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We use the Bellman technique and try to carefully present and motivate our tactics. As an illustration of how these results can be used, we deduce the following result: if a weight $w$ is in $A_2$ then it self-improves to a weight, which satisfies a Reverse H\\\"{o}lder condition."},"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":"1105.4848","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-05-24T18:17:35Z","cross_cats_sorted":[],"title_canon_sha256":"33ce5b14066fe52d97e33cb2c67accf6bd8beccfbc2258385e1f9411d7333685","abstract_canon_sha256":"71566541ebf91e57963fc72c695ac736368b3b062cb1fee29b58e6f28b4a0646"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:21:21.735997Z","signature_b64":"dqQLESQs0qOH6hHGXmN7K5rsdQVJDqBVATLULl+1KLTpJiKrjouQNZnItkqkQvdjV0yobTrpmXXEoiT2K+DmDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b6631f35553397954e330690dfa05f8952e6cd43e2b743fe76de20f07b89553d","last_reissued_at":"2026-05-18T04:21:21.735546Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:21:21.735546Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp weak type estimates for weights in the class $A_{p_1, p_2}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alexander Reznikov","submitted_at":"2011-05-24T18:17:35Z","abstract_excerpt":"We get sharp estimates for the distribution function of nonnegative weights, which satisfy so called $A_{p_1, p_2}$ condition. 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