{"paper":{"title":"Martingale transform and Square function: some weak and restricted weak sharp weighted estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.PR"],"primary_cat":"math.AP","authors_text":"Alexander Volberg, Paata Ivanisvili","submitted_at":"2017-11-28T21:54:39Z","abstract_excerpt":"Following the ideas of A. Lerner, F. Nazarov, S. Ombrosi from [12] we prove that there is a sequence of weights $w\\in A^d_1$ such that $[w]^d_{A_1}\\to \\infty$, and martingale transforms $T$ such that with an absolute positive $c$\n  $\\|T: L^1(w) \\to L^{1, \\infty}(w)\\| \\ge c [w]^d_{A_1}\\log [w]^d_{A_1}$.\n  We also show the existence of the sequence of weights (now in $A_2$) such that $[w]^d_{A_2}\\to \\infty$, and such that the following holds:\n  $[w]_{A_2^d}\\asymp \\|M^d\\|_{w^{-1}}^2$;\n  $\\|S_{w}: L^{2} (w) \\to L^2(w^{-1})\\| \\ge c\\, \\|M^d\\|_{w^{-1}}\\sqrt{\\log \\|M^d\\|_{w^{-1}}}$;\n  $\\|S_{w}: L^{2,1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.10578","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"}