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More precisely, let $\\mathcal{T}$ be either the maximal fractional function $M_\\gamma$ or the fractional integral operator $I_\\gamma$, $0<\\gamma<n$, $1\\leq p<n/\\gamma$ and $1/q=1/p-\\gamma/n$. If $u,v^{q/p}\\in A_1$ or if $uv^{-q/{p'}}\\in A_1$ and $v^q\\in A_\\infty(uv^{-q/{p'}})$ then we obtain that the estimate \\begin{equation*} uv^{q/p}\\left(\\left\\{x\\in \\R^n: \\frac{|\\mathcal{T}(fv)(x)|}{v(x)}>t\\right\\}\\right)^{1/q}\\leq \\frac{C}{t}\\left(\\int_{\\R^n}|f(x)|^pu(x)^{p/q}v(x)\\,dx\\right)^{1/p}, \\end{equation*} holds for every positi"},"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":"1712.08186","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-12-21T19:31:48Z","cross_cats_sorted":[],"title_canon_sha256":"8e91b8501c7de7386add48c6172457e206a76ee3c0d8c28b62762d6bcce05d53","abstract_canon_sha256":"bbc13a1c8317451c4c3599e7c966fd247e4d85a788a57458ec473a6132070841"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:24.224182Z","signature_b64":"tTjBVzKz14xHvs+6w3aQ4bDlILzBXOGiJE2O9UGwk8GDcluXnES90KFpXEYpYimDBgONHsWT+rdkZyKZCoKNBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1f9cbb784508e2590abaabd64d15b351376837d6d24eefcbe92edff148ea952a","last_reissued_at":"2026-05-18T00:27:24.223697Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:24.223697Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mixed weak estimates of Sawyer type for fractional integrals and some related operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fabio Berra, Gladis Pradolini, Marilina Carena","submitted_at":"2017-12-21T19:31:48Z","abstract_excerpt":"We prove mixed weak estimates of Sawyer type for fractional operators. More precisely, let $\\mathcal{T}$ be either the maximal fractional function $M_\\gamma$ or the fractional integral operator $I_\\gamma$, $0<\\gamma<n$, $1\\leq p<n/\\gamma$ and $1/q=1/p-\\gamma/n$. If $u,v^{q/p}\\in A_1$ or if $uv^{-q/{p'}}\\in A_1$ and $v^q\\in A_\\infty(uv^{-q/{p'}})$ then we obtain that the estimate \\begin{equation*} uv^{q/p}\\left(\\left\\{x\\in \\R^n: \\frac{|\\mathcal{T}(fv)(x)|}{v(x)}>t\\right\\}\\right)^{1/q}\\leq \\frac{C}{t}\\left(\\int_{\\R^n}|f(x)|^pu(x)^{p/q}v(x)\\,dx\\right)^{1/p}, \\end{equation*} holds for every positi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08186","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1712.08186","created_at":"2026-05-18T00:27:24.223775+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.08186v1","created_at":"2026-05-18T00:27:24.223775+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.08186","created_at":"2026-05-18T00:27:24.223775+00:00"},{"alias_kind":"pith_short_12","alias_value":"D6OLW6CFBDRF","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"D6OLW6CFBDRFSCV2","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"D6OLW6CF","created_at":"2026-05-18T12:31:10.602751+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/D6OLW6CFBDRFSCV2VPLE2FNTKE","json":"https://pith.science/pith/D6OLW6CFBDRFSCV2VPLE2FNTKE.json","graph_json":"https://pith.science/api/pith-number/D6OLW6CFBDRFSCV2VPLE2FNTKE/graph.json","events_json":"https://pith.science/api/pith-number/D6OLW6CFBDRFSCV2VPLE2FNTKE/events.json","paper":"https://pith.science/paper/D6OLW6CF"},"agent_actions":{"view_html":"https://pith.science/pith/D6OLW6CFBDRFSCV2VPLE2FNTKE","download_json":"https://pith.science/pith/D6OLW6CFBDRFSCV2VPLE2FNTKE.json","view_paper":"https://pith.science/paper/D6OLW6CF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.08186&json=true","fetch_graph":"https://pith.science/api/pith-number/D6OLW6CFBDRFSCV2VPLE2FNTKE/graph.json","fetch_events":"https://pith.science/api/pith-number/D6OLW6CFBDRFSCV2VPLE2FNTKE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D6OLW6CFBDRFSCV2VPLE2FNTKE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D6OLW6CFBDRFSCV2VPLE2FNTKE/action/storage_attestation","attest_author":"https://pith.science/pith/D6OLW6CFBDRFSCV2VPLE2FNTKE/action/author_attestation","sign_citation":"https://pith.science/pith/D6OLW6CFBDRFSCV2VPLE2FNTKE/action/citation_signature","submit_replication":"https://pith.science/pith/D6OLW6CFBDRFSCV2VPLE2FNTKE/action/replication_record"}},"created_at":"2026-05-18T00:27:24.223775+00:00","updated_at":"2026-05-18T00:27:24.223775+00:00"}