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As a consequence, for two fractional integral operators $I_{\\lambda_1}$ and\n  $I_{\\lambda_2}$, we prove a Bloom type inequality \\begin{align*} \\mbox{\\hbox to 8em{}}& \\hskip -8em \\left\\|\\big[I_{\\lambda_1}^1,\\big[b,I_{\\lambda_2}^2\\big]\\big]\n  \\right\\|_{L^{p_2}(L^{p_1})(\\mu_2^{p_2}\\times\\mu_1^{p_1})\\rightarrow L^{q_2}(L^{q_1})(\\sigma_2^{q_2}\\times\\sigma_1^{q_1})} % \\\\ %&\n  \\lesssim_{\\substack{[\\mu_1]_{A_{p_1,q_1}(\\mathbb R^n)},[\\mu_2]_{A_{p_2,q_2}(\\mathbb R^m)} \\\\ [\\sigma_1]_{A_{p_1,q_1}(\\mathbb R^n)},[\\sigma_2]_{A_{p_"},"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":"1907.07292","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-07-17T00:11:58Z","cross_cats_sorted":[],"title_canon_sha256":"000b43a18a5ed8259d21440ea488cce6d7324b867ecd5768f579a6a860007756","abstract_canon_sha256":"70c3bfcea5a7d65ff4479b4a771f194ac6c4c3ce94157b479b91164d0189302b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:22.773165Z","signature_b64":"EdFLeh0AFvb4Z+U7vJBU9i3Te7en6wdhhfZRPkmY8BmhIHKteX2Mqhq2KWAr7rMwnpQ+e7T8TM3w8c+vFfSGDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"060df53e628170b12c4a28d0696025f93d68f1d63f51188dbddf6bb680c430a7","last_reissued_at":"2026-05-17T23:40:22.772549Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:22.772549Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bloom Type Inequality: The Off-diagonal Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Junren Pan, Wenchang Sun","submitted_at":"2019-07-17T00:11:58Z","abstract_excerpt":"In this paper, we establish a representation formula for fractional integrals. As a consequence, for two fractional integral operators $I_{\\lambda_1}$ and\n  $I_{\\lambda_2}$, we prove a Bloom type inequality \\begin{align*} \\mbox{\\hbox to 8em{}}& \\hskip -8em \\left\\|\\big[I_{\\lambda_1}^1,\\big[b,I_{\\lambda_2}^2\\big]\\big]\n  \\right\\|_{L^{p_2}(L^{p_1})(\\mu_2^{p_2}\\times\\mu_1^{p_1})\\rightarrow L^{q_2}(L^{q_1})(\\sigma_2^{q_2}\\times\\sigma_1^{q_1})} % \\\\ %&\n  \\lesssim_{\\substack{[\\mu_1]_{A_{p_1,q_1}(\\mathbb R^n)},[\\mu_2]_{A_{p_2,q_2}(\\mathbb R^m)} \\\\ [\\sigma_1]_{A_{p_1,q_1}(\\mathbb R^n)},[\\sigma_2]_{A_{p_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.07292","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":"1907.07292","created_at":"2026-05-17T23:40:22.772637+00:00"},{"alias_kind":"arxiv_version","alias_value":"1907.07292v1","created_at":"2026-05-17T23:40:22.772637+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.07292","created_at":"2026-05-17T23:40:22.772637+00:00"},{"alias_kind":"pith_short_12","alias_value":"AYG7KPTCQFYL","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_16","alias_value":"AYG7KPTCQFYLCLCK","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_8","alias_value":"AYG7KPTC","created_at":"2026-05-18T12:33:12.712433+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/AYG7KPTCQFYLCLCKFDIGSYBF7E","json":"https://pith.science/pith/AYG7KPTCQFYLCLCKFDIGSYBF7E.json","graph_json":"https://pith.science/api/pith-number/AYG7KPTCQFYLCLCKFDIGSYBF7E/graph.json","events_json":"https://pith.science/api/pith-number/AYG7KPTCQFYLCLCKFDIGSYBF7E/events.json","paper":"https://pith.science/paper/AYG7KPTC"},"agent_actions":{"view_html":"https://pith.science/pith/AYG7KPTCQFYLCLCKFDIGSYBF7E","download_json":"https://pith.science/pith/AYG7KPTCQFYLCLCKFDIGSYBF7E.json","view_paper":"https://pith.science/paper/AYG7KPTC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1907.07292&json=true","fetch_graph":"https://pith.science/api/pith-number/AYG7KPTCQFYLCLCKFDIGSYBF7E/graph.json","fetch_events":"https://pith.science/api/pith-number/AYG7KPTCQFYLCLCKFDIGSYBF7E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AYG7KPTCQFYLCLCKFDIGSYBF7E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AYG7KPTCQFYLCLCKFDIGSYBF7E/action/storage_attestation","attest_author":"https://pith.science/pith/AYG7KPTCQFYLCLCKFDIGSYBF7E/action/author_attestation","sign_citation":"https://pith.science/pith/AYG7KPTCQFYLCLCKFDIGSYBF7E/action/citation_signature","submit_replication":"https://pith.science/pith/AYG7KPTCQFYLCLCKFDIGSYBF7E/action/replication_record"}},"created_at":"2026-05-17T23:40:22.772637+00:00","updated_at":"2026-05-17T23:40:22.772637+00:00"}