{"paper":{"title":"The Linear Bound for the Natural Weighted Resolution of the Haar Shift","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, Maria Carmen Reguera, Sandra Pott","submitted_at":"2013-08-24T17:45:06Z","abstract_excerpt":"The Hilbert transform has a linear bound in the $A_{2}$ characteristic on weighted $L^{2}$, \\begin{equation*} \\left\\Vert H\\right\\Vert _{L^{2}(w)\\rightarrow L^{2}(w)}\\lesssim \\left[ w \\right] _{A_{2}}, \\end{equation*} and we extend this linear bound to the nine constituent operators in the natural weighted resolution of the conjugation $M_{w^{\\frac{1}{2}}}\\mathcal{S }M_{w^{-\\frac{1}{2}}}$ induced by the canonical decomposition of a multiplier into paraproducts:% \\begin{equation*} M_{f}=P_{f}^{-}+P_{f}^{0}+P_{f}^{+}. \\end{equation*} The main tools used are composition of paraproducts, a product "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5349","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"}