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Specifically, if $\\mu$ and $\\lambda$ are biparameter $A_p$ weights, $\\nu := \\mu^{1/p}\\lambda^{-1/p}$ is the Bloom weight, and $b$ is in $bmo(\\nu)$, then we prove a lower bound and testing condition $\\|b\\|_{bmo(\\nu)} \\lesssim \\sup \\| [b, R_k^1 R_l^2]: L^p(\\mu) \\rightarrow L^p(\\lambda)\\|$, where $R_k^1$ and $R_l^2$ are Riesz transforms acting in each varia"},"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":"1701.06526","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-01-23T17:51:08Z","cross_cats_sorted":[],"title_canon_sha256":"4c991402a7ee5110d32acf3492f4d41a2a9a66f3fc190a5fa14b6b0f2f082499","abstract_canon_sha256":"35827701a6e19c6d5ab8f909d96085d536e67e6d22c563cb0ed9a79d2f2c2b71"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:11.538576Z","signature_b64":"31WNkBfa+zDmbb2VL9wVwHTGpUiNIg6avnlI0P1/TmcQarEWZSljT9ECANiqtDsCFCZ4KMZvcpWiOU+mEUdBAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c2528cbd88d08b8adb255fc28439ac08ec6e8f32a797f5dcc5146734ef148ad","last_reissued_at":"2026-05-18T00:14:11.538091Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:11.538091Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weighted little bmo and two-weight inequalities for Journ\\'e commutators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Brett D. 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