{"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 $w$. 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(\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.5523","kind":"arxiv","version":2},"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"}