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We show that $$ \\|S_{\\alpha,\\psi}(\\vec{f})\\|_{L^p(\\nu_{\\vec{w}})} \\leq C_{n,m,\\psi,\\vec{P}}~ \\alpha^{mn}[\\vec{w}]_{A_{\\vec{P}}}^{\\max(\\frac{1}{2},\\tfrac{p_1'}{p},\\dotsc,\\tfrac{p_m'}{p})} \\prod_{i=1}^m \\|f_i\\|_{L^{p_i}(w_i)}.\n  $$ This result extends the result in the linear case which was obtained by Lerner in 2014."},"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":"1502.05490","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-02-19T08:04:51Z","cross_cats_sorted":[],"title_canon_sha256":"0c8d5c49a6e9419cb80bd6cf3e8d7fd155eefb92638a51af6436acbdbe272d4a","abstract_canon_sha256":"54cef80e5ce50345a3380cc339856cc307cc14deeee50acd4ad141c3adb5ab5c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:51.203449Z","signature_b64":"rQn0k9MtPrn8a5fAFArKtjXSnRaQKotyAmMnQTqFmlTgnHmpUF3vPQcM3eeLfwTeBTy2obfepj+/jo8XPD+pAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7eeab6adeb91175d16943519c10e90996b0b9951a6a0f2198329d33ed9ec6420","last_reissued_at":"2026-05-18T02:17:51.202768Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:51.202768Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weighted bounds for multilinear square functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Mahdi Hormozi, The Anh Bui","submitted_at":"2015-02-19T08:04:51Z","abstract_excerpt":"Let $\\vec{P}=(p_1,\\dotsc,p_m)$ with $1<p_1,\\dotsc,p_m<\\infty$, $1/p_1+\\dotsb+1/p_m=1/p$ and $\\vec{w}=(w_1,\\dotsc,w_m)\\in A_{\\vec{P}}$. In this paper, we investigate the weighted bounds with dependence on aperture $\\alpha$ for multilinear square functions $S_{\\alpha,\\psi}(\\vec{f})$. We show that $$ \\|S_{\\alpha,\\psi}(\\vec{f})\\|_{L^p(\\nu_{\\vec{w}})} \\leq C_{n,m,\\psi,\\vec{P}}~ \\alpha^{mn}[\\vec{w}]_{A_{\\vec{P}}}^{\\max(\\frac{1}{2},\\tfrac{p_1'}{p},\\dotsc,\\tfrac{p_m'}{p})} \\prod_{i=1}^m \\|f_i\\|_{L^{p_i}(w_i)}.\n  $$ This result extends the result in the linear case which was obtained by Lerner in 2014."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.05490","kind":"arxiv","version":4},"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":"1502.05490","created_at":"2026-05-18T02:17:51.202878+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.05490v4","created_at":"2026-05-18T02:17:51.202878+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.05490","created_at":"2026-05-18T02:17:51.202878+00:00"},{"alias_kind":"pith_short_12","alias_value":"P3VLNLPLSELV","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"P3VLNLPLSELV2FUU","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"P3VLNLPL","created_at":"2026-05-18T12:29:34.919912+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/P3VLNLPLSELV2FUUGUM4CDUQTF","json":"https://pith.science/pith/P3VLNLPLSELV2FUUGUM4CDUQTF.json","graph_json":"https://pith.science/api/pith-number/P3VLNLPLSELV2FUUGUM4CDUQTF/graph.json","events_json":"https://pith.science/api/pith-number/P3VLNLPLSELV2FUUGUM4CDUQTF/events.json","paper":"https://pith.science/paper/P3VLNLPL"},"agent_actions":{"view_html":"https://pith.science/pith/P3VLNLPLSELV2FUUGUM4CDUQTF","download_json":"https://pith.science/pith/P3VLNLPLSELV2FUUGUM4CDUQTF.json","view_paper":"https://pith.science/paper/P3VLNLPL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.05490&json=true","fetch_graph":"https://pith.science/api/pith-number/P3VLNLPLSELV2FUUGUM4CDUQTF/graph.json","fetch_events":"https://pith.science/api/pith-number/P3VLNLPLSELV2FUUGUM4CDUQTF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P3VLNLPLSELV2FUUGUM4CDUQTF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P3VLNLPLSELV2FUUGUM4CDUQTF/action/storage_attestation","attest_author":"https://pith.science/pith/P3VLNLPLSELV2FUUGUM4CDUQTF/action/author_attestation","sign_citation":"https://pith.science/pith/P3VLNLPLSELV2FUUGUM4CDUQTF/action/citation_signature","submit_replication":"https://pith.science/pith/P3VLNLPLSELV2FUUGUM4CDUQTF/action/replication_record"}},"created_at":"2026-05-18T02:17:51.202878+00:00","updated_at":"2026-05-18T02:17:51.202878+00:00"}