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Lerner","submitted_at":"2012-02-09T00:29:58Z","abstract_excerpt":"Given a general dyadic grid ${\\mathscr{D}}$ and a sparse family of cubes ${\\mathcal S}=\\{Q_j^k\\}\\in {\\mathscr{D}}$, define a dyadic positive operator ${\\mathcal A}_{{\\mathscr{D}},{\\mathcal S}}$ by $${\\mathcal A}_{{\\mathscr{D}},{\\mathcal S}}f(x)=\\sum_{j,k}f_{Q_j^k}\\chi_{Q_j^k}(x).$$ Given a Banach function space $X({\\mathbb R}^n)$ and the maximal Calder\\'on-Zygmund operator $T_{\\natural}$, we show that $$\\|T_{\\natural}f\\|_X\\le c(n,T)\\sup_{{\\mathscr{D}},{\\mathcal S}}\\|{\\mathcal A}_{{\\mathscr{D}},{\\mathcal S}}f\\|_{X}.$$\n  This result is applied to weighted inequalities. In particular, it implies:"},"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":"1202.1860","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-02-09T00:29:58Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"e171fa73c4e13b31d6bc56d96a6dd19bc9034e482e5466828373deaa565bdda0","abstract_canon_sha256":"41f85bfa5ef4cdec99c4e691f583cffecb3162e8aec67bb75a23d5ab9a0fd677"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:02:01.554024Z","signature_b64":"dqwNiVBxN5Pnv1LL9HiTx+ZWkkmPYFw9gtq7PMs71Sjxbaw0OS+hPXQsWZ/1fTeaoRspnTie9jzcuZJc+aNHAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f192f82f46c14894af61d7b83ea9e2e345f3d11b048899fb4ec5482be0e4e64e","last_reissued_at":"2026-05-18T04:02:01.553409Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:02:01.553409Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On an estimate of Calder\\'on-Zygmund operators by dyadic positive operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Andrei K. Lerner","submitted_at":"2012-02-09T00:29:58Z","abstract_excerpt":"Given a general dyadic grid ${\\mathscr{D}}$ and a sparse family of cubes ${\\mathcal S}=\\{Q_j^k\\}\\in {\\mathscr{D}}$, define a dyadic positive operator ${\\mathcal A}_{{\\mathscr{D}},{\\mathcal S}}$ by $${\\mathcal A}_{{\\mathscr{D}},{\\mathcal S}}f(x)=\\sum_{j,k}f_{Q_j^k}\\chi_{Q_j^k}(x).$$ Given a Banach function space $X({\\mathbb R}^n)$ and the maximal Calder\\'on-Zygmund operator $T_{\\natural}$, we show that $$\\|T_{\\natural}f\\|_X\\le c(n,T)\\sup_{{\\mathscr{D}},{\\mathcal S}}\\|{\\mathcal A}_{{\\mathscr{D}},{\\mathcal S}}f\\|_{X}.$$\n  This result is applied to weighted inequalities. 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