{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:VH77CWDLUACJILAG2GTY733DCL","short_pith_number":"pith:VH77CWDL","schema_version":"1.0","canonical_sha256":"a9fff1586ba004942c06d1a78fef6312eae34dad4e8a82a29a2812aa32dce981","source":{"kind":"arxiv","id":"1512.00569","version":2},"attestation_state":"computed","paper":{"title":"On The Boundedness of Bi-parameter Littlewood-Paley $g_{\\lambda}^{*}$-function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Mingming Cao, Qingying Xue","submitted_at":"2015-12-02T04:01:11Z","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"},"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":"1512.00569","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-12-02T04:01:11Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"0ba19186ab4bd309406325b3572a1f52e50008577e72bb32511d881bf863dd02","abstract_canon_sha256":"a5708408cf36ffc06249cf3ac4ce96742127464063c13ed5de1acbf4362026c0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:17.590546Z","signature_b64":"gj0rhczWr/XsGWx/y/SYGIMezBQPnWALqJrPj7qSpXlAz62vY6st+qjNndX9jtl2KAJRsyRzFPN/FhiAZRVcDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9fff1586ba004942c06d1a78fef6312eae34dad4e8a82a29a2812aa32dce981","last_reissued_at":"2026-05-18T01:25:17.589893Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:17.589893Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On The Boundedness of Bi-parameter Littlewood-Paley $g_{\\lambda}^{*}$-function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Mingming Cao, Qingying Xue","submitted_at":"2015-12-02T04:01:11Z","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 $\\mathbb{R}^{m+n}$. 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