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We prove that each constituent operator obtained from this resolution has a linear bound on $L^2(\\mathbb{R}^d;w)$ in terms of the $A_{2}$ characteristic of $w$. The main tools used are a product formula for Haar coefficients, the Carleson Embedding Theorem, the linear bound for the square function, and the well-known linear bound of $T_{\\sigma}$ on $L^2(\\"},"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":"1402.5523","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-02-22T16:11:33Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"1e130d3e86e03e3937f820f5f7b6e37634da6f47d74dac6b15f835014cda95d0","abstract_canon_sha256":"6bd74e9f25916a96133d56929f9d0bf094a329d13c7d9b15e8589c1da637b829"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:16.325628Z","signature_b64":"lUODoeckOPPJLLti70lFAGpzkwX4qtZLNxPPCF1ei9DE6eIUMkIOiuJ7yckDICE/o3jd7kWVwEwZTsU4pjYCBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"17fb15fc2628d2a3ea3a85c32f9392b0c415e879a998fd6334037440577d30bd","last_reissued_at":"2026-05-18T01:21:16.324918Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:16.324918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Linear Bound for Haar Multiplier Paraproducts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.CA","authors_text":"Brett D. Wick, Eric T. Sawyer, Kelly Bickel","submitted_at":"2014-02-22T16:11:33Z","abstract_excerpt":"We study the natural resolution of the conjugated Haar multiplier $M_{w^{\\frac{1}{2}}}T_{\\sigma}M_{w^{-\\frac{1}{2}}},$ where the multiplication operators $M_{w^{\\pm\\frac{1}{2}}}$ are decomposed into their canonical paraproduct decompositions. We prove that each constituent operator obtained from this resolution has a linear bound on $L^2(\\mathbb{R}^d;w)$ in terms of the $A_{2}$ characteristic of $w$. 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