kind":"pith_short_12","alias_value":"VH77CWDLUACJ","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"VH77CWDLUACJILAG","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"VH77CWDL","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:6b8f1add7b54787d51f39d2309a37cb2e3a05cc59a1dc461ceb35bdeccc6a0b1","target":"graph","created_at":"2026-05-18T01:25:17Z","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":"Let $m,n\\ge 1$ and $g_{\\lambda_1,\\lambda_2}^*$ be the bi-parameter Littlewood-Paley $g_{\\lambda}^{*}$-function defined by $$ g_{\\lambda_1,\\lambda_2}^*(f)(x)= \\bigg(\\iint_{\\R^{m+1}_{+}} \\big(\\frac{t_2}{t_2 + |x_2 - y_2|}\\big)^{m \\lambda_2} \\iint_{\\R^{n+1}_{+}} \\big(\\frac{t_1}{t_1 + |x_1 - y_1|}\\big)^{n \\lambda_1}|\\theta_{t_1,t_2} f(y_1,y_2)|^2 \\frac{dy_1 dt_1}{t_1^{n+1}} \\frac{dy_2 dt_2}{t_2^{m+1}} \\bigg)^{1/2}, \\lambda_1>1,\\quad \\lambda_2>1 $$ where $\\theta_{t_1,t_2} f$ is a non-convolution kernel defined on $\\mathbb{R}^{m+n}$. In this paper, we showed that the bi-parameter Littlewood-Paley fu","authors_text":"Mingming Cao, Qingying Xue","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-12-02T04:01:11Z","title":"On The Boundedness of Bi-parameter Littlewood-Paley $g_{\\lambda}^{*}$-function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.00569","kind":"arxiv","version":2},"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":"sha256:298af80d1e2b39b0d60ef3c4b42b5d59a99745a9ded2484530e552332af77caf","target":"record","created_at":"2026-05-18T01:25:17Z","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":"a5708408cf36ffc06249cf3ac4ce96742127464063c13ed5de1acbf4362026c0","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-12-02T04:01:11Z","title_canon_sha256":"0ba19186ab4bd309406325b3572a1f52e50008577e72bb32511d881bf863dd02"},"schema_version":"1.0","source":{"id":"1512.00569","kind":"arxiv","version":2}},"canonical_sha256":"a9fff1586ba004942c06d1a78fef6312eae34dad4e8a82a29a2812aa32dce981","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a9fff1586ba004942c06d1a78fef6312eae34dad4e8a82a29a2812aa32dce981","first_computed_at":"2026-05-18T01:25:17.589893Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:17.589893Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gj0rhczWr/XsGWx/y/SYGIMezBQPnWALqJrPj7qSpXlAz62vY6st+qjNndX9jtl2KAJRsyRzFPN/FhiAZRVcDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:17.590546Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.00569","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:298af80d1e2b39b0d60ef3c4b42b5d59a99745a9ded2484530e552332af77caf","sha256:6b8f1add7b54787d51f39d2309a37cb2e3a05cc59a1dc461ceb35bdeccc6a0b1"],"state_sha256":"111c27bf0ee2ba7ac8476734e05ca16da46d675731ae2365ee6a05debbb534ca"}