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We characterize the two-weight norm inequality, \\begin{equation*} \\lVert T_\\lambda(f\\sigma)\\rVert_{L^q(\\omega)}\\le C \\, \\lVert f \\rVert_{L^p(\\sigma)}\\quad \\text{for every $f\\in L^p(\\sigma)$,} \\end{equation*} for the positive dyadic operator \\begin{equation*} T_\\lambda(f\\sigma):= \\sum_{Q\\in \\mathcal{D}} \\lambda_Q \\, \\Big(\\frac{1}{\\sigma(Q)} \\int_Q f\\mathrm{d}\\sigma\\Big) \\, 1_Q \\end{equation*} "},"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":"1706.08657","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-06-27T03:18:55Z","cross_cats_sorted":[],"title_canon_sha256":"c6b0ccf457a5788e3d87782b5fcbfd1b3605948128810fb73f7bf2d8defe0a80","abstract_canon_sha256":"4a6e1f00c5b70ebc88de70b2dee357656183d9aa370e8105f67e5bbe47947f48"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:31.446460Z","signature_b64":"GkNUz9lSgQDKD5VOisOJH9NQ98e3vWXlthUuk2y8ONro6fxQCXIET4iJe/9g3aReD/mhOndBvb/fO+CAN7wTBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21051e1d7fd23f4934da41a002d30b9fa67aac014c1cd478be2de99cd1e69e0f","last_reissued_at":"2026-05-18T00:41:31.445930Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:31.445930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two-weight $L^p\\to L^q$ bounds for positive dyadic operators in the case $0<q< 1 \\le p<\\infty$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Igor E. Verbitsky, Timo S. H\\\"anninen","submitted_at":"2017-06-27T03:18:55Z","abstract_excerpt":"Let $\\sigma$, $\\omega$ be measures on $\\mathbb{R}^d$, and let $\\{\\lambda_Q\\}_{Q\\in\\mathcal{D}}$ be a family of non-negative reals indexed by the collection $\\mathcal{D}$ of dyadic cubes in $\\mathbb{R}^d$. We characterize the two-weight norm inequality, \\begin{equation*} \\lVert T_\\lambda(f\\sigma)\\rVert_{L^q(\\omega)}\\le C \\, \\lVert f \\rVert_{L^p(\\sigma)}\\quad \\text{for every $f\\in L^p(\\sigma)$,} \\end{equation*} for the positive dyadic operator \\begin{equation*} T_\\lambda(f\\sigma):= \\sum_{Q\\in \\mathcal{D}} \\lambda_Q \\, \\Big(\\frac{1}{\\sigma(Q)} \\int_Q f\\mathrm{d}\\sigma\\Big) \\, 1_Q \\end{equation*} "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.08657","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":"1706.08657","created_at":"2026-05-18T00:41:31.446010+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.08657v1","created_at":"2026-05-18T00:41:31.446010+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.08657","created_at":"2026-05-18T00:41:31.446010+00:00"},{"alias_kind":"pith_short_12","alias_value":"EECR4HL72I7U","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_16","alias_value":"EECR4HL72I7USNG2","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_8","alias_value":"EECR4HL7","created_at":"2026-05-18T12:31:12.930513+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/EECR4HL72I7USNG2IGQAFUYLT6","json":"https://pith.science/pith/EECR4HL72I7USNG2IGQAFUYLT6.json","graph_json":"https://pith.science/api/pith-number/EECR4HL72I7USNG2IGQAFUYLT6/graph.json","events_json":"https://pith.science/api/pith-number/EECR4HL72I7USNG2IGQAFUYLT6/events.json","paper":"https://pith.science/paper/EECR4HL7"},"agent_actions":{"view_html":"https://pith.science/pith/EECR4HL72I7USNG2IGQAFUYLT6","download_json":"https://pith.science/pith/EECR4HL72I7USNG2IGQAFUYLT6.json","view_paper":"https://pith.science/paper/EECR4HL7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.08657&json=true","fetch_graph":"https://pith.science/api/pith-number/EECR4HL72I7USNG2IGQAFUYLT6/graph.json","fetch_events":"https://pith.science/api/pith-number/EECR4HL72I7USNG2IGQAFUYLT6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EECR4HL72I7USNG2IGQAFUYLT6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EECR4HL72I7USNG2IGQAFUYLT6/action/storage_attestation","attest_author":"https://pith.science/pith/EECR4HL72I7USNG2IGQAFUYLT6/action/author_attestation","sign_citation":"https://pith.science/pith/EECR4HL72I7USNG2IGQAFUYLT6/action/citation_signature","submit_replication":"https://pith.science/pith/EECR4HL72I7USNG2IGQAFUYLT6/action/replication_record"}},"created_at":"2026-05-18T00:41:31.446010+00:00","updated_at":"2026-05-18T00:41:31.446010+00:00"}