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We show that the $m$-(sub)linear version below of the Ratnakumar and Shrivastava \\cite{RS1} Littlewood-Paley square function $$T(f_1,\\dots , f_m)(x)=\\Big(\\sum\\limits_{\\ell\\in\\Z^d}|\\int_{\\mathbb{R}^d}f_1(x-\\theta_1 y)\\cdots f_m(x-\\theta_m y)e^{2\\pi i \\ell \\cdot y}K (y)dy|^2\\Big)^{1/2} $$ is bounded from $L^{p_1}(\\mathbb{R}^d) \\times\\cdots\\times L^{p_m}(\\mathbb{R}^d) $ to $L^p(\\mathbb{R}^d) $ when $2\\le p_i<\\infty$ satisfy $1/p=1/p_1+\\cdots+1/p_m$ and $1\\le p<\\infty$. Our pr"},"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":"1504.03424","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-04-14T05:47:53Z","cross_cats_sorted":[],"title_canon_sha256":"f3a6d1881cc79bd5707e733e2fc71d91e0e42748c5448b69116d4ce6f137acf9","abstract_canon_sha256":"fdae02bd20cc72e4e362f3317446431481a63562a27dc85b09a5b404a04c627c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:52.224976Z","signature_b64":"ZuRnXqdgvLac6hZGgYCGxfQNspl+efCCaXRCDqRL96VYpKr6bl30hkAIMxLtQUb0+JFxRCv4U1YKnw5wqrW+AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3dd17e86fecd01d63b08a4acadca314895e66f9c133fc2b4a47d3bfb3333f45c","last_reissued_at":"2026-05-18T02:18:52.224451Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:52.224451Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Certain Multi(sub)linear square functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Loukas Grafakos, Qingying Xue, Sha He","submitted_at":"2015-04-14T05:47:53Z","abstract_excerpt":"Let $d\\ge 1, \\ell\\in\\Z^d$, $m\\in \\mathbb Z^+$ and $\\theta_i$, $i=1,\\dots,m $ are fixed, distinct and nonzero real numbers. We show that the $m$-(sub)linear version below of the Ratnakumar and Shrivastava \\cite{RS1} Littlewood-Paley square function $$T(f_1,\\dots , f_m)(x)=\\Big(\\sum\\limits_{\\ell\\in\\Z^d}|\\int_{\\mathbb{R}^d}f_1(x-\\theta_1 y)\\cdots f_m(x-\\theta_m y)e^{2\\pi i \\ell \\cdot y}K (y)dy|^2\\Big)^{1/2} $$ is bounded from $L^{p_1}(\\mathbb{R}^d) \\times\\cdots\\times L^{p_m}(\\mathbb{R}^d) $ to $L^p(\\mathbb{R}^d) $ when $2\\le p_i<\\infty$ satisfy $1/p=1/p_1+\\cdots+1/p_m$ and $1\\le p<\\infty$. 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