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Namely, given $1<p<q<\\infty$ and a pair of weights $(u,v)$, if the Hardy-Littlewood maximal function satisfies the following two weight inequalities: $$ M : L^p(v) \\rightarrow L^q(u) \\quad \\text{and} \\quad M: L^{q'}(u^{1-q'}) \\rightarrow L^{p'}(v^{1-p'}), $$ then any Calder\\'on-Zygmund operator $T$ and its associated truncated maximal operator $T_\\star$ are bounded from $L^p(v)$ to $L^q(u)$. Additionally, assuming only the second estimate for $M$ then $T$ and $T_\\star$ map continuously $L^p(v)$ into $L^"},"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":"1203.5906","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-03-27T09:33:03Z","cross_cats_sorted":[],"title_canon_sha256":"7907eb00a869910a27f05aaac3d65a5d723b9fdda9b8485d83317b9a9f56b3c3","abstract_canon_sha256":"2004a03e08330cecb3aa859a6df71e40d121fa6470b9871b7e347ba125919760"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:50.767280Z","signature_b64":"cwnfLY+P9PsB6BSc2BDkMHJNJIrgjN+gtojus2HR5lERiwuNxZ+ToDeUmFWLwsp5NnHXv2i+IdJFgYFFb0DuAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62a5255c2a90312c8885a2ba11d161be7f15da075e1ed49bb7890f284defe9f1","last_reissued_at":"2026-05-18T00:03:50.766638Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:50.766638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on the off-diagonal Muckenhoupt-Wheeden conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Carlos P\\'erez, David Cruz-Uribe, Jos\\'e Mar\\'ia Martell","submitted_at":"2012-03-27T09:33:03Z","abstract_excerpt":"We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calder\\'on-Zygmund operators. Namely, given $1<p<q<\\infty$ and a pair of weights $(u,v)$, if the Hardy-Littlewood maximal function satisfies the following two weight inequalities: $$ M : L^p(v) \\rightarrow L^q(u) \\quad \\text{and} \\quad M: L^{q'}(u^{1-q'}) \\rightarrow L^{p'}(v^{1-p'}), $$ then any Calder\\'on-Zygmund operator $T$ and its associated truncated maximal operator $T_\\star$ are bounded from $L^p(v)$ to $L^q(u)$. 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