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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","authors_text":"Alexander Volberg, Paata Ivanisvili","cross_cats":["math.CA","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-28T21:54:39Z","title":"Martingale transform and Square function: some weak and restricted weak sharp weighted estimates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.10578","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:80f71313102fb3778839473951bd0692046d501bf3052a626e14eed0b6210296","target":"record","created_at":"2026-05-18T00:19:03Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b39dd38a004b337f59f448ce28d5ca581ea5dadc139f64d3d3319be98dda76da","cross_cats_sorted":["math.CA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-28T21:54:39Z","title_canon_sha256":"3b8cc707d44e59e24b3849f699c280c9d511ed5a049ab7286bc651a37d8a0dd8"},"schema_version":"1.0","source":{"id":"1711.10578","kind":"arxiv","version":2}},"canonical_sha256":"6c33e73fc8f55487064c7b98b6f3d1c3f1d1e6be090066f76f7a17719964d286","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6c33e73fc8f55487064c7b98b6f3d1c3f1d1e6be090066f76f7a17719964d286","first_computed_at":"2026-05-18T00:19:03.025851Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:19:03.025851Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gIVe6JH/Xz+6X1p+V9D1SHVyxOkrEgO5NSFrk3dj0ldfgTOnF2pXGiJkDtPWCVko5QGfjWOAzi6KbYxZNnF8BA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:19:03.026743Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.10578","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:80f71313102fb3778839473951bd0692046d501bf3052a626e14eed0b6210296","sha256:016ed80026941707dfc71c73bc0d75ce1f2c3bf38e68480d80341f44636d27e6"],"state_sha256":"eec8bfacf96c459e09adb59fcc8cc03f359774df9fc263b5896a7511b952573c"}