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We study multivariable spectral multipliers $F(L_1, L_2)$ acting on the Cartesian product of $X_1$ and $X_2$. Under the assumptions of the finite propagation speed property and Plancherel or Stein--Tomas restriction type estimates on the operators $L_1$ and~$L_2$, we show that if a function~$F$ satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator $F(L_1, L_2)$ is b"},"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":"1512.01607","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-12-05T01:02:56Z","cross_cats_sorted":[],"title_canon_sha256":"b199dcaf811627f3d3df9207d4d9134cfc25d6f5b021b3fcfa434692391ed232","abstract_canon_sha256":"794f3ac264df3c37ce8fa4f09b330573881ae85c5c59df42e9159acb32f46825"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:10.820208Z","signature_b64":"xNyIGENX2+a8BcUmlqZv+jv3pRcL89CBALwqAnP7jWBpZ1Q0hSwvaEqHLp9QVhNCfa0Tx/Iu8pDJH/f31zIsDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa8269316e818e994d100ce0e2b16536413e734df6672f8a2f80a8866e7ca8b0","last_reissued_at":"2026-05-18T01:25:10.819709Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:10.819709Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Marcinkiewicz-type spectral multipliers on Hardy and Lebesgue spaces on product spaces of homogeneous type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ji Li, Lesley A. Ward, Lixin Yan, Peng Chen, Xuan Thinh Duong","submitted_at":"2015-12-05T01:02:56Z","abstract_excerpt":"Let $X_1$ and $X_2$ be metric spaces equipped with doubling measures and let $L_1$ and $L_2$ be nonnegative self-adjoint second-order operators acting on $L^2(X_1)$ and $L^2(X_2)$ respectively. We study multivariable spectral multipliers $F(L_1, L_2)$ acting on the Cartesian product of $X_1$ and $X_2$. Under the assumptions of the finite propagation speed property and Plancherel or Stein--Tomas restriction type estimates on the operators $L_1$ and~$L_2$, we show that if a function~$F$ satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator $F(L_1, L_2)$ is b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01607","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":"1512.01607","created_at":"2026-05-18T01:25:10.819790+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.01607v1","created_at":"2026-05-18T01:25:10.819790+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01607","created_at":"2026-05-18T01:25:10.819790+00:00"},{"alias_kind":"pith_short_12","alias_value":"VKBGSMLOQGHJ","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"VKBGSMLOQGHJSTIQ","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"VKBGSMLO","created_at":"2026-05-18T12:29:44.643036+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/VKBGSMLOQGHJSTIQBTQOFMLFGZ","json":"https://pith.science/pith/VKBGSMLOQGHJSTIQBTQOFMLFGZ.json","graph_json":"https://pith.science/api/pith-number/VKBGSMLOQGHJSTIQBTQOFMLFGZ/graph.json","events_json":"https://pith.science/api/pith-number/VKBGSMLOQGHJSTIQBTQOFMLFGZ/events.json","paper":"https://pith.science/paper/VKBGSMLO"},"agent_actions":{"view_html":"https://pith.science/pith/VKBGSMLOQGHJSTIQBTQOFMLFGZ","download_json":"https://pith.science/pith/VKBGSMLOQGHJSTIQBTQOFMLFGZ.json","view_paper":"https://pith.science/paper/VKBGSMLO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.01607&json=true","fetch_graph":"https://pith.science/api/pith-number/VKBGSMLOQGHJSTIQBTQOFMLFGZ/graph.json","fetch_events":"https://pith.science/api/pith-number/VKBGSMLOQGHJSTIQBTQOFMLFGZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VKBGSMLOQGHJSTIQBTQOFMLFGZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VKBGSMLOQGHJSTIQBTQOFMLFGZ/action/storage_attestation","attest_author":"https://pith.science/pith/VKBGSMLOQGHJSTIQBTQOFMLFGZ/action/author_attestation","sign_citation":"https://pith.science/pith/VKBGSMLOQGHJSTIQBTQOFMLFGZ/action/citation_signature","submit_replication":"https://pith.science/pith/VKBGSMLOQGHJSTIQBTQOFMLFGZ/action/replication_record"}},"created_at":"2026-05-18T01:25:10.819790+00:00","updated_at":"2026-05-18T01:25:10.819790+00:00"}