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We do so by using Bourgain's localization method and solving the finite dimensional factorization problem. In particular, we obtain that the spaces $\\big(\\sum_{n\\in \\mathbb N} H_n^1(H_n^s)\\big)_r$, $\\big(\\sum_{n\\in \\mathbb N} H_n^s(H_n^1)\\big)_r$, as well as $\\big(\\sum_{n\\in \\mathbb N} BMO_n(H_n^s)\\big)_r$ and $\\big(\\sum_{n\\in \\mathbb N} H^s_n(BMO_n)\\big)_r$, $1 < s < \\"},"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":"1610.01506","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-10-05T16:22:05Z","cross_cats_sorted":[],"title_canon_sha256":"ac84532248793ccc93d167648f7a953c19734ef9ca014882ae51401230869f9d","abstract_canon_sha256":"91791bd8371f90683eec4d4f6497d180b5983b0b95f6d53328a53978fea4364e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:24.329251Z","signature_b64":"oiRCAduXevG6ORC5N1Cg1MrbEo5c15W8RFhA7nuhCI6unX2hZh4dKAoUFtZ5GBVp6XZhTAg/YxxZR2AvK2nPCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7085e3c5f1c33223261beba73be69817199b8d2ff3bb36e6258bc96ac848f5ed","last_reissued_at":"2026-05-18T00:04:24.328674Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:24.328674Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Factorization in mixed norm Hardy and BMO spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Richard Lechner","submitted_at":"2016-10-05T16:22:05Z","abstract_excerpt":"Let $1\\leq p,q < \\infty$ and $1\\leq r \\leq \\infty$. We show that the direct sum of mixed norm Hardy spaces $\\big(\\sum_n H^p_n(H^q_n)\\big)_r$ and the sum of their dual spaces $\\big(\\sum_n H^p_n(H^q_n)^*\\big)_r$ are both primary. We do so by using Bourgain's localization method and solving the finite dimensional factorization problem. In particular, we obtain that the spaces $\\big(\\sum_{n\\in \\mathbb N} H_n^1(H_n^s)\\big)_r$, $\\big(\\sum_{n\\in \\mathbb N} H_n^s(H_n^1)\\big)_r$, as well as $\\big(\\sum_{n\\in \\mathbb N} BMO_n(H_n^s)\\big)_r$ and $\\big(\\sum_{n\\in \\mathbb N} H^s_n(BMO_n)\\big)_r$, $1 < s < \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.01506","kind":"arxiv","version":1},"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":"1610.01506","created_at":"2026-05-18T00:04:24.328755+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.01506v1","created_at":"2026-05-18T00:04:24.328755+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.01506","created_at":"2026-05-18T00:04:24.328755+00:00"},{"alias_kind":"pith_short_12","alias_value":"OCC6HRPRYMZC","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_16","alias_value":"OCC6HRPRYMZCGJQ3","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_8","alias_value":"OCC6HRPR","created_at":"2026-05-18T12:30:36.002864+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/OCC6HRPRYMZCGJQ35OTTXZUYC4","json":"https://pith.science/pith/OCC6HRPRYMZCGJQ35OTTXZUYC4.json","graph_json":"https://pith.science/api/pith-number/OCC6HRPRYMZCGJQ35OTTXZUYC4/graph.json","events_json":"https://pith.science/api/pith-number/OCC6HRPRYMZCGJQ35OTTXZUYC4/events.json","paper":"https://pith.science/paper/OCC6HRPR"},"agent_actions":{"view_html":"https://pith.science/pith/OCC6HRPRYMZCGJQ35OTTXZUYC4","download_json":"https://pith.science/pith/OCC6HRPRYMZCGJQ35OTTXZUYC4.json","view_paper":"https://pith.science/paper/OCC6HRPR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.01506&json=true","fetch_graph":"https://pith.science/api/pith-number/OCC6HRPRYMZCGJQ35OTTXZUYC4/graph.json","fetch_events":"https://pith.science/api/pith-number/OCC6HRPRYMZCGJQ35OTTXZUYC4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OCC6HRPRYMZCGJQ35OTTXZUYC4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OCC6HRPRYMZCGJQ35OTTXZUYC4/action/storage_attestation","attest_author":"https://pith.science/pith/OCC6HRPRYMZCGJQ35OTTXZUYC4/action/author_attestation","sign_citation":"https://pith.science/pith/OCC6HRPRYMZCGJQ35OTTXZUYC4/action/citation_signature","submit_replication":"https://pith.science/pith/OCC6HRPRYMZCGJQ35OTTXZUYC4/action/replication_record"}},"created_at":"2026-05-18T00:04:24.328755+00:00","updated_at":"2026-05-18T00:04:24.328755+00:00"}